Hyperspherical Harmonics and Generalized Sturmians

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3 construct the wave functions of more complicated systems  for example manyelectron atoms or molecules. However, it was soon realized that unless the continuum is included, a set of hydrogenlike orbitals is not complete. To remedy this defect, Shull and Löwdin [273] introduced sets of radial functions which could be expressed in terms of Laguerre polynomials multiplied by exponential factors. The sets were constructed in such a way as to be complete, i.e. any radial function obeying the appropriate boundary conditions could be expanded in terms of the Shull-Löwdin basis sets. Later Rotenberg [256, 257] gave the name “Sturmian” to basis sets of this type in order to emphasize their connection with Sturm-Liouville theory. There is a large and rapidly-growing literature on Sturmian basis functions; and selections from this literature are cited in the bibliography. In 1968, Goscinski [138] completed a study of the properties of Sturrnian basis sets, formulating the problem in such a way as t o make generalization of the concept very easy. In the present text, we shall follow Goscinski’s easily generalizable definition of Sturmians. Generalized Sturmian basis sets can be understood from the following simple argument: Suppose that we are able to find a set of solutions to the wave equation

Here

is the generalized Laplacian operator and [ = ^x x xG` are the mass-weighted Cartesian coordinates of the particles in a system. If we are discussing a single particle, then d = 3, while if we are discussing an NSDUWLFOH system, d = N. We let ßv be a set of constants chosen in such a way that the functions all correspond to the same value of energy, ( regardless of their quantum numbers. Then it is easy t o show, using the Hermiticity of the operator

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Acknowledgements I am extremely grateful to the Carlsberg Foundation for travel grants which allowed me to make several visits to the laboratory of Prof. Dudley Herschbach at Harvard University. I would also like to thank the Danish Science Research Council which made possible two periods of research at the laboratory of Prof. Vincenzo Aquilanti at the University of Perugia, Italy, and also a short visit to the laboratory of Prof. Jamie Fernández Rico in Madrid. Prof. Fernández Rico and Dr. Rafael López have generously allowed me to use their computer program for evaluating many-center two-electron integrals over Slater-type orbitals. It has been an extremely great pleasure for me to receive encouragement from two of the most important pioneers of momentum-space quantum theory  Prof. Roy McWeeny and Prof. Carl Wulfman. I am also grateful to Professors Mario Raimondi, Robert Nesbet, Luis Tel, Jens Bang and Jan Vaagen for encouragement and for stimulating discussions. Finally, I would like to thank my co-workers, Dr. Frank Antonsen, Dr. Wensheng Bian, Prof. John Loeser, Cand. Scient. Tom Børsen Hansen, Stud. Scient. Rune Shim, Dr. Cecilia Coletti, and Assist. Prof. Stephan Sauer for their important contributions.

Chapter 1

MANY-PARTICLE STURMIANS Suppose that we are able to find a set of functions øv[ which are solutions to the many-electron equation: (1.1) In equation (1.1),

is the generalized Laplacian operator (1.2)

while [ represents the set of coordinate vectors for all the electrons in the system: [ = {;1, [2, ..., ;1} (1.3) and [^xJ, yJ, zJ` j = 1,2, ..., N (1.4) The potential, V [ , is chosen to be the Coulomb attraction potential due to the nuclei in the system. The simplest case is that of an Nelectron atom or ion, where

(1.5) 7



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9 and so on. In equation (1.6), µ represents the set of quantum numbers ^n, O, m, s`. The functions satisfy the relationships: (1.11) (1.12) and (1.13) We now introduce the subsidiary conditions (1.14) and (1.15) Provided that the subsidiary conditions are satisfied, the product shown in equation (1.6) will be a solution to (1.1), since

(1.16) It can easily be seen that an antisymmetrized function of the form

(1.17)

will also satisfy (1.1), since the antisyrnmetrized function can be expressed as a sum of terms, each of which has the form shown in equation



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11 Combining (1.21) and (1.22), and making use of the orthonormality of the spin functions and the spherical harmonics, we obtain the potentialweighted orthonormality relations: (1.23) where v stands for the set of quantum numbers {µ, µ', µ'', ...}.

The secular equation Having constructed a set of many-electron Sturmian basis functions by the method just described, we would like t o use this basis set t o solve  the Schrodinger equation for an NHOHFWURQ atom. Using atomic units,  we can write the Schrodinger equation in the form: (1.24) where (1.25) Here V  [ is the nuclear attraction potential shown in equation (1.5), while (1.26) is the interelectron repulsion potential. Expanding the wave function as series of generalized Sturmian basis functions, and making use of equation (1.1), we obtain:

(1.27) Multiplying (1.27) from the left by a conjugate basis function, integrating over the coordinates, and making use of the potential-weighted orthonormality relation, (1.23), we obtain: (1.28)



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CHAPTER 2. MOMENTUM-SPACE WAVE FUNCTIONS

31 Table 2.4 Alternative 4-dimensional hyperspherical harmonics

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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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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:

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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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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 and

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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

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159

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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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