3Pe states of two-electron atoms. Feshbach-Rubinow approach

3Pe states of two-electron atoms. Feshbach-Rubinow approach

I. K. Dmitrieva (*) and G. I. Plindov (**)

(*) ^{A. V. Luikov} Heat and Mass Transfer Institute Byelorussian Academy ^of Sciences, ^Minsk 220728, ^U.S.S.R.

(**) ^{Nuclear Power} Engineering Institute, Byelorussian Academy ^of Sciences, Minsk, ^U.S.S.R.

(Reçu ^{le 19 dgcembre 1985,} accepti le 18 avril 1986)

Résumé.

Les expressions analytiques de la fonction d’onde et de l’énergie ^{de l’état 3Pe d’un atome} à deux élec- trons sont obtenues dans le cadre de l’approximation de Feshbach-Rubinow. Les valeurs numériques ^de l’énergie

ainsi obtenues sont

bon accord

les valeurs exactes. On montre que l’approximation ^{FR donne les valeurs}

exactes des deux premiers coefficients du développement ^de l’énergie

^{Z-1 et}

^bonne estimation du troisième coefficient.

Abstract

The Feshbach-Rubinow approach ^is applied ^{to obtain} analytical expressions for the energy and

function of the lower 3Pe state for helium-like atoms. The numerical results provide good estimates of the exact

eigenvalues. ^{The FR} approach ^{is shown to} give ^{the exact values of the two} first coefficients of the Z-1 expansion

for energies ^and

good estimate of the third coefficient.

Classification Physics ^Abstracts

31.90

The problem ^{of bound states} of two-electron atoms has a long history ^{and attracts} continuing ^interest

The variational method of solving ^this problem ^is

well known. In order to obtain accurate results, ^an appropriate trial function is chosen and a large ^number

of variational parameters ^is introduced. The number of parameters may be reduced provided ^{that the} approach used takes account of physical ^{features of}

the problem. One of the most advantageous ^{is the}

Feshbach-Rubinow approach [1] that allows one to reduce a three-body problem ^{to a} two-body ^one.

The modified FR approach ^when applied ^{to the} ground

state of a two-electron atom [2, 3] provided ^rather

accurate analytical energy and ^wave function expres- sions with the aid of but one or two variational parameters.

We shall show here that this approach ^{is also of}

success when applied ^{to excited states. We shall} analytically estimate the wave function and the energy of the lowest 3pe state for helium-like atoms with the

help ^{of a} single parameter.

Let us write down the Schr6dinger equation ^for

two electrons in the field of an infinitely heavy ^nucleus

with a charge ^{Z :}

Here _{ri , r2,} V2 1 ^and V2 2 ^are the radius-vector and the

Laplacian of the first and second electrons; _r12

r2 + ri - 2 rl-r2 ^cos 0 the distance between them.

Atomic units will be used in the paper.

It has been shown recently [4] that for the states with

a certain total angular ^{momentum J and} parity

the six-dimensional equation ⁽¹⁾ might ^{be reduced to}

a system of three-dimensional ones. For -3P’ states,

as well as for S-states, ^this system ^{reduces to one}

equation [4, 5]. ^{The form of the wave} function of the 3pe state

shows that in (1) the external variables, ^Euler angles a,

P ^{and y are} completely separated ^and (1) ^becomes

where

- - -1 - -

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019860047080131500

It can be easily ^{seen that in} (4) the variables are not

separated, ^thus hindering ^{an exact solution. We shall}

use the FR approach [1, 3] ^{in order to} approximately

estimate the eigenvalues ^and eigenfunctions ⁱⁿ equa- tion (3). ^The approach implies ^{that in} searching ^for

the variation of the function

the trial function f,(rl, r2, r12) is ^{chosen as} the function of a single-non-negative variable f = f (R) ^where

R

R(rl, r2, rl2). ^{Then, so} far It ^{is a} function of the variable R alone, integration ^{over two other variables}

may be performed ⁱⁿ (5). ^This approach describes well the ground ^state of atomic and nuclear systems [1, 3].

Direct application of the FR approach ^{to the} doubly-excited ^{pe state would cause} divergent ^inte- grals. ^{This is not} surprising ^{since the} singularities ^of

Hamiltonian (4) ^are omitted. If the singularities ^are

taken account ot ^the trial function _may be written as

Having ⁱⁿ mind that the 3Pe state is symmetric ^relative

to the permutation ’1

r2, it is convenient ^{to choose} the simplest ^form of R(rl, r2, r12) ^as

where a is the variational parameter. ^{As R} > 0, ^the

condition a > - 1 must hold

Performing the variation in (5), provided ^the integral f,(rl, r2, r12) f (ri, r2, r12) > ^is finite, yields

and integrating over r2 and r12, ^we arrive at

The main steps ⁱⁿ derivation of (8) and the values of the coefficients A(a), B(a), C(a) ^and F(a) ^{are cited in}

Appendix.

By writting T(R)

^{R III} 11 (R), ^we get ^the Schr6dinger-like equation ^to obtain the wave function and the energy of the 3Pe state of ^a two-electron atom

This is the wave equation ^of

particle ^{in the Coulomb field with an effective} charge

and an effective orbital momentum lee

7/2.

Note, ^{that the use of the FR} approach ^{for the} ground ^state of helium-like systems results in a similar equation

with an effective orbital momentum lee

3/2 [2, 3].

