Coupling of radial and angular correlations in two-electron atoms

UtemicalPhysics 10 (1975) 67-72 0 North-Holland Publishing Company

CflIJPLUUGQF RADIAL AND ANGULAR CORRELATIONS IN TWO-ELECTRON ATOMS * N. MOISEYEV and 1. KATRlEL Department of Chemists, Technion - Israel Institute of Technology, Hat$a, Iuael Received 27 February 1975 Revised manuscript received 16 May 1975

The nature of interelectronic correlation in theground and low lying excited statesof the helium sequence isinvestigated. It is demonstrated that as a consequence of the coupling between radial and angular conehtion the electrons are actually pulled together rather than pushed apart, in the low nuclear charge end of the isoelecrronic sequence. A uniform treatment of currelation effects in both the ground and the excited states is thus shown to be feMile.

1. Introduction The Hartree-Fock approximation does not account for the dynamic (Coulomb) correlation among the electons. It is, therefore, usually expected that upon introducing the neglected correlation effects the electrons should more effectiveIy avoid each other. This should result in an increasing average interelectronic distance and reduced interelectronic repulsion. This view has gained considerable support in terms of numerical results concerning the shape of the Coulomb hole and the magnitudes of the correlation corrections to two electron expectation values [ 11. Lijwdin’s analysisof this aspect of the correlation problem [2] has established the theorefical foundations for a proper discusrion of correlation in terms of its effect on relevant expectation values,and hasshown how,asa result ofa misleading interpretation of the physical significance of the formal structure of certain perturbation theory type approaches to correlation the contriiution of the interelectronic repulsion energy is overemphasized. However, some concrete examples studied (3--S] seem to have confirmed the assumption concerning the predominant role of the reduction of the interelectronic repulsion. This assumption is implicit in many l

Based on a part of a thesis to be submitted by NM to the senate of the Technion - Israel institute of Technology, in partial fulf--ent of the requirements for the D.Sc. degree.

of the attempts to evaluate correlation effects in terms of a wavefunction constructed out of the HartreeFock wavefunction by multiplying it with a c&reIation factor. The attempt to relate arbitrary improvements in the variational wavefunction and energy to a reduction in interelectronic rcpuhion has &eady been shown to be an oversimplification of the actual situation [6,71. However, as noted in these investigations, his kind of criticism does not apply to correlation proper, defmed in terms of the exact and Hartree-Fock wavefunctions and energies. The results obtained by Boyd and Coulson [8] for the 3S state of helium and by Tatewaki and Tanaka [9] for the ‘P state of beryllium are. therefore. of particular significance. ‘Ihey involve weU defied and genuine correlation effects and are therefbre in direct clash with the traditionid view. In the present communication we investigate the effects of electronic correlation in the lowest InsI? as well as 3S states of the helium isoelectronic sequence. The results are analyzed in a manner which strongly supports the view that the unusual correlation effects mentioned ought to be anticipated in the low nuclear charge end of many isoelectronic sequences. They ate, therefore, the rule, rather than an exception. This investigation is preceded by an analysis of the nature of the paradigmatic evidence concerning the effects of electronic correlation, i.e., the ground state of

68

N. MoLpeycv. J. Katrfcl/Cbupli~

of dial

and at@ar

comddons In two-electron atoms

the He sequence. The discussion indicates the nature and limitations of the simple minded description of correlation in terms of avoided proximity of the electrons involved, and provides the frameworkfor an understandingof the excited state correlation effects. 2- A dirussion of correlation in the ground state of the helium sequence

AL

. -0.02

The nature of interelectronic correlation in the ground state of the helium sequence has served for a long time as the prototype example for the discussion of correlation [lo], Essentially, the most important result is the fact that the correlation energy, i.e., the difference between the exact and Restricted HartreeFock (RHF) energies, is aImost constant over the isoelectronic sequence. Using the Hellmann-Feynman

