- Email: [email protected]
Some comments on the coupled cluster with generalized singles and doubles (CCGSD) ansatz
Chemical Physics Letters 397 (2004) 174–179 www.elsevier.com/locate/cplett
Some comments on the coupled cluster with generalized singles and doubles (CCGSD) ansatz Debashis Mukherjee a
a,* ,
Werner Kutzelnigg
b,*
Department of Physical Chemistry, Indian Association for the Cultivation of Science, Calcutta 700 032, India b Lehrstuhl fu¨r Theoretische Chemie, Ruhr-Universita¨t Bochum, D-44780 Bochum, Germany Received 12 August 2004 Available online 11 September 2004
Abstract In this Letter, we present some comments on the nature of the CCGSD ansatz W = eTU for an n-electron wave function, with T a linear combination of general one- and two-particle operators. Nooijen had conjectured that such a parameterization is possible for the exact eigenstate W. We point out that the essential reason for the invalidity of this conjecture lies in the fact that the basis operators into which the Fock space Hamiltonian can be expanded, are not closed under commutation, i.e. do not span a Lie algebra. We give two proofs, based, respectively, on the variation principle and the method of moments, to show this. The variational proof traces the same ground as in an earlier one by Nakatsuji, which did not get the due recognition. We, however, formulate it in such a way that the key role of the algebra of the operators of T becomes transparent. We also discuss a related proof of Mazziotti. Our second proof sheds further light on why the ansatz, while not exact, might still provide an accurate description of a many-electron state, as has been found in some recent computational studies. We also discuss two recent attempts to disprove the exactness of the ansatz, based on a dimensionality argument. 2004 Elsevier B.V. All rights reserved.
aqp ¼ ayq ap ;
1. Introduction In most numerical studies of the n-electron Schro¨dinger equation one expands everything in a finite one-electron basis, and one tries later (at least in principle) to extrapolate to a complete basis [1]. Starting point is then the finite-dimensional Fock space Hamiltonian H in terms of spin–orbitals, that we give here in a tensor notation with the Einstein summation convention [2] 1 H¼ þ gpq ars ; hpq ¼ hwq j h j wp i; 2 rs pq gpq rs ¼ hwr ð1Þws ð2Þ j gð1; 2Þ j wp ð1Þwq ð2Þi; hpq aqp
ð1Þ
* Corresponding authors. Fax: +49 234 32 14045 (W. Kutzelnigg); +91 33 2483 6561 (D. Mukherjee). E-mail addresses: [email protected] (W. Kutzelnigg); [email protected] (D. Mukherjee).
0009-2614/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.08.100
y y ars pq ¼ ar as aq ap :
ð2Þ
The n-electron problem reduces to an eigenvalue problem for the full CI-matrix H. This matrix consists of the matrix elements H lm ¼ hUl j H j Um i 1 hUl j ars ¼ hpq hUl j aqp j Um i þ gpq pq j Um i; 2 rs
ð3Þ
in terms of all possible n-electron Slater determinants Ul constructed from the given one-electron basis {wp}. The transition density matrix elements like hUl j aqp j Um i are independent of the Hamiltonian. While H is characterized by d parameters, H contains D parameters, with d¼
m 2
m 2
þ 1 =2 ¼ Oðm4 Þ;
ð4Þ
D¼
N
2 m N¼ ; n
¼
m n
m
n
D. Mukherjee, W. Kutzelnigg / Chemical Physics Letters 397 (2004) 174–179
175
þ 1 =2 ¼ Oðm2n Þ;
ð9Þ
y
hU j eT ðH EÞU k eT j Ui ¼ 0: ð5Þ
where m is the dimension of the spin orbital basis, and n the number of particles. The D matrix elements of H are expressible through the d parameters contained in H. This implies that an eigenstate of H is a functional of only d = O(m4) parameters, although the corresponding eigenvector of H has (for n > 4) the larger dimension N = O(mn). The fact, that the Hamiltonian (1) is characterized by only d parameters, poses an important challenge to try to parameterize an eigenstate W by a set of d parameters, rather than (for n > 4) by the much larger set N of coefficients in the full-CI expansion. It is obviously so that the manifold of ground states of all possible Hamiltonians of the form (1) represents only a small subset of all possible eigenvectors of an arbitrary N · N matrix. One may, if one wishes, say that the physically possible ground states constitute a subset of Ômeasure zeroÕ of the full set of linear combinations of N n-electron Slater determinants. Unfortunately, it is not so easy to characterize this small manifold, and this is, of course, the key issue.
