e12) - University of Chicagoberrygroup.uchicago.edu/papers/128.pdf · zation of l/r12 [13-15]_ The...

Chemical Physics 31 (1978) 239-262 Q North-Holland Publishing Company

SPATIAL CORRELATION OF ATOMIC ELECTRONS: He**

Paul REHMUS, Michael E. KELLMAN* and R. Stephen BERRY Department of Chemistry and the James Fran& brsritute. The University of Chicago. cilicago. Illinois 60637. USA

Received 13 October 1977

For anp two-electron wavefunction whose angular dependence is given in terms of the spherical harmonics of the individual electrons and/or 012, where 012 is the interelectronic angle, a transformation is generated which reduces I~~(r,.r*,82,01,812)IZ to Iw1.~21,~12N 2. Geometric aspects of electron correlation arc analyzed in terms of this resulting three-coordinate function, with specific application to a number of wavefunctions for doubly-exited helium. Angular correlation has been studied by integrating 1\I~(r~.rz,0,~)1~ over its radial dependence to yield p(B1z), the probability density function for the interelectronic angle. Trends of p(Ot2) for doublyescitcd states of the helium atom are related to a number of quantities including energies and autoionization widths. These trends can be rationalized in terms of a simple classical model. The full spatial correlation involving IY(r,, I-_, 1 e12)12 is explored by the use of three-dimensional graphs for some of these states.

1. Introduction

The study of electron correlation has always had two objectives, which have not always been equally well served. One is the attainment of high mathemat- ical accuracy in the representation of correlation in wavefunctions, so that computations would give accurate values for observables. The other is the achieve- ment of a graphic, intuitive conception of how electrons move and locate themselves as they influence one another’s behavior. Progress toward the first goal has been impressive through devices such as configuration interaction (CI), the use of interelectronic coordinates in wavefunctions, and explicit consideration of fluctuation potentials [ 1,2] . The second ob- jective has been more elusive, partly because it is loosely defined and subjective, and partly because the tools that have given us great accuracy have often seemed to preclude any simple graphic interpretation of correlation. The accurate wavefunctions are usual- ly the most compiex, and therefore the least amen- able to description by geometric models.

This paper is directed towards the interpretive,

* Department of Chemistry, University of Oregon, Eugene, Oregon 97403. USA.

didactic aspect of the correlation problem, and specifically toward extracting insight regarding the spatial disposition of strongly correlated electrons. To antic- ipate what follows, let us state our findings: given a two-electron atom, whose wavefunctions \Ir(rl ,r$ are based on products of one-electron eigenfunctlons of orbital angular momentum. the density [9(rl ,r2)12 can be integrated analytically over three variables to give the density p(rl, r2z012) and further over the radial coordinates r1 and r2 give the averaged angular density function p(B&_ (The angle e12 is the angle between r1 and ‘2 _) Examination of p(rl, 9, e12) and ofp(Ol?) provides insights into the geometric aspects of electron correlation, some of which we believe have not been apparent previously from other ways of studying correlation. Much of this paper is devoted to discussion of the behavior of P(rl, r2, tl12) and p(B,,), and of physical processes associated with their behavior, for the doubly-excited states of helium.

Interpretation of the geometric effects of electron correlation has long been hampered by the multidi- mensionality of the problem. Even the general two- electron wavefunction, with only six spatial coordinates, has been too unwieldy to permit a straight- forward geometric interpretation of each electron’s relative position. Because of the simplicity of S states,

240 P. Rehrrs et al. jSpatia1 correlarion ofatotnic electrons: He**

where the angular dependence depends ordy on e12, geometric correlation has been studied via graphic techniques, specifically for the ground state of helium [3]. Simplification of one kind, with an attendant loss of angular information, can be obtained by integrating a wavefunction over Its angular coordinates [4]. This leaves the radial correlation expressed in terms of a function of the radial coordinates. A general and convenient method to condense information regarding correlation has been to calculate and analyze the moments of radial or angular coordinates, including rt2, and of simple functions of the coordinates [5-8 ] .Extensions of this approach based on analysis of “correlation functions” [9] , or two-electron distribution functions [IO,1 I], have simplified the evaluation and interpretation of such moments. With the method we report here; spatial correlation in two-electron atoms [nay be studied graphically for any state, without loss of inforpation about either radial or angular correlation. The tool for this study is a transformation which reduces the angular coordinates for two electrons (specifically the spherical polar coordinates Ot , e2, @, and 02) in an II-electron atom into the single relative coordinate 012_ The transformation is independent of the portions of the wavefunction expressed in the coordinates of the remaining 12-2 electrons. These remaining electrons can be ignored by integration over their coordinates. It is also independent of the radial portion of the wavefunction. Specific application is made to 2-electron wavefunctions where the reduction of the complete spatial density function yields the previously mentioned ~(rt , r2, 6,,). A byproduct is a method of generating moments of combinations of rt, r2, and 8~; however, suitable graphing yields much physical insight into electron spatial correlation, without the need to calculate such moments.

In the foliowing discussion, wq illustrate our approach with a set of interesting and reasonably good but not highly accurate wavefunctions. Future com- munications will be based on niore elaborate functions and show how the composition of the wavefunction affects its representation of electron correlation. The choice of functions used here was spurred by the analyses ofWulfman and Kumei [12,13] and Herrick and Sinanoglu [14,15] regarding the geometry of electron correlation. They developed a strikingly sChn- ple method for obtaining fairly accurate, doubly-

excited two-electron wavefunctions of helium-like atoms, particularly when both electrons have the same or nearly the same principle quantum number in the dominant configuration of the state. They use a novel construction of a two-electron basis from hydrogenic orbit& that exploits the well-known SO(4) symmetry of the Coulomb problem. The resulting “doubly- excited symmetry basis” (DESB) [14] is an SO(4) basis comprised of antisymmetrized products of one- electron hydrogenic functions, with each of the electrons restricted to a single principle quantum number. The DESB functions are labeled by two new quantum numbers R and T [ 141, related by K = P + 1 - 12 and T = IQ1 to the more usual Cartan labels [ 131 [P, Q] which label the irreducible representations of the SO(4) group. The variable f~ is the larger of the two principle quantum numbers labeling the hydro- genie basis used to con$uct tke DESB. The O(4) al- gebra is generated by L = it f I2 and B^ = ~?t - 62, where Ii and Ci are the single particle orbital angular momentum and the energy-weighted Runge-Lenz operators that generate the SO(4) degeneracies for the bound states of hydrogen [16]_ Consider the set of functions having fixed principle quantcm numbers rzt and II,. For example, iftzt = tt7 = 2, six independent wavefunctions can be formedfrom a basis of configurations denoted (2s)“, (Xp), and (3~)‘~ For brevity, we shall refer to this set as the “jr = 2 shell”. Within a shell, and for a given total L, possible values for Tare 0,l , . . . . min(L, w-l), where tt is the larger of the two principle quantum numbers. Values of K are restricted to S, S-2&4, . . . . -S, where S = n-T- 1. The classical Runge-Lenz vector points in the direc- tion of the major axis of a particle’s classical elliptical path, and has magnitude e2e, E being orbit’s eccentricity.

Mixtures of intrashell Russell-Saunders hydrogenic states obtained by diagonalization of B2, the squared distance between the individual Runge-Lenz vectors, are approximately those generated by diagonalization of l/r12 [13-15]_ The wavefunctions obtained by diagonalizing B2 and those obtained by diagonalizing 1/r12 with the same intrashell, Russell- Saunders basis have mixing coefficients that differ by less than 1% in many case8. Since intrashell configuration mixing is a reasonable good first approxi-

* Footnote see next page.