f ul -0.031

b hl

-0.04

EC

and the vi&l theorems, it follows from the quasi-con-

stancy of the correlation energy that the correlation correction to the interelectronic repulsion is equal to about twice the correlation energy [ 11,121. Corrclation in this case is therefore very clearly associated with a reduction in the interelectronic repulsion. It was suggested a long time ago, that correlation effects in the helium atom contain roughly additive radial and angularcomponents [ 13,141. The additivity of these components of the correlation energy is rigorous in the leading (second order) term in (l/Z)perturbation theory, but the higher order terms contain an interference between them, The radial limit [IS] can be used in order to effect this type of separation in a well defined manner. Radial correlation, in contrast with the total correlation energy, turns out to be a monotonously decreasing function of the nuclear charge(fig. 1). The decreasein the importance of radialcorrelation, pointed out by Gimarc [16], is a consequence of the increasing penalty paid in terms of reduction of nuclear attraction energy, for the attempt of one electron to avoid the other in a radial manner. By invoking the HeUmannFeynman theorem, which is valid for the radial limit as well as for the exact and Hartree-Fock energies, we conclude that while radial correlation reduces theinterelectronic repulsion, angular correlation actually increases it, at the low nuclear charge end of the isoelectronic sequence, where the dependence of the angular correlation energy on the nuclear charge is steep enough.

:/-Fig. I. Correlation in the ground state of the helium sequence:

Correlationenergy (I?@,Radialcorrelationenergy(ER) and mnrlation content of the EHF approximation (Aa.

lt should be pointed out that angular correlation can only reduce the average interelectronic distance by means of the coupling with the radial correlation, which allows the angularly correlated electrons to approach the nucleus closer than without angular correlation. This is consistent with the fact that the increase in the interelectronic repulsion is observed only at the low nuclear charge end of the isoelectronic sequence, whtzzethe coupling between radial and angular correlation is most significant. The same trend in relative importance of angular and radialcorrelationswas also observedby Banyard and Baker [ 171 on the basis of the magnitudes of the natural expansion coefficients of the correlated wavefunctions of the ground state of the He sequence, and by Banyardand Ellis [18] on the basis of the behaviour of the angular distribution. In this connection we note that whereas the total correlation energy increases

with the nuclear charge, the contribution of all but the leading natural orbital to the wavefunction decreases 117)) indicating a reduction in the effect of correlation on the wavefunction. This feature of the correlation problem is very simply accounted for by

N. MO~SQWV. 1. Katriel/Couplingof tndialand an@v combtions in twodcuon Table I Results for the ground state of H-and He in Killingbeck’s model a

B

69

afoms

I/i?

0.1

0.5

1.0

Oh&

A

-E (au)

0

0.472656 0.483883 0.483883

0.42969

0 0.0623 0.0689

0.5 13303

0.29585

0.516410

0.28969

0.5 16740

0.30895

0

2.847656 2.859186 2.859186

1.05469 1.03182 1.03186

2.875661 2.884691 2.884746

0.99208 0.97415 0.97743

(Ilrl2)

Ez -0005

0.6875 0.6875 0.6879

H-

1.039 1.039 1.034 He

0.1900 0.1898 0.283 0.283 0.3G9

1.6875 1.6875 1.6876 2.183 2.183 2.173

0.0323W 0.032381 1.188 1.188 1.200

the Eckart criterion

0

0.0217 0.0219

0.40776 0.40800

[ 191

in which $0 and $Jare the exact and approximate wavefunctions respectively, AE the energy error and El -E. the difference between the first excited and

ground state energies. For systems without zero order degeneracy [20] the correlation energy can be written asAE=a+P(l/Z)+ ... , whereas the Ieading term in Et - Eo is either quadratic or linear in 2. For large 2, IAE/(E1 -&)I + 0 so that the overlap between the uncorrelated and exact wavefunction approaches unity. To examine the general conclusions obtained we investigate the model wavefunction proposed by Killingbeck 1211 9 = [exp(-orl

-or*)

+ exp(-or2

-13rl)l

x (1 -~coset~). Some of the results obtained are presented in table 1 and in fig. 2. The essential features of angular versus radial correlation are borne out by this computation. The angular correlation energies computed with respect to a radially correlated and a radially uncorrelated wavefunction exhibit the separability of angular and radial correlations in the leading term by obtaining identical values as l/Z + 0. The latter angular correlation energy is larger in magnitude and less Z dependent than

Ei

lz

E.