2. Nakatsuji theorem and Nooijen conjecture The question of the minimal parameterization of a quantum mechanical state has been studied in detail by Nakatsuji [3–5]. In this context an interesting observation plays a central role, that it now usually referred to as Nakatsuji theorem [6]. To formulate it we define the k-particle contracted Schro¨dinger equations CSEk [6,7] (which are necessary conditions for a function W to be exact, and which can be interpreted as the conditions for stationarity of the energy with respect to the variation W ! f1 þ U ðkÞ p gW) hW j ðH EÞU ðkÞ p j Wi ¼ 0;
ð6Þ
with fU ðkÞ p g a set of k-particle excitation operators. Especially for k = 1 or k = 2 we have hW j ðH EÞapq j Wi ¼ 0; hW j ðH EÞapq rs j Wi ¼ 0:
ð7Þ
The content of the Nakatsuji theorem [6,3] is that if (and only if) the CSE2 (which implies the CSE1) is satisfied, (and if W is antisymmetric) then W is an exact full-CI wave function. The proof rests on the simple fact that (7) implies 2
hW j ðH EÞ j Wi ¼ hðH EÞW j ðH EÞWi ¼ 0;
ð8Þ
which is only possible if (HE)W = 0 (in the full-CI basis). Let us express W in (6) through a known reference function U and an unknown Fock space wave operator W = eT
It is tempting to express T by as many (i.e. d) parameters, as there are equations to be satisfied. The simplest possibility to do this, and to satisfy at the same time the separability theorem [8], is that probably first proposed by Nooijen [9], and usually referred to as Nooijen conjecture, which chooses T to consist only of one- and two-particle operators. This implies the claim that the exact parameterization of W, for a given reference function U, is W ¼ W U;W ¼ eT ; T ¼
X p
1 T l U l ¼ T pq aqp þ T pq ars : 2 rs pq
ð10Þ
If one inserts (10) into (6), one gets a nonlinear system of as many (d) equations as there are unknowns Tl – or more specifically T pq and T pq rs . Nooijen regarded it as plausible that these equations do have a solution, which should then – in view of the Nakatsuji theorem – be equivalent to the full CI wave function. One must insist that the operator T is finite, in order not to fall into the trap that W becomes a projector, rather than a wave operator. Such solutions do exist, but have no practical use in the present context, and must hence be avoided. Shortly after the publication of NooijenÕs Letter, a paper by Nakatsuji appeared in print [3], in which the same ansatz was studied, using a single determinant reference U, and in which Nakatsuji came to the conclusion Ôthat this ansatz cannot be exactÕ. Nakatsuji called this ansatz Ôthe coupled cluster with generalized singles and doublesÕ (CCGSD), and this is the nomenclature we would henceforth be using. The proof, which was probably not presented in the best possible form, and which has found astonishingly little attention, is based on the variation principle, more precisely on the comparison of two stationarity conditions, whence we refer to it as the proof by variational comparison. We want to emphasize in this Letter that the essential reason why the Nooijen conjecture is invalid is the fact that the basis {Ul} of one and two-particle excitation operators does not span a Lie algebra. We give two proofs to bolster this point, one based on variation principle and the other on the method of moments. The one using the variation principle rederives NakatsujiÕs proof in a more sharpened form, which transparently brings out the importance of the lack of a Lie algebraic structure of the operator basis {Ul}.
3. Proof by variational comparison An exact wave function of CCGSD type must not only satisfy the CSE2 and CSE1 (9), but it must also make the energy expectation value
176
D. Mukherjee, W. Kutzelnigg / Chemical Physics Letters 397 (2004) 174–179 y
hU j eT H eT j Ui E¼ ; y hU j eT eT j Ui
ð11Þ
0¼
y oeT U UeT ðH EÞ oT p
Ty
stationary with respect to variation of the set of coefficients {Tp}. It must, if it exists, belong to the family of wave functions, which satisfy the stationarity condition y oeT 0 ¼ UeT ðH EÞ U oT p y
¼ hU j eT ðH EÞfU p þ 12½T ; U p þ 16½T ; ½T ; U p þ . . .geT j Ui:
ð12Þ
Since the energy does have a minimum as functional of the Tp, the Eq. (12) do have a solution. In order for a CCGSD function to be exact, it must also satisfy Eq. (9). This differs from the variational stationarity condition (12). Instead of requiring that (12) is satisfied simultaneously with (9), one may also require that (12) holds simultaneously with the difference condition y
0 ¼ hU j eT ðH EÞf12½T ; U p þ 16½T ; ½T ; U p þ . . .geT j Ui: ð13Þ This is satisfied in special situations, e.g. if the operator basis spans a Lie algebra (vide infra), or if the reference function U is equal to its exact counterpart W, but does not hold under the premises of the Nooijen conjecture. This variational proof (of the Ônon-exactnessÕ of the CCGSD ansatz – or the invalidity of the Nooijen conjecture) has two rather subtle aspects. 1. It is imperative to formulate it such that multiple commutators of the Up with T appear, also in the higher order terms in braces in (12), that are not indicated explicitly. This makes it obvious, that what matters is that the set of one- and two-particle excitation operators {Tp} is not closed under commutation and hence not a subalgebra of the Lie algebra of all excitation operators (as it would be required for a consistent formulation of the variation principle in terms of a Lie algebra [10]). Let us note that the single commutator and the double commutator in the Ôextra termsÕ in Eq. (12) beyond CSE2 contain respectively three- and four-particle operators. If we repeat the just given argument for an operator basis that does constitute a Lie algebra, we find that the stationarity of the energy and the CSE1 and CSE2 are compatible, because then [T,Up] and the higher order commutators are elements of the Lie algebra spanned by the Up. Nakatsuji, in his proof, using the variation principle with the CCGSD ansatz, did indeed show that the stationarity condition contains three and higher particle operators, but he did not present these equations in terms of commutators. Rather, he expressed this equation as
¼ hU j e ðH EÞfU p þ 12ðU p T þ TU p Þ þ 16ðU p T 2 þ TU p T þ T 2 U p Þ þ g j Ui:
ð14Þ
Nakatsuji argued that the extra terms arise because of the noncommutativity of the set Up. Since Nakatsuji did not write the stationarity condition in terms of commutators, its compatibility with the CSE for the case of a one-particle Hamiltonian was not immediately obvious, and he showed the compatibility using a different route via the Thouless parameterization. On the other hand, using the alternative form, viz. Eq. (12) it is straightforward to see that the extra terms in it are all expressible in terms of the one-particle operators, and the same argument, as was given above by us to disprove the validity of the Nooijen conjecture for an H with both one- and two-particle operators, can be used for a one-particle H to prove its validity. In both the cases, the non-closed and the closed nature of the Lie algebra respectively played the crucial role. Also, we consider NakatsujiÕs condition on W oW ¼ U p W; oT p
ð15Þ
as being too restrictive for guaranteeing the exactness of W. It is enough for W to satisfy the weaker condition oW X ¼ C qp U q W; ð16Þ oT p q with q running over the sets of the labels p, for W to be exact. Non-closed nature of the Lie algebra of the associated operators, rather than non-commutativity, is the key issue. Even with the one-particle H, the operators Up do not commute, but they form a closed Lie algebra, and hence guarantees satisfaction of the Nooijen conjecture with W parameterized by T1 only. 2. The variational proof is based on a contradiction. Starting from the hypothesis, that a CCGSD ansatz, which makes the energy stationary, can also satisfy the CSE2, we come to the conclusion that this hypothesis cannot be true. If, however, we start from the inverted hypothesis, that a CCGSD ansatz, which satisfies the CSE2, does also satisfy the stationarity condition (12), we do not arrive at a contradiction, since for eTU exact one has ÆUjeT (HE) = 0. The story could have ended here, had Nakatsuji presented his proof involving commutators, and pointed out its subtleties. For the majority of the community working in this field the Nooijen conjecture was – for quite a while – regarded as neither proved nor disproved. This was the motivation for two numerical variational studies [11,12] based on the CCGSD ansatz, which showed high accuracy of the computed energy as compared to full CI.
