Fundamentals of Inorganic Chemistry [2, Second ed.]

Citation preview

t.m lc

ea

th

e/ m

he

ry

ra

ib

yl

Volume 2

Fundamental Concepts of

yl

ib

ra

ry

Cbe

t.m

e/

th

ea

lc

he

m

Revised Second Edition

ry ra ib yl m he

t.m

e/

th

ea

lc

This Page is Intentionally Left Blank

ry ra ib yl m he lc ea th e/ t.m I] CBS

, CBS 'Publishers & Distributors Pvt Ltd New Delhi • Bengaluru • Chennai • Kochi • Kolkata • Mumbai Bhubaneswar • Hyderabad • Jharkhand • Nagpur • Patna • Pune • Uttarakhand

Disclaimer Science and technology are constantly changing fields. New research and experience broaden the scope of information and knowledge. The authors have tried their best in giving information available to them while preparing the material for this book. Although, all efforts have been made to ensure optimum accuracy of the material, yet it is quite possible some errors might have been left uncorrected. The publisher, the printer and the authors will not be held responsible for any inadvertent errors, omissions or inaccuracies. eISBN: 978-93-890-1751-9

ry

Copyright © Authors and Publisher Second eBook Edition: 2019

Fundamental Concepts of

ra

Inorganic Chemistry All rights reserved. No part of this eBook may be reproduced or transmitted in any form or by any means,

ib

electronic or mechanical, including photocopying, recording, or any information storage and retrieval system Revised Second Edition without permission, in writing, from the authors and the publisher.

m

yl

ISBN: 978-81-239-1867-9 Published by Satish Kumar Jain and produced by Varun Jain for CBS Publishers & Distributors Pvt. Ltd. Copyright © Authors and Publisher Corporate Office: 204 FIE, Industrial Area, Patparganj, New Delhi-110092 Ph: +91-11-49344934; Fax: +91-11-49344935; Website: www.cbspd.com; www.eduport-global.com; Revised Second Edition: 2016 E-mail: [email protected]; [email protected]

he

Reprint: 2011, 2012, 2013, 2015,2018

Head Office:2000 CBS PLAZA, 4819/XI Prahlad Street, 24 Ansari Road, Daryaganj, New Delhi-110002, India. First Edition: Ph: +91-11-23289259, 23266861, Fax: 011-23243014; Website: www.cbspd.com; Reprint: 2001, 2003, 2005, 2008, 23266867; 2009

lc

E-mail: [email protected]; [email protected].

All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical. including photocopying, recording, or any information storage and retrieval system without permission, in writing, from the authors and the publisher.

ea

Branches

Published by Satish Kumar Jain and produced by Varun Jain for

th

Bengaluru: Seema House 2975, 17th Cross, K.R. Road, Banasankari 2nd Stage, Bengaluru - 560070, CBS PubHshers a Distributors Pvt Ltd Kamataka Ph: +91-80-26771678/79; Fax: +91-80-26771680; E-mail: [email protected] 4819/XI Prahlad Street 24 Ansari Road, Daryaganj, New Delhi 110002, India.

e/

Ph: 23289259, 23266861, 23266867 Website: www.cbspd.com Chennai: No.7, Subbaraya Street Shenoy Nagar Chennai - 600030, Tamil Nadu Fax: 011-23243014 e-mail: [email protected]; [email protected]. Ph:Office: +91-44-26680620, 26681266; E-mail: [email protected] Corporate 204 FIE, Industrial Area, Patparganj, Delhi 110 092

t.m

Ph: 4934 4934 Fax: 4934 4935 e-mail: [email protected] Kochi: 36/14 Kalluvilakam, [email protected]; Hospital Road, Kochi - 682018, Kerala

Branches Ph: +91-484-4059061-65; Fax: +91-484-4059065; E-mail: [email protected] • Bengaluru: Seema House 2975, Dr. 17thE.Cross, K.R. Road, Moses Road, Worli, Mumbai - 400018, Maharashtra Mumbai: 83-C, 1st floor, Banasankari 2nd Stage, Bengaluru 560 070, Karnataka

Ph: +91-22-24902340 - 41; +91-22-24902342; E-mail: [email protected] Fax:Fax: +91-80-26771680 e-mail: [email protected]

Ph: +91-80-26771678/79

• Chennai: 7, Subbaraya Street, Shenoy Nagar, Chennal 600 030, Tamil Nadu

Kolkata: No. 6/B, Ground Floor, Rameswar Shawa-mall: Road, Kolkata - 700014 Fax: +91-44-42032115 [email protected]

Ph: +91-44-26680620, 26681266

• Koehl:Ph: 42/1325, 1326, Power House Road, Opposite KSEB Power House, +91-33-22891126 - 28; E-mail: [email protected] Ernakulam 682 018, Koehl, Kerala

Representatives

Fax: +91-484-4059065

Ph: +91-484-4059061-65

a-mall: [email protected]

• Koikata: 6/B, Ground Floor, Rameswar Shaw Road, Kolkata-700 014, West Bengal

Hyderabad

Ph: +91-33-22891126, 22891127, 22891128

a-mall: [email protected]

• Mumbai: 83-C, Dr E Moses Road, Worli, Mumbai-400018, Maharashtra

Pune

Ph: +91-22-24902340/41

Fax: +91-22-24902342

a-mall: [email protected]

Representatives Nagpur • Bhubaneswar 0-9911037372

• Hyderabad

• Patna

•

Manipal

0-9334159340

Pun.

0-9885175004

• Jharkhand

0-9623451994

• Uttarakhand 0-9716462459

0-9811541605

• Nagpur

Vijayawada Printed at Goyal Offset Printers, GT Karnal Road, Industrial Area, Delhi, India Patna

0-9021734563

ry

Foreword

lc

he

m

yl

ib

ra

Dr. A.V. Saha, DSc Head of the Department of Chemistry Ramakrishna Mission Residential College P.O. Narendrapur 743508, 24-Parganas (8) West Bengal, India

t.m

e/

th

ea

Present-day inorganic chemistry is no more a collection of unrelated facts. The scenario has undergone a drastic change over the last fifteen to twenty years with the applicati.on of kinetic, thermodynamic and . structural studies to inorganic substances and with newer techniques providing newer information. Often the information gathered, has made theoreticians develop/modify valency theories and'principles. In this situation every teacher and student realise the importance of a textbook that will" help them develop the concepts and understanding ofthe subJect. There are attempts by a few authors at achieving this goal but those seldom cover the whole curricula followed by most of the Indian Universities and Institutes. In many of these books the authors virtually neglect ,the evolutionary developments of the subject, creating undesirable lacunae in the readers' understanding. Here is an honest and sincere attempt at bridging these gaps and presenting a comprehensive textbook on concepts and understanding to the readers. The treatment of every topic is elaborate and is marked by remarkable clarity and the author has not compromised herewith the volume of his work. The book Fundamental Concepts ofInorganic Chemistry covers the inorganic chemistry curricula at the BSc (honours) arid MSc preliminaries. A large number of exercises and problems essential.for modem teaching, have been incorporated meticulously at the end of each chapter to bridge the gaps, if any, in the understanding of the subject. Hope this earnest effort of Dr. Das, a teacher of distinction, will receive well-deserved acclaim from the students and teachers of this subject.

