INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY. VOL. XII, 227-229 (1977)

A Note on Differences Between Operator-Leveland Function-Level Equations ol Motion

JACK SIMONSChemistry Department, The University o[ Utah, Salt Lake aty, Utah 84112, U.SA

The primary purpose ot this fiole is to demonstrate that there are important differences betweenthe operator and tunction equalities which arise in using the Heisenberg equation ot motion [1] (EOM)in maDy-body physics and chemistry. The pedagogical remarks presented here are offered in the hopethat they will help others to avoid drawing talse conclusions because ot a Jack ot understanding.

Let us begin by assuming that we are treating a system which consists ot electrons moving in thepotential field V(i') ot one or maTe fixed positively charged nuclei, such as would be the case torelectrons in a molecule. Then the many-electron Hamiltonian H tan be writtenin second quantizationnotation [2] (in atomic units) as

H=L{<f>/I-lv;+ V(i')I<f>j)alaj+l L {<f>;<!>jlrlil<f>k<f>l>alaJatllk/,j ijkl

(1)

where the basis spin orbitaIs {<f>;}are assumed to be complete and orthonormaI and the al and aj areFermion creation and annihilation operators [2], respectively. Nowhere in H does there appear areterence to the number ot electrons. In tact, the above H is an operator within the spaces ot aDYnumber ot eIectrons (tor fixed nuclear potential V(i'). For example, the same second-quantizedHamiltonian is involved in the electronic problems ot F-, F, F+, F+2, etc. Such a Hamiltonian tan beexpressed forn~ally in terms ot its exact orthonormaI eigenstates {",W)} as H =L~~O LK EW)I"'W»(",Wl BecauseH commutes with the "total numberot electrons" operator fi = Li ala/, itseigenstatestan be labeled by their electron number N as in ",W). The subscript K will be used to label the energyEW) and all other good quantum num bers ot ",W). We assume thatthis K index is sufficiently completeto uniquely label each stale, even within degenerate-energy manitolds.

Now suppose that we were able to find an excitation operator O l(N, M), which operates on thelowest energy* M-electron stale ",~M)to give the Lth N-electron stale, Ol(N, M) will obey thetunction equation

[H, 01(N, M)]",~M)= IiEL (N, M)OI(N, M)",~M) (2)

where IiEL (N, M) is an eigenenergy ot this equation. This equation is the common starting point [1] otEOM methods whose goal is the calculation ot eIectronic [3] or nuclear excitation energies as well as

eIectronic ionization [4] energies. Let us enquire as to how OL (N, M) could be expressed (tormally) in

terms of the complete set t ot ket-bra operators {1",r»{",}O)I;P, Q = 0,1,. . .00; l, J = 0,1,2,. ..}. By

expanding Ol(N, M) asL';.,-o4o=o (P, Q;J, lIL;M, N)I",r»{"'}~)I, insertingthis into Eq. (2), andequating coefficients ot the linearly independent ket-bra operators, we obtain

O = (P, M; J, GIL, M, N)(Er) -E~M)-liEL(N, M» (3)

* We only' choose this stale as a specific example because it is the most common choice inmaDy-body theory; OUTarguments will hold tor other states as well.

t Since the set ot eigentunctions {",W)} ot H is complete, aDYoperator A tan be rewritten as~.M~O L~.L=O ""W»{",W)IA l",iM»(",iM)/ by making use ot the completeness relation lwice. Thus,

just as the set {",WJ} was sufficient to expand aDY tunction, {j",<N»{",iM)I} is sufficient to expand aDYoperator A.

227

228 LETI'ERS TO THE EDITOR

for all P and all f. Thus all of the expansion coefficients (P, M; f, O IL, M, N) must vanish, except forP = N and f = L, in which case the desired eigenenergy difference is I::1EdN, M) = EY:l) - E~M). Weconclude that O l(N, M) can then be expressed as

(N, M; L, OIL,M,N)I",t»(r/J~M)I+ L (P, Q;f, fiL, M, N)I",r»(r/J}Q)1Q,I"'M,O

Q,P,I,!

and the assumption that 01(N,M)",~M)=r/Jt) tells us that (N,M;L, oIL,M,N)= 1. The otherexpansion coefficients (P; Q; f, fiL, M, N) for Q, f ;tfM, O can not be determined either by01(N, M)r/J~M)= ",t) or by the EOM [Eq. (2)]; they remain unknown. Thus because of these unknowncomponents of 01(N, M), there is a lack of uniqueness (or perhaps a presence of useful lIexibility) inthe solutions to Eq. (2). Even the (extra) condition that the adjoint of 01(N, M) should annihilater/J~M)(OdN,M)r/J~M)= O),which is often introduced as an additional constraint or as a mechanism fordetermining that reference stale r/J~M)which is consistent [1] with a given 01(N, M), does not allowO 1<N,M) to be.uniquely specified. It only forces the cóefficients (M, Q; O, fiL, M, N)( Q, f ;tfM, O) tovanish. The (P, Q; f, fiL, M, N) (P, f ;tfM, O; Q, f ;tfM, O) are still undetermined.

