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THE JOURNAL or CHEMICAL PHYSICS VOLUME 57, NUMBER 9 1 NOVEMBER 1972

Energy-Shift Theory ot Low-Lying Excited Electronic States ot Molecules*JACK SIYONS

Chemistry Departnunl, Unil1e1suyoJ Utah, Salt Lake City, Utah 8411Z

(Received 11 April1972)

The present research is directed toward understanding the low-lying excited electronic states of atomsand molecules. Such excited states play essential roles in atmospheric photochemistry, charge transferprocesses, and chemical reaction dynamics. The excitation energies and oscillator strengths between thesestates and the ground state are the principal concerns of this investigation. A new theory, referred to asthe energy-shift theory, is put forth as a means of directly calculating molecular electronic excitationspectra. Within this theory, four computationally tractable approximations are developed, two of whichallow the use of experimental data on higher states to aid the calculations. The method is shown to havesame advantages over other direct calculation schemes currently being used in this and other laboratories,the primary advantage being the separation of low- and high-energy excitations which arises naturally inthe energy-shift formalism. The relation of this approach to Green's function theory is discussed.

INTRODUCTION not of immediate interest. A major disadvantage of theseThe theoretical study of molecular electronic excita- methods has to do with the celebrated N-represent-

tion energies and oscillator strengths is important both ability problem~37: once approximations are introducedin interpreting experimentally observed spectra and in into the theory, how can one be certain that there existpredicting the photochemical behavior of moleculesproperly antisymmetric wavefunctions (even if ap-which have not been thoroughly examined by' experi- proximate) which correspond to the calculated transi-ment. Until qui te recently, the prediction \f excitation tion properties?energies and osciUator strengths required the calcula- In addition to the problem of lY-representability, thetion of the wavefunctions and energies of the individual physical content of the mathematical approximationsstates of interest. The excitation energy is then ob- used in the theories discussed above is somewhat vague.tained aS the difference of the two calculated energies, For example, the self-energy operator, which introducesand the oscillator strengthl is computed from the matrix the effects of electron correlation into the equationelement of the electric dipole operator between the governing the one-pa~ticle GF, is usually expanded in awavefunctions of the two states. A serious disadvantage divergent perturbation series. The resulting expansionof this approach is that it requires a separate variational of the GF is also divergent, even in the Fredholm sense,(or perturbation) calculation for each electronic state and the attachment of physical meaning to the terms inof interest. Moreover, large canceUation errors are this expansion is difficult, to say the least. In anotherlikely to be introduced when the excitation energ)' is formulation of the GF theory, one is faced with thecalculated as a difference of two numbers which are problem of truncating or decoupling a hierarchy ofoften qui te similar. Finally, if one is interested oniy in equations analogous to the BBGKY equations38 ofpredicting the transition properties of the molecule, statistical mechanics. This is usually done\9-%1by ap-e.g., oscillator strengths, dipole moment differences, proximating various two-electron operators as sums ofand excitation energies, then much of the information one-electron operators. The difficulties with this ap-contained in the individual wavefunctions is of no proach are twofold: there is no well-defined methodinterest and adds only to the expense and time involveJ for improving on the first approximation and there is noin the computation. way to estimate the terms which have been neglected

In hopes of overcoming some of the difficulties men- in the decoupling process. The principal disadvantagetioned above, several chemists and physicists have of theEOM approach is that it does not allow the in-recently begun to use and extend theories which were corporation of known facts concerning higher excitedused previously by nuclear%-Il and many-bodyl:i-IS states into the calculation of low-Iying excitations.physicists to calculate transition properties of other The theoretical research presented here overcomessystems. Two notable examples are the many-body one of the primary objections to both the GF and EOMGreen's function (GF) method which has been exploited theories. It provides the chemist with a syslemaliceleverly by Linderberg and Ohm,I9-%\Reinhardt and approach for calculating the excitation energies andDoli,%!and Schneider, Taylor, and Yaris,23 and the oscillator strengths of low-Iyingexcited states, in whichEquations of Motion (EOM) approach used by experimental or theoretical knowledge of higher statesMcKoy2l-28andothers.29-35Inbothoftheabovetheories, can be employed to aid the calculation. A sequence ofcomplete knowledge of the properties of the individual approximations can be defined in which the role of thestates, te., the wavefunctions, is sacrificed to make higher excited states in determining. the low-Iyingpossible the direct calculation of the transition prop- excitations is quite elear. The introduction of a pro-erties, thus minimizing both the problems of cancella- jection operator onto the subspace of low-lying excitedtion and the amount of calculated information which is states allows the elear separation of high- and low-