A.V. Saha

t.m

e/

th

ea

lc

he

m

yl

ib

ra

ry

To My Teachers and Well-Wishers Who Paved the Way to Reach Here

~~~

ra

ry

One who has shraddha acquires knowledge

yl

ib

Preface to the Second Edition

he

m

I do endeavour. He gives the strength and patience.

t.m

e/

th

ea

lc

My beloved readers are requested first to accept my unconditional apology for this long time taken to shape the second edition of the book. Though the second edition incorporates some aspects of coordination chemistry, the readers desire the complete Coordination Chemistry which is expected to appear in the next volumes and the process is in progress. I feel delighted to record the warm· response which the first edition of the book has received from the students and teachers throughout the country. I have tried my best to incorporate all the suggestions received from the readers. The present edition has been thoroughly revised and substantially enlarged and, in fact, no single page has been left untouched. However, I have taken all the measures to retain the features for which the first edition has been so popular. In this second edition of the book, scope of the book has been broadened by adding new topics and revisions of the earlier sections. The new topics are atomic units; NMR active nuclei and principles of NMR spectroscopy;' nucleosynthesis of elements after big bang; nuclear waste management; BoseEinstein condensate (BEC): fifth state of matter; bonding properties of d- and.forbitals: comparison; application of Woodward-Hoffmann orbital symmetry rules; structure and properties of fluoroalkyl radicals and ions; structure and properties of zeolites and clay materials; aluminosilicates and surface acids; heteropoly and isopoly acid anions; hypervalency; molecular orbital diagrams of polyatomic molecules in terms of group orbitals (Le. TASOs); 3c-4e versus 3c-2e bonding systems; geometry of the ~olecules in terms of Walsh diagrams; isolobal fragments; reaction-mechanism of the main group elements; different aspects ofmaterial science; PSPET- Wade's rule, Jemmi's rule; chemistry ofboranes and carboranes; metal clusters and carbonyl clusters; Zintle ions; symmetry elements and point groups; fullerenes; intercalation compounds ofgraphite; band theory ofsolids - band splitting and band bending; electrical and magnetic properties of solids; piezoelectric solids; organic metals and organic semiconductors; superconductivity; supramolecular chemistry; aqueous chemistry of amino acids; surface

viii

Fundamental Concepts of Inorganic Chemistry

t.m

e/

th

ea

lc

he

m

yl

ib

ra

ry

In this second edition of the book, scope of the book has been broadened by adding new topics and revisions of the earlier sections. The new topics are atomic units; NMR active nuclei and principles of NMR spectroscopy; nucleosynthesis of elements after big bang; nuclear waste management; BoseEinstein condensate (BEe): fifth state of matter; bonding properties of d- andforbitals: comparison; application of Woodward-Hoffmann orbital symmetry rules; structure and prop~rties of fluoroalkyl radicals and ions; structure and properties of zeolites and clay materials; aluminosilicates and surface acids; heteropoly and isopoly acid anions; hypervalency; molecular orbital diagrams of polyatomic r,lolecules in terms of group orbitals (i.e.·TASOs); 3c-4e versus 3c-2ebonding systems; geometry of the molecules in terms of Walsh diagrams; isolobal fragments; reaction-mechanism ofthe'main group elements; different aspects ofmaterial science; PSPET- Wade's rule, Jemmi's rule; chemistry ofboranes and carboranes; metal clusters and carbonyl clusters; Zintle ions; symmetry elements and point groups; fullerenes; in~rcalation compounds ofgraphitt;; band theory ofsolidS - band splitting and band bending; electrical and magnetic properties of solids; piezoelectric solids; organic metals and organic semiconductors; superconductivity; supramolecular chemistry; aqueous chemistry of amino acids; surface acids and solid acids; agostic interaction; ionic equilibria and related numerical problems; inorganic photochemistry; lithium battery; fuel cells; solar cells; EMF diagrams - Frost diagram, Pourbaix diagram; Tafel equation; stability field of water; chemistry of explosives; electroanalytical techniquespolarography, coulometry, ampereometry, LSV, CV, polarography; chemical clock reactions and oscillating chemical reactions. In preparing the revised manuscript, I have freely consulted the books and reviews of different authors and I have borrowed their ideas whenever it has been required. I am extremely grateful and indebted to these authors. In fact, I have picked up the flowers from these gardens to prepare the garland to worship the goddess of learning. The golden period of my s.tudent life at Ramakrishna Mission Residential College, Narendrapur, gives the foundation of my career. There I got a gifted teacher, Prof. AV Saha whose teaching in the subject is still remaining vibrant in my memory and his teaching is reflected at many places of this book. The author is thankful to Prof. KL Sebastian, IPC, IISc, Bangalore; Prof. S Basu, Burdwan University; Prof. ON Mukherjee, Sir Rashbehari Qhose, Professor of Chemistry; Dr A Sen, Ramakrishna Mission . Centenary College, Rahara, 24 Parganas; Dr MN Bishnu, City College, Kolkata; Prof. RN Mukherjee, lIT, Kanpur; Dr SP Banerjee, Retired Professor, Vivekananda Mahavidyalaya, Burdwan; Dr S Roy Chowdhury, Dinabandhu Andrews College, Garia, Kolkata; Prof. K Dey, Kalyani University; Prof. A Sarkar, Kalyani University; Dr. G Mukherjee, Kalna ,College, Burdwan; Dr N Koley, Retired Professor, Raghunathpur College, Purulia; Prof. B Pathak, Retired Professor, Presidency College, Kolkata; Dr B Saha, Burdwan University; Dr H Karak, St Paul's College, Kolkata, for their encouragement and suggestion. I am thankful to all my colleagues for extending their moral support. J am specially thankful to my colleague Dr M Seikh for discussion on many topics and procuring some reference materials. I am extremely grateful to Mr SK Jain, Managing Director, CBS Publishers and Distributors, New Delhi for his continued support. I am also thankful to Mr YN Arjuna, Senior Director (Publishing, Editorial and Publicity) and the staff involved in DTP for taking the troubles in processing the revised

ry

ib

ra

Preface to the First Edition

yl

- "No great work can be achieved by humbug. It is through love, a passion for truth; and tremendous energy, that all undertakings are accomplished. "

he

m

- "I hold every man a traitor who, having been educated at their expense pay not the least heed to them. " Swami Vivekananda

t.m

e/

th

ea

lc

As a student and also as a teacher, the author has experienced that for a systematic and comprehensive coverage ofthe present subject, one is forced to consult various books on different disciplines to collect the reading materials. Consequently, the students are specially constrained. Keeping this idea in mind, it has been attempted here to present a complete textbook on the subject. In a logical sequence, the book deals extensively with different aspects such as atomic structure (both classical and wave mechanical) and atomic spectra, fundamentals of quantum mechanics and wave mechanics, nuclear chemistry and radiation chemistry, different theories of valence forces and chemical forces including band and Bloch theory of solids, solid-state chemistry, acids and bases, nonaqueous. solvents and redox potentials. It also covers the different aspects of materials science which is emerging with a great promise. Thus the present book covers the curricula followed by most of the Indian Universities and Institutes at the BSc and MSc levels. It also aims to help the students preparing for competitive examinations like NET, GATE, SLET, etc. Adequate stress on the basic theories and concepts has been given everywhere to rationalise the pre.sentation. The ideas have been very often illustrated through solution of related numerical problems. Each chapter is ended with various types of questions and problems to afford an opportunity to th~ students for self-evaluation. In writing a book of this nature, one accumulates indebtedness to the previous authors of different books. The books which have been c0!1sulted are listed separately and gratefully acknowledged. The author expresses his deep sense of gratitude to Dr AV Saha, a gifted teacher, from whom the author had started to learn and understand the subject in his student life in Ramakrishna Mission Residential College, Narendrapur, for writing the foreword. The author's association for a fairly long period with the said Institute was highly fruitful. in shaping the present ideas and the author must express his indebtedness to his teachers and the authorities of the Institute. The author. is grateful to his wife, Dr M Das for various types of help, assistance and cooperation.

x

Fundamental Concepts of Inorganic Chemistry

t.m

e/

th

ea

lc

he

m

yl

ib

ra

ry

The author is specially grat~ful to Mr SK Jain, Managing Director ofCBS Publishers and Distributors, New Delhi for his earnest interest in publishing the book. Thanks are also due to his colleagues and students. In conclusion, the author's attempt will be amply rewarded, ifit is found helpful to the students and teachers. In spite ofall precautions, some errors might have crept in. Constructive criticism and valuable suggestions from the readers will be most welcome.

Asim K. Das

yl

ib

ra

ry

Contents

m

Foreword Preface to the Second Edition

he

Preface to the First Edition

v vii ix

lc

VOLUME 2

th

ea

9. Introduction to Chemical Bonding and Theories of Covalence: Valence Bond Theory (VBT) and Molecular Orbital Theory (MOT) 542-752 542

9.2 The Theories of Covalent Bond : Born-Oppenheimer Approximation

550

9.3 The Valence Bond Theory (VBT) The Valence Bond Theory for Hydrogen Molecule; The Major Conclusions Drawn from VBT; The Major Drawbacks in VBT

551

9.4 Molecular Orbital Theory (MOT) The LCAO-MO Treatment of Diatomic Hydrogen Molecule Ion (H;); The LCAD-MO Treatment of the Diatomic Hydrogen Molecule: A Two Electron-Two Centre (2e-2c) System; Comparison of VB and LCAD-MD Treatments on H2; The Important Aspects of LCAD-MOT; Proof of Three Basic Conditions (EOS) in LCAD-MO Method

557

9.5 Symmetry and Overlap in Forming Molecular Orbitals in LCAO method

570

t.m

e/

9.1 Historical Background Development of the Concept of Valency; Lewis-Langmuir Concept; The Octet Rule; Sugden's View of Singlet Linkage (2c-Ie) in Favour of Octet Rule; Sidgwick's Rule of Maximum Covalency; Hypervalence and Hypercarbon Compounds; The Main Types of Chemical Forces; Isosteric or Isoelectronic Species

xii

Fundamental Concepts of Inorganic Chemistry

9.6 Sigma (0)-, Pi (x)- and Delta (~)-Molecular Orbitals, Phi (+) and Mu (1) Bonds 9.7 VBT versus LCAO-MOT

574 575

9.8 The United Atom Method in Molecular Orbital Theory

576

9.9 Simple Molecular Orbital Model for Homonuclear Diatomic Molecules

578

Simple Molecular Orbitals and Diagrams; Molecular Orbital Picture of Some Representative Homonuclear Diatomic Molecular Species; Limitations of the Simple Molecular Orbital Model; Simple Molecular Orbital Diagram for the Homonuclear Diatomic Molecules Involving (n-l)d Orbitals

ry

1

ra

9.10 Modified Molecular Orbital Energy Diagram for the Homonuclear Diatomic Molecules (Specially for N 2 and Lighter Molecules)

589

yl

ib

Symmetry Interaction (Noncrossing Rule) among the Molecular Orbitals Obtained in the Simple Model; Molecular Orbital Energy Diagram by Considering the Participation of Hybrid (s-p interaction) Atomic Orbitals in Generating the Molecular Orbitals; Success of the Modified MO Energy Diagram : Reactivity of Some Molecular Species

m

9.11 Molecular Orbital Picture in Heteronuclear Diatomic Molecules: UV Photoelectron Spectroscopy and Photosensitivity in the Light of MOT Molecules (Le. XH) Containing a H-atom; Molecules (XY) in which both the Atoms

601

he

(X, Y) Contain s- and p-Valence Orbitals; Ultraviolet Photoelectron Spectroscopy

(UV-PES): Identification of the Nature of Molecular Orbital Energy Level; Colours and Photosensitivity of the Compounds in Terms of MOT

lc

9.12 Molecular Orbital Treatment of Triatomic Molecules

620

t.m

e/

th

ea

Basic Concepts of Group Orbitals; Delocalised and Localised Molecular Orbitals; Molecular Orbital Treatment for H) and H; Molecular Species; Molecular Orbital Treatment for Linear HydridesXH2 Having Valence Electrons Four; Molecular Orbital Treatment for Bent Hydrides XH2 Having Valence Electrons More Than Four; Illustration of Walsh Diagram for XH2 Species; Molecular Orbital Energy Diagram of Some Representative Triatomic Hydrides in Terms of Hybridisation of the Central Atom; Triatomic Nonhydrides (XY2)-Linear Species Having 16 Valence Electrons; Molecular Orbital Energy Diagram for Some Representative Linear Molecules like CO2 and BeX2 - Use of Hybridised Orbitals by the Central Atom; Molecular Orbital Picture of the Bent Molecule XY2 (X and Y Second Period Elements); Molecular Orbital Energy Diagram for the Trigonal Planar Molecular Species -XY); Molecular Orbital Energy Diagram of Some Representative Polyatomic Species; Characteristics of Three Centre Bonding

9.13 Hybridisation of Atomic Orbitals

669

Linear Combination of Atomic Orbitals in Hybridisation; Stereochemistry of the Hybrid Orbitals; Bonding Potentiality of the Hybrid Orbitals; Energetics of Hybridisation; Participation of d-Orbitals in Hybridisation in Nonmetals; Equivalence and Nonequivalence of the Hybrid Orbitals; Bent's Rule of Hybridisation

9.14 Summary of the Concept of Hybridisation

699

9.15 Conjugated Molecules and Delocalisation Energy

699

9.16 Resonance and Delocalisation

70S

Contents

xiii

Concept of Resonance in VBT; Resonance Energy; Conditions for Effective Canonical Forms; Concepts ofFonnal Charge and Lewis-Langmuir Charge in Resonating Structures: Fonnal Charge and Oxidation Number; Resonance in the Light of Molecular Orbital Theory; Application of the Concept of Resonance in Some Inorganic Compounds

9.17 Limitations of the' Concept of Resonance;and Hybridisation

716

9.18 Multiple Bonds

717

Bond Length and Bond Multiplicity; Prc-P rc Bonding

ry

9.19 Delta and Quadruple Bond

ra

9.20 Odd Electron Covalent Bonds in Odd Electron Molecules

720 723 725

9.22 Molecular Orbital Theory in Explaining Higher Oxidation States of Nonmetals

734

9.23 Molecular Orbital Theory in Explaining the Bonding in Inert Gas Compounds: 3c-4e Bonding Model

735

9.24 Isolobal Fragments

739

m

yl

ib

9.21 Electron Deficient Covalent Bonds and Electron Deficient Covalent Compounds

Exercise-IX

he

Appendix 9A: FMO Approach to Woodward-Hoffmann Orbital Symmetry Rule for Intramolecular Cyclisation and Concerted Intermolecular Cycloaddition Reactions

lc

. 10. Covalent Compounds: Characteristics, Structure and B~nd

Energy

748

753-976 753

th

10.1

ea

Reactivity

742

t.m

e/

Bond Energy in the Diatomic Molecules; Thennodynamics ofBond Energy in the Diatomic Molecules; Average Bond Energy in Polyatomic Molecules; Thermochemistry in Computing Bond Energies; Bond Energies of Different Bonds; Intrinsic Bond Energy; Different Factors Governing the Bond Energy

10.2 Covalent Bond Lengths and Covalent Radii

764

Single Bond and Covalent Radii; Multiple Borid and Covalent Radii; Different Factors Governing the Bond Lengths

10.3 The Inert Pair Effect

770

Meaning and Illustration of the Inert Pair Effect; Interpretation of Inert Pair Effect in Tenns of Bond Weakening Phenomena for the Heavier Congeners

10.4 Covalent-Ionic Resonance: Partial Ionic Character in Covalent Bonds 10.5 Dipole Moment and Molecular Polarity

774 777

Different Contributing Factors to the Molecular Polarity; Prediction of Geometry of the Molecules from Dipole Mo~ent and Calculation of Dipole Moment; Dipolar Repulsive Interaction; Contribution of Hybrid Lone Pair Moments to the Molecular Polarity; Induced Dipole Moments; Application of the Concept of Dipole Moments

10.6 Covalency in Ionic Bonds: Deformation or Polarisation of Ions: Fajans' Rules Polarisation and Covalency; Factors Governing the Degree of Polarisation; Fajan's Rules

791,

xiv

Fundamental Concepts of Inorganic Chemistry in Predicting the Melting and Boiling Points and Solubility of Some Compounds; Application of the Concept of Polarisation

10.7 Effects of x-Bonding (with Special Reference to the Involvement of d-Orbitals) on the Structural Properties and Reactivity

802

ib

ra

ry

Comparison ofCa~bon and Silicon Compounds; Comparison ofNitrogen and Phosphorus Compounds; Comparison of Oxygen and Sulfur Compounds; Pi-Boding in Boron Compounds; Effect of Pre ~ dre Bonding on the Properties of Different Oxyacids and Periodic Variation of Properties ofOxyacids;Pre-dre Bonding Leading to Aromaticity in Inorganic Ring Compounds; Changes in Structure due to Electronegative Substituents; Synergistic Effect due to the 1t-Bonding and Stabilisation of Low Oxidation States of Metals in Complexes; 1t-Bonding in Soft-Soft Interactions; 1t-Bonding and trans-Influence, trans-Effect in Kinetics; 1t-Bonding and cis-Effect in Kinetics; 1t-Bonding and Isomerism in Coordination Compounds

10.8 Structure of Covalent Molecules: Molecular Topologies

824

he

m

yl

Helferich's Rule; Concept of Hybridisation and Bent's Rule; Sidgwick-Powell Theory and Valence Shell Electron Pair Repulsion (VSEPR) Theory; Shape of the Molecules Having Stereochemically Active Lone Pairs Predicted from VSEPR Theory; Concept of Stereochemically Active and Inactive Lone Pairs; Structure of XeF6 and Related Species; Limitations of VSEPR theory

10.9 Stereochemically Nonrigid Covalent Molecules: Fluxional

Molecul~s

848

ea

lc

Atomic Inversion in Amines, Phosphines and Arsines; Berry Pseudorotation and Turnstile Rotation; Twisting Mechanism; Fluxional Molecule : Fluxionality and Time Scale of Observation; 19F-NMR Spectra and Fluxionality of PCI2F] .and CIF]; Examples of Fluxional Molecules

861

th

10.10 Nucleophilic Substitution Reactions (e.g. Solvolysis) in Some Covalent Molecules of Nonmetals

e/

Thermodynamic and Kinetic Aspects of Chemical Reactions; Reaction Pathways of Nucleophilic Substitution Reaction; Energetics of A and D Processes; Some Representative Nucleophilic Substitution Reactions in Covalent Compounds ofp-Block Elements; Some Representative Redox Reactions through Nucleophilic Substitution Reactions

t.m

10.11 Structure of Covalent and Molecular Crystalline Solids

884

Diamond Cubic (dc) Structure; Graphite Structure; Properties of Graphite; Lamellar or Intercalation Compounds ofGraphite; Reactivity of Graphite; Band Structure of Graphite; Comparison Between Graphite and Inorganic Graphite (Le. Boron Nitride); Molecular Crystals; Buckministerfullerene; Carbon Clusters and Allotropic Forms of Carbon

10.12 Structure and Bonding in Higher Boranes and Carboranes

896

Classification and Different Series of Boranes; Nomenclature of Boranes; Structure and Molecular Framework of Hydrides of Boron; Skeletal Electron Pair Counting and Wade's Rules : Polyhedral Skeletal Electron Pair Theory (PSEPT); Calculation of the Number of Electron Deficient Bonds; stxy Numbers and Topology of Boron Hydrides; Molecular Orbital Picture of Higher Boranes; Carboranes and their Structural Features; Calculation of Skeletal Electron Pairs (S) in Carboranes; Jemmis' Unifying Electron Counting Rule for Condensed Polyhedral Boranes

10.13 Synthesis and Reactivity of Boranes and Carboranes Synthesis of Boranes; Reactivity of Boranes; Synthesis and Reactivity of Carboranes

917

Contents xv 10.14 Molecular Symmetry and Point Groups

931

ry

Introduction; Symmetry Elements and Symmetry Operations; n-Fold Rotation Axis (i.e. n-Fold Proper Axis of Rotation) (Cn ) and Rotation About a Symmetry Axis; Plane of Symmetry (i.e. Mirror Plane) arid Reflection at the Plane of Symmetry; Inversion Centre (Le. Centre of Symmetry) and Reflection Through the Inversion Centre; n-Fold RotationReflection Axis and Rotation-Reflection (in either order); Identity Element (E or I); Point Groups and Identification of Point Groups; Special Point Groups with Multiple Highorder Axes (more than one Cn' n > 2); Steps to Select a Point Group (excluding the special point groups with multiple high' order axes); Symmetry Elements and Optical Activity; Symmetry Elements and Dipole Moment; Symmetry Number and Point Groups

ra

Exercise-X

ib

Appendix lOA: Fluoroalkyl Radicals and Carbocations and Fluorocarbenes

m

11.1 Characteristic Properties of Ionic Compounds

yl

11. Structure, Bonding and Properties of Ionic Solids and Solid-State Chemistry

962 972

977-1109 977 980

11.3 Electron Density (ED) Maps and Ionic Radii

982

he

11.2 Different Types of Ions and Electronic Configurations of the Ions Involved in Ionic Bonding

ea

lc

Ionic Radii from X-Ray Electron Density·, Maps; Ionic Radii from Other Methods; Shannon's Crystal Radii; Comparison among Shannon's Crystal radii, Goldschmidt's Radii and Pauling's Radii

989

11.5 Energetics of Ionic Bond Formation: Born-Haber Cycle

994

th

11.4 Factors Affecting the Ionic Radii

e/

Born-Haber Cycle in Ionic Bond Fonnation; Drawbacks in the Concept of Born-Haber Cycle; Factors Favouring the Formation of Ionic Bonds

t.m

11.6 Applications of Born-Haber Cycle 11.7 Theoretical Aspects of Crystal Forces: Lattice Energy of Ionic Crystals

997 1003

Born-Lande Equation of Lattice Energy; Verification of the Born-Lande Equation; Modification of the Born-Lande Equation; Kapustinskii Equation; Factors Affecting the Lattice Energy; Hardness of Ionic Compounds

11.8 Radius Ratio Rule and Preferred Structures of Ionic Compounds

1014

Limiting Radius Ratios for Some Common Geometries; Application of the Radius Ratio Rule; Limitations of the Radius Ratio Rule

11.9 Some Basic Crystal Geometries (i.e. Unit Cells) Involved in the Structure of Crystalline Solids Simple Cubic (se) Unit Cell; Body Centred Cubic (bee) Unit Cell; Face Centred Cubic (fcc) Unit Cell; Diamond Cubic (de) Unit Cell; Graphite Layer Lattice; Close Packing Models of Spheres; The Body Centred (bee) Lattice in Terms ofClose Packing of Spheres; Characteristic Features of hep, fcc and bee Structure

1024

xvi

Fundamental Concepts of Inorganic Chemistry

11.10 Structures of Ionic Crystals

1035

Structures of Ionic Crystal$ of AB "'ype; Structures of Ionic Crystals of AB 2 Type; Structures of Ionic Crystals of AB) Type; Structure of Ionic Crystals of A2B) Type: Corundum (6:4 Coordination Crystals); Structure of Io'nic Crystals Consisting of Polyatomic Ions; Comparison of the Basic Ionic Crystals

11.11 Structures of Mixed Metal Oxides

1042

ry

The Spinel and Inverse Spinel Structure; The Ilmenite Structure (FeTiO); The Perovskite Structure (CaTiO); Mixed Oxide"s - Coloured Minerals and Gem Quality Crystals

1047

11.13 Deviations from Simple Ionic Structures

1049

ib

ra

11.12 Crystal Structure in Relation to Superconductivity and/Ferroelectric Property YBa 2Cu)07_x; BaTiO)

yl

Layer Lattices (Le. Two Dimensional Lattices); Chain Lattices (Le. One Dimensional Lattices)

m

11.14 Structures of Silicates and Isopoly and Heteropoly Acids: Representative Examples of Complex Ionic Crystal

10SI

11.15 Crystal Defects in Ionic Solids

he

Ch,emistry of Silicates and Cement; Isopoly and Heteropoly Acids

1063

Stoichiometric (Le. Intrinsic) Defects; Nonstoichiometric (or Berthollide) (Le. Extrinsic)

lc

D~fects

11.16 Magnetic Materials : Magnetic Properties

1076

th

ea

Terminology in Describing the Magnetic Properties and Different Units; Classification of Magnetic Substances; Cooperative Magnetism - Aniferromagnetism, Canted Magnetism; Ferrimagnetism and Ferromagnetism; Magnetic Properties of Metals; Temperature Dependence of Magnetic Susceptibility of Different Types of Magnetic Materials

e/

11.17 Properties of Ferroelectric Materials and Piezoelectric Materials

1088

Ferroelectric Materials; Piezoelectricity and Piezoelectric Materials

1091

11.19 Glasses and Glass Transition

1092

11.20 Structure of Liquids: General Aspects: Hole Theory of Liquids

1094

t.m

11.18 Carbides

Solved Numerical Problems Exercise-XI

Appendices Appendix Appendix Appendix Appendix Appendix Appendix

Index

I: II: III: IV: V: VI:

1095 1102

A·1-A·12 Units and Conversion Factors A-I Some Physical and Chemical Constants A-4 A-5 Wavelength and Colours Names, Symbols, Atomic Numbers and Atomic Weights of the Elements A-6 Some Useful Mathematical Relationships A-8 Books Co.nsulted A-II

1·1-1·20

Contents

xvii

VOLUME 1 1. Classical and Vector Models of Atoms

1-110 1

1.2 Anode Rays Production of Anode Rays; Properties of Anode Rays; Analysis of Positive Rays by Thomson's Parabola Method; Aston's Mass Spectrograph; Dempster's Mass Spectrometer; Bainbridge Mass Spectrometer

8

yl

1.3 The Neutron as a Fundamental Particle 'The Discovery of Neutron; Properties of Neutron

ib

ra

ry

1.1 Cathode Rays and Electron Electric Discharge through Gases and Production ofCathode Rays; Properties ofCathode Rays; Detennination of Charge to Mass Ratio (elm) ofan Electron by Thomson's Method; Detennination of Charge of an Electron by Millikan's Oil-Drop Method

16 18

1.5 The Hydrogen Spectrum

18

m

1.4 Some Important Properties of the three Fundamental Particles

he

Balmer Fonnula; Rydberg Fonnula; Ritz Combination Principle 20

1.7 Scattering of a-Particles and Rutherford's Nuclear Model of Atom,

21

lc

1.6 Thomson's Plum-Pudding Model of Atom

ea

Rutherford's Nuclear Model of Atom; Rutherford's Theory of a-Particle Scattering; Nuclear Dimension from the a-Particle Scattering Experiment

25

th

1.8 Bohr's .Atomic Model for the Hydrogen-like Systems

t.m

e/

Bohr's Atomic Model; Numerical Values ofRadius and Energy in Bohr's Atom; Correction for the Finite Mass of the Nucleus; Bohr's Theory and the Spectral Lines in Hydrogenlike Systems; Ionisation and Resonance Potential for Hydrogen and Hydrogen-like Systems in the Light of Bohr's Theory; Bohr's Theory and Correspondence Principle; Moseley's Law in the Light of Bohr's Theory; Franck-Hertz Experiments (Nobel Prize in Physics, 1925) in the Light of Bohr's Atomic Model; Merits of the Bohr's Atomic Model; Drawbacks in the Bohr's Model

1.9 Fine Structure of Spectra and Sommerfeld's Atomic Model Sommerfeld's Atomic Model; Penet;ating Power of the Orbits and Quantum Defect in Polyelectronic Systems in the Light of Sonlmerfeld's Theory; A'chievements of the Sommerfeld's Theory; Drawbacks in the Sommerfeld's Model

45

1.10 Vector Model of the Atom Concept of Spatial Quantisation; Concept of the Spinning Electron; Quantum Numbers in the Vector Model; Four Quantum Number System and Pauli Exclusion Principle; Coupling Schemes and Atomic States in the Vector Model; L-S and] -] Coupling Schemes; Spatial Quantisation of the Resultant Vectors (Le. L, S, J) in Atoms or Ions; Detennination of Microstates and Russel-Saunders Terms; A Simple Working Procedure

52

xviii

Fundamental Concepts of Inorganic Chemistry

1.11 Electronic Configuration Scheme for Many Electron Systems

ra

Rules for Placing the Electrons in Different Energy Levels

ry

to Determine the Term Symbols; Hund's Rules to Determine the Ground State Term; Derivation of the Ground State Terms in the Light of Hund's Rules; Spectral Selection Rules in the Vector Mod~l; Intensity Rules for Spectral Transitions in the Vector Model; Fine or Multiplet Structure of the Spectral Lines in the Light of Vector Model; Nuclear Spin and Hyperfine Structure ofthe Spectral Lines; Orbital and Spin Magnetic Moments ofan Electron; Stem-Gerlach Experiment-An Evidence in Favour of Space Quantisation and Spin Quantum Number; Explanation ofthe Magneto-Optic Phenomenon-the Zeeman Effect; Electronic Structure of the Atoms; Merits of the Vector Model

ib

Solved Numerical Problems

yl

Exercise-I

97 105

111-132 111 111

he

2.1 Introduction 2.2 Nature of Black-Body Radiations

m

2. Origin of the Quantum Theory

86

2.3 Different Laws and Theories of Black-Body Radiations

113

2.4 The Photoelectric Effect

lc

The Stefan-Boltzmann Law; Wien's Law; Rayleigh-Jeans Law; Planck's Quantu~ Theory

118

th

2.5 Compton Effect

ea

Characteristics of Photoelectric Effect; Einstein's Theory of Photoelectric Effect; Work Function and Ionisation Potential

.

121

e/

Theoretical Background of Compton Effect; Compton Shifts at Different Cases; Characteristic Features of Compton Effect

t.m

2.6 Pair Formation Solved Numerical Problems Exercise-II

3. Fundamentals of Wave· Mechanics

126 127 130

133-184

3.1 Introduction

133

3.2 Wav'e Particle Duality and de- Broglie's Matter Wave

133

de Broglie Wavelength for Matter Waves; The de Broglie Wavelength for the Macroscopic and Microscopic Bodies; Difference between de Broglie Matter Wave and Electromagnetic Wave; The de Broglie Electron Wave; Relativistic Correction in the de Broglie Electron Wave; Verification of the de Broglie Electron Wave by Davisson and Germer's Experiment; Bohr's Quantum Restriction from the de Broglie Concept

3.3 The Heisenberg's Uncertainty Principle and its Implications Uncertainty Principle; Illustration of the Uncertainty Principle in the Determination of the Position ofa Particle by a Microscope: Compton Effect ,!nd the Uncertainty Principle;

140

Contents xix Applicability of the Uncertainty Principle to Large and Small Particles; Some Important Applications of the Uncertainty Principle

3.4 Schr6dinger's Wave Equation

145

Representation ofSchrodinger's Wave Equation; Physical Significance ofSchrodinger's Equation and Hamiltonian Operator; Eigen Values and Eigen Functions; Physical Significance of the Wave Function ('II); Normalised and Orthogonal Wave Functions

150

ry

3.5 Some Applications of the Sch-r6dinger's Wave Equation

ra

Free Particle; Particle in a One-Dimensional Box; Particle in a Three Dimensional Box; Importance of the Model - Particle in a Box; Some Applications of the Model- Particle in a Box; Quantum Mechanical Tunnelling Effect Variation Method; The Linear Combination Method

yl

Solved Numerical Problems Exercise-III

165

ib

3.6 Approximate Methods of Solving the Schr6dinger's Wave Equation

m

Appendix 3A: Application of Variation Method in Hydrogen Atom

he

4. Wave Mechanical Model of Atom

4.1 Wave Mechanical Model for the Hydrogen-like Atoms or Ions

~

167 177 180

185-235 185

t.m

e/

th

ea

lc

Wave Functions for the Hydrogen-like Systems; Wave Mechanical Genesis of the Quantum Numbers; Important Features of a Dirac Atom Developed from the Relativistic Wave Mechanics; The Radial Wave Function and Radial Probability Distribution of Electron Cloud in Hydrogenic Systems; The Physical and Chemical Significance of the Nodal Points; Angular Wave Functions and Orbital Shapes; Electron Cloud Density Representation of the Orbitals in Hydrogen-like Systems; Symmetry ofthe Orbitals; Concept of Atomic Orbitals in Wave Mechanics in Relation to the Orbits in Classical Mechanics; Energy of the Electron in Hydrogen-like Systems; The Four Quantum Numbers; Pauli Exclusion Principle in the Light of Wave Mechanics

4.2 Wave Mechanical Model for Polyelectronic Systems

219

Self-Consistent Field (SCF) Method; Concept of Slater Orbitals; Concept of Shielding and Quantum Defect; Slater's Rules in Calculating the Shielding Constant (S) and Effective Nuclear Charge (Z*) .

4.3 Bonding Properties of d- and I-Orbitals

225

Solved Problems

226

Exercise-IV

234

5. Atomic Nucleus and its Structure 5.1 The Atomic Nucleus Composition of the Nucleus; Size of the Nucleus; Mirror Nuclei and Nuclear Radius; Nuclear Density; Shape of the Nucleus; Total Angular Momentum of the Nucleus :

236-308 236

xx

Fundamental Concepts of Inorganic Chemistry Magnetic Properties ofthe Nucleus and Nuclear Magnetic" Resonance (NMR) Frequency and NMR Spectra of Some Representative Examples; Classification of the Nuclides; Nuclear Spin Isomerism in Diatomic Molecules: Ortho- and Para-Hydrogen: Intensity of the Rotational Lines in the Band Spectra of Diatomic Molecules

~.3

ry

5.2 The Isotopes Discovery and Characterisation of Nonradioactive Isotopes; Isotopic Shift in Optical Spectra; Characteristic Features of the Isotopes; Isotopic Composition of the Elements; Uses of Isotopes; Separation of Isotopes Nuclear Stability

253

268

ib

ra

Even-Odd Nature of the Nucleons (Harkins' Rule); The Neutron to Proton Ratio and Stability of the Nuclides; The Neutron to Proton Ratio and Different Modes of Decay; Packing Fraction and Nuclear Stability; Mass Defect and Nuclear Binding Energy (NBE)

278

5.5 Nuclear Models Fermi Gas Model; The Liquid Drop Model; Nuclear Shell Model; The Optical Model

282

5.6 The Fundamental Particles

296

Solved Numerical Problems

lc

Exercise-V

he

m

yl

5.4 The Nuclear Forces Nature ofthe Nuclear Forces; Exchange Force as the Nuclear Force; The Nuclear Potential and Nuclear Potential Barrier

ea

305

309-382

th

6. Radioactivity and Radiation Chemistry

299

6.2 Types of Radioactive Emanations

310

6.3 Properties of Alpha, Beta and Gamma Ra-diations Alpha (a)-rays; Beta (~)-Rays; Gamma (y) Rays; Comparison among the Different Types of Radioactive Emanations

311

6.4 Detection and Measurement of Radioactivity

315

6.5 Units of Radioactivity

320

6.6 Different Modes of Radioactive Decay

322

6.7 The Fluorescence and Auger Effect

323

6.8 Disintegration Chain and Soddy-Fajan Group Displacement Law

324

6.9 The Decay Kinetics and Parent-Daughter Growth Relationships Case I : The Daughter Nucleus is Stable; Case II : When the Daughter Element is Radioactive .

325

t.m

e/

6.1 Discovery of Natural Radioactivity

309

6.10 Radioactive Equilibrium

332

6.11 Energetics of the Disintegration Processes

336

6.12 Some Important Aspects of Alpha Decay

338

Contents xxi 6.13 Some Important Aspects of Beta Decay

340

6.14 Some Important Aspects of Gamma Emission

341

6.15 The Disintegration Series

342

6.16 Artificial or Induced Radioactivity

345

Discovery of Artificial Radioactivity; Production of Artificial Radioactivity

6.17 Applications of Radioactive Isotopes

347

ib

6.18 Some Important Aspects of Radiation Chemistry

ra

ry

Application of Radioisotopes in the Field of Chemistry; Application of Radioactivity in Biological Fields; Application ofRadioactivity in Agriculture; Application ofRadioactivity in the Medical Field; Application ofRadioactivity in Industry; Application ofRadioactivity in Age Detennination

363

yl

Radiation Chemistry; Characteristics of Interaction of Medium Energy Charged Particles with Matter; Interactions ofNeutrons with Matter; Interaction of Gamma-Radiation with Matter; Radiation Dosimetry; Radiolysis of Water

m

Solved Numerical Problems

he

Exercise-VI

lc

7. Nuclear Reactions

371 379 383-447 383

7.2 Bethe's Notation of Nuclear Reactions

384

7.3 Nuclea"r Reactions Versus Chemical Reactions

385

7.4 Classification of Nuclear Reactions

386

7.5 Conservation Laws in Nuclear Reactions

390

th

ea

7.1 Artificial Transmutations and Nuclear Reactions

t.m

e/

Conservation of Energy; Conservation of Linear Momentum; Conservation of Angular Momentum; Conservation of Neutrons and Protons

7.6 Threshold Energy of a Projectile for a Nuclear Reaction

392

7.7 Nuclear .Reaction Cross-Section

392

7.8 Mechanisms of Nuclear Reactions

393

The Compound Nucleus Theory; Oppenheimer-Phillip~ Mechanism in Stripping Reactions; Direct Nuclear Reaction Mechanism

7.9 Charged Particle Accelerators

397

The Linear Accelerator; The Cyclotron

7.10 Filling in the Gaps in the Periodic Table: The Man-Made Elements

401

7.11 Transuranic and Translawrencium Elements: The Man-Made Elements

402

7.1l Nuclear Fission

406

Historical Background; Characteristic Features ofNuclear Fission; Bohr-Wheeler Theory ofNuclear Fission: Liquid Drop Model; Charge Distribution in Fission Products: Principle of Equal Charge Displacement

xxii

Fundamental Concepts of Inorganic Chemistry

415

7.14 Nuclear Fusion:· Thermonuclear Reaction Characteristic Features of Nuclear Fusion; Hydrogen and Cobalt Bomb; Stellar Energy; Controlled Fusion Reactions: An Innocent and a Never-Ending Source of Energy; Cold Fusion

424

ra

ry

7.13 Fission Chain Reaction: A Source of Atomic Energy Factors Controlling a Fission Chain Reaction; The Fertile and Fissile Nuclides as the Nuclear Fuels; Atom Bomb; Fermi's Four Factor Formula and Nuclear Reactor or Atomic Pile for the Controlled Nuclear Fission; Breeder Reactor; A Natural Fission Reactor: Okla Phenomenon; Nuclear Reactors in India; Re.covery of Uranium and Plutonium from Spent' Fuel

ib

7.15 The Origin and Evolution of Elements: Nucleosynthesis of Elements Nl:lcleosynthesis of Elements from the Primordial Element Hydrogen in Stars; Cosmic -Abundance of Elements

yl

Solved Numerical Problems

m

Exercise-V-))

he

8. Periodic Table and Periodic Trends of Different Properties of Elements

435 444

448-541 448

ea

lc

8.1 Historical Background of the Development of Periodic Classification -Law of Triads; Pattenkofer's Rule of Integral Multiple; Chancourtois Law of Telluric Screw; Newlands' Law of Octaves; Mendeleev's and Lothar Meyer's Periodic Laws

429

451

8.3 Characteristics and Usefulness of the Mendeleev's Periodic Table Systematic Classification of the Elements; Correction of Atomic Weights; Prediction of the Missing Elements

455

t.m

e/

th

8.2 Mendeleev's Periodic Law and Periodic Table Characteristics of Mendeleev's Short Periodic Table

8.4 Defects in the Mendeleev's Periodic Table .

456

8.5 Long Form of the Modern Periodic iable Important Segments of the Long Form of the Modem Periodic Table; Advantages of the Long Fonn of Periodic Table; Defects of the Long Form Periodic Table

458

8.6 Extended Periodic Table: Superheavy Elements: Postactinides and Superactinides

462

8.7 Classification of the Elements Based on the Electronic Configuration and Chemical Affinities Classification Based on the Position ofthe Differentiating Electron; Bohr's Classification Based on Electronic Configuration; Goldschmidt's Geochemical Classification ofElements Based on Chemical Affinity 8.8 Characteristics and Position of the Border Line Elements Position of the Zn-Family; Positions of Lu, 1n and Lr; Position of the Lighter Actinide~; 4fSeries (Lanthanides) vs. 5fSeries (Actinides)

464

467

Contents xxIII 8.9 Lanthanide and Actinide Contractions (i.e. I-Contractions) Nature ofthe}Contraction; Explanation offContraction; Effects ofLanthanide Contraction

469

. 8.10 Periodic Trends of Z* and Size of the Atoms and Ions Two Important Factors (n and 2*) to Determine the Size of Atoms and Ions; Periodic Trends of 2* (Effective Nuclear Charge); Variation of Size along the Periods; Variation

473

of Size in a Group

478

ry

8.11 Variation of Atomic Volumes .in the Periodic Table

ra

Atomic Volume: Definition and Limitation- of the Parameter; Nature of Variation Trend ofthe Atomic Volume with the Atomic Number; Explanation ofthe Atomic Volume Curve

480

ib

8.12 Variation of Ionization Energies in the Periodic Table (cf. Sec. 9.11 for Ionisation Energies of Molecules and Photoelectron Spectroscopy)

t

yl

Periodic Variation of the Ionisation Energies; Factors Governing the Ionisation Energies and Explanations of the Periodic Trends; The Cases which Flout the General Periodic Trend; Ionisation Energy of Some Typical Molecular Species

m

8.13 Electron Affinity and its Variation Trend in the Periodic Table

486

8.14 Concept of Electronegativity

he

Concept, of Electron Affinity; Factors Governing the Electron Affinities; Variation of Electron Affinity in the Periodic Table

491

e/

th

ea

lc

Pauling's Electronegativity; Mulliken's Orbital Electronegativity; Jaffe-Hinze Electronegativities (Mulliken-Jaffe electronegativity); Allred-Rochow's Electronegativity: Sanderson's Electronegativity; Calculation of Charge Separation from Electronegativity: Concept of Electronegativity Equalization; Electronegativity Not an Inherent Property of the Elements; Examples Illustrating the Variation of Electronegativity Depending upon the Structural Characteristics; Group Electronegativity; Concepts ofAbsolute Electronegativity; Optical Electronegativity and Spectroscopic Electronegativity

t.m

8.15 Periodic Trends of Electronegativity

507

8.16 Applications of the Electronegativity Concept in Different Fields of Chemistry

507

8.17 Periodic Trends of Cohesive Energy (i.e. Melting and Boiling Points)

511

8.18 The Diagonal Relationship

511

Causes of the Diagonal Relationship; Comparison of the Physical Constants of the Three Diagonal Pairs; Similarities in the Pair, Li and Mg; Similarities in the Pair, Be and AI; Similarities in the Pair, Band Si

8.19 Basic Difference in Group Trends Between the Representative and Transition Elements

517

8.20 Secondary Periodicity : Anomalies among the Post-transition Elements and Postlanthanides

517

Secondary Periodicity and Consequences of Secondary Periodicities; Explanation of the Anomalies in Properties of the 4-th and 6-th Period Posttransition Elements; Relativistic Effects on Chemical Properties of the Fifth and Sixth Row Elements

xxiv Fundamental Concepts of Inorganic Chemistry 8.21 Position of Hydrogen in the Periodic Table

524

8.22 Peculiarities of the First and Second Row Elements Anomalou~ Properties of Fluorine : Superhalogen or Subhalogen? Anomalous Properties of Lithium

525

8.23 Comparison of Transition (i.e. d-Block) and Inner Transition (i.e. I-Block) Metals

528

8.24 Variation of Oxidation States of s- and p-Block Elements

530 532

ry

Solved Numerical Problems Exercise-VIII

I: Units and Conversion Factors

A-I

ib

Appendix

A-1-A-12

ra

Appendices

535

A-4

Appendix III: Wavelength and Colours

A-5

yl

Appendix II: Some Physical and Chemical Constants

A-II

he

Appendix VI: Books Consulted

m

Appendix IV: Names, Symbols, Atomic Numbers and Atomic Weights of the Elements A-6 Appendix V: Some Useful Mathematical Relationships A-8

1-1-1-20

lc

Index

ea

VOLUME 3

1110-1249

e/

th

12. Bonding in Metals and Metal Clusters: Electrical Conductivities of Solids: Semiconductors and Superconductors

1110

12.2 Crystal Structure of the Metals and the Effects of Lattice Structure on their Properties

1111

t.m

12.1 General Properties of Metals

12.3 The Free Electron Theory of Metallic Bonding Drude-Lorentz Classical Free Electron Theory: Electron Sea Model; Sommerfeld's Quantum Mechanical Free Electron Theory and Fermi Dirac Distribution Function; Quantitative Aspects of Some Properties of Metals; Electrical Conduc~ivity and Hall-Effect

1114

12.4 The Bloch Theory (i.e. Zone Theory) of Metallic Bonding Basic Concepts of Bloch Theory; Brillouin Zone, Forbidden Zone and Conductivity

1131

12.5 Molecular Orbital Theory or Band Theory of Metallic Bonding Formation ofEnergy Bands and Brillouin Zones; Band Structure ofNontransition Metals; Band Structure of Transition Metals; Valence and Conduction Bands; Explanation of Electrical Conductivities of Solids in the Light of Band Theory; Optical Properties of Solids in' the Light of Band Theory; Magnetic Properties Including the Temperature Independent Paramagnetis (TIP) (Le. Pauli Paramagnetism) of the Metals in Terms of

1136

l,;ontents xxv Band Theory; Ferromagnetism in Transition and Inner-transition Metals; Band Structure of Some Inorganic Solids and their Properties

1154

12.7 Solid Solutions: Alloys: Intermetallic Compounds Interstitial Alloys; Substitutional Alloys, Intermetallic Compounds and Hume-Rothery Rule

1156

12.8 Metal-Metal Bonds in Metal Clusters Binuclear Compounds; Trinuclear Compounds; Octahedral Clusters; Chevrel Phase; Octahedral Metal Cluster; Carbonyl Clusters; Metal-only Clusters (Zintle Cluster Ions); Metal-Metal Bonds in Stacked Polymeric Structure; Conditions for Metal-Metal Bonding

1161

12.9 Semiconductors and their Properties in the Light of Band Theory Characteristic Features of Intrinsic Semiconductor; Characteristic Features of Extrinsic or Doped Semiconductors; Temperature Dependence of Semiconductor Conductivity; Optical Properties of Solids; Uses of Semiconductors : Direct and Indirect Gap Semiconductors; Some Useful Semiconductor Materials and their Properties; Characteristics of p-n Junctions; Characteristic Features of p-n-p and n-p-n Junction Transistors; Some Applications of Semiconductor Devices

1181

he

m

yl

ib

ra

ry

12.6 Valence Bond Approach of Metallic Bonding , Valence Bond Approach for the SholtPeriod Metals; Valence Bond Approach for the Long Period Metals; Merits ofthe Valence Bond Approach; Demerits ofthe Valence Bond Theory

12.10 Semiconductivity and Metallic Conductivity of Metal Oxides: Hopping Mechanism of Conductivity

lc

1209 1214

ea

12.11 Electrical Conductivity in One Dimensional Solids : Peierls Distortion Electrical Conductivity in KCP [Ptossium tetracyanoplatinate(II)] and Peierls Distortion; Electrical Conductivity in Polythiazyl (SN)x (Le. ~ [SN] - a one dimensional metal); Electrical Conductivity in trans-Polyacetylene (C2H 2 )x or (CH)x -an Organic Semiconductor

e/

th

12.12 Conducting Organic Substances: Organic Metals, Organic Semiconductors, Molec-~Iar Inorganic Superconductors IntrodUction; Some Representative Examples ofOrganic Salts dConducting Substances: Organic Superconductors

t.m

12.13 Superconductivity Concept ofSuperconductivity; Characteristic Features ofSuperconducting Materials; BCS Theory of Superconductivity; Superconducting Materials; Structural Features of Some Copper-containing Superconducting Ceramic Materials (Warm Superconductors); Application of Superconductivity 12.14 Electrical Conductivity of Solid Electrolytes

1218

1222

1232

Solved Numerical Problems

1236

Exercise-XII

1241

13. Hydrogen Bonding and Other Weaker Chemical Forces Including Supramolecular Systems

1250-1342

13.1 Hydrogen Bonding Types of Hydrogen Bonding Depending on the Nature of Electron Clouds on Donor (D); Types of Hydrogen Bf?nding Depending on the Position of Donor (D); Participation of

1250

xxvi

Fundamental Concepts of Inorganic Chemistry

C-HBonds in H-Bonding; Symmetrical and Unsymmetrical Hydrogen Bonding; Hydrogen Bond Energy and Bond Length; Effects of Intermolecular H-Bonding; Effects of Intramolecular H-Bonding; Detection of H-Bonding; Theories of Hydrogen Bonding; Agostic Interaction· vs. Hydrogen Bonding Interaction; Comparison among H-bridge Bond, H-Bond and Agostic Interaction

13.2 Hydrates and Clathrates or Cage Compounds

1279

ry

Hydrates; Clathrates or Cage Compounds; Configurational Entropy and Thermodynamic Aspects of Clathrate Compounds

13.3 The Nature of Intermolecular Forces: van der Waals Forces: Lennard-Jones Potential: The 6-12 Potential

1285

ib

13.4 Interaction Between Ionic and Covalent Compounds

ra

The Attractive Intermolecular Forces; Total Intermolecular Attractive Forces (van der Waal~ Forces); The Repulsive Intermolecular Forces; Lennard-Jones Potential

1295

yl

Ion-Dipole Interaction; Ion-Induced Dipole Interaction; Metal-Ligand (neutral) Interaction in Terms of Ion-Dipole Interaction and Polarisation ofthe Metal Ions and Ligands; HardHard Interaction (Electrostatic Interaction) and Soft-Soft Interaction (Mutual Polarisation)

m

13.5 Noncovalent Interactions in Supramolecular Systems and Molecular Recognition

1298

e/

th

ea

lc

he

Interactions in Supramolecules; The Lock and Key Principle in Supramolecular Assemblies; Representative Examples of Supramolecular System; Double Helical Structure of DNA; Macrocyclic Systems (Synthetic Molecular Receptors for Cations); Molecular Recognition of Specific Metal Ions by Macrocyclic Systems; Crown Ethers as Receptors; Spherical Recognition of Metal ions; Molecular Recognition of NHt and RNH; (Primary Alkyl Ammonium Cation) by [18]-Crown-6; Selective Perching or Nesting of NHt or RNH; on Crown-6; Cryptands; Synthetic Molecular Receptors for Different Types of Substrates - Cations, Anions, Neutral Moiecules; Application of Macrocyclic Crown. Ethers and Cryptands; Polyazamacrocycles and Polyazacryptands as Anion Receptors (Synthetic Receptors); Molecular Recognition by Macrocyclic Systems; Calixarenes as Receptors; Cyclodextrins (CDs) as Natural Receptors; Cyclophanes-as Synthetic Receptors for Apolar Guests; H-Bond Directed Molecular Assembly; Metal Coordinated Self-assembly; Molecular Switch; Catenanes and Rotaxanes; Inverse Crown Ethers : Novel Anion Receptors

1324

t.m

13.6 Energetics of Dissolution Thermodynamic Aspects and Conditions ofDissolution; Born Equation and Ion Solvation; Energetics of Solubilities of Ionic Salts in Polar Solvents; Solubilities of Ionic Salts in Nonpolar Solvents; Solubilities of Nonpolar Solutes in Nonpolar Solvents; Solubilities of Nonpolar Solutes in Polar Solvents ,

