THEORY OF ELECTRON CORRELATION IN GROUND AND EXCITED STATES WITH APPLICATION TO ATOMIC PROPERTIES

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THEORY OF ELECTRON CORRELATION IN GROUND AND EXCITED STATES WITH APPLICATION TO ATOMIC PROPERTIES

O. Sinanoğlu

To cite this version:

O. Sinanoğlu. THEORY OF ELECTRON CORRELATION IN GROUND AND EXCITED STATES

WITH APPLICATION TO ATOMIC PROPERTIES. Journal de Physique Colloques, 1970, 31 (C4),

pp.C4-83-C4-88. �10.1051/jphyscol:1970414�. �jpa-00213869�

JOURNAL

THEORY OF ELECTRON CORRELATION IN GROUND

AND EXCITED STATES WITH APPLICATION TO ATOMIC PROPERTIES

Dept. of Chemistry alid Molecular Biophysics, Sterling Chemistry Lab.

Yale Uiiiversity, New-Haven, Conn. 06520, U. S. A.

Résumé.

Diverses approches pour l'introduction de la corrélation électronique dans l'étude des structures atomiques sont examinées rapidement ; on montre en particulier que la théorie de corrélation de paires de Sinanogi~i et des ses collaborateurs s'applique surtout aux couches fermées et au inie~ix à quelques états fondaiiientaux comportant des couches ouvertes. La corrélation élec- troniqiie, aussi bien pour l'énergie que pour la fonction d'onde d'un état excité d'~in atome

plusieurs électrons, a un coiiiporteriient trcs dinërcnt, qui peut cependant être également étudié à l'aide d'une théorie récente. Lcs points essentiels de cette théorie des états excités sont exposés.

On en présente des applications aux affinités électroniq~ies, aux énergies d'excitation et aux pro- babilités de transition, et les résultats obtenus sont comparés aux données expérimentales les plus récentes.

Abstract.

Vario~is approaches for incl~rsion of Electron Correlation in Atomic Structures will bc exari~ined briefly, showinç in particular that the pair correlation theory by Sinanoglu and co-workers applies iiiainly to closed shells and at best to soinc other ground states. Electron corre- lation both in the cnergy and wave fiinction of an excited state many electron atoni displays a very diffèrent behavior for which however a recent theory is also available. This excited state theory will be s~iiiimarired

applications to electron afinities, excitation energies, and transitioii probabilities will be given, and the results will be compared witli the results of especially recent experiineiital techniqiies.

1. A More General Definition of Correlation Energy.

We shall define the « Correlation energy

of an atom of arbitrary Z , niorc geiierally tlian wliat is customary, by

The EHELRHk is tlie relutici.stic Hrrrtrc~c>-I;o(k eiiergy obtained fiorn the Hamiltonian

where 11: is tlie Dirac Hamiltonian for electron i and the Rreit interaction.

Tliiis defined EH,i~o, incorporates relativistic efrects on correlation. Past sodium atom relativistic energy corrections and correlation eiiergy are comparable in magnitude so tliat we expect an iiifluence 01' tlie two effects on cach otlier. These are incliided in ERELc,R whicli may be obtained by ail extension of the corre- lation theory below.

Only where 1 E*,, 1 ⁶ ( ECU,, 1, caii we furtlier appro- ximate eq. (1) and write

a n d

For atoms up to about Z - ^{I I} we assume as is customary eq. (3) and (4), and discuss tliis case below.

II. Various Approaches for Treatment of Electron Correlation in Atoms.

Thcre are corrclatioii c f i c t s of a quite different nature iri cxcited statcs (geiierally non-closed sliells) than in closed shells. Tlic tlieory of correlation efîects for nori-closed sliells lias becii developed by tliis writer aiid CO-workers [ l , 31 :iiid applied to various atomic properties [3, 51. It allows systcmatic treatment of correlation e f i c t s in general excited states. Thcre is of course also the traditional configuration-interaction (C. 1.) series tliat in prin- ciple could be used. However witliout additional tlieory, tliis technique rcquires too large a calculatiori (usually beyond tlie capacity of preseiit cornputers) and does ~ i o t indicate Iiow to select al1 the significant correlation elrects.

For Ground States tliree modern approaches Iiave been developed (al1 since about 1960) and given a quantitative, yet physical understanding of clcctroii correlation in atoms. These are :

1 ) Tlie pair correlations and Many-Electron Tlieory of Atoms and Molecules (« MET))) by Sinanoglu [6, 9, IO].

2) Tlie non-uiiiforni electron gas approacli o f Brueckner aiid M a [II], and.

