WHY THE HARTREE-FOCK Fk INTEGRALS ARE TOO LARGE

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Submitted on 1 Jan 1970

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WHY THE HARTREE-FOCK Fk INTEGRALS ARE TOO LARGE

M. Fred

To cite this version:

M. Fred. WHY THE HARTREE-FOCK Fk INTEGRALS ARE TOO LARGE. Journal de Physique

Colloques, 1970, 31 (C4), pp.C4-161-C4-162. �10.1051/jphyscol:1970426�. �jpa-00213881�

JOURNAL DE PHYSIQUE Collocl~ie C4, sirppk;tllct~t air 110

1 1- 12. Totllc 3 1. N o r . DPc. 1970,

C4-161

WHY THE HARTREE-FOCK Fk INTEGRALS ARE TOO LARGE (*)

M. FRED

Chemistry Division, Argonne National Laboratory, Argonne, Illinois 60439

RCsume. -

Les integrales FGvaluees a partir des fonctions propres radiales Hartree-Fock sont d'ordinaire superieures d'environ 50 % a celles qui sont tirees de I'experience

cela vient de ce qu'on neglige I'interaction de configuration et d'autres effets dans le calcul HF. Puisqu'il est difficile de faire une thkorie complete, il semble utile de considerer les proprietes des solutions HF plus directement. Slater a discute la relation non hydrogeno'ide entre la valeur propre et la grandeur de la fonction

le

effectif pour la fonction est plus grand que le Z effectif pour l'energie. Le rapport des <

> evalues pour les 2 valeurs de

reproduit le rapport Fk calculC/FL-' experimental.

J1 en resulte que les fonctions ne peuvent pas Ctre utilisees directenlent comnie mesure de I'Cnergie mais qu'elles peuvent Ctre corrigees assez simplement.

Abstract. -

The F t integrals evaluated from Hartree-Fock radial eigenfunctions are usually about 50 per cent larger than those derived experimentally,

fact ascribed to neglect of configuration interaction and other effects in the

calculation. Since a complete theory is difficult it appears useful to consider the properties of the

solutions more directly. Slater has discussed the non- hydrogenic relationship between the eigenvalue and the size of the function

the effective Z for function is larger than the effective

for energy. The ratio of <

> evaluated for the two Z values reproduces the

ratio. Hence the fitnctions cannot be used directly as a measure of energy but may be corrected rather simply.

This paper is concerned with the relationship between Hartree-Fock eigenvalues and eigenfunc- tions. The content is not at all new

it is discussed, for example, by Slater [I]. But it seems to m e that the consequences are usually ignored.

I t will be recalled that the usual Hartree-Fock calculation is made for the center of gravity of a configuration. T h e term structure is then described in terms of the Slater radial integrals F h n d Gh which are evaluated by first order perturbation theory as the integral of the electron-electron repulsions over the zero order radial functions. There are several reasons why the F' are not accurate. In the first place the different terms have different energy

as one goes u p in energy the radial functions would be expected to expand, and s o the

for the highest term are smaller than for the lowest term. This efyect is, however, small

of the order of a few per cent. A more serious limitation is the fact that the term structure is not at all a small perturbation

the repi~lsion energy is compa- rable to the binding energy and so first order perturba- tion theory is not very good. One has t o go t o high orders of perturbation, which is tedious and gets to be unphysical. But the most important limitation, it seems

(*)

Based on work performed

auspices of

S. Atomic Energy Commission.

to me, is the fact that for the Hartree-Fock radial func- tions distance a n d energy are not inversely related. T o compute the Coulomb repulsion we integrate

But this does not work for the interaction energy of each electron with the nucIeus

Some numerical values are listed in the table. So I don't see how the radial functions can be used directly t o evaluate the electron-electron repulsion energy.

Slater discussed this in terms of the effective nuclear charge Z* as a function of r, the distance from the nucleus. F o r hydrogen-like functions the scale of length is proportional to I/r and s o for the Hartree- Fock we can define Z* as < ^r >,,/< ^r

We can also define Z* in terms of energy as E

(Z*)2/n2.

For hydrogen-like functions the two Z's are the same

for Hartree-Fock functions they are not because of outer screening. Z* for the scale of the function is larger than Z* for energy, that is, the function is compressed toward the nucleus compared with a hydrogen-like function with the same energy.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970426

C4- 162 M . F R E D

In order to compute F"s which are in fact propor- tional to energy it would seem more reasonable t o use radial functions having a scale of length which does correspond to energy. We can define new radial functions which have the same Z* for both <

>

and E. This can be done graphically (figure 1) from plots of z:,, ^and z,* as functions of r. ( I t could be argued that the new functions are not legitimate because they are not eigenfunctions of the Hartree-

tionship < r > Fh

constant. The constant can be evaluated from the original Hartree-Fock < r >

and F" and the revised F h is then obtained from the new < ^r >.

Table I gives data for two examples, one good, one not so good. Evidently the reason the two cases do not give comparable results is that on ionization the eigenvalue increases quite a bit whereas the 5 f radial functions change very little, so the eigenvalue is really not a proper measure of how to adjust the functions. A better criterion would be Fo(ff) but it is not evident how it can be used. But I think these examples illustrate some deficiencies in the radial functions which seem to be responsible for errors in the F ~ .

5 f eigenvalue 0.75 1.01 a. u.

< r >,; ^{3.61 2.99}

Hartree-Fock results for Pu I 5 ^{f 5} 6 d 7 s2. The points plotted

are values of the effective nuclear charge against the square

~ i - ~ ^{85 422. 82 100.} cm-'

root of the expectation value of r for each kind of electron

(r* is used instead of r merely to give a more convenient slope).

< ^r

~ f i - ~

The two sets of curves refer to Z* for scale of radial function

100 456. 99 834.

and for energy. The point A indicates

<

>

having a Z* corresponding to the 5 f eigenvalue (see text).

27 827. 33 389.

F 2 experimental 24 448. 51 236.

Fock equations, but the eigenfunctions are not

legitimate, either, for energy calculations.) It is I am indebted to D r Michael Wilson for pointing unnecessary to obtain the complete new function out to me the reciprocal relationship between < ^r >

because < ^r > is sufficient owing to the rela- and F k .

[l]

(J.

Quantum Theory of Atomic Structure

McGraw Hill Book

New York, 1960.