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International Journal of Quantum Chemistry

Extension of the cumulant definition for low particle number systems

Liguo Kong

Published Aug 3, 2010• 8 citations
Publisher

Abstract

Abstract Cumulants are usually interpreted as the connected components of density matrices, but this interpretation fails and practical problems arise when the rank n of cumulants is larger than the number of particles (N) in the system. In that case, cumulants defined in the traditional way become disconnected. To solve this problem, the definition of cumulants is extended by introducing a simple modulation factor. The modified cumulants reduce to the conventional definition, but they vanish when N < n. Using the modified definition also eliminates the error in the approximation of density matrices by low‐rank cumulants, when N < n. The problem assumes a slightly different form when we work with active space–based theories, and it can be solved by a similar approach. Another problem with cumulants, due to spin coupling [Herbert, Int J Quantum Chem 2007, 107, 703], can be solved via the introduction of a similar modulation factor. A related yet more serious issue, termed as the local particle number constraint problem, is also discussed. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

DOI
10.1002/qua.22832

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Abstract Cumulants are usually interpreted as the connected components of density matrices, but this interpretation fails and practical problems arise when the rank n of cumulants is larger than the number of particles (N) in the system. In that case, cumulants defined in the traditional way become disconnected. To solve this problem, the definition of cumulants is extended by introducing a simple modulation factor. The modified cumulants reduce to the conventional definition, but they vanish when N < n. Using the modified definition also eliminates the error in the approximation of density matrices by low‐rank cumulants, when N < n. The problem assumes a slightly different form when we work with active space–based theories, and it can be solved by a similar approach. Another problem with cumulants, due to spin coupling [Herbert, Int J Quantum Chem 2007, 107, 703], can be solved via the introduction of a similar modulation factor. A related yet more serious issue, termed as the local particle number constraint problem, is also discussed. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011
Abstract Approaches to the solution of then‐electron problem in terms of the reducedk‐particle density matricesγkor their cumulantsλkare critically reviewed. The advantages of a cumulant‐based theory are outlined, especially its extensivity, which implies a connected‐diagram theorem. A perturbative analysis of the hierarchy ofk‐particle approximations, based on the irreducible contracted Schrödinger equations (ICSEk) (which are conditions for the stationarity of the energy), reveals that the three‐particle approximation is necessary to obtain the energy correct to the MP2 (second‐order Møller–Plesset) level. This disqualifies the naive use of thek‐particle hierarchy, in a linearly converging iterative scheme, as ineffective (compared, e.g., to coupled‐cluster theory). A promising alternative is, however, a quadratically convergent iterative scheme based on successive unitary transformations in Fock space. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
Abstract The use of cumulant expansions for the approximation of reduced density matrices (RDMs) is reviewed. It is pointed out that the oft‐cited theorems of Nakatsuji and Rosina are insufficient to guarantee either N‐representability or that an N‐representable RDM has a unique antecedent wave function. Approximations involved in Mazziotti's recently proposed “formal solution for reconstruction” are identified. The behavior of his scheme, and of an older one in which all cumulants beyond the second are neglected, are illustrated by detailed examination of a model problem. The limited experience provided by this example casts doubt as to the probable effectiveness of Mazziotti's reconstruction scheme. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002
Abstract The methods of quantum chemistry and solid state theory to solve the many‐body problem are reviewed. We start with the definitions of reduced density matrices, their properties (contraction sum rules, spectral resolutions, cumulant expansion, and N‐representability), and their determining equations (contracted Schrödinger equations), and we summarize recent extensions and generalizations of the traditional quantum chemical methods, of the density functional theory, and of the quasiparticle theory: from finite to extended systems (incremental method), from density‐to‐density matrix (density matrix functional theory), from weak to strong correlation (dynamical mean field theory), from homogeneous (Kimball–Overhauser approach) to inhomogeneous and finite systems. Measures of the correlation strength are discussed. The cumulant two‐body reduced density matrix proves to be a key quantity. Its spectral resolution contains geminals, possibly the solutions of an approximate effective two‐body equation, and we sketch the idea of how its contraction sum rule can be used for a variational treatment. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
Abstract The cumulant 2‐matrix χ is that part of the two‐body reduced density matrix γ2, which cannot be reduced to products of the one‐body reduced density matrix (1‐matrix) γ. This irreducible part χ is calculated perturbatively for the high‐density electron gas (EG) in its ground state, such that the pair densities and the interaction energy are correctly reproduced in their high‐density limits, which are exactly known and summarized here. From the thus available cumulant 2‐matrix the pair density in momentum space can be derived and used for a fluctuation analysis and compared with the analog analysis in position space, where it is concluded that “correlation suppresses fluctuations”. The perturbatively available cumulant 2‐matrix χ can be used also for the high‐density electron gas to start the iterative solution of the Yasuda integral equation of density matrix functional theory (DMFT), which is a nonlinear functional relation between the cumulant 2‐matrix χ and the 1‐matrix γ recently derived from the contracted Schrödinger equation approach: χY[γ]. From the perturbatively determined χ one can find another functional χ[γ] as an alternative approximation for a DMFT. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002
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.
We propose a novel interpretation of the reduced density matrix (RDM) and its cumulant that combines linear and exponential parametrizations of the wavefunction. Any n-particle RDM can be written as a weighted average of "configuration interaction" amplitudes. The corresponding n-particle cumulant is represented in terms of two types of contributions: "connected" (statistical averages of substitution amplitudes) and "disconnected" (cross-correlations of substitution amplitudes). A diagonal element of n-RDM represents the average occupation number of the orbital n-tuple. The diagonal elements of 2- and 3-cumulants take particularly elegant forms in the natural spin-orbital basis: they represent the covariances (correlated fluctuations) of the occupation numbers of the orbital pair and triples, respectively. Thus, the diagonal elements of the cumulants quantify the correlation between the orbital occupation numbers. Our interpretation is used to examine the weak to strong correlation transition in the "two electrons in two orbitals" problem.