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The Journal of Chemical Physics

On the definition of the spin-free cumulant of the second-order reduced density matrix

Luis Laín, Alicia Torre, Roberto C. Bochicchio

Published Aug 30, 2002• 30 citations
Publisher

Abstract

This note deals with the appropriate definition for the spin-free cumulant of the second-order reduced density matrix. Our approach leads to a direct derivation of one of the proposals reported by Kutzelnigg and Mukherjee [J. Chem. Phys. 116, 4787 (2002)] and it points out its suitability.
DOI
10.1063/1.1499955

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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.
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.
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.
Abstract More than fifty years ago, in a couple of seminal works (Kubo 1962 J. Phys. Soc. Japan 17 1100–20; Kubo 1963 J. Math. Phys . 4 174–83) Kubo introduced the important idea of generalized cumulants, extending to stochastic operators this concept implicitly introduced by Laplace in 1810. Kubo’s idea has been applied in several branches of physics, where the result of the average process is a Lioville operator or an effective time evolution operator for the density matrix of spin systems or the reduced density matrix for boson-fermions etc. Despite this success, the theoretical developments in these Kubo works pose problems that were highlighted many years ago by Fox and van Kampen and never solved. These weaknesses and errors, in particular concerning the factorization property of exponentials of cumulants and the explicit expressions that give generalized cumulants in terms of generalized moments and vice-versa, caused some perplexity (and confusion) about the possible application of this procedure, limiting its use, in practice. In the present paper, we give a sound ground to the approach to cumulant operators, working in a general framework that shows the potentiality of the old Kubo idea, today not yet fully exploited. It results that, for the same moment operators, different definitions of generalized cumulants can be adopted. A general Kubo–Meeron closed-form formula giving cumulant operators in terms of moment operators cannot be obtained, but the reverse one, cumulants in terms of operators, is given and, noticeably, formally it does not depend on the specific nature of the moments, but just on the definition of the generalized cumulants.