P. Rehnur et aI.&atial correIation of atomic electrons: He** 241

mation in the lower shells where E? is diagonal and where the DESB is a good approximation to intrashell (CI) results, we make the pragmatic choice to carry out our first investigations of geometric correlation with the intrashell DESB states. These states are related to values of B2 and the K, T predictions for energy orderings and autoionization selection rules. We have analyzed several states based on the intershell states with 11 t, fz2 = 2,3 but will report here only on the intrashell states which diagonalize B’ in the II = 2,3, and 4 shells. Higher states with ftl = IIT are increasingly spoiled by mixing with intershell con- figurations_

In section 2 we present the simple reduced transformation of 19(rl,r2,0t,62,~1,0~)12 yielding the three-coordinate joint density function ~(rj, r2,8t2) with which our study of spatial correlation will be

$ Although our concern here is primarily with interpreting electron correlation, some comments on the DESB functions are in order. One advantage of the [K. 71 labeling scheme over the [P, Q] labels is that K and T remain ap- prosimately good quantum numbers when both intershell and intrashcll hvdrogenic contigurations are mixed. This stimulated Her&k and Sinano&t to develop empirical rules for level sequences and autoionization rates based on K and T [ 151. However states characterized by K and Tare not in general eigenfunctions of thc_Cssim]r operators of the O(4) aIgebra generated by the .Ci and Bf. So far as we know, the meaning of the K, T scheme is not yet completely un- derstood either mathematicttlly or physically. Significant steps toward both kinds of interpretation have been made by Herrick [17,18] and a physical interpretation ofR and T as asymptotic channel designations has been made in the dipole representation [ I$]_

Nonetheless, it is evident that in the K, T scheme, the configuration mhing coefficients are close to their true values, and are best when n r = rz2. In the DE%, the eigenvalues of@ are rigorously equal to 0s +-@ - 1 f T2 - L(L + 1) with n the larger of the two principle quantum numbers. SinanoElu and Herrick pointed out that the eigenval- ue ofB2 is intuitively related to angular correlation in intrashell states [ 141. Without additional CI, the DESB gen- crates less accurate coupling coefficients as the difference between the two principle quantum numbers increases [13, 14,201. Recent discussions by Nikitin and Ostrovskv 1201 have shown that when jr, b pi2 another O(4) coupling-of - the individual electron’s angular momenta and Runge-Lenz operators can be used to generate good miring coefficients for states very near ionization thresholds. However, B* is not diagonal for these new O(4) coupled wavefunctions nor is it diagonalized after CI mising of DES6 intershell and intrashell states.

done. Details of the actual derivation of the transformation appear in the appendix_ From the three-coordinate joint density, we generate p(Bt7), the density function for the interelectronic angle in a two-electron atom, and present graphs ofp(Bt$ for many of the DESB functions of helium in the /I = 2,3, and 4 shells. We provide the complete set of graphs of Russell-Saunders and DESB functions in the II = 2 shell, and we also include the DESB functions in the II = 3 shell_ General observations which will be noted for these two lower shells are also made for the II = 4 shell, and we have provided selected graphs of p(Bt2) for these states. These observations pertain to various trends noted in the graphs as functions of quantities like energy, B2, and moments of O,, and cos0t7. In section 3 we relate the shapes ofp(i12) to the partial autoionization decay widths for these states.

Classically, the conservation of the squared difference of the individual Runge-Lenz vectors implies that the electron orbits precess together_ This geometric constraint, and the conservation of angular mo-

mentum and energy restrict the relative positions of the two electrons. Our investigation explores how the geometry of this classical motion relates to the angular correlation, and more generally to the spatial correlations of two-electron atom. In section 4, a previously reported classical interpretation of the DESB [ 141 is expanded to account for a number of trends observable in the graphs of p(B12)_ In section 5, we explore the geometric information contained in&-r, ~,0t~) by plotting this function for DESB states in the II = 2 shell of helium.

2. The transformation and its application

We present here the transformation which allows the reduction of any six coordinate, squared, two- electron wavefunction to the three-coordinate, joint probability density in terms of rl, ‘2, and 6t2, the interelectronic angle, where

012= cos-l [cosf?l,cos62 +sinO1 sine, cos(@l-92)1. (1)

We have implicitly summed the squared wavefunction over its spin coordinates_ The reduced form of the transformation that appears here follows from the full transformation if \k is a two-electron function

242 P. Rebus et al_/Spatial correlatiotl of atomic electrons: He**

which is an eigenfunctian of .L2 and&_ Full details may be found in the appendix, with the result P(‘1”2’(&)

012+4

=+, J d6219(r,.‘2.01re7re17_)l 2

0 en-01

[ (

cos e,, - Xl- -

cos0tcose, z -112

_ -)I * sin 0, sin 8, (3

Examples of wavefunctions which can be easily reduced this way are Russell-Saunders, DESB, and Hartree-Fock functions expanded in two-electron products. The only angular information lost by in- tegmtion over the laboratory coordinates in eq. (2) is the orientation of the plane which contains the nucleus and the two electrons. The integrand does not contain exphcit reference to 9r and 02 because if 9 is an eigenfunction of L,,.then the dependence of the squared wavefunction on QI and O7 is contained in its dependence on 0t, Oz. and 012 ;hrough eq. (1). The result of implicit integration over an irrelevant dummy variable is aheady contained in eq. (2). The jacobian of the space ofrt. 1-2 and e12 is $nrTrisin0t2. Therefore

0 0 0

x sine,2dr1.~2,012) and the rrth moment ofany functionf’(rl, r7,012) is given by

x c-2

(f”) =j drl s drz j dfI,z (r/2) 0 0 0

X sine12f’rP(r1,‘2.01,). (3)

One finds for instance that (l/r,,) becomes a relative- Iy simple integral that eliminates the need for a mul- tipole expansion of l/rrz.

The density function of BL2: p(B &, is given by ca m

p(ell) =J dr! S tiZr:rGp(r1,r2.er2), 0

and the jacobian is (n/2) sinBrz_ As an example of the application of eq. (2) we

have the following joint density function for the Russell-Saunders helium lDe state based on the (2~)~ configuration*:

where Rli) . 13 1s the usual hydrogenic radial function, R,,, for eikctroni. Averaging over both radial coordinates we have

’ de12hDe = ST -‘( 1 - ~sin20,2).

Fig. 1 shows the calculated density functionp(B,2) for the DESB and Russell-Saunders states for helium in the II = 2 shell. These graphs include the jacobian, except for the sin(dlz) factor. Figs. la and Ib graph p(0rz) for the (2~)~ and (2~)~ configurations respectively. They are mixed in the DESB to yield two new states, ISe K, T = 1 ,O and K, T = -1,O, whose density functions are graphed in figs. lc and lh, respectively. Note the pronounced angular correlation introduced by this limited configuration interaction. Energies for this and other !I = 2 shell DESB states may be seenintablel.ThelSeK,TT=--1,Ostateisthehigh energy state in the shell; the angular structure of its density function is similar to that of the (2~)~ Russell- Saunders lSe function. That portion of the (2~)’ state which does not contribute to the density for the high energy must appear in the low energy I.9 state since the configurations based on two electrons in the 11 = 2 shell generate only two singlet S states. The low energy ISe state is dominated by the (2s)” configuration. Examples of otfler graphs of p(0t2) for states based on hydrogenic functions of the higher shells are given in Bgs. 2-8. Several tendencies, summarized below, may be discerned in these graphs. There are occasional exceptions which will be mentioned; fortunately they are marginal cases that only deviate slightly from the generalizations.