D. Mukherjee, W. Kutzelnigg / Chemical Physics Letters 397 (2004) 174–179
The story continued with two theoretical Letters [13,14], published back to back, and similar in content, which tried to disprove the Nooijen conjecture on the argument that the subspace of the allowed functions U for which NooijenÕs conjecture is valid is of Ômeasure zeroÕ, and it is highly unlikely that one can hit upon such a U easily. Finally a paper by Mazziotti appeared [15], with both a theoretical and a numerical part. In the former, a variant of the proof by variational comparison in a more convincing form than NakatsujiÕs original was given and it was pointed out that this proof does not exclude the possibility that in some instances the Nooijen conjecture may be valid by chance. It was further shown that a product of two exponential operators is superior to a single one. Mazziotti did respect the right order of assumptions in the proof, but he did not stress the role of the lack of a Lie-algebraic structure. In the numerical part he found cases both where the CCGSD result is numerically indistinguishable from exact and where it is just Ôhighly accurateÕ.
4. Use of the method of moments Let us now give an alternative proof of the non-validity of the Nooijen conjecture, that does not make use of the CSE2 and the Nakatsuji theorem, but that, starting from the hypothesis that a CCGSD wave function is exact, directly leads to a contradiction. This is possible by making use of the method of moments like in traditional coupled-cluster theory. Let us expand both H and T in a basis of operators in normal order with respect to the reference function U [16]. This means we use the following basis of particle number-conserving operators ~apq ¼ apq cpq ; cpq ¼ hU j apq j Ui; ð17Þ pq p q ~ apq as cqs ~ apr þ cps ~ aqr þ cqr ~ aps cpq rs ; rs ¼ ars cr ~ pq pq crs ¼ hU j ars j Ui; . . .
ð18Þ
U is taken as a single Slater determinant with the spin orbitals wi,wj,. . . occupied and wa,wb,. . . unoccupied. In this basis the operator H becomes [16] H ¼ E0 þ fqp ~ aqp þ 12gpq ars pq ; rs ~
fqp ¼ hpq þ gpiqi :
ð19Þ
The assumption that eTU is a full-CI wave function implies that hU j Xp eT H eT j Ui ¼ 0;
ð20Þ
for all (non-constant) basis operators Xp of arbitrary particle rank in normal order with respect to U, that are constructable in the given finite-dimensional Fock space. Actually one need only consider those Xp, which, when applied to the left, do not annihilate U. This means we choose the Xp as the k-particle deexication operators (k = 1,2, . . ., n)
~aijab ;
~aia ;
~aijk abc ; . . .
177
ð21Þ
As a consequence of the normal ordering [16] all the equations (20) are independent. The number of equations (counted by the Xp) to be satisfied is much larger than the number of unknown coefficients Tp in (10). Since the similarity transformed Hamiltonian [2] L ¼ eT H eT ¼ H þ ½H ; T þ ;
ð22Þ
contains operators of particle rank 3 and higher, the Eq. (20) for Xp of higher particle rank are not trivially fulfilled (this would be the case if L contained only 1and 2-particle operators) we arrive at an overdetermined system of equations. This is a strong indication, though not a definite proof, that the system of Eq. (20) does not have a solution. The definite proof will result from the perturbative analysis to be presented in the following section. The key point is again that the {Up} do not constitute a Lie algebra. If we had an operator basis, that spans a Lie algebra, and in terms of which H can be expanded, eT H eT would be an element of this Lie algebra, and we would get as many equations as there are unknown coefficients. Let us compare Eq. (20) with that appearing in the traditional coupled cluster theory (TCC) hU j Xp eS H eS j Ui ¼ 0:
ð23Þ
The difference is that the operator S in (23), is expanded in the k-particle excitation operators ~aai ;
~aab ij ;
~aabc ijk ; . . . ;
ð24Þ
i.e. in the Hermitean adjoints of the deexcitation operators (21), while T is expanded in the CCGSD basis (10). It is interesting to note that, for an exact W, the variational stationarity condition, Eq. (12), the CSE1 and CSE2, Eq. (20) as well as the TCC would all be compatible not only for the full-blown theories, but also at each order of the corresponding expressions using perturbation theory.
