Chemical Physics 14 (1976) 145-158t9 North-Holland Publishing Company

ANALYSIS OF THE EQUATION-OF-MOTlONTHEORYOFELECTRONAFFINITlES AND IONIZATIONPOTENTIALS

Tsung-Tai CHEN * and Jack SIMONS **Departmento[ Chemistry, Universityo[ Utah,Salt Lake City, Utah 84112, USA

and

KD. JORDAN ***Department o[ Chemistry, Massachusetts Institute o[ Technology, Cambridge, Massachusetts 02139,and Department o[ Engineering and Applied Science, Mason Laboratory, Yale University, New Haven, Connecticut 06520, USA

Received 3 November 1975

An analysis of the equation-of-motion (EOM) method for computing molecular electron affinities and ionization paten-tials is presented. The method is compared with the Dyson equation approach of Green function theory. Particular empha-sis is devoted to c1arifying the similarities between these twa theories when carried ou t to second and to third order. TheEpstein-Nesbet hamiltonian and the fiction of diagonal scattering renormalization have been used to systcmatize this com-parison.

1. Introduction

Until recently, most of the theoretieal studies ofmolecular electron affinities (EA) wecebased on sep-arate calculations of the energiesof the neutral andjon species; the EA would then be obtained erom thedifference of these twa quantities. As discussedbyone ofus in ref. [1] (hereafter referred to as EOM-I),simplecalculations based on Koopmans' theorem [2]or on separate Hartree-Fock (HF) treatments ofboth the neutral and the jon frequently result in poorEAs due to the inadequacy of such approaches indealing with charge redistribution and the change inelectron correlation energy associated with adding the"extra" electron. The conventional approach for over-coming these difficulties bas been to perform moceaccurate calculations by same form of perturbation

. Present address: Department of Computer Science. Uni-versity of WaterIoo, Ontario, Canada.

.. Alfred P. Sloan Foundation Fenow.... National Science Foundation Predoctoral Fenow, 1970-

1973. Present address: Department of Engineering andApplied Science, Yale University, New Haven, Connccticut06520.

theory or configuration interaction (CI) on the twaspeciesand to then compute the energy difference.As a result, one frequently bas to subtract Iwo num-bers of almost equal magnitude. Furthermore, it isfound that many identical terms appear in the neutraland jon energy expressions and therefore cancel whenthe difference is considered [3]. Therefore, a directcalculation technique which is aimed at the significantaspects of the problem is to be preferred. The EOMmethod as developed by Rowe [4] and applied byMcKoy [5] and other authors [6-13] to atomie andmolecular problems provides a means for directly cal-culating excitation energies, transition moments, andother quantities ofinterest. This method is capable ofprovidinghighly accurate results without explicitly re.quiring calculations of correlated ground and excitedstale wavefunctions.

RecentIr , several papers [l, 14, 15] have appearedin which the EOM theory bas been developed and ap-plied to the problems of molecular ionization poten-tials (IP) and EAs. In additio,n to these ion-neutralenergy differences one can illsoobtain other usefulinformation including the one-particIe density matrix

146 T.-T. ClIelI et al./EOM tlleor)' of elcetroTl affillities alld iollizatioTl potelltials

ofthe paleni cIosed-shell species. Further,just as in.the ease of eleetronic excitation energies, there is acIose eonneetion between the Green ruBelion [16-27]and EOM treatments. One of the goals of Ibis paperis to establish Ibis eonneetion.

In seetion 2, we present expressions for the varioushamiltonians used in the development of OUTtheory.In seetion 3, we summarize the well-known results of

the Dyson equation approaeh. Finally, in sections 4and 5, we establish the eonneetions between the EOMand the perturbational treatment of one-particieGreen funetions [16, 18,22-25,28,29].

2. The hamiltOllian

The hamiltonian deseribing the electronic structureof moleeules (negleeting relativistie effects and in-voking the Born-Oppenheimer approximation) is

H= H2r + WHr + <OIHIO), (I)

whele

H~r =~ ciN[Ct Cd,l

(2)

and

WHr =l L) (ijlvlkI)N[Cjet qcd .i,j, k,l I

(3)

The c/s ale spin-orbital energies in the Hartree-Foek(HF) approximation and WHFis the perturbationaJpart of the hamiltonian. Note that we have expressedthe zeroth-order approximation HRF and the residualperturbation part WHFin normaJ produet form [30].The third term in eq. (I) is the expeetation value ofthe hamiltonian for the zeroth-order approximationto the ground stale of the neutral moleeule. Wehaveadopted the eonvention for two-eleetron integraJsthat (ijlvlkI) ==<ijlvlk[)- <ijlvllk),Le., the integralsinvofvedireet and exehange terms. The partitioningof H deseribedin eqs. (I )-(3) is widely used in theeurrent literature il}the application of perturbationtheory to eorrelation problems. There is another par-titioning scheme, proposed by Epstein and Nesbet[31] (EN) which is alsoused in ibis article, and whichwe naw deseribe.