Trend I Within a shell, a high value of B2 is associated with high density around O,, = rr, and low Bz with high density around O,, = 0. 6g. 2a demonstrates this trend for states with ti1 = jr2 = 4. As a measure of where the density concentrates we have used (cos~?~$, which is easily generated from eq. (3). Moreover, an approximate ordering of the energy in these states can be made on the basis of where the maximum ofp(BI,) lies. This correspondence is shown for the II = 2 shell in fig_ 2b. Fig. 6 presents graphs for all of the states with tzl = ~2~ = 3. The trend is generally followed but the plots show that there are

f In general, p(r,. r2,012) is independent of ml, so the transformation yields the same resuIt with all five ‘De functions.

r-

0

(e) “, ,“, i 2.2 MSB ‘PC K,T=O,l =Russel-Sounders 12PlZ

I I I I I I.@ I

05 IO 1.5 6

2.0 2.5 30 I pO.S- 12

0 05 I.0 1.5 30 77 EJ

2.0 2.5

I2

I.51 (4 n, ,n,=2,2 DEE ‘SC K,T = I,0

I (9) n,,l=2,2 DESEi ‘P” K.T=O.I =Rwsell-Saders 2s2p

I$ , //y---j Pill\

0 0.5 I.0 l%* 2o 25 3.0 =

0 05 IO 15 e

20 25 ,

I2

(4 n,,n,= 2,2 OESB 3po KJ:I,O +7esell-Scmdss 2s2p IO

PO.5-

Table 1 Energies for II = 2 shell wavefunctions (eV above He ground state)

state DESB b) Hylleraas-QHQ Espt.

rse (2p)2 63.705 62.134 c) 62.15 9) ‘PO 2s2p 62.226 60.186 d) 60.130

+ 0.015 h) rDe (2~)’ 61.046 59.902 e) 59.9 0,

60.0 j) are (2p)2 59.930 59.680 f) 59.680 h) ape 2s2p 59.037 58.300 d) 58.34 9) rse (2s)2 58.871 57.842 c) 57.82 g)

a) For a survey of theoretical approaches from 1963 to 1970 see ref. [29] _

b) For the Rydberg state we have used ,P = 13.60535 eV. c) Ref. [28] _ d) Ref. [30]. e) Ref. 1311. f) Ref. [32]. g) Ref. [33]. h)Ref. [34]. F) Ref. [35]. j ) Refs. [36,33]. k) Ref. [37].

P. Rehnus et d/Spatial correlation of atomic ekctrons: He** 243

Fig. 1. ~(0 ,a) versus 812 for helium DESB and Russell-Saunders wavefunctious For the rr = 2 shell. (c)-(h) are ordered from low

(al 0.4&0 ’

I I I I_

o2’;!,o~e90

<co56 > o- - 8 0 %2- o eo go” 0 0 -04- oa B

- -06-

O0 00

-08- I I I I I 0 IO 20 *a 30 40 48

Fig. 2. (a) (cost?ra) versus ei~envnlues ofB2 For helium DESB

wavefunctions in then = 4 shell. (b) Energy (eV above He ground state) versus Wra) for helium DESB wavefunctions in the 12 = 2 sllelI.

244 P. Refrmus et al.&atiat correlation of atomic electrons: He**

exceptions, such as the tD0 and 3Fo states of figs. 6m and 6n. States which do not conform to the trend tend to be those with higher total _L, and are often DESB states which are equivalent to the Russell-Saunders states. The energy ordering as a function of either @t2) or<cos 0t2) for fixed 2s+t_LR, where 71 is the parity, have no exceptions however.

The correlation of energy with expectation values of e12 (it makes little difference whether one uses U12) or (COSTS;)), and ofOt2 withB2, suggeststhat and B2 are roughly equivalent measures of the energies of these doubly excited states. A direct comparison [2 11 of energy versus B2 showed that adding the small quantity LylO to B* made this relationship smooth_ We find that the spread seen in graphs such as fig. 2a dimishes when the abscissa is increased from B2 to B2 +L2/10. SmallLdependent anomalies will be seen again in trends 3 and 4. We can see why addi- tion of L2 improves the picture. Diagonalization with respect to B* is nearly diagonalization with respect to l/r121 some of the difference lies with some small angular momentum dependence in l/r12 which is not included in B2 alone. This is made more specific in Wulfman’s expansions of (l/r12) [13], where, after several approximations, (I/rt2) is expressed in terms of the eigenvalues of B’. The first neglected_ noncon- stant term in the final approximation is f(L: +Lz). The coefficient I/ 10 was apparently chosen heuristic- ally.

Fig. 3. Ln(Da,e) versus ,I and L (where Dave is the avenge of D for StdteS with constant n and L) for helinm DESB wavc- functions in shell 1, 2 and 3. D for any state is the maximum peak height in graphs of p(012) minus the lowest though height.

Before presenting the other trends we define what we mean by a strong angular correlation, as opposed to a weak one. The weakest angular correlation between two electrons is that in which the angular probability density for one electron is not influenced by. the presence of the other. In a situation where each electron has zero angular momentum, and where there is not angular correlation, p(0t2) is constant. An example is the (2s)’ state graphed in fig. la. Strong correlation is represented by fig. Ic, one of the DESB states that is peaked at s radians. In this case the electrons are most often about x radians apart. We find that one simple and useful measure of the strength of the angular correlation in a state, which we denote by D, is the maximum peak height minus the loweSt trough height for that state’s ~(012).

Trend 2 If L, S, and parity are held constant, then as II increases, the angular correlation becomes stronger in the sense that D increases. Fig. 3 shows this trend over L as well as 11, where averaging has been done S and parity. In each of the three shells, the high energy state is a tSe state. An alternative demonstration of trend 2 is given in fig. 4a showing ~(8~2) for the 1Se state of highest energy from each

(a) I I I I I

4.0 =4shell KS=-30 B*=O

30 n=3 shell KJz-8 $=O

2

2 p;;

* l2

0.5 1.0 1.5 2.0 2.5 3.0 7T e 12

Fig. 4. (a) ~(0~2) for the highest state of each of then = 2,3, and 4 shells. @) ~(6712) for the lowest energy state of each of the II = 2, 3 and 4 shells.

P. R.&mus et al./Spatial correlation of atomic electrons: He** 24.5

shell. A similar comparison is provided by the low energy states in each shell, fig. 4b.

Tretzd 3 For decreasing L and constant II, the angular correlation becomes more pronounced in the same sense as in trend 2, i.e. that for fixed fr, D grows as L diminishes. Fig. 3 diagrams the trend as an average overS and parity. It is apparent that the trend holds only approximately. It frequently happens that an individual state has a larger D than some state with lower L. The quantum numbers K and T tend to influence the strength of the correlation. In particular, states with higher IKI tend to have larger values of D than states with values of IK[ near zero. This effect can dominate the apparent effect due to angular momentum alone, though in general the ]K] dependence appears to be less important. On the other hand, fig. 5 provides a set of graphs from the tz = 4 shell which demonstrates the trend very clearly. In regard to trend 2 as well as 3, we note that the parity of these states does not appear to be a factor. There is also no suggestion that triplet states are more strongly correlated than singlet states.

Trend 4 For constant n and L, the high energy states are more peaked than the low energy states.

.:~

0.5 1.0 1.5” 2.0 2.5 30

P 0 0.5 1.0 1.5 B 2.0 2.5 3.0

l.Ol(d’ 12

n,.n,=4,4 ‘,Ge K,T=Z.O I

P&

0’ I I I I I I I 0.5 1.0 I.5

0 2.0 2.5 3.0

II

Fig. 5. Comparison of p(B12) for helium DESB wavefunctions of differing L from then = 4 shell.

I_.Jsing D again as the measure-of correlation sharpness, one finds the clearest demonstration of this trend with a comparison ofp(0t2) for the high and low energy lSe states in figs. 4a and 4b. Exceptions occur In high angular momentum states where angular correlation is not very pronounced, in accordance with trend 3. In anticipation of the results of later com- munications, we note that this trend will not appear in the analysis of nearly exact Hylleraas wavefunctions.

3. Angular correlation and autoionization widths

Herrick and Sinanoglu [ 151 have given “selection rules” for autoionization based on the well-known criteria oi existence of valence states, proper ex- change symmetry, and core penetrability [Z--24] . Their rules I and II are based on the first two criteria and will not concern us; rule III supposes that high B’ is associated with high electron density near 0t2 = rr, and therefore with core penetrability. It is expected to be most applicable to partial widths for decay to the l&I continuum channel. Fig. 7 shows how the partial widths, rt,, of the DESB states in the 12 = 3 shell vary with p(n). All of the curves in fig. 6 show ~$6,~) for the entire II = 3 shell, with attendant values for K, T, f@, energy, and rP, the calculated partial width_ Autoionization of these states is allowed by rules I and II, and therefore their widths should be governed by rule III. We make two observations.