3787

3788 JACK SIMONS

energy excitations in the proposed formalism. A newenergy-shift function, which contains the effects of thehigher excited states on the states of interest, can beapproximated by using experimentaIly measuredspectra.

THE ENERGY-SHIFT THEORY

In statistical mechanics one is often interested informulating the equations of motion of dynamicalproperties in such a manner that a elear separation oftime scales is realized. For example, in hydrodynamictheories39the number, momentum, and energy densitiesvary on a (slow) macroscopic time scale. In deyelopingreasonable approximate hydrodynarnic equations, it is

- very advantageous if the exact equations governing thetime and spatiaJ behavior of the hydrodynamic vari-ables display a elear separation of slow varying andrapidly varying terms. If this is the case, one can oftenintroduce rather simple approximations to the rapidlyvarying terms without appreciably affecting the ac-curacy of the resulting solutions.

In connection with this time-scale separation prob-lem, Mori40has derived exact equations for the equilib-rium-averaged time correlation functions <..1;(1)..1/)of any set of operators 1..1;1. In these equations aseparation of time scales appears naturally. Choosingthe set of operators to be4' ICr+Co, Co+Crl, and takingthe zero-temperature limit, we can use Mori's resultsto obtain new equations of motion for the followingmatrix elements:

Xij",,(t) = (g ICi+(I)Cj(t)C,+Ck!g), (1)

where the index pairs i,j and k,l are eitherU a, Tor T,a.Using the definition of the Heisenberg tin:e-dependentoperators, it is easily shown that the poles and residuesof the one-sided Fourier transform of Xij,kl(l) aredirectly related to the excitation energies and oscillatorstrengths between the exact excited electronic states andthe ground state Ig}.

The Mori equations40discussed above are written inmatrix form below .

d' . I

;[/(/) + iX(t)S-IQ+ i ~ dTX(T)S-IM(t~T)=O,(2)withG

Q;j,kl= - <gI [H, C'+Cj]C,+C" , g), (3)

Sij",,= <gI Ci+C,C,+C"Ig}. (4)

The matrix M(t) is defined in terms of the projector Ponto the subspace of the operators {Co+Crand Cr+Col

PB(t)= :E:E <gIB(t)C,+C" Ig)Skl,mn-IC...+Cn, (5)k,l m,n .

as foIlows:

M'""mn(t)= (i/fi) (g I

X {exp[(i/fi)t(1-P)H~J(l-P)[H, C,,+C,JI

X (l-P)[H, Cn+CmJIg). (6)

H~ is used to represent the commutator with theHamiltonian

HzB=[H, BJ. (7)

Denoting (l-P)[C,,+Cr, HJ by Ikr allows us to writeEq. (6) more briefly as f

Mkl,mn(t) = (i/fi) <gI

X {exp[(i/fi)t(l-P)H~J/k"ln..1 g). (6')

The content and properties of M(t) can be seen byinserting a complete set of eigenstates of H to the rightof the braces in Eq. (6'). Jt has been assumed thus farthat the low-Iying excited states consist mai7l1yof singleexcitations (Cr+Co Ig) and C..+Cr Ig» of the groundstate, whereas the higher excited states are composedmainJy of double and higher excitations. Analyzing theresult of inserting the complete set mentioned above,one can see that single, double, and higher excitationsof Ig} contribute only terms of high-frequencytimedependence to M(t). In addition, these terms are of smalimagnitude unless Ig}contains a great deal of configura-tion interaction (CI) involving single, triple, and higherexcitations of the single-determinant approximation toIg}. The only remaining function in the complcte set,Ig) itself, also contributes a term of high-frcquencytime dependence to M(t). The magnitude of this term isdetermined by the importance of doubly excited con-figurations in the CI expansion of Ig}.