5. Perturbative analysis The perturbation expansion of the exact S is known [2,17,18]. Let us decompose the Hamiltonian (19) as H0 + lV and let us further choose the one particle basis such that fqp ¼ dpq ep H 0 ¼ E0 þ ep ~app ;
1 ~ars : V ¼ gpq 2 rs pq
ð25Þ
We expand S in powers of l. S¼
X k
lk S ðkÞ ;
1 S ð1Þ ¼ ðS ð1Þ Þpq ars pq : rs ~ 2
ð26Þ
178
D. Mukherjee, W. Kutzelnigg / Chemical Physics Letters 397 (2004) 174–179
Then we get for the lowest order in l ð1Þ
0 ¼ hU j Xp f½H 0 ; S þ V g j Ui;
ð27Þ
or explicitly 0 ¼ hU j ~ aijab fðea þ eb ei ej ÞðS ð1Þ Þijab þ gijab g~ aab ij j Ui;
ð28Þ
ðS ð1Þ Þijab ¼ ðei þ ej ea eb Þ1 gijab ; S ð1Þ ¼ V H ;
ð29Þ
where the subscript H symbolizes the inversion of a commutator with H0 [2]. All other contributions to S(1) vanish. Obviously the perturbation expansion of (20) leads ð1Þ to the same result for that part T N of the leading term (1) T , that is expandable in the basis ~ aab ij . The contribuð1Þ tions, called T Y expandable in the combined basis ~ ade amc kc and ~ kl are not determined by (28), because such operators annihilate U. Unlike the corresponding contributions to S they do not vanish, and they contribute indirectly, namely in the terms T2, T3, etc. that arise from the formal expansion of the exponential. Actually a term TY TN does not annihilate U and one rather gets e.g. a deb ~ ade ade aab aab akij þ dbc ~ adea kc ~ ij U ¼ ½~ kc ; ~ ij U ¼ fdc ~ kji gU:
ð30Þ
On this way 3-particle excitations show up. It is somewhat tedious, but elementary, to show that from the assumption that W = eTU solves the full CI equað1Þ tions, one is lead to the following condition for T Y ð1Þ
ð1Þ
ð1Þ
hU j Xp f12½H 0 ; ½T Y ; T N þ ½V Y ; T N g j Ui ¼ 0;
ð2Þ
ð32Þ
the solution of which is ð2Þ
S 3 ¼ ½V Y ; V H H ;
ð33Þ
which involves two energy denominators, one associated with V, i.e. with a two-electron operator, the other associated with [VY,VH], i.e. with a 3-electron operator. This appearance of cumulative energy denominators is an essential feature of many-body perturbation theory. In its time-dependent formulation [17,18] this was the result of successive time integrations. It is interesting that we have here to solve D3 equations for d3 unknowns with mn n mn d 3 ¼ nðm nÞ ; D3 ¼ ; ð34Þ 2 3 3 and that this formally overdetermined system does have a solution expressible in terms of d3 parameters. Note that D3 differs from d3 by a factor of O(n2). Let us now go back to the system (31). The solution ð2Þ of this system amounts to expressing the exact S 3 given by (33) in the factorized form ð1Þ 1 ð1Þ ½T ; T N : 2 Y
ð1Þ
ð1Þ
Rð1Þ ¼ R2 ¼ T N ; ð2Þ
ð1Þ
ð2Þ
ð2Þ
Rð2Þ ¼ R2 þ R3 ;
ð1Þ
R3 ¼ 12½T Y ; T N :
ð36Þ ð2Þ S3 ,
The energy is affected to fourth order in l by so the error in the CCGSD energy starts with an incorrect representation of a part of E(4). In a variational CCGSD ansatz one tries to make the error in E(4) as small as possible.
ð31Þ
with Xp any three-particle excitation operator of type (21). We can compare this with the equation satisfied ð2Þ by the 3-particle part S 3 of the exact 2nd order opera(2) tor S . hU j Xp f½H 0 ; S 3 þ ½V Y ; S ð1Þ g j Ui ¼ 0;
On this way one can represent triple excitations, but one cannot represent the cumulative energy denominator corresponding to these, which are present in (33). Only energy denominators with respect to a single vertex are possible. The system (31) has hence no solution. Consequently the CCGSD ansatz cannot be equivalent to full CI. It is not flexible enough to be exact. It does simulate 3-particle- and higher excitations, and even reproduces the diagrams of many-body perturbation theory, just with wrong prefactors and energy denominators. This does not invalidate the possibility to use the CCGSD ansatz in a variational context. The corresponding Euler equations will have a solution. We shall elaborate this analysis elsewhere [19] in more detail. Let us just point out that it is possible to reformulate the operator eT in the form eR, with R expanded in the basis (24), with, of course, R expressible through T. In a perturbation expansion one finds
ð35Þ
6. Dimensionality arguments Let us now discuss the two recent attempts to prove the invalidity of the Nooijen conjecture by Ronen and Davidson [13,14], which hardly contribute to an understanding of the failure of this conjecture. Both Davidson [13] and Ronen [14] tried – successfully – to prove that the Nooijen conjecture does not hold for U a completely arbitrary reference function. The proof is rather elementary and essentially based on the fact that the set of all possible reference functions is of dimension N, while the searched-for ground state wave function is characterized by d parameters, and belongs hence (like the CCGSD wave operator) to a subset of measure zero. One would, of course, never have expected that the Nooijen conjecture holds in such a general sense, like one would never have tried to formulate traditional coupled-cluster theory starting from a completely arbitrary reference function. In both cases a natural and sensible choice of U is a single Slater determinant or a linear combination of a small number of Slater determinants. Such a reference function belongs to a subset of measure zero of the full-CI space as well, and dimensionality arguments are not helpful at all in order to prove or disprove the Nooijen conjecture. Here the arguments put forward in the main body of this Letter must be evoked.
D. Mukherjee, W. Kutzelnigg / Chemical Physics Letters 397 (2004) 174–179
7. Conclusions After all, what makes the Nooijen conjecture so exciting, is not its formal invalidity, contrasted with its apparent plausibility, but rather the fact, that in spite of its lack of rigor, it may lead to surprisingly accurate numerical results. So in order to really understand the implications of this conjecture, it is essential to explain, why the CCGSD ansatz can perform so well in a variational calculation. This, in our opinion, is best understood in terms of perturbation theory.
[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
Acknowledgements The authors thank the DFG and INSA for financial support of this work. References [1] B. Klahn, W.A. Bingel, Theor. Chim. Acta 44 (1977) 9, 27.
