RELATIVISTIC ATOMIC WAVE FUNCTIONS FOR OPEN SHELLS

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RELATIVISTIC ATOMIC WAVE FUNCTIONS FOR OPEN SHELLS

D. Mayers

To cite this version:

D. Mayers. RELATIVISTIC ATOMIC WAVE FUNCTIONS FOR OPEN SHELLS. Journal de Physique Colloques, 1970, 31 (C4), pp.C4-221-C4-224. �10.1051/jphyscol:1970435�. �jpa-00213890�

JOURNAL DE PHYSIQUE Colloque C4. .supplément au /;" 11-12. Tome 31. Nov.-Déc. 1970. page C4-221

RELATIVISTIC ATOMIC WAVE FUNCTIONS FOR OPEN SHELLS

D . F . M A Y E R S

Oxford University C o m p u t i n g L a b o r a t o r y Oxford, England

Résumé. — Cet article décrit une méthode simple permettant d'adapter l'approximation rela- tiviste de Hartree-Fock à des configurations comprenant des couches incomplètes. Une couche de nombre quantique angulaire /se sépare par effet relativiste en deux sous-couches pour lesquelles j = / + i et j = / — V, et les électrons peuvent se distribuer de plusieurs façons différentes entre ces sous-couches, donnant ainsi naissance à plusieurs configurations différentes, ayant des énergies presque égales. On étend la notion de moyenne de configuration et l'on donne une moyenne pon- dérée de tous les termes apparaissant dans ces configurations. Les fonctions d'ondes de Hartree- Fock sont alors déterminées pour cette configuration moyenne ; la matrice de l'hamiltonien complet peut alors être déterminée et les niveaux d'énergie calculés. Des résultats sont donnés pour la configuration de base de l'azote (2 p)-'.

Abstract. — This paper describes a simple method of adapting the relativistic Hartree-Fock approximation to configurations involving open shells. A shell with angular quantum number / splits relativistically into subshells with j — I ~ i and j — I — -I, and the electrons can distribute themselves between these subshells in several different ways, giving rise to several different confi- gurations, having very nearly equal energies. The idea of an average of configurations is extended to give a weighted average of all the terms arising from all these configurations. The Hartree Fock wave functions are then determined for this average configuration ; the full Hamiltonian matrix can then be determined, and the energy level scheme found. Results are given for the (2 p)-1

ground configuration of Nitrogen.

I. Introduction. — Recently there have appeared a number of reports of calculations of atomic wave functions using the relativistic Hartree-Fock equations.

Most of these calculations deal with configurations involving closed shells, or one electron outside closed shells, since open shells present a problem in the relativistic formulation. When using relativistic one- electron wave functions, one is forced to use /-/

coupling, so for light atoms in which LS coupling gives a good approximation to the energy levels the relativistic scheme is in difficulties.

As a simple example, consider a (2 p)3 configuration, as in the ground state of the Nitrogen atom. In the relativistic formulation the 2 p shell divides into two subshells, which we shall represent as the (2 p*) subshell, having / = 1/2, and the (2 p) subshell having j = 3/2. The 3 electrons can then be distributed between

these subshells, giving rise to the scheme

The three terms with / = 1 will have nearly the same energy, since relativistic effects are likely to be small in such a light a t o m , and hence there will be considerable mixing between them.

II. The average of configurations. — As a first step towards a full mixed-configuration calculation it therefore seems useful to extend the idea of the

« average of configurations » developed by Slater [I]

to such a situation. In the ordinary non-relativistic case the (2 p)3 configuration gives rise to three terms,

4S , 2D and 2P . The average of configurations gives these terms the weights 4, 10, and 6 respectively, and then takes the weighted means of the energies. In general the terms are weighted by the number of corresponding levels (2 L + 1 ) . ( 2 S + I). The same idea can be used to define an average for the (2 p*) (2 p")2 configuration, giving each term the weight (2 7 + 1). Finally, it is a simple matter to define a weighted average of all the terms, the total weight being 20, as before.