From these observations, it follows that the one-sidedFourier transform of M(t)

M(E) = ~.. exp (~/E) M(t)dt (8)

possesses only high-energy poles corresponding toappToxll.'Ultehigher excitation energies of the molecule.As will be seen shortly, this important property of thee71eTgy-shiltma/rix M(E) aIlows us to obtain reasonableapproximate equations governing the Jow-frequencybehavior of X(t). Because the poles of the one-sidedFourier transform of X(t) correspond to electronicexcitation energies, this implies that Eq. (2), togetherwith a reasonable approximation to M(t), can be usedto probe low-Iying excited states of molecules.

Taking the one-sided Fourier transform of Eq. (2)gives the fundamental equation of the energy-shifttheory

X(E) [E1-S-IQ-S-IM(E) J=ifiS, (9)

where 1 is the unit matrix and X(E) is the transfonnof X(t). The energy-shift matrix M(E), which possessesonly high-energy poles, is weakly dependent on E forvalues of E much less than the higher excitationenergies. Because we are interested in only the low-energy poles of X(E), the E-dependence of M(E) ~be treated in approximate fashion.

The lowest approximation which can be made is toneglect the M(E) matrix. The resulting X(E) is givenin terms of the eigenvalues "i and eigenvectors" U;, Vi

i

f",'""

,>'

..

j ELECTRONIC STATES OF MOLECULES

of the matrices S-IQ and (S-lQ)+

S-lQUi= ~iUi,

(S-lQ)+Yi= ~Ni,as follows:

X(E) =iltS L (E-'Aj)-lUNj+.j

In this approximation, the excitation energies are equalto the 'Aj,and the residues of X(E), which determine thet~ansition strengths, are equal to the elements of SUjY/.The physical content of Eq. (12) is most easily under-stood by examining the matrix S-IQ, whose eigenvaluesand eigenvectors determine X(E). By rewriting Eq. (3)in the form

{lkl ,,= (g I Ck+C,(H-Eg)C"+C,,.1 g),

it is elear that the eigenvalues and eigenvectors ofS-l Q are identical to the energies and expansion co-efficients which would arise in a linear variationalproble~ with Hamiltonian (H-Eg) and nonorthogonalbasis functions C,,+C...Ig). Thus, the excitation energiesarrived at by neglecting the energy-shift matrix are thesame as the energy differences obtained in a CI calcula-tion using the fC,,+C...1g> and Ig)} as basis functions.The above remarks are only true if the function I g) isthe exact ground state. In the research presented here,we have not yet discussed appropriate choices of Ig).In the application of the energy-shift thcory to atomieand molecular systems, one eventually is faced with theproblem of calculating various ground-state expectationvalues, e.g., the first- and second-order density matricesof Ig). For practical calculations, we have in mind tworeasonable altematives: if accurate configuration inter-action approximations to Ig) are available, they can beused in our theory; otherwise, one will probably haveto use a single-determinant approximation for Ig).The latter choice would, of course, greatly reduce thecomplexity of the computational problem. Clearly, acomplete subhierarchy of energy-shift theories can bedeveloped by making various approximations to Ig).Because we do not wish to unnecessarily complicatematters in this introductory paper, we shall not discussthe choice of Ig) any further.

To make progress beyond the results presented above,more reasonable approximations to the energy-shiftmatrix must be constructed. Some especially promisingpossibilities are now presented.