Solutions of (9) ^are well-known. Using ^{one of them,}

^{find now the} energy and wave function of the lowest

lpe state in an analytical ^{form as functions of a :}

Before minimizing (10) ^{over a, we} shall consider (10) ^{at a}

0. As it follows from R estimation, a

^{0 cor-} responds ^{to an effective} one-particle problem. Equation (10) implies ^{that at a}

^{0 the FR} approach yields ^the

energy of the 3pe state of a two-electron atom

that coincides with the value obtained within the screened Coulomb approximation.

Minimizing E(a) ^over

^{for a} given Z, ^{we obtain}

Table I.

Energies of the lowest ’P’ states for ^{He-like atoms.}

Having ^{found the roots of} (13) ^{in a} physically per- missible region - ^{1 a ,} 0, ^{we obtain the} energies

of He-like atoms presented ^{in table I. The} comparison

with the results of precision calculations shows that

(10) finely reproduces the energy of the ^{3pe state of}

two-electron atoms beginning from He. These data

resulting ^{from the} application ^{of a} single variational parameter ^{are more accurate} than the results of calculations based on the truncated diagonalization

method with hydrogen basis with thirty variational parameters [6].

High accuracy in describing the _{binding energy}

is due to a good quality ^{of the wave function on the}

average. It seems of interest to study the behaviour of the wave function in the vicinity ^{of the} points ^of

double electron-nucleus and electron-electron colli- sions. Equation (11) yields ^the expression ^at ri

0,

r2

0 or ri2

0

Here ^{is the root of} (13).

Comparison ^{with the} expansion ^{of the exact wave} function, 4PO, ^{of the 3pe state} [4] ,

shows that the trial function f,(rl, r2, r12) ^{well des-}

cribes the domain in the vicinity of the electron- nucleus collision point (ri -+ 0 or r2

0), being

Table II.

Expansion coefficientsof3pe wave function

at ri(r2) 0 ^and rl2 -+ 0’

satisfactory ^{in the} vicinity of the electron-electron collision point (Table II). ^The description ^accuracy

for double collisions increases, ⁱⁿ general, ^with increasing ^{Z. However,} for electron-electron collision,

in contrast to electron-nucleus one, the ultimate value

(at ^{Z -+} oo) of the coefficient 1/9 Zef(am).am ^accounts only for about 54 % ^{the exact value of} 1/4.

The analytical ^function E(a, Z) ^and f,(a, Z) ^allows

one to present ^the energy and wave functions of the 3pe state of helium-like ions in the form of a Z-’

expansion. Indeed, the solution of (13) ^for large ^Z

may be given ^as

Two terms of (16) ^ensure am estimation with ^{an error} less than 1 % ^{even for Z}

^1. Substituting (16) ^into (10)

and expanding (10) ^{over Z-1} gives

The zeroth and first terms of (17) coincide with exact values [7], while the second-order term, ^{due to the} variational character of the estimate, gives ^an upper limit for the value of 82 constituting ^94.5 % ^{of the}

exact value - 3.939

10-2 [7]. ^{It is worth} noting

that the first three expansion ^terms (17) yield ^more

accurate values of the _energy of the 3pe state than total variational estimate (10) (Table I). ^{This is} explain-

ed by ^the positive coefficient ~3 in the Z - ’ expansion.

We shall show now that the FR approach may

provide ^a reliable estimate for the leading ^correla-

tional contribution to the _energy.

The Z-1 expansion ^{similar to} (17) may also be obtained for the energy Eo ^{of the} ground ^{state of the}

He-like ions. Using ^the analytical ^function Eo(a, Z)

obtained within the FR approach [2]

and repeating ^the procedure ^{similar to} derivation of

(17) ^{we arrive at}

As for the 3Pe state, the ^{first two terms} of (19) ^coincide

with the exact values [7], ^{the third one} yields ^92.7 %

the exact value of 82. The first three ^terms in (19) ^once

more yield ^better energy values than the direct varia- tional estimate (Table III). ^An approximate ^estimate

of the leading correlation contribution, AE,., ^to

the ground ^state energy within the FR approach ^is equal ^to the difference of 8R(a".) ^and 8’(a

0) ^values

obtained for the effective one-particle consideration

(at

0) :

Table III.

Energies of ^the ground ^state of ^He-like

atoms.

This value is very close ^to the exact one

In (21), e2 is the ^exact value of the perturbation theory

second-order coefficient of Z - 1 obtained by very tedious variation-perturbation calculation [7]. t!1F ^is

the second coefficient value calculated within the Hartree-Fock method [8]. ^{It is seen that the error in} åEcor estimation within the FR approach ^{is of the}

same order as in 4t.

For the ’P’ _state, we fail to find the value of t!1F ⁱⁿ

the literature. By comparing (17) with the energy estimate in the framework of effective one-particle

consideration (12), ^{we find an} approximate ^{value of the} leading correlation contribution to the energy of the 3Pe state

We hope, ^that (22) gives ^{the value of} ð-Ecor, ^{as it is the}

case in (20).

So, ^{the FR} approach provides ^not only ^a fair estimate for the _energy of doubly ^{excited state} of helium-like

ions, but also reliable values of the first three coeffi- cients of the Z-1 expansion.

Appendix.

Now, substitute the variables

As f ^{is the function of the} variable R, ^we may integrate ^over r2

and _r,2

in (7). ^Because ft(R) is symme-

trical in r1 ^and r2, the range ri > r2 is only integrated The normalization integral ^{for a} half-volume

where

In a similar way ^we find

where

References

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188.

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