The EN deeomposition of His defined as follows:

1/ =HEN + WEN + <OIHlO), (4)

where

HEN = HI~r + Vd ' (5)

and

Vd =~ ~(ijIVlij)N[CrCJCjC;J1,1(6)

and

WEN=WHF- Vd . (7)

The zeroth order of the EN hamiltonian differs erom

the H2F in eq. (2) by the diagonal seattering term Vd'The perturbation term WENis the same as WHFwhenthe diagonal seattering term is omitted. The operator

HEN bas the following diagonal properties. For a gen-eral p-particie h-hole vector lA ),

lA) = C~llC~12 .., C~pCO:1CO:2... CO:hIO>'

The expectation value of HEN is given by

p h

<AIHENIA)= ~ cm; - ~ cal1=1 1=1

(8)

+ . ~ aij(ijlvlij) .-1,/E(m;. aj)

(9)

where

aij = I if i, i ale both particIe states or bothhole states, (10)otherwise ,= -I

and

<AIIHENIAZ) = O, ifIAI):#=IAZ)' (11)

For eonvenienee in using the EN hamiltonian in sub-

sequent seetions, we introduee the folIowing shortband notation for diagonal scattering terms:

(ijlvlij) ==[ii] , (12)

and for non-diagonal seattering terms:

(ijlvlkI) ==[ijlklj , (i :#=k. j:#= I), (13)

and in general (without distinguishing the diagonaland non-diagonal):

(ijlvlkI) ==(ijlkI) . (14)

I

II!I

T.- T. Chen et al./EOM theory ol electron alfinities and ionization potentials

In the EOMmethod, as described in ref. [1] andbriefly outlined later, HEN is treated as the zeroth or-der approximation and WENis considered as the per-turbation. However,in the Green function develop-ment, the introduction of HEN and WENis avoidedin the perturbation expansion, but the notion ofdiag-onal scattering terms is found to be very useful (seerefs. [32, 40]). With this discussion of the twa decom-positions of H completed, let us naw tum to a reviewof Green functi~mtheory and then to the connectionswith OUTEOM theory.

3. Many-body Green functions

The perturbational approach to Green functions isnaw well-established[16, 18, 22- 25, 28, 29] .Th~Dyson equation may be formuIated as a pseudo-eigenvalueproblem [33],

6 [dj..+M-. (M )]X.=MX.. } I} I} } I 'I -

where the matrix elements Mij(M) of the mass oper-ator or self-energyM(AF.)ale represented by the setsof Hugenholtz diagramsin figs. 1 and fig. 2 in secondand third orders, respectively [23]. The rules for as-signingcontributions to each of these diagrams dependon whether the diagonal scattering renormalization(DSR) [32] is used. For the case where renormaliza-tion is neglected, the rules ale well-known [34], andale given,along with their DSRgeneralizations as fol-lows:

Rule (J): To each vertex in a diagram,

(15)

:Xl

.~. I~'(I) (2)

Fig. 1. Second order Hugenholtz diagrams.

~I

~8 iY a

m 'fJ. m

.~, ; ~ I(5)

Fig.2a.

~~(7) (8)

~~

(10) Ul)

Fig.2b.

147

(3)

(6)

(9)

(12)

p~' .~. ~'\t01i W1; pfjVa Ii

(14) (15)(13)

P

~a i

~a m I

lO yp

i I(161 (17)

Fig.2c.

(18)

Fig. 2. Third order Hugenholtz diagrams.

::

.8

Yi

(I) (2)

Table lSecondorder contribution to Green function and EOM

Green function Eq uation-()f -fiction

....~00

DiagramI

2

*

a) For the energy denominators in tables l and 2, the following notation is employed:

e~~::~1lfl=(ea + eiJ +... + e'Y + eó) - (em + en + u. + €p + eq)'

E:;;r.:::fl.q = (ea + eiJ +...+ e'Y + eó) - (em + e" +...+ €p + eq) + {sgn("v-",,)} AJ:' + ~ biiliil ,1,/ '

where i and i run through the set of indiccs {m, ", u., p, q, a, iJ, u., 'Y,ó}, "p and II" are the number ol' particie and hole lines, rcspectively, and

bij=-l= l

if i, i are both particie or both hole statcs,otherwisc.

Table 2Third order contributions to Green function and EOM

Green function

Numcrical

Diagram factorl II

Denominators

III IVTermV

(l) [? ~J {3}(3 3') {3'}~,?J

(2) [? ~J {3}~ n

(3) ~~J {3}(3S){s}~?J

(4) fi~] {3}~?]

- (S) ~~J {S}(S 3){3}~?J

(6) ~~] cn (~3) {3'} ~'~J -1/4

1 1/4

1/42

3 1/4

4 -1/4 Ell/'Yo E~

. ~.,,- ..-""

(ial-yó) ('Yó Imn) (mn !ja)*

(imlO<{J) [0</3l-yo] ho lim)* *

Numerical Numericalfactor Denominator a) Term factor Denominator a) Numerator Case l Casc II Casc IIIli III IV V VI VlI VIlI IX X

1/2 E :;111(1) [8 J {3} [J

1/2 E::n (io<lnVl)(mnVo<)* * *

-1/2 E(2) J rn[?J -1/2 E (imlaiJ)(aiJlim) * * *

(3) [ J {S} .J -1/8 Emii .aiJ

:-3::,g'":::

'",....t:;;-

;;....c

'";;;-

c:::

'[.

..'":::

.

