AN EXTENSION OF THE HYLLERAAS EXPANSION FOR THE HELIUM GROUND STATE

HAL Id: jpa-00213873

https://hal.archives-ouvertes.fr/jpa-00213873

Submitted on 1 Jan 1970

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

AN EXTENSION OF THE HYLLERAAS EXPANSION FOR THE HELIUM GROUND STATE

W. Somerville

To cite this version:

W. Somerville. AN EXTENSION OF THE HYLLERAAS EXPANSION FOR THE HE- LIUM GROUND STATE. Journal de Physique Colloques, 1970, 31 (C4), pp.C4-107-C4-109.

�10.1051/jphyscol:1970418�. �jpa-00213873�

JOURNAL DE PHYSIQUE C o ~ ~ o q l l e C4. sllpl,~c'r)?ent a11 1 1 ~ 1 1 - 12, To117c 3 1. Nor.-Dkc. 1970. page C4-107

AN EXTENSION OF THE HYLLERAAS EXPANSION FOR THE HELIUM GROUND STATE

W. B. SOMERVILLE

Astronomy Department, University College London, U. K.

Resume. - Dans le travail de Hylleraas, on a utilise des expressions variationnelles contenant (rl + rz), ( r ~ - r2) et r12 seulement aux puissances entieres. H. M. Schwartz a introduit des puis- sances demi-entikres. Nous avons ici discutk l'utilisation de nombres reels generaux. Les premiers rksultats obtenus suggerent que cette methode peut servir.

Abstract. - In the work7of Hylleraas, variational terms containing only integer powers of (rl + ^{rz), (rl - r2) and rl2} were used. H. M. Schwartz introduced half-integer exponents. The use of general real-number exponents is discussed here. The first results suggest that the method could be of value.

I n the original work of Hylleraas [I ] on the use of variational trial functions of the form

for the helium atom, the exponents I, n1 and ¹¹ were restricted t o be positive integers. I n later investiga- tions the same form was used, with greater numbers N of terms in the variational expansion. This work with integer exponents reached its climax in the calculations of Pekeris [2]. I t has been reviewed by Stewart [3]

and Somerville [4]. Kinoshita [5, 61 introduced nega- tive exponents. H. M. Schwartz [7, 8, 9 ) and C . Schwartz [lo] showed that the use of terms with half-integer exponents increased considerably the rate of convergence with increasing N. I n particular, the 164-term function of C. Schwartz gave the same variational energy as the 1078-tern1 f ~ ~ n c t i o n of Pekeris. H. M. Schwartz also considered some terms with quarter-integer exponents.

The present investigation considers the use of terms with general real numbers as exponents. This could be expected to effect a further improvement in the rate of convergence. When real numbers are to be used, the problem of how to select the numbers arises.

One could choose the real-number exponents to be simple rational fractions, o r one could take the values of a set of pivots used in Gaussian quadrature, for example. Such schemes could give a simple prescrip- tion for increasing the number of terms N. It was decided, however, t o try to find the real-number exponents which are best t o use, in the sense of giving the lowest variational energy. The quality of a wave function may be judged in various ways ; here simply the energy criterion is used. Consideration is restricted to trial functions of the form (I).

It might be felt that the use of irrational numbers is in some way less 'physical' than the use of integers.

This is not so. A wave function with asymptotic form

say, for large r , , is n o more unphysical than one with asymptotic form e-"'I

":.

I n previous calculations, terms o f various forms were added to the basic expansion in terms with integer exponents. Here, we consider whether the function might be improved by replacing some of the integer exponents by non-integers, for some of the basic terms ^- rather than add new terms, we consider the form of the terms already present, to see how they can be improved. The exponents become effecti- vely additional variable parameters. By examining each term (1, 171, 11) for variational functions with small N, it is hoped to learn what terms should be included in functions with larger N, where it is not practical to vary all these real-number exponents directly.

The theory is the same as for the case of integer exponents, discussed in detail by Bethe and Sal- peter [ I l l . The computational methods used are discussed elsewhere [4]. It is found rather important to minimise properly with respect to the non-linear parameter, the scale factor k. This can be done most conveniently by interpolation.

The exponent ⁿⁱ must be an even integer, if the wave function is to remain real. There is no reason why complex wave functiorls should not be considered,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970418

C4- 108 W. B. SOMERVILLE but that would make the algebra much more compli-

cated and to start with it seems best to restrict conside- ration to real functions. Preliminary computations suggested that variations in tz are more important than variations in I. In the work reported here, the terms (I, ^{1 1 2 ,} n ) ⁼ (0, 0 , 0) and (0, 0, 1) have been considered as they occur [4] in the best integer- exponent wave functions with N = 1 to 7, and the effect of changing the n values from 0 and 1 examined, in the three cases :

Case A : vary n in (0, 0, 0) only ; Case B : vary rz in (0, 0, 1) only :

Case C : vary n in (0, 0, 0) and in (0, 0, 1).

Results are presented in Tables I, I1 and 111, for the energies, the optimum values of n and the optimum values of the scale factor k. The energy and the optimum k are continuous functions of n ; this is illustrated in the figure.