(a) For fixed L and S, as p(r) increases, so does the partial width. (Although fig. 7 suggests a sys- tematic variation in p(n) with L, in general this is not the case. For example, in the II = 3 shell, Russell- Saunders states which are unmixed in the DESB show no simple progression with L_) Not surprisingly, this trend can also be observed as a function of B’, kos 812), or (I?&. However, the clearest correspondence is seen in fig. 7.

We have attempted to see whether there is any support in these graphs for an autoionization mechanism where one electron repels the other in a direct, close collision. Enhancement of autoionization by such a mechanism would be correlated with large p(O). The scatter in plots of rt, against p(O) is large, indicating little direct support for such a mechanism. Rather, our correlation analysis suggests that one

P. Relrmrls et aI.&atial correlatiorr of atomic ekctrons: He** 246

(a) n,,nz=3,3 ‘Se K,T=-2,O ~p=2.14x10-6E=-6.781

jzrj

‘.:i 0.5 IO 15 2.0 2 5 3.0

05 IO I5 20 25 3.0

[c) n,,n,:3.3 jpe K,Tz-1.1 r,=O E=-7709

p.*_ / 0 I I I I

05 LO I5 20 25 3’0

(4 n,.n,=3.3 ‘Oe K T=O,O rp=4.39xlC4 E=-7.799

IO (e’ n,,n,=3,3 ‘F” K T+ 1 rp=785x10-4 E=-8.245

P 0 I 1

05 10 1; I I I

2.0 2.5 3.OT

n&=3,3 K,T=Z;O &=l.Ol x10+ E=-8356

P 0.5 IO 15

P

IO (h] n,.n,=3.3 3Fe K.T=l.i r,=O C-8.405

P/

,_l !i!

I I I 0.5 I.0 1.5 3.0 s

n&=3,3 *Oe K,T=0,2 l’p=6.78x10-4 E--8.517

P 0 ' I

0.5 I.0 I.5 2.0 2.5 3.0rr

812

(j) 4 &=3,3 ‘Se K.T=O.O r&92x164 E=-8.533

n,.n,=3.3 ‘0’ k.T=0.2 IYp=O E=-8.539

n,,nz=3,3 3De K,T=l,l ~p=l.12x164E=-89Z5

l,o (ml n,,n2=3,3 ‘D°K,TzI,I r,=O E=-8.925

P

1.0 ( n fl,,flz=3,3 3~o ~,T:2,0 r~=2.37M4E=-9.034 ) P

0 I I 05 10 1;

I I I 2.0 2.5 3.0 B

n,&=3,3 ‘PC K,T=l,l rpp’o E=-9.184

(S] n&=3.3 ‘5.’ K.TQ.0 Tp=632xl@E=-9624

Fig. 6. Density graphs, p(Otz) versus 812 for helium DESB wavefunctions of the rz = 3 shell. Data provided includes the eigenvalues for 8’. Energies are in CV and are measured from the double ionization threshold. Partial autoionization widths, r are given in eV. Both the cnergics and widths are t&en from ref. LL.51.

P’

P. Rehmus et aI./Spatia[ correlntion of atomic e-lctrons: He** 247

Fig. 7. Ln(l’$ versus p(x) for states mised in the DESB in then = 2 shell. Excluded states are Russell-Saunders srates in the DESB, or are “parity disallowed” toward partial autoionization.

electron approaches the nucleus on one side, driving the second electron out of the other side, in correlation-driven autoionization. To substantiate this, one must of course examine what parts of the correlated bound-state function contribute most of the bound- free transition amplitude.

(b) Ifp(n) is zero, then the transition to the ground state of He+ is a “parity disallowed” transition [25] , and the partial width is therefore zero. States for which this is true are those with odd L and even parity, or vice versa. For example, autoionization of the first doubly-excited 3Pe state to the ground state of He+ results in a % configuration for the helium ion and a p-wave free electron. The parity of the free electron’s wavefunction in this case, and the parity it must have in order to conserve the parity of the entire system are incompatible, and the decay cannot occur.

The states with parity-forbidden partial autoionization widths have nodes in ~(0~~) at e12 = 0 and 0,~ = ST, independent of the radial distances of the two electrons. These nodes are a result of the vector coupling that generates II,, Z,. LM from the coupling of two angular momenta II and 12, for any values of el, 02, 4,, and &_ Recalling that

we specifically have the identity for B,, = 0 [26].

where Q is a proportionality factor. If Or2 = H we have the same result but Q includes a factor (-1)L-l1-~2 Now since

=(-1)L+~l+~zu1i7-“21 -nz,lIIZ,.L-M), - _

we see that (IL &OO]Z, I?, LO) = 0 if L + II + 12 is odd. This is just the condition that L be even and the parity odd, or vice versa, and for such states we conclude that I II I,. _LM) = 0 where rI X ‘2 = 0. We observe that other states have non-zero partial widths and non-zero values for p(a).

4. A classical interpretation of p(BrJ

In classical mechanics, B’ measures the distance between the two individual Runge-Lenz vectors. Herrick and Sinanoglu have represented the role of B2 by the two elliptical systems in figs. 8a and 8b [ 141. In the case of fig. 8a, two electrons move in the same classical orbit but in opposite directions_ Therefore L = 0, and B =a1 - u2 = 0. Here one effect of the coulombic repulsion on such a constrained system is to increase the energy of the system because the electrons must spend considerable time near each other. Fig. 8b represents two electrons in oppositely directed elliptical orbits. Again L = 0, but B* is equal to the maximum value for fixed values of a1 and o2 _ One effect of the electron repulsion is again to raise the energy of the system. However, because the orientation of the Lenz vector keeps the electrons out of each other’s way for a greater part of their orbital _ motion, the energy is not raised as much as in fig. 8a. We would like to expand on the insight provided by these classical figures in order to give some accounting for many of the gross observations and trends previously noted. Figs. Sa and 8b will be viewed as bigb and low energy prototypes for L = 0 systems with and without-the consideration of l/rrz repulsion. Classically, B' is not strictly conserved when full account is made for the l/r12 interaction; quan- turn-mechanically, l??- is approximately diagonal for

248 P. Refvnus et d&aria1 correlation of aromic electronr He**

(bl

_I~ 0.5 I.0 1.5 2.0 2.5 3.0

8 12

Fig. 8. Classical systems with zero angular momentum [ 14]_ (a) has aI = 82. and intrinsically has higher repulsive energy than in (f~) where the Runge-Lenz vectors are spread H radians. (c) graphs the classical density function of 812 for differing values of the eccentricity corresponding to (a). Both electron paths have the same eccentricity. (d) graphs the classical density function of 012 For differing values of the eccentricity corresponding to (b). A_eain, both electron paths have the same eccentricity.

these intrashell states. To simplify what follows, we make two assumptions. First, we ignore the precession introduced by l/r12 in the classical two-electron problem, and assume that B’ is conserved. Secondly, we assume for this part of the discussion that the two electrons have equal energies.