-0010

E2

Fii. 2. Correlation energy (EC) and angular correlation with respect to a radially unmrrelated (f$) and to a radially correlated (&A) wavefunction in Killingbeck’s model.

the former. These differences can easily be interpreted in terms of the very different screening constants involved in the closed- versus open-shell descriptions. The interrelationship between radial and angular correlation is exhibited in an even clearer way by a comparison of the effect of angular correlation for a radially frozen wavefunction, to the results obtained if the radial wavefunction is allowed to relax upon introducing angular correlation. It is only by allowing for this relaxation that the electrons get closer together, in H- ,on introducing angular correlation, on top of the radially correlated wavefunction. This occurs by means of a contraction of the more diffuse orbital which accompanies the introduction of angular correlation. A similar contraction is observed in the radially uncorrelated wavefunction as we& but its effeet is much smaller and does not result in an actual increase in the interelectronic repulsion, in comparison with the closed shell uncorrelated result. The simplest way of introducing a certain amount of radial correlation is the use of the EFJF approximation [$ = e(l) x(2) + 442) x(l)]. The results in fig. 1 indicate that this approximation accounts for a considerable portion of the radial correlation. Therefore, most of the remaining correlation is angular, so that it is to be expected that the increase in the interelectronic repulsion brought about by angular correlation will also be exhibited by the residual correlation with respect to the EHF approximation. This is coni%med by the behaviour of exact - (r&m (n = -1 ,1,2), presented in table 2 for the low nuclear charge members of the helium sequence for which the exact results are

70

z 1 2 3

N. M~kyev.

Wl2k4lrx

0.0125 -0.0439 -0.0536

1. KatdcllCoupling of rmiial and anguh ctwebtions in rwo4ectron atoms

z2 (~1akorI

23 (‘:gcorr

-0.4872 0.1348 0.1553

-6.6300 0.7084 0.8133

available [22,23]. The corresponding computed by the present authors.

3. Correlation sequence

EHF values were

in the excited states of the helium

In order to investigate the nature of correlation effects in the excited states of the helium isoelectronic sequence, the Roothaan-Hartree-Fock wavefunctions for the lowest 3S and *B~Pstates were evaluated in a basis of STOs. The energies obtained are in satis. factory agreement with those of Davidson [24] for the neutral atom as well as with those of Ali and Dasgupta [25] for the first six members of the tP isoelectronic sequence.

Actually, the excited state Hartree-Fock wavefunctions have a structure which is in some sense comparable with the EHF wavefunction of the ground state. The two electrons are radially separated in the HF level already, so that correlation is probably niostly angular. It is therefore reasonable to expect a behaviour similar to that of the angular correlation in the ground state, rather than that of the total ground state correlation. In a sense, therefore, the unexpected correlation effects in the excited states could have been anticipated on the basis of the behaviour of the ground state. This conclusion is much more satisfying than the initiat temptation to discuss correlation effects in excited states as if they were fundamentally different from those in the ground state. A contraction of the outer orbital as a consequence of the introduction of a correlation factor has recently been observed by Kagawa and Murai [27], employing minimal basis set ST0 wavefunctions. This is in complete agreement with the results and discussion presently reported.

4. Analysis in terms of (I/Z)-perturbation theory

For the 3P sequence our results are almost

uniformly lower than those of Ah and Dasgupta by about 0.0004 au. The Hartree-Fock wavefunctions obtained were used for the evaluation of the two electron expectation values (l/r,,), (rt2) and (r$. The value of (I/r,,) in the 3P state of He is identical with that given by Davidson 1241. A slight deviation in the lP state is inconsequential from the point of view of the present investigation. The differences between the exact 1261 and Hartree-Fock expectation values, i.e., the correlation corrections, are presented in fig. 3. lle sign reversal of most of these quantities for the Iow nuclear charge members indicates that the previously obtained results exhibiting similar effects [8,9] are neither accidental nor rare. The physical origin of this behaviour is, no doubt, associated with the fact that upon approaching the nuclear charge at which the outermost electron becomes unbound, the related orbital becomes very diffuse. The effect of correlation is therefore essentially to bind that outermost electron by pulling it back towards the nucleus. This is exhibited by a reduction in the average distance between this electron and the inner electron(sb. <