;::;.'"IIIXII-*

jS.!:;'*

*

*

*

*

Equations-()f-motion

Numericalfactors DenominatorsVI VlI VlIl

1/4 E::n ECq

1/4 E:;m emIl'Yó

1/8 Enn Ell/ni'Yó

1/4 envl E::n'Yó

E{:" ECq

E::n Emnj'Yó

Emni E::"'Yó

Emij Emij'Yó O<{J

CaseNumerator I 11IX X Xl

(ialmn}(mnlpq] (pq!ja)* *

(ialmn)(mnl'Yó )('YóIja) * *

. . ._"--

Table 2 (continued)

Green function Equations-<Jf-motion

Numerical Numerical CaseDiagram factor Denominators Term factors Denominators Numerator I II IIII II III IV V VI VII VIII IX X XI XII

(7) [?J {j}(J5){S}[?J -1/8 E;J; Emij *C't/J

(8) [? J {S}(S 3) {3}[?J -1/8 Emij E * :-i'Yb :-i

(9) [? J {S}(S S) {S}[?J -1/16 Emij Emij * g'Yb ojJ '"

(10) [? g en;:s

Emij Er;:f -1/4 E g (imlpq) (pqlC't/J)(C't/J!jm) * * * '"S -1/4 ...aif3 ..,...

(11) [? {3}(jS){S}?J -1/8 EZ/J Ej * t>J

(12) [? J {S}(S S') {S'} ,?J -1/16 Emij Eg7/* ......C't/J '"c

-1/4 Ei Emij(13) [?J {j}[t -1/4 EZ/J (imIQ(3) (aif3lpq) (pqljm)

* * *6Q(3 .s;

(14) [? 2J {S}(S 3) {j}[?J

'"-1/8 Ei Em * fi;',.,Q(3

(1S) [? J {S}(S S'){S'} ,?J -1/16 Ei Emij * :::aif3 :=;';::j

7 l E:;m Emp(16) [?J {3}(33'){3'},?J

1 E::zn Emp (iQlmn) [n'YlpQ] (mpl/'Y) * * * .'Y 'Y ,,'

(17) [? g] {3}:J

.,

8 I E::zn ppnj 1 Ezm pn (iQlmn) (pnlQ'Y)(m'Yljp) * * * §'Q'Y Q'Y

(18) J {3}(3S){S}?J Epnjc'

1/2 E::zn * ;:sQ'Y if

9 I Epni E::zn (19) Q J {3}l pn E;:m (ipl111Y)(Q'Ylnp) (mnljQ) * * * c'

:::Q'Y Q'Y

(20) [? J {S}(S 3) {3} ?J Epnic

1/2 E::zn * ...'""Q'Y :::

EPij Emij(21) [? J {3}(j j'){n ,?J

s.10 -l -I EPEZ/J (imIQ(3) [(3pl'Ym] (Q'Yljp)

* * * O;"Q'Y Q{3 Q'Y

(22) [7 {j}(j S) {s} ?J -1/2 EP Emij *Q'Y C't/J

(23) J {S}(S 3) {3} ?J -1/2 EPij EZ/J*

Q'Y

(24) [? J {S}(S 5) {5] ?]-1/4 EPij Emij *

Q'Y Q{3

11 -l EmijER!!,j (2S) {j} -l

EZ/J (i'YIQP) (mpl'Y(3) (Q{3ljm)* * *C't/J ....

.".'D

Table 2 (oontinued)VIo

Green function Equations-of-motion

Numerical Numerical CascDiagram factor Denominators Term factors Denominators Numerator I II IIII II III IV V VI VII VIII IX X XI XII

(26) {!}(Js) {S} -1/2 EJ:b E/hmj*

(27) [J {S}(S S') {S'}.,?J -1/4 Emij E/hmj* :-i

a{J :-i12 -1 E/trmi EW (28) [ {!} -I eft.rm EJ:b (imlCl/J) (-YPlmp) (aplj-y) * * * Q'"::

(29) [ {S}(S 3) {!} ?J -1/2 EG!;li EJ:b* ..,...

(30) [? {S}(SS'){S' , -1/4 EGi Emij * h1esCl/J

13 1/2 EPs; E'/Ji (31) D ij 1/2 e t4J (ipljq)(sPICl/J).(a{Jlsq)a)* * * :;.'"ap c

(32) D {s} ?J Epsi EW1/4 (ipljq) (sp 1a{J)(a{JIsq) *ap '"

(33) [? {S} J

..1b'

1/4 FP,i Egp.* '"

'ixjJ ...o(34)[? {S}(S S') {S'},?]

Epsi EW::

1/8 *ap

Si14 -1/2 EGi EPqj(3S) D G -1/2 e epq (ialj{J)('Yalpq) (pql'Y{J)b}

* * * :;'ary ary .

(36) D {s} ?J..

-1/4 ef1i EPqj [ialj{J] (-yalpq)(pql'Y{J) * ::I:>..a'Yc'

(37)[?' {S}j ERii epq * ::-1/4 ..°ary

(38) [? {S}(S S') {S'} ).?J EGi EPqjc'

1/8 * ::ary "1:3c

IS 1/2 Emj EPqj(39) [? {3}rn n 1/4 Emj epq (ialjm)(amlpq) (pql'Ya)

* * * ::a 0l'Y Ol ary ...ISo

(40) [? iJ 1/4 E EPq * * ;;;-ary

(41) [? {3}(3 S) {S} ?]1/4 Emj EPqj *

Ol ary

16 1/2 EPqi Emi(42) D {3}?J 1/4 epq Emi (imIjOl) (pql'Ym) ('YOllpq)

* * *a'Y Ol ary Ol

(43) U?J 1/4 EPq EJ: * *0l'Y

(44) [? {S}(S3) {3} ?J 1/4 EPqi E:;li *ary

, . ---""'_."""""'"

O)

~u-x

c,:2CE~?~,:2""::3O"[.LJ

c;u'c '"O) ...E .s::3u-Z";':>

:aO)::3

,S

C~<'IO):c~

co';Iuc,EcO)O)

c3

"jJ';:::O) ...e o::3 t;

Z";': =

T.-T. Chen et al./EOM theory ol electron alfinities and ionization potentials151

-a~ * * * * *

, :.~1:;l::". - <:0-

I

~&C71 C""I :::::: ~

I o:'" o-, '%I:>..