It follows from our earlier discussion of the E-dependence of M(E) that the energy shift matrix can bewritten in the following form:

M(E) = L (E-E"O)-lM", (14)"

where the E"O are approximate higher excitationenergies of the molecule, and the M.. are expansioncoefficients. We can make Eq. (14) practical byemploying a truncated' expansion in which we useexperimentally determined excitation energies45and

3789

(10)

(11)

theoretically calculated coefficients M". The method ofcomputation is derived and outlined below.

, Expanding Eq. (14) asymptotically in powers of E-lgives

M(E) =E-l L M,,+E-2 L E"OM..(12) " "

+E-3L: (E..O)2M,,+ (15)"

We can also use a theorem relating the expansion ofM(E) in powers of E-l to the Taylor series expansion of

M(t) to writ~M(E) = L: E-(l+1)A(I+1),

, 1-0(16)

where

(13) Apq,.,(I+1)= (-1) /+1(g I f[(l-P)H"'J+ICp+Cq}

X (l-P)H"'C,+C Ig). (17)

For a given ground wavefunction, the matrices A(I+1)can be calculated, although for higher l values, theeffort involved would probably be quite substantial.For this reason, we have in mind truncations of Eq. (14)involving only a few terms. By equating the coefficientsof E-l appearing in Eqs. (15) and (16), we obtain a setof algebraic equations relating the unknown expansioncoefficientsM.. to the theoretically calculated A(l+I)andthe E..owhich are taken from experiment. For example,if the expansion in Eq. (14) is limited to two terms,Eqs. (15) and (16) imply

A(I)=M1+Mz,

A(2)= ElOM1+ E2°Mz.

which can easily be solved to give

M1= (E1°-E2O)-I(Am-E2°A(I),

(18a)

(18b)

( 19a)

(19b)and

M2= (E1°-E2O)-I(E1°A(I)-A(Z».

Once the M.. have been computed, Eq. (14) gives anexpression for M(E) which can be used in Eq. (9) tocalculate X(E). The inelusion of more and more termsin Eq. (14) thus provides a systematic series of approxi-mation to M(E). In this scheme, the poles of X(E)are located by searching (with the aid of a computer)for values of E for which thc clemcnts of [El-S-1Q-S-IM(E)]-1 diverge. To expedite the calculation, theeigcnvalues of S-IQ can be used as initial guesses in thepole-scarching process. The residue at the pole Ek,which determines the transition strength to excitedstate Ik), is iltS[1 +S-IL:..(Ek-E,.°)-2M..]-l.

The amount of numerical work involvcd in the abovesearching procedure may become prohibitively largefor large molecules such as those commonly studied bysemiempirical molecular orbital methods. Therefore,it is important to consider a further approximation ofthe theory described above. As was implied earlier, theenergy-shift matrix is strongly dependent on E onlywithin neighborhoods of the poles E..o. Thus, in cal-

3790 JACK SIMONS

culating the low-energy poles of X(E) it is reasonableto neglect the E-dependence of Eq. (14) and write

M(E)':::::.-:EM"/E,,o=M(O)...

The resulting X(E) is a generalization of that gh'en inEqs. (10-12), in which the matrix S-IQ is replaced bys-ln+S-lM(O). Once the elements M.. and E,.° con-stituting M(O) are determined, the excitation energiesand residues are easily obtained by solving two matrixeigenvalue problems analogous to Eqs. (10) and (11).This is elearly a major computational advantage overthe preceding more general approximation.

In the event that the A(l+l)beyond A(2)are verydifficult to compute, a geometrie approximation ofM(E), analogous to that used by Linderberg and Rat-ner«' for Green's functions, may prove useful. This ap-proximation is easily obtained by first re,,'riting Eq.(16) as

M(E) =E-IA(I) [1+E-I(A(I)-IA(2)

+E-2(A(I»-lA(I)+...] (21)

and then approximating the power series geometrically:

M(E)':::::.E-IA(l)[1-E-l(A(I»-lA(2)]-1

=A(I)[El- (A(l)-IA(2)]-1. (22)

Equation (22) provides a simple parameterization ofthe energy shift involving only A(l)and A(2),which canbe used in Eq. (9) to calculate X(E). This approxima-tion is also useful when no information about higherexcitation energies is available. In this theory, the polesof X(E) are determined by searching for values of E atwhieh elements of [El-S-1Q-S-IM(E)]-l diverge.The residue of X(E) at the pole Ek is

ifiS{l +S-IA(I)[E"l- (A(I)-IA(2)]-2j-l.