The simplicity of the average of configurations approach, of course, arises from the fact that it is not necessary to determine the energies of the terms separately and then take their weighted mean. The same result is achieved by taking the mean of all the single-determinant wave functions arising from the configuration ; in this case there will be 20 of them.

When summing the interactions between one electron and all possible states of a second electron, the sum is independent of the q u a n t u m number m of the

C'J-211 D. F. MAYERS

lirst clcctron. Hence we find simple rules for deter- mining thc energy of the average configuration.

For interactions between difrerent shells, we take thc rcbult for closed shells, and reduce i t in proportion to thc numbers of electrons occupying the shells. If C is thc interaction for the closed shells. containing (2,j, + I ) and (Li,

+

\vill be C . q I ( I ? / ( ? ,jl

+

I n tllc same way. for interactions within a shell, the result is proportional to the number of interacting pairs of electrons. tlerc we simply multiply the result I'or a closed shell by the factor q,(rl, - 1)/2.j1(2jl + ^1).

A general result is not difficult to derive, but here we shall consider only the simple (2 p)3 configuration.

The expression for the energy for the closed shell configuration (2 p*I2 (2 p)4 contains the terms

I Fo(2 P*. 2 p*)

+

+

2 4

- 5 F2(2 P, 2 P) - 3 G2(2 p*, 2 p)

.

These coefficients are given by Grant [2].

We then easily obtain the results

(2 P * ) ~ (2 p) weight = 4 : average energy =

- . - . . . . -

1 f o(2 p*, 2 p*)

+

(2 p*)(2 p)2 weight _.- = 6

+

+

energy =

(2 P)3 weight = 4 : average energy = --

The weighted average then becomes

1 8 6

- 5 Fo(2 P*, 2 p*)

+

+

2 4

- B F2(2 P, 2 P) - 5 G2(2 p*, 2 p)

.

In the non-relativistic limit the functions 2 p* and 2 p become the same, and this result reduces to

As we should expect, this agrees with the usual non- relativistic formula for the average of the (2 p)3 configurations.

The coeficients of the other integrals in the expres- sion for the energy are easily obtained in the same way.

For example, the coefficient of 1(2 p*) is simply the number of 2 p* electrons ; the required value is therefore the weighted average of the coefficients 2, 1 and 0, giving 1. For completeness we can now givc the t'i111 expression for the average energy of the ground state of Nitrogen.

1 8

+ 4 Fo(2 s, 2 P)

+

+

2 4

- j GI(' s, 2 p) - 25 G2(2 p*, 2 p) .

111. Energy levels. - From such an expression for the average energy it is a simple matter to set up and solve the relativistic Hartree-Fock equations. This computation uses the standard methods as applied to configurations of closed groups, such as are des- cribed by Desclaux, Mayers and O'Brien [3]. Having obtained approximate wave functions, the various Slater integrals are easily computed, and we can obtain values for the various energy levels.

We have already obtained expressions for the J = .$

terms of (2 P * ) ~ (2 p) and (2 P ) ~ , since these are the only terms which arise from these configurations.

The (2 p*) (2 p)2 is slightly more complicated, and the simplest method here seenls to be to construct one eigenfunction of each term, then forming the full Hamiltonian matrix. In such a simple case, it is sufficient t o use the standard methods involving cc step-down ⁾⁾ operators as described by Condon and Shortley [4] ; in the relativistic case the calculation is somewhat simpler, as we need only eigenfunctions of the total angular momentum J, instead of simulta- neous eigenfunctions of L and S.

We denote the single determinant wave functions by giving the values of twice the quantum number rnj as a subscript. The required eigenfunctions are then easily found to be :

(2 P*) (2 pI2

RELATIVISTIC ATOMIC WAVE FUNCTIONS FOR OPEN SHELLS C4-223

The Hamiltonian matrix then takes the form

2 ^' A , , 0 0 0 0

0 A,, 0 0

- J $ P : ) ( ~ P ~ ) ( ~ P - , ) .