I

1° "'I l2..::j ~.

~r"-, ~ ~$~ , ~~

1

'-:..,1 C) 10:..,1 M ~:'510MI M ,OMI '" ,E?-,g

r"-, ;:z:. r"-, ;:z:.:-:-~ ~ ~ ~,!:J,!:J

r:::::;:jr::t ~ r:::::::;:;]r:I r::-:-::l :2:2OM <'1- OM OM 0- O'" 5 51°-11°-1,°-'1--11<'1-11°-1 .;;;.;;;

~~~~~~ ......'" 'D r- 00 0\ o 00'1' '1' '1' '1' '1' '" ........~~~~~~ """... ...

O) O)e e::3 ::3C CO) O)-5-5c:: c::,:2 ,:2t) t)c c,E,Ec cO) O)O) O)

...

""".....

o o~ ~13 13

=~ * * * *

* *

~""...O)e

z~

~

'"('-E&~.9-'"$E'--;g

EE,'"$~'"('-~&~::::-:S

"'=B>'"c'soc-O) -o>

E"" E"" 'E'"" '- '-1:>..('- 1:>..('- 1:>..('- E~ E~ E~'" '" t.Q t.Q '" t.Q

', ';:;- E"" E"" 'E""E~ E~ ~~ 1:>..('- 1:>..('- ~('-t.Q '" t.Q '" '" ....

:!.-I

:!.-I

:!.-I

:!.-I

:!.-I

e...O)1-<>

~i::a'soc0)-0=

'E'oo1:>..('-t.Q

'E~t.Q

'E' ~t.Q

'E",1:>..('-t.Q

~-1

~-I

e'"~

,!110- r-- 00-

*

:!.-1

O) O)

-5-5.5.5'2:0'

assign a factor (ijJkl) for the case without DSR; a fac-tor (ijlkl] with DSR.

Rule (2): To fach vertical cu t between twa neigh-bering vertices, assign an energy denominator equal to

L; (hole Jine energies) - L; (particIe Jinea p

energies)+ C

where

C= {sgn(np-nh)} M,

for the case without DRS, where np and nh arf thenumber of particIe and hole Jines, respectively; and

C= ~bij[ij] + {sgn(np-nh)}M,I,/

for the case with DSR, where i, jare line indices be-longing to this vertical cut, and

b" =-1I/ if i, jare both particIe indicesar both hole indiees,otherwise,=

Rule (3): Assign a sigo factor (-l)nh+l where l isthe number of closed loops [35],

Rule (4): Assign a factor 2-e, where e is the num-ber of pairs of equivalent Jines in the given diagram, .

Rule (5): Multiply alt the factors assignedabove,and sum over al! the internat Jine indiees,

For convenience, we have written the contributionsto the second and.third-order Green function dia-grams in the tables l and 2. To aid in OUtcomparisonwith the EOMmethod, discussed in the next section,we have alsoJisted the contributions of fach secondand third order diagram to the effective hamiltonian

Hi/M) of OUtEaM theory. Ascan be seen eromtables l and 2, theprincipal differences, between theiwo approaches is the assignmentof energy denomi-nators to fach diagram and the overal!numerical fac-tor multiplying fach diagram. Beforegoing further in-to the comparison, we naw givea brief outline of vari-ous approximations which arf made to generale work-ing EOMtheories of molecular electron affinities.

4. The equation-of-motion method

Following the formulation of Rowe [4] and

152 T.-T. Chcn et al./EOM t/wory o! electron aJfinitics and ionization potentials

McKoy [5], we consider a Fermion operator niwhich is constructed erom products of odd numbersof creation and annihilation operators, describingtheadditionof an electronto the groundstale IgN)of anN-electron cIosed-shellmolecule. As shown by Roweand in ref. [1], one caDobtain the basic EOMin sym-metrized [36] "double commutator" form,

(gNI{ónA,H, n!J }lgN) = MA (gN, {ónA; ni}lgN),. (16)

where the commutators are Gefined as

{x,y, z} ==~ {x, [y, z]} + 4 {[x,y],z}, (17)

{A,B} ==AB+ BA, (18)

[A,B]==AB-BA, (19)

and MA ==E-t-Er:+1 is the vertical EA (Er:+1is theelectronic energy of the Athstale of the negativejonand E-t the ground stale energy of the neutral species.)Generally, the Fermion operator nl is taken to be atruncated linear combination of products of odd num-bers of creation and annihilation operators,

ni ~ xct + yct cct + zctcctcct + .... (20)

The operator ónA is then the adjoint of aDYoperatorwithin the srace in which ni is expressed

ónA~c,cctc,cctcctc, (21)

In practice, the criteria by which theimportant com-

ponents in the truncated operator ni are chosen andthe requirements for the input ground stale wavefunc-tion IgN) to becomparablein accuracyto a chosenni are not elear. This is an inherent difficulty of theEOMapproach. Furthermore, the dimensionality ofthe EOMshown in eq. (16) is often very large. Theproblem of1arge dimensionality caDbe avoidedbyusing the partitioning technique and perturbationtheory [38] to cast the originaleigenvalueprobleminto a new effective pseudo-eigenvalueproblem ofmuch smaller dimension. Such an approach caDbevery convenient provided the convergenceof the per-turbation expansion of the elements of the new effec-tive hami1tonianmatrix is fast enough to permit theincIusion of only second and third order terms (theprobabIe limit of OUTpresent computing facilities).Basedupon the assumption that calculationswhichare carried out through third order will be adequate to

yield EAs to an accuracy of:!: 0.2 eV, we ghalIuse thenotion of order in perturbation theory to determinewhether various components in ni and in IgN)aregoing to eontribute to the effective eigenvalueprob-lem through third order. Following this !ine ofthought, we have found that, in order to inelude allthe second and third order effects in the effectivepseudo-eigenvaJueproblem, we must choose the trun-