The perturbation expansion in l/Z can in principle be expected to be less reliable in the low nuclear charge range in which we are presently interested, than straightforward variational computations. However, the combined use of the two approaches does lead to some further insight into the subtleties of the reversed correlation effects presently discussed. The l/Z expansion for the 3S and t13P states of the helium sequence has been carried out by Sharma and Coulson [28], Perrin and Stewart [29] and by Scherr and coworkers [30-321. The HF approximation to these states has been considered within the same formalism by Linderberg [33] and Layzer [34] to second order and by Perrin and Stewart [29] to third order. The agreement, to second order, among the three sources, as well as the agreement, to third order, between the l/Z expansion coefficients extracted from the HF energies reported in the previous section and those of Perrin and Stewart [29], is an indication that the HF expansion of the latter authors is reliable. This is in contrast with their exact results which are in obvious disagreement with those reported in refs. [30-321. Using the Scherr and Knight exact energy expan-

N. Moiseyev,J. Katriel/Couplingof radialand at&u

71

cwrehions in hwelectron atoms

-15 t

t

-2001 0 01

02

Ia) Fig. 3. Corrcl~tion

0.3 0.4

t35,

-loo-0

-lcoO~ , 0.1

cwrcctions

to two-clcc~ron

cxpeclation

03

04

0

c

I/Z

(b)

I/i!

0.2

QI

I

02

I

I

a4 05

I/Z

(Cl

values for the 23S and 2’lP

I

03

states of the helium scqucncc:

(~1 (l/r12),

(bj (r12), (c) tr:$. sion and Perrin and Stewart’s Hartree-Fock expansion one obtains for the correlation energies in the 1*3P sequences Ec(‘P) = -0.0104

+ 0.02256 (l/Z),

Ec(3P) = -0.00512

t 0.00665 (l/Z),

For the 3S sequence Sharrna and Coulson [28] obtained EJ3S) = -0.00205

+ 0.00258 (l/Z).

The intercepts and slopes of the variational correlation energies plotted in fig. 4 are in complete agreement with these results. Moreover, the agreement of the values obtained from the intercepts and slopes of C, versus l/Z, (fig. 3a) with those predicted from the correlation energy on the basis of the virial and HellmannFeyrunan theorems is a useful indication of the reliability of the variational computations. One of the simple features of correlation, which the t*3P states of helium violate, is the fact that the existence of Fermi type correlation in the HartreeFock level already results in a smaller Coulomb correlation in the triplet state than in the corresponding singlet [35]. The situation becomes normal for the higher members of the isoelectronic sequence, as indicated in fig. 4. This is also reflected by the fact that

the leading term in the l/Z expansion of the correlation energy is indeed larger in the singlet than in the triplet state. The reversed ordering of correlation energies is therefore a consequence of the higher order corrections. The leading term in the correlation energies of the 1g3Psequences presented by Perrin and Stewart [29] are in conflict with this result, i.e., they seem to suggest a (slightly) larger correlation energy in the 3P sequence than in the lP even for highly positive ions. I/Z

0

Fig. 4. Correlation helium sequence.

01 02

03

0.4

05

-l

energies in the 2% and 2’e3P states of the

72

N. Motseycv, 1. Katricl/Couphg

of r&11 and an&u

lhi.9 is a consequence of their erroneous results for the exact energies, and on the basis of the present discusaion seems to have been able to indicate the existence Of some inaccuracy in their correlation energies.

5. Conduaions A study of the correlation correction to two electron expectation values in excited states of the helium sequence has been undertaken. The results, which ap pear to be in conflict with a simple minded interpretation of the nature of interelectronic correlation effects, have been discgsed in comparison with those related to the ground state of the same sequence. One easily realizes that the residual correlation in the singly excited states is mostly angular correlation. Correlation effects in these excited states should, therefore, be compared with those associated with angular correlation in the ground state, rather than with the total correlation effects. An analysis of angular correlation in the ground state from the point of view of its effect on two electron expectation values exhibits the same unusual features observed in the excited states as well as in other few electron systems. The physical reason for these correlation effects is clarified. They are shown to be very intimately associated with the physics of the system in the low nuclear charge range, in which the outermost orbital becomes very diffuse and weakly bound. For systems with a larger number of electrons one may expect to observe very similar correlation effects as far as the values of iErii> and (Er$), are concerned, becuase the contributions involving the outermost electrons are likely to be dominant in these quantities. However, (I2 l/Q is probably more sensitive to the contributions of the inner electrons and therefore less likely to exhibit correlation effects of the “reversed” type-

Acknowiedgement This work is part of a research project supported by the U.S. - Israel Binational Science Foundation.

conrhtkws

in tnv-ckctmn

atom8

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