Energiesfor the groundstate of the Heliirm atom, in a. u., with N terms in the variational expansion

- N 1 2 3 4 5 6 7 8 Exact

Integer n - - 2.847 656 -2.891 121

- 2.902 432 - 2.902 695 - 2.903 302

- 2.903 370 - 2.903 480

- 2.903 529 - 2.903 724

Case A

-

- 2.880 384

- 2.891 186 - 2.902 724

- 2.902 941 - 2.903 377

- 2.903 441

- 2.903 490

Case B

- -

- 2.891 161

- ^{2.902 744}

- 2.902 967

- 2.903 429

- 2.903 487

- 2.903 497

Case C

-

/

- 2.891 187 - 2.902 754 - 2.902 973 - 2.903 429

- 2.903 487 - 2.903 497

TABLE I1 Values of the exponents n for the energies in Table I

Case A Case B Case C

- -- -

n1 ¹¹²

0.163 - - -

0.012 0.949 0.01 5 1.017

0.027 0.826 0.01 1 0.881

0.026 0.835 0.009 0.878

0.018 0.741 0.000 0.741

0.016 0.773 0.000 0.773

0.007 0.857 0.000 0.857

TABLE I11

Values of the scale parameter k

N Integer n Case A Case B Case C

The dependence of the energy E and the optimum scale factor k on the exponent n, for the 1-term function (0, 0, 12).

The improvement in the energy is most spectacular for the 1-term case. The small amount of correlation introduced by having the wave function

instead of

improves the energy considerably beyond the Hartree- Fock value - 2.861 7. The importance of correlation decreases as Z increases ; this is clearly seen from the results in Table IV. The 2-term integer-exponent function (0, 0, 0) + c(0, 0, 1) shows the same beha- viour. For large 2, the optimum n behaves as 112.

TABLE 1V Variational energies for d~fferent Z , in a. u.

Integer n, Integer ^11, Variable n,

Z 1 term 2 terms 1 term n

- - ^- - -

1 - 0.472 7 - 0.508 8 - 0.503 3 0.430 2 - ^{2.847 7} - 2.891 1 - 2.880 4 0.163 3 - 7.222 7 - 7.268 2 - 7.255 6 0.099

A N EXTENSION OF THE HYLLERAAS EXPANSION FOR TI-IE HELIUM G R O U N D STATE CJ-10')

The best ?-term integer-exponent function is (0, 0, 0) + c(0, 0, 1) and it is this which has been used as the basis for the entries in the tables. However, a much better energy, - 2.897 031 a t k = 3.516, comes from the function (0, 0, 0.135) + ^c(0, 2, 0.057).

This particular mixture of (r, - ^r2) and ^{r I 2} repre- sents the true function much better than d o terms with r , , only. For larger N , the best integer function already contains the term (0, 2, 0).

In general, the optimum 17 depends on N and on how many exponents are allowed to vary. The results suggest that, for (0, 0, 0), while departures from ^{iz =} 0 are important if n o other n is non-integer, it would seem better for large N t o leave this term with n = 0.

This is not a surprising conclusion, for the basic nature of the wave function is spherical symmetry with correlation distortions. F o r functions with N 3 5, the term (0, 0, 2) is present. I n the case of (0, 0, l), it seems that it is in general good to replace ¹¹ = 1 by n 1.0.8. This conclusion is still tentative. From considerations of this sort we (nay hope to see what are the best terms to include in more con~plicated trial functions.

In several cases, by varying one exponent we have obtained a n energy better than the best integer-

exponent function which has one term more. So we have the result that a cc~ntinilously-variable exponent can be an effective variational parameter.

It is important that the value of k should be fairly close to its o p t i n ~ u m value. For a given N, the various k in the table are quite close together, and this is seen also in the figure, but there is some change in k as N increases. The accuracy needed in k is less than that in E. The wave function is less accurate than the energy. It is also slightly undefinedin the calculation ; the interpolated k for the best E can depend on the original trial values of A-, beyond about the fourth decimal place.

The problem of obtaining accurate wave functions for the helium atom was solved with the work of Pekeris. There is still some interest in trying to obtain good accuracy with a smaller number of variational terms, both as a problem in itself and as a basis for calculations on more complex atoms. This must involve examining the form of each term which is included in the variational expansion, to make sure that the choice of terms is as efficient as possible.

While the present results are not absolutely conclusive, they suggest that the use of terms with real-number exponents could lead to a significant improvement.

References

[I] HYLLERAAS (E. A.), 2. Phys., 1929, 54, 347. [7] SCHWARTZ (H. M.), Phys. Rev., 1956, 103, 110.

[2] PEKERIS (C. L.), Phys. Rev., 1959, 115, 1216. [8] SCHWARTZ (H. M.), Phys. Rev., 1960, 120, 483.

[3] STEWART (A. L.), Adv. Phys., 1963, 12, 299. [9] SCHWARTZ (H. M.), P I ~ s . Rev., 1963, 130, 1029.

[4] SOMERVILLE (W. B.), Jourtlal of P I ~ i c s , to be publi-

shed. [lo] SCHWARTZ (C.), Phys. Rev., 1962, 128, 1146.

[5] KINOSHITA (T.), Plzys. Rev., 1957, 105, 1490. [ll] BETHE (H. A.) and SALPETER (E. E.), Handbuch der [6] KINOSHITA (T.), Phys. Rev., 1959, 115, 366. Physik, 35, Springer, 1957.