3) The diagrammatic many-body (field tlieoretic)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970414

C4-84

SINANOGLU perturbation techniques as implemented first by

Kelly [I21 and in tlle excellent recent works of Das and Amusla.

The three approaches and earlier work are discussed In a recent book by Slnanoglu and Brucckner [I31 wlilch includes also key papers reprinted from the ori- ginal literature. A combination of techniques used by Kelly wit11 the closed form methods in the theory by Slnnnoblu would be par tici~larly fruifful, but has not been attempted.

111. Different Pair Correlation Approximations and Types for Ground States.

Pair correlations intro- duced in Approach 1 were defined in several forms.

Diflcrent levels of approximations for them as well as different techniques for their calculation and stirdp of their validity were also presented.

As there are a number of alternatives given in this theory and different ones of these have been used by different authors in tlie initial calculations and in the many more, recently, we list below the different types of pairs and approximation methods. We also call reattention t o the lirnitdtion s of the pair theory which were discussed in the initial papers, but which may need to be reconsidered in view of the recent new calculations in the literature.

We confine ourselves t o closed (or quasi-closed,

i. e. single determinantal, ground) states in this sec- tion. Pair correlations for these were derived by solving the first order Schrodinger equation formally, thereby obtaining the equations

where !-) denotes variational expressions and

The first order pair correlation functions z:,?' ^are

rigorously

orbital-orthogonal

IC

1, 2, ..., N }

occupied Hartree-Fock spinorbi-

tals. Eq. (7) is used in proving that each pair can be minimized separately. For further details, proofs and deviations of tlie results stated in this sections, the reader may consult the original papers [6, 91. T o avoid repetition, we slial! list below, only the different types of pairs, their basic equations a n d references for the theory and recent applications.

Table I Summarizes different levels of pair approxi- TABLE I

A Brief Guide to the Pair Correlation Theory

Type of pair 1

Main Equations approximation

( References References for the

for ~

1 derivation applications

2. Lower bound

< 2 B(d)

+- ^{Be :} Ref. [8] Derived first as a n intermediate approximation

pairs ( Ir;,1"> between first order pairs and the He-like exact pairs.

I +

Lster found to provide lower bounds to pair correla-

L

i tions Csuld b: very useful in pinpointing actual

~ i j

< e l

Zj values of each pair correlation.

1. First order

R ' - ³ and more electron

/ correlation energy remainders 1 1

pairs correlation theory below. However a s such for some

types of pairs not accurate enough. Better to use the more general pairs below where possible.

elf' < zj:'

. He-like pairs

Ref. [6, 101

, - r

Eij

<

2 B(d)

(

) +

- -

+l+<;ij1&,>

1 _12.-

I

+

+

¹

l +

N

-,

Eeorr

+ 1

^but

i : . j

Be : Ref [8]

Ref. [7, 8, 9, lo].

Provided the basic notions of the more general pair

12, 14, 15, 161.

and E. Recently applied by Inany authors mostly using the C. I. technique. Care needed in not over- looking details of the original derivation [lo] ; e. g.

each pair is a n upper bound to its own, but their sum not a n upper bound for E. Additional coupling terms a r e involved in non-closed shell (even ground state) which have not been taken into account in some

I

I ^I recent calculations by several authors (cf. 22) and related recent papers; also (26) and (20) especially p. 252 concerning the method used in ref. 1221.

Be, B, C, ---Ne [8, 9,

Derived by subvariational principles of Sinanoglu.

Starting with the exact form of the N-electron

f.

THEORY OF ELECTRON CORRELATION IN GROUND C4-85 mations derived in the papers by this writer showing

also some recent applications and remarking on the useful and limited features of each.

Within a given approximation (e. g. first order pairs, He-like pairs, etc.) several types of pairs were intro- duced, all related by unitary transformations to each other. These are summarized in Table 11. The localized orbitals are of particular use in the analogous theory for molecules.

An important point to recall with the pairs of Table II is that in variational theory, the sum of He-like pairs is not invariant, only the sum of these and the more-electron remainders is. It would be desi- rable therefore to investigate which type of pairs

A' , .

maximize the £ c,

part relative to the remainders, as remarked earlier [10].

For each type of pair or approximation, different standard techniques like C. I., r

method and some newer ones may be used. These are summarized and commented upon in Table III.

In all the material of the Tables, the starting point

is the derivation of the exact form of the iV-electron wave function (w. f.) in finite form. Then approxima- tion of the w. f. (for closed shells) to the overall w. f.

of MET which contains the pairs and their products is made. Then certain indicated portions of the total energy are varied (subvuriational methods, « varied- portions » approach) to get the different methods and definitions of pairs, etc., as well as to get estimates of more than two electron effects. These methods should also be useful if generalized to meet the different physical situation caused by the hard core, in finite nuclei. But except for Peirls and al. to first order only, (28) they have not been applied there.