Of course, if the pole-searching process proves to be toocostly for a specific moIecuJe,one can, further approxi-mate M(E) by M(O)and proceed as was described in thepreceding paragraph.

This eoneludes our discussion of the approximationsto the energy-shift matrix. Although we have alreadymade considerable progress, future research effortsmust be directed toward making the above approxima-tions as computationally useful as possible.

RELATION OF ENERGY-SHIFT THEORY TOGREEN'S FUNCTIONS

The two-body, energy-dependent Green's functionmatrix is definedl. in the following manner:

Gii,kl(E)= ~i f~ tp(~Et)X(g I TC,+(t)Ci(t)Ck+C,1 g)dt, (23)

,.

(20)

where T is the Wick time ordering operator. By usingthe definition of T and the identity

(g I C.+C,C;+(t)Ci(t) I g}

= (g ICk+(-t)C,( -t)C,+Ci Ig), (24)

one can show that the Green's function matrix is cIoseh'related to X(E) : .

ifiGii,kl(E) = Xii,lk(E) +Xkl.ii(- E),

whereE is consideredto be a complexvariable

E= ReE+i8,

(25)

(26)

whose (positive) imaginary part is allowed to approachzero. Xij,lk(E) has poles along the lower side of thepositive real E axis at values of the electronie excitationenergies, and XU.;i( -E) has corresponding poles alongthe upper side of the negative real E axis. Thus, it iselear from Eq. (25) that Gii,kl(E) possesses poles alongboth the positive and negative real axes. This result is,of course, well known in Green's function theory.

The principaJ differences between the Green's func-tion approach and the energy-shift theory are containedin the equations of motion used to calculate the G(E)or X(E) matrices. In the Green's function formalism,the Heisenberg equation of motion

iMC;+(t)Ci(t)/dt= [Ci+(t) Ci(t) , H]

is used to obtain the following starting equation:

(27)

EGi;./o,(E) =8jk(g I C,+CII g)-8i1(g I Ck+CjIg}

-i i-

(i

)X T -.. exp fiEt (g I T[C,+(t)Ci(t),H]

X Ck+CI I g }dt. (28)

To arrive at a elosed equation for G(E), one can(approximately) expand the commutator appearing inEq. (28) in terms of single excitations:

[C,+(t) Ci (t) , HJ~ L A ii.kzCk+(t)C1(t).k,l

(29)

At least two reasonable possibilities exist for thechoice of the expansion coefficients Aij,kl. One canapproximate the exact commutator [C,+(t) C;(t) , H]by its projection onto the space of the operators[Ccr+(t)C,(t)and C,+(t)Co(t) l. which results in

Aii,lI= L (gI[C,+C;,H]c..+C..1g)S k,-l. (30)"."

Another criterion for determining A'j,k' has been putforth by Linderberg and Ohm.l. They choose the ex.pansion coefficients to minimize the sum of the norms ofthe two functions

[C,+Cj, H] I g)- L Aii,kzC,,+CII g}k,l

(31)

v

ELECTRONIC STATES OF MOLECULES 3791

and[Ci+Cj,H]+ I g)- L Aii.kl*CI+CkIg).

k.1

This procedure gives the following expression:

(32)

Aii.kl= (g I [Ci+C;, H]Cl+Ck+CI+Ck[Ci+C;, H] I g).

(33)

:BecauseEqs. (30) and (33) are not identical, we con-dude that the Linderberg-Ohrn truncation schemeneglects some components of [Ci+C;,H] which lie inthe space of single excitations. Of course, this does notimply that the truncation generated by Eq. (33) isinaccurate. In fact, this truncation has been success-fully applied to a wide range of molecular problems.19.21It should also be mentioned that Linderberg and Ohrnhave developed more sophisticated truncation schemes19involving higher-order Green's functions.