[13] [14] [15] [16] [17] [18]
[19]
179
W. Kutzelnigg, J. Chem. Phys. 77 (1982) 3081. H. Nakatsuji, J. Chem. Phys. 113 (2000) 2949. H. Nakatsuji, E.R. Davidson, J. Chem. Phys. 115 (2001) 2000. H. Nakatsuji, 115 (2001) 2465. H. Nakatsuji, Phys. Rev. A 14 (1976) 41. D. Mukherjee, W. Kutzelnigg, J. Chem. Phys. 114 (2001) 2047, Erratum 114, 8226 (2001). H. Primas, in: O. Sinanog˘lu (Ed.), Modern Quantum Chemistry, Academic Press, New York, 1965. M. Nooijen, Phys. Rev. Lett. 84 (2000) 2108. W. Kutzelnigg, Theor. Chim. Acta 83 (1992) 263; W. Kutzelnigg, Theor. Chim. Acta 86 (1993) 41. T. van Voorhis, M. Head-Gordon, J. Chem. Phys. 115 (2001) 5033. P. Piecuch, K. Kowalski, P.-D. Fan, K. Jedziniak, Phys. Rev. Lett. 90 (2003) 113001. E.R. Davidson, Phys. Rev. Lett. 91 (2003) 123001. S. Ronen, Phys. Rev. Lett. 91 (2003) 123002. D.A. Mazziotti, Phys. Rev. A 69 (2004) 012507. W. Kutzelnigg, D. Mukherjee, J. Chem. Phys. 107 (1997) 432; D. Mukherjee, Chem. Phys. Lett. 274 (1997) 561. J. Goldstone, Proc. Roy. Soc. (London) A 239 (1957) 267. W. Kutzelnigg, in: D. Mukherjee (Ed.), Applied Many-Body Methods in Spectroscopy and Electronic Structure, Plenum, New York, 1992. W. Kutzelnigg, D. Mukherjee, Phys. Rev. A, submitted.
Some comments on the coupled cluster with generalized singles and doubles (CCGSD) ansatz Debashis Mukherjee a
a,* ,
Werner Kutzelnigg
b,*
Department of Physical Chemistry, Indian Association for the Cultivation of Science, Calcutta 700 032, India b Lehrstuhl fu¨r Theoretische Chemie, Ruhr-Universita¨t Bochum, D-44780 Bochum, Germany Received 12 August 2004 Available online 11 September 2004
Abstract In this Letter, we present some comments on the nature of the CCGSD ansatz W = eTU for an n-electron wave function, with T a linear combination of general one- and two-particle operators. Nooijen had conjectured that such a parameterization is possible for the exact eigenstate W. We point out that the essential reason for the invalidity of this conjecture lies in the fact that the basis operators into which the Fock space Hamiltonian can be expanded, are not closed under commutation, i.e. do not span a Lie algebra. We give two proofs, based, respectively, on the variation principle and the method of moments, to show this. The variational proof traces the same ground as in an earlier one by Nakatsuji, which did not get the due recognition. We, however, formulate it in such a way that the key role of the algebra of the operators of T becomes transparent. We also discuss a related proof of Mazziotti. Our second proof sheds further light on why the ansatz, while not exact, might still provide an accurate description of a many-electron state, as has been found in some recent computational studies. We also discuss two recent attempts to disprove the exactness of the ansatz, based on a dimensionality argument. 2004 Elsevier B.V. All rights reserved.
aqp ¼ ayq ap ;
1. Introduction In most numerical studies of the n-electron Schro¨dinger equation one expands everything in a finite one-electron basis, and one tries later (at least in principle) to extrapolate to a complete basis [1]. Starting point is then the finite-dimensional Fock space Hamiltonian H in terms of spin–orbitals, that we give here in a tensor notation with the Einstein summation convention [2] 1 H¼ þ gpq ars ; hpq ¼ hwq j h j wp i; 2 rs pq gpq rs ¼ hwr ð1Þws ð2Þ j gð1; 2Þ j wp ð1Þwq ð2Þi; hpq aqp
ð1Þ
* Corresponding authors. Fax: +49 234 32 14045 (W. Kutzelnigg); +91 33 2483 6561 (D. Mukherjee). E-mail addresses: [email protected] (W. Kutzelnigg); [email protected] (D. Mukherjee).
0009-2614/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.08.100
y y ars pq ¼ ar as aq ap :
ð2Þ
The n-electron problem reduces to an eigenvalue problem for the full CI-matrix H. This matrix consists of the matrix elements H lm ¼ hUl j H j Um i 1 hUl j ars ¼ hpq hUl j aqp j Um i þ gpq pq j Um i; 2 rs
ð3Þ
in terms of all possible n-electron Slater determinants Ul constructed from the given one-electron basis {wp}. The transition density matrix elements like hUl j aqp j Um i are independent of the Hamiltonian. While H is characterized by d parameters, H contains D parameters, with d¼
m 2
m 2
þ 1 =2 ¼ Oðm4 Þ;
ð4Þ
D¼
N
2 m N¼ ; n
¼
m n
m
n
D. Mukherjee, W. Kutzelnigg / Chemical Physics Letters 397 (2004) 174–179
175
þ 1 =2 ¼ Oðm2n Þ;
ð9Þ
y
hU j eT ðH EÞU k eT j Ui ¼ 0: ð5Þ
where m is the dimension of the spin orbital basis, and n the number of particles. The D matrix elements of H are expressible through the d parameters contained in H. This implies that an eigenstate of H is a functional of only d = O(m4) parameters, although the corresponding eigenvector of H has (for n > 4) the larger dimension N = O(mn). The fact, that the Hamiltonian (1) is characterized by only d parameters, poses an important challenge to try to parameterize an eigenstate W by a set of d parameters, rather than (for n > 4) by the much larger set N of coefficients in the full-CI expansion. It is obviously so that the manifold of ground states of all possible Hamiltonians of the form (1) represents only a small subset of all possible eigenvectors of an arbitrary N · N matrix. One may, if one wishes, say that the physically possible ground states constitute a subset of Ômeasure zeroÕ of the full set of linear combinations of N n-electron Slater determinants. Unfortunately, it is not so easy to characterize this small manifold, and this is, of course, the key issue.