IV. Pair Correlations in Excited States and other Non-closed Shell Systems. — The additive, decoupled pair correlations theory applies strictly to closed shells (Be and Ne). In non-closed systems whether ground or excited state, the correlation theory shows three distinct kinds of electron correlation. For detailed definition and calculations of these we refer again to the earlier papers [1-3] ; we simply state

TABLE II

Types of Pairs by Symmetry

Types of Pairs

Reducible pairs (B(//)-type pairs)

Irreducible pair

Localized Pairs

Pairs with Sym- metry Configu- rations

Main Equation

P R H F = A(l 2 . . . ij... N)

s.

B ( y ) -»- uij -»-««

e. g. a(2 s„, 2 p

P) i/(2 p

ITT ^ITT ITT

<Pl -+ III - > £ i

e. g. «(2 p

- ; P3)

^ i/(2 s 2 p, Pi)

; P

) etc.

P R H F = A(12 3...N)=--

= A ( « l 111 ••• » N )

m = localized spin-orbitals

e. g. tetrahedral ones for Ne,

etc.

N N

V Eij = V e„

for exact solu-

' > j i > j

tion but not for variational result

A ( £ i / S ' , ( N - 2 ) ) ( p ( J L " S " 2 ) )

<p -> u(L" S", 2)

Pair of electros of L" S" coupled to (N — 2) e- part with V S' to get overall symmetry

References for the derivation

Ref. [6, 7]

Ref. [6, 7]

Ref. [7, 10, 19]

Comments

Convenient for closed shells, but bring about additional terms in E, « cross-pairs )) etc. which invalidates calculations when omitted e. g. in open-shell ground states. (Referred to as « stan- pard pairs » by F. Harris in the Paris Conference; these pro- cedings.)

Each pair belongs to a 2 e- irrcd. repr. of 0(3) ® Su(2) just like various Helium atom states. These pairs are the most clear-cut to use (least coupling terms, etc.) in closed and non-closed shells.

Also most convenient pairs for semi-empirical use. Values determined for atoms B through Na (3). Not used in the recent non-empirical calculations by other authors.

These and other pairs in this table are related to each other by unitary transformations [20]. However the sum of pair energies is invariant only for the exact result. In the variational theory only the sum of pairs and the n > 3-electron remainder terms is

! invariant (10). Thus depending on the type of pair used, the ratio or the pairs vs. the remainder part varies. Not clear a priori

! which type of pairs maximizes the pairs part.relative to the i remainder.

Lowrey (The- sis, Yale, 1965) Harris, Nes- bet, Schaefer (refered to as

« symmetry adapted pairs ») (these proceedings)

Not as convenient in calculation or for physical interpretation as

the other types above. Different distinct pair correlations are

mixed together in this method. They are not therefore truly

decoupled pairs as the others and may include also some un

specified part of the remainder. Thus results with these may not

be as accurate as with the others and differ from i w r more.

0 . STNANOGLU

TABLE 111

Techniques for Calculating eaclz Pair Correlation and More Electroiz Remainders

Technique References Comment

- A

Uij-trial function with rij-coordinate (Hylleraas type

w. f.

orbital-orthogonali- zed

a s indicated in MET

A

Uij-trial function with r <

r >coordinates (otherwise like the rij-method)

Pair-C. I. Additional larger C. I. for estimates of remain- ders mi+ 2

^{B(kI) in}

i . > l > N

the sub-variational methods of MET

A

Uij with continuum summa- tion

Pseudo-natural orbitals (PNO)

i

1 to each other implicitly which had been shown [22c] to be like a total all-electron C. I. rather than the recent approxin~ations [7a] used [26, 231.

Be Ref. [8, 151 l ~ i r s t accurate calculation of pairs. Convenient for atoms, but not tried for

/ molecules where method may become quite difficult.

~

I

Technique used by el-Ivariational equations [6, 71 could also evaluated by this tccliniquc. Not ly in diagrammatic used this way. Potentially powerfill technique for getting the more- manybody p r t a r b a - l electron remainder ternls e s t i ~ ~ ~ a t e s .

tion theory Ref. 12 Direct numerical solution of

pair-open-Schrodinger Equations

MCSCF

multi-confi. SCF

Ref. [16, 211 for first order and (several hi- gher orders) of pairs in Be

Step-wise C. I. proce-1) dure for obtaining in- dividual pair correla- tions plus estimating remainders derived

Kzauss, Weiss, Kutzel- From a C. I. o n one pair a n a t ~ ~ r a l orbital basis is dcrivcd, then used to nigg, Davidson and d o C. 1. on others.

others [24] I

May be one of the most convenient and powerful methods. Not used widely a s yet.