A simple classical rationalization demonstrating trend 1 can be obtained just by spreading the Lenz vectors from 0 radians in fig_ 8a to ‘R as in fig. 8b. It is interesting ?o derive the classical probability density function for the two prototypes with the implied symmetric initial conditions (or choice of initial

phase) of figs. 8a and Sb*_ These classical density functions appear to be independent of the initial choice of phase. The result for a system with L = 0, B2 = 0, and without Coulomb repulsion is:

p(B1*) = [( 1 -- &3”/2a1

x [(l--ECPS~~~2)-2+(1+ECOS~et2)-2]. (4) where E is the eccentricity of the individual electron

* See for instance ref. [27! and interpret the distribution function for ff’& as the time 6 12 is between 0 degrees and 0 12 divided by the total pfriod.

orbits. The analogous function for the case where the Lenz vectors are spread by ?r radians is:

p(0 12 ) = [(l - E’)3”/27r]

X [(l-.5sin~012)-2t(l+~sinf0ta)-2]. (5) figs. 6c and 6d show p(B12) based on eqs. (4) and (5) for varying eccentricities. Because these classical systems have L = 0, both eccentricities must be equal. Note first the rough correspondence between the density distributions of the classical systems, figs. 8c and 8d with the analogous graphs for the two quantum- mechanical systems, figs. lc and 8h. Qualitat&ely we note that spreading the two Runge-Lenz vectors from 0 to ‘II radians will shift the density from 0 to ‘II. Trend 1 can be seen as a consequence of the physical

spreading of the Lenz vectors. In the II = 3 shell there are three 19 states, figs. 6a, 6j and 6s. The state with intermediate B2, energy, and autoionization width, fig. &j, has oscillations in its angular distribution that prevent its simple interpretation, but Ml21 is nearly n/2.

Well-known results relating eccentricity to energy for a particle with negative energy experiencing a central force are [27]

P. Rel~mus et al.fSpatial correlation of atomic electrons: He** 249

l?!= [El-‘, p cc IEl-“2, where E is the energy, OL is the major axis of the el- lipse and /3 is the minor axis. If we increase the energy of the system, as one would by increasing the principle quhntum numbers 11~ and 1z2, then we see that Q! increases faster than @, and hence the eccentricity increases. For our classical L = 0 system with B2 = 0, the effect on ~(0,~) is seen in fig. 8c and the quantum-mechanical results for states with B2 = 0 and increasing N can be seen in fig. 4a. A similar comparison can be made between figs. 8d and 4b for systems with maximal B*, Such comparisons constitute a consistent rationalization for trend 2. As noted be- for&, the systems with lower B2 intrinsically have somewhat higher energy than those with higher B’ due to Coulomb repulsion effects. This means that the individual eccentricities in fig. 8a are somewhat greater than the corresponding eccentricities in fig. 8b. Since the peaking of p(Br2) seen in figs. Xc and 8d increases with E, we expect classically, and see quantum-mechanically as in figs. 4a and 4b, that the higher energy lSe state will always be more peaked at 8171 = 0 than the low energy ISe state is peaked at e12 = 51. This comparison is made of course for states for constant ltl and r22. Within this limited classical model we thus have a reasonable basis from which to infer trend 4. An alternative way to view this is to note that a system which puts the electrons in close proximity most of the time is a system where the relative momentum of the electrons is low for a great proportion of time. Therefore, on a time average, there are repulsive interactions as well as the “elliptical” effects from coupled precession that enhance p(O) for the high-energy states. With respect to the low energy states,p($ cannot reflect this repulsive effect as strongly.

tion of the individual orbit eccentricities increase with L. This accounts for the increase of angular correlation with decreasingL noted in trend 3.

5. The mutual dependence of radial and angular correlation

By concentrating on ~(0~~) one effectively ignbres radial correlation by averaging over both Y, and r7. We now turn to &I-,. r2,0L2) to provide a graphic demonstration of the dependence of the interelectronic angle and the two radial distances. Since the requisite four-dimensional graph is unconstructable, we compromise and use the conditional probability density function ofB12 and one of the radial distance, say ‘2, with the other radial distance rI Gsed, e.g. ~1 = a. This function will be denoted p(1-1: 012 lrl C(U) and is given by

The conditional probability density function (6) is convenient because it can be graphed in three di- mensions, and it allows the same scale to be used for different values of a. Moreover, it retains the charac- ter of a probability density since

1 =_fdr2/dO12(ir/2) sinB12~V2,~121~1 = 4,

for any at. Unfortunately,p(r,,0121’1 = IY) does not treat the two electrons equivalently; however, multi- plication of (6) by S will regenerate this equivalence.

These classical prototypes have been L = 0 states, We note that since S is the marginal probability yet qualitatively, the attendant rationalizations can density for rl being equal to ot, then S actually is a be generalized for states of any L. ForL > 0, the direct measure of the relative probability that r1 = (Y. classical system is not restricted to a plane, nor are We have attempted to make this clear by providing the ellipses necessarily of the same eccentricity. This S along with each graph of (6). For a given state, the tends to shift t52 density toward 7~/2. Since the maximum value of S is attained when 01= (rl), and if jacobian for three-dimensional systems involves a for instance two successive graphs have respective val- factor of sinfI12, peak heights must be reduced if nor- ues of S which are in a 10: 1 ratio, then the relative malization of p(B12) is to be maintained. The effect amounts of time that the electrons have these prob- increases withL, since the twisting or bending of the ability densities is 10: 1. Equivalently, if electron 1 is ellipses with respect to one another, and relative varia- improbably close to the nucleus, then WI’ will reflect

P. Rehmus et al./Spntial correlntion of atomic electrons: He** 250

r, zO.7

S10.0524

0 08 16 2.4 T

Ql2

0 GB

Fig. 9. p(r2,~?&-~ =cz)versusr~ and 0 ~2 for the helium DESB tSe K. T = I, 0 wavefunction in the n = 2 shell. The swle factors is the value of the denominator in eq. (6). When its value is compared between differentgraphs for differing (Y, it serves as a relative measure of the probability that the electrons have a distribution corresponding to the compared graphs. When Q islarger than the largest value given, (e.g. CY = rt = 2.875 ingraph (e)) thegraph does not change qualitatively from graph (e). Distances are in atomic units.

P. Rehnucs et at./Spatiat correlation of atomic electrons: He** 251

this be being very small, and S will be corresponding- ly small.

Figs. 9a-9e show plots ofp(r2,B1$r1 = a)) for the low energy lSe (= (2~)~) in the II = 2 shell. In the exact wavefunction, the asymptotic density for one electron as the other collapses to the nucleus should approach the hydrogen 2s rather than the He+ 2s COII-

figuration. The fact that fig_ 9a is basically the He+ 2s configuration is a direct consequence of the in- flexibility of the basic set. For He, the DESB utilizes radial functions which are solutions to the He+ problem. The small admixture of the (2~)~ configuration, allowing for the electron correlation, can only contribute appreciably to the wavefunction when both electrons are far enough from the nucleus so that the ~-dependence of the (2~)’ configuration does not cause this contribution to vanish. Hence, for this state, when one electron is (improbably) close to the nucleus, only the (2s)z configuration with its He+ radial function contributes appreciably to p_

As rt increases from small improbable values to a! = 0.7 ati (note the relatively small value for S compared with that of fig. 9e for instance) the density develops a bulge at O,, = 0 for r? = 2.2 au (Fig. 9b). As r1 continues to increase, the density bulge under- goes a rapid shift to r radians (figs. 9c and 9d). For this state, the density does not go to zero between a12 = 0 and 012 = H, as it would if there were a node at some 0t2 in 9(rt ,r2)_ The most probable value of either of the radial distances for this state is 2.875 au, and the graph in fig. 9e does not change qualitatively for larger rt than this. The spatial density function has the general appearance of figs. 9d and 9e for 90% of the time. This accounts for the distinct bulge at B radians in ~(0~2) as shown in fig. lc. For this state we note in summary that if both electrons are at least about 0.8 or 0.9 au from the nucleus, then the angle between the two is nearly always close to s radians. If one electron is farther than 0.8 au and the other is closer, then the angle tends to zero. Finally, if both electrons are within 0.8 au then there is only slightly more probability for 012 to be x than any other value. These observations are interesting in light of the fact that much of the density for the ground state of He+ lies within 0.8 au. The relation between p(rZ, O&, = CY) and the autoionization width for this state is not clear. This state has a calculated autoionization width of 0.124 eV [28], the maximum

for states in the 81 = 2 shell. In providing contrast we examine the analogous

set for the high energy tSe state (= (2~)‘) in the II = 2 shell, figs. IOn-10f. While not perfectly symmetric, the graphs for the R, T = - I,0 state appear to be qualitatively the same as those for the low energy IS2 state, except that all densities are reflected through 8t2 = ~12. Again, the small amount of (2s)? in this state dominates for small CY. Note that fig. 111, which reflects only angular information about the system, can only suggest that the electrons spend about 85% of their time in the situation graphed in fig. 1Of. For the higher energy *Se state, when both electrons are near or both are far from the nucleus, then ~(0~2) tends to be large at 0t2 = 0, while when r1 and rZ are rather different, it becomes likely for the electrons to be R radians apart. Figs. 10e and lOfare represen- tative of the densities when both electrons are near their expected radial distance of 2.625 au. One sees that with both radial and angular coordinates ac- counted for, the two electrons are very close to each other much of the time. This state of affairs can be noted easily by the calculation of moments ofrlz, but then a separate accounting must be made for the locations of the electrons relative to the nucleus [ll] .