From" the above discussion, it is elear that the ap-proximation given in Eq. (29), when substituted intoEq. (28), learls to a elosed equation for the elements ofG(E), regardless of which expression for A ii,kI is used.The final equation governing G(E) is written below:

EGii.U(E) =~ik(g I Ci+C11 g)-~il(g I C,,+Ci I g)

+ L Aii""nG",n,kl(E). (34)"'.n

In comparing Eq. (34) to Eq. (9), one notices thatEq.(9) contains the additional E-dependent termS-IM(E). Because the energy-shift matrix reflects,through lts high-energy poles, the presence of higherexcited states, we feel that the absence of an E-depend-ent matrix in Eq. (34) is a disadvantage of the Green'sfunction method.

The lowest approximation to the energy-shift theory,which is obtained by neglecting M(E) in Eq. (9), canbe shown to be equivalent to Eq. (34) with the coeffi-cients A ii.kIgiven by Eq. (30). Because the Linderberg-Ohrn truncation [Eq. (33) J differs from Eq. (30), weconelude that Eq. (33) neglects some of the termscontained in S-IQ, while ineluding, in an energy-independent fashion, some contributions to S-IM(E)from higher excited states. This observation points outanother potential advantage of our energy-shift method:in our formalism,all effectsof higherexcitedstates are "

isolated in M(E). Because M(E) has only high-energypoles, it will, perhaps, permit rather simple approxima-tions of its low-energy structure to be reasonably ac-curate. Moreover, Eqs. (14)-(17) provide a well-defined means of generating approximations to theenergy-shift matrix.

Finally, there exists the possibility of employing ex-perimental knowledge about higher excitation energiesto aid the construction of the energy-shift matrix. Thiscan be especially advantageous for molecules whosehigher states have been thoroughly studied but whoselow-lying states are in question.

TREATMENT OF DOUBLE EXCITATIONS INLOW-LYING EXCITED STATES

For some molecular systems, our assumption thatthe space of low-lying excited states is spanned by singleexcitations {Cr+CaIg) and Ca+Cr Ig)1 may not bevalid. In this event, the energy-shift theory must beextended to properly inelude the presence of doubleexcitations {Cr+CaC.+CpIg) and Ca+Crcp+c.1g)1 inthe excited states of interest. This extension of the

above theory is acc~plished by decomposing theenergy-shift matrix in Ducha manner that the contribu-tions of double excitations are isolated from the termsarising from triple and higher excitations. The techniquefor realizing this decomposition is now presented.

Because the electronic Hamiltonian H consists ofone- and two-partiele operators, the f,.1

/kl==(l-P) [C,,+CI,H] (35)

contain only double excitations, the single excitationshaving been annihilated by the operator (1- P).Therefore the operator I P, definedby

'PB== L L (g I Bfkl+Ig)'Skl""n-1j",n,",1 m.n

(36)

with'S,.I,mn==(g Ifklfmn+ I g), (37)

is a projector onto a subspace of double excitations.In his definitive works on statistical mechanical time

correlation functions, Mori developed a formalisminvolving the projectors P and I P which can be used toobtain a elosed equation for the energy-shift matrixM(E). In this equation there existsa elear isolationofthosecontributionsto M(E) whicharisefromoperatorsIying within the double-excitation space spanned by the{f,.d. The final matrix equation can be written in aform which is analogous to that of Eq. (9) as follows:

M(E) [El-/S-I/Q-/S-lIM(E) ]=ili'S, (38)

where

'P.ij,kl= - (gI {(1-P)[fij, H]lfkl+ Ig). (39)

The matrix element' NIij,u(E) is the one-sided Fouriertransform of "

'NI,UI(t) == (i/Ii) (gIX {exp[(it/Ii) (l-P-' P)Hz](1- p_' P) [fi;' H]I

X (1_P_'P)[fkl+, H] Ig). (40)

The presence of the projector (1-P-' P) in Eq. (40)implies that neither the single excitations {Cr+CaI g),Cr+CrI g)1 nor the double excitations {/kt! g)1 con-tribute to 'M(E). Moreover,it is elear from Eq. (39)and the above discussion both that the single excitationsdo not appear in 'Q and that aUcontributionsof the(je I) to M(E) are containedin 'Q.