cated operator ni as follows [39]

nt =1; XiCl + 6 YnamcJcac~I m<na

+ 6 y crm(JclCmcJa«Jm

+ 6 Yróq-ypcJ CócJ C'YC;p<q<r-y<ó

==6 Xicl + 6 Yncrmd~am + 6 yQ:1n(Jd!m(Ji m <n a«Ja m

(22)

+ p<~r Yróa-rPd;óq-yp'-y<ó

and the ground stale wave function IgN), apart eroma normalizationfactor,as*IgN) =10)+11>+12>, (23)

where we have introduced the shorthand notation

) - I ~ (pqIJlv)ctctc C lO),11 =- LJ eP.a p a v }J.4 }J.,v }J.v

p,q

(24)

12)==!(

6 (rmlpq)(qplrQ)2 m, a cm cpq

\p,q, 'Y a C<-y

~ (pQlrÓ)(rÓIPm)

)ct Cala).-LJ mpm m

m, a Ca c'Yóp, 'Y,Ó

(25)

* In EOM-I, monoexcited configurations were not includedin the ground state; as a consequence of Hus EOM-I ne-glects the terms (40), (43), (46), and (49) shown in tatle 2of this paper. There ale also second order bi-excited com-ponents which ale not included because they do not contri-bute until fourth order to the ionization potential or dec-tron affinity.

I~i;~~"

~;l!

j) In theseequations 10) is the HF ground-state wave-function, 11) is the fjrst order bi-excited component,12) is the second order monoexcited component, and

€~~::~f:l={€c< + €(J+ 000 +€~ + €v)-{€m +€n +... +€p +€q).

By using eqo (22) for nt and successively choosing

on as Ci' CmC!Cn, C{JClCc<,and CpC~CqCl Cr' oneobtains erom eq. (16) a set of equations for the ex-

pansion coefficients Xi> Yna<m' y a<m{3'and Yróq-yp'which in matrix form becomes

T.-T. Chen et al./EOMtheory ofelectron affinities and ionization potentials

(: :)C)=~t:)(:) (26)

Here, the coefficients Xi and Yn<P'!'y a<m{3'Yróq-yp'appear in column v~ectorform as X and Y, respectively.The elements of the submatrices in eqo(26) are .

Aij =(gN I{Ci' H, C} }lgN), (27a)

Bi,x = <goNI{Ci' H, dl }lgN),

(x = nOiJ11,0iJ11(3,roq-yp) ,

Dx,y = (gNI {dx' H, d~}lgN),

(27b)

(x, y = nOiJ11,0iJ11(3,roq-yp) , (27c)

Ri,x = (gNI{Ci,dl}lgN), (27d)

Sx,y =(gNI {dx,d~}lgN) o (27e)

The dl have been introduced to represent cJ cc<cl,etco The partitioning of eqo(26) to yield a pseudo-eigenvalue problem of smaller dimension is easily per-formed by rewriting eq. (26) in the form of simulta-neGus equations and eliminating the vector Y in favorof Xo In this manner the following equations are ob-tained:

H{M)X=MX, (28a)

where

H{M)=A + (a-MR){MS-D)-l (at-MAt).(28b)

Eq. (28) is a pseudo-eigenvalue equation which issolved iterativelyo It should be noted that the dimen-

sion of H (M) is equal to the dimension of the spin':'-orbita! basis used to generale the HF orbitaIso The

153

most convenient war of eva!uatingthe matrix elementsof A, a, D, S and R in eqso(27)-{28) is to employWick's theorem and diagrammatic techniques [41] ofmany-body theoryoThis greatly simplifies the book-keepingo

Two difficulties remain at this stage of the analysisoThe calculation of the overlap matrices S and R in-volve the evaluation of 1-,2-,3- and 4-partic1edensitymatricesoIt can be shown that R is at least of second

order and S differs erom the unit matrix 1 by secondand higher order termsoThe replacement of S by 1causes a fourth order error in the second term on theright-handside of eq. (28b) since a is at least first or-der while D is at least zeroth order. Since aur objectiveis to obtain results accurate through third order, theapproximationS ~ 1 ismade.Further,asin EOM-I,we make the approximation that R ~O, turning laterto the consequences of this approximationo

HI{M)X= MX.

HI{M)=A+a{M1~D)-lat o

(29a)

(29b)

In EOM-I, we have made one additional approxima-

lian, namely, the replacement of H by HEN to renderthe D matrix diagonal. This was dane in order to avoidthe very laborious ca!culation of the inverse matrix

{M1- D)-lo As an improvement over this approxi-mation, we employ the following expansion of the in-verse matrix:

{M1-D)-1 =={M1-DEN)-1

+ (Ml-DEN)-l Dl (M1-DEN)-1

+ {M1-DEN)-1 Dl (M1-DEN)-1 Dl (M1-DEN)-1(3D)

where D is decomposed as

D= DEN+ Dl'

with

(31)

{DEN)X,y= °x,y<gA'1{dx,HEN,d~}lgN), (32)

(Dl)x,y=(1-oX,y)(gNI{dx' WEN,d~}lgN). (33)

In eqs. (32)-{33), the diagonal properties [eqso(9)-(ll)] ofthe EN hamiltonian have been fully ex-ploitedoKeepingin mind that we are only interestedin keeping terms in H(M) through third orderwe

154 T.. T. Chen et al.!EOM theory ot electron atfinities and ionization potentials

need only retain the first and second terms on theright band side of eq. (30). The improved effectivehamiltonian operator H(~E) thus becomes *

H(M)= A+B(M1 -DEN)-lBt

+B(M1-DEN)-1 Dl (M1-DEN)-lBt . (34)

To aid in the description of the various termsin eq.(34) we define the following quantities:

[~:]==<al{Ci'WHF' C}}lb),(35a)

[~~J ==<al{Ci' WIIF' dl}Ib>,(35b)

{x} ==[<Ol{dx,HEN,dl}10)j-1 ,

(xy) ==<Ol{dx, WEN' d~}IO>,

(35c)

(35d).~

[~~J=[~~J* ==<bl{dx'WHF,Cit}la),(35e)