-

~

-

13.7 Energetics of Phase Transitions - Melting,' Boning, Sublimation

1332

Phase Transition in Covalent Compounds; Ionic Compounds; Ionic-Covalent Compounds (Le. Fajans Type Compounds); Ease of Phase Transition Depending on the Nature of Involved Chemical Forces

Exercise-XIII

1336

14. Acids and Bases and Ionic Equilibria in Aqueous Solutions 1343-1554 14.1 Definitions of Acids and Bases

1343

Contents xxvii 14.2 Arrhenius Theory of Acids and Bases

1343 1344

14.4 Mechanism of Buffer Action: Buffer Capacity

ib

14.5 Selection of Acid-Base Indicators in Acid-Base Titrations

ra

ry

14.3 Brtinsted-Lowry Concept: The Proton Donor-Acceptor Concept: Protonic Concept: The Conjugate Acid-Base Theory Basic Concepts of Brl>nsted-Lowry Theory; Relative Strength of Acids and Bases: Differentiating and Levelling Solvents in the Light of Brl>nsted-Lowry Concept; Merits ofthe Brl>nsted-Lowry Concept; Limitations ofthe Brl>nsted-Lowry Concept; Cosolvating Agents and Acid-Base Strength; Application of Brl>nsted-Lowry Concept in Calculating pH ofAqueous Solutions: A Generalised Approach to Calculate the pH Values ofDifferent Types ofAqueous Solutions; Acid-Base Behaviour ofAm"ino Acids in Aqueous Solution: Is~electric Point; The Chemistry of Proton in Water: Proton Transport Process

1372 1382

m

yl

14.6 Hammett Acidity Function: Superacids Concept of Hammett Acidity Function (If); Concept of Superacids and Examples of Superacids; Application of Superacids

1370

1386

ea

lc

he

14.7 The Lewis Concept: The Electron Pair Donor-Acceptor System Basic Concepts and Characteristic Features ofthe Lewis Theory; Different Types ofLewis Acids; Classification ofthe Lewis Bases; Lewis Acid-Base Reactions and Redox Reactions; Some Typical Examples of Lewis Acids and Bases, Lewis Acid-Base Adducts; Covalent and Ionic Contributions in the Stabilities of the Adducts : Drago-Wayland Equation; Demerits of Lewis Acid-Base Theory; Solvent Properties in the Light of Lewis AcidBase Concept; Change of Bond Length in Lewis Acid-Base Interaction

1414

t.m

e/

th

14.8 Strength of Brtinsted (i.e. Protonic) Acids and the Gover~ing Factors Involved and the Properties of Brtinsted Acids and Bases Strength of Different Protonic Acids and Conjugate Acids of Some Bases; Born-Haber Cycle and Thermodynamic Parameters to Determine the Acid Strength: Acid Strength of Hydracids and Carboxylic Acids: Gas Phase Acidity; Factors Governing the Strength of Oxyacids; Hydrolysis and Aqua Acid Strength of Metal Ions in the Light of Brl>nstedLowry Concept; Basicity ofthe Anions; Periodic Variation ofAcidic and Basic Character; Nucleophilicity and Basicity; Acld and Base Catalysis 14.9 Some Important Factors Governing the Acid-Base Strength (both Br6nsted and Lewis Acid-Base Systems) . Inductive Effect : Entropic Effect, d-Orbital Participation; State of Hybridisation of the Central Atom; Steric Factors: F-Strain and B-Strain; Solvation Effects (cf. H-bonding); Resonance and 1t-Bonding arid the Possibility of d-Orbital Participation in the Reacting Species; Effect of Resonance Stabilisation in the Product; Effect of H-Bonding on Acid Strength

1437

14.10 Lux-Flood Concept

1466

14.11 Solvent System Definition

1468

14.12 Usanovich Concept 14.13 A Generalised Concept of Acids and

Bas~s

1472 1472

xxviii

Fundamental Concepts of Inorganic Chemistry

14.14 Concept of Ultimate Acids and Bases 14.15 Hard and Soft Acids and Bases (HSAB)

1472 1473

Basic Concept and Principle of Hard and Soft Acids and Bases (HSAB) Theory; Basis of Classification of the Acids and Bases as Hard or Soft; Characteristics of Hard and Soft Species; Symbiosis in Hardening or Softening a Centre; Acid-Base Strength versus HSAB Principle; Application of HSAB Principle; Theoretical Background of HSAB Principle

ry

14.16 Surface Acids and Solid ~cids : Heterogenous Acid-Base Reactions 14.17 Aqueous Solutions 01 Salts: Hydrolysis of Salts: Solubility Product of Sparingly Soluble Salts

ib

I

yl

Solved Numerical Problems Exercise-XIV

15.2 Classification of Solvents

1555 1555

he

15.1 Introduction

1510 1541

1555-1576

m

15. Nonaqueous Solvents

1494

ra

Hydrolysis of Salts; Solubility Product and Activity Product of Sparingly Soluble Salts; Application of the Principle of Solubility Product --.

1492

1556

1.5.4 Liquid Ammonia as a Solvent

1558

lc

15.3 Characteristics of Ionising Solvents

1564

ea

15.5 Liquid Hydrogen Fluoride as a Solvent

1565

15.7 Acetic Acid (CHJC02H) as a Solvent

1566

15.8 Sulfuric Acid as a Solvent

1567

th

15.6 Liquid Hydrogen Cyanide as a Solvent

15.9 Fluorosulfonic Acid (HSOJF) as a Solvent and Superacids

e/

1568 1569

t.m

15.10 Liquid Sulfur Dioxide as a Solvent 15.11 Bromine Trifluoride as a Solvent

1570

15.12 Oxyhalides as Solvents

1571

15.13. Molten Salts (i.e. Ionic Liquids) as Solvents

1572

Exercise-XV

1575

16. Redox Systems and Electrode Potential: Application of Electrode Potentials: Electroanalytical Techniques

1577-1783

16.1 Some Preliminary Aspects of Redox Reactions Ion Electron Method of Balancing Redox Reactions; Oxidation Number and Rules for ·Calculating Oxidation Number; Oxidative Addition and Reductive Elimination Reaction; Equivalent y.Jeights ofOxidants and Reductants; Complementary "and Noncomplementary Redox Reactions; Redox and Acid-Base Reactions; Disproportionation and Comproportionation Reactions

1577

Conter ,ts xxix

16.2 Electrode Potentials: Standard Potentials

1596

Electrical Double Layer and Genesis of Electrode Potential; Determination of Electrode Potential; Standard Electrode Potential : Nernst Equation: Stoichiometric Standard Potential and Formal Potential : Concept of pE; Sign Conventions of the Electrode Potentials; Standard Electrode Potentials of Metals and Electrochemical Series; Standard Reduction Potentials; Reversible Chemical Cells: Cells with and without Transference; Galvanic vs. Electrolytic Cell: Ohmic Potential and Observed Cell emf

ry

16.3 Formal (Conditional) Potentials (cf. Conditional Stability Constant vs. Thermodynamic Stability Constant)

1610

ib

ra

16.4 Effects of Different Factors on Electrode Potential: Application of Such Effects in Analytical Chemistry Effects of pH on Electrode Potentials due to the Participation of Jr or O~ in Electrode Process (cf. Pourbaix diagrams); Effects of pH on Electrode Potentials due to the

1607

he

m

yl

Formation of Sparingly Soluble Hydroxo-species (cf. Pourbaix diagrams); Effects of Precipitation (other than hydroxides) on Electrode Potentials;· Effects of Complexation on Electrode Potentials; Chemistry of Aqua Regia and Dissolution of Noble Metals: Enhancement ofOxidisability of Metals through Complexation; Attack ofNoble Metals by HF and by a Mixture of HF and HN03 : Enhancement of Oxidisability of Metals through Complexation; Dissolution of Antimony by a Mixture ofNitric Acid and Tartaric Acid : Enhancement of Oxidisability of Metals through Complexation

1640

lc

16.5 Periodic Trends of Electrode PQtential

ea

Variation of Standard Oxidation Potential of the Metals; Variation of Oxidising Power of the Oxyanions in a Group

1643 1644

16.8 EMF Diagrams

1647

e/

th

16.6 Function of Zimmermann-Reinhardt Solution (in Titrating Fe(II) by KMn04 in the Presence of Chloride) 16.7 Instability of Some Species in Aqueous Solution

t.m

Latimer Diagram; Frost Diagrams (Ebsworth Diagram); Pourbaix Diagram or PotentialpH (E-pH) Diagram

16.9 Equilibrium Constant from the Standard Electrode Potentials

1661

16.10 Disproportio.nation and Com proportionation Reactions

1663

16.11 Potential Profile in a Redox Titration

1672

16.12 Selection of a Redox Indicator

1677

16.13 Potentiometric Titrations

1680

16.14 Practical Applications: Electrochemical Cells and Batteries Secondary Cells, Fuel Cells: Corrosion

Primary Cells,

Basic Requirements of an Electrochemical Cell to Act as a Power Source; Representative Examples ofPrimary and Secondary Cells; Some Representative Fuel Cells; Electrochemistry of Corrosion; Electrolytic Cell vs. Galvanic Cell : Electrolysis of Solutions and Electrodeposition; Decomposition Voltage of Electrolysis and Electrolytic Separations of Metals; Some Other Practical Applications of the Knowledge of Electrode Potential

1686

xxx

Fundamental Concepts of Inorganic Chemistry

16.15 Kinetic Factors in Electrode Process and Electron Transfer Reactions Importance of Kinetic Factors; Importance of Overpotential (in Electrode Process and Cell Potential): Tafel Equation; Redox Reactions Through Electron Transfer; Redox Reactions Through Atom Transfer

1704

ry

16.16 Photochemical Reactions: Photoredox Reaction and Photochemical Splitting of Water 1708 Some Representative Photochemical Reactions Illustrating the Characteristic Features; Photochemical Splitting of Water and Photochemistry of Ru(bpyX+; Ti02, an Important Photocatalyst; Direct Photochemical Reduction ofDintrogen; Charge Transfer Band and Redox Stability 1719

16.18 Hydrometallurgy

1722

ib

ra

16.17 Ellingham Diagram: Reduction of Metal Oxides: Carbon - A Potential Reducing Agent 16.19 Electrode Potentials in Nonaqueous Systems

1723 1724

16.21 Redox Activity in Relation to Explosive Action

1726

m

yl

16.20 Examples of Some Common Catalysed Redox Reactions

1727

1752

lc

Solved Numerical Problems

he

16.22 Chemistry of Some Important Electroanalytical Techniques Polarographic Method of Analysis; Cyclic Voltammetry (CV); Amperometric Titration; Coulometric Analysis

1770

Appendix 16A: Chemistry of Explosives

1773

Appendix 16B: Chemical Clock Reactions: Oscillating Reactions

1777

Exercise-XVI

th

ea

Numerical Problems

e/

Appendices Appendix

I: Units and Conversion Factors

1763

A-1-A-12

A..:J A-4

Appendix III: Wavelength and Colours

A-S

t.m

Appendix II: Some Physical and Chemical Constants

Appendix IV: Names, Symbols, Atomic Numbers and Atomic Weights of the Elements A-6 Appendix V: Some Useful Mathematical Relationships A-8 A-II Appendix VI: Books Consulted

Index

1-1-1-18

ry ra ib yl

lc

he

m

VOLUME 2

th

ea

9. Introduction to Chemical Bonding and Theories of Covalence: Valence Bond Theory (VBT) and Molecular Orbital Theory (MOT)

t.m

e/

10. Covalent Compounds: Characteristics, Structure and Reactivity 11. Structure, Bonding and Properties of Ionic Solids and Solid State Chemistry Appendices Appendix Appendix Appendix Appendix

I: II: III: IV:

Units and Conversion Factors Some Physical and Chemical Constants Wavelength and Colours Names, Symbols, Atomic Numbers and Atomic Weights of the Elements Appendix V: Some Useful Mathematical Relationships Appendix VI: Books Consulted

9

yl

ib

ra

ry

Introduction to Chemical Bonding and Theories of Covalence: Valence Bond Theory (VBT) and Molecular Orbital Theory (MOT) 9.1 HISTORICAL BACKGROUND

m

9.1.1 Development of the Concept of Valency

t.m

e/

th

ea

lc

he

At the early dawn of nineteenth century, several distinguished chemists like Berzelius, Leibig, Dumas, Laurent, Gerhardt, Frankland and many others attempted to account for the driving forces responsible for chemical bonding. It was established that the combining power of the atoms is different for different elements and for a particular element it is fixed. This is why the combining capacity was referred to as atomicity by a·erhardt and the term was replaced by quatrivalency by Hoffman (1865). This was shortened into valency or valence by Wichelhaus in 1868 and it is popularised by this name. After a long period of time, in the later part of the nineteenth century, a more or less qualitative idea of the mechanism of combination of the atoms got crystallised from the contributions of a number of thinkers like Kekule, van't Hoff, Ie Bel, Werner and others. To explain the structure of the saturated hydrocarbons,. van't Hoff and Ie Bel (1874) independently proposed the quadrivalency of carbon and these are projected to the corners of a regular tetrahedron. This idea of tetrahedral quadrivalent carbon yielded an outstanding achievement in explaining the structure, optical activity and isomeric configurations of different organic compounds. Another important prediction on the nature of bonding in unsaturated organic compounds made by Kekule (1857) somewhat earlier could also be interpreted by the tetrahedral carbon. Thus the Kekule's theory to explain the structure and properties of benzene got a sound support from van't Hoff and Ie Bel. In this connection, the theory of partial valency proposed by Thiele (1899) to explain the behaviour of the organic compounds having conjugated double bonds made a significant contribution. On the other hand, to explain the structure of the coordination compounds, Werner ( 1891 ) proposed the idea of primary or principal valency and secondary or residual or auxiliary valency. This also made an outstanding contribution in understanding the bonding mechanism in coordination compounds. All these ideas were developed by the chemists based on the factual observations of the compounds. But the development of the atomic structure by Rutherford and Bohr provides an 542

Introduction to Chemical Bonding and Theories of Covalence 543

m

yl

ib

ra

ry

understanding of the inner mechanism of the bondiQg. Actually, the knowledge of electronic configuration of the atoms gave a reasonable interpretation of the bonding mechanism. This is often referred ~o as electronic theory of bonding. The foundation-stone of the electronic theory of valency was laid independently almost at the same time by Kossel (1916) in Germany and Lewis (1916) in the USA. Kossel pointed out that the elements (e.g. halogens) just preceding the inert gases are strongly electronegative while the elements (e.g. alkali metals) which are immediately following the in~rt gases are highly electropositive. The electronegative atoms tend to attain the nearest inert gas configuration by accepting the requisite number of electrons in the valence shell, and the electropositive atoms lose their outermost electrons to attain the nearest inert gas configuration. Thus they tend to attain the stability by acquiring the inert gas configurations. In this way, through the electron transfer process, the electronegative atoms form the anions while the electropositive atoms generate the cations. These oppositely charged ions are bound through Coulombic forces. Thus the Kossel's idea gives the mechanism of ionic bonding. The number of electrons involved in the transfer process to attain the inert gas configuration g~ves the valency ofthe element. Thus, the Kossel's idea gives an interpretation of Abegg's theory ofcontravalency (or simply, the rule of eight), an empirical relationship between the valericy of an element and its periodic group number.

he

The Kossel's theory consi~ered the chemical forces as the electrostatic attractive forces which were also proposed by Berzelius in his dualistic theory much earlier. But the nature of this force can hardly be rationalised for the nonpolar molecules, specially for the homonuclear diatomic molecules like H2, e12, N2, etc. The existence of cations and anions of the same element appears unthinkable.

th

ea

lc

Thus,the Kossel's ide~ is only applicable for the ionic compounds. To explain the chemical force involved in forming the homopolar molecules, Lewis in the same year i 1916) proposed a new idea. According to him, to attain the stable nearest inert gas electronic configuration, the complete transfer of electrons is not necessarily required. This can be attained by sharing the ele~tron pairs by the combining atoms.

e/

9.. 1.2 Lewis-Langmuir Concept

t.m

When the atoms or other species combine throllgh the sharing of an electron pair (2c - 2e, i.e. two centre-two electron bond) to attain the inert gas configuration (ns 2 or ns 2 np6), the corresponding linkage (very often represented by a bar - ) is called covalent bond. The electrons in the pair residing in a particular orbital must be of opposite spin. This concept is known as the Lewis-Langmuir concept. The sharing electron pair (a common property between the combining atoms) may be constituted by each of the two combining species or by only one of the two species. In the second case, the species which provides t~e electron pair is called donor (Le. Lewis base) and the one which accepts the pair is called acceptor (Le. Lewis acid). Classically, this type of bond is called coordinate covalent bond. The linkage is very often represented by an arrow (~) pointing towards the acceptor species. For example, in the case ofammonia-boron trifluoride adduct formation, the lone pair residing on nitrogen ofammonia is donated to boron in boron trifluoride' as H3N: ~ BF3 • This idea of coordinate covalent bond is extensively utilised in explaining the Lewis acid-base interaction (see Sec. 14.7) and complex compound formation. However, in the present status of knowledge, it is established that there is no basic difference between an ordinary covalent bond (where each of the species provides one electron) and a coordinate covalent bond. This indistinguishability is justified from the experimental facts. For

544

Fundamental Concepts of Inorganic Chemistry

example, in the classical concept, NH; should have three ordinary covalent bonds and one coordinate covalent bond, but experimentally all the four bonds are found identical to hold the four identical hydrogens. The number of covalent linkage between the combining species may be single or double on triple depending upon the situation to reach the inert gas configuration.

9.1.3 The Octet Rule

ib

ra

ry

In most of the cases, through the covalent linkage formation, each of the combining species attains the octet, Le. ns 2 np6, configuration. If the species already bears the octet configuration, it will not display any tendency to form any covalent linkage. In the case of hydrogen, the inert gas configuration of helium, i.e. ns 2, is attained. This is very often referred to as the rule of duplet. There are a lot of examples in which the rule of octet and duplet have been satisfied. Some representative examples are

t.m

e/

th

ea

lc

he

m

yl

H2 (H-H),N2 (:N=N:),02 (Q=Q), CO2 (Q =C= Q), H20 (H-Q-H), :NH3 ,CCI4 ,etc. Deviation from the Octet Rule : A number of cases are known where the combining species or atoms have less than eight (Le. incomplete octet) or, more than eight (Le. expansion ofoctet) electrons in the covalently bonded molecules. Incomplete Octet: In the molecules such as BeCI2, BCl3 and NO (: N = 0), the central atoms, Le. Be, B, and N, bear four, six and seven electrons respectively. Here it is nOfeworthy that other atoms except the central ones in the above compounds maintain the octet rule. As a matter of fact, a large number of compounds formed by Be (quartet), M (sextet) where M = B, AI, Ga are known to have incomplete octets. These compounds are very often referred to as electron deficient comRou'nds (see Sec. 9.21) characterised by a profound tendency to receive back a lone pair (for Gr 13 or IlIA elements, e.g. B, AI, Ga, etc.) or two lone pairs (for Gr 2 or IIA elements, e.g. Be) to attain the octet. This is why, the electron deficient compounds act as Lewis acids. Expansion of Octet : In the compounds like, PCls, CIF3, SF6 , SiF62-, etc. the central elements, Le. P, CI, Sand Si, are bearing ten, twelve, and twelve electrons respectively to display the expansion of octet. Similarly, in OsFs' there are sixteen electrons around Os. Here, also except the central atoms, all other atoms satisfy the octet rule.

9.1.4 Sugden'S View of Singlet Linkage Bond (2c-1 e) in Favour of 9ctet Rule To explain the expansion of octet which apparently contradicts the rule of octet, Sugden proposed that to maintain the octet rule, some of the bonds may arise by sharing a single electron (Le. 2c-1 e, two centre-one electron bond) instead of 2c-2e bonds. This single electron bond (Le. 2c-1 e) is very often referred to as singlet linkage or half-bond. This singlet linkage is represented by a half arrow (~) pointing towards the acceptor. Thus, it is believed that in PCls, there are two singlet linkages in addition to the three ordinary covalent (2c-2e) bonds leading to the octet (Le. 2 x 1 + 3 x 2 = 8) at P. Similarly, in SF6, there are four singlet linkages along with the two ordinary covalent bonds to give the octet (Le. 4 x 1 +. 2 x 2 = 8) around S. This concept of singlet linkage is not supported from the experimental facts. In pels or Sf"6 , all the bonds are found equivalent. Definitely, 2c-Ie bonds are weaker than the 2c-2e bonds, but no such indication is supported by the fact. This is why, the Sugden's concept of singlet linkage to defend the octet theory is of no practical importance.

Introduction to Chemical Bonding and Theories of Covalence

545

9.1.5 Sidgwick's Rule of Maximum Covalency

ib

ra

ry

According to Sidgwick, it is proposed that the number of maximum electrons which can be accommodated around an atom in the valence shell depends on the periodic position of the concerned element. The empirical rule states as : the element, i.e. hydrogen, in 1st period can have maximum 2 electrons (i.e. 1 bond), for each of the elements in the 2nd period (i.e. Li to F) the maximum number of electrons to be accommodated is given by 8 (i.e. 4 bonds), similarly for the elements in the 3rd and 4th periods the maximum capacity to accommodate the electrons is 12 (Le. 6 bonds), while for the elements in higher periods it is limited by 16 (i.e. 8 b:onds). Thus, the maximum capacity to accommodate the electrons around an atom in its valence shell in a compound is mainly dependent on the number of orbitals available for bonding. This is why, the rule of octet is limited only for the elements in the second period, and it is nothing but a special case of Sidgwick's more generalised rule. Here it is worth mentioning that because of steric factors and other factors, all the available orbitals for a particular atom may not always participate in bonding to attain the maximum possible covalency.

yl

Tetravalence and trivalence of oxygen - typic.al examples

)~

o

o

''\. - \ N40 ,~

o

or

''\. - \ C-R ,~

o

th

ea

lc

he

m

Basic beryllium nitrate [Be 4 0(N03 )6] and basic beryllium carboxylates [Be 4 0(RCOO)6] where R == H, CH3 , C2H s etc. are the typical examples where the central O-site shows 4 coordination number and 4 covalence (in terms of2c-2e bonds).

e/

Fig. 9.1.5.1. Structure of basic beryllium nitrate or basic berrylium carboxylate.

t.m

In the said compounds, there is a regular tetrahedral arrangement offour Be-centres. The six bidentate ligands (i.e. NO; or RCO:;) span along the six edges of the tetrahedron; and the basic oxygen present at the centre ofthe tetrahedron coordinates the four Be-centres. In this structure, each Becentre is tetrahedrally surrounded by four oxygens. • In the crystal of ice; each O-site is also tetrahedrally surrounded by four hydrogen sites. Two of these bonds are considered to be H-bonds. In H 30+ (hydronium species), oxygen shows trivalence (three 2c-2e bonds).

• Hypervalent and subvalent compounds: In hypervalent compounds, the number ofelectrons around an atom (generally the central atom) exceeds an octet as in SF6 , PCIs etc. In subvalent compounds, the number of valence electrons around an atom is less than an octet as in PH2 , CH2, NO, N0 2, etc. The concept of hypervalence is discussed in detail below. 9.1.6 Hypervalence and Hypercarbon Compounds (cf. Sees. 9.12.12 and 9.13.5) It has been mentioned (Sec. 9.1.5) that in the formation of covalent compounds by the p-block elements, the octet rule is limited for the Period 2 elements but expansion ofoctet for the Period 3 and subsequent

546

Fundamental Concepts of Inorganic Chemistry

elements is very much common. The common examples are: PCls' SiF62- , SF6 , AIF;- , XeF2 , XeF4 , etc. Such compounds for which Lewis structure (may be an important. resonating structure) needs more than an octet ofelectrons at least for one atom (generally the central atom) are described as hypervalent compounds. Thus expansion of octet for the p-block elements leads to hypervalence. The ex-istence of hypervalence is controlled by the following factors .

ra

ry

• Availability and participation of the postvalence nd-orbitals (cf. Sec. 9.13.5) along with the ns and np-orbitals in bonding; for participation of nd-orbitals in bonding, the atom should link with the highly electronegative substituents. • Size of the atom showing hypervalence should be sufficiently large to accommodate more than 4 pairs of electrons (bonding + non-bonding).

t.m

e/

th

ea

lc

he

m

yl

ib

The Period 2 elements do not possess the 2d orbitals and they fail to show hypervalence. The hypervalent molecules (where the central atom shows the hypervalence) are generally represented by . N-X-L notation where N denotes the number of electrons present at the valence shell of the central atom (denoted by the symbol X) in the compound and L denotes the coordination number (C.N.) of the central atom in the compound. Thus in terms of the notation, XeF2 is represented by 1O-Xe-2; similarly, SF6 i~ represented by 12-S-6. (A) Bonding mechanism to show the hypervalence (cf. Secs. 9.12.12, 9.13.5) : In terms of VBT, participation of the postvalence nd-orbitals in bonding leading to the hybridisations like sp 3d, sp 3cJ2, sp 3d3 is essential for showing the hypervalence. For participation ofd-orbitals in such bonding, the atom should combine with the highly electronegative substituents. This causes the contraction of d-orbitals to facilitate the participation of d-orbitals in bonding (cf. Sec. 9.13.5). Here it should be pointed out that participation of these postvalence ndorbitals in bonding for the p-block elements is now called in question. By using the concept of multicentre bond like 3c-4e bond, the four atomic orbitals are sufficient to show the higher valence like 5, 6 (see Sec. 9.13.5). Thus participation ofnd-orbitals is not an essential condition for the hypervalence. In fact, theoretically existence of the hyperval~nt species like NF6-, CFZ- (in which there is no question of nd-orbitals) have been predicted and they might be isolated in future. Expansion of octet (Le. hypervalence) has been explained by using the concept of the multicentre bond in Sec. 9.13.5. Here it is important to mention that for showing the hypervalence, the linking with the electronegative substituents is also supported in the 3c-4e bonding process. Here the nonbonding molecular orbital (NBMO) which bears the electron pair is basically enriched with the character of the atomic orbitals (AOs) of the substituents (i.e. outer atoms). Thus, the NBMO pushes the excess electron density to the outer atoms. Thus electron pushing towards the outer atoms is favoured if the outer atoms are highly electronegative. In the light of 3c-4e bonding process, hypervalence is redefined as the 3c-4e bonding that leads'to an expansion of octet. Sometimes, the 3c-2e bonding model can lead to ~lectron deficient hypervalent compounds where no expansion of octet occurs (Sec. 9.13.5). (B) Hypercarbon compounds: There are many examples in which carbon attains the coordination number (C.N.) more than four. Such hypercoordinated carbon compounds are simply described as hypercarbon compounds. However, they should not be considered as hypervalent compounds of carbon as the hypervalence is generally defined in terms of 3c-4e bonds

Introduction to Chemical Bonding and Theories of Covalence

547

t.m

e/

th

ea

lc

he

m

yl

ib

ra

ry

leading to an expansion of octet. The hypercarbon compounds are generally found in carbido complexes (Fig. 9.1.6.1) where carbon is bounq to only metal atoms. The alkyl or aryl bridged bonds (M-C).l-M) also lead to hypercarbon centres. The common example is AI2(CHl )6 (Sec. 9.21) where the bridging carbon shows 5 coordination number. The methyl bridging bonding is explained by considering the 3c-2e bond (cf. Fig. 9.21.6). The 6 coordination number (C.N.) of carbon in ILi(CHl )]4 has been explained by considering the 4c-2e bond constituting each Li3C centre (Fig. 9.21.7). In carboranes (Sec. 10.12), organometallics and cluster compounds (Ch. 12), such hypercoordinated carbon centres are well documented. It is also argued that in generating the carbocations in superacids, the five coordinated carbon centres are the intermediates. In H; (cf. Fig. 9.12.2.2) and other related species (e.g. H-bridge bonds in B2H6) (Sec. 9.21), the 3c-2e bond can lead to hypercoordination at the central H-atom. The valence of carbon is restricted to four by the availability of four valence orbitals. In hypercarbon compounds, the carbon centre is linked to more than fOUf atoms. In such cases, the electron deficient multicentre bonds (e.g. 3e-2e bond) are prevailing. The case of hypercoordination number through the formation of electron deficient multicentre bonds is not restricted to carbon only. It happens for other elements also. The common examples are: H;, H-bridge bond as in IJ;H6, azoborane NB 11 H 12 where N is attached to 6 neighbouring. atoms, aurophilic cation [N(AuPPh3 )5 ]2+ where 'N is attached to 5 Au-centres. Note: In terms of the rigid definition of hypervalence, the compounds like C(AuL);, C(AuL)~+, CH~+, CH;, N(AuL);+ etc. cannot be considered as the hypervalent compounds, but these are very often described as hypervalent electron deficient compounds. The bonding is explained in terms of 3c-2e bond model (cf. Sec. 9.13.5). However, according to some authors, it is not absolutely necessary to retain the original idea ofhypervalence causing an expansion of octet. The expansion of octet is generally explained by considering the 3c-4e bonding scheme for the main group elements without involving the d-orbitals. In the compounds like CH;, CH~+, N(AuPR3 );+ (Sec. 8.20.3) etc., the bonding has been explained by considering the 3c-2e bonds by using the p-orbitals. In such cases, expansion of octet does not occur. This is why, such compounds are described as electron deficient hypervalent compounds (cf. Sec. 9.13.5). Examples 0/ hypercarbon: Carboranes - in C2Bl0H12 carbon shows the 6 coordination number; cubic coordination of carbon in theantifluorite structure of Be 2 C, 5 coordinated carbon in iron carbonyl carbide [Fe5(CO)15C]; 6 coordinated carbon in [Fe 6(CO)16C]2-, hedgehog cation (an aurophilic cation) [C(AuPPh 3)6]2+ (cf. Sec. 8.20.3); 8 coordinated carbon in [CoS(CO)lSC]2-etc. Metallic carbides (e.g. Fe 3C) also show hypercoordinated C-centres. Structure of some representative hypercarbon compounds are shown in Fig. 9.1.6.1. (C) Bonding mechanism in hypercoordinated carbon compounds (cf. Sec. 9.13.5): In organic chemistry, carbon always shows tetravalence by using the spn hybrid orbitals and pure p-orbitals (if remaining, Le. n < 3). Such orbitals make a very good overlap. In the carbido complexes when the carbon orbitals interact with the metal orbitals, then no such good overlap occurs. This is why, carbon shows the tendency to increase the coordination number (C.N.) through the/ormation o/multicentre electron deficient bonds (e.g. 3c-2e bond). For this purpose, the 4 valence orbitals of carbon are sufficient (cf. Sec. 9.13.5C, 9.21, 9.23).

548

Fundamental Concepts of Inorganic Chemistry

ry

2+

th

ea

lc

he

m

yl

ib

ra

2-

(iii)

M~~~M

·,·

/'~

tM,- ---:/_?::,--- ,-M 1 ,1/ , I /

',1//

/1' /1'

/

1

,

M~IC/M M

(iv)

t.m

e/

Fig. 9.1.6.1. Structural representation of some carbido complexes (in which carbon bil)ds to only metal centres) illustrating hypercarbon compounds. (i) [Fes(CO)sCj. (ii) [Au6(PPh3 )6Cj2+ (an aurophilic cation). (iii) [Fe6 (CO)16Cj2-. (iv) metal carbide core of [COaC(CO)1S]2- where the CO groups are not shown.

9.1.7. The Main Types of Chemical Forces The chemicalforces (called valence forces) are responsible to hold the atoms and molecules together through the chemical bonds. This combination is called bonding. There are different types of chemical forces to hold the species together.. Except the ionic and covalent forces, other chemical forces leading to combining ofdifferent species are called weak chemicalforces. These are important in aggregation of species. These weak chemical forces are discussed in Chapter 13. 9.1.8 Isosteric or Isoelectronic Species Originally, it was defined as : isoelectronic species contain the same number 01electrons and the same number olatoms. Such examples are CH4 , NH; and BHi (5 atoms and 10 electrons) which are also

Introduction to Chemical Bonding and Theories of Covalence 549 Table 9.1.7.1. Different types of chemical forces

(i) Ion-ion interaction (Le. ionic bond) (ii) Covalent bond (exchange interaction)

Example

Equilibrium distance (pm)

*Dissociation energy (kJ mol-I)

Na+·····F-

230

H-H

74

670 458

240

84

+

.(iii) Ion-dipole interaction

Na

/

H

----~

,

H

H

"

(v) Hydrophobic interaction

/

(vi) Van-der Waals (dispersion force)

"0 / ' H

CH - -- -H C / 2

Ne····Ne

280

20

- 300

-4.0

- 330

-0.3

ra

O----H,

/

ib

H

2"

yl

(iv) Hydrogen bond (dipole-dipole interaction)

ry

Type of force

~

he

m

* Dissociation energy denotes the energy required to dissociate the combined species into the units, e.g. H-H H + H; Na+----OH 2 ~ Na+ + OH 2, Ne····Ne ~ Ne + Ne.

ea

lc

isostructural (tetrahedral). Now the definition is broadened as : species having the same number of valency electrons are called isoelectronic species. Such examples are: (i) CH4 and SiH4 ; (ii) CO2, +N02 and CS2; (iii) 03 and S02; (iv) BF3, NO;, and COJ-; (v) NX3, PX3 (X= halogen) and C/O;; (vi) BN and c(jl+.

t.m

e/

th

The isoelectronic species are isostructural. Examples: (i) XY2 (with valence 16 electrons, linear structure) : CO2, CS2, +N02, HCN etc. Note: (i) With more than 16 valence electrons the species are angular (e.g. N0 2, NOi., etc.); (ii) XY3 (24 valence electrons, trigonal planar): BF3, NO;, CO;- etc.; (iii) XY3 (26 valence electrons, pyramidal) : NX3, PX3, C/O;, etc. The isoelectronic groups are :

..

..

(i) -CH3, - NH 2 , (it)

)CH

(iii)

" "

2•

/CH,

-

)NH'

..

QH, -~.: (7 valence electrons)

)9. (6 valence electrons)

/ N: (5 valence electrons)

The following series of compounds illustrate the replacement among the isoelectronic groups. H 3C-C(= O)CH3 (acetate), HII-C(=O)NH2 (urea), HO-C(=O)OH (carbonic acid), F-C(=O)F (carbonyl fluoride). Note: Isostoichiometric species are having the same molecular formula with different electronic configurations, e.g. H;, H 2' H;; and O2 , 0;, 0;- etc.

0;

550 Fundamental Concepts of Inorganic Chemistry

9.2 THE THEORIES OF COVALENT BOND: BORN-OPPENHEIMeR APPROXIMATION

ra

ry

The quantitative aspects of a covalent bond can be revealed in terms of wave mechanics. A covalent bond can be expressed by a suitable wave function describing the electrons under the influence of all the involved nuclei. Thus, we are to consider the Schrodinger equation for the wave function expressing the covalent linkage in the molecule and then it would be solved to obtain the eigen values and eigen functions. The Schrodinger equation for the wave function of a covalent linkage containing N nuclei and n electrons is given by : ... (9.2.1) H\V = E\V -.

yl

ib

where the Hamiltonian operator, H is given by :

... (9.2.2)

he

m

The terms M; and me denote the masses of the i-th nucleus and an electron respectively. The terms Vne , Vee and Vnn represent the nucleus-electron, electron-electron and nucleus-nucleus interactions respectively. Thus, the wave function \V depends on the coordinates of both the electrons and nuclei. The implicit Hamiltonian operator in Eqn. 9.2.2 involves both the nuclear and electronic motions.

t.m

e/

th

ea

lc

They can be separated by considering the Born-Oppenheimer approximation. According to this approximation, it is considered that the electrons being much lighter than the. nuclei move faster than the nuclei. Compared to the electronic movement in a particular time interval, the nuclei can be considered to remain almost stationary with the constant internuclear distances. Thus, the translational energy of the whole system can be considered separately as the sum of translational energies of the electrons and nuclei. Thus, the total wave function (\V) of the system can be regarded as the product of the electronic wave function (\Ve) which depends on both the coordinates of electrons and nuclei and nuclear wave function (\Vn) which depends only on the coordinates of nuclei. Thus \V = \Ve x \Vn. Similarly, the Hamiltonian operator can be split as follows:

where,

H= He+H n Hn=

N h2 - L -2- V 2 ;=181t

M;

I

h2 2 V H e = L-2- j +Vne + Vee +Vnn 81t

... (9.2.3)

N

and,

j=1

me

... (9.2.4)

where V stands for the potential energy for the electrostatic interactions. Thus, the electron wave function (\Ve) which is important to our present problem is given by, ... (9.2.5) The eigen values of the electronic wave function can be obtained by solving the above Schrodinger wave equation, i.e. Eqn 9.2.5. In the case of polyatomic systems, Eqn. 9.2.5 cannot be solved exactly

Introduction to Chemical Bonding and Theories of Covalence 551

without any approximation. l~o have the approximate solution, the two methods, Le. valency bond method (VBn and molecular orbital method (MOn, are well documented. The approaches ofthe two methods are totally different. But, both the methods after a series ofrefinement produce more or less the same results.

9.3 THE VALENCE BOND THEORY (VBT)

yl

9.3.1 The Valence Bond Theory for Hydrogen Molecule

ib

ra

ry

To explai-n the covalent linkage formation, valence bond theory (VBT) is well known. In VBT, the interactions among the various valence shell electrons are considered when the combining species approach close together from infinity. The equilibrium distance is attained when the potential energy drops to a minimum value. The VBT was proposed by Heitler and London (1927) and it was extended by Pauling and Slater.

he

m

Heitler and L"ondon proposed the theoretical framework ofthis method by utilising the wave mechanical concept of atom just one year after the fonnulation ofSchrOdinger's famous wave equation. Let the two separate hydrogen nuclei be represented by A and B and their corresponding electrons be labelled by 1 and 2. Let us consider the process offonnation ofa hydrogen molecule starting from these two isolated hydrogen atoms. The wave functions of the two separate hydrogen atoms in the ground state are given by,

lc

1

\VA (1) = \V B(2) =- exp (- r),

. (in atomic units) • ... (9.3.1.1)

ea

1t

where r is the distance of the electron from the corresponding nucleus in Bohr unit.

that,

Here,

t.m

e/

th

The energies of the isolated species are given by, EA = EB = EH and the corresponding Hamiltonian operators are : 2

h

2

--V1 + v:1 2 81tm

... (9.3.1.2)

... (9.3.1.3)

VI =

2

-~, (in CGS);

v.2 =

(in CaS);

rAJ 2

__e_ x _1_, (in SI); 41tE o rAl

2

e 1 - - - x - (in SI), 41tE O

rB2

'

Here, rAl denotes the distance between the electron 1 and its nucleus A. Similarly, nucleus-electron distance in the iSQlated B atom.

r B2

denotes the

• In atomic units, the distance is expressed in bohr and energy in hartree. These are defined as : 1 bohr = ao = 52.9 pm, 1 hartree = e2/a o = 27.1 eV.

552

Fundamental Concepts of Inorganic Chemistry

The simplest wave function of the two electron system is given by : \V' ~ \V A (1)\V B (2)

... (9.3.1.4)

where the respective electron is fully associated with its own nucleus. The Hamiltonian operator of the system is given by : ... (9.3.1.5) e

2

=- -

H

r A2

2

e

e

2

e

2

- - + - + ---;, (in CGS) R

rl2

r BI

ry

-,

where,

ra

1/47tEo factor is to be incorporated in SI system. Here, 'A2 denotes the distance between ~he electron 2 and the nucleus A, and r BI bears the similar meaning;"12 and R represent the distances between the

ib

electrons and between the nuclei respectively. The energy of the wave function (\II') is given by, E' = f\jl 'H\jI,d't, which leads to :

... (9.3.1.6)

f\jl A (l)\jI B (2) fl' \jI A (l)\jI 8(2) d't

m

Q=

where

yl

E' = E A + E B + Q

. 2 2

- f\jl A(l) -=-\jI A(l) d't - - f\jl 8(2) ~\jI8(2) d't

he

~I

~2

2

2

r l2

R

lc

e e + f\VA(I)\VB(2)-\VA(I)\VB(2)dt+-

... (9.3.1.7)

t.m

e/

th

ea

Here, in Eqn. 9.3.1.7, the first term denotes the Coulombic interaction energy between the electron 1 around the nucleus A and the nucleus B via the Coulombic potential; the second term denotes the similar interaction energy between the electron 2 around the nucleus B and the nucleus A; the third term represents the Coulombic interaction energy between the electrons and the fourth term shows the energy ofCoulombic interaction between the nuclei. As all the terms in Qcan be expressed in terms ofCoulombic interaction, Q is referred to as Coulomb integral. The binding energy of the hydrogen molecule is 'given by, where,

E:binding=

EA

E'-(EA +EB )=E'-2EH =Q

... (9.3.1.8)

= EB = E H

The solution of the wave function leads to Ebinding = 24 kJ moI- 1 at R o (equilibrium internuclear distance) = 90 pm compared to the experimental value 458 kJ mol- I at R o = 74.1 pm. Thus the calculated binding energy can only account for 5% of the experimental one. This poor agreement can be modified as follows. This binding energy is simply called Coulombic energy (J). In the real molecule, it is reasonable to consider that at the molec'ular distance, the electron 1 is not necessarily confined with the nucleus A, and the same thing is true for the electron 2. As a matter of fact, after the bond formation, the electron 1 has got a. certain probability to be associated with the nucleus B. Similarly, the electron 2 also spends a certain fraction of its time in the vicinity of the nucleus A. Considering this electron exchange phenomenon we get the two states, H A(1)HB(2), (I) ; and, HA(2) H B( 1), (II). This idea of electron exchange phenomenon was introduced by Heitler and London. The corresponding wave functions are described as Beitler-London wave functions.

Introduction to Chemical Bonding and Theories of Covalence 553 The wave functions \VI and \Vn for the states I and II are given by :

and,

\V I = \V A(I) \VB(2)

... (9.3.1.9)

\VII =\VA(2)'VB(I)

... (9.3.1.10)

Thus, the true wave function must be a combination of the two, Le.

= c1\V I + c2\V II = c1\V A(I) \V B(2) + c2\V A(2) \VB(I) \V a = c3 \V I - c4 \V II = c3\V A(I) \V B(2) - c4 \V A(2) \V B(I)

... (9.3.1.11)

\V s

ry

and,

... (9.3.1.12)

ib

ra

The" H2 molecule is a symmetrical one and gets an equal contribution from \VI and \VII' Le. c 1 = c2 and c3 = c4 • To evaluate c 1 and c3' the conditions/or normalisation of the respective wave functions are utilised as follows. .

c: f'l'; d't = I; or, c: f('I'[ + '1'11)2 d't = I; c)

+ c)2 + 2Ct2rJ \V I\V II d t

=I

yl

2

or,

f'l'[ 'l'lI d 't can be treated as follows:

m

Now,

he

f'l' J'I' II d't = f'l' A(1)'1' B(2) 'I' B(1)'1' A(2) d't]d't 2 (in which the general volume element dt is replaced by dt) and dt 2 for the electron I and 2 respectively).

f'l' J'I' IId't = f'l' A(1)'1' B(l)d't l f'l' A(2)'1' B(2)d't 2 =S AB x S AB

lc

or,

=S~B

ea

The two integrals, i.e. f'l' A(1)'1' B(I)d't] and f'l' A(2)'1' B(2)d't 2, are equal as they only differ in

2

th

labelling the electrons. The integrals denoted by SAB are called overlap integrals. 2

e/

Therefore, 2cI (I + S AB) = I, or, c. = ~

t.m

Similarly it can be shown that, c3

1

2

... (9.3.1.13)

2(1 + SAB)

=~

1 2

... (9.3.1.14)

2(1- SAB)

Thus, the two possible wave functions are :

... (9.3.1.15)

... (9.3. 1.16) Here, \VS is symmetric as it remains unchanged by the exchange of the electrons while \Va is antisymmetric as it changes its sign by the exchange of the electrons. The results are shown in Fig.

9.3.1.1. The symmetric combination leads to stabilisation while the antisymmetric one destabilises the system.

554

Fundamental Concepts of Inorg,anic Chemistry

+Ve

\ 'V' = 'II A(1) 'IIs(2)

o

Coulomb Energy

,,

,,~'IIcov

"

"

J:

w N

-Ve

I

,,

"-

,. "

Exchange Energy

""

ra

\

=C1['IIA(1) 'Vs(2) + 'VA(2) 'IIs(1)]

,,"#1'

.I~('IIH2 = (1 - A)'V cov + 'Ilion

,,

.I

,

......-.1_ _

m

---+

Covalent -Ionic Resonance energy (after considering Z*)

yl

"

ib

1 ,,,,

,,"

ry

,~

Internuclear Distance

he

Fig. 9.3.1.1. Valence bond treatment of H2 molecule. The energy of the system can be evaluated'as usual,

lc

Es = I\jI" H\jIsd't, and Ea = I\jIaH\jI ad't

ea

where,

and,

t.m

e/

th

It leads to

E.f =2EH +

Ea =2EH +

Q+A 2

... ·(9.3.1.17)

1 + SAB

Q-A 2

... (9.3.1.18)

I-S AB

where, Q is referred to as the Coulomb integral, and A is referred to as the exchange integral (A = J\jI A(l) \jI 8(2) i/' \jI A(2) \jI 8 (l)d't). The exchange integral appears due to the exchange of the electrons between the nuclei. This phenomenon provides a larger space for the electrons to move around both the nuclei. Thus according to the principle of the model, particle in a box, the energy of the system gets lowered down. This modified wave function leads to the binding energy,

Ebinding

= E.'t - 2EH =

Q+A 2

... (9.3. 1.19)

1 +SAB

The calculated value of the binding energy is found to be 303 kJ mol-I at Ro = 86.9 pm. The observed corresponding values are: 458 kJ mol-I and 74.1 pm. Thus, the calculated value can account

Introduction to Chemical Bonding and Theories of Covalence 555

for -- 66.6% of the observed binding energy. In the calculated value, Coulombic energy (.I) contributes about 10% only while the exchange energy (K) contributes about 90%. The wave function 'lis represents the covalent interaction and this is why, it is very often referred to as 'V COy' Le.

I'V

s

= 'V COy =ct ['V A(1) 'V B(2) + 'V A(2) 'V B(1)]

I

... (9.3.1.20)

\Vion

m

yl

ib

ra

ry

A further modification of the result is obtained by considering the effective nuclear charge z* instead of Z. This was introduced by Wang. The consideration of T is reasonable. In the isolated atoms, there is no possibility of electron shielding but'in the molecule, the two electrons are now prevailing and consequently, the electron screening phenomenon will result. It leads to the binding energy 365 kJ mol- 1 at Ro = 74.3 pm. Thus the agreement is enhanced to -- 80%. Another modification leading to the possibility of the existence of both the electrons to a particular nucleus improves the result further. This probability will introduce the consideration of, H~ H"B or . H~ H;. Due to the electron-electron repulsion, the electrons will try to avoid each other, but still there is a small but finite probability for the electrons to coexist. Thus the wave function, 'Ilion describing the ionic state is given by :

= Cs \V A (1) \V A (2) + c6 \V B(I) \V 8(2)

... (9.3.1.21)

Hz =

(1 - "') 'V COy + "''Vio"

lc

I'V

he

Thus, the resultant wave function of the actual hydrogen molecule is given by :

I

... (9.3.1.22)

t.m

e/

th

ea

where A « 1) denotes the mixing coefficient measuring the extent of mixing of the ionic wave function with the covalent wave function. This phenomenon introduces the idea of covalent-ionic resonance. The solution of Eqn. 9.3.1.22 yields, Ebinding = 388 kJ mol-I, at Ro = 74.9 pm (i.e. the agreement with the observation is 85%). Eqn. 9.3.1.22 indicates that the actual hydrogen molecule is some sort of hybrid of the following canonical forms which differ in the pattern of the distribution of the electrons. ionic ~ Jr:I1, -H: Jr covalent ~ H: H; No single one of the above canonical forms can explain the behaviour of the ~ctual molecule but the hybrid can. Table 9.3.1.1. Energy values and equilibrium distance for the ground state of H2 °molecule in vaJence bond approach

. Type ofwave function and modification

Energy

Uncorrected Heitler-London (incorporation of exchange energy) Consideration of shielding and covalent-ionic resonance Experimental value

(kJ mOll)

Internuclear distance (pm)

24 303 388 458

90.0 86.9 74.9 74.1

Incorporation ofspin wave and Pauli's exclusion principle: Sor far, we have only discussed the orbital wave function of the electrons. To get the total wave function ('Vtotal)' both the orbital wave function ('Vorbital) and spin wave function ('Vspin) must be considered. Thus, 'V totaI = 'Vorbital

x 'Vspin

•..

(9.3.1.23)

556

Fundamental Concepts of Inorganic Chemistry

Now let us consider the spin wave function of the electrons. It is known that only tWp types of spins exist. Let, a. and P be the corresponding spin wave functions. If both the electrons (1 .~nd 2) in H2 are having the same spin, Le. a. or p, then the corresponding combined spin wave functions are given by :

'Vspin(l) = 0.(1) 0.(2)

(9.3.1.24)

'V spin(Il) = P( 1) P(2)

(9.3.1.25)

= J2 [0.(1) 13(2) -

1

J2 gives the normalising constant

ra

"'spin (IV)