3792 JACK SIMONS

Because we are interested in probing low-lyingexcited states which are assumed to be spanned by thefunctions {Ca+CrIg), Cr+Ca Ig),Jar Ig) and ira Ig)1we can hopefully obtain reasonable results for theenergy-shift matrix by employing rather simple ap-proximations for 'M(E) in Eq. (38). The lowest ap-proximation which can be made is to neglect 'M(E).This leads to an expression analogous to Eq. (12),giving M(E) in terms of the eigenvalues (/Xj) andeigenvectors (/Uj, 'Vj) of ('S-II Q) , and ('S-l/Q)+;

M(E) =iJi/S L: (E-/Xj)-l/U/Vj+.j

(41)

This approximation for M(E), when substituted intoEq. (9), yields a theory for X(E) which incorporatesthe effects of double excitations (ikl lg» in the low-lying excited states of interest.

The computational difficultYof the above scheme fortreating double excitations isdetermined by the amountof effort needed to generate elements of the 's and 'Qmatrices.46 Because this effort will probably be sub-stantial for all but very simple molecular systems, wefeel that more sophisticated approximations to 'M(E)are impractical at present.

In principle, we can develop a well-defined series ofapproximations for 'M(E) by proceeding in a fashionanalogous to that employed earlier in deriving approxi-mate expressions for the energy-shift matrix. However,due to the practical difficulties discussed above and theinherent complexity of 'A/ij.kl(t) as given in Eq. (40),

. we shall not treat this matter further. Rather, we planto concentrate our efforts in the immediate future onmaking the energy-shift theory presented here ascomputationally useful as possible. This will, of course,involve a considerable amount of calculational researchin which the approximations discussed in this paper aretested on molecular systems whose low-lying excitedstates are well understood. We believe that the theo-retical framework described in this paper will enable usto make significant advances in both interpreting andpredicting the electronic excitation energies and oscil-lator strengths of low-lying excited states of molecules.

* Acknowledgment is made to the donors of the Petroleum

Research Fund, administered by the ~Ameriean Chemieal Society,for support of this research. .

1 The oscillator strength between excited stateI

j) and theground state Ig) is given by /;=2/3AE;! Ulr g)12, wheret:.Ej is the excitation energy to state Ij) and all quantities aregiven in atomie units.

2M. V. Mihailovic and M. Rosina, Nuci. Phys. A130, 386(1969) .

2 K. lkeda, T. Udagawa,and H. Yamaura,Progr. Theoret.Phys. (Kyoto) 33,22 (1965).

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12 J. Da Provid~ncia, Phys. Letters 21,668 (1966).12 D. J. Rowe, Rev. Mod. Phys. 40, 153 (1968).H D. J. Rowe, Phys. Rev. 175, 1283 (1968).16 J. Schwinger, Proc. Nat!. Acad. Sci. U.S. 37, 452 (1951).16 P. C. Martin and J. Schwinger,Phys. Rev. 115, 1342(1959).17L. P. Kadanoff and P. C. Martin, Phys. Rev. 124,670 (1961).12J. ~oldslone, Proc. RRY. Soc. (London) A239, 267 (1957).18J. Lmderberg and Y. Ohrn, J. Chem. Phys. 49, 716 (1968);

Chem. Phys. Letters l, 295 (1967); Proc. Roy. Soc. (London)A285,445 (1963); Phys. Rev. 139, AIO63 (1965).

20J. Linderberg and M. Ratner, Chem. Phys. Letters 6, 37(1970).