Despite the accuracy of its eigenvalues, the DESB is not adequate to represent the full correlation given by more nearly exact wavefunctions. (A comparison of DESB and accurate wavefunctions will be given in a later publication.) The absence of a Coulomb hole for this high energy lSe state emphasizes one basic qualitative limitation of the basis set used to generate the DESB. Additionally, the DESB wavefunctions are not orthogonal to the infinity of states of the same symmetry below the first ionization threshold. This allows for a possible “sag” of the calculated energy of a doubly-excited quasi-bound state, below the true energy. As table 1 shows, the effect of this non-orthogonality, if appreciable, is more than com- pensated by the misrepresentation of the Coulomb hole. We note that the error in the energy increases with the energy of the wavefunctions in a single shell and with increasing coincident electron density.

The spatial correlation of the II = 2 tSe wavefunction has a rough correspondence to the radial and angular dependence in the classical prototypes for L = 0 states, figs. 8a and 8b. In fig. 8a we can make the same set of qualitative observations as were made

252 P_ Rehms et d/Spotid corrdation of atomic elecrrons: He**

i

0.4

P ‘r = 0.01

(a) 0.2 S=9.6x10-6

I, i111lY4.4 0 OIS I.6 2.4 P

0.5

!t

tizo.7 P 0. s=zoxlo-*

0 08 I.6 24 ‘IT

1 0.5

0.3

P

06

(d)

0 08 16 2.4 r

0 0.8 I.6 24

e 12

Fig- 10. (d-(0 dot &z.012lr1 = Q) for the *Se K. T= -1,O wavefunctionin the n =2she~_ Also see comments for fig. g.

P. Rehmus et d/Spatial correlation of atomic electrons: He** 253

(a)

r, a3

fc) WO654

0 08 1.6 2.4 r

42

0.6

0.4 P r,=l.2

k) '.' S=O.l36

0 OE 1.6 2.4 r 0 08 1.6 2.4 71

42

0.6

0.4 P I, = I.0

(dJ 0.2 s= 0.0594

8 12

0.6

0.4

P r,=2

(f) O.* s=o.342

Fig. 11. (a)-(0 plot&,$2lq =o$ for the 3Po K, T= I,0 wavefunction in then = 2 shell. Thesegraphsare shared by the ‘PO K. T = 0,l wavefunction in the same shell, but each graph is reflected through 8 12 = r/2. Also see comments for fig. 9.

for figs. IOa-1Of: when both electrons are very near or far From the nucleus, then 19~2 is near zero, and at

some relatively improbable intermediate distances (consistency with quantum mechanical results suggests that about 0.S au would be appropriate), their angle tends to open up to x radians. Conversely, we can compare qualitative observations made for figs. 9a-9e with those following from classical fig. 8b: when both electrons are very near or far From the nucleus, then 0t2 is about in, but when one is far and the other is close to the nucleus, then the angle closes to zero radians.

Similar threedimensional graphs describe the remaining states in the tz = 2 shell, which are equivalent to the Russell-Saunders states. Figs. 1 la-l If show the ;Po state which is energetically the second lowest state in the shell. The degree of angular correlation apparent in Fig_ Id is not provided by configuration interaction but rAther by the exclusion principle forcing one eIectron into a 2s orbital and the other into a 2p orbital. The resulting Fermi hole however is only a partial realization of the Coulomb hole because of the spatial overlap of the 2s and 2p orbitals. The 1p state shares the same set of graphs in the DESB, but the density is reflected through 0t2 =7-r/2. In comparing these two PO states with the W- states, note that the same qualitative radial and angular

t 0.6

0 0.8 1.6 2.4 r

012

Fig. 12.p(r2,fJt2lt-t =P) for the ‘DeK, T= 1.0 wavefunction. This graph applies for all values of ~1.

relationships exist, but that the pronounced trough which forms in the progression of graphs for the S states is missing. Since this trough occurs at about 8t2 = ir/2, we expect that the overall correlation is much less pronounced in accordance with trend 3. An early study of the radial correlation of the 1p function characterized it as exhibiting simultaneous in-out motions of the two electrons [22] _ The inclu- sion of the effects of anglllar correlation demonstrates

how this motion would have to be accompanied by rapid and gross shifts in the electron’s angular orientation when either enters the effective radius of the He+ orbital.

Finally, figs. 12 and 13 present analogous graphs for the nearly uniform lDc state and the “parity disallowed” 3Pe state with its characteristic nodes at 0~ = 0 and 012 - - pi for all radial values. Again, like the 3@ state, the 3Pe state forces the electrons into different orbit&, (this time p orbitals) and the electrons spend their time distributed at 0t2 =x/2. The node at 0t2 = 0 insures that the Fermi hole qualitatively describes the principle characteristic of the Coulomb hole, whereby there is no probability for the electrons to coincide.

A partial analysis of similar three-dimensional graphs for states in the higher shells appears to indi- cate that the gross features of spatial correlation is patterned after that seen in the tz = 3 shell.

c 0.6

0 OR 1.6 2.4 r

Fig. 13.p(Q,8121rI =a) for the ‘peK, T=O,l WVefUnC- tion. This graph applies for aU values of a.

P_ Rehmus et al./SpatiaI correlation of atomic electrons: He** 255

Acknowledgement

We would like to thank Michael Strand for helpful discussions, especially concerning the nodal behavior

of me wavefunctions, and Ugo Fano for his insight- ful comments. This research was supported by a Grant from the National Science Foundation.

Appendix: The angular transformation from 0, , 02, @I and $J~ to 012

In what follows G(O, ,&,@t ,&) is the angular portion of a squared Iz-electron wavefunction, in terms of the spherical angular cooniinates of electrons 1 and 2. The other variables in the problpm, e.g. the radial variables for electrons 1 and 2 and the coordinates of the other n-2 electrons, are extraneous and are not involved in the transformation. Thus they may be ignored for instance by assuming that the requisite summing and integration has occurred over them. Because radial coordinates are not involved in the transformation, their form, and the form of the radial portions of the wavefunction are completely irrelevant. When the radial variables are discussed they will be referred to as the usual rl and r2 of the spherical coordinate system, but in this appendix this is only a notational convenience_ A number of reductions occur for the rz-electron result when II = 2. These reductions are also independent of these extraneous variables.

We seek to solve the quadruple integral:

jdOtjd02~dQI~dQ2G(B 1’ 2’ 1’ L? 0 Q Q )sin0tsin02, (A-1) 0 0 0 0

by a transformation of variables so that one of the new variables is

e12 = cos-1 [cosf$ cos02+sinB1sin0,cos(~1-~,)]. (A.2)

We require that the transformed integral has the form

9 = i d0,2 /;;;;’ drI [I;’ dvZ [;;;;:;I;’ dy~lJIG(T-1(8,,82,91,92))sin(T-1eI)sin(T-102), 0 2 4 * 6 7 ,’ (A.l’) c

where T-’ is shorthand notation for the appropriate inverse transformation of the indicated arguments and IJI is the jacobian. TheJpi’s are dummy variables selected later, such that integration over theyi’s leaves ~(0,~). The integral (A-1’) is solved - i.e., reduced to a function of 0t2 - by the folfowing steps:

,( 1) by transforming from @1, #2 to new variables ql, q2; (2) by obtaining 0t7(e1 .02, ql. q2); (3) by transforming from O,, 02, ql, q7 to 81,e2,012,~2; (4) by trivial integration over (12;

.