0.(2) 13(1)

... (9.3.1:26) ... (9.3.1.27)

yl

where 1/

= J2 [0.(1) 13(2) + 0.(2) 13(1)]

ib

and,

1

"'spin (III)

ry

If the spins are different, then it leads to spin wave functions a( 1)P(2) and a.(2)P( 1) which are equally probable. Their linear combination yields:

m

Here, \V spin(l), \V spin(Il) and \11spin(IIl) are symmetric while \V spin(IV) is antisymmetric. Now to combine the orbital wave function (\Vorbital) and spin wave function ('Vspin)' we must consider the Pauli's exclusion principle (see Sec. 4.1.12). It can be stated as: the total wave function in any

he

system (atomic or molecular) must be antisymmetric with respect to the exchange of the electrons.

ea

lc

This is why, the antisymmetric spin wave function can only combine with the symmetric orbital wave function and vice versa. Therefore only the following four combinations are feasible.

th

Ground state {"'" x

... (9.3.1.28)

\V a

X

[P(l) P(2)]

... (9.3.1.29)

\V a

X

[0.(1) 0.(2)]

... (9.3.1.30)'

J21 [0.(1) 13(2) + 0.(2) 13(1)]

... (9.3.1.3 1)

'" a X

t.m

e/

Excited state

~[0.(1) 13(2) - 0.(2) 13(1)]

It has been pointed out that the antisymmetric orbital wave function 'Va leads to only repulsive energy of the excited state. Hence Eqns. 9.3.1.29 to 9.3.1.3 1 represent the excited states. Thus the excited state has three spin wave functions leading t~ the triplet state. The symmetric orbital wave function \V S leads to the ground state whose total function is represented by Eqn. 9.3.1.28 in which there is only one spin wave function leading to the singlet state. Thus, in the ground state the two electrons are paired up with their opposite spins. Thus, in VBT, the spin pairing phenomenon automatically arises to make the total wave/unction antisymmetric. This is why, this method is very _often referred to as the method of electron pairs. Thus, this also supports the quantum mechanical extension of Lewis theory of the electron pair bond.

9.3.2 The Major Conclusions Drawn from VBT (i) The covalent bond (2c-2e) is formed due to the pairing oftwo unpaired electrons with the opposite spins. The number of unpaired electrons gives the maximum possible number of covalent linkage.

Introduction to Chemical Bonding and Theories of Covalence 557

= 'Vorbital (symmetric)

x 'V(spin) (antisymmetric)

m

'Vtotal

yl

ib

ra

ry

If the combining species do not have any unpaired electron, then before to participate in the bond formation the paired electrons must be unpaired. The paired electron pair can itselfparticipate in bond formation as in coordinate covalent bond. (ii) The exchange energy contributes significantly in stabilising a covalent bond. This quantum mechanical phenomenon can be rationalised from the principle of the model, particle in a box. (iii) The covalent-ionic resonance is also important even in the homonuclear diatomic molecules. This factor becomes more important in the case of heteronuclear diatomic molecules where the electronegativity of the combining species differs significantly. (iv) The antisymmetric orbital wavefunction being combined with the symmetric spin wavefunctions leads to the excited states characterised by the triplet state. The symmetric orbital wave function combines with the antisymmetric spin wave function and produces the ground state characterised by the singlet state. Here, the electron pairing automatically arises as a demand of Pauli 's exclusion principle. Thus, the total wave function describing the ground state is given by,

9.3.3 The Major Drawbacks in VaT

t.m

e/

th

ea

lc

he

(i) In VBT, it is assumed that each of the combining species will provide an unpaired electron with opposite spin to form a covalent linkage. But in the case of a coordinate covalent bond, both the electrons are provided by one species. This aspect remains apparently inexplicable. (ii) In 02' all the 2c-2e bonds along with the lone pairs, support it to be a diamagnetic one. But, in fact, O2 is a paramagnetic molecule. This aspect also remains inexplicable. (iii) In VBT, N 2 is supposed to contain a triple bond (Le. N == N) as in the case of acetylene (HC == CH). In spite of this close similarity, their activities differ drastically. Nitrogen is very much inert whiie acetylene is fairly active. This difference gets no explanation. (iv) It cannot explain the odd electron bond in the molecules (e.g. He; , H;). As a matter of fact, bonding in odd electron molecules cannot be explained by VBT which mainly considers the pairing of electrons in 2c-2e bonds. To deal with the odd electron molecule in the light of VBT, some special types of bonds are invoked (see Sec. 9.20).

9.4 MOLECULAR ORBITAL THEORY (MOT)

The molecular orbital theory was p.foposed and developed mainly by Mulliken, Lennard-Jones and Hilckel. In this approach, we are to start from the skeleton of the molecule with its nuclei in fixed positions, then we are to develop the molecular orbitals (MOs) into which the electrons are to be fed as in the atomic orbitals (AOs) in the atoms. In placing the electrons, it follows the aufbau principle, Hund's rule arid Pauli's exclusion principle. T~e MOs are polycentric as they involve more than one nucleus while the AOs are monocentric as t~ey involve only one nucleus. There are two well known methods of developing the molecular orbitals .(MOs). These are: (i) Linear Combination ofAtomic Orbitals (LCAO) method, and (ii) United Atom method. The LCAO has got a good deal of similarities with the VB method. The LCAD method will be discussed here in different cases.

558

Fundamental Concepts of Inorganic Chemistry

9.4.1 The LCAO-MO Treatment of Diatomic Hydrogen Molecule Ion (H;) A. One Electron-Two Centre (1e -2c) System

ib

\11 MO = c 1\11 A + c2\11B

ra

ry

The entitled system contains two protons (i.e. two nuclei) and one electron, and it is the simplest molecule. In the ground ARB state,' the atoms will utilise the lowest atomic orbitals (Le. 1s) Fig. 9.4.1.1. Schematic represento form the MOs. For convenience, the two nuclei are tation of hydrogen molecule ion represented by A and B (see Fig. 9.4.1.1). When the electron (H; ) skeleton. spends most of its time to the nucleus A or B alone, its behaviour is controlled by the atomic orbital wave function \11A or \11 B alone in the ground state. Thus, in the molecule, the molecular orbitals can be regarded as a linear combination of the individual atomic orbital wave functions \11A or \11 B as follows: •••

(9.4.1.1)

he

m

yl

The combination between A and B will be effective if the following three conditions are satisfied. (i) Energy: The combining wave functions should have comparable energy. (ii) Overlap: The AOs representing the electron cloud must undergo overlapping considerably to produce a significant extent of net overlap. (iii) Symmetry: The lobes of combining AOs must have the same symmetry with respect to the bond axis. It indicates that the lobes of AOs of the same symmetry can only overlap.

ea

lc

These three conditions are very often abbreviated as EOS conditions where, E stands for the energy condition, 0 stands for the overlap condition and S stands for the symmetry condition. The corresponding SchrOdinger wave equation is given by :

h

th

-.

H\I1 MO

2

H=--2- V 81t m

t.m

e/

where,

2

... (9.4.1.2)

= E\I1 MO 2

2

2

-e IrA -e IrB +e IR

... (9.4.1.3)

(to express in SI units, _1_ is to be incorporated). 41tE o

To evaluate the energy, we are to consider the following expression: I\JI MO H \JI MO dt

E=

... (9.4.1.4)

J\V~o dl: I(C1\JIA +C2\J1B)H(C1'lJ A + c2\J1B) dt I(C1\JIA +C2\J1B)2 dt 2

(C 1

2

a. A + C2 a. B + 2c1C2 2

C1

PAB )

2

+ C2 + 2c1 C2 S AB

... (9.4.1.5)

where, a A == I\JI A H \JI A d't, a B == J\JIB H \JIB d't, 13 AB == I\JI A H \JIB d't and, S AB == I\JI A \JIB d't; and

J\JI~ dt == J\JI~ d't = I, (from the consideration of normalised AOs).

Introduction to Chemical Bonding and Theories of Covalence

559

2

2

ib

ra

ry

The integrals a.A and a. B are referred to as Coulomb integrals. These are very close to the energies of the electrons in the isolated respective atoms, i.e. a.A ~ EA and a. B ~ EB • Thus in the present system, the Coulomb integral actually gives the measure of energy released when an electron separated from the nucleus by an infinity \VA 'VB occupies the Is orbital ofa hydrogen atom. Thus a. is negative and Fig. 9.4.1.2. Overlap of two from the definition, the ionisation potential is given by - a.. The atomic orbitals to produce' the integral PAB is referred to as resonance integral which gives the overlap integral (SAB)' measure of exchange energy ofan electron. It is a negative quantity indicating stabilisation through the exchange phenomenon. The integral SAB is called overlap integral representing the extent of overlap between the atomic orbitals (AOs) as shown in Fig. 9.4.1.2. To evaluate c 1 and c2' we are to follow the variation principle to find out the condition to minimise the energy. Thus, aElac 1 = 0 and aElc2 = O. From Eqn. 9.4.1.5 we get: 2

2

yl

E(c1 + C2 + 2c Ic2SAB) = c1 a. A + c2 a. B + 2c]c 2 PAB

... (9.4.1.6)

m

Thus, the differentiation of Eqn. 9.4.1.6 with respect to c 1 gives,

=

-

= 2c1u A + 2c 2PAB

2E (C1 + c2 S~B)

cf + c; + 2ctcZS

AB

ea

Ocl

"2c1a. A + 2c2 PAB

lc

aE

or,

he

2 2 aE E(2c 1 + 2c2SAB) + (C1 + C2 + 2c1c 2S AB) aCI

th

By putting the condition, aElac 1 = 0, we get: C1a. A +c2PAB -E(c1 +C2SAB )=0

... (9.4.1.7)

t.m

e/

Similarly from the condition, aElac2 = 0 we get: c1P AB + c2U B

-

E(C1S AB + c2 )= 0

... (9.4.1.8)

Eqns. 9.4.1.7 and 9.4.1.8 lead to the following pair.

and,

E) + c2 (J3 AB - ES AB) = 0

... (9.4.1.9)

C1(P AB --ES AB )+c2 (U B -£)=0

... (9.4.1.10)

c1(a. A

-

The above pair of secular equations gives,

I;~B-_EESAB ~:B_-:SABII:: 1=0 The energy value can be obtained from the following secular determinant,

560

Fundamental Concepts of Inorganic Chemistry

Thus we get:

(a A

E) ( a B

-

E)

-

= (P AB -

ES AB )

2

... (9.4.1.11)

For the hydrogen molecule ion, a A = a B = a (say), and thus Eqn. 9.4.1.11 becomes, (a - E)

2

= (P AB -

ES AB )

2

;

or, a - E

= ± (P AB -

ES AB )

Thus, the two values are :

a + PAB = a + PAB - asAB 1+ SAB 1+ SAB

E =a-

and,

-

PAB

=a _

PAB -

asAB 1- SAB

1- SAB

ry

=

+

ra

E

... (9.4.1.12) ... (9.4.1.13)

\V MO(-)

+ '" B)' E +

- C ( - - \V A

)

\V B'

a+P AB

=1

m

and,

= C+ (", A

+

E - a- -

he

'"MO( +)

yl

ib

By using the energy values in the secular Eqns. 9.4.1.9 and 9.4.1.10, we get, c 1 = ±c 2• Therefore we have the eigen functions and eigen values as follows. S

... (9.4.1.14)

PAB

... (9.4.1.15)

AB

1_ S

AB

ea

lc

It is evident that the \V MO (+) stabilises the system while the \V MO (_) destabilises the system through the repulsive energy. Generally, the wave function which leads to destabilisation is called an antibonding molecular orbital (ABMO), represented by \V ~o. The wave function which imparts stabilisation is called a bonding molecular orbital (SMO). These are simply represented as follows:

e/

th

... (9.4.1.16)

t.m

and,

.

... (9.4.1.17) ~

(Using the convention, \V MO = \V MO(+)' c+ = c, \V MO(-) = \V MO' C_ = C C

, E+

=

E

and

E_

=

E

. )

and c· can be evaluated by imposing the condition of normalisation as follows:

J\V Mod'! = I; or, J(\V 2

c2

A

+ \V B) 2 d'! =. I;

or,

c (1+1+2S AB )=I; or, c=1/~2(I+SAB)

Similarly,

c·

2

= 1/~2 (1- SAB) 1

c+ =----;:====-

... (9.4.1.17a)

J2(I±SAB)

1f the overlap integral is negligibly small, then

~e get c =c· =

Jz'

and

Introduction to Chemical Bonding and Theories of Covalence 561

\IIMO

1

=12 (\II A + \II B ), E = a. + PAB

I ~O ="* \II

(\II A - \IIB)' E·

... (9.4.1.18)

=a. - P I

... (9.4.1.19)

AB

This result is shown in Fig. 9.4.1.3.

1

1

JJ2 (\II A + \IIB) J2 (\II A -

\IIB) d't

ra

J\II MO \II ~d't =

ry

Here it is worth mentioning that \V MO and \V MO are mutually orthogonal, i.e.

J"2 (\II A - \IIB)d't ="21 J\II Ad't -"21 J'¥Bd't 2

2

2

1/2 - 1/2, (\V A and

H;

'VB

2"

ib

1

are individually normalised).

t.m

e/

th

ea

lc

he

m

yl

The solution for produces bond energy = 1.77 eV = 170.8 kJ mol- 1 and bond length 132 pm, while the observed corresponding values are: 2.79 eV, i.e. 269 kJ mol- 1 and 106 pm. Thus the calculated bond energy only accounts for about 60% ofthe observed one. Thus the agreement is not good. However it can be improved by a number of modifications. 'VMO Effect of SAB on the energies of MOs; Existence of He l : Here it must be remembered that the overlap integral (SAB) can never be zero in forming a real covalent linkage. Here, it has been neglected (in a mathematical sense) 8iliA compared to the other terms. Thus, in the cases where c: \ A I W \... I strong covalent bonds are formed, SAB cannot be \ I -Va I neglected. 'VMO In actual calcu.lation without neglecting SAB' it is found Fig. 9.4.1.3. Energies of bonding and that the destabilization induced by the antibonding molecular anti-bonding molecular orbitals. orbital (\V ~o) is greater than the stabilization procured by the corresponding bonding molecular orbital ('liMO). If it were not so, we would get He 2 along with the isolated He atoms, because in He 2 the stabilization could get balanced by an equal amount ofdestabilisation giving rise to no net stabilisatioll or destabilisation with respect to the starting He-atoms. 'Thus it (i.~. He 2) would have an equal probability of its existence as for the isolated atoms, i.e. 2He. But He 2 does not at all exist in the ground state. It is due to the fact that in He'], destabilisation by the two electrons in the 'V ~o is greater than the stabilisation yielded by the two electrons in the 'l'Moe Here it may be pointed out that entropically the system of 2He is also favoured over the He 2 system. Graphical representation ofthe MOs : Fig. 9.4.1.4a, b, d show the wave functions of the individual atomic orbitals and the molecular orbitals of The antisymmetric combination in \V MO shows its zero value in the midway between the nuclei. The symmetric combination in \V MO shows a high value in the range between the nuclei. In Fig. 9.4.1.4c, e and Fig. 9.4.1.5, the electron probability distribution functions for the AOs and MOs are represented for It is revealed that the bonding molecular orbital (HMO) leads to an accumulation of electron density between the nuclei to minimise the nucleus-nucleus repulsion. On the other hand, the antibonding molecular orbital (ABMO) removes the electron density

H;.

H;.

l+ve //T'\

~lO

\--,-----[--:+-1-'V-

562

Fundamental Concepts of Inorganic Chemistry

'P S (1S)

ib

ra

ry

\jJ A(1 s)

(c)

he

m

yl

(b)

B

lc

A

ea

(e)

e/

th

Fig. 9.4.1.4. Qualitative representation: (a) \II A(IS) and \II S(ls) of the isolated (Le. noninteracting H-atoms HA and Hs; (b) \liMO (Le. bonding molecular orbital) of H;; (c) probability of finding electron density in \liMO of H;; (d) \II~ (Le. antibonding MO) of H;; (e) probability of finding electron density in 'V~o.

t.m

,

Note: Compared to (4), the higher electron density for (3) between the nuclei is expected from the expression of 'liMO' For 'liMO' electron density between the nuclei is increased while for \v ~o, electron density between the nuclei is decreased.

\v~o = \v~ + 2'11 A \II S + \v~; 'V~o = \II~ ~ 2'11 A'VS + 'V~

(1 )/\

2

'PA

./

1\ I

(2)

\.---'P~ \

\

'-----tI

\

/ I I

,, \

I I / / ~

~

~

A

B

H;

Fig. 9.4.1.5. Qualitative representation of electron density distribution (relative) curves for and isolated H-atoms HA and Hs. (1) and; (2) for each atomic orbital separately; (3) for the bonding molecular orbital of H;; (4) sum of probabilities of finding the electron in the isolated H-atoms; (5) for the antibonding molecular orbital of

H;.

Introduction to Chemical Bonding and Theories of Covalence 563

from the region between the nuclei giving rise to a repulsive interaction between the nuclei. Thus the MOs give the pattern of charge distribution as, 'VMO : + - - +, 'V ~o : - + + Bonding MO arising from the positive overlap of 'VA and 'VB brings a higher electron density between the nuclei than the sum of electron density ofthe two isolated H-atoms. This is why, in Fig. 9.4.1.5, the curve (3) runs above the curve (4). PAS

and SAS for H;

ry

Evaluation ~f a,

h2

=-

2

81t 2m V -

(

1

lIe ) 2

r + r - R 41tl,o A n

ib

-

H

ra

H in SI unit is given by :

yl

(cf Eqn. 9.4.13)

It can be represented in terms of atomic units (where m = I, e2 == i,

= I , 41tEo ==

m

problem 4 in Chapter 4) as follows:

h

-21t

he

H=- !2 V2 - (~ + ~ - -!.) r r R A

...(9.4.1.20)

I; cf Solved

(9 4 1 21) ... · · ·

B

H'111.f(A»)'

ea

= a A == ('II1.f(A)

t.m

e/

('II1.f(A) -

('II1.f(A) -

(assuming the electron in Is orbital in ground state)

~2 r: - r: + ~ '111.f(A»)

th

a

lc

(i) The term a (= energy of the electron in' the isolated H-atom) is given by :

-

~2 r: 'IIt.«A») +('III.«A) 1- r: I'111.f(A) ) +('II ...(A) 1+ ~I '111.f(A») . -

(Note: Hamiltonian ofan isolated H-atom is: HO leads to : ('II ...(A)

=- ..!- V 2 - ~ (i~ a.u.). Thus the Ist integral 2

IHol 'V ...(A») = ('V1.f(A) IEHI 'V1s(A») = EH')

rA

The first term of the above equation denotes the energy of Is orbital of an isolated H-atom. It is denoted by EH. The second integral considers the electrostatic attraction between the 1s electron of A and the nucleus B and it is known as Coulomb integral denoted by J. It represents the electrostatic attraction and is a negative quantity. The third integral is independent of the nuclear coordinates and consequently it is independent of R. It simply equals to I/R where R is the internuclear distance (fixed). Thus we have:

I

a = EH + J +

*I

..

(9.4.1.22)

Fundamental Concepts of Inorganic Chemistry

PAB called resonance integral is given by :

PAB =

H'Vb(B») = ('Vb(A)

('VIs(A)

('Vb(A) -

~2 r~ - r~ + ~ 'Vb(B»)

-

-

~2 ~ 'Vb(B) ) +(\l1J.v(A) 1- r~ 1'Vb(B) ) +('Vts(A) 1+ ~I 'Vb(B») -

IP

+~ I

(Note: The first integral represents :

ib

IH" I'Vb(B») ::: (\JIb(A) IEHI\JIb(B») ::: EH ('VtS(A)'Vb(B») ::: EHSAB)

yl

(\JIb(A)

... (9.4.1.23)

ra

AB =EHSAB + K

or,

ry

(ii) The term

he

m

Here, K gives the exchange integral which considers some sort of exchange of electron between the nuclei A and B. Physically, it depends on the overlap of the orbitals. It is a negative . quantity. It is favoured maximum when the overlapping orbitals are of the same energy. Now, E+ and E_ may be expressed as follows:

=

(in atomic units)

1 J - K =EH +-+--R 1- SAB

ea

E =E_

PAB

a. -

... (9.4.1.24)

1- SAB

... (9.4.1.25)

th

and

•

lc

E -- E -- a. + PAB -_ EH+ ~ + J + K + 1+ SAB R 1+ SAB

e/

The above expressions are given in a.u. For the values of R = 2.5 a.u., I/R = 0.4, J= - 0.39, K= - 0.287, SAB = 0.458 : E+ = - 0.564 and E_ = - 0.290 in a.u. (Le. hartree)

t.m

564

These are calculated by taking E H = - 0.5 hartree. Note: Here, it should be mentioned that the integrals 1, K and SAB can be expressed in terms of R. In atomic units, the wave function of Is for H-atom is expressed as follows:

1

\jib

= .In. exp (- r)

The integrals are expressed as : 1 J ::: - R {I - (l + R) exp (- 2R)} ; K ::: - (l + R) exp (- R);

SAB ::: (1 + R + ~2

)

exp (- R)

(R expressed in atomic units)

... (9.4.1.26)

Introduction to Chemical Bonding and Theories of Covalence

565

9.4.2 The LCAO-MO Treatment of the Diatomic Hydrogen Molecule: A Two Electron-Two . Centre (2e-2c) System The Hamiltonian operator for the title system consisting of two nuclei and two electrons (see Fig. 9.4.2.1) is given by : 2

2

2

2

2

'i2

rAI

'BI

if =_ _ h_ (V I2 + V22 ) + ~R + ~ _ ~ _ ~ _ 2 81t

m

-=- _-=2

2

'A2

r B2

... (9.4.2.1)

m

yl

ib

ra

ry

(in SI unit, 1/47tEo factor is to be incorporated properly in the expression)

he

Fig. 9.4.2.1. Schematic representation of the hydrogen molecule (H2 ) skeleton.

ea

lc

It is a two electron system. Hence the total orbital wave function is the product of the wave function for each electron. For the electron I, it will occupy a molecular orbital given by Eqn. 9.4.2.2, in the net Coulombic field provided by the two nuclei and the electron 2. 'VI

= c I'VA (I) + c2'VB(I)

... (9.4.2.2)

th

Similarly, the electron 2 will occupy the molecular orbital given by Eqn. 9.4.2.3.

(2) + C2'VB (2) ... (9.4.2.3) Thus, 'VI and 'V2 are the linear combination of the two atomic hydrogen orbitals (i.e. Is). Thus the net molecular orbital 'V MO is given by : = c I 'VA

t.m

e/

'V2

'V MO = 'V I x 'V 2 = [c I \VA (I) +

C2\VB ( I )]

[c I \V A (2) + c2\V B (2)]

·.. (9.4.2.4)

By using the condition to nonnalise the'll MO' we get, J'II~d't = 1. It produces the normalising constant 1/[2(1 + SAB)] (under the condition, c I = c2) for the bonding molecular orbital. Therefore, we get the wave function of the bonding molecular orbital as follows,

'V MO

=

1

2(1 + SAB)

'

['V A(I) 'V A (2) + 'V B (1) 'V B (2) + 'V A(I) 'V B (2) + 'V A,(2) \V A(1')] !

... (9.4.2.5)

I

The first two terms, Le. 'VA (1) 'VA (2) and 'VB (I) 'VB (2) represent the ionic forms, Le. H~ H; and H~ H B respectively. The other two terms represe,nt the contribution of the covalent canonical forms originated from the electron exchange phenomenon. Thus compared to VBT, MOT gives an excessive weightage to the contribution coming from the ionic canonical forms. In MOT, the ionic and covalent forms get equal weightage. This unexpectedly high weightage to the ionic form results due to the fact

566

Fundamental Concepts of Inorganic Chemistry

that in obtaining the molecular orbital the mutual repulsion of the electrons was not taken into account. As a matter of fact, the simple LCAO-MOT always introduces an excessive ionicity in the molecules. The energy of the system can be evaluated by using the variation principle from the following Schrodinger wave equation.

E'V MO

~ J'l'MO H'l'MO dt = H'V MO; or, E = 2

J'V

Mo dt

ry

H;

ra

The present mathematical task is more complicated compared to the system because of the additional terms in the Hamiltonian operator. The solution after consideration of shielding yields the binding energy (Ebinding = E min 2EH ) ~ 337 kJ mol- l at Ro ~ 73 pm compared to the observed value 458.3 kJ mol- t at Ro = 74 pm. Thus the calculated value accounts for 73% of the observed one. Further modifications can improve the result. Compared to the H;, the bond energy in H2 is greater. It is reasonable becaus~ the more electron clouds between the nuclei (Le. + - - +) in the bonding molecular orbital of H2 gives a better stabilisation compared to the distribution, + - + in the bonding molecular orbital of

H;.

m

yl

ib

'"'J

9.4.3 Comparison of VB and LCAO-MO Treatment on H2

ea

lc

he

The fundamental approaches in the two treatments are different. In the VBT, we start from the individual atoms to interact with each other as they proceed together. The progress of the process is comparable with that of ,a chemical reaction. The orbital wave functions of the isolated atoms are 'VA (I) and \liB (2) where A and B are the nuclei with the electron I and 2 respectively. When they approach within the bonding distance, the electrons being indistinguishable may exchange the nuclei with an equal probability. Thus the following two wave functions become equally probable.

th

'V/ = 'VA(I) 'VB(2), and 'VII = 'VA(2) 'VB(I)

e/

Thus the resultant VB covalent wave function is given by :

\II VB (cov) = ct['VA(I) 'VB(2) + \IIA(2) \IIB(I)]

t.m

Then after incorporating the contribution of the ionic canonical forms, we get the modified VB wave function as follows.

\IIVB = (I - A) cl['VA(I) \IIB(2) + 'VA(2) \IIB(I)] + A[(\IIA(I) \IIA(2) + \IIB(I) \IIB(2)]

On the other hand, the MOT starts with the nuclei at fixed positions in the molecule. The wave function of the electron 1 in the field of the two nuclei and the electron 2 is given by,

'VI = calVA (I) + Cb'VB (I) Similarly for the electron 2, we have: \112 = Ca'VA (2) + Cb'VB (2)

Thus the MO wave function is given by :

'V,

'V MO = 'V t x 'V2 = c2 ['VA( 1) \II A(2) + 1) \V B(2) + \V A( 1) \V B(2) + 'VA(2) 'VB( 1)] (the present case being symmetrical, ,ca = cb = C say)

Introduction to Chemical Bonding and Theories of Covalence 567 In \V MO' the coefficients ofthe four terms are identical. It indicates that each term contributes equally. In \V MO' the first two terms arise due to the ionic canonical forms of the molecule while the -third and fourth terms arise from the covalent structures. In \V VB' both the covalent and ionic parts are also existing as in \V MO' but they do not contribute equally while in \V MO both the ionic and covalent forms are expected to contribute equally. This is the basic difference between the theories. As a matter of fact, the simple LCAO-MOT overestimates the contribution coming from the ionic forms. But, through a series of refinements, both the methods lead to almost the same result.

ry

9.4.4 The Important Aspects of LCAO-MOT

ra

The basic principles are very similar to those of the self consistent field method (SCF method) (see Sec. 4.2.1). The important features of the treatment are mentioned here.

yl

ib

(i) The molecular orbitals (MOs) are developed from the skeleton of the molecule containing the nuclei at the fixed positions through the LCAO method. Thus, the MOs are polycentric (involving more than one nucleus) while the AOs are monocentric (involving only one nucleus). (ii) The numb~r of molecular orbitals are the number of starting atomic orbitals.

he

m

(iii) For an effective combination of the atomic orbitals they must satisfy the EOS conditions (see Sec. 9.4.1) according to which the energy ofthe orbitals should be comparable; they must overlap and the overlapping lobes must have the same symmetry.

ea

lc

(iv) Depending upon the overlap integral (SAB)' the MOs are classified as bonding MO (SAB > 0), antibonding MO (SAB < 0) and nonbonding MO (SAB = 0). (v) The energy of each molecular orbital can be calculated by solving the appropriate Schrl>dinger wave equation.

th

(vi) All the MOs are normalised, i.e.

I"'~odt = I; or, I(", ~O)2 dt = I

The corresponding bonding and antibondingMOs are mutuallyorthogonal, i.e.

I'"

MO x

",~dt = 0,

t.m

e/

(vii) After construction of the MOs, the electrons are placed in the MOs by following the aufbau principle, Hund's rule of spin multiplicity and Pauli's exclusion principle. (viii) The molecular orbitals are also defined by the four quantum numbers as in the case of AOs. Generally the n and I are retained from the starting AOs, and they bear the same significance. The magnetic quantum number of the MOs are represented by A. In a_diatomic molecule, the a-bonding axis, Le. the axis containing the nuclei, is taken as the reference direction. The corresponding quantised angular momentum is expressed in h/27t units with respect to the reference axis. It can have values, A. = -I, - (/- 1), ... ,0, ... , (/- 1), I. For A = 0, the orbitals are symmetrical around the reference axis and the orbitals are called a-orbitals. For A = ± 1, the orbitals are referred to as 'It-orbitals, and for A = ± 2, the corresponding orbitals are termed as O-orbitals. The fourth quantum number is the spin quantum number having the same significance as in the atoms. (ix) The different MOs have different symmetry elements. Based on these symmetry elements they can be classified as gerade (g) and ungerade (u), 0'- and 'It-orbitals, bonding and antibonding orbitals (see Sec. 9.6). (x) The bonding molecular orbitals concentrate the electron cloud between the nuclei to stabilise the

568

Fundamental Concepts of Inorganic Chemistry

system while the antibonding orbitals remove the electron cloud from the space betweeJ! the .... nuclei to destabilise the system. ·(xi) It can explain the odd electron bond (e.g. H;). (xii) Approximately, the stabilisation created by a bonding molecular orbital (BMO) is balanced by the destabilisation induced by the corresponding antibonding molecular orbital (AMBO). This is why, in simple diatomic molecules, the bond order is roughly calculated as follows,

ra

ry

number of electrons in BMOs - number of electrons in ABMOs bond order =- - - - - - - - - - - - - - - - - - - - - 2 (xiii) In the case of homonuclear diatomic molecules, both the atomic orbitals contribute equally in forming the MO, but in the heteronuclear diatomic molecules they contribute unequally. Thus, \II MO = N(\II A + A'\II B); for homonuclear diatomic molecules A' = 1 while for the heteronuclear diatomic systems A' 1.

ib

*

yl

9.4.5 Proof of Three Basic Conditions (EOS) in LCAO-MO Method

lc

he

m

In Sec. 9.4.1, it has been already mentioned that in an effective LCAO, the three basic requirements are (i) energy ofthe overlapping orbitals should be comparable, (ii) overlap should be as much as possible, (iii) the lobes ofthe orbitals having the same symmetry relative to the molecular axis will only combine. These three conditions are simply described as EOS conditions (E stands for energy, 0 stands for overlap and S stands for symmetry). To justify the above conditions, we can select the following trial wave function.

ea

\liMO

= c) \II A + c2\11 B

•.•

(9.4.5.1)

th

Combination of two atomic orbitals \II A and \liB will produce two molecular orbitals. The coefficients and energies of the MOs will be given by the following equations.

O}

(cf. Eqns. 9.4.1.9,10)

e/

c (u - E) + C (A - ES ) = ) A 2 PAB. AB _ c) (P AB -ES AB )+c2 (u B -E)-O

by

t.m

(u A -E)(u B -E)=(P AB -ESAB )2, (cf. Eqn. 9.4.1.11) The integrals, u A' u B' PAB and SAB have been already discussed in Sec. 9.4.1. The integrals u A and u B denote the energies of the electrons in the isolated atoms; PAB is called the resonance integral given

f\!l A if \!I B d't;

SAB

is called the overlap integral given by

f\!l A \!I B d't.

A. Proof of the condition of energy (E) • Let us consider u B » u A (e.g. in the case ofheteronuclear diatomic molecules). Consequently, the size of \II A will be much smaller than that of \liB as the energy of an orbital depends on its size. It leads to very small values of SAB and PAB . Consequently, the R.H.S. of the above equation (Eqn. 9.4.1.11) will be very small. To make the L.H.S. of the same equation small, either (u A E) or (u B - E) will be small. Ifwe assume (uA - E) to be small, then the approximate solution of Eqn. 9.4.1.11 can be obtained by putting E = U A in the terms (u B - E) and (PAS - EA.')AB). This leads to :

Introduction to Chemical Bonding and Theories of Covalence

or,

569

... (9.4.5.2) 2

~aA -

13 AB , when SAB is very small u B -u A

2

~ aB +

... (9.4.5.3)

ra

ry

Similarly, for the condition E ~ UB' we have:

ib

13 AB , when SAB is very small u B -u A

E+ and E_ denote the energies of the bonding MO and antibonding MO respectively. Since, u B

(PAR -

c)

= -

c2

(

c

+

.

,(for the 'V+ orbital)

(P AB -E_SAB ) (u B -

.

E ) ' (for the 'V- orbital)

lc

- 2 =-

and

uA

E+S AB )

_ E)

he

-

m

yl

» u A' Et ~ u A and E_ ~ u B' Le. no effective combination. • Now let us consider the magnitudes of mixing coefficients. We have (cf. Eqns. 9.4.1.9, 10) :

c)

_

ea

Under the conditions when E+ ~ u A' c)fc2 is very large for 'V+ orbital and when E_ ~ us' c 2fc) is also very large for 'V_orbital. It indicates that mixing between \VA and \11 B is very small. This proves- the condition (i).

th

• Under the condition, u A ~ u B (Le. homonuclear diatomic system), and very high. For, u A = u B =

t.m

e/

U,

we can write;

5.- ~ ± 1, c

the mixing is

2

\VA ±\VB

\jI±

= ~2(l tSAR)'

E =

U

+

(cf. Eqn. 9.4.1.14, IS)

+ PAB -uS AB and E ]

+ SAB'

= U _ -

~

-uSAB ~ - S AB

AB

(cf. Eqns. 9.4.1.12, 13) E+

~ U

+ P AB and E_

~ U - P AB

when

SAB«

1 (cf. Eqns. 9.4.1.18,19)

• Thus for, u A = Us and SAB« 1, the splitting of the MO's occur roughly by 2p. In such cases, the molecular orbital wave functions are independent of p. For, u B » u A ' due to the MO formation, the energy of u A is lowered approximately by p~H/(aH -u A) neglecting SAB and the upper level is pushed up by the same amount. This is shown in Fig. 9.4.5.1. For, u A ~ us' the AOs are shifted by ± p, while for u A "# UB' the shift is approximately proportional to p2 (Le. second order interaction).

570

Fundamental Concepts of Inorganic Chemistry lJI MO

\

~i

/

~

e> Q)

~A

w

A

c

'lIMO

~ p2 {__ ~~,

\

\

, 1 '

I

\

\}J

'---1---"

I

\

\

B I I

I

, 1'

\

f3

\

"

I I

1 \ ____ 1

~s B

I I I I I I I I

I

B

1

\

1 1 I

, ~. ------.,L -},., /32

A

ry

"'--r-'

'------J

\fJMO

'1 J MO

ra

I

(b)

(a)

(b) as » a A·

yl

B. Proof of the condition of overlap

y

he

m

'lJ B

x

ea

lc

If the atomic orbitals do not overlap, then both ~ AB and SAB will have zero values. This leads to E+ == u A and A'_ = u B' i.e. no effective combination betw een 'VA and 'VB. For poor overlapping, ~AB and SAB are small and then the mixing between \VA and \VB is small (as argued in the proof of condition (i).

= as,

ib

Fig. 9.4.5.1. LCAO for 'V A and \V s· (a) a A

th

c. Proof of the condition of symmetry

Fig. 9.4.5.2. Overlapping between s (Le. \VA) and Py (Le. \Vs) orbitals when the molecular axis is x-axis. The marked area denoted by 1 and 2 are equal but of opposite sign (Le. their . d S 0) . sum IS zero an AB = .

t.m

e/

The integral SAB becomes zero, if the d' ff . \VA an d \VB are h aVlng I erent symmetries about the molecular axis. For example, if 1s orbital (say \VA) overlaps with the 2py orbital (say, \VB perpendicular to the molecular axis), then SAB becomes zero (Fig. 9.4.5.2). In such cases, no effective mixing between \VA and \VB occurs.

9.5 SYMMETRY AND OVERLAP IN FORMING MOLECU.LAR ORBITALS IN LCAO METHOD The overlap integral S AB

=

f

\jI A\jI BdT.

is extremely important in characterising the different types of

molecular orbitals, Le. bonding MO, antibonding MO and nonbonding MO. It actually gives the measure of electron density between the nuclei. This electron density is measured with respect to the isolated atoms. The conditions are: SAB > 0, bonding MO; SAB == 0, nonbonding MO; SAB < 0, antibonding MO. The magnitude of SAB gives the measure of bond strength of the covalent bond formed between A and B. With the increase of positive value of SAB' the bond strength increases. The value of SAB largely depends on the symmetry properties of the orbitals.

Introduction to Chemical Bonding and Theories of Covalence

571

Combinations of different atomic orbitals leading to the formation of different molecular orbitals are shown in Fig. 9.5.1. It is evident that overlapping of the lobes with the same sign produces a bonding interaction while overlapping of the lobes with opposite sign produces an antibonding interaction. Molecular Orbitals

Atomic Orbitals

E® E®

)

s

s

m m Px

(-)

00 00 P P x

0Q) I

(+)

agley Nodal plane

~

)

)

Og(PJ

e!E(V

)

ea

lc

00 Px

~

)

m

x

(j~(S)

oo:B

)

@i8

)

c::Effi

)

Px

(+)

crg~ Nodal plane

ry

(-)

E9

)

ra

)

s

ib

s

yl

-e- -e-e- -e-

(+)

e/

th

Fig. 9.5.1. (a) Formation of sigma-type molecular orbitals from atomic orbitals through LCAO. Note : Px + Px leads to a-type antibonding interaction while Px - Px leads to a-type bonding interaction. Here it may be pointed out that many authors suggest the following conventions. Px + Px => a-BMO; Px - Px => a*-ASMO; They use the following pictorial representations.

t.m

EXB Px

+ ~ => a-SMO;

Px

EXB Px

+ ~ => a*-ABMO

Px

Here in drawing the same Px orbital, different conventions are followed. hi one case, the lobe of Px orbital in the +x direction is given the + sign while in the other case, the lobe in the -x direction is given the + sign. Th.e practice is not justified. Symmetry Elements in the MOs

(i) Cylindrical symmetr.v (Coo) : If the orbital is rotated around the bonding axis, Le. molecular axis, and it remains unchanged with respect to the sign at any angle of rotation, then the orbital is called cylindrically symmetrical. The corresponding symmetry element is denoted by Coo' The molecular orbitals having Coo are referred to as sigma (cr) molecular orbitals. These are associated with the magnetic quantum number, A == O. (ii) Centre ofsymmetry or centre ofinversion (Ci ) : The centre of symmetry indicates that for every point there is an equivalent point (in all respects including the sign) in the just opposite direction at an equal distance measured from the centre of symmetry. Such orbitals having Ci are called

572

Fundamental Concepts of Inorganic Chemistry

gerade represented by g. The orbitals lacking in C; are termed ungerade and represented by u. Here it must be mentioned that only in the case ofhomonuclear diatomic molecules providing identical AOs to form MOs, the question ofe;, i.e. the classification ofMOs by g and u, arises.

c:!::> Py

Py

Py

Nodal Plane

~

-

m

+

---~)

he

-

(-)

~

ea

~~

.

Nodal

I~Plane symmetry elements (i.e. cylindrical symmetrical) but the a-bonds developed by the donation of elongated x-electron cloud lacks in Cex:>' Such bonds are formed in different alkene complexes (e.g. Zeise's salt, Sec. 14.7.5 viii) and this type of a-bonds is sometimes described as Jl (mu)-bonds.

th

9.7 VBT VERSUS LCAO-MOT

e/

Similarities

t.m

(i) Both the theories explain the covalent bond. (ii) Both the theories demand that to form a covalent linkage, the corresponding orbitals must overlap. (iii) For stabilisation ofthe covalent linkage, both the theories demand the concentration of electronic charge between the nuclei. (iv) Both the theories consider the quantum mechanical exchange phenomenon ofthe electrons among the nuclei and the covalent-ionic resonance interaction to explain the stabilisation. (v) Both the theories predict the directional properties of the covalent linkage. (vi) Both the trial 'VVB and 'liMO can be utilised in the same way to evaluate the eigen functions and eigen values. Differences

(i) In VBT, the individual atoms are gradually brought together from infinity to interact through the overlapping of the atomic orbitals. On the·other hand, in LCAO-MOT, the MOs are developed through the LCAO from the skeleton of the molecule containing the nuclei at fixed positions. Then the electrons are placed in the polycentric MOs by using the aufbau principle, Hund's rule and Pauli's exclusion principle as in the case of atoms.

576

Fundamental Concepts of Inorganic Chemistry

yl

ib

ra

ry

(ii) In LCAO-MOT, the combining atoms lose their individual identities, while in VBT, the combining atoms retain their individual identities to a large extent. (iii) In LCAO-MOT, the odd electron bonds are explainable while these cannot find any explanation in VBT. (iv) In VBT, 02 is considered to be diamagnetic, but it is not supported by the experimental fact. The LCAO-MOT can explain the paramagnetic behaviour of 02. (v) VBT appears easier tp explain the polyatomic molecules in terms of the localised covalent bonds between the nuclei in terms of hybridisation leading to the stick- and- ball models ofthe molecules. On the other hand, in MOT, the molecular orbitals are delocalised and consequently, to correlate with the stick- and- ball model in terms of MOT, it appears difficult. However, the concept of delocalised molecular orbitals is very important in some cases to explain the molecular properties. MOT has been found more promising to explain the molecular properties, at least qualitatively, in the case of simple diatomic molecules. But the VBT in terms of Linnett's double quartet theory (it will not be discussed here) produces more or less the identical results as expected from the simple MOT.

m

9.8 THE UNITED ATOM METHOD IN MOLECULAR ORBITAL THEORY

t.m

e/

th

ea

lc

he

In the LCAO-MO treatment, we start with the AOs while in the united atom method, we start from a hypothetical atom formed by the coalescence of the two nuclei A and B. This hypothetical united atom can be considered as a single atom in which the orbitals can be assigned to accommodate the electrons. The separation is carried out from zero to infinity at which the two isolated atoms exist. Thus in passing .from zero separation to infinity separation, the system will pass over through the diatomic molecule at some intermediate internuclear distance. Thus the journey starts from the hypothetical united atom (r = 0) to the isolated atoms (r == (0) through the diatomic molecule (r == internuclear distance between the nuclei in the molecule). In this process, the atomic orbitals of the united atom will be converted into the molecular orbitals and finally (at r == (0) into the atomic orbitals of the isolated atoms. Consequently, the energies of the atomic orbitals of the united atom will pass through the energies of the molecular orbitals to the energies of the atomic orbitals of the isolated atoms. This gradual change of energies of the orbitals of the united atom to those of the isolated atoms through the molecular orbitals of the diatomic molecule is represented by the correlation diagram developed by Mulliken. In this correlation diagram, the energies of the different atomic orbitals of the united atom and the isolated atoms are shown at two extremes at r == 0 and r == 00 respectively. Before to draw the correlation diagram, we are to consider the conversion suffered by the united atom orbitals in making the journey, r == 0 to r == 00 (Le. isolated atoms). The state of transformation at different stages of the journey is shown in Fig. 9.8.1 (assuming the z-direction for movement) for some orbitals of the first three principal quantum numbers. From the nature of transformation suffered by the united atom orbitals in the above mentioned process, they can be :Iassified in two groups. (i) There are S6me united atom orbitals which retain the same principal quantum number (n) and azimuthal quantum number (I) on separation. These are (assuming z-axi~ as the bond axis) : Is, 2s, 2px' 2py, 3px' 3py' 3dxy and 3dx2 _ 2. The 3d; of united atom gets converted into two 3s orbitals of the isolated atoms. Here, th~ principal quantum number remains the same, but the azimuthal quantum number changes. The energy of these orbitals increases

Introduction to Chemical Bonding and Theories of Covalence Normal Bond Distance

Slightly Separated

United Atom

sn

1s

@) +

1'. \ \_~

ra

G)

m

ffiffi

lc

3dxz.yz

he

~ ~~

yl

ib

ffi 88 8

ry

2s

nip

1

-12('4Jx,y)

1

12 ('4Jx,y)

ea

Fig. 9.8.1. Generation of some representative molecular orbitals in the light of united atom method. (Adopted from K.L. Kapoor, "A Text Book of Physical Chemistry", Vol. 4, Macmillan India Ltd. Reproduced with permission).

t.m

e/

th

with the increase ofr. It is reasonable as the separation leads to bifurcate the nuclear charge of the atom while the principal quantum number of the orbitals remains unchanged with the increase of the separation. It is well established that with the increase of Z in the nucleus, the energy of a particular orbital decreases. (ii) There are some united atom orbitals in which the principal quantum number decreases by unity along with a change ofazimuthal quantum number, t1l = ± 1 with the increase ofseparation (r). These are: 2pz' 3s, 3pz' 3dxz and 3dyz when the z-axis denotes the course of separation. The changes from r = 0 to r = 00 are as follows: 2pz ~ Is, Is; 3s ~ 2pz' 2pz; 3pz ~ 2s, 2s; 3dxz ~ 2px' 2px; 3dyz ~ 2py, 2py. In these cases, the energy of the united atom orbitals decreases as the separation increases because of the decrease of the principal quantum number. The correlation diagram for a diatomic homonuclear molecule is shown in" Fig. 9.8.2. The exact order of energy depends on the extent of separation, i.e. r. There are some cross-over regions, but in crossing the orbitals there is a rule referred to as noncrossing rule according to which two orbitals having the same symmetry can never cross each other. In preparing the correlation diagram, this noncrossing rule is to be satisfied everywhere. Here it is worth mentioning that the energy changes in passing from r = 0 to r = 00 may not always occur smoothly. This is why the changes are not always shown by the straight lines.

578

Fundamental Concepts of Inorganic Chemjstry

1t u

3p au

35

"

-~ ;",

0'9

Og

">---25 ~~

ry

2p

2p

25

e; Q)

m

yl

ib

ra

c: W

United Atom

he

Molecule

Isolated atoms

---. x-axis

lc

Fig. 9.8.2. Correlation diagram for a homonuclear diatomic molecule (united atom method).

th

ea

9.9 SIMPLE MOLECULAR ORBITAL MODEL FOR HOMONUCLEAR DIATOMIC MOLECULES 9.9.1 Simple Molecular Orbitals and Diagrams

H;

t.m

e/

We have already discussed the MO-theory for and H 2 in detail. Here we shall pay attention in a general way for the elements from hydrogen tOl)eon a~d shall consider the formation ofMOs from the AOs, i.e. Is, 2s and 2p. By considering the x-axis to be the molecular axis, according to the EOS conditions (see Sec. 9.4.1), we get the following appropriate combinations (cf. Fig. 9.9.1.1).

•

= N1['Vb(A) + 'Vb(B)]' O'u(b) = N 2 ['Vb(A) - 'Vb(B)] • g(2s) = N 3 [\II2s(A) + \112s(B)]' (Ju(2s) = N 4 [\II2s(A) - 'V2s(B)]

0' g(b) (J

CJ g(2P.)

= N s[\jI2P.(A) - \jI2P.(B»)' CJ:(2P.) = N 6 [\jI2P.(A) + \jI2py (B)]

1t u {2py )

=N, [\jI2p

1t u(2P:)

=N 9 [\jI2p,(A) + \jI2p,(B»)' 1t:(2Pz) = N IO [\jI2Pz(A) -

y

(A)

+ \jI2py (B»)' 1t:(2Py )

= N 8 [\jI2p

y

(A) - \jI2py (B)] \jI2Pz(B)]

where N's stand for the normalising constants. The shapes of the MOs are shown in Fig. 9.9.1.1. Here, to denote the two separate ato~, A""anq B are used but they are of the same kind in the problem of a homonuclear diatomic molecule. .

Introduction to Chemical Bonding and Theories of Covalence

579

Nodal plans

'OiOO~(25)

o

G)G):;;: 25 (B)

°g(25)

yl

ib

ra

ry

25 (A)

Node

t.m

e/

th

ea

lc

he

m

Node

Fig. 9.9.1.1. Formation of MOs through the LCAO. The AOs are provided by the atoms A and B. The molecular orbital diagram for the homonuclear diatomic molecules is shown in Fig. 9.9.1.2. Here, no interaction between the MOs of the same symmetry, and no hybridisation among the AO~ participating in the formation of the MOs are considered. In the MOs, the electrons are placed. in order of increasing energy (Le. aujbau principle). Hund's rule ofspin multiplicity and Pauli's exclusion principle are also applicable in placing the electrons in the MOs as in the case of atoms. The extent of splitting between the corresponding bonding and antibonding MOs depends on the extent ofefficiency ofoverlapping (Le. the magnitude ofoverlap integral SAB). In the sigma interaction, the better overlapping produces more splitting compared to that in the pi-interaction. Thus, if x-axis is considered to be the bond axis, then the' sigma interaction between the 2px orbitals produces • more splitting between CIg( 2px) and CI u (2px) compared to that between the 7t -MOs. In this case, 2py ± 2py

580

Fundamental Concepts of Inorganic Chemistry

and 2p= ± 2p= interactions produce two sets of 7t-MOs perpendicular to the sigma skeleton. The two sets of 7t-MOs developed from 2py and 2p= are energetically degenerate but they are mutually perpendicular. We have already considered that approximately, the stabilisation induced by a bonding MO gets balanced by the destabilisation introduced by the corresponding antibonding MO. Thus to measure the net stabilisation in the molecule formed, we are to consider the net effect earned by the electrons residing in the bonding and antibonding MOs. By considering the covalent linkage as a pair of electron (Le. 2c - 2e bond), we can calculate the bond order parameter as follows :

ib

lc

he

m

yl

Here it is worth remembering that to stabilise the system, the electrons in the bonding MOs are not to be necessarily paired up. The Hund's rule of spin multiplicity along with the Pauli's exclusion principle and autbau principle will decide whether the electrons will be paired up or not. It is evident that after He 2, the inner electrons coming from n = I (Le. K-shell) are contributing nothing to stabilise the system. In representing the electronic configuration, very often the symbol of the inner shell is used to represent the electrons originated from that inner shell. Thus in Li2, for the MOs, 0g(ls) and 0:(1.\')' the symbol KKis used. Similarly

ra

ry

No. of the electrons in the bonding MOs - No. of electrons in the antibonding MOs bon d order = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 2

for n = },.J.he symbol fL is used. Thus, for Na 2 , the configuration is : KKLL g(3.\'). The electronic configurations for the molecules H2 to Ne 2 are given in Table 9.9.1.1.

((f~(1S)

ea

°

1s ~:_,

"---

•

",,,,....,,,'~--1 s

",'"

\. CT (1s)

t.m

e/

th

