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The Journal of Chemical Physics

An algebraic proof of generalized Wick theorem

Liguo Kong, Marcel Nooijen, Debashis Mukherjee

Published Jun 16, 2010• 69 citations
Publisher

Abstract

The multireference normal order theory, introduced by Kutzelnigg and Mukherjee [J. Chem. Phys. 107, 432 (1997)], is defined explicitly, and an algebraic proof is given for the corresponding contraction rules for a product of any two normal ordered operators. The proof does not require that the contractions be cumulants, so it is less restricted. In addition, it follows from the proof that the normal order theory and corresponding contraction rules hold equally well if the contractions are only defined up to a certain level. These relaxations enable us to extend the original normal order theory. As a particular example, a quasi-normal-order theory is developed, in which only one-body contractions are present. These contractions are based on the one-particle reduced density matrix.

DOI
10.1063/1.3439395

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The multireference normal order theory, introduced by Kutzelnigg and Mukherjee [J. Chem. Phys. 107, 432 (1997)], is defined explicitly, and an algebraic proof is given for the corresponding contraction rules for a product of any two normal ordered operators. The proof does not require that the contractions be cumulants, so it is less restricted. In addition, it follows from the proof that the normal order theory and corresponding contraction rules hold equally well if the contractions are only defined up to a certain level. These relaxations enable us to extend the original normal order theory. As a particular example, a quasi-normal-order theory is developed, in which only one-body contractions are present. These contractions are based on the one-particle reduced density matrix.
A generalization of normal ordering and of Wick’s theorem with respect to an arbitrary reference function Φ as some generalized “physical vacuum” is formulated in a different (but essentially equivalent) way than that suggested previously by one of the present authors. Guiding principles are that normal order operators with respect to any reference state must be expressible as linear combinations of those with respect to the genuine vacuum, that the vacuum expectation value of a normal order operator must vanish (with respect to the vacuum to which it is in normal order), and that the well-known formalism for a single Slater determinant as physical vacuum must be contained as a special case. The derivation is largely based on the concepts of “Quantum Chemistry in Fock space,” which means that particle-number-conserving operators (excitation operators) play a central role. Nevertheless, the contraction rules in the frame of a generalized Wick theorem are derived, that hold for non-particle-number-conserving operators as well. The contraction rules are formulated and illustrated in terms of diagrams. The contractions involve the “residual n-particle density matrices” λ, which are the irreducible (non-factorizable) parts of the conventional n-particle density matrices γ, in the sense of a cumulant expansion for the density. A spinfree formulation is presented as well. The expression of the Hamiltonian in normal order with respect to a multiconfiguration reference function leads to a natural definition of a generalized Fock operator. MC-SCF-theory is easily worked out in this context. The paper concludes with a discussion of the excited configurations and the first-order interacting space, that underlies a perturbative coupled cluster type correction to the MCSCF function for an arbitrary reference function, and with general implications of the new formalism, that is related to “internally contracted multireference configuration interaction.” The present generalization of normal ordering is not only valid for arbitrary reference functions, but also if the reference state is an ensemble state.
We propose a spin-free approach to the cumulant decomposition of reduced density matrices of singlet and spin-rotation or SU(2) invariant ensemble of nonsinglet states as in [W. Kutzelnigg and D. Mukherjee, J. Chem. Phys. 110, 2800 (1999); 116, 4787 (2002)]. We provide a simple recursive procedure to obtain expressions which relate different spin components of spin-orbital reduced density matrices and cumulants of such states to the spin-free counterparts. These results are used to set up a spin-summation procedure to arrive at a definition of spin-free cumulants of any order. Alternatively, an analytic formula for the spin-free form resulting from a spin summation involving product of two spin-orbital cumulants is derived and its utility in spin-free cumulant decomposition of reduced density matrices is demonstrated. This leads to suitable definitions of spin-free analog of multireference normal ordering and the associated Wicks theorem. The results of this formulation are expected to be useful in investigations of spin-free multireference internally contracted coupled-cluster methods where cumulant approximations to the active reduced density matrices are employed.