where a, b = O,1,2 represent the zeroth, first, andsecond order components of IgN) respectively. In ad.dition, we shalluse 1,3,3, and 5 to denote collective-Iy the illdicesi, nam, am{3,and roq"lPrespectivelyasshown ~eqs. (27a)-(27c). With these notational sim-plifica~ions,we caDnaw write down the non-zerocontributions to the matrix elements Hi/M) throughthird order. In the tables we have specialized to thecase for which i and jare both particIestates, that isthe results are presented for an electron affinity cal-culation. For the sake of organization and clarity, wehave presented the second- and third-order contribu-tions together with the Green function results intables 1 and 2. In zeroth order we obtain the Koopmans'theorem result erom the Aij term:

Ai,j =(gNI{Ci'H, C}rlgN)

=e/iii + (third and higher order terms). (36)

* The referee has pointed out that this expression for H(~has improper ana1ytica1 properties as a function of t:..Eandmay give non-physical resuIts if it were applied to "shake-up" processes. For a discussion of the use of GF methodsas applied to shake-up processes, see: LS. Cederbaum, J.Chem. Phys. 62 (1975) 2160, and LT. Redmon, G. Purvisand Y. Ohm, J. Chem. Phys. 63 (1975) 5011.

In second order, as listed in tatle 1, the contribu.tions come erom the sum of the twa Hugenholtz dia.grams shown in fig. 1. To compute the contributionof each diagram to the self.energy of Green functiontheory (or the effective hamiltonian of the EOMthe.ory) one musi form the product of the numerical rac.tors listed in columns II and VII (V and VII) anddivide by the energy denominator listed in column III(VI). The numerical factors for the Green functioncontributions are a result of applying the diagramrules discussedin section 3, whereas the factors listedfor the EOMcase arise erom using the diagram rulesof section 3 to evaluate the symmetrized matrix ele.ments givenin eq. (27).

The flrst point to be made concerning the relation.ship of EOMand Green function theories is that De.

glect of the CJCÓCJC"'(CJterms in nt leads toequivalence between the twa approaches through sec-and order (see tatle 1).

The third order terms listed in labIe 2 caDbe re-presented by the 18 Hugenholtz diagrams shown infig. 2. For the Green function (EOM) theory, the con.tribution of each diagram is the product of the num.erical factors givenin columns II and IX (VI and IX)divided by the product of energy denominators givenin columns III and IV (VII and VIII). The diagonalscattering form of the Green function is given;themaTefamiIiar form without renormalization of theenergy denominators is obtained by omitting all twae1ectroninteraction terms appearing in the denomina-

tors (Le.,the e':fi~::~~qformof the energydenomina.tors should replace the E':fi~::~q).In order to facili-'tate comparison between the twa theories inthird order, let us naw specificallyconsider the fol-lowing special cases within the general EOMtheorydiscussedearlier in this section (recalI that we havetakenR "'" O and S"'"1).

O1seI: Only the first order component of IgN) isretained:

IgN) "'" 10)+ II),

and nl is truncated by eliminating dtÓq"'(p:

nl =P XiCl + ~ YncxmdZcxm + ~ y cxml3d!m13.l m<n 01<13Ol m

CaseII: The fulllgN) of eq. (23) is used:

IgN)~ 10)+ II>+ 12), (37)

T.-T. Chen et al./EOM theory o{ electron a{{initie!Iand ionization potentiab 155

'; but ut is truncated:

ut =6XicJ + 6 Yncxmd~cxm+ 6 y cxml1d!ml1'A i I m <n a<11 (38)a m

Case III: The fuli second order IgN) is used:

IgN) ~ 10) + II> + 12) , (39)

and ut is not further truncated:

ut=6x.d+ 6 Yncxmd~cxm+6Ycxm~!m11A i I I n<m a<11a m

~Y dt .

+ LJ róq-yp róq-ypp<q<r -'Y<ó

(40)

In columns X, XI and XII of Labie2, we have markedwith an asterisk each term which contributes to theoverall self-energyor effective hamiltonian undereach of the above three specjalcases of the EOM theo-Ty.From this tabulation it isseen that Case II of theEOMbears the most similanty to the maDy-bodyGreen function (MBGF) approach; the only differencebetween EOMCase II and the MBGFapproach beingthe form of same of the energy denominators. Forexample, for term 2 of Labie2 the MBGFdenominatorisE;:n E~nj wbiJethe corresponding denominator inthe EOMisE;:n €~n. Usingthe definitions givenineq. (12) we find

E;:n =-€m-€n + €a-[mn] + [ma] + [na] + M ,(41)

E;;,nj = -€m-€n-€j+ €'Y+€ó- [mn] - [ni] - [mi)(42)

- [-y6] + [m-y] +[m6] + [n-y] +[n6]+IJ-y] +1J6] + M,

€mn = -€ -e n + €", + €ó .-yó m , (43)

From comparison of eqs. (42) and (43) we see thatfor term (2) of Labie2 the EOMand MBGFenergydenominators differ in twa ways. First, two-electronmatrix elements appear in eq. (42) but not in eq. (43)and secondly eq. (42)also contains a M-€j contri-bution lacking in eq. (43). From expansion of theenergy denominator

(€m+€n+€j-€'Y-€ó ~M- 6bxy[x,y] )-1

x,y

~ (€m + €n + €j - E-y- Eó - M)-1

( ~ bxy[x,y] )X l+LJxy € +E +€.-€ -€ ó -M. m n J 'Y