2. Nakatsuji theorem and Nooijen conjecture The question of the minimal parameterization of a quantum mechanical state has been studied in detail by Nakatsuji [3–5]. In this context an interesting observation plays a central role, that it now usually referred to as Nakatsuji theorem [6]. To formulate it we define the k-particle contracted Schro¨dinger equations CSEk [6,7] (which are necessary conditions for a function W to be exact, and which can be interpreted as the conditions for stationarity of the energy with respect to the variation W ! f1 þ U ðkÞ p gW) hW j ðH EÞU ðkÞ p j Wi ¼ 0;
ð6Þ
with fU ðkÞ p g a set of k-particle excitation operators. Especially for k = 1 or k = 2 we have hW j ðH EÞapq j Wi ¼ 0; hW j ðH EÞapq rs j Wi ¼ 0:
ð7Þ
The content of the Nakatsuji theorem [6,3] is that if (and only if) the CSE2 (which implies the CSE1) is satisfied, (and if W is antisymmetric) then W is an exact full-CI wave function. The proof rests on the simple fact that (7) implies 2
hW j ðH EÞ j Wi ¼ hðH EÞW j ðH EÞWi ¼ 0;
ð8Þ
which is only possible if (HE)W = 0 (in the full-CI basis). Let us express W in (6) through a known reference function U and an unknown Fock space wave operator W = eT
It is tempting to express T by as many (i.e. d) parameters, as there are equations to be satisfied. The simplest possibility to do this, and to satisfy at the same time the separability theorem [8], is that probably first proposed by Nooijen [9], and usually referred to as Nooijen conjecture, which chooses T to consist only of one- and two-particle operators. This implies the claim that the exact parameterization of W, for a given reference function U, is W ¼ W U;W ¼ eT ; T ¼
X p
1 T l U l ¼ T pq aqp þ T pq ars : 2 rs pq
ð10Þ
If one inserts (10) into (6), one gets a nonlinear system of as many (d) equations as there are unknowns Tl – or more specifically T pq and T pq rs . Nooijen regarded it as plausible that these equations do have a solution, which should then – in view of the Nakatsuji theorem – be equivalent to the full CI wave function. One must insist that the operator T is finite, in order not to fall into the trap that W becomes a projector, rather than a wave operator. Such solutions do exist, but have no practical use in the present context, and must hence be avoided. Shortly after the publication of NooijenÕs Letter, a paper by Nakatsuji appeared in print [3], in which the same ansatz was studied, using a single determinant reference U, and in which Nakatsuji came to the conclusion Ôthat this ansatz cannot be exactÕ. Nakatsuji called this ansatz Ôthe coupled cluster with generalized singles and doublesÕ (CCGSD), and this is the nomenclature we would henceforth be using. The proof, which was probably not presented in the best possible form, and which has found astonishingly little attention, is based on the variation principle, more precisely on the comparison of two stationarity conditions, whence we refer to it as the proof by variational comparison. We want to emphasize in this Letter that the essential reason why the Nooijen conjecture is invalid is the fact that the basis {Ul} of one and two-particle excitation operators does not span a Lie algebra. We give two proofs to bolster this point, one based on variation principle and the other on the method of moments. The one using the variation principle rederives NakatsujiÕs proof in a more sharpened form, which transparently brings out the importance of the lack of a Lie algebraic structure of the operator basis {Ul}.
3. Proof by variational comparison An exact wave function of CCGSD type must not only satisfy the CSE2 and CSE1 (9), but it must also make the energy expectation value
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D. Mukherjee, W. Kutzelnigg / Chemical Physics Letters 397 (2004) 174–179 y
hU j eT H eT j Ui E¼ ; y hU j eT eT j Ui
ð11Þ
0¼
y oeT U UeT ðH EÞ oT p
Ty
stationary with respect to variation of the set of coefficients {Tp}. It must, if it exists, belong to the family of wave functions, which satisfy the stationarity condition y oeT 0 ¼ UeT ðH EÞ U oT p y
¼ hU j eT ðH EÞfU p þ 12½T ; U p þ 16½T ; ½T ; U p þ . . .geT j Ui:
ð12Þ
Since the energy does have a minimum as functional of the Tp, the Eq. (12) do have a solution. In order for a CCGSD function to be exact, it must also satisfy Eq. (9). This differs from the variational stationarity condition (12). Instead of requiring that (12) is satisfied simultaneously with (9), one may also require that (12) holds simultaneously with the difference condition y
0 ¼ hU j eT ðH EÞf12½T ; U p þ 16½T ; ½T ; U p þ . . .geT j Ui: ð13Þ This is satisfied in special situations, e.g. if the operator basis spans a Lie algebra (vide infra), or if the reference function U is equal to its exact counterpart W, but does not hold under the premises of the Nooijen conjecture. This variational proof (of the Ônon-exactnessÕ of the CCGSD ansatz – or the invalidity of the Nooijen conjecture) has two rather subtle aspects. 1. It is imperative to formulate it such that multiple commutators of the Up with T appear, also in the higher order terms in braces in (12), that are not indicated explicitly. This makes it obvious, that what matters is that the set of one- and two-particle excitation operators {Tp} is not closed under commutation and hence not a subalgebra of the Lie algebra of all excitation operators (as it would be required for a consistent formulation of the variation principle in terms of a Lie algebra [10]). Let us note that the single commutator and the double commutator in the Ôextra termsÕ in Eq. (12) beyond CSE2 contain respectively three- and four-particle operators. If we repeat the just given argument for an operator basis that does constitute a Lie algebra, we find that the stationarity of the energy and the CSE1 and CSE2 are compatible, because then [T,Up] and the higher order commutators are elements of the Lie algebra spanned by the Up. Nakatsuji, in his proof, using the variation principle with the CCGSD ansatz, did indeed show that the stationarity condition contains three and higher particle operators, but he did not present these equations in terms of commutators. Rather, he expressed this equation as
¼ hU j e ðH EÞfU p þ 12ðU p T þ TU p Þ þ 16ðU p T 2 þ TU p T þ T 2 U p Þ þ g j Ui:
ð14Þ
Nakatsuji argued that the extra terms arise because of the noncommutativity of the set Up. Since Nakatsuji did not write the stationarity condition in terms of commutators, its compatibility with the CSE for the case of a one-particle Hamiltonian was not immediately obvious, and he showed the compatibility using a different route via the Thouless parameterization. On the other hand, using the alternative form, viz. Eq. (12) it is straightforward to see that the extra terms in it are all expressible in terms of the one-particle operators, and the same argument, as was given above by us to disprove the validity of the Nooijen conjecture for an H with both one- and two-particle operators, can be used for a one-particle H to prove its validity. In both the cases, the non-closed and the closed nature of the Lie algebra respectively played the crucial role. Also, we consider NakatsujiÕs condition on W oW ¼ U p W; oT p
ð15Þ
as being too restrictive for guaranteeing the exactness of W. It is enough for W to satisfy the weaker condition oW X ¼ C qp U q W; ð16Þ oT p q with q running over the sets of the labels p, for W to be exact. Non-closed nature of the Lie algebra of the associated operators, rather than non-commutativity, is the key issue. Even with the one-particle H, the operators Up do not commute, but they form a closed Lie algebra, and hence guarantees satisfaction of the Nooijen conjecture with W parameterized by T1 only. 2. The variational proof is based on a contradiction. Starting from the hypothesis, that a CCGSD ansatz, which makes the energy stationary, can also satisfy the CSE2, we come to the conclusion that this hypothesis cannot be true. If, however, we start from the inverted hypothesis, that a CCGSD ansatz, which satisfies the CSE2, does also satisfy the stationarity condition (12), we do not arrive at a contradiction, since for eTU exact one has ÆUjeT (HE) = 0. The story could have ended here, had Nakatsuji presented his proof involving commutators, and pointed out its subtleties. For the majority of the community working in this field the Nooijen conjecture was – for quite a while – regarded as neither proved nor disproved. This was the motivation for two numerical variational studies [11,12] based on the CCGSD ansatz, which showed high accuracy of the computed energy as compared to full CI.