Total C. I. with all pairs at the same w. f. either too inaccurate a s in early calculation [22u] o r if large enough basisset used usually beyond coni- puter capacity. (Nesbet, Paris Confe.)

2) 90 Y: or brtter estimates of E,... becomes possible, with closed shell pair-C. I. of MET. Proposed initially a s a means of estimating highc,

McKoy and al, Musher and al. (tobe publi- shed)

with MET [22] remainders in MET [7a]. Recently applied a s a i~seful method by Nesbet [22b] Schaefer, F. Harris, E. Davidson, A. C. Wahl, A. Weiss and others the

'

recent. Nesbet, version includes one-electron effectsf,

one electronexci-

I

tations

which had b-en shown by S i n a n o l i i ~ and Tuan [27] to be small

1 in closed shells. Caution needed in non-closed shells ; not applicable

1 without the additional equations of NCMET. Also not to be confused

i with Brueckner theory 113, 231, in which all pairs are strongly coupled

Applied to first row atoms. Powerfill and new technique.

them here. They are : 1)

Internal correlations

2)

Semi-internal correlations

(including orbital

/ ~ a r t r e c , Hartree and

polarization) and 3)

all-external correlations

Used in general to get the H. F. Wave function in the several detern~inanta.

In closed shells, the first two are rigorously zero.

Thus the pair correlations of the previous section are just the

all-external correlations

the only effect possible in closed shells. These being d u e mainly to

,

the short range parts of the coulomb repulsions (llr,;),

Swirles, A. Jucys, A.

they were called

dynamical

before [7-91.

approximation for non-closed shells. However it may be used also to gene-

We have in terms of the reducible pairs

(like

~ ( p b ! ) , etc.)

N

ECoRR

1 &Ij.

(closed i > j

shell

only)

C. Wahl Roothaan rate a rapidly convergent basis for pair C . I. o n one pair, then used in others.

and others [25], 1 Possibly n ~ o s t proniising method for n~olecules.

where

&ij

(all-ext.)

( i n

(8b)

closed shells)

or in terms of the

irreducible pairs

(like ~ ( 2 p2, 3P),

~ ( 2 s 2 p, lp), etc.)

where

E =

~i;' (all-ext.) .

(in

(9 b)

closed shells)

However for a non-closed shell ground state, even though eq. (8a) and (9a) still apply, we now have

&ij ⁼ cij (int.) + cij (semi-int.) +

(all-ext.) (10a)

&jrr'

= EV ^(int.) + EV (semi-int.) +

(all-ext.) (lob)

defining total pair correlations each containing the

three contributions. It is the all-external part

THEORY O F ELECTRON CORRELATION IN GROUND C-I-87

(all-ext.) o f a pair correlation that can be transfe-

red from ion to ion, state to state, being insensitive t o the rest of the N-electron environment.

I n a n arbitrary non-closed shell state there does not seem t o be any particular advantage to the dealing with the internal a n d semi-internal correlations in pairs. Rather, just as easily and more completely, including three and inore particle terms, they are caiculable by an overall C. I. So we split u p into pairs only the cr all-external

correlations [2, 31.

As examples of the different types of pairs above, we give in Table 1V the irreducible pairs for

comparing the

experimental

values of 0ksiiz and

~ i n a n o i l u [3] which are derived from a collection of different 1 sZ 2 sn 2 pn' configurations with the purely non-empirical values of several workers.

I n Table V, also for Ne ground state, we show the

((

reducible pair

values obtained by 0ksziiz and

~ j n a n o i l u from their irreducible pairs by a unitary transformation. Localized pairs for tetrahedral orbitals have also been obtained which show the importance of interorbital correlations no matter what types of orbitals are used 1191.

For an arbitrary excited o r ground state atomic property, one calculates the internal & semi-internal w. f. added to R H F totally. These parts of the w. f.

alone, yield quite accurate transition probabilities (cf. paper by C. Nicolaides and 0. Sinanoilu in this conference proceedings) and other one-electron type properties. T o get energetics, the EL,,, (all-ext.) is calculated from the table of pair values [3] and added to the

+ E,",. + E\c,,,,-i",.+,ol,,,LJ.

Examples of the wave functions are given in the next paper where application to electron affinities is also discussed.