21J. Linderberg and E. W. Thulslrup, J. Chem. Phys. 49, 710(1968).

22 W. P. Reinhardtand J. D. Doli,J. Chem.Phys.50, 2767(1969).

22 B. Schneider, H. S. Taylor, and R. Yaris, Phys. Rev. A l,855 (1970).

UT. Shibuya and V. McKoy, J. Chem. Phys. 53, 3308 (1970)26 T. Shibuya and V. McKoy, J. Chem. Phys. 54,1738 (1970).26 T. Shibuya and V. McKoy, Phys. Rev. A 2, 2208 (1970).27T. H. Dunning and V. McKoy, J. Chem. Phys. 47, 1735

(1967).26 T. H. Dunningand V. McKoy,J. Chem.Phys. 48, 5263

(1968).29 R. A. Harris, J. Chem. Phys. 47, 3972 (1967).30P. L. AItick and A. E. Glassgold, Phys. Rev. 133, A632

(1964).. .31 N. Ostlundand M. Karplus,Chem.Phys.Letters 11, 450

(1971).32A. Herzenbcrg, D. Shcrringlon, and M. Suveges, Proc. Phys.

Soc. (London) 84,465 (1964).32M. A. Bali and A. D. McLachlan, Mol. Phys. 7, 501 (1964).3. B. Lukman and A. Azman, Mol. Phys. 16, 201 (1969).36J. Simons, J. Chem. Phys. 55, 1218 (1971).26For an excellent review of this problem see the Ph.D. thesis

of M. B. Ruskai which can be obtained as University of WisconsinThcoretical Chemistry lnstitute Report WIS-TCI 346.

37See for example J. Simons, Phys. Rev. A 2, 1034 (1970),Chem. Phys. Lctters lO, 94 (1971), and Ref. 35 for a discussionof N-representability in EOM theoncs.

38Sce, for example, G. E. Uhlenbeck and G. W. Ford, Lecluresin Statistical Meckanics (American. Mathematical Society,Providence, 1963), Chap. VII.

28See, for example, B. V. Felderhof and 1. Oppenheim, Physica31, 1441 (1965).

.0 H. Mon, Progr. Theoret. Phys. (Kyoto) 33, 423 (1965); 34,399 (1965).

4l The C.+ and Cj are fermion creation and annihilationoperators, respectivcly. The indices a and {3label single-particiefunctions (spin orbit aIs) which are occupied in the single- or few-determinant approximation to the ground state wavefunction.Similarly, the indices r and s label unoccupied Sfin orbit aIs. Thetrue ground state wavefunction is designated as g).

42From now on, the index pairs i, j; k, l; and m, n will be eitherof type a, r or r, a. Terms involving a, {3or r, s will not occur.

.3 The significance of the O matnx is easily understood forthe following simple Hamiltonian H ="};.E.C.+Ci. In this case!}ij.kl= (Ej-E.) (g I Ci+CjCI+Ck Ig). Thus, O contains somecontributions to the excitation energies. The remaining contribu.tions to the excitation energies are contained in the matnx M(t).The presence of M(t) in the equations of motion causes thepredicted excitation energies to be skifted from the values cal.culated by neglecting M(t). This aIteration of the energy spectrumis the origin of the nam e of the energy-shift theory. S is simply theoverlap or metric matrix for the space of functions ICI+Ck Ig) I.

.. Because 5-10 is not symmetric, we must introduce theeigenvectors of S-la and its Hermitian conjugate (5-10)+. Therow vector Vi+ is formed by taking the Hermitian conjugate ofthe column vector Vi.

.6 We have in mind molecules for which some or the higherexcitation energies are known but for which uncertainty existsconcerning the ordering of the lower states.

.6 Notice that the dimensions of the matrices appearing inEq. (38) are identieal to those occurring in Eq. (9). Thus, theextension of the energy-shift theory to include the effects ofdouble excitations does not alter the amount of effort neededto carry out the matrix algebra in a practical calculation.