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International Journal of Quantum Chemistry

Methods for electronic‐structure calculations: Overview from a reduced‐density‐matrix point of view

P. Ziesche, Ferenc Tasnádi

Published Jan 1, 2004• 2 citations
Publisher

Abstract

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

DOI
10.1002/qua.20129

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The reduced-density-matrix method is an promising candidate for the next generation electronic structure calculation method; it is equivalent to solve the Schrödinger equation for the ground state. The number of variables is the same as a four electron system and constant regardless of the electrons in the system. Thus many researchers have been dreaming of a much simpler method for quantum mechanics. In this chapter, we give a overview of the reduced-density matrix method; details of the theories, methods, history, and some new computational results. Typically, the results are comparable to the CCSD(T) which is a sophisticated traditional approach in quantum chemistry.

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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
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, NNN-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 quasi-particle 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, being possibly the solutions of an approximate effective two-body equation, and the idea is sketched of how its contraction sum rule can be used for a variational treatment.
Abstract Recently, it has been shown, that the pair density of the homogeneous electron gas can be parametrized in terms of 2‐body wave functions (geminals), which are scattering solutions of an effective 2‐body Schrödinger equation. For the corresponding scattering phase shifts, new sum rules are reported in this paper. These sum rules describe not only the normalization of the pair density (similar to the Friedel sum rule of solid state theory), but also the contraction of the 2‐body reduced density matrix. This allows one to calculate also the momentum distribution, provided that the geminals are known from an appropriate screening of the Coulomb repulsion. An analysis is presented leading from the definitions and (contraction and spectral) properties of reduced density matrices to the Kimball‐Overhauser approach and its generalizations. Thereby cumulants are used. Their size‐extensivity is related to the thermodynamic limit. See erratum Ann. Phys. (Leipzig) 13, 624 (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
Through the 3,5-contracted Schrödinger equation (3,5-CSchE) quantum energies and 3-particle reduced density matrices (3-RDMs) are determined directly without wave functions. Since the 3,5-CSchE involves the 5-RDM, its solution is indeterminate without N-representability conditions. However, the indeterminacy of the 3,5-CSchE may be removed through a reconstruction strategy for building the 4- and 5-RDMs from the 3-RDM. We present a systematic procedure for obtaining corrections for Valdemoro's reconstruction functionals from two complementary approaches, the particle–hole duality and the theory of cumulants. With the cumulants we are able to demonstrate that we have obtained all terms in the reconstruction functionals which may be written as antisymmetric products of the lower rdms. The cumulants allow us to understand the reconstruction functionals in terms of a renormalized many-body perturbation theory. The reconstruction functionals also lead to a natural generalization of Wick's theorem for evaluating expectation values of fermionic annihilation and creation operators with respect to correlated reference states. Previous work [Phys. Rev. A 57, 4219 (1998)] has explored the determination of correlation energy and 2-RDMs through the 2,4-CSchE, also known as the density equation. Because the reconstruction functionals employed with the 3,5-CSchE depend only on the antisymmetric products of lower RDMs in constrast to those used with the 2,4-CSchE, the 3,5-CSchE method presented here does not require the solution of systems of linear equations during reconstruction or the storage of the reconstructed RDMs. Application of the 3,5-CSchE technique to a quasi-spin model generates ground-state energies and 2-RDMs similar in accuracy to single–double configuration interaction (SDCI). We employ a simple iterative procedure for the solution of the 3,5-CSchE without traditional diagonalization. The CSchE techniques offer an approximate solution of the N-representability problem and a new approach to electron correlation. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 557–570, 1998
Variational approach for the electronic structure calculation on the second-order reduced density matrices and the NNN-representability problem
The reduced-density-matrix method is an promising candidate for the next generation electronic structure calculation method; it is equivalent to solve the Schrödinger equation for the ground state. The number of variables is the same as a four electron system and constant regardless of the electrons in the system. Thus many researchers have been dreaming of a much simpler method for quantum mechanics. In this chapter, we give a overview of the reduced-density matrix method; details of the theories, methods, history, and some new computational results. Typically, the results are comparable to the CCSD(T) which is a sophisticated traditional approach in quantum chemistry.