I n conclusion we remark that

1) additional lion-empirical calculations of the

Co~nparison oj'the

exj~erinie~ital)~ NCMET [3]

all-external irrec/ucible pairs of Neon ~zlirlz non-enipiricul ones of'seceral workers (in eV)

Electron Pair

-

2 p2 ZP 2 p l ID 2 pz

2 s 2 p IP 2 s 2 p

2

1s 1 s l 1s

NCMET ['I

-

.I82 ,563 1.149 .596 .la0 [dl .275 1.268

Viers and Weiss [.I Nesbet [b] al. [''I

-

-

.272 .449 1.186 .444 .070

.285 .294 ,307

1.072 1.085 1.061

( I s - 2 s ) correlation 0.170 0.131 0.140 0.136 ( I

2 p) correlation 0.612 0.498 0.541 0.517 (2 s

correlation 2.688 1.962 2.220 2.158

2 p) correlation 5.676 5.880 6.117 5.050

["I Wrrss (A.),

Rev. (To be published).

["

NESBET (R. K.),

Zbid. 1968,

175, 2 ; cf. Ref. [26] in text.

[r] VIERS (J. W.), HARRIS (F. E.) and SCHAEFER (H. F.),

["I The total 4 2

may be more accurate than the separate (2 s 2 p) pairs (cf).

ref. [3] for details').

[(,I Reference [3] in text.

diKerent types of pairs above, especially for excited states and their comparison with the

experimental

ones above, and

2) calculations by the stepwise C. I. procedures [7] of the three and more electron remainders in both ground and especially excited states, would be of great usefulness in putting the theory on a more quantitative basis. The magnitudes of these higher effects are still not well established as they would be by direct calculation.

((

Reducible pair

energie for Ne 's. Tliey are obtained from the irreducible pairs (3) by a unitary trans- formation. All-external pair and cross-pair correlation energies are in the (+, 0, -) description. For defi

nition and derivation o f these pairs cf. ref. [20]. All values are in (- ev).

Continued

Akn~wled~ment. - Discussions by Mr. C. Nicolaides a n d support from a U. S. National Science Foundation

grant are aknowledged with thanks.

C1-88

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(O.), J. Chem.

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[Z];SINANO~LU (0.) and

(I.), P / I ~ . Y . Rev. Letts, 1968, 21, 507.

(I.) and S I N A N O ~ L U (O.), Phys. Rev., 1969, 181, 42.

[3] O ~ s u z (I.) and S I N A N ~ ~ L U (O.), Pllys. Re\!., 1969, 181, 54.

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[5] NICOLAIDES (C.), WESTHAUS (P.) and

(0.) (to be published).

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[7ci]

(O.), J. Cllem. Pl~ys., 1962, 36, 706.

[701 SINANO~;LU (O.), J. Cl?enlL Pl~ys., 1962, 36, 3198.

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[9] McKov (V.) and

(O.), J. Cl~em. Pl~ys., 1964, 41,2689.

[lo] S I N A N O ~ L U (O.), A ~ V O I I C C S Chem. Pllys. (Interscience Wiley, N. Y., 1964, VI, 315.

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[12] KELLY (H. P.), Phys. Rev., 1964, 136, B 896 ; Phys.

Rev., 1966, 144, 39.

[13] S I N A N O ~ L U (0.) and BRUECKNER (K. A.),

Three Approaches to Electron Correlation in Atoms

Yale University Press, New Haven, and London (1970).

[I41 KELLY (H. P.), Phys. Rev., 1963, 131, 684.

[I51 GELLER (M.), TAYLOR (H. S.) and LEVINE (H. B.), J. Chem. Phys., 1965, 43, 1727.

1161 BYRON (F. W.) and JOACHAIN (C. J.), Plzys. Rev.

1967. 157.

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[21] SCHWARTZ (C.) in

Atomic Physics

Plenum Press N. Y., 1969 and references there in.

[22n] NESBET (R. K.), J. C11ein. P l ~ j s . , 1960, 32, 1

14.

[22bl NFSBET ( R K.),

Clleln. PIIJ'\., 1968, XIV.

[22cl NLSBET (R. K.), PIIYS. Rev., 1958, 109. 1632.

[23] See also footnote p 252 of artlcle by 0 . S.

Adv.

Clletn. Pliys. XIV, 1968.

[24] KUTZELNIGG (W.), J. Cl~em. Plys., 1964, 40, 3640, DAVIDSON (E. R.), J , Cl~ein. Pl~ys., 1962, 37, 577, EDMISTON (C.) and KRAUSS (M.). J. Clzetn.

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