(5) by an anaiysis of the jacobian when we expand the remainin, m integrand as a terminating power series in sin 01, cos 8,) sin O2 and cos O2 with 012 appearing only in separable factors and in the jacobian;

(6) by integrations by parts over O2 and finally, (7) by implicit integration over B,. First consider the transfonnation:

41 = cos ($1 - G,), (A.3)

92 =+I +c2- (A.4)

This transformation is not 1 to 1. The following 4 regions in the @I, & plane will overlap to some degree in the

ql. 42 plane:

region I:

256 P. Rehmus et al./Spnrial correlntion of atomic electrons: He**

region II: 0d$2-@lGr,

region III: l&&-9*<2”,

region N: “G$~-$ i27i.

hr region I, from (A-3) the inverse transformation is

$51 = i(q* f cos -Iqr ),

92 = $(Q - cos-‘q&

However,

(A-6)

(A.7)

(A-8)

(A-9)

(A-10)

iu region II cos-Iql = 9, - I& f 27i, (A.1 1)

in region III CO&l = @, - $I1 + 2a, (A.12)

in region IV Cos-IqI = G2 - 9,. (A.13)

These lead to four distinct inverse transformations to be summarized below. With regard to the following boundary equations of region I, we have the respective boundary equation in the qt. q2 plane:

@* = 0 0 q2 = CC&r, (A.14)

9, =2iieq, =4n-cos -1

9, =Q+r&r =-I,

ql, (A.15)

(A.16)

+ =+,*qr = 1. (A-17)

Table 2 summarizes the boundary transformations. Fig. 14 diagrams the compIete mapping of ff I, 192 space into ql, q2 space. Thus

c 4r-cxx-‘q~ 4?i-cos-‘q,

x L-5& dq,W--l 11)) + s dq,W-l [II))

cos-‘ql

2n+cos-‘ql 27i+cos-lql f

J dq#(T-1 {III}) f $ dq,G(T-‘CIV)) ,

Zn-cos-57, Zrr-ms-lq, I (A-18)

where it is recognized that the absolute value of the jacobian is equivalent in all regions, and is equal to l/2( 1 - q”)?

Now c&tsider the transformation:

YI = 0,s y2 =8,, y3 =t2, y-812 =cos~1(cos~~,cos8,~sin8~ sinB*qI). (A-19)

This transformation has only one inverse transformation and is thus 1 to 1. The inverse transformation is

01 ‘Yl’ 0, =y2* 42 =373, q1 = (cosy - cosy, cosy2)/siny1 sinyz. (A-20)

In a strict sense this is a four-dimensional transformation but q2 does not enter the transformation to any greater extent than 8, and 82 did in the previous one. (Remember 0r and 0, were ignored because they were not being transformed.) Therefore we focus on the three-dimensional transformation involvingy,yl andy2. The three- dimensional space in ql, &, e2 will be called 0 space which maps into the correspondiigy space. Because of the simplicity of the inverse transformation foryl and yz, the following boundary planes in 0 space .yield the associated planes iny space:

P. Rehmus et al.jSpatial correlation ofatomic electrons: He*’ 257

Table 2 Summary of QI, $2 to 41.42 transformations

Region in $ space Transformation T-1

I

II

III

Iv

41 = cm (@1 - &I

42 =ol+Q2

q1=cosb,-dz)

q2 =91+@2

(II =c0s(01-@2)

q2=01+02 -

41 =cos(QI -02)

42 =@I+@2

@I = $42 + cos-‘qd

02 = &2 - cos-%I)

91 = $Cs* - cos-+?t)

Qz = $42 f cos-‘q1)

91 = f(Q - cos-‘q1) + T

62 = $(q2 + cos-‘q,) - P

01 = i(q* f COS_Ql) - II

02 = $(q2 - cos-‘q1) f 71

Boundary in q space

*2 = cos-‘q, 41= 1;

q2 =4n - cas-‘ql q1= -1

as in region I

q* = 2n f cos-‘ql 41 = -I;

q2 = 27-r - cos-‘ql

as in region III

e1 =I?=-y, =rr, e1 ‘o-y1 =o, e2=7i*y2 =I?, 8, =ooy, =o. The boundary equation Q = 1 is significantly more complicated_ 4 I = 1 implies

cosy = cos 13~ cos e2 + sin @I sin B2 = cos (9, - 02)_

(A.21)

This is a 1 to 2 transformation of one plane in B space to two planes in y space. For 0 1 - o2 > 0 the equation COSy=COS(~~-82)~Y=Y~-Y2bUtfOr~ 1-e2< 0, ~osy=cos(O~ -02)*y=y2-yl_ Analogously,the boundary plane q1 = -1 maps into two planes with the result:

(AX)

These planes (A-22) iny space bound a regular tetrahedron with vertices (y, yl, yz) = (O,O,O), (O,a,~r), (l,O,n) and (1,x, n). (See fig. 15.) The set of planes (A-21) do not bound a closed volume in three-space; hence, because of the countable subadditivity of the measure, these planes have measure zero, and wilI not be considered further. For y between 0 and n/2, say y = Q the region for integration in the yL,yZ plane is the crosshatched region of

Fig. 14. Diagram of the mapping from @ space into q space. The crosshatched region represents the 4: 1 mapping from regions III, IV and part of regions I and II. The rest of regions

m Kegion in q space mapped 2:l from $4 space. Iand II map 2: 1 Into the hatched region of the figure.

258 P. Reirmus et al./Spatial correIatiorz ofatomic electrons: He*’

i I

__________ _____= $rJl-,Tl)

I __-- e- -0 I

_-

I~~o,o);K-__ ___-__ ______ ** .- :

I :

t \ I

;

I

.ii

\; ;

1

I t I

I I .

1 I

I I I

1 :_,.-‘~~

: __= - i0.rr.m I _-- _-___-_-_ _____ A-‘-- -----,

(0.0.0) Icl.lT.0) ?,

Fi:. 15. Diagram of the boundary planes inY-space, eq. (A.22), that enclose the tetrahedron (solid lines) in which integration takes place.

fig. 16. This is of course a slice of the tetrahedron of tig. 15 at a height ofy &a[. The boundary Iines for this rect- angle at a height ofy = cr are: y 2 =y, -4y2 =y1 +cu,yz =‘-)‘I +CY,y? =2n-o-yyl.

There are several ways to solve the yr.y? (plane) integral. One way i break it into three parts:

y1 -cx<y,<yf +cr, fora=Syf<7r--_; -yl + (Y Gy2 dy, + Q, forO<y,<a; .F_.-.’ -y1 + 2Tr - (Y <yz <y, - (Y, fors-cr<yfGn; fory=o, O<czdr7/2.

Integration over they plane gives three terms:

for 0 <y < 7r12.

Analogously, fory greater than rr/2, the three terms are:

-J+2r-y

J s dy1 _,‘I+..

dy2 -tj-Y dyl j’+Y dy2 +j dyf [Y’i?dy2. x-J, 0 -Y1 +Y Y Yl -Y

A glance at the set of eqs. (A.20) reveals that the absolute value of the jacobian is merely: sin (y)/sin (yl) sin &) for all valid y, y1 and ~17. Substituting for the variables in 8 space, we have the final result:

9 =[“[dy j-’ dyl p-” dy., +s’ dyl p” dy7 t i 0 Y Y1-Y 0 Y -Y1 “-,

dyl T-‘-” dy z] Y1-Y

X f siri_v 1 - [ ( cosy - cosyl cosy, 2 -112

_ -)I siny, siny,

‘lrt-y -t Pt~d~~G().~,~~,t~~-7)fn,t(y3fy)--ir))+ j

2n-y dY3GCYl.Y2&+7r) - T;CY--Y) +.rr)) ,

2n-y I

fory G n/Z, where 7 = cos-’ [(COSY - cosyI cosy,)lsiny, siny,] , and (A.23a)

259

Fig. 16. A horizontal slice of the tetrahedron of fig. 15 at n height of y = a.

