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The Journal of Chemical Physics

Cumulant decomposition of reduced density matrices, multireference normal ordering, and Wicks theorem: A spin-free approach

K. R. Shamasundar

Published Nov 4, 2009• 49 citations
Publisher

Abstract

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.

DOI
10.1063/1.3256237

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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.
The k-particle reduced density matrices γ k for a singlet state or a spin-orientation averaged ensemble are invariant under the spin rotation group SU(2), i.e. they are independent of spin, even if they are expressed in terms of spin orbitals. The γ k contain the same information as their spinfree counterparts Γ k expressed in terms of spinfree orbitals. However, the relation between the two quantities is, by no means, elementary. The summation over spin is particularly demanding and surprisingly complicated for the k-particle density cumulants λ k and for the formulation of a generalised normal ordering with respect to an arbitrary reference state, including the generalised Wick theorem. Plausible extrapolations from small k turned out to be unjustified and some published expressions for k = 4 need to be revised. A compact general result, valid for all k has not yet been found.
Recently [W. Kutzelnigg and D. Mukherjee, Chem. Phys. Lett. 317, 567 (2000); D. Mukherjee and W. Kutzelnigg, J. Chem. Phys 114, 2047 (2001)] the irreducible k-particle Brillouin conditions IBCk and the irreducible k-particle contracted Schrödinger equations ICSEk were derived. These permit the definition of a hierarchy of k-particle approximations for the direct calculation of the cumulants λk of the k-particle density matrices. Now, the spin-free form of these conditions, appropriate for a spin-free Hamiltonian, is given. This is particularly useful for open-shell states. The definition of the cumulants of the reduced densities has to be generalized for these anyway, making use of irreducible tensor operators with respect to SU2. There are two alternative definitions of spin-free cumulants, of which the one in terms of spin-free reduced density matrices appears to be preferable. Alternatively to the straight spin-free formulation, we also present a theory in terms of spin-free operators adapted to the symmetric group. Partial trace relations that relate the elements of the cumulants of different particle rank are derived. There are partial trace relations for “exchange elements,” which are determined by the total spin quantum number S. From these relations the individual exchange elements of Λ2 can be obtained in special cases. This allows a simple formulation of the stationarity conditions for open-shell states of any spin multiplicity.
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.
k -particle cumulants λk (for 2⩽k⩽n) corresponding to the k-particle reduced density matrices γk for an n-fermion system are defined via a generating function. The two-particle cumulant λ2 describes two-particle correlations (excluding exchange), λ3 genuine three-particle correlations etc. The properties of these cumulants are analyzed. Conditions for vanishing of certain λk are formulated. Necessary and sufficient for λ2=0 is the well-known idempotency condition γ2=γ for γ≡γ1. For λ3=0 to hold, a general necessary condition is Tr{2γ3−3γ2+γ}=0, for three special forms of the wave function (arbitrary two-electron state, antisymmetrized product of strongly orthogonal geminals on antisymmetrized geminal power wave function of extreme type) 2γ3−3γ2+γ=0 turns out to be necessary and sufficient. For a multiconfiguration self-consistent field wave function the only nonvanishing matrix elements of the cumulants are those where all labels refer to active (partially occupied) spin orbitals. Spin-free cumulants Λk corresponding to the spin-free reduced density matrices Γk are also defined and analyzed. The main interest in the density cumulants is in connection with the recently formulated normal ordering and the corresponding Wick theorem for arbitrary reference functions, but they are also useful for an analysis of electron correlation.
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.