This paper describes Wick&d, an implementation of the algebra of second-quantized operators normal ordered with respect to general correlated references and the corresponding Wick theorem [D. Mukherjee, Chem. Phys. Lett. 274, 561 (1997) and W. Kutzelnigg and D. Mukherjee, J. Chem. Phys. 107, 432 (1997)]. Wick&d employs a compact representation of operators and a backtracking algorithm to efficiently evaluate Wick contractions. Since Wick&d can handle both fully and partially contracted terms, it can be applied to both projective and Fock-space many-body formalisms. To demonstrate the usefulness of Wick&d, we use it to evaluate the single-reference coupled cluster equations up to octuple excitations and report an automated derivation and implementation of the second-order driven similarity renormalization group multireference perturbation theory.
We study perturbation expansions for quantum systems which are nonlinear in the sense that they may contain derivative couplings. The characteristic feature of these systems is the occurrence of products of noncommuting operators in the interaction Hamiltonian. In order to perform perturbation expansions in the Dyson-Wick manner it is in general necessary to order these operators according to some ordering scheme. We determine the most general scheme of this sort which allows a Dyson-Wick expansion with pair contractions. It involves a set $\ensuremath{\lambda}$ of three dimensionless ordering parameters which are related to the three aaa priori undetermined equal-time contractions. Different $\ensuremath{\lambda}$-ordering schemes are related via an "equal-time" or "static" Wick theorem. In order to be able to contract operators which may be taken either at equal or at different times we introduce a time-ordering operation ${T}_{\ensuremath{\lambda}}$ which extends Dyson's time-ordering operation to equal times in a fashion dependent on the $\ensuremath{\lambda}$ ordering. We arrive at a "dynamic" Wick theorem which enables us to decompose the ${T}_{\ensuremath{\lambda}}$ product of a set of operators taken at arbitrary times into a sum of ${\ensuremath{\lambda}}^{\ensuremath{'}}$ products, where the most important case is that in which ${\ensuremath{\lambda}}^{\ensuremath{'}}$ represents normal ordering. Contrary to the usual Wick theorem ours works also in the case when two or more times coincide. Applying this theorem to the scattering operator expressed as a ${T}_{\ensuremath{\lambda}}$ product we obtain a class of Dyson-Wick expansions for nonlinear systems. Different expansions correspond to different choices of the ordering $\ensuremath{\lambda}$ and lead to different Feynman rules. The resulting diagrammatic technique for nonlinear systems is seen to have the following special features: (i) The interaction vertices include terms which arise from "quantum corrections," which are dependent on the ordering \ensuremath{\lambda} chosen for the noncommuting factors in the interaction Hamiltonian, and (ii) the equal-time contractions which occur in the closed-loop diagrams must be assigned corresponding $\ensuremath{\lambda}$-dependent values. In this way individual diagrams are $\ensuremath{\lambda}$ dependent. Nevertheless their sum is $\ensuremath{\lambda}$ independent, confirming the fact that the complete scattering operator is independent of the particular ordering chosen. The ordering multiplicity is also reflected in functional versions of the Dyson-Wick expansion. In the path-integral version this multiplicity lurks in the different possible discretization prescriptions. We discuss the relevance of our considerations to the question of the invariance of a quantum theory with respect to nonlinear point transformations. For the special case of a reducible system we present explicit calculations which show how this invariance is maintained in perturbation theory, for any ordering scheme belonging to the $\ensuremath{\lambda}$ class. In any particular scheme the invariance is ensured by a more or less intricate cancellation mechanism between the individual diagrams.
Reduced density matrix cumulants play key roles in the theory of both reduced density matrices and multiconfigurational normal ordering. We present a new, simpler generating function for reduced density matrix cumulants that is formally identical with equating the coupled cluster and configuration interaction ansätze. This is shown to be a general mechanism to convert between a multiplicatively separable quantity and an additively separable quantity, as defined by a set of axioms. It is shown that both the cumulants of probability theory and the reduced density matrices are entirely combinatorial constructions, where the differences can be associated with changes in the notion of "multiplicative separability" for expectation values of random variables compared to reduced density matrices. We compare our generating function to that of previous works and criticize previous claims of probabilistic significance of the reduced density matrix cumulants. Finally, we present a simple proof of the generalized normal ordering formalism to explore the role of reduced density matrix cumulants therein. While the formalism can be used without cumulants, the combinatorial structure of expressing RDMs in terms of cumulants is the same combinatorial structure on cumulants that allows for a simple extended generalized Wick's theorem.