g The highest occupied molecular orbital (HOMO) and --+ x-axis the lowest unoccupied molecular orbital (LUMO) are jointly Fig. 9.9.1.2. Simple molecular orbital referred to as frontier molecular orbitals (FMOs) of the diagram (for the homonuclear diatomic is its LUMO. The molecule. In F 2, 7t: is HOMO and molecules). singly occupied molecular orbital is called SOMO (e.g. 7t: orbitals of 02). Depending on the condition, the SOMOs of O2 can act as both the HOMOs and LUMOs. Thus HOMOs and LUMOs of 02 are half-filled.

a:

Note: According to many authors, the MOs are named as follows : ag(b)

••

••

as 109; aU(b) as 20 u ; O"g(2.\') as 3a g ; 0U(2s) as 40 u '

••

0g(2px)

as 5a g ;

7t u (2py ,z)

as 17t u ;

•

1t K (2py.:)

•

as 21t g ; 0u(2px) as 60 u • The energy order (according to MO diagram given in Fig. 9.9.1.2) is : 109 < 20: < 30"g < 40: < 5a g < 17t u < 27t: < 6a: • For the ~econd period elements, some authors start naming of the orbitals from 0g(2s) as lag; as 20 u and so on. This convention is also reasonable in the. case where the inner MOs arising from the K-shell do not contribute in the bond order determination. 0U(2s)

Introduction to Chemical Bonding and Theories of Covalence 581 Table 9.9.1.1. The electronic configurations of some homonuclear diatomic molecular species and some characteristic points according to the simple molecular orbital model (see Fig. 9.9.1.2)

• O'g(h)

0.5

2 O'g(b) 2

·1 0' g(b} O'I/(I.t)

105

Paramagnetic

1.0 (0)

458.0

74

Diamagnetic

0.5 (0)

251.4

108

Paramagnetic

·2

1.0 (0') ·2

2

[Be 2 ] 0' g(2P%} 2

4 1tu (2py. 2pz} 3

1.0 (0'), 2.0 (1t)

160

(a)Paramagnetic

600

125

(a)Diamagnetic

941.7

110

Diamagnetic

842

112

Paramagnetic

1.0 (0), 1.5 (1t)

·2 [ N 2 1t g (2py. 2pz}

1.0 (0), 1.0 (1t)

493

120

Paramagnetic

1. 0 (0), 1·.5 (1t)

625

112

Paramagnetic

142

Diamagnetic

he

[Be 2 10' g(2P%} 1tu(2py. 2p,}

l

••

lc

[ N 2 l 1tg (2py. 2pz } ·4 [ N 2 1t g (2p". 2pz}

l

1.0 (0)

ea

Ne 2

1.0 (0), 1.0 (1t) .

m

l

Diamagnetic

290

1.0 (0')

. 2 2 [Be2 0' g(2P'C} 1tu(2py .2pz}

266

108

0.0

0'I/(2.t}

2 [Be2 lO'g(2P.'C}

ra

[He 2 ] 0'g(2.t}

(0')

0.0

2

Magnetic properties

255.5 .

0' g(b} O'I/(b}

2 [He 2 ] 0'g(2.t}

Bond length (pm)

ib

2

Bond dissociation energy (kJ mol- J)

yl

He;

Bond order

ry

Molecular Electronic configuration species

·4 ·2 [ N 2 1t g (2py. 2pz} °U(2px}

l

155

± 30

0.0

th

Note: x-axis is taken as the bond axis (a) Experimental result contradicts the theoretical prediction from simple molecular orbital diagram

t.m

e/

9.9.2 Molecular Orbital Picture of Some Representative Homonuclear Diatomic Molecular Species

Now let us consider the predictions from the simple molecular orbital picture in connection with the homonuclear diatomic molecules. (i) H]. and He]. : The bond order in H 2 is one due to cr~(lS). The electron pair bond is comparable in valence bond theory, it is with the idea of the valence bond theory. But to explain the required to invoke a special type of bond (Le. one electron bond in which electron pairing does not occur). The stabilisation in the system arises due to the resonating structure, H· J-r ~ J-r ·H Similarly, to explain the stability of He; and HeH, another typical bond, Le. three electron bond, is to be considered in VBT. Again the stabilisation can be explained from the following resonating structures.

H;

He

:·He+~+He·

: He; He

:·H~+He·:~

However, in VBT, no sound support in favour ofthese typical bonds is available, but in MOT, no special consideration is required to explain the species. They automatically arise. From Table 9.9.1.1, it is interesting to note that the bond order in both H; and He; is 0.5 and their bond

582

Fundamental Concepts of Inorganic Chemistry

lengths are 105 pm and 108 pm respectively. Their bond energies are also very much comparable as expected from the molecular orbital picture. In He 2, the bond order is zero. Because of this fact, it is nonexistent. Here it is worth mentioning that actually the antibonding orbital, Le. O':~IS)' destabilises more than what the bonding orbital, Le. 0'~(1s)' stabilises. The actual energy expressions (see Eqns. 9.4.1.16-17) are given by :

E=u+P 1 +SAB u -

P

ry

E. =

... (9.9.2.1)

yl

ib

ra

... (9.9.2.2) l-SAB If it were really, stabilisation by BMO = destabilisation by ABMO, then there would be a finite probability for the existence of He 2 because under this condition, no net destabilisation in He 2 occurs compared to the system of isolated atoms (Le. He + He). But, in fact, the destabilisation is more than the stabilisation. This is why, He 2 is nonexistent. However, for practical purposes (SAB« 1), we can approximately consider E = u + P and g = a. - p. (Note : In the process, He2 ~ 2He, Mf ~ 0, M = + ve, the entropy factor favours the

•

.

he

m

decomposition of He z molecule into the He-atoms.) In this connection, it is worth mentioning that He 2 does not exist in the ground state but if it is * excited sufficiently to promote at least one electron from O'u(ls) to the next higher MO, Le. 0'g(2s)' then the bond order becomes unity and He; exists. 2

*1

I

O'u(ls) O'g(2s);

lc

He2 (excited): O'g(ls)

bond order = 1

e/

th

ea

(ii) Liz: KK 0'~(2.\")' It shows the bond order unity. It really exists as a diatomic species in the gas phase. Note: In the above electronic configuration, K stands for the 1st shell K (n = 1). The inner shell electrons do not contribute anything in the net bonding. This is why, very often the inner shell electrons are denoted by the shells like K (n = 1), L (n = 2), etc. In this convention, the electronic configuration of Na 2 is given by : 2

KKLL cr g(3$)

t.m

(iii) Be z : KK 0'~{2.f) cr:~2.~). Here the bond order is zero, and it really exists as a monoatomic species in the gas phase. 2

*2

2

2

*2

2

2

(iv) 8 z and Cz : For B2 , KK crg(2.~) cr u{2.\') O'g{2px) and for C2 , KK O'g(2s) O'u(2s) O'g(2px) 1t u(2p y .2pz>' The magnetic behaviour predicted from these electronic configurations is not satisfied by the experimental facts. It indicates the requirement of modification ofthe simple modeL

(v) N 2 : KK 0"~(2.') 0":~2.') 0"~(2P.) 7t~(2PY. 2pz)' It shows the existence of two pi-bonds along with a sigma bond as in acetylene. Hence, from the knowledge of acetylenic linkage, a high re~ctivity in N2 is expected. But unfortunately, it is very much inert. Besides this, from the ionisation potential of N 2 (i.e. N 2 ~ N; + e), it is established that the highest level filled (Le. HOMO) is a a-orbital in character while it is predicted as a 7t-orbital from this model. Thus, the simple model can explain the observations in N z qualitatively, butforfiner analysis ofthe data, it requires a modification which will be discussed later. • Ionisation energies of atomic nitrogen vs. molecular nitrogen: In spite ofthe above mentioned drawback, the simple model can explain some important observations. Here it is interesting to

Introduction to Chemical Bonding and Theories of Covalence

583

note that the first ionisation energy of the atomic nitrogen is less than that of the molecular nitrogen Le. IE(N2) = 1503 kJ mol-I> IE(N) = 1402 kJ mol- 1 (cf. oxygen where the reverse is true, for 0: 1314 kJ mol-I; for O2 : 1164 kJ mol- 1). N

~ N+ + e, 1402 kJ mol-I; N 2 ~ N; + e, 15Jl3-k.J mol--I

In the case ofN 2, the. electron is to be knocked out from the HOMO, a bonding MO (7t u) which is stabler than the corresponding AO of nitrogen. This is why, N 2 requires higher energy for its

ionization.

ra

ry

In this connection, the ionisation of NO (which contains one electron excess compared to N2 ) leading to NO+ (i.e. nitrosyl cation, an isosteric species with N 2) is worth considering. NO ~ NO+ + e, 894 kJ mo]-I

Its ionisation energy is much less than those of oxygen atom (IE = 1314 kJ mol-I) and nitrogen atom (IE = 1402 kJ mo.-I). In .NO, the odd electron to be removed resides in the HOMO, an antibonding MO (i.e. 7t g ) which is destabilised more compared to the corresponding AOs of both nitrogen and oxygen. Because of this low ionisation energy, NO+ is easily stabilised in many compounds such as NO+ H80i ' NO+ BF4- , etc.

yl

°

2 ·2 2 4 ·2 2 : KK a g (2.\') au(2.\") a g (2Px) 1t u (2p y ,2pz) 1t g (2py,2pz)··

lc

he

m

It produces one sigma (a)-bond along with a pi (7t)-bond. The two unpaired electrons in Ttg are responsible for its paramagnetic behaviour. The unpaired electrons result from the Hund's rule ofspin multiplicity. These electrons ! are. unpaired but they contribute to maintain the bond order. Thus to yield the bond order, the :. electrons are not to be necessarily paired up (cf. VBT). Thus the MOT does not give any special weightage to the paired electrons in determining the bond order. It simply reckons the number ofbonding versus antibonding electrons irrespective oftheir pairing or unpairing. VBT could not produce any unpaired electron in O2 to explain its paramagnetic properties. It was explained by MOT (Mulliken, 1928) from the electronic configuration designed above. The • two unpaired electrons in the HOMO 7t g (2p) (i.e. SOMOs) produce the triplet state (28 + 1 == 3), designated by symbol 3~;, Le. 3°2 . From the molecul~r orbital picture, two excited singlet states L; and I l1g ) were predicted. In fact, three years after Mulliken's classic explanation, Child and Mecke established spectroscopically the I~; state, 155 kJ mol-I above the ground state. In 1934, Herzberg reported the existence of It!g state, 92 kJ mol-I above the ground state. It has got a relatively long half-life period. The properties of both the ground triplet and excited singlet states are given in Table 9.9.2.1. In this connection, it is worth mentioning that the

e/

(I

th

ea

0 > H. In F 20 2, because ofthe very high electronegativity of F, very little amount of additional electron density is pushed into the 7t~ of 02 through the 3c-2e F-O-O bond formation t9 weaken the 0-0

586

Fundamental Concepts of Inorganic Chemistry

bond. On the other hand, in H 20 2, because of the higher electronegativity of oxygen than that of hydrogen, the electron cloud is attracted into the 7t~ of 02 to weaken the 0-0 borf!. It leads to almost double bond order in 0-0 linkage (as in free 02) of F 20 2 while it becomes 'of unity bond order in H 20 2• • Ionisation energies of atomic oxygen vs. molecular oxygen: Lastly, it is interesting to compare the ionisation energies of atomic oxygen and molecular oxygen, Le. ~ 0+ + e, 13 14 kJ mol-I; 02 ~ 0; + e, 1164 kJ mol-I. The first ionisation potential of atomic oxygen is greater than that of ~,he triplet molecular oxygen. It is due to the fact that in the molecular ox;,gen, the electron to be knocked out resides in the HOMO (Le. SOMO), an antibonding MO (7t g ) which is destabilised compared to the outermost AO of the free oxygen atom. It may be noted that the reverse situation arises for Nand N 2 . • Colour of 02 : In gaseous state it is colourless but it is blue coloured in solid and liquid state (see Q. 135 for explanation in terms of MOT). .. ) 2 ·2 2 4 ·4 Th e bon d or d er .IS unity . w h·IC h·IS a Iso (Vll F2 : KK a R(2s) a u (2.\") a g(2px) 1t u (2py , 2pz) 1t g (2py, 2pz)· expected from VBT. Here, it is worth noting that all the halogens exist as X 2 (X == F, Cl, Br, I). The trend of variation of colour of the halogens (i.e. X 2 ) can be nicely explained from the molecular orbital picture. The visible light being absorbed actually excites the electron from the • • HOMO 7t g (np) to the next higher level a u(np) i.e. LUMO. The energy gap between these levels (i.e. HOMO and LUMO) determines the colour of X 2 • The observations are:

he

m

yl

ib

ra

ry

°

~ a:(2P); Cl2 (yellow) : 1t:(3P) ~ a:(3P); Br2 (reddish brown) : 1t:(4P) ~ a:(4P); /2 (violet): 1t:(5P) ~ a:(5P)· 7t:(2P)

lc

F 2 (pale yellow) :

t.m

e/

th

ea

The excitation energy lies in the sequence, F 2 > Cl2 > Br2 > 12• Gaseous F2 actually absorbs the highest energy light, i.e. violet light, and therefore, it appears pale yellow; whi Ie gaseous 12 absorbs the lower energy light, Le. yellow-green light, and it appears violet. The energy gap between 1t:(np) (HOMO) and a:(np) (LUMO) decreases with the increase ofthe principal quantum number n. With the increase of n, the larger p-orbitals get more diffused and as a result, the effectiveness in overlapping to form the MOs decreases. This gradual decrease in the efficiency of overlap reduces the difference of energy between the bonding and antibonding MOs. It is known that a better overlap produces a higher splitting between the MOs. Thus the better overlap condition stabilises the bonding MOs more and destabilises the antibonding MOs more. This along the given series. explains the sequence of energy difference between 1t~ and For F 2, the absorption occurs virtually in the UV region, for Cl2 the absorption band spans both UV and visible region, for Br2 and /2 the band spans in the visible range. The present transition, HOMO (7t.) to LUMO (a·) is forbidden (Fig. 9.9.2.3) because of two 1 i.e. singlet) and excited state (7t~ Le. triplet different spin states of the ground state (1t~4 state) and this is why, the intensity of colour, in general, is less. However, with the decrease of energy gap, the probability of transition also increases because of the better spin-orbit coupling to mix the spin states and this is why from F 2 to /2' the intensity of colour increases. In fact, when the energy gap between the excited state and ground state is small, the spin-orbit coupling can mix the singlet ground state with the triplet excited state. Consequently, the excited state bears the character of singlet state to some extent. This partially relaxes the spin forbidden transition. This relaxation is most efficientfor /2 where the energy gap is minimum. This is why, intensity of colour of iodine is so high.

a:

a:

a:

Introduction to 'Chemical Bonding and Theories of Covalence

(l~ (LUMO)

.

~

~

~

587

----{;--

hv (allowed)

(forbidden)

w

1l 1

lt

n

g(HOMO)

1l 1

(Singlet)

(Singlet)

ry

(Triplet)

1l

ra

Fig. 9.9.2.3. Spin allowed and spin forbidden transition between HOMO (7t~) and LUMO (cr~) of halogen.

he

m

yl

ib

2 ·2 2 4 ·4 ·2 (vii) Ne 2 :KKcr K(2 ...·) cr u (2.\·) cr K(2px) 1t u (2py ,2P:) 1t K (2py .2pz) cr u (2px)" Here the bond order is zero. This is why, it is monoatomic (cf. the case of He 2 vs. 2He). Because of the same~ fact, all other inert gases are monoatomic. Removal of one electron from the highest antibonding MO produces the bond order 1/2 and the positively charged species should exist. We have already mentioned the existence of He;. Similarly, Xe; ion is advocated to exist in strongly acidic solvents which may provide a high solvation energy. Similarly, to stabilise Xe; in the solid state, the lattice energy should be high.

lc

9.9.3 Limitations of the Simple Molecular Orbital Model

t.m

e/

th

ea

The simple model discussed above can explain a lot of facts but still, it is not satisfactory specially for the lighter elements, e.g. B2 , C2 , and N 2 ,. Some serIous drawbacks are mentioned below. (i) Magnetic properties of 8 2 and C2 : The model predicts B2 to be diamagnetic and C2 to be paramagnetic with respect to two unpaired electrons. But experimentally, B2 is found to be paramagnetic while C2 is found dialnagnetic. (ii) Properties of N 2 : The bonding pattern, in N 2 is comparable with that of an acetylenic linkage. Hence it should be reactive like the acetylenic compounds. But, in fact, N2 is very much inert. Besides this, experimentally, it has been established that in N2, the highest occupied orbital (Le. HOMO) is sigma (0') in character not pi (7t) in character. The above discrepancies arise because the order of energy of the MOs for the lighter species like B2 , C2 etc. is different from that of the relatively heavier species like 02' F2 • Fig.,9.9.3.1 describes the variation of energy order of the MOs for Li 2 to Ne 2 • In terms of the Fig. 9.9.3.1, the ele~tronic configurations (ignoring lcr~ Icr:2 ) of the species Li 2 to Ne 2 are in conformity with the experimental observation. These are given below. Li2 : 2cr~; Be 2 : 2O'~ 20':2 (i.e. Bond order = 0); B 2 : 2cr~ 2O':2I7t~ (i.e. paramagnetic) C 2 : 20"~ 2a:217t~ (i.e. diamagnetic); N 2 : 20"~ 20":217t~30"~ (Le. HOMO is a 0" - BMO) ·2·

2

·2

2

4

2

.?

?

4·4

.

02 : 2cr g 2O' u 30 g 17t u 27t g (I.e. paramagnetic) 2·2

2

4

·4.

·2

.

F2 : 20'g 2O' u- 30'; IJt u 27t g ; Ne 2 : 20' g 2O' u 30' g I7t" 27t g 3O' u ; (I.e. Bond order = 0)

588

Fundamental Concepts of Inorganic Chemistry 30~(2p)

3a~(2p)

11t~(2p)

11t~(2P)

~

~

Q)

c

en ~

......

3cr g(2p)

W

Q)

-- -- -_ . _----

_

.... ......~-....

11t u(2p)

======= . . ------

....

c

W

11t u(2p)

..... _ - - - -

ry

0> ~

ra

2cr~(2s)

2o~(2s)

20 g(2S)

ib

2ag(2s)

30 g(2p)

(a) Applicable for Li 2 to N2

(b) Applicable for 02 to Ne2

m

yl

Fig. 9.9.3.1. Variation of energy order of the MOs for the species Li2 to Ne2 of the 2nd period elements. Note: For Li2 to' N2 , the energy order is : 17t U(2P) < 3crg(2P) while for 02 to F2 the energy order is : 17t U(2P) > 3cr9(2P)·

Li2

Bond order: Bond length (pm) :

Be2

he

The experimental findings are :

B2

C2

N2

°2

F2

0

th

ea

lc

1 0 1 2 3 2 1 267 159 124 110 121 142 Bond energy (kJ mol-I): 110 272 602 941 493 138 Ma-gnesium : D D P D D P D = diamagnetic, P = paramagnetic

Ne 2

e/

9.9.4 Simple Molecular Orbital Diagram for the Homonuclear Diatomic Molecules Involving (n-1)d Orbitals

t.m

For the d-block elements, in many cases, gaseous diatomic molecules have been characterised. The bonding in such species of the 1st transition series (Le. 3d series) is complicated in terms of MO diagram as the 3d orbitals being contracted can give only poor overlap with the 3d orbitals of th~ adjacent centres. However, for the early members of the series (e.g. Ti, Vetc.), the radial extension of the 3d-orbitals is relatively larger compared to that of the late members of the series. It happens so due to gradual contraction of d-orbitals along the period with the increase of effective nuclear charge. Thus

overlap between the 3d orbitals is relatively favoured for the early members ofthe series. For the heavier congeners (Le. members of the 4d and 5d series), because ofthe relativistic expansion of the d-orbitals (cf. Sec. 8.20.3), the overlap becomes better to produce the MOs. It is illustrated for the 4d and 5d-series. If z-axis is taken as the bond axis, the d-orbitals can combine as follows:

d=2 ±d=2 (cr-MOs);dx= ±dx= (1t-MOs); dy= ± d y= (1t - MOs); d xy ± d xy (8 - MOs); d x 2 -y 2 ± d x 2 -y 2 (8 - MOs); The stability sequence runs as : crg >

1t u

> 8g •

2

4 4 1t u (J) 8.l:(J)

2 (J K(.\')

ra ib yl m Fig. 9.9.4.1. MO diagram of M2 molecule of the 2nd (Le. 40) and 3rd (Le. 50) transition series where the valence orbitals are ns and (n - 1)d. Here z-axis is taken as the molecular axis.

lc

(J K(J)

he

In addition to the (n-l )d, the ns orbital also participates in the MO formation. The energy difference between the ns and (n-l)d orbitals depends on many factors. This is why, the relative positions of the MOs obtained from the (n-l)d orbitals and ns orbitals cannot be always predicted. Fig. 9.9.4.1 gives the simple MO diagram for the homonuclear diatomic molecular species of the 4d and 5d series. In the light of the MO-diagram given in Fig. 9.9.4.1, the electronic configuration of M0 2 species of 4d-series can be given as follows: Electronic configuration of 42 Mo is [Kr] 4cP 5s 1; the 12 valence electrons (2 x 6) are distributed in the MOs in the following way:

589

ry

Introduction to Chemical Bonding and Theories of Covalence

th

ea

The 12 electrons are in the bonding MOs to give 6 bonds (20 + 21t + 28). It is diamagnetic.

t.m

e/

9.10 MODIFIED MOLECULAR ORBITAL ENERGY DIAGRAM FOR THE HOMONUCLEAR DIATOMIC MOLECULES (Specially for N2 and lighter molecules) In the previous section, the limitations of the simple molecular orbital energy diagram (Fig. 9.9.1.2) in explaining the properties of N2 and lighter molecules have been discussed. Actual variation trend of energy of the MOs is shown in Fig. 9.9.3.1 It suggests some modifications required. This modification can be attained in two possible ways: (i) incorporation of symmetry interaction (i.e. noncrossing rule) among the molecular orbitals obtained in the simple treatment; , (ii) participation ofhybrid orbitals (mixing ofs- and p-orbitals) instead ofpure atomic orbitals to generate the MO energy diagram. Both the approaches lead to the same conclusion. This aspect will be discussed and illustrated. 9.10.1 Symmetry Interaction (Noncrossing Rule) among the Molecular Orbitals obtained in the Simple Model According to the noncrossing rule ofquantum mechanics, the two orbitals ofthe same symmetry mutually repel. Consequently, the lower energy level gets lowered down more while the higher one is raised

590

Fundamental Concepts of Inorganic Chemistry ~ 3a~(2Px) (=: (1)_2) " " " ..3(J~(2Px) " J" "

----------_.

1

1t g(2p

(I

Y·Z

)

1'ru(2py. Z)

--(

;'

" -------.,.:=::::_-_. ,.,,""""

7' 30 g(2PK) (= _1)

r

l1t U (2Py,z)

"

ry

(1 *

~ l1t g(2P y,z )

' ..... ..........

..........

..........

..........

--'-----------

42CT g(2s)(= . c2 ; and for _i and _2' c 2 > c1• +1

+1

= = =

can be expressed as follows:

yl

The wave function

+ 0/2s(B)}] + c2[N{'¥2Px(A) - '¥2Px(B)}]

tC1[N{'¥2s(A)

c; {\V2s(A) + A\V2Px(A)} + ci {\V2s(B) - A\V2Px(B)} c; {\V '\Px(l)(A) + \V .\Px(I)(B)} ' (Le. in phase, combination)

m

+1

c;

he

For

ib

±\

where, =c1N,c; =c2NandA,=c;lci c2 for +1' Thus it is evident thatJor the hybrid orbitals sPx(I)(A) and sPx (I)(B) the contributionfrom the respective 2s orbital is more. Similarly, the can be expressed as : _1 =

.

lc

_1

c1[N{ 0/2.\,(A) + 0/2s(B)}] - c2[N{ \V 2Px(A) - \V 2Px(B)}]

ea

wave function

c; {A\V2'\'(A) -

\V2Px(A)}] +

c; {A\V2'\'(B) + \V2Px(B)}

= c2N, c; = c1N and A, = c; I c;

e/

where c;

th

c;{\VsPx(2)(A) + \V.fPx(2)(B)}' (Le. in phase combination)

c

< 1 as 2 is greater than c1 for

_1' Consequently, the hybrid

orbitals \V'\Px(2) (A) and \V'\Px(2) (B) are more enriched with the respective 2px orbital character.

t.m

(Note: Thus the two sp-hybrid orbitals, i.e. sPx(l) and sPx(2) of a particular atom, are not equivalent. The hybrid orbital. sPx( 1) is enriched more with the 2s-character while the other hybrid orbital sPx(2) is enriched with the more 2px character. Thus these sp-hybrid orbitals are different from the equivalent sp-hybrid orbitals discussed in Sec. 9.13). The wave function +2 is given by : ~2 = c1[N{\V2s(A) - \V2s(B)}] + c2[N{'¥2Px(A) + '¥2Px(B)}]

c; {\V2.f(A) + A\V2Px(A)} - c; {\V2s(B) -

A\V2Px(B)}

c; {\VsPx(l)(A) - 'V.fPx(l)(B)}' (Le. out ofphase combination) _2 =

c1[N{'¥2s(A) - 'V2s(B)}] - c2[N{'¥2Px(A) + '¥2Px(B)}]

c; {A\V2s(A) -

\V2Px(A)} -

c; {\V.\Px(2)(A) -

c; {A\V2s(B) + 'V2Px(B)}

\VsPx(2)(B)}' (Le. out ofphase combination)

•

+

A

0~(25)

0~25)

°g(25)

°g(25)

~

+ I

-

..

~

+

I

(J~(2P .--)

+ I

-

-°g(2px)

A

B

. + I

•

0g(2p)()

+ I I

0~(2s) - a~(2p)()

--+

--+

----+

B

If

-

~

+ I

(+2)

c:{)tp

I +

(+1)

d+h

A

(_2)

ry (}{)

ra

0~(25) + a~(2Px)

ib

yl

11-.+1 -6+11

°g(2s) - °g(2px)

-

+

--+

and 0g(2PIl) and between 0~(2S) and 0:(2Px)O

-----+

-----+

Gg(2s)

II +.

m

----+

he

0g(25)

lc

°g(2px)

Y\- ~

B

ea

th

e/

r"I

t " .m +

A

Fig. 9.10.2.1. Symmetry interaction between

•

+

B U)

C1I

~

"

~

Cir .....

3

CD

=r

(")

c:r

~

Q)

ca

0

:;-

~

0

.....

~ CJ)

0 CD

~

~ (") 0

~ .....

CD

3

~

0.

c

~

Introduction to Chemical Bonding and Theories of Covalence

595

2s

ra

ry

A_I is more stable than sPx(2) while _2 is less stable than sPx(2); (b) +1 (or +2) concentrates more electron density between the nuclei compared to that of its main constituent 0g(2s) [or 0:(2S)]' It makes

he

CI> +1 and CI> +2 more stable compared to 0 g(2s) and 0u(2s) respectively; (c) similarly _1 and _2 are less • stable compared to 0 g(2p,:> and 0u(2p,,) respectively. All these aspects are shown in Figs. 9.10.1.1, 9.10.2.1 and 9.10.2.2.

t.m

e/

th

ea

lc

It is evident that the MO energy diagram shown in Fig. 9.10.2.2c is identical with the one obtained from symmetry interaction (cf. Fig. 9.10.1.1 and Fig. 9.10.1.3). Possibility ofparticipation ofhybrid orbitals in MO formation: The possibility of hybridisation to produce the orbitals sPx ( 1) and sPx(2) can be analysed by considering the energy difference between 2s and 2px orbitals. The energy difference between the 2s and 2p-orbitals reaches from ca. 200 kJ mol1 to ca. 2500 kJ mol-I in moving from lithium to fluorine. From Fig. 9.10.1.2, it is evident that for the heavier atoms (like 0, F), the energy difference between the 2s and 2p-orbital is quite large and consequently the mixing between these orbitals to produce the hybrid orbitals is not much important. On the other hand, for the lighter elements, the energy difference is quite small (cf. Fig. 9.10.1.2) and the mixing between the 2s and 2p orbitals is quite significant. This is why, it is reasonable to consider the fact that for the lighter elements, the hybrid orbitals (rather than the pure atomic orbitals) participate in generating the MOs. However, here it is worth mentioning that there is no experimental evidence to deny the fact that in the case of highly electronegative elements like fluorine or oxygen the hybrid orbital does not participate in forming the MOs. If the hybridisation is insignificant, then the MO energy diagram shown in Fig. 9.10.2.2c will automatically be converted into the simple MO energy diagram shown in Fig. 9.9.1.2. Thus the MO energy diagram given in 9.10.2.2c is of much general character. The electronic configurations of some representative homonuclear diatomic molecules are given in Table 9.10.2.1 in terms of the modified energy diagram.

Note: The concept of hybridisation before molecular orbital fomlation simplifies the treatment as shown in Fig. 9.1 0.2.2c. But this VBT concept is not essential. LeAO of four orbitals, Le. 2s and 2px (x-direction being the bond axis) orbitals of A and B can produce 4 MOs, Le. \II MO = cl \112s(A) + c2 \112Px(A) + c3\II 2s(B) + c4 \112Px (B)

596

Fundamental Concepts of Inorganic Chemistry

A

A

B

.

B

°g(2s)

~

0-

~_(}_+r=D

ry

-~-v--v (c])_1)

°9(2s)

r1J~_r=D-(}V

ra

~~ ~

+~~-~-VV (J~2S) + a ~2p.)

°u(2px)

$px(1 )(A) sPx(1 )(B)

(

I I

I I

Fig. 9.11.2.3b. The characters and shapes of the MOs of CO with reference to the Fig. 9.11.2.3a.

Introduction to Chemical Bonding and Theories of Covalence, 611

,

\ 1t.

• ,

(LUMO)

I'll

"

\

'\

)'

w

1~

),.;"

,,

1

AC's of C

\

'"

(HOMO)

c.;"\

(-19.5 eV)',

',\

1~

~

,

~

""

\ \

~.

'\ ,

,

(-16 eV)

~,

~~,

\

"

~

",

1~1tb ~~ "

l' --

\

,

1~(J

1~nb

I

,

ra

2s

\)

'\

ib

~

Q)

,, \

,1 \,

(-10.7 eV) \ "" 0nb c:

Note : This diagram is also approximately applicable for other heteronuclear diatomic molecules like NO

',,,

- - - - -

1~

yl

·2p

,

\

I

ry

,

0·

,

m

MO's of CO

(-32.4 eV) AO's of 0

Fig. 9.11.2.3c. Approximate MO energy diagram of CO (energy values not in scale).

he

Note : a 1,2,3 = ci \V 2s(c) + c2 \112px (c) + c3\112px (0)

lc

and a l ,2,3 = denotes a, a nb and a·; x-axis taken as the molecular axis.

ea

high dipole moment which is in contradiction to the experimental fact (-- 0.1 D). This is why, the following cannonical forms are proposed. :C

= Q(I) ~ :c ~ Q+ (II) ~ : C+ -

Q:- (Ill)

t.m

e/

th

Here, the dipole moments of (I) and (Ill) are supposed to oppose the dipole moment of (II), and as a result, the dipole moment is significantly reduced (see Sec. 10.5.3 (iv) ). But it is difficult to realise the relative contributions of (I), (II) and (III). On the other hand, in the MO theory, it is easy to explain the high bond order in CO and other properties of co. • Properties of co and N 2 and other isosteric species as ligands : Now we shall pay a special attention to the molecule CO. It can act as a potential pi-acid ligand, specially to stabilise the low oxidation states of different transition metal complexes. Interestingly, though N 2 is isosteric with CO, N 2 has got almost no tendency to act as a ligand. We have already discussed CO in the light of the general molecular orbital diagram (see Fig. 9.11.2.2c). But, here the electronegativity difference between the combining atoms is sufficiently large to keep \V (.\"P}I of 0 and \V (!;ph of C as the nonbonding orbitals. The more appropriate molecular orbital diagram of CO is given in Fig. 9.11.2.3a. The nonbonding orbitals carry the lone pairs. The lone pairs reside in the nonbonding \V(SP}I of 0 and \V(.\"ph of C. The nonbonding \V (.\1'}1 of 0 is enriched in the 2s character (see Fig. 9.11.2.3b) of oxygen while t~e nonbonding \V(sph of C is enriched in the character of 2p AO of carbon (see Fig. 9.11.2.3b). The nonbonding \V (SP}2 of C is the 110MO and 7t* is the LUMO. CO to act as a 7t-acid ligand, the HOMO electron pair is to be used in a-donation and LUMO is to be used in receiving the metal electron through 7t-boniding. Thus when CO acts as a ligand, it first donates the lone pair residing in \V(.\"ph on carbon and then takes back the electron cloud in H* which is again enriched

612 ·Fundamental Concepts of Inorganic Chemistry

in the properties of 2p orbital of carbon. The lone pair in the nonbonding \V (sp). on oxygen cann~t be preferably donated as it is tightly bound to the more electronegative oxygen centre through an orbital which is almost 2s in character. If CO is argued to bind through the a-end by donating the lone pair on O-end then x-acceptance in the 7t. (Le. LUMO) is disfavoured. Because, the 7t. is mainly concentrated on the C-end. Thus ligation through the O-end of CO is disfavoured compared to ligation through

e/

(c) SF VB. CO, N2

th

ea

lc

he

m

yl

ib

ra

ry

the C-end because of both unfavourable ligand to metal a-donation and metal to ligand x-back bonding (called retro-bonding). If the picture is compared to that of dinitrogen, it is seen that it has got no nonbonding pair to be donated to the metal ion. In N 2, the HOMO is a a-BMO i.e. the electron pair to be donated is concentrated between the nuclei. It is difficult to use this HOMO pair for a-donation. The metal ~ ligand re-bonding involves the partication of 7t. MO (which is the LUMO) of N 2 and this 7t.-MO is equally concentrated on both ends. Thus the overlapping in the said 7t-back bonding does not experience any difficulty but this x-back bonding is not so efficient compared to the case of CO. Because, in CO, presence of the more electronegative atom oxygen favours the electron flow from metal to ligand. No such driving force works for N 2 as a ligand. However, it has been proved that the marginal stability of the metal-dinitrogen linkage arises mainly due to the very weak metal-.+ ligand x-back bonding. Thus the difference between co and N 2 as ligands arises because of the difference in their MO picture. In N 2, there is no difference of electronegativity between the combining atoms. This is why, N2 cannot produce a molecular orbital diagram which is present in CO. Thus among the isosteric species, e.g. ClV, CO, NO+, N 2 , except N 2 , all other species can act as potent pi-acid ligands. Except N z in all other cases, there is a HOMO (which is nonbonding or weakly bonding) enriched sufficiently in 2p character of the less electronegative element. The electron pair residing in that orbital is donated (Le. a-basicity) easily and then the electrons from the metal centre are accepted back (Le. x-acidity) in the vacant 7t. MO (which is LUMO) enriched in the 2p character of the less electronegative atom. Presence of the more electronegative heteroatom acts to favour the electron flow from the metal to ligand. Both the a-basicity and x-acidity act synergistically.

t.m

These are isoelectronic species. Ifwe construct the MO diagram for BF as in CO (cf. Fig. 9.11.2.3), the bond order in BF molecule becomes 3 (1 cr + 27t). But experimentally, the bond energy of BF has been found very low, 548 kJ mol-I (cf. bond energy of CO = 1070 kJ mol-I). Thus, in BF molecule, it is reasonable to consider the existence. of a single bond only. In fuct, the molecule BF strikingly differs from its isoelectronic species CO and N 2 • This can be explained by considering the relative energy difference among the combining orbitals. In N 2 (which is a homonuclear diatomic species), the question· of energy match does not arise. In BF, the electronegativity difference is maximum among these species. The energy values (in eV) of the orbitals are as follows: COB F 2s: -19.5eV -32.4eV -14eV -40.2eV 2p : - 10.7 eV - 16.0 eV - 8.3 eV - 18.6 eV Thus in BF, the good match (in terms of energy condition) only prevails between 2s(B) and 2p(F) orbitals. Thus the MO diagram (Fig. 9.11.2.3c) for CO is automatically converted into the diagram

shown in Fig. 9.11.2.4.

Introduction to Chemical Bonding and Theories of Covalence a* \

I

I I

I

,, , \

I I

2p I ~----

~

~ w

25 (-14 eV)

,

U ~

//

b../

"

,,

U ", '" ' ,,'\

/

CJnb

\ 1tb

\

----~~

It U

'

// (-18.6 eV)

\

\

I

\

I

\

/

,

/

flUb

I

ib

\

2p

ry

~

'"

n

,

ra

.

(-8.3 eV)

1t

61 a

25

CJnb

MOs of BF

m

AOs of B

- - - - - ---+4U~

yl

ft

Fig. 9.11.2.4. MO energy diagram for SF (cf. Fig. 9.11.2.3c) where 1,2,3 denotes 0b' 0nb and 0* respectively.

AOs of F

0),2,3

=C)\V(2S)(S) + C2'V(2Px)(S) + C3\V(2Px )(F)

he

°

(-40.2 eV)

ea

lc

Note: For CO (Fig. 9.11.2.3a, c), the p-orbitals of C and 0 differ by about 5 eV in energy and they can participate in forming the 7t-MOs. On the other hand, for SF (Fig. 9.11.2.4), the 2p orbitals of S and F differ by about 10 eV in energy and consequently they fail to participate in forming the 7t-MOs. In fact, they remain as nonbonding x-orbitals {7t ntJ.

In terms of the diagram (Fig. 9.11.2.4), the electronic configuration of BF is :

th

(cr nb)2 (cr b)2 (7t nb )4 (cr nb)2 (Le. bond order = 1)

e/

(cf. for CO: (cr nh )2 (cr h )2 (1t h )4 (cr nh )2 and bond order = 3, in terms of Fig. 9.11.2.3a, c)

t.m

This is why, BF does not survive and undergoes disproportionation as follows:

3BF(g) ~ 2B(s) + BF3(g)

(d) C2 , BN, BeD and LiF

All these are isoelectronic species, but their thermodynamic properties differ widely (in gaseous phase): Bond length (pm) : Bond energy (kJ mol-I) :

C2

BN

BeO

LiF

124 602

128 385

133 444

156 568

This change can be understood by constructing their MO diagrams. In moving from C2 to LiF, there is decreasing trend of energy match of the combining AOs.

(e) CIF, ICI The MO energy diagram of such interhalogen compounds can be roughly drawn as in 9.11.2.3a and c for co. The.~-orbital of the more electronegative centre may be roughly kept as a nonbonding one

614

Fundamental Concepts of Inorganic Chemistry

because of the energy reason. But the p-orbitals of both halogens can participate in MO formation because their energy difference is not too large (cf. 1st I.P. values: 18.6 eV, 12.95 eV, 10.44 eV for F, CI, I respectively). Their 14 valence electrons are distributed as follows:

cr~bcr;1t:cr~h7t*4 (cf. Figs. 9.11.2.3c and 9.II.2.3a)

ry

The bond order is I. Here it may be noted that if the general MO energy diagram (as given in Fig. 9. I 1.2.2c) involving all the valence orbitals to participate in MO formation is considered, then the bond order is also I. In fact, if the energy difference among the valence orbitals is not too large, the general diagram of Fig. 9.11.2.2c is applicable.

ib

ra

9.11.3 Ultraviolet Photoelectron Spectroscopy (UV·PES) : Identification of the Nature of Molecular Orbital Energy Level

m

yl

It may be noted that ordinary electronic spectroscopic technique does not lead to ionisation of the species. It simply leads to excitation of an electron from one energy level to another. The important transitions are: cr ~ cr*, nb (nonbonding) ~ cr*, 1t ~ 1t*, nb ~ 1t*

he

When a molecule (say M) interacts with the photons of high energy UV-radiation, the photon absorption may lead to the ejection of electrons from the molecule, Le. M+hv~Ar+e

th

ea

lc

The energy of the photon will be utilised to provide the ionisation energy (I) of the bound electron and rest ofthe energy will impart the kinetic energy (EK ) ~c: UJ~ g0 to the ejected electron. A small part of the energy (Ev~ib) _ e ~~ will be utilised in exciting the vibrational level of the ~Q) Q) .s ionised species (Le. Ar). Thus the energy balance is: c: 0

e/

hv = 1+ EK + Evib ' or, EK = hv - Evib - I

Q)

Q) .J::.

c: -

~o

t.m

In this technique, the monochromatic UV radiation of21.2 eV [He(I) emission, 2p ~ Is] is generally used. For much higher energy, radiation of 40.8 eV [He(II) line obtained from He+] mdY be used. The ejected

Q).J::.

o a.

+:

electrons are detected according to their kinetic energies (EK ) and the peaks in a PES correspond to the E K

hv

values of the photoelectrons ejected from different energy levels (cf. Fig. 9.11.3.1) E Vib (which is quantised) is much less than the ionisation energy (I) and thus E K value (~ hv - I) corresponds to the ionisation energy from a particular level. In a molecule, there are several energy levels and ionisation may occur from different levels and these different levels require different ionisation energies' (Le. representing different peaks in PES).

fJ Fig. 9.11.3.1. Principle of photoelectron spectroscopy. /1 and /2 denote the ionisation energies from the respective energy levels. hv denotes the energy of the photon. The spectrometer records the electron counts with different kinetic energy.

Introduction to Chemical Bonding and Theories of Covalence

615

'b V1

=_1

27t

if V;

m

v

yl

ib

ra

ry

The ionised molecule (say Ar) uses some of the energy ofthe incident photon to excite the vibrational level and the required energy is quantised (EYib = nhv~ib' n = 0, 1, 2,... which gives the change in vibrational quantum number). It leads to the reduction of E K (= hv - nhv yib- I). This is why, for a particular I-value, the photoelectron appears with different kinetic energies separated by the quantised energy (= nhv yib ) required for vibrational excitation. In actual experiment, intensity (measured by electron count of some certain EK value) versus hv EK (= I + nhv vib) is plotted. The lowest energy end ofa band corresponds to the ion~ation energy (Le. n = 0 leading to EVib = 0). It indicates that each band corresponds to a particular energy level from which the photoelectron is ejected and the band at its lowest end gives the ionisation energy for that particular energy level. The highest energy end of the band indicates the ionisation followed by the dissociation of the ionised molecule. Thus, the width of a band depends on nhv vib. The highest energy end of the band indicates the limiting value of vvib leading to dissociation of the excited ionised molecule. The vibrational frequency (v vib ) is given by :

he

where f = force constant which increases with the increase of bond strength (Le. bond order), I.l = reduced mass. In terms of wave number, it is given by :

lc

VVib

=_1_ 27tc

{7

V;

t.m

e/

th

ea

The yibrational frequency of a diatomic species depends on its bond order. Comparison of the vibrational frequency of the ionised species (M+) and unionised species (M) can tell us the nature of the molecular orbital from which the electron is lost. If a nonbonding electron is lost, the bond order ih Nr does not change and consequently vvib does not change remarkably compared to that of the unionised species. If a bonding electron is lost during ionisation, the bond order decreases and as a result v vib decreases; if an antibonding electron is lost, the bond order increases to increase the v vib' It is illustrated for NO (stretching frequency = 1890 cnl 1, bond order = 2.5). • NO+ (produced for the loss of a 7t-bondi~ electron from_the bonding 7t~li MO of NO, cf. Fig. 9.11.2.2a) : It reduces the bond order and v vib decreases. v vib = 1200 cm < 1890 cm- 1, 1= 18 eVe • NO+ (produced for the loss of an antibonding_electron from the HOMO 7t;p MO of NO cf. FIg. 9.11.2.2a) : It increases the bond order and Vvib increases. VVib = 2260 cm -1 > 1890 cm -1, 1= 9.3 eVe Let us now illustrate the UVPES for determination of energy levels in some molecules. 8 2 : The lowest energy end of the UVPES (Fig. 9.11.3.2) band is 15.5 eV (with the separation of 2260 cm- 1) and it gives the ionisation energy (I crb ~ (0). The vibrational separation in the UVPES band gradually decreases as the band moves towards the dissociation limit 18 eVe Here it is important to mention that for atomic bydrogen, the ionisation energy is 13.6 eV which is less than that of molecular hydrogen. In the bonding MO, the electrons are stabilised more compared to those in the atomic hydrogen. The stabilisation is (15.5 - 13.6) eV = 1.9 eVe The vibrationalfrequency 2260 cm- 1 for H; is less than

616

Fundamental Concepts of Inorganic· Chemistry } Vibrational energy levels ~

13.6 eV

(H) // ,

i

10 eV

0'*

·15

(1)

c

,

15.5 eV (ab)

-·crb~"""""'"

18

17

ra

Ionization energy

5eV

ry

11

'ecn

16

(h\.I -

15

~

ib

18 eV

EK ) eV

(b)

yl

(a)

Fig. 9.11.3.2. (a) Energy level diagram of H2 and its UV-photoelectron spectrum. (b) The fine structure of UV-photoelectron spectrum of H2 due to excitation among the vibrational energy levels in the cation

m

H2·

he

the vibrational stretching frequency 4200 cm- 1 of H 2 • It also supports that the bond in H; is weaker than that in H 2 • Ordinary electronic spectroscopy gives the measure of cr ~ 0* transition.

th

ea

lc

N 2 : UVPES (Fig. 9.11.3.3) of N 2 shows three ionisation energies: 15.6 eV, 16.7 eV and 18.8 eV (cf. 1== 14.5 eV for atomic nitrogen). The lowest ionisation energy (15.6 eV) corresponds to thel-IOMO Sag level (cf. Figs. 9.9.3.1 and 9.10.2.2c, d), the second value (16.7 eV) corresponds to the'rem?val of an electron from the 7t-bonding MO (l7t u)' and the third value (18.8 eV) corresponds to the 40 u level. These level assignments are also in good conformity with the vibrational separation.

+---

e/

2

Vibrational energy levels

t.m

11

~

~

Q)

:;:

c


ec::!), enriched with the character of metal orbital. Thus LUMO is basically a \ metal orbital (cf. Fig. 9.11.4.1) and it LUMO ~ \ Q) acts as the acceptor of electron. The c: W "6> e and their combinations. (b) MOs of H3t (c) Wave patterns of the MOs of H3 in terms of the model particle in a box. Note : If a vertical nodal plane runs through a nucleus, then the orbital of that atom does not participate in the MO formation as in Gnb in the present case.

Introduction to Chemical Bonding and Theories of Covalence a*g I

, \

I I

\

I I

,,

I

\

I

\\ (Nonsymmetrical)

I I

--(au)

I I

anb

( \

\

\

,, \

" ,,,

\ \ \

\

,

+H(+)

ADs of2H

(TASOs) of 2H

ab

yl

(MOs)

1s

ry

"

ra

\

\"..----',.,,,»--1S

, ,, , (Symmetrical)

\ \

+H(-) ----------.." ',HA

------~ \,

ib

1s

623

m

Fig. "9.12.2.2. Molecular orbital energy diagram of the linear triatomic molecule H3 .

he

Note: Without considering the concept ofH-group orbital, the simple LCAO of the three Is orbitals of H A , H B and He can also generate the three a-MOs. These are:

± \II b( He)

- ~ Is (Ha)

=\112 : \II Lv (H A) -

Fig. 9.12.2.3 and (cf. Fig. 9.12.3.1)

\Ills (He) + \Ills (Ha)

= \112 ::- \II b (H A) + 'V Lv ( He) - \II Lv (Ha)

th

CJah

= \113 : \II b (H A)

ea

a nh • a ah •

or

lc

a h = \111 : \lib (H A) + \IIb(Hc ) + \lib (Ha )

e/

O+O+Q dd,a,-p,ane O+O-G) · ·:0 0-0-0 80 0-0+0 ·~0!O!G -O+O-G) ~O!Q!G

t.m

=>

=>

+

I

=>

HA

He

He

CJb

= '1'1

°nb

=~3

. 'l'nb

= '1'3

}

a~b = '1'2

a~b

= '1'2

MOs

Fig. 9.12.2.3. Schematic representation of MOs. in H3 .

Weakly .antibonding

624

Fundamental Concepts of Inorganic Chemistry "

For the sake of simplicity, the mixing coefficients are not shown. Apparently, the expression for crnb in two concepts~are different, but they lead to the same results. The bonding interaction between H A and He is counterbalanced by the antibonding interaction between the H B and He. The three electrons in H] are distributed as cr~ cr~h (i.e. total bond order = 1). In fact, 'l'J (leading to G nb ) containing an unpaired electron is weakly antibonding. This antibonding effect between the (cr~) where there is no electron in the cr~b. In terminal atoms destabilises the system compared to fact, in spite of two electrons in the lowest SMO, the species H] is not stable and 'it is transient. It is evident from the following fact.

ra

H] ~H2 +H,tUf=-34kJmol- 1 2H] ~ 3H2 , ~H =- ve (i.e. exothermic)

ry

H;

he

m

yl

ib

In the product (i.e. 3H2), the total bond order is 3 while in the reactant (i.e. 2HJ ) the total bond order is 2. It makes the process exothermic. Existence of this transient species H] has been postulated to explain the exchange reaction with deuterium (D). D+H2 ~[DHH]~DH +H It is evident that \II] (bearing an unpaired electron) is of antibonding character between the terminal atoms (HA and HB ). If the molecule tends to bend, the terminal atoms come closer and consequently, this bending is opposed by the antibonding electron. This is why, H 3 is linear. The molecular species, having no antibonding electron does not experience any such opposition in bending the geometry. In fact, H; gives the geometry ofan equilateral triangle (cf. Fig. 9.21.4). It can be shown that for this triangular molecule, the energy of the lowest MO (bonding) is lower than that of the crb of the linear geometry. It must be mentioned that in the triangular species, because of the absence of cylindrical symmetry the notation of cr-a*-(Jnb is not strictly applicable. The absence of any antibonding electron makes the species more stable than H J •

e/

th

ea

lc

H;

H;

H; ~ H+ + H 2 , Ml = + 441 kJ mol-

H; ~ 2H + H+ , Ml

1

= + 857 kJ mol- 1

t.m

It is important to note that in H; (i.e. cr; cr~h) both the bonding and nonbonding MOs are filled in (i.e. 3c - 4e bonding mcdel) and the species is linear. The antibonding effect of cr~b destabilises the species when it tends to bend.

Stability order: H; > H] > Hi, (cf. antibonding effect of crnb ). It indicates that the number of valence electrons plays an important role to determine molecular shape and stability. This concept will be illustrated for XH2 (linear and bent shape, cf. Walsh diagram), XY2 (linear and bent) species in the next section. 3c--4e bonding gives a linear segment - why? 3c-2e bonding gives a bent segment - why? Bond order: The isostoichiometric species H], H; and Hi are having the electronic configurations 0 an d cr h2 cr nh 2 respective • Iy. Th· · h·lometnc . specIes · are h · th &lornu ' . I a b ut cr h2 cr 1nh' cr b2 cr nb e ISOstOIC aVlng e same different valence electrons.

Introduction to Chemical Bonding and Theories of Covalence 625

. .. Total bond order • Bond order per InteratomIC lInk = - - - - - - - - - Number of interatomic links Each interatomic link connects a pair of nuclei. Total bond order =

I

"2

(No. of bonding electrons-

No. of antibonding electrons).

%). Thus

ry

In the linear H 3 molecule, there are two interatomic links with a total bond order is 1( =

the bond order per link is 0.5. In the same way, the bond order per link in H; and Hi is also 0.5.

ib

ra

In the hypothetical linear H IO molecule, the total bond order is 5 with 9 interatomic links. It leads to bond order 5/9 per H-H link.

yl

9.12.3. Molecular Orbital Treatment for Linear Hydrides XH 2 (e.g. BeH 2, BH;, etc.) Having Valence Electrons Four

lc

he

m

The central atom X may be considered as a second period element and it can provide 2s and three 2porbitals for MO formation. If z-axis is considered as the molecular axis for the molecule, HA-X-HB (terminal H-atoms are denoted as HA and HB ), then the central atom (X) can provide 2p= and 2s orbitals for a-MO formation. For this purpose, the terminal H-atoms provide their Is orbitals. Thus LCAO of these four orbitals 'V2,v(X)' 'V2Pz(X)' 'Vb(H A ) and 'Vb(H B ) will produce four a-MOs. To visualise the generation of these four a-MOs, let us consider the H-group orbitals (i.e. TASOs) from th~ terminal H-atoms. The Is orbitals of HA and HB can give the following H-group orbitals. cPH(+)

ea TASOs

= 'V b(H ) + 'V h(H A'

th

{ cPH(-) = 'Vl.\'(H

A

\VIs(H

) -

)'

(symmetrical)

)'

(nonsymmetrical)

B

B

(cf. Fig. 9.12.1.1 a)

t.m

e/

The symmetrical H-group orbital will combine with the symmetrical 2s orbital ofthe central atom to generate two a-MOs (I bonding and I antibonding). Similarly, the nonsymmetric H-group orbital will combine with the nonsymmetric 2p= orbital of the central atom to produce two a-MOs (I bonding and I antibonding). Formation of these four cr-MOs is shown below. 'VI = cr,\, =lcr g = c1'V2s(X) +c2 cPH(+)

c1\II2s(X) + c2

{\II Is(H A)

+ \IIIs(H B) }

C3'V2s(X) - C4 cPH(+)

= C3\112s(X) •

\11

'1'3

=

00* Pz

= a Pz

+ \II 1.H(-)

yl

\', ''1.\