= L dy f d~$-“-~’ dyz +j; dyl p” dy2 +j dy,T-“-’ du-, “-Y Yl -Y Y-Y1 Y-Y1

X i siny 1 [ I(

cOsy--cOsyl cosy7 2 -l/2

_ -)I siny, siny, [ IY > fory > 42. (A23b)

This transformation applies to an Iz-electron system, but it cm be simplified immediately for a two-electron system because /9[2 will not depend onyg. Therefore, for a two electron wavefunction, the density function for BL2 is given by

n-42 &+@I2

~6’~~) =l do1 j. d02 +j” dBL p+0’2 d8, + i dO,~-e’z-el 47r~‘l’(B,.02_012)~2

012 f31-412 0 ~12-~1 77412 Ql--42

X sineI 1 - [ f

COSO12 - COSOl coso2 2 -112

sin 8: sin O2 )I > for 012 G x/2

X sinO12 I - c ( cos o,, - COSOl cos e, 2 -112 sin 0, sin O2 )I 9 fory>7&!,

(A24a)

(A24b)

where we have substituted for y. y, and y7 _ In associating the result with ~(0~2) we implicitly integrate and suin over the radial and spin dependence. (If the radial dependence is left unintegrated, then the result is that referred to as p(rl. rz,e,,) in the text.) The full angular jacobian,J, is given by

“0 COSO12 - cOsel c0s e2 2 -112

2 sin 12 c ( I1 - sin 8, sine, )I .

Further reductions are possible, though they do not become evident without an analysis of the jacobian. It must be determined if the product of a multiterm wavefunction with this jscobian can be integrated through the integrals over Q1 and Q2 in such a way as to keep the two contributions from the jacobian separate from the con-

260 P. Rehmus et al. fSpatia1 correlatibn of atomic electrons: He**

tribution due to the squared wavefunction. Because the jacobian places Or2 under a radical, one must check to see if the averaging process done by integrating over 6, and O2 might scramble these two contributions. We want the fttnction of Or, contributed by [@[z, and interpretation of p(Br2) would be much more difficult if each term in ~(tYr2) involved inseparable or varying quantities due to the jacobian. We will show however that when integration over 8, and 8, is completed, the result is equivalent to an integration by parts where the contribution from the jacobian is separable and independent of the wavefunction. Since the following argument depends only on the functional dependence of Iq12 on 01, S2 and 012, explicit reference to other variables will be dropped.

The density function of Or2 (A.24a, and A.24b) is a sum of terms of the form

Pt ;8nJ dOI cs+@zI df?21~(01,02,~12)~2f sinB12[l -(case~~~:~~~~:O18?)3-1’2, (A.2.5)

bt - t where we are assuming that lU[” is a two-electron eigenfunction of L2 and L,. The quantities nt, b,. tzt and mt are appropriate to a particular term in p(6,,). The k sin (et,) portion of the jacobian may be slipped through the integrals and will be ignored for the moment_ We restrict attention to

+9~~W~+27rmt)

J- s do1 de2 iwel,02,f+,)t2 I- cos 012 - cos 81 cos e2 “1-W

bt +(e,*--B,+2nmt~ IL ( _ 1 sin e1 sin 19, J ’

where the underscoring will be drawn under that portion of an integrand due only to [\k12. Later it will also ap pear under those portions of a completed integral due to I*[’ after integration by parts.

The squared wavefunction in terms of Or and 8, can be written as aft&e sum

(A.26)

We will work only with one term of thisseries, though the results will clearly apply to all of them. The e12 dependence is contained in the Qi_ Though computational algorithms using the results of this appendix are best given in terms of spherical harmonics (algorithms and computational theorems will be made available elsewhere) the following proof is easier in the sines and cosines. It can be shown that the powers of the sine functions are even, and analysis is then of the function:

i-(Or2+e,+2nnt) I=rdQI s de2 co? O1 sinti 0, siti3 0, cosp 0,

-s ( 1 -

~0~8~~ - cOsel ~0~8~ 2 -112

bt +(012-OLf2nmt) , sin O1 sin e2 )3 ‘(A-27)

Make the following substitutions: x = cos 0, and y = cos 0, with the result:

COS-‘Q, ~ws(et2+c0s-?r) I=1 dxj du x”( 1-x2)myp(l-y2)~ (a + by - _Y~)-I’~,

COS-'bt cos(e~~-cos-bx)

(A.28)

where Q = 1 -x2 - COS~O~~ and b = 2x cos 012 . It is clear at this point that the inner integral is independent of which term in p(Br2) we are evaluating. Therefore, for all termspt, the innermost integral reduces to the consideration of i

P cos(e~*+cos-'x)

I'=ZS dy a.$’ (Q + by - JJ~)-I’~

II , (A.29) cos@r~-cosx)

where ki = q + i and oi is an appropriate binomial coefficient multiplied by some function of x and f?,,. Again we will consider onby one term though it will be clear after analysis that the results will apply to all terms in the sum. We consider then one term of the inner integral:

P. ReI~mus et aLfSpatia1 correlatiotl of atomic electrons: He**

61 I” =s

Q dyd (n f by - y’)-I”,

261

(A.30)

Integrate I” by parts to yield:

dy (-k)gk-l ’

cos-l t, (A.3 1)

where 5 = (-Z!y f b) (b’ + 4a)-@. Now make th e s u bstitution: 4 =y + s2 where CZ = -? ZI. This yields

dq (-k)(q-a)“-’ cos-1 4q, (A.32)

where A = -2(b’ + 4a)- U2 Aoain for the right-most integral, expand the quantity whose parentage is IW’ and . b consider one term:

C+R I”’ = 1 dq&cos-’ 4q. (AX)

8,+Q.

Integration by parts yields:

I”’ = rl IC+1

- cd’4q ,x-+l,

The right-most integral is zero if k f 1 is odd and equal to

-@+I)! 6,+n

[(k+l)/‘P] +A++ ‘OS-’ Aq 6,+n ’

if k + 1 is even. Therefore after numerous resubstitutions we have that:

(A.34)

(A.39

where fli is an appropriate sum of coefficients, each of whose terms is a product of functions ofx and 0t2_ How- ever, COS-~~, = 0 and cos-102 = n so for the inner integral we have:

I=$nsine12 CB,Yilo2, (A.36)

remembering the 4 sin B,, factor. The jacobian therefore has served as an integrating factor whose contribution to that integral, as measured by the factor it introduces after integration by parts, is just $ R sin .9,, _ Specificalty for an S state which has no dependence on 61 and e2, I@ merely slips through the integral uponktegration by parts, and &he jacobian left behind integrates to t x sin 0,, as it must.

For the outer integral overt we find that the jacobian is 1 and hence ifp(fI& is the probability density of 012, then a factor of it equal to *n-sin812 is due to the jacobian and the rest is attributable to l\k12. Looking specifically at the sum of terms involving integrals overx we see that for each domain of 8,,

(A-37)

Further we note that (A.37) yields the same functional dependence on BI,, regardless of whrther 0 I, is greater

262 P. Rehmus et al_fSpatiat correlation of atomic e!ectrons: He**

or less than IT/~. Therefore for two-electron wavefunctions, the transformation reduces to a single term:

(A.38)

where the resulting quantity, after integration, inc!udes 1 rrsin e12 as the jacobian. From the form of (A-26) we note additionally that the exact reduction of I@[’ to ~(0~2) can always be achieved in closed form.

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141

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171

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I101

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