(2 P*I2 (2 P) A34 A44 A 4 5

--

0 0 A35 A45 A55

J = 5 (2 P:) (2 P*- 1) (2 ~ 3 ) ^, In calculating the elements of this matrix it is impor- tant t o remember that we must include all terms involving the incomplete groups ; for example, the coefficients of 1(2 p) and Fo(l s, 2 p) are not the

J = - 3 (2 P3) (2 PI) (2 P- I)

-

2 displayed in the following table.

A34 = A35 =

J2

A 4 5 = - 3 G2(2 p*, 2 p) In this table the colun~ns give the coefficients of the

various integrals corresponding to the diagonal ele- ments of the matrix. The off-diagonal elements each contain only one integral, and are given at the bottom.

In the last column appear the numerical values of the integrals, calculated from the Hartree-Fock wave functions ; they are given in atomic units. The matrix then becomes

Average

-

2 2 1 2 1 4 2 4 1 2 4

115 815 615 - 2/25 - 2

- 113 - 213

- 113 - 213 - 4/25

Value

- 24.445 7 - 5.321 5 - 4.773 6

- 4.770 0 4.129 3 0.975 19 0.932 78 0.931 83 0.687 33 0.662 49 0.662 19 0.642 02 0.641 72 0.641 43 0.286 48 0.064 72 0.084 33 0.084 17 0.406 15 0.406 01 0.286 54

from the diagonal terms. It is a curious fact that in this case the energy of the J = 5 term is equal to the average energy. We can now find the eigenvalues and eigenvectors of this matrix t o determine the energy levels.

Two of the eigenvalues are, of course, 0 and 0.068 7.

The other three are found t o be 0.068 8, - 0.000 1 and - 0.103 1. We thus obtain the level scheme

where we have subtracted the average energy although the fourth decimal place is not reliable. As - 54.327 6 a. LI. , might be expected for such a light atom, this sclien~e

C4-221 I). F. MAYERS

corresponds closely to that obtained from a non- matrix. I t is also a simple but tedious matter to relativistic treatment. which gives a ", a 'D and include the Breit interaction, if required, by use of the a 4S term. The important point to notice is that in methods described by Grant ['I. In nitrogen the efTect this relativistic treatment the lowest term does not is small but not negligible, of the order of 0.005 a. 11.

correspond to a single determinant : the eigerivcctor It is now hoped to extend these methods to larger

has elerncnts iitorns, and to more complicated configurations, in

(0.00 0.00 0.75 - 0.47 - 0.47) . which departures from LS coupling arc more inipor- tant.

The relativistic efrects in this example are too small Acknowledgements. - It is a pleasure to ackno\\;- to be shown at all accurately. However, the results do ledge the contiti~~ed encouragement and support of give some confidence in the correctness of the treat- Professor Moser in this work, and also the help and nient. C o n i p ~ ~ t e r programs are now available to calcu- criticism of M . J. P. Descla~~x. The opportunity to late wave functions and Slater integrals by solving conipare the results from our similar relativistic the Hartree-Fock equations, and also for computing Hartree-Fock programs has enabled me to eliminate the coefficients of these integrals in the Hamiltonian a number of serious errors.

References

[ I ] SLATER ( J . ) , Q~rnrrtrtm Theory of'Alatnic S I ~ I I C I I I I . ~ , VOI. 1 [4] CONDOK (E. U.) and SHORTLEY ( G . U.), The TJ1c~o1.j.

and I[, (New York, McGraw-Hill), 1960. of Atomic Spectra (Cambridge University Press), [2] GRANT (I. P.), Proc. Roy. Soc., 1961, A 262, 555-576. 1953, Chapter 8, section 5.

[3] DESCLAUX (J. P.), MAYERS (D. F.) and O'BRIEN (F.), To be published.