\\ \\ \\ \ \ \ \

~

'

( \ \ \ \ \

\'

H20 :

experimentally supported by UV-PES

t

A/(

25

y

ry

\\

\ \ \ \ \ \ \

ib

'112 is approximately non-bonding

\

ra

\', I \ , I \\ 'I .\\ ), \\ I'

15

,.,##

4>H(+)

(lASOs) (AOs of 2H

of 2H)

"

/' ('1'1 )

(MOs of XH2 )

t.m

e/

Fig. 9.12.4.2. MO energy diagram for the bent molecule XH2 (e.g. BH2 , CH2 , NH2 , OH2 , etc.). MOs (i.e. as' a Pz ' a PJI ' a ~z and a ~JI) are named after the major contributing atomic orbitals of the central atom (X).

Note : • The relative energy values of the combining AOs are different for different molecular species. • Because of the lack of cylindrical symmetry (i.e. cao)' strictly a-a* notation is not applicable.

Walsh's rules and molecular geometry of XH2 The XH2 molecular species having 3-4 valence electrons are linear while the XH2 molecular species having valence electrons 5-8 are bent. The explanation has been already given in terms of MO energy diagrams (Figs. 9.12.3.2 and 9.12.4.2).

Species : No. of valence electrons: Geometry :

BeH; BeH2 BH;

3

4 Linear

4

BH2

5 ----------------Angular

Introduction to. Chemical Bonding and Theories of Covalence

631

9.12.5 Illustration of Walsh Diagram for XH 2 Species

0

+H

'VI, bent = lal = c3'V2px(X) + c4 'V2s(X) + cS

+

H (+);

(H -X -H

=90

(cf. Fig. 9.12.4.1a) +H(+)

0

and 2s(X) orbital does not participate)

denotes the symmetric H-group orbital Le. (TASO).

he

Here,

(bond angle between 90° to 180°)

m

= C6 '112P.(X) + c7

(+)'

and 2px(X) orbital does not participate)

ib

=as =lag =CI'II2s(X) + C2+H(+); (H -X -H =180

yl

'III, linear

ra

ry

VSEPR theory (Sec. 10.8.3) can predict qualitatively the shapes of the molecular species. In terms of MOT, the electron count also determines the molecular shape. To accommodate the valence electrons, the energy of the MOs may change. As the energy of the MOs changes, the molecular geometry also changes simultaneously. All these aspects have been analysed by A.D. Walsh (1953). Here we shall illustrate the approach by considering the correlation of the MOs of the linear and bentXH2 molecules (where X is a second period element). In this regard, we can compare the MO energy diagrams of the linear and bent molecules of the type XH2 (cf. Figs. 9.12.3.2 and 9.12.4.2). Now, let us examine the change ofMOs in changing the bond angle from 180° to 90°. Effect on +H(+) T ASO : If the linear XH2 molecule (z-axis is the molecular axis) bends in the xzplane, then the change can be expressed as follows:

+H ( +)

= 'V b( H A) + 'V b( H B )

t.m

e/

th

ea

lc

Thus in changing the bond anglefrom 1800 ·to 90°, the contribution of2s orbital of X decreases and in fact, it becomes completely off in bonding interaction when the 90° bond angle is attained. At the same time, contribution of the 2px orbital of the central atom increases in a-bonding interaction (cf. Fig. 9.12.4.1). The energy of the 2s orbital is less than that of the 2p-orbital, and consequently the energy of ",increases in bending the molecule. In the bent molecule, it is denoted by la l , while in the linear molecule, it is denoted by la. Effect on +H(-> T ASO : Now let us consider the interaction between the nonsymmetric H-group orbital (TASO) and 2pz-orbital of the central atom. It is expressed as : 'V2

= cS'V2pz(X) + C9 +H (-) (cf. Fig. 9.12.4.1 b)

[+H(-)

= 'V1s(H A) -

'Vls(H B)]

In bending the molecule, bonding interaction of the 2pz orbital of the central atom (X) gradually decreases. Consequently, the H-group orbital, +H(-> produces an antibonding interaction between the terminal atoms. Thus '1'2 .becomes more stable when the terminal H-atoms are kept more apart. Definitely, the linear geometry gives the most stable condition for 'V2. In other words, on bending, energy of'V2 gradually increases. Conventionally, 'V2 (for linear molecule) is denoted by la u while it is denoted by b2 for the bent molecules. In linear geometry (along the z-axis), the 2P:x and 2py orbitals remain as nonbonding ones (Fig. 9.12.3.2) but on bending in the xz-plane, energy of the 2P:x orbital (contributing in 'VI MO) decreases and 2py orbital remains unaffected. Thus the 2py orbital remains as a nonbonding one in both the linear and angular geometry. In the limiting situation (Le. H-X-H = 90°), the 2s orbital becomes nonbonding and this nonbonding 2s orbital is denoted as 2a l •

632

Fundamental Concepts of Inorganic Chemistry

Thus it is evident that in the linear geometry, there are two nonbonding orbitals which are pure, p-orbitals (i.e. 2px and 2py, cf. Fig. 9.12.3.2) of the central atom. On the other hand, in the bent geometry (90° bond angle), the nonbonding orbitals are the 2s and 2py orbitals of the central atom.

ry

Thus in both the geometries, the common nonbonding orbital is one p-orbital (i.e. 2py orbital ofthe central atom), and consequently its energy remains unchanged in changing the bond angle from 180° to 90°. The other nonbonding orbital in the bent geometry, Le. 2s orbital (for 90° bond angle) correlates with the nonbonding pure p-orbital (Le. 2px orbital) of the linear geometry (bond angle 180°). This is the most noticeable change in effecting the change ofbond anglefrom 180° to 90°. On increasing the

he

m

yl

ib

ra

bond angle, energy of this orbital increases sharply until it is converted into one o~.the degenerate porbital at 180°. The change of energy of the MOs with the change of bond angle (180° to 90°) in XH2 is shown in Walsh diagram (Fig. 9.12.5.1). It has been already discussed that the species (e.g. Bell 2' BH;, etc.) havingfour valence electrons will adopt the linear geometry. On the other hand, the species having valence electrons five to eight (e.g. BH2,CH2, OH2, etc.) will adopt the angular geometry. Thus the number of valence electrons determines the shapes of the molecules and consequently the energy of the MOs. This aspect will be illustrated for XY2 species in the next section. It will be seen that the XYz species having valence electrons 16 are linear, but the species having valence electrons 17 to 20 are angular, and the species with 21 or 22 valence electrons again adopt the linear structure.

lc

Note: Predict the geometries of the species, CH2' CH; , BH2' BH; , BHi. ' NHi. .

ea

9.12.6 Molecular Orbital Energy Diagram of s.ome Representative Triatomic Hydrides In Terms of Hybrldlsatlon of the Central Atom

e/

th

The molecules XH2 (both linear and bent) have been already described in general in terms of nonlocalised 3-centred MOs. Here, we shall discuss two specific molecules BeH2 (linear) and H20 (angular) in terms of hybridisation of the central atom. By using the hybridised orbitals of the central atom,

t.m

formation oflocalised (I.e. 2-centred) a-MOl will be considered. These localised MOs on combination can generate the delocalised MOs. Moreover, the VBT concept of hybridisation is not absolutely necessary. It just simplifies the treatment.

Note: The MO energy diagram involving the localised 2-centred MOsfaiis to explain the ionisation energies indicated by UVPES i.e. photoelectron spectrum. (I) BeH2

It is a linear molecule and the central atom (Be) undergoes sp-hybridisation by using the 2s and 2p=-orbitals (assuming the z-axis as the molecular axis). The two terminal H-atoms (say HA and HB) can combine with the spz-orbitals of Be as follows to produce the local'ised a-MOs. CJ Be - HB

= 'III,~ = cl'llsPz(I)(B~):r c2'11Is(H B)

(bonding)

cr"ae- HII = '" 2, loe = c\ '" SPz (\)( Be) - C2", b( HII ) (antibonding) CJ Be-H If = '113, loe = C1'V sPz(2) (Be) + C2 'V1s(H A) (bonding) cr~e- H A

='" 4, loe =C\ '" spz (2)( Be) -

C2'" b( H A)

(antibonding)

Introduction to 'V2py(x) t----------..;..--~

Ch~mical

Bonding and Theories of Covalence 633

.... \112p(x)

x +z~---A----

i.e. 2px' 2py

(b 1 )

HA

.., I

Hs

ry

+x (Bent molecule, xz-plane)

\112. (linear) (1o u)

ib

\111, (linear) (1 Og)

~----------~ 18~

X-

I I I

'¥

+x

(Linear molecule)

yl

9~

--+ H -

ra

HA X Hs +z ~- - .. --,--~- -

H (bond angle)

ea

lc

he

m

(a)

_r---

\112, bent.(b2 )

th

.. (Nonbonding) ~

(Nonbonding)

e/

)

t.m

'V2s, (28 1)

r - 2px(x)

2s(x) \

)

(b) (ct. Fig. 9.12.4.1)

Fig. 9.12.5.1. (a) The Walsh diagram (i.e. qualitative variation of the energies of the MOs with the change of bond angle) of XH2 • (b) Composition of the MOs at two extreme geometries Le. H-X-H =90° and 180° (i.e. correlation diagram).

634

Fundamental Concepts of Inorganic Chemistry

The two sp-hybrid orbitals of Be are denoted as sPz( 1) and sPz(2). These are given by : 1 0

'II"Pz

= ..J2 ['II2s(Be) ± '112Pz(Be)].

It leads to :

'VI, loe = c;

['V2s(Be)

+ 'V2Pz(Be)] + c 2 'VI s (H B )

= ~ ['112s(Be) + '112Pz(Be) ]

he

'IISPz (\)

m

yl

ib

ra

ry

Similarly 'V2. loe' \113. loe and \II4. loe can be expressed. These 2-centred localised MOs are confined between the respective nuclei. The other two p-orbitals (i.e. 2px and 2py) lying perpendicular to the molecular axis remain as degenerate nonbonding 7t-orbitals. The MO-energy diagram is shown in Fig. 9.12.6.1. The other linear molecules like BH;, HgH2 , etc. can also be treated in the same way. Now let us correlate these localised MOs (Fig. 9.12.6.1) with the delocalised MOs given in Fig. 9.12.3.2. The degenerate localised MOs can combine to generate the delocalised MOs. These are shown below. 0' Be-H A ± 0' Be-H B i.e. \III.loe ± \113.loe produces two delocalised 3 centred a-MOs (bonding) i.e. \III and \114' • • °a Be-H A ± a Be-H B i.e. " \1121oc ± \II410e produces two delocalised 3 centred a-MOs (antibonding) i.e. 'V2 and \113' Note: By taking \II SPz -orbitals of Be, as follows:

lc

It can be shown that :

and 'II sPz (2)

= ~ ['112s(Be) -

'112p;(Be) ]

= 'VI.loe + \113.loc = c 3\112s(Be) + c 4 [\IIIS(H + \IIls(H =c 3\112s(Be) + c 4+H (+) ~ as \114 = \VI.loe - \113.loc = c 3\112Pz(Be) - c4 [\IIls(H A ) - \IIls(H B )] = c 3\112Pz(Be) - c 4+ H (-) = a Pz \112 = \112.loc + \II'4.loc = Cs\ll2s(Be) - C6 [\IIh'(H A ) + \IIls(HB )] = Cs\ll2.\'(Be) - C6~H(+) = a: • \113 = \V2.loe - \114.loc =Cs\II 2pz (Be) + C6 [\IIIs(H \IIls(H =Cs\ll2Pz(Be) + C6~H(-) = a Pz A

th

ea

\III

)

B

) -

e/

A

B

)]

(cf. Figs. 9.12.1.1, 9.12.3.1 and

t.m

Here ~H(+)andcl»H(_)denotetheTASOsorH-grouporbitals 9.12.3.2 and Sec. 9.12.3).

)]

(ii) H2 0

In this angular molecule, the central atom oxygen undergoes sp3 hybridisation. Two of these sp3-hybrid orbitals [say sp3(1) and sp3(2)] can be utilised to produce the localised a-MOs ~ follows: O'O-H A =C I\Vsp3(lXO) O'O-HB

•

+ c2'Vh'(H A );O'O-H A = CI'V sp3(IXO)

•

= cI\V sp3(2XO) + c 2'VIs(HB ); O'O-H B

- c2'VIs(H A )

= cI'V sp3(2XO) -

c 2\11Is(HB )

The other two sp3-hybrid orbitals of the central atom remain as nonbonding ones. The MO energy diagram of H 20 is shown in Fig. 9.12.6.2. Here it may be mentioned that the combination a O-H A ± a O-H B can produce two bonding delocalised • • (3-centred) a-MOs 'VI and \114. Similarly, crO- HA ± a O- HB produce two antibonding delocalised (3centred) cr-MOs \115 and \113' The MO-energy diagram given in Fig. 9.12.6.2 may be compared with the diagram given in Fig. 9.12.4.2.

Introduction to Chemical Bonding and Theories of Covalence 635

. Ha 1s (Ha)

ib

ra

ry

sPz (2) (Be)

m

yl

x

he lc

,,

The MO energy diagram based only on the localised 2c-MOs cannot explain the UVPES

t.m

e/

th

ea

-"

2s ----41~~

,,

y

AOs of Be

Localised MOs of BeH 2

\

Nonlocalised MOs of BeH 2

Fig. 9.12.6.1. The energy diagram of localised MOs of BeH2 and relationship between the localised (Le. 2 centred) and nonlocal,ised (Le. 3 centres) MOs of BeH2 (cf. Fig. 9.12.3.2 for the nonlocalised 3 centred MOs). Note : Localised MO energy diagram involving 2-centred MOs fails to explain UVPES. Localised MO. : 'Vl, Ioc' 'V2, Ioc' 'V3, Ioc' 'V4, Ioc Nonlocallsed MO. : 'Vl' 'V2' 'V3' 'V4 (cf. Fig. 9.12.3.2).

636

Fundamental Concepts of Inorganic Chemistry x '113

~~'

;Jf~==~' I ~~.. I

'115

,

I

"

,

I

"

" I

-l--------+ Z

. . . ,#

I

/

I

I

I

2sp3(nb)

I

2 --t--J

, ' " \

I

I

aO-H

\

I

s

(J

\ \

Pure

\

Hybridised

,1' ,.,

I

~ '114 ',......,--

1/, ~

,

The MO energy diagram based only on the localised 2c-MOs fails to explain the experimental results of

'111

UVPES

' ,

......

.......

AOs (0)

0- Ha.

2sp3(nb)

~ T------'

• -----

MOs (Localised)

MOs (Non-localised)

lc

1

I

AI

\

II

,

yl

I

f.::FI=

m

~

,/

==F!=

ib

2sp3,'

ra

I

.... ~{

C)

(i) c W

I

I

~"''''

~t~ (AOs)

he

2p

y

ry

I

ea

Fig. 9.12.6.2. The energy diagram of localised (Le. 2 centred) MOs of H2 0 and generation of nonlocalised (Le. 3-centred) MOs from ·the localised MOs of H2 0.

Note : MO energy diagram involving the 2-centred localised MOs cannot explain UVPES.

th

Non/oca/ised MOs : \111' \114' \115 and \113 (cf. Fig. 9.12.4.2)

e/

9.12.7 Triatomic Nonhydrldes (XY2)-Linear Species (e.g. etc.) having 16 Valence Electrons

Be~,

CO2, NO;, N2 0, N3-, OCN-

t.m

In the linear molecule XY2, both X and Yare considered to be second period elements. Thus both X and Yatoms can provide their 2s and 2p orbitals for MO formation. Let us consider the z-axis as the molecular axis (YA-X-YB). Yis more electronegative thanX. The 2s orbitals of Y-atoms are of much low energy (cf. Fig. 9.10.1.2). For the sake ofsimplicity, the 2s orbitals ofthe terminal atoms (YA and YB ) may be kept as nonbonding ones. Thus for the a-type interaction, the central atom (X) can provide 2s and 2p= orbitals while the terminal atoms (YA and YB) can provide their 2pz-orbitals. These four orbitals 2s(X), 2pz(X), 2Pz(YA) and 2Pz(YB) can combine to generate 4 (J-MOs (3-centred). To understand the formation ofthese delocalised (3-centred) a-MOs, let us first consider the Y-group orbitals (i.e. T ASOs) produced through the combination of 2Pz(YA) and 2Pz(YB). a-type TASOs

='V 2p (Y ) - 'V 2p (~ ) (a-type symmetrical) A B • (cf. Fig. 9.12.1.1b) { cj)y(z+) = 'V2pz(Y ) + 'V2P (Y ) (a-type nonsymmetrlcal) A z B cj)y(Z-)

Z

Z

Formation"ofa-MOs: The group orbital, ~Y(z-) being symmetrical can combine with the symmetrical

Introduction to Chemical Bonding and Theories of Covalence 637 2s orbital of the central atom (X) to generate two (J-MOs (one bonding and one antibonding). These are shown below: =

= c l \J!2s(X) + c2~(z-)

(J2s

cl'V2s(X) + c2 {\J!2Pz(YA) C3\J!2.f(X) - c 4+r(z-)

Similarly, the nonsymmetrical group orbital, orbital of X.

•

\J!3 = (J2Pz =

~ C4 {'V2Pz(YA) -

cl»Y(z+)

can combine with the nonsymmetrical 2pz

+ c 6 +r(z+)

c S \J!2Pz(X)

'V2Pz(YB )}

ra

C3'V2s(X)

ib

= CJ;.t =

\112

'V2Pz(YB )}

ry

'VI

= CJ 2Pz =

c 7 \J!2pz(X) -

c~+r(=+)

m

\J!4

yl

cS'V2pz(X) + c6 {'V2Pz(YA) + 'V2Pz(YB )}

- Cs {'V2PZ(YA)

+ 'V 2pz(YB ) }

he

c7'V2pz(X) These are shown in Fig. 9.12.7.1.

Formation of'It-MOs : Now let us consider the formation of 7t-MOs. For the formation of 7t-MOs,

lc

the available orbitals are:

ea

2px(X), 2py (X), 2px(YA )' 2px(YB )' 2py (YA ), 2py (YB )

t.m

e/

th

These Px' Py orbitals are perpendicular to the molecular axis. These will combine to produce 67tMas. Let us first consider the Px orbitals of the central atom and terminal atoms (YA and YB). The Px orbitals of the thermal atoms YA and YB will combine (7t-interaction) to produce the following 1t-group orbitals (Le. TASOs).

'It-type TASOs

+r(x+)

= \J!2p (Y ) + \J!2p x

A

(Y );

x

A

{ ~(x-) = \J!2px(Y

) -

(7t-type, nonsymmetrical)

B

•

\J!2px(YB );

(cf. Fig. 9.12.1.1c)

(7t-type, symmetrIcal)

The nonsymmetrical group orbital cl»Y{x+) can combine with the nonsymmetrical 2px orbital of the central atom (X) to produce one bonding and one antibonding MO.. But the symmetrical group orbital

+Y(X-> remains as a nonbonding one because the central atom cannotprovide any symmetrical orbital for 'It-bonding. Thus, the 7t-MOs are : \lis

= 7tx '=

C9 \J!2px(X)

c9'V2P.(X) + clO {'V2P.(yA) + 'V2P.(yB )}

•

'116 = 1t x =

c ll 'V2px(X) - c 12 +r(x+)

clI 'V2P.(X) 7t nb(X)

+ c10+r(x+)

==

+r(x--)

-

cI2 {'V2P.(yA) + 'V2P.(yB )}

= \J!2px(YA )

-

'V2px(YB

>' (weakly antibonding)

Fundamental Concepts of Inorganic Chemistry

he

m

yl

ib

ra

ry

638

Node

I

.

r

MOs

e/

x

Node

TASOs

th

AO of central atom X

ea

lc

I

z

t.m

Ij1nb(x)

= c!lY(x-)

i.8.

8- ---~

y

Fig. 9.12.7.1. Formation of different MOs in the linear molecule XY2 by considering the participation of group orbitals of terminal (YA and Yal. Generation of 1ty (= 'V7) and 1t;(= 'Ve) can be shown accordingly. Note : It is evident that for the nonbonding MO, a vertical nodal plane passes through the nucleus of the central atom and its atomic orbital Le. pxfX) does not participate in the MO formation.

Note: Formation of these three 7t~MOs can also be understood through the following simple LCAO scheme (Fig. 9.12.7.2).

•

or,

7t x : 'V2PZ(YA) - 'V2px(X) 7tnb(x) : 'V2pz(YA)

+ 'V2px("YB )

+ 'V2px(X)

- 'V.2px(YB )

7t nb(x) : 'V 2pz (YA ) - 'V 2px (X) - 'V 2px(YB )

Mixing coefficients are not shown for the sake of simplicity

639

2Px(YA)

2p~YB)

2py(x)

n-MOs

ra

ry

Introduction to Chemical Bonding and Theories of Covalence

ib

Fig. 9.12.7.2: The delocalised three-centre x-molecular orbitals of the linear YA-X- Y8 molecule.

'V7 =

m

yl

In the same way, three 7t-MOs can be generated from the 2py orbitals of the central atom and terminal atoms. These are as follows: 7ty = c 13 'V2py (X) + c 14 +r(y+)

he

c13'V2py (X) + cl4 {'V2Py (YA ) + 'V2Py (YB )} - c 16 +r(y+)

lc

'Vs = 7t~ = c1S 'V2py(X)

1t nb (y)

ea

c1S 'V2py (X) - cl6 {'V2Py (YA ) + 'V2Py (YB )}

== +r(y-) =

'V2py (YA )

-

Y2py (YB )

t.m

e/

th

Generation of the different MOs is shown in Figs. 9.12.7.1 and 9.12.7.2. The MO-energy diagram of the linear molecule XY2 is shown in Fig. 9.12.7.3. The MOs (both bonding and antibonding) are described according to the name ofthe major contributing atomic orbital of th-e central atom (X). The non-bonding MOs of x-symmetry are basically enriched with the character of p-orbitals of the terminal atoms and they introduce an antibonding interaction between the terminal atoms. (i) MO picture of CO2 : Now let us consider the electron distribution in CO 2 (with valence electrons 16) in terms of the MO energy diagram given in Fig. 9.12.7.3. 2

[2S(OA)] [2s(OB)]

222

as a P:

4

4.

[7t x ' 7t y ] [7t nb(x,y)] (cf. FIg. 9.12.7.3)

Thus CO2 houses 8 bonding electrons (4 a-bonding + 4 7t-bonding) and each C-O link is considered to be double bonded (I cr + 17t bonds): It is evident that up to valence electrons 16, there is no requirement to utilise the higher energy MOs which are strongly antibonding. In fact, the triatomic species having valence electrons 1.6 or less than 16 possesses the linear geometry and their MO-energy diagram is similar to that given in Fig. 9.12.7.3. For example, the transientmolecules NCO, NCN, CCN and C3 having the valence electrons 15, 14, 13 and 12 respectively -are linear as expected.

640

Fundamental Concepts of Inorganic Chemistry

x

z

lc

he

m

yl

ib

ra

ry

y

Note: 0- and n-TASOs are not of the same energy. For the sake of simplicity, this energy difference is not shown here.

- - - - -- -

ADs (X)

ea

MOs

CO2 : [2s( 0 A)]2[2s( 0 B)]2

C1: C1~z [1t x,y]4[7tnb(x ,y~4 ...

lASOs 2s AOs (VA' VB)

th

Fig. 9.12.7.3. MO energy diagram~ of the linear molecule XY2"

e/

(ii) MO picture of BeXz : Now let us try to explain the structure and bonding in BeXz (valence electron = 16 for X = F, Cl). The distribution of electrons in terms of the MO energy diagram given in Fig. 9.12.7.3 is : 2

222

4

4·

t.m

[2s(X A )] [2s(X B )] a,v a pz [7t x ,7t y ] [7t nb (x,y)] (cf.Flg.9.12.7.3)

Thus it is expected that as in CO2, in each Be-X linkage there should be a sigma-bond and a pi-bond (i.e. bond order = 2). But in reality, in BeX2 like BeF2, BeCI2, the 7t-bonding interaction is very weak. The 7t-bond in c-o linkage is quite efficient as the energy difference between the 2p-orbitals of C and 0 is quite small, but in Be-X linkage, the energy difference between the 2p-orbitals of Be and 2p orbital of F or 3p orbitals of CI is quite large (cf. Fig. 9.10.1.1) to make 7t-bonding interaction insignificant. Consequently, the group orbitals (i.e. T ASOs) +X(x ±, y ±>' X = F, CI, etc. remain non bonding. Thus though the diagram in Fig. 9.12.7.3 is generalised for the linear XY2 molecules, but for some molecules there may be some specific interactions (as in BeF2 and BeCI2) (see Fig. 9.12.7.4). In fact, the 7t-bonding orbitals in BeF2 or BeCl2 are majorly concentrated towards the X-nuclei. Thus though the generalised MO-diagram (Fig. 9.12.7.3) indicates one 1t-bond in each Be-X linkage, but in practice, it is insignificant.

Introduction to Chemical Bonding and Theories of Covalence 641

x

z

~

Q)

y

c

(z+), }

a-type

v TASOs (2X)

ea

lc

he

m

yl

ib

ra

'x

ry

W

AOs (Be)

iL

1L

---------------

th

MOs

iL

1L

Two 3s AOs (2X)

e/

Fig. 9.12.7.4. MO-energy diagram of BeX2 (X = F, GI)

t.m

'VI

= as = CI 'V2S(88)

+ c2+X (z-) == CI 'V2SCB8) + c2 {'V p,(XA )

'V P,(Xs )}

-

= a: = c; 'V 25(88) - C;+X(Z-l = C; 'V2s(Be) -:-.C; {'V Pz(X 'V4 = a p, = C3'V2Pz(88) - C4+X (Z+) =C3'V2P,(88) - C4 {'VPZ(X

'V2

'V3

A

'VPz(Xs )}

) -

A

)

+ 'Vp,(Xs )}

= a~z =C;'V2Pz(88) + C~+X(Z+) =C;'V 2Pz (88) + C~ {'V Pz(X + 'V Pz(Xjl)} A)

(cf.· Fig. 9.12.7.1 for these combinations)

BF] vs. CO; XYz vs. BeFz' BeC/z : The bonding pattern in BF is different from its isoelectronic

species eo. The generalised MO diagram is not applicable (cf. Fig. 9.11.2.3c vs. Fig. 9.11.2.4) for BF. The energy difference keeps the 1t-bonding orbitals as nO}lbonding ones. Because of the same reason, the generalized MO-energy diagram cannot be utilised for BeC/2 and BeF2• (iii) MO picture of XYz species having 16 valence electrons :Other species having 16 valence electrons are N 20, N], es2 , oes, OCN-, NO;, etc. where the Pn-orbitals do not differ in energy

642

Fundamental Concepts of Inorganic Chemistry

ib

ra

ry

significantly and these can be explained in terms ofthe MO energy diagram given in Fig. 9.12.7.3 and all these species are linear as expected. (iv) MO picture of the XY2 species having more than 16 valence electrons: If more than 16 valence electrons are to be accommodated in the MO energy diagram of the linear molecule XY2' then some electrons must be placed in the strongly antibonding orbitals. This is an unfavourable condition. In such cases, the molecule undergoes bending so that the excess electrons can be housed in relatively stabler MOs having bonding character. A similar situation arises fo~ XH2 where the species is linear for the valence electrons up to 4, but the species uQdergoes bending if the number of valence electrons exceeds 4. This aspect has been already discussed in the earlier sections. The MO-energy diagram for the bent species like N02 (17 valence electrons, bond angle 135°), N0i. (18 valence electrons, bond angle 115°), 03 (18 valence electrons, bond angle 117°) has been discussed separately in the next section (Sec. 9.12.9).

yl

9.12.8 Molecular Orbital Energy Diagram for some Representative Linear Molecules like CO2 and BeX2 - Use of Hybridlsed Orbitals by the Central Atom

lc

he

m

By using the MO energy diagrams given in Figs. 9.12.7.3 and 9.12.7.4 which consider the 3-centred delocalised MOs, the structure and bonding in the linear molecules like CO2 and BeX2 can be explained. It has been already pointed out that in BeX2, the pi-bonding is quite inefficient and the appropriate MO-energy diagram for BeX2 is given in Fig. 9.12.7.4. In this section, the MO energy diagrams will be developed by using the hybrid orbitals of the central atom. The MO picture based on the localised 2c-MOs cannot explain UVPES. .

ea

A. CO2

t.m

e/

th

Now let us illustrate the structure and bonding of CO2 by considering the jp hybridisation ofthe central atom C. If the molecular axis is the z-axis, then two sPz hybrid orbitals of C can combine with the 2pz orbitals of the terminal oxygen atoms to produce 4 localised 2c-a-MOs as followed :

(cf. Fig. 9.12.8.1a)

CJ C - OB

='III.loc = cI'V'fPz(lXC) -

c 2 'V2Pz(OB);

(bonding)

•

= '112, loe =CI'V sPz(lXC) + C2'V2Pz(OB); (antibonding) 0c-oA = '113, loe =C3 'V sPz(2XC) + C4 'V2PZ(OA); (bonding) C1C - OB

•

CJ C - OA

= 'II 4, loe =

C3 'V'\'Pz(2XC) - C4 'V2PZ(OA);

(antibonding)

The two sp= hybrid orbitals of carbon are denoted as sPz{l) and sPz(2). These are: \V sPz(l) ~h:, bonding orbitals

: \V2s(C)

+ \V2pz(C); \V sPz(2)

: 'V2s(C) - \V2pz(C)

CJ c - o and CJ c - o are degenerate, and similarly the antibonding orbitals A B and CJ C - OB are also degenerate. These are the 2-centred localised MOs (cf. Fig. 9.12.8.la). The combination, CJ C - OA ± CJ C - OB can. produce. two delocalised (3-centred) a-bonding MOs (Le. \VI and '114); similarly, the combination, CJ C - OA ± CJ C - OB can generate two delocalised antibonding aMOs (Le. '112 and '113)' The delocalised CJ-bonding MOs have been discussed in Sec. 9.12.7 (cf. Figs. 9. 12.7. 1 and 3). • •

CJ C - OA

Introduction to Chemical Bonding and Theories of Covalence 643

....

ry

O'C_ OA

ra

2spz (1 )(C)

yl

ib

(a)

x

* * (O'C-OA'O'C-O S

Combination

m

\

I~~------------('

"

"

---/t\-----y

'V5

,

I

----""",'{ I

2px

n*

~

I"

~,.,

-r "

-~2-py--""\~ .,

I'

,

Z

ry

,,"

~

~ I

~'

1~'1

I~

"

I~ ~

I

I I I

,

~

cr* 2

II II , ' ,

MOs

-------------

-----------

(s - S)

.....

" " ..... ,,> ., ,

(S + S)

}2s 2AOs (YA , Ya)

TASOs

Fig. 9.12.9.5. MO energy diagram of the bent molecule XY2.

the 19th and 20th electrons will have to occupy the 1t*-MO (cf. Fig. 9.12.9.6). In the case ofOF2, there will be 3 pairs of bonding electrons 6 pairs of nonbonding electrons and one anti-bonding pair of electrons. The net effect leads to 4 bonding electrons pver the three nuclei. It leads to bond order unity in each D-F linkage. The o-F bond energy (189 kJ mo.-I) is quite small. In VBT, this small bond energy is explained in terms of repulsion among the nonbonding electrons. In terms of MOT, this bond weakening phenomenon arises from the antibonding electron in 7t *.

Fundamental Concepts of Inorganic Chemistry

t.m

e/

th

ea

lc

he

m

yl

ib

ra

ry

652

AO-s (X)

0'b(1)

(nb)

=============== S(YA ), s(YB ) MO-s (XY2 )

Two 2s

AO-s (YA , VB)

Fig. 9.12.9.6. Simplified representation of MO-energy diagram of the bent molecule XY2 like N02 obtained from Fig. 9.12.9.6 without showing the TASOs. O'b(l)' O'nb' 0'1} *

LCAO of 2s( X), 2pz( X), 2py ( X), 2pz(YA ) and 2pz (YB )

0'b(2)' 0'2

7tb • 7too • 7t*} LeAO of 2pAX). 2pAYA ) and 2px eYe) • MO picture of the XY2 species having 22 valence electrons: Now let us consider the species like /3' KrF1 , XeF1 (which will be discussed separately in Sec. 9.12.11) having 22 valence electrons. If such species are supposed to have the bent structure, the last two electrons will be bound to occupy the antibonding orbitals (0;) and this is highly unfavourable. This is why, such species do not adopt

Introduction to Chemical Bonding and Theories of Covalence 0*

, ,

1 I

\ \

1t(x)

I I , I'

I

,z

\

I

I

,

, \

,

I

,

\ \ ,\

\)

1\\

\\ \ \\

\

~~ \

\\

sp2 \ \ .... \ ... (s + Py + Pz) \ \ \

C1 nb , 1tnb(x)

,,\ \

(nb)

\

\ \

\

~~

\

'.

Itb(x) , . /

\

,

lc ea th

e/

I

,

==== 2p(YA ) and 2p (Ya)

I

I

\

\

- Hybrid AOs (X)

UVPES.

I

\

L---y--J

~

~I

~~,

,/

he

" \

\\

:"

,py(YA , VB)

\

\\

' ..........'~~,\~ 1...------------.. .

\" \

ADs (X)

'"

c:-,',

YB

The MO-energy diagram based only on the localised 2c-MOs cannot explain the results of

\'

\ \

2p(X)

1 1 1

YA

ry

\\ ,

--_...I,,'

L---y--J

1 \

\ \

II I

\

~\

;

"

,

ra

,

\

yl

,

1- - - - - _. 1 Y I

\

\

II "

x (X)

--7T\_x

\

m

"

\

I

\

ib

---- P

1

\ \

II

======~, , 2p (X)

653

,

I

I

--- -------- ---or

MOs (YA , VB)

t.m

Fig. 9.12.9.7. MO-energy diagram of the bent molecule XY2 (e.g. CI2 0, 03' N02-, OF2 etc.) after considering the hybridisation of the central atom (X). Note: For 03' the terminal atoms and the central atom are the same. Consequently, the diagram will be slightly modified.

the bent structure. If, such species adopt the linear structure (cf. MO energy diagram for linear XY2' Fig. 9.12.7.3), then 3 pairs of electrons are also to be placed in antibonding orbitals (two'1t· and one cr·), but probably this effect is not so destabilising compared to the case of bent geometry. In the linear structure, the 22-valence electrons are distributed as : 4 bonding pairs + 4 nonbonding pairs + 3 antibonding pairs (cf. Fig. 9.12.7.3). This leads to only 2 bonding electrons over the 3 nuclei, Le. bond order = ~. In XeF2, this is reflected in bond strength (= 130 kJ mol-I) and bond length (= 200 pm). It may be compared with IF (having 2c-2e bond) : bond length = 191 pm, bond energy = 278 kJ mol-I. Thus the MO energy diagram/or 1), XeF1 , KrF1 is similar to that o/COz. The difference is that, 1t-bonding is possible for CO2, but it is not possiblefor the species with 22 valence electrons, because the p-orbitals (perpendicular

654

Fundamental Concepts of Inorganic Chemistry

to the molecular axis) to form the 1t-MOs are filled in. In other words, both the 1t-BMOs and 1t-ABMOs are completely filled in to cancel their effect mutually.

9.12.10 Molecular Orbital Energy Diagram for the Trigonal Planar Molecular Species XY3 (e.g. SF3 , 503 , NO~ , COi-)

ra

ry

Without any mathematical treatment, we shall try to understand the fonnation of a-MOs and then the 7t-MOs. The a-MOs lie in the molecular plane (say xy-plane) while the 7t-MOs will lie above and below the molecular plane. The tenninal atoms YA , YB, and Yc are more electronegative than the central atom X. Thus it is quite reasonable to assume that the 2s orbitals of the tenninal atoms are of low energy· and they remain as

nonbonding ones.

m

yl

ib

Let us first consider theformation of a-MOs. If the molecule is supposed to lie in the xy-plane, then the central atom (X) can provide 2s, 2px and 2py orbitals for a-bonding. For the a-bonding interaction, each terminal atom can provide one p-orbital (say px-orbital, assuming the x-axis of each terminal atom is directed to the central atom). These six atomic orbitals, 2s(X), 2px(X)' 2py(X), 2px(YA )' 2px(YB )' 2px(Yc ) will combine to generate six 4c-a-MOs (3 bonding MOs + 3 antibonding MOs). The combinations are : Os pair: Combination of 2s orbital of the central atom (X) and 2px-orbitals of the terminal atoms (YA , YB and Yc) (cf. Fig. 9.12.10.1 a). .. Py - Py pair: Combination of 2py orbital of the central atom and 2px-orbitals of the tenninal atoms (cf. Fig. 9.12.10.1 b). (J P -(J~x pair: Combination of2px orbital of the central atom and 2p orbitals ofthe terminal atoms x x (cf.9.12.10.1b). If the molecule remains in xy-plane and the px-orbitals of the terminal atoms are directed towards the central atom, then the Py-orbitals ofthe terminal atoms remain directed perpendicular to the respective X-Y bonds in the molecular plane (i.e. xy plane). Definitely, these 2py orbitals of the terminal

he

0:

th

ea

lc

° °

t.m

e/

Y-atoms remain as nonbonding ones. Now let us consider the formation of x-molecular orbitals in XY3 • Each of these four atoms is associated with a 2p=-orbital (perpendicular to the molecular xy-plane). These four pz-orbitals will undergo LCAO to generate 41t-MOs (2 nonbonding + 1 bonding + 1 antibonding). The bonding 1t-MO (of lowest energy) is of the following fonn (mixing coefficients not shown) : 'V 7t h : 'V 2 Pz (X) + 'V 2 Pz (YA ) + \V 2 Pz (YB ) + \V 2 Pz (Yc ) The antibonding 1t-orbital (7t.) can be obtained as follows (mixing coefficients not shown) : \V 7t.

:

'V2pz(X}- 'V2pz(YA }

-

\V2Pz (YB }

-

'V2p z (Yc }

Generation of these bonding and antibonding 1t-MOs is shown in Fig. 9.12.10.1 c. The other two 1tMOs are nonbonding (i.e. 1tnb ). The MO energy diagram (considering the 4-centred delocalised MOs i.e. 4c-MOs) of XY3 is shown in Fig. 9.12.10.2 which accounts for all 16 atomic orbitals of XY). For the trigonal planar species like BFJ, NO; , coi-, S03' etc. with 24 valence electrofJs, the electron distribution pattern can be given as follows: [2s(YA )2s(YB)2s(Yc )]

62222 CJ s CJ px CJ py 1t pz

[2p y (YA ) 2Py (YB) 2py (Yc )]

6

[1t nb (pz)]

4

•

(cf. FIg. 9.12.10.2)

Introduction to Chemical Bonding and Theories of Covalence

655

y

y

I X /

y

\

y

-----~x

yl

ib

ra

ry

(Coordinate axes for the central atom X)

m

(a)

he

.

yi

x

th

ea

lc

--- -----. y

px

t.m

e/

CJ

(b) Y

Y" y

X- y

/ x

7t z

n*z

(c)

Fig. 9.12.10.1. Formation of different MOs in the planar molecule XY~ (a) Formation of as - a: pair; (b) Formation of a p• - a~. and a py - a~y pairs; for a'·orbitals, the sign of the lobes of the p-orbital of the central atom is to be reversed; (c) Formation of

7t pz -

7t~z pair.

Fundamental Concepts of Inorganic Chemistry cr~ (X, y)

,

I

I

I

\

n*(pz}

I

\

\

'''' ,,"

,

~

,

\

\ \

\ \ \

,

I

,~"

~==3r = f...... 2p (Y)

\

t

.

'

I

nnb(pz}

I

"

I

\\

2s ( X ) '

"

~

'" '~)=== ~

I

,

\

'\ \,

,\ \ \

,,

\,

, ,

I

crs

I

I

2p orbitals of YAt Ya and Ye

BF 3: [2s(FAt Fat F c )6 [0'.J2 [O'px. pt [1t pz ]2

[2p~FAt FBt Fe )]6 [1tnb(pz~4

I I

ea

"

ADs

e/

(X)

th

====-------"""""=-- - - 2s Y )

~

2p (Ye )

I

I

\

lc

\

cr p(x, y)

2p (Ya)

/1,' I I

he

\

1tb(pz}

I

I I I I ,I " I I I I I I I I

1

m

\

II~ 2p (YA)

III

1,'

\\ ' \

\

,\\ ~

"

\

"\

~-----+x

\'

2py (YAt Yat Ye )

\\ , \

_ _ _.-.I'

\' \ \

",

~

I

) ~,\

I

'\'",\ ,

"..... I..... .....

~I

I

'

I

.....

,

,

\

ry

,,", I" I

x \

ra

I

\

ib

I

I

I

I

y

\

_-----s::o....-~\

I

I

\

cr*

II

yl

656

(YA , YB , ~e

MDs (XY3 )

---~-

~s (YA ), 2s.za) and 2s (~)

ADs (YA , Y& Yc )

t.m

Fig. 9.12.10.2. MO-energy diagram of the planar molecule XY3.

It is evident that to acconimodate these 24 valence electrons, the highest antibonding orbitals are not required. Out ofthese 24 valence electrons, there are 4 bonding pairs (i.e. 30' + In) and 8 nonbonding

pairs. It indicates that in each X-Y linkage, the bond order is

l~ ( =1(J+~1t). All these MOs are delocalised

(4-centred) over the whole molecule. Sometimes, the said XY3 planar species can be treated by considering the Sp2 hybridisation (s + Px + Py' assuming the molecular plane to be xy-plane) of the central atom. These three sp2 hybrid orbitals can be considered to combine with the 2px-orbitals of the tenninal atoms (assuming the x-axis of each tenninal atom to be directed towards-the central atom). These will produce 6 localised (J-MOs (2-centred i.e. 2c-MOs). 2 2 2 sp (1)(X)±px(YA)'sp (2X X )±px(YB )'sp (3XX)±px(Yc ) These combinations will produce three localised a-Mas which are degenerate, and three degenerate localised cr·-MOs.

Introduction to Chemical Bonding and Theories of Covalence aX-YA' a X - YB • a X - Yc

•

657

(bonding, 2-centred localised MOs, degenerate)

•

•

a x -YA ' a x -YB ' a x -Yc (antibonding, 2-centred localised MOs, degenerate) Combination among the three localised a-MOs can lead to three delocalised 4c-a-MOs. Similarly, ~combination among the localised 0'. -MOs can also generate three delocalised 4c-0'·-MOs.

I

\

\

he

2p z (X)

"

\

\ \

\

1\ "

I , I ,

,\

, sp2 (X)

th

\

"

\ \

\\\\

",

ea

\

\ \ \

\

lc

I! ~~

\

" \

e/

,,

Ttnb (Pz)

t.m

\\

\\ \\

,

\\

,

'

,\ '''2p \

'------~x

\ \ \ \ \\

~

" "

28 (X)

\

\

\

----I

\\

,

I' "

" (s + Px + Py )"

,

\

I

====~,, = 2p (X)

\

Tt(pz)

y

\ \ \

m

I I • I I I , I' I'

"\

\

I

UVPES.

,

a*

yl

,"

The MO-energy diagram based only on the localised 2c-MOs cannot explain the results of

ib

ra

ry

The 2py orbital on each terminal atom remains perpendicular to the respective X-Y bond in the molecular plane (Le. xy plane). Such three 2py orbitals remain as nonbonding_ ones. The p~-orbitals of the four atoms are perpendicular to the molecular plane and these can combine to produce 47t-MOs. (1 bonding + 1 antibonding + 2 nonbonding) as discussed earlier. Thus 7t-MOs are considered delocalised over the whole molecule. The MO-diagram considering the formation of the localised 2 centred a-MOs (3 bonding + 3 antibonding) by using the sp2-hybrid orbitals of the central atom (X) and Px orbital of the t~rminal atoms is shown in Fig. 9.12.10.3. Both the diagrams Le. Figs. 9.12.10.2 and 9.12.10.3 lead to the same conclusion.

"

\\~

-----~,=======

Ii "-

y (YA' YB' YC)

I~'

'

-

II I

\ ' \ ' \\ ',\,..

\

,

I

,

I II

..." I

Ttb(Pz}

\,

I

I

\

,,

'

"'

=======----------" '- ) 2s (YA , VB' Yc

AOs (X)

~------_.----J

2p (YA ), 2p (YB ) and 2p (Yc )

~---------'J

'---v---J

'---v---J

~(YA)' 2s (YB ) and 2s (~) .......,.

Hybrid AOs (X)

MOs (XY3 )

AOs (YA , YB , Yc )

Fig. 9.12.10.3. MO-energy diagram of'the planar species XY3 like BF3,S03, coj-, NOi,etc. after consideration of the hybridisation of the AOs of the central atom (X).

Fundamental Concepts of Inorganic Chemistry

/

/

/

//

//

~

~--_.

np"

Lfi

, \

__

n~b

,"

. as

,

"

'\ \

,', ,

~

\

"

\,

"

"

.-------

\

......

ry

~

,, ,

X

yl AOs·of

'

\

/'------

/

//," , '

31s

'

/

,"

Ox,Oy"

------,

9.12.11 Molecular Orbital Energy Diagram of Some Representative Polyatomlc Species

,

..... ~ /

"===== /

ra

MO energy level diagram for planar XHJ (e.g. BHJ ) : In XH3 , the terminal H-atoms provide only Is-orbital while the central atom (X) provides both ns and np orbitals for bonding. Thus the problem is simplified and there is the no 7t-bonding. The 2P:x, 2py and 2s orbitals of X (central atom) and Is orbitals of three terminal atoms combine to produce 6 a-MOs Le. • ax' cry, cr• • as' as, x and cry. The a-MOs can be constructed as done for XY3 (cf. Fig. 9.12.10.1). The MO-energy level diagram is shown in Fig. 9.12.10.4.

ib

658

MOs of XH~

, ,,

AOs of 3H

t.m

e/

th

ea

lc

he

m

Fig. 9.12.10.4. Molecular orbital energy diagram B 2H 6 : Each B is sp3 hybridised and each Bfor the planar XH3 species like BH3 . centre is linked with two terminal hydrogens (Note: • Compare this Fig. with Fig. 9.12.10.3 for (Ht ) through 2c-2e bonds. There are two planar XY31ike BF3 and Fig. 9.12.11.6 for pyramidal XH3 Hke NH3 .) bridging hydrogens (Hb ). Each bridging • The MO diagram of a planar XH3 species (e.g. hydrogen makes a bridging bond (3c-2e). The BH3 ) can be constructed by considering three delocalised MOs for the bridging bonds and the TASOs constituted by 1s orbitals of three H-atoms complete MO-diagram are shown in Sec. 9.21. (cf. Fig. 9.12.11.7). The complete MO-diagram shows that the aMOs (4 bonding + 4 antibonding) are localised (Le. 2 centred to explain the 4 B-Ht linkages). The three centred MOs (i.e. delocalised) are considered to explain the two B-Hb-B bridging interactions in terms of 3c-2e bonding. ·Such three centred MOs are: 2 bonding + 2 nonbonding + 2 anti bonding. Fig. 9.21.3 describes the complete MO energy diagram of B2H6 • I; and lei; : They possess 22 valence electrons. It has been already mentioned that their linear structures can be explained by considering the general MO energy diagram (Fig. 9.12.7.3) of linear XY2 molecule (e.g. CO2). In VBT, to explain their structures, participation of d-orbitals is required. It is believed that the central atom is sp 3d hybridised and the lone pairs are placed in the three equatorial places to give the linear structure (see VSEPR). However, formation of I; (and the related species like lei; with 22 valence electrons) can be explained by considering the 3c-4e bonds without the involvement of s- and d-orbital. In these species, all the 7t-MOs are filled in (cf. Fig. 9.12.7.3) and 7t-bonding has no effect. Only a-bonding stabilises the species. In the case of I; ,combination of5px orbitals oftwo terminal I-atoms and the central I-atom (assuming the x-axis as the molecular axis) generates three 3-centred delocalised a-Mas (cr, a nb and a·). The four electrons are placed in the lowest two MOs (Le. 0' and a nb ). Thus, it leads to 3c-4e bonding system (Figs. 9.12.11.1-2). The net bond order is 1/2 in each I-I linkage (i __e... 2 bonding electrons are delocalised over the three centres). This 1/2 bond order is consistent with the experimental observation (stretching

Introduction to Chemical Bonding and Theories of Covalence

659

m

yl

ib

ra

ry

frequency, v = 215 cm- I for 12 ; v = 113 and 135 cm- 1 for I;). The same thing appears for ICIi. (3c4e bond, and bond order = 1/2) and the stretching frequencies are: v = 384 cm- I for ICI; v = 267 cm1 and 222 cm- I for leli.. The participation of d-orbital in VBT is objected. But in the MOT, the 3c-4e bonding system does not require the participation ofany d-orbital. This is an added advantage of MOT compared to VBT. XeFz' XeF4 , XeF6 : Without any involvement of d-orbital of Xe, by considering the general MO-energy diagram of linear molecule XY2' the bonding and structu~e of XeF2 and KrF2 (with 22 valency electrons) can be explained (cf. Sec. 9.12.9 and Fig. 9.12.7.3). This aspect has been already discussed. By considering the p-orbitals only through the 3c-4e bonding system (as in /; discussed above), the bonding in x~non compounds has been discussed in Sec. 9.23. Fonnation of the linear segment F-Xe-F can be explained by considering the formation of TASOs or LCAO of the involved AOs. Note: Bending of the 22 valence electron species (XY2) will place the cr~b pair into a cr·-MO (cf. Sec. 9.12.9. and Fig. 9.1~.9.6). Thus the angular structure of such species is not favoured. No requirement of s- and d-orbitals of the central atom

I

lc

I I

th

\,

\\

,,

\

e/

\

\

\

t.m

\

,

I I

1b

I

z

[IA - Is - lc]-

\

~

la

Ie

(ab)

\

\ ... \ \

---*1~~--°nb

(

5Px (I)

\

ea

I

I I

(Central I-atom)

,

\

\

I

~x 80-08+08

he

0*

I I

\

- - - ~)=== 5p x (I) ,' I' (Terminal I-atoms) I

00-08-00 00+08+00 «(Jnb)

(0'*)

Fig. 9.12.11.1. Simple MO-energy diagram of I; without the involvement of s- and d-orbitals. Formation of 3-centred MOs is shown at the right hand side.

Note : • Formation of the 3-centred MOs can also be understood through the formation of group orbitals (Le. TASOs) from the 5px orbitals of the terminal atoms. These are illustrated in Figs. 9.12.1.1 , 9.12.7.1 and 9.12.7.3. In terms of TASO, G nb is nothing but a group orbital developed by the '-' combination of the 5px orbitals of the terminal I-atoms. It is evident that though 0b and 0* are the 3-centred MOs, 0nb is basically a 2-centred MO. • °nb: 'Vnpx(lA) ± \Vnp.(/c ) - 'Vnpx(ls) == \Vnpx(lA) - \VnPx(/c )· Thus CJ is weakly bonding between the terminal nb atoms. If the molecule is bent, O~b electron pair will be placed into a o*-MO and it will destabilise the system (cf. Sec. 9.12.9 and Fig. 9.12.9.6). • Complete MO picture of I; (22 valence electrons) can be explained in terms of the general MO energy diagram given in Fig. 9.12.7.3 without the participation of d-orbitals.

660

Fundamental Concepts of Inorganic Chemistry cr· I

I

\

I

, \

I

,

I

,

I

, ,

I

\

>e> -5-p~x(-I)----(~ ... ... c \ W \, "

z [CI-I-CI]-

,

"

Q)

'_....;~~L__

iP

,

",

Cfnb

'\

No requirement of s- and d-orbitals of the central atom

\

",\

\,

,...-==:r=-

"

" 3px (2CI) \

ra

I

ry

I

I \

I I \

ib

\ I

yl

"U " crb

he

m

Fig. 9.12.11.2. Simple MO-energy diagram of leI; without the involvement of s- and d-orbitals. Note: leI; with 22 valence electrons can be explained in terms of the general MO energy diagram of Fig. 9.12.7.3. A

cr~.

'"'=====

B CfC. -H ' -H . ' { ~_~,ac_~'

",

*

'*

*

...-.::::

0p ' CJpy' CJpz x

''''----

cr*

--

e> Q)

\

\

I

~ \ Hybrid AOs (C)

,.SIi!I~§§

11s(HA ,HB , \ \ \ \

" I

I I

\

\

"

\

\

I

\\

I

I'

1:

II

aC-HA,aC-Ha

He' HD) ~

AOs (4 H-atoms)

I

, ~;

Combination

11 ,=al'=S::!!::::

a px ' apy' a pz

"

' , '"

aC-Hc,crC-Ho ~

MOs (CH 4 )

Delocalised. MOs

Fig. 9.12.11.3. Simple MO-energy diagram for CH 4 after consideration of sp3-hybridisation of the central atom. Note : This simple diagram involving the localised 2-centred MOs cannot explain the UVPES. However, combination of the localised MOs can generate the 5-centred delocalised MOs as given in Fig. 9.12.11.4b.

Introduction to Chemical Bonding and Theories of Covalence 661

CH4 : The central atom carbon is sp3 hybridised. These four sp3 hybrid orbitals can combine with four Is orbitals of terminal H-atoms to generate 4 localised (Le. 2 centred) a-MOs and 4 localised a·-MOs. sp3(IXC) ± Is(HA ), sp3(2)(C) ± Is(HB ), sp3(3XC) ± Is(Hc ); sp3(4XC) ± Is(HD )

m

yl

ib

ra

ry

The eight valence electrons are placed in the 4 localised 2-centred bonding a-MOs which are degenerate..The MO-energy diagram is shown in Fig. 9.12.11.3. This simplified MO energy diagram fails to explain the UV-photoelectron spectrum. Without considering the hybridisation of the central atom (Le. carbon), formation of the 5-centre~ MOs in CH4 can also be understood. T~e available orbitals are: . 2s, 2px' 2py and 2pz orbitals of C; four Is orbitals of 4 hydrogens (say H A , H B, H c' H D) which are described as terminal atoms. The hydrogen atoms can produce/our group orbitals (i.e. TASOs) which combine with the orbitals of the central atom. Without going into the details of symmetry interaction, formations of the 5-centred bonding MOs, a,\, and a Px are illustrated in Fig. 9.12.11.4a. Other 5c-MOs can be constructed in the same way. The 5-centred MOs are:

0" s and 0":; three 0" p (i .e. 0" P. ' 0" Py and 0" pz ) and three 0": (i .e. 0":. ' O":y and O":z )

he

a:

e/

th

ea

lc

The 5-centred 0' s and MOs are obtained in the interaction between the symmetric 2s orbital of C and the symmetric H-group orbital. Other three nonsymmetric H-group orbitals interact with the Px' • Py and Pz orbitals to give the six 5-centred MOs (Le. 5c-MOs) designated as a p and a p' • The a p MOs are triply degenerate. Similarly a pMOs are also triply degenerate. The MO-energy diagram is shown in Fig. 9.12.1'1.4b. The UV-photoelectron spectrum (UVPES) supports the MO diagram shown in Fig. 9.12.11.4b. The photoelectron spectrum shows two absorption bonds at about 23 eV and 14 eV which correspond to ionisation from O's and a p (which is HOMO) respectively. It may be noted that the oversimplified MO-diagram given in Fig. 9.12.11.3 fails to explain the photoelectron spectrum.

t.m

Note: Formation of the 5-centred MOs in CH4 can be realised by considering the formation of/our H-group orbitals. 0" H (s)

= \111.f( H A) + \II Is( H B) + \IIls( He)

aH(x)

= \J!ls(H A ) + \Vb(H B ) -

aH(y)

=

a H (z)

= \V Is( H A)

\J!ls(H

A

) -

-

\J!b(H

B

)

\V b( H B)

\V1s(H c ) - \Vls(H D )

+ \J!1s(Hc ) -

+ \111.f(HD) } Symmetric (matching with 2s-orbital of C)

- \V1s(H

D

)

Non -symmetric (matching with 2p orbitals of C)

\V Is( He) + \J! b( H D)

These H-group orbitals combine with the orbitals of C-centre as follows (mixing coefficients not shown) in terms of appropriate symmetry. \J!2.\'(C)

• • ± +H(.\,) => as and O's; \J!2px(C) ± +H(x) => a Pll and a Px • •

\J!2py(C)

± +H(y) => 0' Py and a Py; \V,2Pz(C) ± +H(z) => a Pz and 0' Pz

Fundamental Concepts of Inorganic Chemistry

2s(C) +1 s(HA ) + 1s(HB ) + 1s(HC ) + 1s(HD )

(Jpx: 2px(C) +1 s(HA ) + 1s(HB )

1s(HC ) - 1s(HD )

-

ib

Os :

ra

ry

662

I

I

'*

I

he "

"

"

a;

I I

th

,. \

I

(

e/ t.m

/

,, '

,

I

.1.1

I

\

'

a px ' apy' 0pz

I

c

W

"

"

e>

,1!iE===="

y.l', .1\ : ' .I , ; I' . I ) " " Four 1s orbitals

I ,

Q)

',+H(x, y, z) "

,I

"

(Nonsymmetric)

, "

I

I

r

\

",

I

,,

,

',',

ea

2p

,

\

-..

lc

"

,

\

/

, ,.

::::::ii,;l:l====::::::((

'*

0px' apy' a pz

/ /

'*

m

yl

Fig. 9.12.11.4~. Formation of 5-centred MOs inCH4 • as and a p are shown. Other Mas can also be constructed in the similar way because all the p-orbitals are equiv~lent here.

"

~

I

,

---ilII1.... ~ _ _-oJ' 2s \,

,,

+-

of H-atoms - (Symmetric) Ionisation energy: 14 eV +H(s)

I I \

I

I \

I

~ Ionisation energy: 23 eV

"

\~I Os

'--v--J

"--v-----.J

AOs (e)

LGOs

Fig. 9.12.11.4b. MO energy diagram of CH 4 involving the 5-centred delocalised Mas. Note: Fig. 9.12.11.4b is also applicable for other t~~.rahedral species like, BHi. For B~-, CC/4 etc. similar diagram may be constructed.

Introduction to Chemical Bonding and Theories of Covalence

663

NHJ : Here N is sp3 hybridised and it uses three of its four sp3 hybrid orbitals to combine with the Is orbitals of three H-atoms (HA, HB and He); and one sp3 hybrid orbital remains as a nonbonding one. The MO energy diagram in Fig. 9.12.11.5 shows three degenerate localised (2-centred) bonding aMOs and three localised a--MOs. The 8 valence electrons are placed in three a-bonding Mas and in one nonbonding MO which is basically a sp3 hybrid orbital localised on nitrogen. This simple model

e/

th

ea

lc

he

m

yl

ib

ra

ry