D. Mukherjee, W. Kutzelnigg / Chemical Physics Letters 397 (2004) 174–179
The story continued with two theoretical Letters [13,14], published back to back, and similar in content, which tried to disprove the Nooijen conjecture on the argument that the subspace of the allowed functions U for which NooijenÕs conjecture is valid is of Ômeasure zeroÕ, and it is highly unlikely that one can hit upon such a U easily. Finally a paper by Mazziotti appeared [15], with both a theoretical and a numerical part. In the former, a variant of the proof by variational comparison in a more convincing form than NakatsujiÕs original was given and it was pointed out that this proof does not exclude the possibility that in some instances the Nooijen conjecture may be valid by chance. It was further shown that a product of two exponential operators is superior to a single one. Mazziotti did respect the right order of assumptions in the proof, but he did not stress the role of the lack of a Lie-algebraic structure. In the numerical part he found cases both where the CCGSD result is numerically indistinguishable from exact and where it is just Ôhighly accurateÕ.
4. Use of the method of moments Let us now give an alternative proof of the non-validity of the Nooijen conjecture, that does not make use of the CSE2 and the Nakatsuji theorem, but that, starting from the hypothesis that a CCGSD wave function is exact, directly leads to a contradiction. This is possible by making use of the method of moments like in traditional coupled-cluster theory. Let us expand both H and T in a basis of operators in normal order with respect to the reference function U [16]. This means we use the following basis of particle number-conserving operators ~apq ¼ apq cpq ; cpq ¼ hU j apq j Ui; ð17Þ pq p q ~ apq as cqs ~ apr þ cps ~ aqr þ cqr ~ aps cpq rs ; rs ¼ ars cr ~ pq pq crs ¼ hU j ars j Ui; . . .
ð18Þ
U is taken as a single Slater determinant with the spin orbitals wi,wj,. . . occupied and wa,wb,. . . unoccupied. In this basis the operator H becomes [16] H ¼ E0 þ fqp ~ aqp þ 12gpq ars pq ; rs ~
fqp ¼ hpq þ gpiqi :
ð19Þ
The assumption that eTU is a full-CI wave function implies that hU j Xp eT H eT j Ui ¼ 0;
ð20Þ
for all (non-constant) basis operators Xp of arbitrary particle rank in normal order with respect to U, that are constructable in the given finite-dimensional Fock space. Actually one need only consider those Xp, which, when applied to the left, do not annihilate U. This means we choose the Xp as the k-particle deexication operators (k = 1,2, . . ., n)
~aijab ;
~aia ;
~aijk abc ; . . .
177
ð21Þ
As a consequence of the normal ordering [16] all the equations (20) are independent. The number of equations (counted by the Xp) to be satisfied is much larger than the number of unknown coefficients Tp in (10). Since the similarity transformed Hamiltonian [2] L ¼ eT H eT ¼ H þ ½H ; T þ ;
ð22Þ
contains operators of particle rank 3 and higher, the Eq. (20) for Xp of higher particle rank are not trivially fulfilled (this would be the case if L contained only 1and 2-particle operators) we arrive at an overdetermined system of equations. This is a strong indication, though not a definite proof, that the system of Eq. (20) does not have a solution. The definite proof will result from the perturbative analysis to be presented in the following section. The key point is again that the {Up} do not constitute a Lie algebra. If we had an operator basis, that spans a Lie algebra, and in terms of which H can be expanded, eT H eT would be an element of this Lie algebra, and we would get as many equations as there are unknown coefficients. Let us compare Eq. (20) with that appearing in the traditional coupled cluster theory (TCC) hU j Xp eS H eS j Ui ¼ 0:
ð23Þ
The difference is that the operator S in (23), is expanded in the k-particle excitation operators ~aai ;
~aab ij ;
~aabc ijk ; . . . ;
ð24Þ
i.e. in the Hermitean adjoints of the deexcitation operators (21), while T is expanded in the CCGSD basis (10). It is interesting to note that, for an exact W, the variational stationarity condition, Eq. (12), the CSE1 and CSE2, Eq. (20) as well as the TCC would all be compatible not only for the full-blown theories, but also at each order of the corresponding expressions using perturbation theory.
