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International Journal of Quantum Chemistry

n‐Electron problem and its formulation in terms ofk‐particle density cumulants

Werner Kutzelnigg

Published Jan 1, 2003• 17 citations
Publisher

Abstract

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

DOI
10.1002/qua.10751

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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 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
Two alternative conditions for the stationarity of the energy expectation value with respect to k-particle excitations are the k-particle Brillouin conditions BCk and the k-particle contracted Schrödinger equations, CSEk. These conditions express the k-particle density matrices γk in terms of density matrices of higher particle rank. The latter can be eliminated if one expresses the γk in terms of their cumulants λk, but this is not sufficient to make the BCk or CSEk separable (extensive), i.e., they are not expressible in terms of only connected diagrams. However, in a formulation based on the recently introduced general normal ordering with respect to arbitrary wave functions, the irreducible counterparts IBCk and ICSEk of the BCk and CSEk can be defined. They are easily evaluated explicitly in terms of the generalized Wick theorem for arbitrary wave functions, and they lead to equations for the direct construction of the cumulants λk, which are additively separable quantities and which scale linearly with the system size. The IBCk or the ICSEk are necessary conditions for γ and the λk to represent an exact n-fermionic eigenstate of the given Hamiltonian. To specify the desired state, additional conditions must be satisfied as well, e.g., the partial trace relations which relate λ2 to γ and γ2. The particle number and the total spin must be specified and n-representability conditions enter implicitly. While the nondiagonal elements of γ and the λk are determined by the IBCk or the ICSEk, the additional conditions mainly serve to fix the diagonal elements. A hierarchy of k-particle approximations is defined. It is based on the fact that the expansion in terms of cumulants λk can be truncated at any particle rank, which would not be possible for the density matrices γk. For closed-shell states the one-particle approximation agrees with Hartree–Fock.
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 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
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.
In this paper we address a number of topics regarding the contracted Schrödinger equation (CSE): (i) The cumulant expansion of reduced density matrices (RDMs), and the reduction of the CSE to a connected form. (ii) The comparison of the resulting formalism to coupled cluster theory. (iii) The direct calculation of the two-particle RDM in terms of the cumulants, without explicitly imposing any N-representability conditions. We explore a number of approximations, the most elaborate of which is the 3,5-CSE in which the cumulant expansion is carried out to the three-particle level, and the only approximation made is to equate the four- and five-particle cumulants to zero. Overall the numerical results we present for selected small model systems are found to be disappointing and we provide a critical analysis of the perceived shortcomings of our cumulant approach to the CSE.