fails to explain the UV-photoelectron spectrum which shows three different first ionisation energies for three different energy levels: 11.0 eV, 16 eV and 23 eVe This can be explained by considering the formation of multi-centred delocalised MOs as in the case of CH4 . This modified MO energy diagram is given at 9.12.11.6.

'"--...r--J

AOs(N)

Hybrid AOs (N)

t.m

'"--...r--J

Fig. 9.12.11.5". Simple MO energy diagram of NH3 after consideration of sp'3 hybridisation of the central atom N. Combination of these localised 2-centred MOs can generate the 4-centred MOs that can explain the UVPES.

H-group orbitals (Le. LGOs) can be generated (cf. Figs. 9.12.11.6a, 7). By considering the Fig. 9.12.11.7, the LGOs are as follows (mixing coefficients not shown) : at

-symmetry {+'(H)

e-symmetry

{

: \IIls(HB )

+ \IIls(H,4) + \IIls(Hc )

~(H) : 'Vls(HB ) ± 'V1s(H... ) cI>.J(H} : - 'V1s(HB )

'Vls(Hc )

+ 'V H1s(H... )

== 'Vls(HB )

-

'Vls(H c )

- 'Vls(Hc )

It is evident that in terms of Fig. 9.12.11.7, ~l(H) can combine with the 2s and 2pz orbitals of N; ~2(H) can combine with the Px orbital of N, and ~3(H) can combine with the 2py orbital of N.

664 Fundamental Concepts of Inorganic Chemistry 2s(N)

m

yl

ib

ra

ry

15

"'-

lc

he

~ , _ ~ J

a;z

~

ea

/1

/ / I,.

//",,,,,'

th

/~'"

2p

/,.

~(JX' ay ',' Ionisation energy = 11.0 eV

e/

\, "

I

J, I' 1\' I / 1\ ' I / 1\ ~, I \ 1

" :-._ +3(H) a n b ' ' , ,' ... +2(H)' _---. sz ,'/ ' ,~,)5fI!E= / / ') ~ 1S /

t,

//,

: X(' , I I, I I, \

_ ....~.. s_-e:~

,

* '

1

:=:iE!'==tl."';;;;i r - - I.,. - /.,

t.m

*

//,' ax'

,

Cfy

\

"

+

~1(H), ~

LGOs

AOs of 3H

'

" "

\

,

"

\

'

Ionisation energy = 16.0 eV ,

~ Ionisation energy

" \

\\~'.,

=24.0 eV

°sz ~

~

AOs of N

MOs of NH 3

Fig. 9.12.11.6. (a) Overlapping interaction in pyramidal NH3 . (b) Approximate MO energy diagram of pyramidal NH3 molecule.. Note: This diagram is applicable for other pyramidal species like H3 ()+. For NF3 , similar diagram may be obtained by using its 2p-orbitals.

Introduction to Chemical Bonding and Theories of Covalence 665 y

HA I

\. \

I I

@] ~

\

..-......----.x

I

Hc./~/J-_~~HB Projection ofN

ry

@.HAS ]

ra

e-symmetry

ib

Fig. 9.12.11.7. Formation of ligand group orbitals (LGOs) from the 15 orbitals of H-atoms of NH3 .

m

yl

Combination of the LGOs Le. ~1(H)' ~2(H) and ~3(H) with the orbitals of nitrogen to generate the MOs has been illustrated i~ Fig. 9.12.11.6a. The MOs are generated as follows (with the appropriate signs of c's) :

he

c\ \jI2s(N) + c2 \j12pz(N) + C3~(H)' crsz ' cr:: and cr:z Le. 3cr - MOs (for appropriate signs of c 1• c2 and c3) c4 \jI2P.(N) ± C5~(H) : cr x and cr: Le. 2cr - MOs; c6 \jI2py (N) ± C7~(H) : cry and cr: Le. 2cr - MOs

~F(X+) = 'V2p

t.m

{

e/

th

ea

lc

HF2: This hydrogen bonded linear (F-H-F)- symmetrical species can be explained by considering the 3c-4e bonding system. Very low energy 2s orbitals of fluorine do not participate in bonding. If x-axis is considered to be the bond axis, then for a-type interaction, the available orbitals are: 2P:x orbitals of terminal atom (say FA and FB) and Is orbital of the central atom hydrogen. They combine to produce 3a-MOs (one bonding + one nonbonding + one antibonding). Generation of these a-MOs can be explained by considering the formation of group orbitals (Le. TASOs) from the 2P:x orbitals of the terminal atoms (FA and F B ). TASOs

cl>F(x-)

(F ) x

+ 'V2p (~ ) (nonsymmetrical and antibonding a-interaction) x

A

= 'V2Px(F

A

) -

B

'V2Px(F

B

• )

The symmetrical group orbital atom (H) to generate two a-MOs.

• • •

, (cf. Fig. 9.12.1.1)

(symmetrical and bonding a-Interaction) ~F(:X-)

.a b

can combine with the symmetrical Is orbital of the central

= 'VI = c I'V1s(H

A

)

a· = 'V2 = c 3'Vl s (H A )

+ c2~F(x-) -

c4~F(x-)

The nonsymmetrical group orbital remains as a nonbonding one because it cannot interact with the symmetrical Is orbital of H. a nb = 'V3 = ~F(x+) : 'V2Px(FA ) + 'V2Px(FB ) == 'V2Px(FA ) ± 'VIs(H A ) + 'V2Px(FB ) Thus, a"b leads to an antibonding interaction between the terminal atoms because a"b is weakly

antibonding.

666

Fundamental Concepts of Inorganic Chemistry

The other p-orbitals (Le. 2py and 2pz' perpendicular to the molecular axis) remain as nonbonding ones. The 2s orbitals of very low energy of the F-atoms also remain as nonbonding. The MO-energy diagram involving 0b' 0nb and 0* is shown in Fig. 9.12.11.7a. Fig. 9.12.11.7a, b describe the 3c-4e bonding portion only while Fig. 9.12.11.7c describes the complete diagram.

t.. .__~)X

ra

ry

cr*

MOs (HF"2) .

lc

AO(H)

he

m

yl

ib

1s (H)

....

TASOs

AOs

(FA' Fa)

(FA' Fa)

a*

t.m

e/

th

ea

Fig. 9.12.11.7a. MO energy diagram of HFi (after consideration of group orbitals, Le. TASOs). Note : The nonbonding MO bears an antibonding Interaction between the terminal atoms. The electron pair in this orbital experiences -minimum antibonding interaction when it adopts the linear structure. It is the characteristic feature of 30-4e bonding.

i.8. 'V3: '112PX = 1) and the wave function 'V2 lies in the xz-plane (Le. eI> = 180°, cos 180° = -1), we get: \jI2

=.!. +.!. sin 9 +J2 cos9 2

2

For the maximum value of'V2' we have the condition: d'V2/dO = 0 Le.

d\jl2

dO

=.!. cos9 - J2 sin 9 =0; or, tan 9 =0.354 =tan 19°28' 2

The value of 0 gives the angle between the z-axis and 'V2 axis. Thus the angle between 'VI and 'V2 becomes 90° + 19°28' = 109°28'. Note: For the equivalent s-p hybrid orbitals, the bond angle (a.) can be calculated by using t.he relationship: coso. =

s

P -100 where sand p denote the respective % of the sand p

p orbitals respectively (see Table 9.13.6.1). s -100

Bonding potentiality 0/ the hybrid orbitals: The bonding potentiality of an orbital is measured by its angular strength. For 'VI' it has the maximum value along the x-axis, for 0 = 90°, = 0°,

680

Fundamental Concepts of Inorganic Chemistry i.e.

\j!l(max) =

\j!3(max)

~+~= 2 2

2

= y4(max)

(cf. it is 1 for s orbital, and 1.732 for p-orbital). Similarly for,

I = 2. These are measured with reference to the unit 41t

\j!2(max)

=

= I.

• Relative values ofangular strength and bonding potentialities ofdifferent s-p hybrid orbitals: The relative values of angular strength are: Angular strength: sp3 (2.0) > sp2 (1.99) > sp (1.93)

ry

Bonding potentiality should also foll~w the same sequence but the reverse is true. This aspect has been explained in Sec. 9.13.3.

ra

Bonding potentiality: sp > sp2 > sp3 (cf. Sec. 10.1.7)

ib

9.13.2 Stereochemistry of the Hybrid Orbitals

m

yl

The stereochemistry attained by the different types of hybridisation is given in Table 9.13.2.1. It is evident that in the combination of s + p + d, depending on the nature ofthe d-orbitals participated, the geometry differs significantly. For example, in sp 3d if the d-orbital is d z2 then the combination Table 9.13.2.1. Stereochemistry of the hybrid orbitals

he

Combining atomic orbitals

Spatial orientation

sp sp2

2.00 2.24 2.69

(n - 1)dxy' (n - 1)dxz' (n - 1)dyz' ns

cPs

Linear Trigonal planar Tetrahedral (Td ) Tetrahedral (Td)

(n -l)dx2 - y2, ns, npx' npy ns, npx, npy' ndx2 -y 2

dsp 2

Square pla~ar

sp 2d

Square planar

(n - l)di , ns, npx' npy' npz

dsp 3

ns, npx' npy' npz, ndi

sp 3d

(n - l)dx2 - y2, ns, npx' npy' npz ns,nPx,npy,nPZ,ndx2 -y 2

dsp 3

Trigonal bipyramidal (TBP) Trigonal bipyramidal (TBP) Square pyramidal

sp 3d

Square pyramidal

(n - l)dx2 - y2, (n -l)dz2' ns, npx' npy' npz ns, npx' npy' npz' ndx2 - y2, ndz2

cPsp3

Octahedral (Oh)

sp3J2

Octahedral (Oh)

(n - l)dx 2 - y2, (n -l)dxy' (n - l)dz2, ns, npx' npy' npz

tPsp3

Pentagonal bipyramidal

t.m

e/

th

ea

ns, npx' npy; (bonds in xy plane) nS,nPx,npy,npz

lc

ns, npx; (bond axis along x axis)

Angular strength·

Common notation

sp3

(n - l)di , (n -l)dxy' (n -l)dyz' (n -l)dzx, ns, npx' npy' npz

cfsp3

Dodecahedral

(n -1)dx2 _ y2' (n -l)dxy ' (n -l)dyz' (n -l)dzx ' ns, npx' npy' npz

cfsp3

Square antiprismatic

(n - 2)1xyz' (n -l)dxy' (n -l)dyz' (n -l)dzx, ns, npx' npy' npz

Id3sp3

Cubic

* Assuming unity for ns and 1.73, 2.24 for np and nd respectively.

1.93 1.99

2.92

Introduction to Chemical Bonding and Theories of Covalence 681

yl

ib

ra

ry

produces a trigonal bipyramid (Le. s + p x + Py; p = + d _2); but participation of the d x 2 _ y2 orbital in the combination produces a square pyramid (Le. s + Px + Py + d x2 _ 2; Pz). It is very difficult to predict whether the trigonal bipyramid geometry or square pyramid ge~metry will be preferred. It is expected that they are not energetically greatly different. But, most of the ADs type molecules have been found to adopt the trigonal bipyramid geometry. In some cases, different combinations lead to the same geometry. For example, the tetrahedral geometry of MnOi and CrO;- probably constitutes both the combinations sp3 and d 3s. It is believable as the energies of 3d, 4s and 4p do not differ significantly. In the combination of d + s + p, if the d-orbital comes from the penultimate (Le. n - 1) shell, the corresponding combination is very often referred to as inner orbital type while participation of the d-orbitals from the outermost valency shell (i. e. n-shell) produces an outer orbital type hybridiSaion. Such possibilities mainly exist for the transition metal complexes. The occurrence of the inner orbital type combination is energetically more favourable but there are many other factors which may govern the process. This aspect will be discussed in Sec. 9.13.5.

m

9.13.3 Bonding Potentiality of the Hybrid Orbitals: Angular Strength and Overlap Integral

t.m

e/

th

ea

lc

he

The hybridisation produces the orbitals which concentrate the electron clouds more in some particular directions compared to the starting atomic orbitals. This is why, the hybrid orbitals in some particular directions give a better overlapping with the suitable orbitals of other atoms. This is why, the bonding potentiality of the hybrid orbitals is always greater compared to that of the pure atomic orbitals. For example, the s-orbitals are spherical having no directional property and the p-orbitals concentrate their electron clouds in two equal lobes projected in opposite directions but their hybridisation products mainly concentrate the electron clouds in some specified directions to produce a better overlap integral in the specified directions. The bonding potentiality is very often measured by the parameter, angular strength as proposed by Pauling. It is measured by the' maximum value of the angular wave function I I I I (cf. Sec. 9.13.1). It gives a measure of the 1.-1-.1 concentration of the electron cloud of an orbital I I I I along the bond axis. Thus the parameter is directly Fig. 9.13.3.1. Angular strength of the hybrid proportional to the length ofthe axis ofthe orbital orbitals. measured from the nucleus to the periphery. It is measured with respect to the s-orbitals. . The spherically symmetrical s-orbitals being nondirectional produce the same overlap integrals in all directions. The p-orbitals having the directional properties (x, y and z axis) can overlap better than the s-orbitals in their projected directions. The comparison among the orbitals must be confined for a particular principal quantum number. If the radius of ns orbital is r, the length of the axis of the np orbital measured from the nucleus to the periphery is given by 1.73 r (see Fig. 9.13.3.1). Thus, according to the definition of angular strength measuring the bonding potentiality, if it is unity for the ns orbital then it is 1.73 for the np orbital. Similarly, in the same scale, it will be 2.24 for the nd orbital. The scale of angular strength can be understood from the angular wave function of the orbitals

682

Fundamental Concepts of Inorganic Chemistry

(cf. Table 4.1.6.2, Sec. 9.13.1). For the Px' Py and Pz orbitals, the maximum value of the angular wave function becomes

~%7t

s-orbital, it is given by

along the x, y and z-axes respectively, while for the spherically symmetrical

Y.ji;;,

in all directions. Taking

Y~

as unity, the maximum value of the

Ji.

angular wave function of the p-orbital is This aspect has been discussed in Sec. 9.13.1 to derive the hybrid wave functions. The angular strengths ofthe hybrid orbitals are shown in Table 9.13.2.1. It is evident that compared

ra

ry

to the pure orbitals, the hybrid orbitals are more promising for the bonding purpose. Actually, the angular strength gives the measure of overlap integral to be earned in bond formation. Thus the bonding efficiency and angUlar strength of the s-p hybrid orbitals should follow the same

ib

order.

Angular strength: sp3(2.0) > sp2 (1.99) > sp (1.93). But, in reality, the bonding efficiency follows

yl

the opposite' trend.

he

size of the lobes varies in the following order.

m

Bonding efficiency: sp > sp2 > sp3.(cf. Fig. 10.1.7.1). This can be explained by considering the relative size of the major lobes (i.e. bonding lobes) of the s-p hybrid orbitals~ The shapes of the lobes are the same for sp, sp2 and sp3 hybrid orbitals but the

lc

Minor lobe size: sp3 > sp2 > sp; Major lobe size: sp > sp2 > sp3. It explains the highest overlap integral for the sp-orbital. The following C-Hbond strength supports this prediction.

th

ea

C-H bond energy: C2H 2 > C2H 4 > C2H 6• The overlap integral also depends on the internuclear separation, e.g. the internuclear separation between two sp-C centres is less than between two sp3--e centres (cf. Sec. 10.1.7).

e/

In/act, overlap integral does not solely depend on the angular strength ofthe s-p hybrid orbitals (cf. Sec. 10.1.7.1).

t.m

9.13.4 Energetics of Hybridlsatlon

The energy difference between the unhybrid system and hybrid system gives the measure of hybridisation energy required. As the hybrid orbitals are produced from a linear combination of the atomic orbitals, the energies ofthe hybrid orbitals are given by the weighted average of the energies of the combining atomic orbitals. To clarify the fa~ let us- oonsider- the possible combination of sand p-orbitals of carbon. The energies of the 2s- and 2p-orbitals of carbon are - 1878 and - 1028 kJ mol-I respectively. .~. · OJ."C : '2s22p 2 exeitation and 2( ~12( ~12 1 1 sp hy brlulSatlon h b °do ° ) SPJ SPJ Py 2Pz y rt lsatlon

E2(sp)

= ~ (E2s + E2px ) = -1453 kJ mol-I

Therefore, the energy required for hybridisation is given by :

Ehy = (2E2(sp) + 2E2p ) - (2E2s + 2E2p) = (- 4962) - (- 5812)

= 850 kJ mol- 1

Introduction to Chemical Bonding and Theories of Covalence 683

ee

e

2

Sp 2 hybrldlsatlon ofC: 2s 2p

hybridisation

3

2

I

I

=.!..4 (Eh + 3E2p ) =-1240.5 kJ mol-I"

=(4E2(si»

J

) 2(sp 3)4

- (2Eh + 2E2".) =(- 4962) - (5812) =850 kJ mol-I

ib

Hence, Ehy

I

(2E2s + 2E2p ) =(- 4961.99) - (5812) =850 kJ mol-

e OJa'-C : 2 S 22p 2 excitation and sp3 hy brledelSat,on

2¥)

2

=} (E2s + 2E2p ) =- 1311.33 kJ mol-I

=(3E2(sp2) + E2p ) -

E (

2 I

) 2(sp ) 2(sp ) 2(sp ) 2pz

ry

Hence, Ehy

...

hybndlsatlon

ra

E2(Sp)2

2 excitation and

3)

3~

. . ) 3' ( sp3)2 ( sp 3)1 '(sp3)1 ( sp 3)1·I.e. 3(sp3)5

and excitation

m

E(

spJ hybridisation

1 1 _I =-(E3s +3E3p )=-[-1806+3(-981)]=-1187.25kJmol

4

4

he

3 S23P3

yl

Sp3 hybridisation of P : Now let us take the case of phosphorus, where E3s = - 1806 kJ mol-I, E3p = - 98i kJ mol-I.

Hence, Ehy =(5E3(si» - (2~s + 3E3p ) =(- 5936.25) - (- 6555) l'l:l619 kJ mol-

,

J

e/

th

ea

lc

In the above calculations, E hy gives the measure of requirement of energy to carry out the hybridisation. Hence to make the process energeticallyfeasible, the energy released in theformation of bonds by utilising the hybrid orbitals must be greater than Ehy• It is evident that if the energy difference between the combining orbitals is very high then Ehy becomes very much large, and the energy released in the bond formations may not be able to compensate the Ehy. Variation ofstructuralfeatures of the hydrides of Group VA (Le. 15) and Group VIA (Le. 16) : To clarify the effect of Ehy (Le. cost of hybridisation energy), let us take the hydrides ofthe Group VA

t.m

(15) and VIA (16) elements (see Table 9.13.4.1) in which the bond angle gradually decreases to . . ., 90° with the increase of atomic number. It means that the heavier members prefer pure p-orbitals

keeping the lone pair in the s-orbital. It occurs so, because the energy released in forming the bonds cannot compensate the hybridistion cost (i.e. Ehy ) for spJ hybridisation. Actually the larger size of the heavier members reduces the bond energies and as a result, it becomes difficult to meet the hybridisation cost. The observation can also be qualitatively rationalised in terms ofstericfactorse The lighter members of the groups being small find the besOt way to minimise repulsion amOl)g the bonding and nonbonding electron pairs by projecting the orbitals to the vertices of a regular tetrahedron. But, for the heavier members, because of their larger sizes, the repulsion among the bonding and nonbonding pairs is itself less pronounced. Hence to avoid the repulsion it does not require essentially to undergo o

hybridisation. In VSEPR theory, it is believed that the bonding pair goes away from the central atom with the decrease of electronegativity of the central atom and as a result, the repulsion among the bonding and nonbonding pairs gets automatically reduced for the heavier members (acting as the central atoms

684

Fundamental Concepts of Inorganic Chemistry

which are relatively less electronegative). In other way, the lone pair (which is more diffusedfor the less electronegative central atom) on the central atom causes more distortion to contract bond angles. Table 9.13.4.1. Bond parameters in the hydrides of Group VA(15) and VIA(16).

Bond angle·

Molecule

Bond energy (kJ mOll)

Atomic tadius ofthe central atom (in pm)

Electronegativity of the central atom

389

74

318

110

91 °48'

247

121

91 °18'

255

141

1.9

104°28'

459

74

3.5

}{2 S

92°15'

363

104

H2Se H2 Te

91°

276

89°30'

238

3.0 2.1

2.0

2.5

117

2.4

137

2.1

m

yl

ib

ra

ry

106°45' 94°

NH) PH) AsH) SbH) H 20

* Controlling factors: X-Hbond energy and cost of hybridisation; steric factor; electronegativity of the central

he

atom (i.e. VSEPR).

lc

9.13.5 Participation of d-Orbitals in Hybridisatlon In Nonmetals (cf. Sec. 10.7 for 7t-bondlng): A Debate

ea

A. Barriers for d-orbita' participation in bonding

t.m

e/

th

We have already mentioned that the participating atomic orbitals in the process of hybridisation should not differ significantly in their radial distribution functions. In other words, they should not differ largely in energy since the energy of an orbital is proportional to its mean radial distance. If the orbitals differ largely in energy, the cost of hybridisation energy becomes large. For the lighter nonmetals, (e.g., Si, P, S, etc.), the nd (i.e. 3d) orbitals differ both in radial size and energy significantly from the ns- and np-orbitals. For example, in phosphorus, the most probable radial distances for the 3s-, 3p- and 3d-orbitals are 47 pm, 55 pm and 240 pm respectively. Hence, though the hybridisation between 3s- and 3p-orbitals may be possible, the participation of the 3d-orbitals in the hybridisation with the 3s- and 3p-orbitals is not expected because of their (i.e. 3d-orbital's) much higher energy. But, if the energy of the 3d-orbitals can be brought down to the comparable range of the 3s- and 3p-orbitals then their participation in the hybridisation, i.e. 3s + 3p + 3d, may be permitted. There are number of ways to contract or lower down the energy of the d-orbitals. These are discussed below. Besides the large promotion energy (reflected in the cost of hybridisation) required for d-orbital participation, the highly diffuse character of the d-orbital disfavours its participation in bond formation. The d-electrons are shielded by the inner electrons and consequently, the d-electrons experience the less nuclear charge. This leads to their diffuse character which does not allow them to make a good overlap with the other combining orbitals. Thus the high promotion energy and diffuse character ofthe d-orbitals are the basic barriersfor the participation ofd-orbitals in bonding in nonmetals.

Introduction to Chemical Bonding and Theories of Covalence

685

The above mentioned barriers can be interpreted as follows. The mean value of radial distance for nd orbitals of the p-block elements lies far away from the typical covalent bond distanc~t.Thus it

needs the contraction of d-orbitals for covalent bond formation. In fact, contraction of d-orbital can reduce the required promotion energy and diffuse character and the radial distance of the orbital matches with the covalent bond distance.

B. Possible ways to allow the contraction of d-orbitals

1

yl

ib

o

E

ra

When the atom under consideraJion binds with a highly electronegative atom such as F or 0 or CI, the bonding pair will be shifted to the more electronegative atom giving rise to +.charge on the central atom (say, P, Si or S). With the increase of positive charge on the element, the orbital electrons will be pulled more towards the nucleus, but the effect is transmitted more towards the outer electrons. As a result, the energy of

ry

(i) By increasing the effective nuclear charge

he

m

the nd-orbitals falls more sharply compared to that of the ns- and np-orbitals with the increase of positive charge on the central atom under consideration (Fig. 9.13.5.1). ·For example, in spJtP configuration, on accumulation of + 0.6 charge on S, the most probable radii for 3s-, 3p- and 3d-orbitals

----. Positive charge Fig. 9.13.5.1. Qualitative represen-

tation of the decrease of energy of ns, np and nd orbitals with the increase of .positive charge on the atom (cf. Fig. 1.11.1.2).

t.m

e/

th

ea

lc

become 87 pm, 93 pm and 140 pm respectively; but in the neutral atom for the spJtP configuration, Le. 3s 13p33cP, the 3d-orbitals possess very high energy (cf. the corresponding values of neutral S are: 88 pm, 94 pm, and 160 pm). Ip SF6 , the d-orbital (for sp3cP hybridisation) is heavily contracted so that its radial distribution maximum lies at 130 pm distance while in free sulfur (3s 23p43J», it occurs for the 3d-orbitals at about 300-400 pm away from the nucleus. In free sulfur, the electronic configuration is : 3s 23p43J> Le. d-orbital is unoccupied. The s- and p-orbitals of sulfur in SF6 are contracted but not to such an extent. Thus the radius of the maximum probability for the d-orbitallies within the S-F bonding distance. In other words, in SF6, both the energy and diffuse character of the d-orbital are significantly reduced. Because of this fact, to effect the d-orbital participation (e.g. spJd, spJtP, etc,), it generally requires to bind with the highly electronegative elements (e.g. F, CI, 0). This is why, PHs does not exist but PCls, PFs exist through the formation of sp3d hybridisation. Similarly, the existence of SiFi- ~ SF6 and the nonexistence of the hydrogen analogues can be explained by considering the possibility of occurrence of sp 3cP hybridisation. Table 9.13.5.1. Effect of charge and extent of population in 3d orbital of sulfur on of 3d orbital of sulfur

Electronic configuration sl p

cP

3

slp 3d2 s2p 4JJ s2p 3d 1 slp 3d2

Charge on S

(in pm) for 3d of S

+ 0.6 inSF6

140 130

o o o

300-400 246

160

686

Fundamental Concepts of Inorganic Chemistry

(ii) By increasing the population in d-orbitals (cf. Table 9.13.5.1)

m

yl

ib

ra

ry

If the number of electrons occupying the d-orbitals can be increased, then size of the d-orbitals can also be shrinked. Thi~ fact is well illustrated for the transition elements where it has been found that after some population density in the (n - 1) d-orbital, its energy becomes lower than that of the ns orbital (cf. Fig. 1.11.1.2). In the case of S, it has been observed that if only one electron is placed in the d-orbital (i.e. 3s23p33d) i.e. d)-configuration), the most probable radial distance for the 3d-orbital is 246 pm, but on placement of two electrons in the d-orbital (Le. 3cP for 3s) 3p33cP configuration), the distance falls to 160 pm from 246 pm (cf. for 3s23p43cfJ, it is about 300~00 pm). In fact, in spltP configuration of neutral S, the average (Le. the most probable) radial distances for }s-, 3p- and 3d-o~bitals become 88 pm, 94 pm and 160 pm respectively. S (neutral) : Most probable radial distances for 3d orbital are 300-400 pm (tf>-configuration), 246 pm (d)-configuration) and 160 pm (cP-configuration) (Table 9.13.5.1). Excitation of electron from 3s and 3p levels to 3d level reduces the shielding experienced by the 3d electron because sand p electrons are more efficient than the d-electrons in shielding mutually. It reduces the energy and of the 3d-orbital. (iii) By the coupling of spins of the electrons

e/

th

ea

lc

he

A slight shrinkage in the size of the d-orbitals may be carried out through the coupling of the spins of the electrons occupying different orbitals. This can be explained by considering the origin of nonCoulombic Pauliforce (arising from the Pauli exclusion principle) which states that the electrons of the same spin repel mutually. To avoid this spin-spin repulsion, electrons of the same spin will try keep themselves apart as far as possible. This will lead to an expansion of the orbital (say d-orbital) in which they are remaining present. When the electrons undergo pairing through spin-spin coupling, the Pauli force does not operate for such paired electrons with opposite spins. The spin pairing reduces the number ofelectrons with parallel spins experiencing the Pauli force. This is why, spin-spin coupling leads to a slight shrinkage of the orbital.

c. Objection against the concept of d-orbital participation in hybridisation and an

t.m

alternative bonding model without d-orbital participation In VBT, the concept of hybridisation like sp 3d, sp 3cP, etc. is- very much useful in explaining the bonding and geometry of many compounds of main group elements (Le. p-block elements). But, theoretical calculations supported by experimental verifications indicate that in many cases, the extent of d-orbital participation is almost negligible. In fact, their contribution in bonding is very. much less significant than the s- and p-orbitals. Thus the s-p hybridisation like sp, sp2~ sp3 is quite reasonable while the hybridisations like sp 3d, sp 3cP etc. are called in question. • Introduction of ionic bonds (Pauling) : Without considering the participation of the d-orbital, Pauling had attempted to explain the structure of the species like PCls which may exist in the following resonating structures. '

"'4Eo--~~.

yl CI-PZCI CI

61

...4i------:i~~

3 more equivalent resonating structures

--------------

Introduction to Chemical Bonding and Theories of Covalence

687

yl

ib

ra

ry

Here, the central atom P uses its four valence orbitals·{s + p) to produc.e 2c-2e four P-CI bonds to attain the octet ~onfiguration while the fifth CI is ionically bound. Thus the total species is PCI;CI-. Each P-CI bond possesses 20% ionic character and 80% covalent character by considering the existence of 5 equivalent resonating structures. When the electronegativity difference between the combining atoms is significantly large, the existence of such resonating structure is only meaningful. The extra bonds formed after the attainment of octet or arganic structure (in hypervalent compounds) are described as trans-arganic bonds which are weaker than the normal or arganic bonds (e.g. average P-CI bond energy: 326 kJ mol- 1 in PCI) and 270 kJ mol- 1 in PCI s). However, the said trans-arganic bonds become stronger with the increase of electronegativity difference (e.g. difference of average PCI bond energy for the pair PCIs and PCI) is 56 kJ mol- 1 while this difference reduces to 40 kJ mol- 1 for the pair PF) and PFs). . According to Pauling concept, in SF6 there are two trans-arganic S-F bonds and the corresponding resonating structures are :

lc

he

m

• 3c-4e bonding model (i.e. aia;b): Alternatively, by considering the 3c-4e bonding model (cf. Sec. 9.12.11 and 9.23) along with the 2c-2e bonding model, the bonding in the said geometries like trigonal bipyramid, square pyramid, octahedron, etc. can be explained. It should be mentioned that in 3c-4e bonding model (cf. Figs. 9.12.11.1-2), only one bonding pair is spread over the three nuclei and consequently the bond order in each linkage is 0.5. Such bonds (with bond order = 1/2) (Le. trans-arganic bonds in the Pauling concept) are weaker than the normal2c-2e bonds (with bond order = 1). In the 3c-4e bonding model, the nonbonding pair is mainly localised (cf. Figs. 9.12.11.1-2 and Sec. 9.23) on the terminal

e/

th

ea

atoms which are more electronegative. • Correlation between the Pauling concept and 3c-4e bon~ing model: The 3c-4e bonding model leads to placement of 2e in crnb MO which is mainly localised on the terminal atoms. It leads to an accumulation ofnegative charge on the terminal atoms (Le. separation of charge to develop the ionic character, cf. Sec. 9.12.12). Thus the 3c-4e bonding model supports the idea of Pauling model which considers the partial ionic character in such compounds.

t.m

• Correlation between VBT concept and 3c-4e bonding model: It is important to note that in VBT, for expansion of the octet through the participation of d-orbitals, the central atom is required to link with the electronegative terminal atoms. On the other hand, in the light of 3c-4e, bonding model though the participation of d-orbital is not required, it demands that the terminal atoms should be sufficiently electronegative to stabilise the nonbonding pair (Sec. 9.12.12). Thus both VBT and 3c-4e MOT suggest that the terminal atoms should be highly electronegative to stabilise the hypervalence showing an expansion of octet. Note: • Hypervalence is classically defined as expansion of octet through the formation of 3c-4e bonds. See Sec. 9.1.6 for hypervalence. • In explaining hypervalences, all the theories, i.e. the Pauling concept, VBT concept and MOT concept nicely agree with the demand: the terminal atoms must be electronegative. • In 3c-4e bonding model, the terminal atoms should be electronegative while in the 3c-2e bonding model (i.e. electron deficient bonding), the terminal atoms should be electropositive (e.g. H;, CH; etc.). This aspect has been discussed in Sec. 9.12.12. This

688

Fundamental Concepts of Inorganic Chemistry

3c-2e bonding is required to explain the electron deficient hypervalent compounds. These are discussed later. • In both 3c-4e bonding model and 3c-2e bonding model, the bond order in each linkage is 0.5.

• 3c-MOs to explain the geometries : By considering the 3c-2e bonds and 3c-4e bonds, the molecular geometries like trigonal bipyramid, square pyramid, octahedron, etc. can be explained without the involvement of d-orbitals. These are illustrated in 'some representative examples.

ra

ry

The linear segment B-A-B (say, along the x-axis) can be explained by the 3c-4e bonding model (cf. Fig. 9.12.11.1-2 and Sec. 9.23) in which the orbitals, Px(B), Px(A) and Px(B) combine to produce three a-MOs (Le. a b , a nb , a·) in which the bonding (a b ) and nonbonding (a nb ) MOs are filled in.

ib

(i) PCIS' PFs (i.e. PXs of D3h symmetry) .tt'

(3s 2

DO

lc

P:

he

m

yl

Here the central atom P 3p 3) is considered to undergo sp2 hybridisation to bind with the three equatorial X-atoms (Le. X eq ) through the 2c-2e bonds. The fourth orbital (Le. one p-orbital) filled with two electrons participates in 3c-4e bonding to bind the two axial X-atoms (Le. Xax ). If the equatorial plane lies in the xy-plane then Px and Py orbitals participate in sp2-hybridisation and the p:; orbital participates in 3c-4e bonding.

3p

sp2-hybrid orbital~

DO Pz-orbital

ea

3s

Three Sp2 hybrid orbitals: To bind three X eq atoms through 2c-2e bonds as usual.

P:

t.m

e/

th

(Xax) + p; (P) + P: (X ax ): To produce the linear segmentXax-P-Xax through the 3c-4e bonding. It explains the bond angles (Xeq-P-Xeq = 120°, Xax-P-Xeq = 90°, Xax-P-Xax = 180°), bond lengths (axial P-X bond having bond order 0.5 is weaker than the equatorial P-X bond having bond order 1.0) and the trigonal bipyramid geometry without the d-orbitals.

,

Fig. 9.13.5.2. 30-4e bonding in the Iinear•. Xax-P-Xax segment along the z-axis (cf. Figs. 9.12.11.2, 9.12.11.7).

Introduction to Chemical Bonding and Theories of Covalence

689

(ii) AB4(lp) like :SF4 (C2v symmetry)

DO

Dill]]

3s

Dill]]

~

ra

s:

ry

It has got a distorted trigonal bipyramidal structure in which one equatorial position is occupied by a lone pair (Ip). If the equatorial plane is considered to lie in the xy-plane, then the sp2 hybridisation is carried out by the s, Px and Py orbitals of sulfur. Two of these sp2-hybrid orbitals bind two equatorial F-atoms through the normal 2c-2e bonds as usual and one sp 2-hybrid orbital houses a lone pair. The p;-orbital of S is used to produce the axial segment Fax-S-Fax along the z-axis through the 3c--4e bonding (expected axial bond angle Fax-S-.Fax = 180°, found 179°).

sp2-hybrid orbitals to house the lone pair and to make two 2c':'2e equatorial bonds

filled pz-orbital to make the Fax-S-Fax segment through the 3c-4e bonding

yl

ib

3p

DO

lc

he

m

If one does not consider the sp2-hybridisation, then the p~ and P~ orbitals may be considered to participate in 2c-2e bonding to bind two equatorial F-atoms, and the lone pair is housed in the pure s-orbital (Le. "stereochemically inactive lone pair). This should lead to F'eq-S-F'eq = 90° but the experimentally found bond angle is 103°. It supports the approximate sp2 hybridisation in the equatorial plane where the lone pair is stereochemically active and the deviation in bond angle (120° ~ 103°) can be roughly explained by VSEPR. The average bond order in each S-F linkage in SF4 is given by :

ea

. SF -1 (2 x. 1 + 2 X 0.5,_..,) =-3 ( cf. bond order In 6 4 eq 4

=-64 =-32) .

th

UA.

e/

(iii) ABs (Ip) like : BrFs (C4v symmetry) It has got the square pyramidal structure. If the basal plane lies in the xy-plane then the axial A-Bax bond is considered to be a normal2c-2e by using the p~ orbitals of A and Box. The two equatorial

t.m

segments Beq-A-Beq at 90° angle are produced through the 3c-4e bonding by using the fi lied orbitals of the central atom.

Br:

p; and p;

Px Py Pz

DO ~ 4s

to make the 2c-2e axial bond

In BrFs' the 4s orbital houses the lone pair (Le. stereochemically inactive), the p~ orbital makes the normal 2c-2e Br-Fax bond. The linear Feq-Br-Feq segments in the equatorial plane are produced as follows:

2p~(Feq) + 4p;(Br) + 2p~(r:q): r:q-Br-Feq (along the x-axis) through the 3c--4e bonding. 2p~(r:q) + 4p;(Br) + 2p~(r:q): Feq-Br-F'eq (along the y-axis) through the 3c--4e bonding.

690

Fundamental Concepts of Inorganic Chemistry

(iv) ABJlp)2 'ike :CIF3 (C2v symmetry) The molecule is T-shaped. The central atom Cl undergoes sp2 hybridi~ation (say in xy plane) and one sp2-hybrid orbital binds the equatorial F atoms through the normal 2c-2e bond, other two sp2-hybrid orbitals house the lone pairs in the equatorial plane. The -orbital makes the linear axial segment Fax-Cl-Fax through the 3c-4e bonding. It gives the T -shape of the molecule.

p;

DO

DO

3s

3p

Pz

ry

C/ :

[ to make the linear axial F-C/-F segment through the 3c-4e bond

ra

~

yl

ib

If one ignores the sp2-hybridisation then the singly occupied p-orbital makes one 2c-2e bond in the equatorial plane; one filled p-orbital makes the axial F-Cl-F segment (3c-4e bond); one lone pair is housed in the 3s orbital and the second lone pair is housed in a p-orbital in the equatorial plane.

(v) Octahedral molecule AB6 'ike SF6 (Oh symmetry)

he

m

In SF6 , two sp-hybrid orbitals (say, s + Px) bearing the unpaired electrons make the linear segment F-S-F "along the x-axis through the normal 2c-2e bonds. The other two filled p-orbitals (i.e. P~ and p;) of S participate to produce two mutually perpendicular segments F-S-F along the yand z axes respectively through the 3c-4e bonding.

3s

lc

DO

s:

ea

3p

The bonding interactions are:

sp~(S) + p~(F) 1

e/

and

th

sp~(S) + p~(F) } 2

1

t.m

py(F) + py(S) + py(F)

p~(F) + p;(S) + p~(F)

Two S-F linkages (2c-2e bond) to produce the linear segment F-S-F along the x-axis Linear F-S-F segment along the y-axis (3c-4e bond) Linear F-S-F segment along the z-axis (3c-4e bond)

Thus the three F-S-F segments lie mutually in perpendicular directions to give the octahedral geometry. In this model, two S-F linkages are having bond order unity while the other four S-F linkages are having bond order 0.5. Thus the average bond order in S-F linkage is (4 x 0.5 + 2 x 1.0)/ 6 = 4/6 = 0.66. Thus the S-F bond is expected to be longer than normal 2c-2e S-F bond. But, in SF6 , the S-F bond is relatively shorter than the estimated single bond length by about 20 pm. It is suggested that the partial ionic character developed in 3c-4e bonding model stabilises the S-F .bond to some

extent. (vi) ABI'p) like :XeF6 , :SbCI:-, :SnCI:- etc. If we consider the formation of three mutually perpendicular B-;-A-B segments (through the 3c-4e bonding interaction), then it would lead to the perfectly octahedral geometry. In this 3c-4e bonding interaction, the filled Px' Py and p= orbitals of the central atom participates. This aspect has been discussed

Introduction to Chemical Bonding and Theories of Covalence 691 in detail in Sec. 9.23 in connection with the structure of XeF6 • This three centre bonding model keeps the lone pair in the pure s-orbital of the central atom and thus the lone pair becomes stereochemically inactive. If the lone pair is not stereochemically active, then it will have If distorted octahedral geometry. Whether the lone ,pair will be stereochemically active or not depends on many other factors. However, the 3-centre model cannot predict this. All these aspects' have been discussed in Sec. 10.8.5. Note: Explain the bonding in the hypervalent elusive molecular species NF6-, CF62- by taking the help of 3c-4e bonding interactions.

ry

(Viii) Bonding in hyperva/ent electron deficient molecular species

he

m

yl

ib

ra

The compounds like CH;, CH~+, C(AuL);, C(AuL)~+, N(AuL);+ (L = PR3 ), etc. are sometimes described as electron deficient hypervalent compounds. This is explained by considering the 3c-2e bonds in addition to the nonnal 2c-2e bonds. The 3c-2e bonding model is very much similar to the 3c-4e bonding model. In3c-4e bonding model, the nonbonding MO is filled in by two electrons while in 2c~3e bonding model the nonbonding MO remains vacant (cf. Sec. 9.12.12). In 3c-4e bonding scheme, the tenninal atoms should be electronegative while in the 3c-2e bonding model, the terminal atoms should be electropositive. In both models, the bond order in each linkage is 1/2. It may be noted that the segment (say B-A-B) is bent for 3c-2e bonding. All these aspects have been discussed ~ earlier (cf. Sec. 9.12.12). Let us illustrate bonding in CH; (which was characterised in superacid matrices) and CH~+.

lc

Sp3

ea

C:

t.m

e/

th

C:

en;

three equatorial C-H bonds (xy-plane), 2c-2e bonds Le. C-Ha , C-Hb and C-Hc

(4 C-H bonds, 2c-2e bonds) pz

rn

Bent HaC-He axial segment, 3c-2e bond

10.....-_1

possesses a distorted structure rather than the symmetrical trigonal bipyramidal.structure.

It indicates that three equatorial C-H bonds are produced by thr~e sp2 hybrid orbitals through the nonnal 2c-2e bonding interaction while the other two axial H-atoms· are linked through the 3c-2e bonding interaction by using the pure pI orbital of..carbon. This axial HaC-He segment is angular~

c[IIJ sPz 2c-2e bonding C-He and C-Hf along the z-axis

px py 3c-2e bonding interactions, Le. bent Ha-C-Hd and Hb-f;-Hc segments in xy-plane

692

Fundamental Concepts of Inorganic Chemistry

D. Merits and demerits of the 3-centre bonding model without the d-orbital participation in the compounds of main group elements

ra

ry

It has been discussed that without using the d-orbitals, the molecular geometries like trigonal bipyramid, square pyramid and octahedron can be explained. It needs the consideration of three centre bonding model (Le. 3c-2e, 3c-4e). This concept requires the participation of s- and p-orbitals only. In some cases (e.g. octahedral system, square pyramidal system, etc.), participation of p-orbitals only is sufficient. Thus the participation of s- and d-orbitals in such cases may be ignored though there is no objection in s-p mixing. To ignore the contributio~ of sand d-orbitals, it is argued that the s-orbital is of very low energy (Le. deeply seated) and the d-orbital is of very high energy. Consequently, they remain as nonbonding ones. However, complete omission ofs- and d-orbitals in these bondings is a

t.m

e/

th

ea

lc

he

m

yl

ib

ruthless approximation. . • Geometry ofthe molecules: The observed bond angles like 90 0 and 1800 in the geometries like squat:e pyramid and octahedron can be explained by using the p-orbitals only in 3-centre bonding model. However, this 3-centre MOT cannot explain whether the lone pair will be stereochemically active or not. • Bond length parameter : The 3-centre bonding scheme without involving the d-orbitals is a qualitative model. It fails to explain the structural properties in many cases. For example in :SF4 (trigonal bipyramidal geometry having two equatorial and two axial F-atoms, and the lone pair in the equatorial plane), the linear segment F-S-F in the axial direction involves the 3c-4e bonding (Le. bond order in each S-F linkage along the axial direction = 0.5). On the other hand, the equatorial S-F bonds are normal 2c-2e bonds (Le. bond order in each linkage = 1). It is expected that the axial S-F bond should be much longer than the equatorial S-F bond. But in reality, the bond lengths differ by 10 pm only. Similarly in SF6 , the bond is expected to be longer than the single S-F 2c-2e bond. But in reality, in SF6 , S-F bond shorter than the S-F bond by about 20 pm. It is believed that the ionic character in S-F-F (3c-4e) bonds (cf. Sec. 9.12.12) shortens the bond length. In XeF2 , the Xe-F bond (200 pm) is longer than the calculated single Xe-F 2c-2e bond. It supports qualitatively the 3c-4e bonding scheme in XeF2• • Possibility of d7t-P7t bonding (cf. Sec. 10.7) : We have already discussed the possibility and objection of participation of d-orbitals in the a-bonding schemes of main group elements. But, in many cases, participation of d-orbitals in 7t-bonding has been strongly argued (Sec. 10.7). The d7t-p7t bonding is well documented in Si-X, Si=O, P=O, P = N, .S= Nbonds (cf. Sec. 10.7). In the tetrahedral species AB4 like SiF4 , 8iO:-, PO~-, the a-bonding framework is constituted by the sp3-hybrid orbitals of the central atom A, but the empty d-orbitals of A can participate in pi-bonding with the filled p-orbjtals ofthe terminal atoms B. The filled p-orbitals of B are mutually perpendicular and these are also perpendicular to the A-B linkage. Among the d-orbitals of A, the d x2 _ 2 and d.. 2 are more suitable than the other d-orbitals for the said d7t~ P7t bonding interacti~. This it-interaction using the d x 2 -y2 orbital of A is illustrated in Fig. 9.13.5.3. Among the oxyanions, the efficiency of 41t-p7t bonding increases as : CIO; > 80I- > PO~- > 8iO:This aspect has been discussed and its effect on the extent of polymerisation has been pointed out in Sec. 10.7.

693

yl

ib

ra

ry

Introduction to Chemical -Bondi~g and Theories of Covalence

he

m

Fig. 9.13.5.3. d1t-p1t interaction in a tetrahedral species AB4 through the overlap of the d 2 2 orbital of A with one p-orbital of B. (The axes of the coordinates pass through- the centres of theYfaces of the cube).

ea

lc

The structural features and Lewis basicity of the species N(CH3)3 (pyramidal), N(SiH3 )3 (planar), N(GeH3)3 (planar), P(SiH3)3 (planar) can be explained by considering the d 7C-P7C bonding. The drt-P7C bonding by using the filled 2p-orbital of N and vacant d-orbital of Si and Ge makes the Si)N and Ge)N arrangements planar. The pi-bondings are: 7t

) Si(3d) and N(2p) _7t---+) Ge(4d) .

th

N(2p)

e/

In fact, ~ecause of this 7t-bonding, the bonds are shorter than the single bonds. In the case of (CH3)3N , there is no question ofsuch d 7C-P7C bonding as C does not have any d-orbital. In P(SiH3)3'

t.m

the P(3p)~ Si(3d) bonding is inefficient because of more diffuse character of the 3p-orbita/s ofP. It makes, P(SiH)) pyramidal. However, some authors have attempted to explain these observation in terms of steric crowding not in terms of dft-Pft bonding. When steric crowding around the central atom increases, it tends to attain the planar geometry (Le. bond angle increases from 109° to 120°). Thus compared to N(SiH3 )3' in P(SiH3 ) the steric repulsion is less important because of the larger size of P. Sin:tilarly, among the C3N, Si 3N and Ge 3N units, the steric crowding around the N-centre is minimum in C 3N. This aspect has been discussed in Sec. 10.7(i). In the cases like, -S-N==S-, F 3S ==N where the S-N bonds are very short, the d 7C-P7C bonding is probably more important.

9.13.6 Equivalence and Nonequlvalence of the Hybrid Orbitals The hybrid orbitals in most of the cases are equivalent. These are: sp, sp2, sp3, sp3tfl, etc. But there are sp 3d and 3 3 sp d hybridisations. '

~_cases in which the hybrid orbitals are nonequivalent. This nonequivalence is found in

694

Fundamental Concepts of Inorganic Chemistry

t.m

e/

th

ea

lc

he

m

yl

ib

ra

ry

• sp3d (trigonal bipyramid Le. TBp) : Here, three equivalent hybrid orbitals developed from (s + Px + p y ) lie in the equatorial plane (Le. xy-plane) at 120°, and other two equivalent hybrie orbitals developed from (p z + d.,.2) project mutually at 180° in the axial directions. The axial hybrid orbitals are devoid 0/ any s-character. Thus there are two sets of hybrid orbitals, Le. a set of three equivalent equatorial hybrid orbitals and a set of two equivalent axial hybrid orbitals. This idea has been established from the experimental findings in PCls in which the three equatorial chlorines are different from the two axial chlorines. The equatorial P-CI bond (204 pm) is shorter than the axial P-CI bond (219 pm). If different atoms of different electronegativity are bound to the central atom, then Bent's rule (see Sec. 9.13.7) will determine the positions of the different atoms of different electronegativity in the trigonal bipyramidal geometry. Note: In a TBP geometry, in terms of VSEPR, each axial electron pair experiences repulsions at 90° from 3-equatorial pairs while each equatorial pair experiences two such repulsions from the axial pairs. It makes the axial bond relatively larger. • sp3d (square pyramid) : The combination can be split as : s + Px + Py + d x2 _ 2 Pz • Thus the four equivalent hybrid orbitals are lying in the basal plane (Le. xy-plane) while th: fifth one is the pure p= orbital lying perpendicular to the xy-plane. Here also the Bent's rule is applicable to determine the positions of different atoms of different electronegativity. For 5-coordination, there are two possible hybridisations leading to trigonal bipyramid (TBP) and square pyramid (SP). Generally, the nonmetals prefer the TBP-structures. Some coordination compounds are known to have SP-geometry. However, the energy difference between these two configurations for the compounds like AXs is very small. As a result, in some compounds ofAXs type, there is an equilibrium leading to TBP ~ SP. In such cases, the structural properties are the weighted average ofthe two configurations. Some examples ofthis type ofcompounds are CdCl;-, P(C 6 H s)s' Ni( CN)~- , etc. As a matter of fact, this low energy difference between TBP and SP gives the key factor in Berry pseudorotation (see Sec. 10.9.2). • sp3tP hybridisation (pentagonal bipyramid) : This combination can be split into two sets (s + Px + Py + dxy + d x2 ~ 2) and (Pz + d z2 ). The first set produces 5 equivalent orbitals in the xy-plane to produce the basal Yplane while the'second set produces two equivalent hybrid orbitals (having no s-character) perpendicular to the basal plane along the axial direction (Le. +z and -z directions). • Presence 0/ nonequivalent group and lone pairs (cf. Bent's rule) : The above examples show that these hybrid orbitals are nonequivalentfrom the standpoint oftheir genesis. But the equivalent hybrid orbitals (e.g. sp3 hybrid orbitals) may become nonequivalent, if the atoms or groups binding with the central atom differ significantly in electronegativity. Even in some cases when some of the hybrid orbitals are occupied by the lone pairs, the equivalent hybrid orbitals may lose their equivalent character in the compounds formed. To clarify the above points, let us take the following cases. In CH4 and CCl4, all the bond angles are 109°28' as expected from the sp3 hybridisation but in CH3Cl or CHC!.J the bond angles are different. In CH3Cl, H - C- H angle is 110° which is different from H - C - Cl; similarly in CHCl3 , the Cl- C- Cl is different from Cl- C- H. These variations can be nicely explained from principle of Bent's rule based on the electronegativity difference among the combining atoms or groups.