5. Perturbative analysis The perturbation expansion of the exact S is known [2,17,18]. Let us decompose the Hamiltonian (19) as H0 + lV and let us further choose the one particle basis such that fqp ¼ dpq ep H 0 ¼ E0 þ ep ~app ;
1 ~ars : V ¼ gpq 2 rs pq
ð25Þ
We expand S in powers of l. S¼
X k
lk S ðkÞ ;
1 S ð1Þ ¼ ðS ð1Þ Þpq ars pq : rs ~ 2
ð26Þ
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D. Mukherjee, W. Kutzelnigg / Chemical Physics Letters 397 (2004) 174–179
Then we get for the lowest order in l ð1Þ
0 ¼ hU j Xp f½H 0 ; S þ V g j Ui;
ð27Þ
or explicitly 0 ¼ hU j ~ aijab fðea þ eb ei ej ÞðS ð1Þ Þijab þ gijab g~ aab ij j Ui;
ð28Þ
ðS ð1Þ Þijab ¼ ðei þ ej ea eb Þ1 gijab ; S ð1Þ ¼ V H ;
ð29Þ
where the subscript H symbolizes the inversion of a commutator with H0 [2]. All other contributions to S(1) vanish. Obviously the perturbation expansion of (20) leads ð1Þ to the same result for that part T N of the leading term (1) T , that is expandable in the basis ~ aab ij . The contribuð1Þ tions, called T Y expandable in the combined basis ~ ade amc kc and ~ kl are not determined by (28), because such operators annihilate U. Unlike the corresponding contributions to S they do not vanish, and they contribute indirectly, namely in the terms T2, T3, etc. that arise from the formal expansion of the exponential. Actually a term TY TN does not annihilate U and one rather gets e.g. a deb ~ ade ade aab aab akij þ dbc ~ adea kc ~ ij U ¼ ½~ kc ; ~ ij U ¼ fdc ~ kji gU:
ð30Þ
On this way 3-particle excitations show up. It is somewhat tedious, but elementary, to show that from the assumption that W = eTU solves the full CI equað1Þ tions, one is lead to the following condition for T Y ð1Þ
ð1Þ
ð1Þ
hU j Xp f12½H 0 ; ½T Y ; T N þ ½V Y ; T N g j Ui ¼ 0;
ð2Þ
ð32Þ
the solution of which is ð2Þ
S 3 ¼ ½V Y ; V H H ;
ð33Þ
which involves two energy denominators, one associated with V, i.e. with a two-electron operator, the other associated with [VY,VH], i.e. with a 3-electron operator. This appearance of cumulative energy denominators is an essential feature of many-body perturbation theory. In its time-dependent formulation [17,18] this was the result of successive time integrations. It is interesting that we have here to solve D3 equations for d3 unknowns with mn n mn d 3 ¼ nðm nÞ ; D3 ¼ ; ð34Þ 2 3 3 and that this formally overdetermined system does have a solution expressible in terms of d3 parameters. Note that D3 differs from d3 by a factor of O(n2). Let us now go back to the system (31). The solution ð2Þ of this system amounts to expressing the exact S 3 given by (33) in the factorized form ð1Þ 1 ð1Þ ½T ; T N : 2 Y
ð1Þ
ð1Þ
Rð1Þ ¼ R2 ¼ T N ; ð2Þ
ð1Þ
ð2Þ
ð2Þ
Rð2Þ ¼ R2 þ R3 ;
ð1Þ
R3 ¼ 12½T Y ; T N :
ð36Þ ð2Þ S3 ,
The energy is affected to fourth order in l by so the error in the CCGSD energy starts with an incorrect representation of a part of E(4). In a variational CCGSD ansatz one tries to make the error in E(4) as small as possible.
ð31Þ
with Xp any three-particle excitation operator of type (21). We can compare this with the equation satisfied ð2Þ by the 3-particle part S 3 of the exact 2nd order opera(2) tor S . hU j Xp f½H 0 ; S 3 þ ½V Y ; S ð1Þ g j Ui ¼ 0;
On this way one can represent triple excitations, but one cannot represent the cumulative energy denominator corresponding to these, which are present in (33). Only energy denominators with respect to a single vertex are possible. The system (31) has hence no solution. Consequently the CCGSD ansatz cannot be equivalent to full CI. It is not flexible enough to be exact. It does simulate 3-particle- and higher excitations, and even reproduces the diagrams of many-body perturbation theory, just with wrong prefactors and energy denominators. This does not invalidate the possibility to use the CCGSD ansatz in a variational context. The corresponding Euler equations will have a solution. We shall elaborate this analysis elsewhere [19] in more detail. Let us just point out that it is possible to reformulate the operator eT in the form eR, with R expanded in the basis (24), with, of course, R expressible through T. In a perturbation expansion one finds
ð35Þ
6. Dimensionality arguments Let us now discuss the two recent attempts to prove the invalidity of the Nooijen conjecture by Ronen and Davidson [13,14], which hardly contribute to an understanding of the failure of this conjecture. Both Davidson [13] and Ronen [14] tried – successfully – to prove that the Nooijen conjecture does not hold for U a completely arbitrary reference function. The proof is rather elementary and essentially based on the fact that the set of all possible reference functions is of dimension N, while the searched-for ground state wave function is characterized by d parameters, and belongs hence (like the CCGSD wave operator) to a subset of measure zero. One would, of course, never have expected that the Nooijen conjecture holds in such a general sense, like one would never have tried to formulate traditional coupled-cluster theory starting from a completely arbitrary reference function. In both cases a natural and sensible choice of U is a single Slater determinant or a linear combination of a small number of Slater determinants. Such a reference function belongs to a subset of measure zero of the full-CI space as well, and dimensionality arguments are not helpful at all in order to prove or disprove the Nooijen conjecture. Here the arguments put forward in the main body of this Letter must be evoked.
D. Mukherjee, W. Kutzelnigg / Chemical Physics Letters 397 (2004) 174–179
7. Conclusions After all, what makes the Nooijen conjecture so exciting, is not its formal invalidity, contrasted with its apparent plausibility, but rather the fact, that in spite of its lack of rigor, it may lead to surprisingly accurate numerical results. So in order to really understand the implications of this conjecture, it is essential to explain, why the CCGSD ansatz can perform so well in a variational calculation. This, in our opinion, is best understood in terms of perturbation theory.
[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
Acknowledgements The authors thank the DFG and INSA for financial support of this work. References [1] B. Klahn, W.A. Bingel, Theor. Chim. Acta 44 (1977) 9, 27.
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