Hyperspherical Harmonics and Generalized Sturmians 9780306469442, 0306469448

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Table 2.2 Hydrogenlike atomic orbitals and their Fourier transforms W N—[, W N —U

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CHAPTER 2. MOMENTUM-SPACE WAVE FUNCTIONS

31 Table 2.4 Alternative 4-dimensional hyperspherical harmonics

32

CHAPTER 2. MOMENTUM-SPACE WAVE FUNCTIONS

([HUFLVHV 1. Calculate the integral in equation (2.6) and show that it yields the result shown (2.7). 2. Starting with J calculate the integrals J and J by differentiating with respect t o kµ , as shown in equations (2.8) and (2.9). 3. Starting with J, use the recursion relation of equation (2.11) t o generate J, J and J. 4. Use the integrals JVO, in Table 2.1 to evaluate the Fourier transform of' the direct-space hydrogenlike orbitals and Show that the transforms correspond to the solutions of Fock, equation (2.15).

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 Then from (6.44), (4.53), and (6.46) and we have: (6.47) where R

X2

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With the help of (6.44) and (5.74), we can also obtain the terms which are off-diagonal in the indices a1 and a:

(6.49) while from (5.14) and (5.15),

(6.50)

A curve representing =

as a function of R = s/kµ can be generated by substituting these values of K² and K into equation (6.32). Such a curve, with 3 basis functions on each center, is shown in Figure 6.1, compared with the nearly exact, results of Koga and Matsuhashi [190-192]. Comparison shows that equation (6.32) yields a more accurate result with 3 basis functions on each center than could be obtained from (6.30) with 15 basis functions on each center. The numerical values of the ground-state energies for obtained using equation (6.32) with 3 basis functions on each center are shown in column a of Table 6.1, compared with the extremely high-precision results of Koga and Matsuhashi (column b ) and the best available direct-space results (column c).



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Chapter 8

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Solution

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Solution

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Exercise 4.3 From equation (4.38) and the associated Laguerre polynomials in Table 4.2, calculate the parabolic hydrogenlike orbitals shown in Table 4.3. Express these functions as linear combinations of

Solution Let t

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= r–z



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The inequality K ² (K)² is due to the fact that the basis set is truncated, so that the sum

does not run over all possible values of

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using Table 5.2 and the definition

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([HUFLVH  Use the results of Exercise 7.1 to calculate the Sturmian overlap integral

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([HUFLVH    Calculate the integral I of Exercise 7.2 using the ellipsoidal coordinates = (ra + = (ra – rb) /R and ø where = [ j . [ j and = . (x j + 5) (x j + 5 ) and where ø has its usual meaning. In ellipsoidal coordinates, the volume element is given by

Compare your answer with the results of Exercise 7.2. Could ellipsoidal coordinates be used to calculate Shibuya-Wulfman integrals?

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160

Ellipsoidal coordinates also offer an alternative method for evaluating Shibuya-Wulfman integrals.

Exercise 8.1 Show that for j = 1/2, l = 0, and M = 1/2, the 4-component solution to the hydrogenlike Dirac equation can be written in the form:

What is the form of the solution corresponding t o j = 1/2, l = 0, and M = –1/2?

Solution From (8.8) we have:

while from (8.9) with l = j – l = 1,

161 Therefore (8.5) yields

Similarly, when

= _ 1/2

Exercise 8.2 Letting bµ = 1, find the values of k , nr , to (8.1) in the n = 1 and n = 2 shells.

N , and

for the solutions

Solution From equations (8.12)-(8.15) and (8.19) we obtain:

Exercise 8.3 The energies calculated in Exercise 8.2 include the electron rest energy mc² and are expressed in units of mc². Subtract the rest energy from the calculated values, and express the results in Hartrees.



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