and

Introduction to .Chemical Bonding and Theories of Covalence

695

The observed bond angles, H - 0 - H (= 104° 28') and H - N- H~(= 106° 48') are different from the tetrahedral bond angle (= 109° 28') in H 20 and NH3 respectively. This is due to the different effects (see VSEPR theory) of the lone pairs and bonding pairs occupying the hybrid orbitals. This difference

in bond angles arises due to the nonuniform distribution ofthe character ofdifferent atomic orbitals in the bonding and nonbonding hybrid orbitals. For example, in the combination of sand p, with the

ra

ry

increase of p-charact~, the bond angle decreases. This is illustrated in Table 9.13.6.1. Hence in H 20, the decrease of H - 0 - H angle by ~ 5° from the tetrahedral bond angle indicates that the orbitals holding the hydrogens are containing more p-character (Le. p-character > 75% and s-character < 25%) and the nonbonding orbitals housing the lone pairs bear p-character less than 75% (Le. s-character > 25%). Table 9.13.6.1. Bond angles in (s + p) combination. % character

ib

Hybridisation

p

sp (linear)

50

sp2 (trigonal)

1 333

2

Angle

180 120

25

75

o

100

1090 28' 90

m

66-

he

sp3 (tetrahedral) p2 (Le. two pure p-orbitals)

50

yl

s

3

ea

lc

The higherpercentage ofs-character in the hybrid orbitals housing the lone pair can be explained by Bent's rule (Sec. 9.13.7). The percentage ofp- or s-character in two adjacent and equivalent hybrid orbitals can be determined from the knowledge of the corresponding bond angle (9) as :

e:= (p -

100)/p = s/(s -100)

th

cos

where p and s denote the percentage of the corresponding orbital. Thus in water,

t.m

e/

cos 104°28' = - 0.25 = (p - 100)/p; or, p = 80%; and, s = 20%. Thus the percentage of p- and s-character in the two nonbonding orbitals bearing the lone pairs are obtained as follows: I -I P = 2" (3 - 2 x 0.80) x 100 = 70%, and s = 2" (1 - 2 x 0.20) x 100 = 30% Now, the angle (cP) between the axes of the nonbonding orbitals housing the lone pairs can be obtained as follows, . cos cP = (p -IOO)/p = (70 - 100)/70 = - 0.428 = cos I i5°38'; or, cP = 115° 38' 9.13.7 Bent's Rule of Hybridlsatlon (see Sec. 10.8 for some applications of the rule)

Selection ofdifferent hybrid orbitals (bearing different extents ofdifferent pure atomic orbital characters) by different atoms or groups differing significantly in electronegativity is .governed by Bent's rule of isovalent hybridisation. The rule has been formulated as follows: (i) The more electronegative atom or group will withdraw the bonding pair more from the central atom to itself. This can be facilitated if the electronegative group or atom shares such a hybrid orbital in which the central atom has utilised less s-character. This poor s-character indicates that

696

Fundamental Concepts of Inorganic Chemistry

the central atom has got no strong influence to hold the electron cloud to itself by using this type of hybrid orbital. Because ,it is known that by using the s-orbitals, the electron cloud can be most tightly bound. This is why, the electronegative moiety always prefers the hybrid orbital ofthe

yl

ib

ra

ry

central atom in which the s-character is less. Eventually, it leads to more ionicity in the covalent bond. (ii) If the electronegativity difference between the bonding moieties (Le. central atom and other bonding atom or group where the central atom is less electronegative) is significantly large, then the corresponding bond will possess some ionic character. This will lead to the shifting of bonding electron pair to the more electronegative terminal atom or group. This electron shifting is favoured nlore if the central atom utilises a hybrid orbital having less s-character. Thus, with the increase of difference of electronegativity between the combining species, the covalency decreases. In such cases, the central atom uses the hybrid orbitals having less s-character towards the more electronegative bonding moiety, because utilisation of such hybrid orbitals favours the generation of more ionic character.

he

m

Thus the more ionic character (i.e. poorer covalency) in a bond (due to the more electronegative substituent to the central atom) leads to the utilisation of hybrid orbitals containing more p-character (i.e. less s-character) of the central atom. In other words, the more covalency in a particular bond encourages the central atom to utilise its hybrid orbitals which contain more

lc

s-character. This poor covalency may result not only due to the electronegativity difference, but it may also originate due to the poor overlap, as a result ofeither steric hindrance or mismatch of the overlapping orbitals.

t.m

e/

th

ea

Thus, the Bent's rule can be stated as follows: (a) The central atom projects the hybrid orbitals ofgreater p-character (Le. less s-character) towards the more electronegative substituent. It leads to more ionicity in the covalent bond. This is 'he classical form ofBent's rule. (b) The central atom projects the hybrid orbitals ofgreater p-character (Le. less s-character) towards the more ionic bond and the hybrid orbitals ofgreater s-character towards the more covalent bond. (c) The central atom projects the hybrid orbital of more s-character towards the bond where covalency is very strong. The strong covalency arises from the factors like small, electronegativity difference, good overlap and existence ofn-bonding. On the other hand, the hybrid orbitals enriched with p-character are projected towards the directions where covalency is weak. This is why, the Bent's rule is restated as follows: s-orbital enriched

hybrid orbital ofthe central atom is concentrated along the strong covalent bonds. Note: The overlap integral (which determines the covalent bond strength) for the s-p hybrid orbital is a complicated function of the s- or p-character (cf. Fig. 10.1.7.1). It may be mentioned that the angular strength (cf. Sec. 9.13.3) is not the sole factor to determine the overlap integral. For example, sp3 hybrid orbitals having angular strength 2.0 possess smaller overlap integral in the C-C and C-Hbonds compared to the sp hybrid orbitals of angular strength 1.93 (cf. Fig. 10.1.7.1). It is due to the fact that the size of the major lobe of the sp-hybrid orbital is higher than that of the sp3 hybrid orbital (Sec. 9.13.3).

Introduction to Chemical Bonding and Theories of Covalence 697

Illustration of the Bent's Rule

m

yl

ib

ra

ry

(i) H]CF: In a perfectly tetrahedral system (i.e. sp3 hybridisation) in each bond, C projects 75% p-character and 25% s-character. The bond angle is 109°28'. Increase ofs-character and decrease ofp-character widen the bond angle (cf. 180° for sp, 120° for sp2, 109°28' for sp3). Here, the C-F bond should have less than 25% s-character of C and consequently the C-H bonds should have more than 25% s-character of C. Thus, the less p-character (i.e. < 75%) in the C-H bonds tends to extend the H -C -H angle greater than 109°28'. Besides, the greater s-character in the C-H bonds tends to shorten the C-H bonds. This enhances the bond pair - bond pair repulsion to widen the H -C -H bond angle. (ii) PC/]FZ : The molecule has got a trigonal bipyramid (TBP) structure. The two fluorine atoms preferably are positioned in the axial directions. The axially orientated hybrid orbitals (Le. pd; p= + d:: 2 ) having no s-character bind the more electronegative substituent, i.e. F. The tendency of more electronegative substituents to occupy the axial or apical pd-hybrid orbitals of low electronegativity in TBP geometries is referred to as apicophilicity. In MePF4, Me 2PF3 the more electronegative F-atoms tend to occupy the axial positions while the Me-groups of low electronegativity occupy the equatorial positions.

he

CH] radical vs. CF] radical (cr. Sec. 10.8.3)

t.m

e/

th

ea

lc

Bent's rule: The CH3 radical is planar and carbon is sp2 hybridised to house the odd unpai.red electron in a pure p-orbital. On the other hand, CF3 radical is pyramidal i.e. here carbon is sp3 hybridised and the unpaired electron is housed in a sp3-hybrid orbital. This difference can be " This explained by considering the Bent's rule. Fluorine is more electronegative than hyqrogen. is why, fluorine directs carbon to project the s-p-hybrid orbital having more p~character. Between sp3 (75% p-character) and sp2 (66.6% p-character) hybrid orbitals, fluorine prefers the sp3 hybrid orbital of carbon having relatively smaller s-character. ./,pure-p ,

y-Sp3

0r /c ..

tJ. .

"H

H--O~H Sp2 (C) \"

F

Sp3 (C)

\""F F

VSEPR theory (cr. Fig. 10.8.3.4) : The C-H bond length is shorter than the C-F bond length (cf. C-H in CH4 = 109 pm, C-F in CF4 = 136 pm). Because of the higher electronegativity of fluorine, the bond pair of C-F bond is attracted more towards fluorine. Because of the slightly higher electronegativity of carbon (with respect to hydrogen), the bond pair of C-H bond is slightly attracted towards carbon. Thus, the bond pair-bond pair (bp-bp) repulsion in eH] is much higher than that in CF]. To minimise this bp-bp repulsion in CH3, it adopts the sp2_ hybridisation where bond angle is higher than that in the pyramidal structure having sp3 hybridisation. In fact, in CH3, bp-bp repulsion operates at 120° and the odd electron-bond pair repulsion which is relatively less operates at 90°. In CF3, bp-bp repulsions are less and it adopts the sp3 hybridisation and all the repulsions occur at about 109°28'.

698 Fundamental Concepts of Inorganic Chemistry (iii) Position of a lone pair: The Bent's rule is very much important in predicting the position of a lone pair. There is a basic difference (see Fig. 10.8.3.3) between the lone pair and bond pair. The bond pai~ remains under the control of two bonding nuclei while the lone pair remains under

ib

ra

ry

the influence of the central nucleus alone. Thus the central atom tends to hold the lone pair cloud tightly and it can be attained if the central atom can utilise the orbital enriched in scharacter in housing the lone pair. This is why, in H 20, the lone pairs are housed in the orbitals where the s-character is greater than 25 % • Similarly in elFl (sp 3d, TBP hybridisation), the two lone pairs are housed in the equatorial hybrid orbitals where the s-character is maximum and the more electronegative F atoms go preferably to the axial directions where there is no s-character. Of course, the third F atom having no other option will have to occupy the third equatorial position. The results predicted from the Bent's rule are in good conformity with those obtained from VSEPR theory.

m

yl

Bent's Rule vs. VSEPR Theory VSEPR rule (dealing with the effect of electronegativity of the substituents) and Bent's rule . give the same conclusions in most of the cases.

th

ea

lc

he

(iv) Effect ofthe strength ofcovalency and pi-bond on hybridisation : When the bonding atoms are of comparable electronegativity, the pi-bonding in a particular direction may shorten the bond length in that direction giving rise to a better covalency compared to the direction having only sigma bond. Thus, the central atom pr_ojects more s-character in the direction where both the pi- and sigma-bonds are formed to ensure the better covalency. In the phosphate compo1Jnd, (RO)2P(==O)Br, phosphorous is tetrahedrally hybridised where the least electronegative substituent is Br. Hence, in terms of the classical form of Bent's rule, it is expected that the P-Br bond should contain the highest amount ofs-character of P but the P == 0 bond is maximum enriched in the s-character of P. This is reasonable. The orbitals of Br are larger and more diffuse and

t.m

e/

consequently the overlapping of the Br-orbitals with those of the relatively smaller P-atom is inefficient. This weakens the covalent interaction in the P-Br bond. The 7t-bond shortens the bond length in the P == 0 bond giving a better covalency and as a result, the central atom uses its maximum s-character in that direction. In fact, the bond energy runs in the sequence : P = 0 ) P - 0) P - Br and this is why the P-atom concentrates s-character maximum along the P = 0 bond. 110 pm

Strongest bond

(maximum s-character of P)

0

~II~

/P ~weakest bond

RO

1

OR

Br

(minimum s-character of P)

~

146 pm

~fc

1

C

105 pm H

(H 1100 1 4 ' \ '-120 pm

a < 109°28'

s-character concentrated by sp'3-C to form the strong C-C covalent bond

The relatively smaller sp-C tends to form a stronger covalent bond with the Spl-e, atom. In fact, the bonding potentiality of the sp-hybrid orbital is maximum (Sec. 9.13.3) among the sp3,

Introduction to Chemical Bonding and Theories of Covalence 699

and sp hybrid·orbitals (cf. Sec. 9.13.3). It directs the sp3_C atom to project the s-character enriched hybrid orbital towards the sp-C. This explains the structural features of the above compound. Here, it may be noted that in terms of simple electronegativity effect, the more electronegative sp-C should direct the sp3_C to project the hybrid orbital having less s-character. But the covalent bond strength factor opposes this demand. Other applications of the Bent's rule will be discussed in the next chapter (see Sec. 10.8). Note: The classical form of Bent's rule cannot explain the experimental findings of the above two examples.

ra

ry

Sp2

ib

9.14 SUMMARY OF THE CONCEPT OF HYBRIDISATION

t.m

e/

th

ea

lc

he

m

yl

(i) The atomic orbitals of comparable energy'on the same atom can hybridise. If the starting atomic orbitals differ significantly in energy, the cost of hybridisation energy (Ehy) will be high to make the process insignificant. (ii) The hybridisation occurs through the LCAO of the combining orbital wave functions. The ~umber of hybrid orbitals is given by the number of participating atomic orbitals. (iii) Promotion of the electrons is an essential step to carry out the hybridisation but the promoted electrons do not necessarily participate in the hybridisation. It occurs in the case of multiple bonding. Generally, the sigma bonding electrons and lone pairs (i.e. nonbonding pairs) occupy the hybrid orbitals while the electrons occupying the unhybrid c,rbitals participate. in pi-bonding. For the nonmetals, the d-orbitals can only participate when the central nonmetallic atom combines with the highly electronegative moieties. The hybrid orbitals are having much greater bonding potentiality compared to ·the starting pure atomic orbitals, as the hybrid orbitals concentrate their electron clouds mainly in some particular directions of covalent bond formation. The hybrid orbitals may be equivalent or not. The hybridisation defines a geometry of the molecule. It depends on the angular distribution function of the combining atomic orbitals. Bent's rule: The central atom projects more p-character to the more electronegative substituent, and to the bond having more ionic character (Le. poorer covalent character); on the other hand, the central atom projects more s-character to the lone pairs, less electronegative substituents, predominant and strong covalent bonds and multiple bonds.

9.15 CONJUGATED MOLECULES AND DELOCALISATION ENERGY In the conjllgated polyenes, the carbons are sp2 hybridised to provide the sigma skeleton ofthe molecule. If the skeleton lies on the xy-plane, then each carbon possesses a Pz orbital perpendicular to the c-e chain. These Pz orbitals may undergo a pi-type interaction through LCAO to form molecular orbitals. Here we shall pay attention only to the 7t-MOs. The procedure is known as Hackel molecular orbital method to honour the scientist who contributed as a pioneer in this discipline. The calculation of

700

Fundamental Concepts of Inorganic Chemistry

energy is done by the variation r.nethod through the establishment of n (when n number ofPz orbitals are involved in LCAO) number of simultaneous linear equations. The molecular orbital wave function is expressed as: \jIpj-MO

n

n

;=1

;=1

= LCj\jl2pz(i);and LC; =1

In the present system, the overlap integrals (Sij) are assumed to be zero in the HUckel's approximate method.

ry

= f'V2 Pz (.) 'V2 Pz(.)J dt = 0 orthogonal. Similarly, S.. = f'V2 P (.) 'V2 P (') dt =1, z z S..IJ

I

Ii \jI2pz(i) dot;

13 = J\jI2pz(j)

Ii \jI2pz{j) d't

yl

a. = f\jl2pz(i)

ib

ra

It appears that the AOs are since the AOs are II I I normalised. Thus the energy ofthe 1t- MOs is always expressed in terms of Coulomb integral (a) and resonance integral (~) as defined below:

he

m

Here, i-th and j-th carbon atoms are directly bonded..The significance of a and Phas been already discussed in Sec. 9.4.1. Now let us ill~trate the method in some representative cases. (a) Ethylene Molecule (C2H 4) : Here two adjacent C-atoms provide two parallel p= orbitals for LCAO. It leads to two pi-MOs, one of which is bonding and another one is antibonding.

= N 1 ['V2pz(C + lV2pz(C E = a +~; \jI:-MO =N2 [\jI2pz(C.) -\jI2Pz(c )],E* =0.-13 'Vn-MO

)

lc

1

)]'

2

ea

and

2

th

The two electrons will reside in the bonding 1t-MO. Thus the ground state electronic energy of the 1t-system is given by, 2 (a + ~). The 1t-electrons are distributed as : 1t~1t·0

t.m

e/

The energy of the two isolated electrons in two 2pz orbitals is 2a. Thus on LCAO the energy is decreased (~ is a negative quantity). The orbital picture of ethylene is given in Fig. 9.15.1.

C1- - C2 "===:'

O-H OH Hypophosphorous acid, minor tautomer

~"

P =0

/\

H

OH

Phosphlnlc acid, major tautomer

t.m

e/

th

(c) Similarly, tautomerism in sulphurous acid is also reported.

(ii)

(iii) (iv) (v)

Q\

~"

5=0 /~ HO 0

Ho/II o-H o

All the examples of tautomerism cited above involve the shifting of a proton and this type of tautomerism is referred to as prototropic tautomerism which is very important in organic chemistry (e.g. keto-enol tautomerism). Contribution of the canonical forms: The contribution of a canonical form to the resonance hybrid is inversely proportional to its energy. This is why, the high energy canonical forms contribute insignificantly. Comparable energy of the canonical forms: The resonating structures should be ofcomparable energy. Equivalent canonical forms: If the resonating structures are equivalent as in the case of coi-, the resonance stabilisation is very high. Covalent ionic resonance :The covalent-ionic resonance becomes important in the cases where the combining atoms differ significantly in electronegativity.

708

Fundamental Concepts of Inorganic Chemistry

-N=N+=O(I,linear)~N==N+-O-(II,linear)~ /

ry

(vi) Number of covalent linkages: The canonical forms should have the maximum number of covalent linkages. The reduction in the number of covalent linkages in drawing a canonical form reduces its contribution. In the case of carbon dioxide, Le. 0 = C = 0 (I) ~ 0 = c+ - 0- (II), the canonical form (II) being deficient of one covalent linkage contributes little. (vii) Geometry of the canonical forms: The canonical forms should not differ in geometry or bond angle drastically. In the case of nitrous oxide, the cyclic (3-membered) structure (III) is not at all promising.

o

\

(III,cyclic)

ra

N===N

he

m

yl

ib

(viii) Number of unpaired electrons: The canonical forms should have the same number of unpaired electrons. (ix) Charge separation: In the case of charge separation, if the adjacent atoms bear the same charge then the electrostatic repulsion will destabilise the structure. On the other hand, placement of ~pposite charge on the adjacent atoms stabilises the system through an electrostatic interaction as in the ionic compounds. In the case ofundissociated hydrazoic acid, the structure (II) contributes less compared to the structures (I) and (III). (I) H - N= tv+" = ~ ~ (II) H-tv+" == tv+"-Nl- ~ (III) H - ~ - ~ == N

t.m

e/

th

ea

lc

(x) Placement of charge : The canonical forms in which the negative charge resides on the electronegative atoms contribute more. On the other hand, placement of the negative charge on the electropositive centres destabilises the system. (xi) Number of covalent linkages : The canonical forms in which a larger number of covalent linkages exists are more contributing to the resonance hybrid. In BF3 , the P1t - P1t bonding places a positive charge on F which is more electronegative than B, but this disfavour is compensated due to the formation ofan additional n-bond.

In fact, due to this type of pi-bonding, B - F bond has got the double bond character to some extent (see Figs. 9.16.5.1 and 10.7.6). (xii) Requirement of coplanarity : For the delocalisation of the pi-electron clouds, coplanarity of the involved skeleton is required. This is why, sp and sp2 hybridisations facilitate the process. This coplanarity is exclusively essential when the pi-bonds made ofp-orbitals are involved.in the process. But,for the d-orbitals (very oftenfound in inorganic compounds), this coplanarity is not to be maintained rigidly (see Sec. 10.7 (vi»). The d-orbitals being diffused can participate in pi-bonding even in nonplanar systems (e.g. 80;-, C/O;, etc.). (xiii) Pauling's electronegativity principle and the stable resonating structure: This aspect has been illustrated in Sec. 9.16.4.

Introduction to Chemical Bonding and Theories of Covalence 709 9.16.4 Concepts of Formal Charge and Lewis-Langmuir Charge in Resonating Structures : Formal Charge and Oxidation Number

• Formal charge without considering the effect of electronegativity difference: Conventionally, in the formation of an additive coordinate bond, the donor atom is believed to bear a positive charge while the acceptor one bears a negative charge. Ifthe involving atoms were ofalmost the same electronegativity, then the above consideration would be reasonably correct. However, in distrib';1ting the charge, no consideration of electronegativity difference is taken into account. This is why, in calculating the

ry

formal charge (qF) of an atom in a compound, all the constituent atoms are considered to be of the same electronegativity. To calculate the qF of an atom in a compound, the number of electrons

~

- nip -

n hp

I

yl

qr = n A

ib

I

ra

actually owned by the atom in the compound is compared with the number of electrons possessed by the isolated atom in the neutral state. Thus qF is given by,

where, nA= number of electrons in the valence shell of the atom at the isolated .and neutral condition, = number of lone pair electrons which are completely owned by the atom under consideration, nbp = number of bond pair electrons which are equally shared by the two combining atoms.

m

nip

:S: 2-.

1 qr (Stenninal)=6-4-"2 x4 =O

S

~/II'\g~ :0:

ea

•

1 2

q F (0) = 6 - 4 - - x 4 = 0

qr (0-) = 6 - 6 _.!. x 2 =-1

th

S20 J

1 qF (Scentre) = 6 - 0 - - x 12 = 0 2

lc

I

he

Some examples of formal charge calculation are given below :

t.m

e/

2

:~B /

-t:-

••

· "r.:

NH4+·•

-

1

qF (B)=3-0--x6=0 2

q F (F) = 7 - 6 -

1

~X

2

2 =0

1 qF (N) = 5 -0 -- x 8 = + 1 2

1

qF (H)=1-0--x2=0 2

710

Fundamental Concepts of Inorganic Chemistry

SF4 : q~S)

.

= 0, q~F) = 0; HIV- OH: q~N) = 0, qF(O) = 0; ope/3

(forO~ +PCI3 ):qIP) = +1,

q~C/) = 0, q~O) =-1; 0 =PC/3 : q~P) = q~O) = q~Cl) = 0; NO; (nitronium, O=~= 0): q~N)

= +1, q~O) = 0; PH3 : q~P) = 0, q~H) = 0; PH;: qlP) = + 1, qlH) = 0; BF(:B == F:) : qlB) = -2, q~F) = +2. The formal charge on the constituting atoms depends on the structure of the resonating form. This aspect is illustrated in the following examples. The formal charges on the respective atoms are shown;' for 0 (zero) formal charge, nothing is shown. ..

+

..-

.. '- B- f;: +---+ .. / . .

:t

:0 II

+

.. ,

../

S= F:

- .. / :9

ry

+..

:f;

N+

,- .. -

ra

-

:~=N=9'-"N=N-9

:f;

:t

Q=

+- "" .... ~

1.9. - - "" .... yg

yl

1.9.

ib

(3 EQ.ivalErt (3 EQ.ivalErt structures, structures, • 1~~. 2~~

3 00 EBtl F)

3 on each 0)

m

~~-C=N: ~ ~=C=N:~>:S=C-N:2(b)

(c)

he

(a)

(most favourable)

lc

2+ 3- . . + + :~=N=g: +-+ :g-N=O:

Isomeric fulminate and cyanate (cf. Sec. 9.16.6)

th

ea

(a)

_

+

(b)

(c)

(b)

(c)

e/

(a)

..

+-+:C=N-Q:-

t.m

For CND- (i.e. fulminate), (c) form bears the most favourable charge distribution while the other two forms experience the more unfavourable situations. On the other hand, for OCfY (i.e. cyanate ion), both the (a) and (b) forms experience the favourable charge distribution and between these two forms, (b) contributes better where the negative charge lies on the more electronegative atom. Thus in terms of the resonating structures, it is evident that OCfY is more stable than CNlr. It is also experimentally verified (cf. Sec. 16.21 for explosive characters of fulminates).

:0:s~-:

+ 8 -:0:/1' :0:.. _ :0: .. .. 12

(a)

sol-,

:·0:

11+

8 :0:/ 1'-:0:

4-+

-··-.:9: .. (b) (4 equivalent structures)

0: II

8 :0#1':0:

..-.

··-.:9: .. -

(c) (6 equivalent

structures)

:0:I

-8

0:

M2-

:0#1I~:0 .. :0:··

:0:#II~:0

(d) (4 equivalent structures)

(e)

:0:

..

the formal charge on 0 is different in different cannonical forms. This is as follows: 3 1 1 (a) q~O) = -1; (b) q~O) = - 4"; (c) Q~O) = -"2; (d) q~O) = - 4"; (e) q~O) = O. For

Introduction to Chemical Bonding and Theories of Covalence 711 Considering the 6 equivalent structure for the c-form of

o (zero) and _..!2

50;-, the formal charges on Sand 0 are

(on each 0, i.e. average value) respectively.

What are the formal charges on Al and 0 in Al(OH2 )~+ ? Ans : -3 for Al and + 1 for 0

ry

In the above mentioned discussion, to calculate the formal charge, the effect of electronegativity difference between the combining atoms has been ignored. It has been assumed that the bonding electron pair is equally shared between the combining atoms.

yl

ib

ra

Pauling's electroneutrality principle: Now let us consider the Pauling's electroneutrality principle which states that in the stable resonating structure, the electrons are distributed in such a way that the atoms can carry the minimum charge tending to zero. In other words, it should lead to charge separation minimum. Thus, for SO;-, the resonating structure a is the most unfavourable condition while the structure c is the most favourable case. Through the formation of multiple bonds (Le. 7t-bonds), the charge separation can be minimised to satisfy the demand of electroneutrality principle.

m

The important predictions ofPauling's electroneutrality principle are:

ea

lc

he

-. minimum charge separation; • if charge separation - electropositive centres will bear the partial positive charge and electronegative centres will bear the partial negative charge; • 7t-bonding to minimise the charge separation when possible; maximum number of covalent bonds.

th

• Lewis-Langmuir charge considering the .effect of electronegativity difference: Considering the effect of electronegativity difference, Allen (J. Am. Chem. Soc., 111, 9115, 1989) has suggested to calculate the charge (called Lewis-Langmuir charge) on the atoms. Thus the Lewis-Langmuir charge

t.m

e/

(qL-L) considers the effect ofelectronegativity difference whUe theformal charge (q F) does not consider the effect 0/ electronegativity difference. When, there is no electronegativity difference, qL-L becomes qF. The Lewis-Langmuir charge,' qL-L on A in the ABx molecule is calculated as follows:

L (XA XA+ XB )

qL-L(A)=nA -nIP -2

. hontk

For, B-A-B molecule, the last term considers the sum of 2 single bonds; for A = B molecule, it also considers over 2 bonds for the double bond. Obviously for XA = XB' qL-L and qF become identical. Calculation of qL-L is illustrated in the following examples (for x-values, see Pauling's values, Chapter 8).

.. I H

/N"

(a) H

H

qL-L(N) = 5 - 2 - 2 x3 (_3_) 3+ 2.1

~ -0.53

J

qL_L(H)=1-0-2xl ( -2.13+ 2.1 X(H) = 2.1, X(N) = 3.0

~+0.18

712

Fundamental Concepts of Inorganic Chemistry

..

qL_L(H)=1-0-2XI{

(b) H-9.1:

2.1 }=1-0.82=+0.18 3.0 + 2.1

qL_L(CI)=7-6-2XI{

3.0 3.0+2.1

}~-0.18

X(H) = 2.1, X(CI) = 3.0

2.5 )-2Xl( 2.5 ) 2.5+ 2.5 2.5 + 2.1

ry

(c) H-C=C-H QL_L(C)=4-0-2X3(

ra

= 4 - 3 -1.08 = - 0.08; taking X(H) = 2.I,X(C) = 2.5 •• -

~

+

:N,= Nc

(I)

••

~) [x(N)

(II)

= 3.0; x(O) = 3.5]

yl

+

(d) N 20 (Le. :N,=='Nc-~:

ib

Note: To get the more correct result, electronegativity of sp-C should be used.

N20 possesses two stable resonating structures as shown above.

=5 -

~

m

qL-L (N,)

2 - (2 x 3){3 3} =0

he

Structure-/:

= 5 - 0 - 2 x 3 {_3_} - 2 x I (_3- ) 3+3 3 +3.5

th

ea

qL-L (Ne )

1){~}=-1.07 3.5+3

lc

QL-L(O) =6 -6-(2x

3.0 } =-1 3.0+3.0

t.m

e/

Structure-II: QL-L (N,) = 5 - 4 - (2 x 2) {

QL-L(Nc ) = 5 - 0 - 2 x 2 {_3_} - 2 x 2 (_3- )

3+3

qL-L(O) = 6 - 4 - (2 x 2)

~ 1.07

3 + 3.5

~ 1.15

{~} = -0.15 3.5 +3

Experimentally it has been found that the Structure I and II contribute in the approximate ratio 2 : 1 to represent the real N 20 molecule. This is quite reasonable because in structure I, negative charge is carried by the most electronegative centre while in structure II, negative charge is placed on the relatively less electronegative centre N It leads to the average value of qL-L as follows: qL-L (N,)

= 2 x 0 +3(-1 x I) =- 0.33

qL-L (N ) e

= 2 x (+ 1.07); 1x (+ US) = 1.09

Introduction to Chemical Bonding and Theories of Covalence 713 qL-L (0)

= 2 x (-1.07); (t x -0.15) =_ 0.76

Thus, N 20 should be represented as follows: -0.33

- 0.76

N:::':N~O

(in tenns of qL-L)

+ 1.09

ry

Note: In the above mentioned calculations, electronegativity ofnitrogen has taken as 3.0 irrespective of its hybridisation state in different cannonical forms 7 But electronegativity of an atom depends on its state of hybridisation.

ra

Formal charge (q,J, Lewis-Langmuir charge (qL-J and Oxidation number (O.H.)

he

m

yl

ib

In calculating qF' polarisation ofa covalent bond due to electronegativity difference is not considered, Le., the compounds are considered to have the 100% covalent character. In calculating the qL-L' development of partial ionic character in the covalent bonds due to the electronegativity difference is considered. In calculating oxidation number, it is assumed that the bonding electron pair is completely.transferred to the more electronegative atom giving rise to 100% ionic character. Thus the concepts of formal charge and oxidation number consider the two limiting cases i.e. 1000k covalent character and 100% ionic character respectively. In covalent compounds, practically these are hypothetical suggestions. On the other hand, qL-L considers the real situation of covalent bonds where the electronegativity difference is not too large or too small.

lc

NHJ : ql.N) = 0, QL-L(N) = - 0.53, O.N.(N) = - 3

ea

H 20: qF(O) = 0, qL-L (0) = - 0.5, O.N.(O) = - 2

th

HC/: qF(CI) = 0, qL-L (CI) = - 0.18, O.N.(CI) =-1

9.16.5 Resonance in the Light of Molecular Orbital Theory

t.m

e/

The concept of resonance is originated in the valence bond theory (VBT). In VBT, the concept of localis~d bonBs is well established. On the other hand, the polycentric molec~lar orbital leads to the delocalisation ofthe electron cloud through a wide range. The concept of resonance in VBT leads to the delocalisation of the electrons. Thus the delocalisation of the electron cloud through the polycentric molecular orbital is equivalent to resonance in VBT. For the planar ABJ species, e.g. BFj, CO;- , NO;, etc. (Sec. 9.12.10) the construction of4-centred pi-MOs can nicely explain the 1/3 pi-bond order in eachA-B bond. Ifthe molecule lies in the xy-plane, each of the four atoms of AB3 can provide their p= orbitals for LCAO to generate four centred 7t-MOs out of which there are two nonbonding 7t-MOs along with a bonding and an antibonding 7t-MO (Fig. 9.16.5.1). In these species, besides the 18 sigma-electrons (including 6 lone pairs)

, ,, ,

,

"

n*

,

\ \ \ ~

\

~

\

-P-z(-A-)~\" l...

"

"'~11', 1tnb

\

\

"\

",

"\ \

,

"

\

',.

1tb

Q) ~

\

.. It

1

Three Pz

,, ' ( 3 8 ) ,

Fig. 9.16.5.1. The 4c pi-molecular orbitals in the ABa type molecular species (e.g. B~, 00;-, NO:;, etc.).

714

Fundamental Concepts of Inorganic Chemistry

placed in different sigma-orbitals, there are 6 pi-electrons (provided by three B atoms in AB3) which are distributed in the pi-MOs. These are discussed in detail fn Sec. 9.12.10(cf. Figs. 9.12.10.2 and 9.12.10.3). The four pi-electrons placed in the two nonbonding pi-MOs have nothing to do with the bond order. The two electrons placed in the bondingpi-MO produce the total bond order unity which is distributed in three A-B bonds. Thus the pi-bond order in each A-B bond is 1/3. The electronic configuration (cf. Fig. 9.12.10.2) is: ( 0' b )

6

( 0' nb )

12

( 1tb )

2

4

( 1tnb) •

ra

ry

Such delocalised pi-molecular orbitals have been discussed (see Secs. 9.12 and 9:15) in some specific cases, e.g. NO;, conjugated molecules, etc. The 7t-bonding in 0 3 and related molecular species have been discussed in Fig. 9.12.9.4. Here three p-orbitals are available for 7t-bonding and these will produce three 3-centred 7t-MOs (one bonding + one nonbonding + one antibonfing). Four electrons are distributed in the 7th and 7tnb MOs (i.e. 3c--4e 7t-bonding) to pr~duce 1/2 7t-bbnd in each linkage (cf. Fig. 9.16.5.2). In 3,18 valence electrons are distributed as: (O'nb)10 (O'b)4 (1t b )2 (1t nb )2 (cf. Fig. 9.12.9.6). In S02' the same bonding scheme works. These aspects have been discussed in detail in Sec~ 9.12.9.

m

yl

ib

°

he

" '"

" .."

ea

\

I

\

I

\

I

\

I

\

I

\

I

---, \

1b

( \

I

\

I

\

(OAt Os)

I

\

I

\

th

2pz

I

\

I

\

I

\

I

\

I

\

e/ t.m

II

\

I

/

lc

" "

\

I

:..

~

\

I

Lxy-p,ane

..

,

I I

\

,

I

1~

I

--

Fig. 9.16.5.2. Three centred 1t-MOs in 03.

9.16.6 Application of the Concept of Resonanoe In Some Inorganic Compounds

Carbon monoxide (CO) : In CO, the bond length (113 pm) lies between the bond lengths ofC = 0 (122 pm) and C == (110 pm). This intermediate value can be rationalised from the resonating structures Le. : •• + :c= Q (I) ~ :C..- 0: (II)

°

In the structure (II), the positive charge on the more electronegative oxygen atom destabilises the structure, but this disfavour is 90mpensated due to the formation ofan additional covalent linkage. It is interesting to note that in the structure (I), the bond moment operates towards 'oxygen, while in the structure (II), it works in the opposite direction. Probably, because o/these two opposing moments, the dipole moment ofthe system is very small. The whole picture including the lone pair moments will be discussed in Sec. 10.5.3.

Introduction to Chemical Bonding and Theories of CovalencEl 715

• Carbon dioxide (C0 2) : Here the C-O bond length (= 115 pm) again lies between the C = 0(122 pm) and C= 0 (110 pm) bond lengths. This increased bond order arises due to the following contributing resonating structures, Le. ••

••

2===C. Q (I)

-..

+

~ :~-e==O: (II)

+

•• -

..-. :O==C-g: (III)

The structure (I) is symmetrical, and the two unsymmetrical structures (II) and (III) contribute equally. This is why, the molecule overall shows no dipole moment • Nitrogen dioxide (N0z) : The canonical forms can explain the partial double bond character of the two equivalent N (119 pm) bonds which are intermediate between N - 0 (136 pm) and N = (115 pm). The canonical forms are:

~O O=N,

o

°

0, 0

~N.

bond length (121 pm) can be explained from the

yl

• Nitrate ion (NO;): The observed N contribution of the canonical structures, Le.

H

m

o

~N.

ib

~

~

/0-

~ O=N~

f-) O.-N,

he

0

°

ra

ry

°

~

_

#0 O-N,

0

0

ea

lc

• Hydrazoic acid ion (HN]) and azide (N]): In the undissociated acid, the bond lengths are as follows: H - N (107 pm), middle N - N (124 pm), terminal (Le. re~ote from H) N - N (113 pm). Thus, the N - N bond distances .in HN] are unequal. But in azide all the.N - N bond lengths are equal (115. pm). The above facts can be explained by considering the resonating structures of both HN) and Ni. In HN) the corresponding structures are :

th

••

+

H-N=N

-

U:

(I)

~

_..

+

H-'!.-N==N:(l1)

t.m

e/

Here, the middle N - N has got both the double and single bond character while the terminal N - N has got both the double and triple bond character. This is why the N - N distances in HN) are" not equal. In Ni, the corresponding resonating structures are : +

2-..

+

+

•• 2-

N. : ' -. ' . :N-N==N:'-' :N==N-N: ... ..

:N=N

Here, all the N - N bonds are equivalent. The enhanced stability 0/ the ionic azides compared to that 0/ 'he covalent azides can also be explained/rom the consideration o/resonance energy. The covalent azides have the structure like that of the undissociated acid with two nonequivalent resonating structures. On the other hand, in the ionic azides, the azide ion is having three canonical forms in which two forms are equivalent. Thus the resonance stabilisation in the ionic azides is greater than that in the covalent azides. • lsosters (N 2 0, Ni, OCN-, CNO-, SCN-, Ni; cf. Sec. 9.16.4): In this connection, it is worth mentioning that the isosteric species, e.g. nitrous oxide (N20), azide (Ni), cyanate (OClV), fulminate (CNo-), thiocyanate (SClV), are also resonance stabilised. The species having identical outer electronic configurations are called isosters. All are linear. Because of this structural similarity, they can form the isomorphous alkali salts. The resonating structures of SCN"" can be shown as:

716

Fundamehtal Concepts of Inorganic Chemistry

..

:S-C=N:(I)

~

••

•• -

:S=C=N:(II)

~

•• 2-

+

..

:S==C-N: (III)

Here, the structure (III) contributes less, as the opposite charges are placed at the largest distance. The most favourable structure is (II) where the most electronegative atom bears the negative charge. • ,CND- (fulminate) vs. OCN- (cyanate) : The resonating structures of these two isomeric and isosteric species are : 2=:)

+ :~=N=g:(I)

..

_

+

•• _

.....- -.. ~ ~==N-g: (II) .....~-~~

3 - ••

+ + :~-N==O:

(III)

ry

CNO-

+ •• 2:O==C-!i: (III)

ra

OCN- =>:()=.C=N:-(l) .....~......; .. ~-:Q-C==N: (II) ... ~--.....

m

yl

ib

In tenns ofcharge placement, for CNo-, the most favourable structure is II. In other two structures, charge distribution patterns are relatively unfavourable and these are not stable. For OCN", the structures I and II place negative charges on the electron~gative centres (Le. N, 0). Obviously, structure II is more favourable than I, because in II, n'egative cHarge is placed on the most electronegative centre. It is evident that for CNo-, the resonating structures are less stable compared to those for OelV. This is why, OCN- is more stable than CND- (cf. Sec. 16.21 for explosive character of fulminates).

he

9.17 LIMITATIONS OF THE CONCEPT OF RESONANCE AND HYBRIDISATION

ea

lc

For shifting of the pi-electrons (specially the p-electrons) through resonance, the condition of coplanarity is to be maintained. Thus to effect the resonance, the state of hybridisation of a particular atom in a polyatomic species may vary in different canonical forms. The actual molecule

t.m

e/

th

is the resonance hybrid ofall the canonical fonns. Hence it is difficult to define the state of hybridisation ofthe constituent atoms in the actual molecule. The above fact can be clarified in the following examples. In constructing the structures, it must be kept in mind that generally the pi-bonds can be fonned by using the suitable p-orbitals. • HNJ : The states of hybridisation of the different nitrogen atoms are given below. + + H-N=N=N .......-.. H-N-N=='N (sp2) (sp) (sp2) (Sp3) (sp) (sp)

• ~: The states of hybridisation of the nitrogen atoms are shown below : + 2- + + 2N=N=N (Sp2) (sp) (Sp2)

H

N-N=='N (Sp 3) (sp) (sp)

~

N=='N-N (sp) (sp) (Sp3)

• B(OCHJ)J (trimethoxy boron) : In drawing the canonical fonns, oxygen changes from sp3 to sp2, so that in tht; structure II, the p-orbital can participate in Pft- Pft back bonding with boron. The bond angle B-O-C is 113 0 and it indicates that the O-atom remains in a state between spJ and sp2. Me~.. B - OMe (I) Meo/ (~p3)

~

Me~_ f/ B = OMe (II) MeCY (- spl)

Introduction to Chemical Bonding and Theories of Covalence 717

• NzOs: The states of hybridisations of different ,atoms in the possible canonical forms are given below.

O~Sp2) •• (sp2)/O #N-Q-N~ 0"

0

(Sp3)

H

O~Sp2)"to (sp2)~ O~sp2) 2+ (sP)/O /N=O-N" ~ N=O=N, Q (Sp2) 0 ~ (sp) l!.

yl

ib

ra

ry

• HzN - PHz : Here to allow the N(PJ --. P(dJ bonding, N-centre should adopt the sp2-hybridisation state so that the lone pair can be housed in a pure p-orbital. In fact, N-centre remains in a hybridisation state between the spz and spJ. The bond angle (H - N- P) and the bond length (N - P) support the proposition. Thus it is evident that ifthe concept of resonance is taken into account, then the consideration ofthe state of hybridisation of a particular centre becomes ill-defined. Hence, the prediction ofgeometry of the molecule by considering the hybridisation becomes uncertain, because the actual state of hybridisation depends upon the relative contribution ofthe canonicalforms to the resonance hybrid.

9.18 MULTIPLE BONDS

ea

lc

he

m

Generally, the multiple bonds arise due to the pi-bonding in addition to the sigma-bonding. We h~ve already discussed the properties ofthe sigma- and pi-bonds. Before to enter into the mechanistic pathways in developing the pi-bonding, we should highlight the effect of bond-multiplicity on the properties of the molecule. The pi-bonding strongly influences a number of properties such as: structure, aciditybasicity, stability and reactivity (e.g. cis- and tran~-effects), stabilisation of unusual oxidation states (e.g. pi-acid ligands to stabilise the very low oxidation states oftransition metals), tendency to polymerise, etc. These phenomena will be discussed separately in the next chapter (Sec. 10.7). Here we shall pay attention only to the effect on bond length. 160

th

9.18.1 Bond Length and Bond Multiplicity

t.m

e/

In determining the bond length, the bond multiplicity (Le. bond order) is extremely important. With the increase of bond order, the bond length decreases (see Table 10.2.3.1). Here it is worth mentioning that besides the bond order, many other factors (see Sec. 10.2.3) can govern the bond length. These factors are : nature of hybridisation, nearest environment in the molecule, electronegativity difference between the combining centres, steric factors, etc. But in spite of the facts, it can be concluded that the bond length decreases with the increases of bond order. For C - C system, an empirical relationship between the bond length (r in pm) and the bond order (n) has been developed by Pauling. The relationship is : r n =r t -71Iogn,

.-.. 150

E

~

.r::.

1»

j

140

"C

c

o

CD

(,) 130 I

(,)

""C2 H2

120 ........- - - - - -.......- - . . . . . - - -

1

2

3

C - C Bond order

Fig. 9.18.1.1. The variation of carboncarbon bond length with the bond order.

where rn denotes bond length (in pm) for the bond order n. The variation of bond length with the bond order in C-C system is shown in Fig. 9.18.1.1.

718

Fundamental Concepts of Inorganic Chemistry

9.18.2 Px-Px Bonding (A) VB Description of pi-Bonding In Unsaturated Hydrocarbons Let us start with the series of hydrocarbons, C2H6, C2H4 and C2H2 for which the conventional structural representations are :

rr

H

H - C - C ; - H (C H)o

I

H

H

(C2H4);

/C= "-

26'

H

H-C::::='C-'-H (C2H 2)

ry

I

H

C/

"

H

he

m

yl

ib

ra

In the light ofVBT, in the saturated hydrocarbon C2H6 , each carbon is tetrahedrally (sp 3) hybridised and the four hybrid orbitals are utilised in forming four covalent bonds. In C2H 4, each carbon is sp2 hybridised to form three sigma bonds in a plane. If the sigma-skeleton lies on the xy-plane, then each carbon atoJ:1l contains a pure Pz orbital.with an unpaired electron perpendicular to the plane. These two' p=-orbitals laterally overlap to form api-bond (see Fig. 9.15.1). Similarly, in C2H2, each sp carbon contains two pure p-orbitals (say, Py and Pz' assuming the. x-axis to be the molecular axis) which are mutually perpendicular (see Fig. 9.18.2.1). Here the lateral overlap of Py - Py and Pz - Pz produces two 7t-bonds leading to the triple bond, C iii C. Here it must be remembered that the x-bonds formed due to the lateral overlap of the orbitals are weaker than the sigma-bonds where the head-on overlap occurs. This is why, a double bond containing one sigma- and one pi-bond is weaker than the sum of two isolated sigma or single bonds. It explains the higher reactivity ofthe unsaturated compounds compared

H

H

I

I

I

I

ea

lc

to the saturated ones where no pi-bond exists.

H

e/

H

th

H-C-C-H (C 2 Ha):

t.m

(:)--------A H,V --------V/H H/(jC -------(jC'H -------.-

H

H

H

H

)c = c( z

Pz y

------x

-

Fig. 9.18.2.1. The

(C 2H4 ): H - C!!!! C -

Pz

~-------~

p

v3J;~Vt:J ~tJC---~k)C-" H -

pj.bondin~

H (C 2 H2 )

------- -------

. interactions in C2 H2 and C2 H4 -

To explain the reactivity ofthe unsaturated hydrocarbons and the origin of multiple bond formation, an alternative approach of bent bond concept has been introduced by Linnett. In this concept, it is believed that the carbon atoms in the unsaturated compound, e.g. C2H 4, are also tetrahearally hybridised. Here, two of the four sp3 hybrid orbitals on each carbon atom are involved to unite two hydrogen atoms through sigma-bonds as usual to form the CH2 groups. The remaining two hybrid orbitals on each carbon atom undergo an angular overlap sharing the common edge of the two different tetrahedra

Introduction to Chemical Bonding and Theories of Covalence

around the adjacent carbon atoms to produce two bent or banana bonds, above and below the plane

H,

~H

H

H

H

H H

H,~/H

c·

H

(b)

(a)

C

H/ " ' . / 'H

~

H-C~C-H

~

ry

strain, the molecules are believed to be more reactive comparedto the saturated hydrocarbons where no such strain exists. In the same way, the

/1T'\

)40f( \J/

ofthe rest ofthe molecule (see Fig. 9.18.2.2). Due to the bent bonds the molecule remains in strain which is measured by the deviation of bond angle from the tetrahedral bond angle. Because 0/ this

719

t.m

e/

th

ea

lc

he

m

yl

ib

ra

(c) (d) triple bond, C ==C can be explained through the bent bond. The bent bonds are very often Fig. 9.18.2.2. The bent or banana bonds in C2 H4 (a, b, c) and C2 H2 (d). represented as shown in Fig. 9.18.2.2. This bent bond concept originating the strain in the molecule has been invoked in explaining the higher reactivity of the tetrameric forms of the elements like P4' As4, Sb 4' ~tc. The M4 species is tetrahedral (see Fig. 9.18.2.3) having the bond angle M -M- M = 60° where three M-atoms occupy the corners of a triangular face of the tetrahedron. Pure p-orbitals can make the minimum bond angle 90°. Hence, to shrink the (8) (b) bond angle more, it requires the participation of Fig. 9.18.2.3. The angular orbital overlap the d-orbitals through the hybridisation with the interaction in the tetrameric forms of the elements p-orbitals. But such a hybridisation involving the like P4' As4 , Sb4 (a), only three atoms occupying higher energy d-orbitals is not energetically the corners of a triangular face of the tetrahedron possible when the element remains in zero are shown); and in cyclopropane (b). oxidation states. Thus no suitable hybrid orbital can be energetically generated to explain this bond angle. This is why, it is believed that the p-orbitals on each M undergo an angular overlap instead of overlap along the internuclear axis present in the sigma-bonds. This type of angular overlap to form the bent or banana bonds introduces a strain in each M - M bond. This strain makes the species reactive. Similarly, bent or banana bonds are also existent in cyclopropane (see Fig. 9.18.2.3).

(8) liD-description of the Unsaturated Hydrocarbon By considering the concept of localIsed MO (i.e. 2-centred MO), we reach almost the same conclusion drawn in the VBT. In C2H4 , according to MOT, each sp2 carbon atom forms two sigma-type localised MOs with two hydrogen atoms (providing Is orbital) along with a sigma-type localised MO between the carbon atoms by using the sp2 hybrid orbitals. These five sigma-MOs will contain ten electrons. The corresponding five antibonding sigma-MOs (i.e. a*-MO) will remain vacant. In addition to this,. the two Pz orbitals (assuming the molecule to remain in the xy-plane) of the two carbon atoms fonn a bonding 7t-MO and an antibonding 7t-MO, in whi~h the bonding 7t-MO contains tw