it is elear that the twa electron renormalization termscontribute in fourth and higher orders of the inter-

action. If M is algOreplaced by Ejthen we haveagreement between the Case II EOMand MBGF re-presentations of term (2) through third order. The re-

placement of M by Ejis precisely the fieststep to aniterative solution of the EOM.If a similaranalysis isperformed on the other entries in Labie2, we findthat through third order the MBGFand Case II EOMapproaches agree. Consequently, CaseII EOMrepre-sents a moce consistent theory than do the Case Iand CaseIII EOM approaches.

The extension of Dur°rerator set to inelude termsof the sort C: eó et eóCp causes the EOMto differfrom the MBGFa~proach in second, third and higherorders. Thus, care must be taken to ensure that theoperator set is consistent with the ground stale ap-proximation and other approximations introduced in-to the EOM.In fact, the discrepancy between theMBGFand EOMapproaches that arisesin second andthird order when et eó et c et terms are ineluded inh .

dr

ha-yP..

ROt e operator 15 ue to t e approxlmatlOn ~ .Eq. (28) may be rewritten

[HI(M) + m (M)] X =M X,

where HI(tlE) is given by eq. (29) and

m(tlE) =B(MS-D)-IRt + R(MS-D)-IBt

(44)

(45)

(46)-MR(MS-D)-IRt.

WhenC: eó eJ eó c; terms are included in theoperator set they givense to second and third ordercontributions to m(tlE). After considerable algebrainvolvingexpansion of energy denominators and theregroupingof terms, we find that these new termscancel through third order the terms involvingtheet eet eet operators in tables 1 and 2. ApparentIy,we again have agreement between EOMand MBGF.However, it algOturns out that the et eet operatorscontribute to m(M) in third order due to the pre-

156 T.. T. Chen et al./EOM theory ot electron atfinities and ionization potentials

sence of singleexcitations in the ground stale givenineq. (23). These new terms ale represented by diagrams15-18 in fig. 2 and their contribution causes a slightdisagreementbetween the MBGFand EOM theories inthird and higher orders. For example, let us considerthe terms contributing to diagram 15. The new contri.bution flam m(AE) enters with a coefficient of -1/8which means that the overallcoefficient is 3/8 ratherthan the 1/2 of the MBGFapproach [remember thatterm (41) containing the contribution flam the fiveoperator is cancelled through third order by the fiveoperator contribution to m(AE)]. This disagreementis not serioussince the contribution of these diagramsto molecular electron affinities and ionization poten-tials is smali [23,42].

5. Discussion

In this paper we have established the connectionsbetween the EOM and the one-particie Green functionapproaches to ionization potentials and electron affin-ities. The EOM derivation presented herc bas em-ployed the symmetrized "double commutator" definedin eqs. (16)-(17). We note that in general

<g.N1{Cj> [H, CJ CQCl] }lgN)

*<gNI{[C;,H],c~CQCl}lgN),

since the exact ground stale is not employed [42]. Inthe development of this paper we have employed thesymmetrized double commutator

Bi.ncrm=H<.gNI([Ci,H], CJCQCl}lgN)

+ <.gNI{Ci' [H,CJCQCl] }lgN)}.

It is not obvious that the use of the symmetrizeddouble commutator results in a mOlestraightforwardor consistent formulation of the EOM.In order to in-vestigate this matter, we have also carried out the der-ivations of this paper using Bi, ncrm=<.gNI{Ci' [H,CJCQCl] }lgN). The differencesbe"tween this development and that of the text arisesinthe coefficients of the terms contributing to diagrams15-18 of LabIe2. To illustrate the resulting changes,we wiJ]briefly discuss the various contributions todiagram 16 with the above unsymmetrized definitionof B. Term (42) naw appears with a coefficient of 1/2

rather than 1/4. Terms (43) and (44) stil! have coef.ficients of 1/4, and the contribution of the five Oper-ators in term (44) againis cancelled through third or-der due to the contribution erom m(AE). As before,the presence of the singleexcitations in the approxi-mation to IgN) also givesrise to a third order tenndue to m(AE), however the coefficient is naw -1/4rather than -1/8. Thus, through third order, we nawhave agreement with the MBGFapproach. Thus, ffOUTobject were to obtain a theory correct throughthird order, there is no oecd to include either the

operators CJClict C ct or the"singleexcitations inthe ground stale (kN5.rhis finding suggests that theunsymmetrized version of the EOM approach maybe preferable.

As with the EOM for excitation energies [43],maur questions rem:rinconcerning the effects of vari-ous choices of IgN)and ni- The role of terms suchas Cl C CJ, Cl CpC!, and Cl Cnct bas not beenestablished. Simi1arlythe significanle of second-orderdouble excitations in IgN)bas not been explored.OUTexperience with the CJClict C ct operators in-dicates that these questions do n~t ha~e unambiguousanswers. Whether a certain operator or ground stalecomponent contributes to a givenorder in the intel'action depends on the other terms and ground stalecomponents present as well as to approximationsmarle to R and S and whether symmetrized commuta.tors ale employed.