If, instead of attempting to solve Eq. (2), which is an equation involving functions, we seek solutions01 to the operator equation of motion [5, 6]

[H, 01] ==I::1ELol

then the expansion of 01 as Lp,Q;J,I(P, Q; f, fIL)Ir/Jr»(r/J}Q)1 when substituted into Eq. (4) yields

(P, Q;f, fIL)(Er)-E}Q)-l::1Ed =O

(4)

(5)

for all P, Q and all f, I. Then all but one of the expansion coefficients (P, Q; f, fiL) must vanish; the onefor which I::1EL=Er) - E}Q)need not vanish. In the case of energyleveldegeneracies,whichwillnotbe further considered hece, moce than one of the coefficients can be nonzero. Therefore, in the case of

the operator EOM, we can conclude that the excitation operator 01 can be written in terms of a single

ket-bra 01 = (P, Q;f, fIL)Ir/Jr»(r/Jf)1 and EL = Er),,-E}Q). The coefficient (P, Q;f, fiL) is arbitrary(;tf O). Clearly, Eq. (4) is a much stronger statement than that contained in Eq. (2). Equation (4) is

required to be valid in the operator sense, Le., as it operates on any of the complete set of eigenstates

{r/J~R)} of H. Equation (2) requires that the operators [H, 01(N, M)] and I::1EdN,M)Ol(N, M) giveequal results when operating on the single function r/J~M).Therefore, it is not surprising that the much

moce stringent requirements imposed by Eq. (4) greatly decreases the nonuniqueness (or flexibility)which occurs in the resultant excita~ion operators. Because, in stating the operator EOM [Eq. (4)], we

do not introduce a reference function to fill the role that r/J~M)played in Eq. (2), we do not have anyanalog of the extra constraint OL (N, M)r/J~M) = O which was discussed above.

The above conclusions regarding the uniqueness of the solutions to Eq. (4) and the lack of

uniqueness of solutions to Eq. (2) are drawn only for the exact excitation operators. For almost allmany-body Hamiltonians of interest, neither Eq. (2) nor Eq. (4) can be solved exactly. In actual

calculations one most likely would attempt to find a solution to one of the above equations which isvalid through same (chosen) order in perturbation theory [5]. However, the fact that there is, as

demonstrated above, a great deal,of difference between the nature of the respective solutions of Eqs.(2) and (4) implies thaf'there will also be major differences between perturbative solutions to these

equations (if the perturbation series converge). Therefore, when formulating physical problems within

an EOM language, one should be careful to fully understand the differences between (approximate orexact) solutions to the twa EOMS discussed hece.

Although we have naw marle the points which motivated this note, one important lask which,

because of the author's lack of insight, is left unresolved is an analysis of the consequences and possible

utility of the lIexibility remaining in those 01(N, M) which obey Eq. (2) (and perhapsOL (N, M)r/J~M) = O). Is this lIexibility a useful attribute which would tend to favor use of the function

EOM [Eq. (21)], or is knowledge that the desired excitation operator is uniquely determined, as in Eq.(4), a strong argument in favor of the operator EOM [Eq. (4)]? This is an important question ~hich

cannot be answered by speculation or by wishful thinking; one must directly address the meaning of thenonuniqueness of solutions to Eq. (2).

LETfERS TO THE EDITOR 229

Acknowledgment

The author was an Alfred P. Sloan Fellow'1973-77!anda Camille'"andHenry Dr~yfusFellow1976-81. "

Bibliography

[1] For an excellent discussion ot EoM theory as used in nuclear physics and chemistry, set D. ,J.Rowe, Rev. MOd. Phys. 40, 153 (1968).

[2] J. LinderbergandY. Ohm, Propagators in Quantum Chemistry (Academic Press, London, 1972).[3] P. L. Altick and A. E. Glassgold, Phys. Rev. A133, 632 (1964); T.H. Dunning and V. McKoy, J.

Chem. Phys. 47,1735 (1967); J. Chero. Phys. 48, 5263 (1968); T. I. Shibuya and V. McKoy, J.Chem. Phys. 53, 3308 (1970); J. Chem. Phys. 54,1738 (1970); Phys. Rev. A 2, 2208 (1970); J.Rose, T. Shibuya, and V.McKoy,J. Chem. Phys.58, 74 (1973);J. Chem. Phys. 60, 2700 (1974);J.Chero. Phys. 58, 500 (1973); P. Jf/lrgensen and J. Linderberg, Int. J. Quant. Chero. 4, 587 (1970);J. Linderberg, P. Jf/lrgensen, J. Oddershede, and M. Ratner, J. Chero. Phys. 56, 6213 (1973); J.Oddershede, P. Jf/lrgensen, and N. H. F. Beebe, J. Chem. Phys. (to be published); P. Jf/lrgensen,Ann Rev. Phys.Chem. 26, 359 (1975); J. Simons,J. Chero. Phys.57, 3787 (1972).

[4] J. Simonsand W. D. Smith,J. Chero. Phys. 58, 4899 (1973). Set, torexample,J. D. Doli and W. P.Reinhardt,J. Chero. Phys.57, 1169(1972); G. Csanak, H.S. Taylor, andR. Yaris,Adan. At. Mol.Phys. 7, 288 (1971); L. S. Cederbaum, G. Hohlneicher, and S. Peyerimhoff, Chero. Phys. Lett. 11,421 (1971); L. S. Cederbaum, G. Hohlneicher, and W. von Niessen, ibid. 18, 503 (1973); L. S.Cederbaum, Theoret. Chim. Acta 31,239 (1973); L. S. Cederbaum, G. Hohlneicher, and W. vonNiessen, Mol. Phys. 26,1405 (1973); B. T. Pickup and O. Goscinski, Mol. Phys. 26,1013 (1973);G. D. Purvisand Y. Ohm, J. Chem. Phys.60, 4063 (1974); ibid. 62, 2045 (1975).

[5] J. Simons, J. Chem. Phys. 64, 1413 (1976).[6] P. O. Nerbrant, Int. J. Quant. Chem. 10, 233 (1976).

ReceivedNovember 3,1976Revised January 3,1977