In this paper wehave discussedvarious approximateEOM theories for ionization potentials and e1ec~ronaffinities. Wehave presented an EOMtheory which iscorrect through third order and sums a large numberof terms to all orders in Rayleigh-Schrodinger per-turbationtheory. This summation of a large numberof terms to all orders is a property shared with theGreen function approaches {23].

It is significant to note that even the EOM-Imeth-od, which neglects certain third order terms, givesgood ionization potentials (usually within 0.2 eV O,fthe experimental value). For perspective, in labIe 3we compare IPs calculated via EOM-I [14,15,44]with the experimental values for HF, BH, BeH-, OH-,and CN- (for the BeH- , OH- , and CN- we ale re-porting the vertical detachment energies). Wehave al.so inc1udedOUTtheoretical predictions [45,46], asobtained from EOM-I,for the EAs of UH, NaH, BeO,and UF. Webelieve that these calculated EAs ale cor-

T.. T. Chen et al./EOM theory of e1ectron affinities and ionization potentials

reet to 0.1 eV. These studies on the negativeions ofUH, NaH, BeO,and UF were undertaken in order toinvestigate the nature of the binding of electrons tohighly polar moleeules. Expenmental electron affini..ties are not availablefor BeO,NaH, and UH, and ourstudies have led us to conc1ude [46,47] that the ex.penmental EA of UF is in erraTby about 0.9 eV.

Weare eurrent1yapplying the EOMmethod to thecalculation of IPs and EAs for vanous polyatomiemolecules. Although, the energies given in LabIe3 COl.respond to vertical proeesses, the method is capableof describingthe geometrical rearrangements that oc.eur upon ionization or electron attachment. So far,we have restricted OUTattention to positive electronaffinities. Weplan to investigate the usefulnessofEOMmethods for describing temporary negativeionsof molecules such as ethylene and butadiene [48].

Acknowledgement

Acknowledgement is made to the donors of thePetroleum Research Fund, administered by theAmencan ChemicalSocjety, for support of this re.seareh. Wehave also benefited erom discussionswithProfessors RP. Kelly, T.T.S. Kuo, V. McKoy,W.P.Reinhardt, RJ. Silbey, H.S. Taylor, and R. Yans.K.D.J. wishes to thank the Universityof Utah for itsfinancial support and for making his star there mostpleasant.

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157

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Table 3

Equation-of'motion and Koopmans' theorem estimates ofelectron affinities and ionization potentials (eV)

Species EA or IP EOM -f LUMO Exptl (14,15,44]

BeO EA 1.769 1.414UH EA 0.299 0.200NaH EA 0.362 0.290UF EA 0.464 0.425HF IP 15.87 17.79 16.01BH IP 9.53 9.30 9.77BeH- IP 0.77 0.51 0.74OH- IP 1.76 3.06 1.825CN- IP 3.69 5.21 3.82

158 T.- T. Cl1en et al_/EOM t/wory of eIectron affinities and ionization potcntials

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4063.

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[33] T.T.S. Kuo, Reaction Matrix Theory for Nuclear Struo-

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[35] There is an ambiguity in assigning the phase faCJaTof(- )11+1in the HugenhoItzdiagrams.To avoid this am-biguity the phase factors are determined by usingGoldstone diagrams. Sce rcfs. 116],129] and 143].

[36] The consequence of using an unsymmetrized form inEOM-I is that the terms labcled (39), (42), (45), (48)in labIe 2 appeal with an ovcrall numerical facjaT of:!:1/2 instead of:!: 1/4 onthe upper triangular blockand vanish on thc laweT triangular block of the effectivehamiltonian matrix H(t.E). Thesc terms were ignored inEOM-I.

[37] J. Simons, Chem. Phys. Lctters 25 (1974) 122.[38] P.O. Lowdin and O. Goscinski, Int. J. Quantum Chem.

5 (1971) 685;P.O. Lowdin, Thc Ca1culation of Upper and LowerBounds of Encrgy Eigcnvalucs in Pcrturbation Thcoryby Means of Partitioning Tcchnique, in: PerturbationTheory and !ts Application in Quantum Mechanies, cd.CH. Wilcox (Wiley, New York, 1966).

[39] The indicesa, p, 'Y,and ó(m,n,p,q) ale uscd to labclHartree-Fock spin orbitaIs which ale occupied (unoo-cupied) in 10).The indices i, i, k, and 1refer either tooccupied or unoccupied spin orbitaIs. From naw on weneglect the subscript A.

[40] H.P. KelIy, Advan. Chem. Phys. 14 (1969) 129.[41] See, for example, J. Paldus and J. Cizek, Advan. Quan-

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592;K.M. Griffing and J. Simons, J. Chem. Phys. 62 (1975)535; ibid, in press.

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(1975) 594. The C2H4" and C4H6" anion s live ~lO.,..ISseconds.