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International Journal of Quantum Chemistry

Recent developments in the contracted Schrödinger equation method: Controlling the N‐representability of the second‐order reduced density matrix

Carmela Valdemoro, Diego R. Alcoba, Luis M. Tel

Published Jan 1, 2003• 22 citations
Publisher

Abstract

Abstract In this article a new procedure aimed at controlling the N‐representability defect of an approximated second‐order reduced density matrix is reported. This procedure should prove to be effective in enhancing the convergence of the iterative solution of the contracted Schrödinger equation besides being of direct applicability in other methods that are reduced density matrices oriented. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 93: 212–222, 2003

DOI
10.1002/qua.10553

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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

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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

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Theoretical and applicative properties of the correlation and G‐particle‐hole matrices

Carmela Valdemoro, Diego R. Alcoba, Luis M. Tel +1 more

Abstract We have recently (Valdemoro et al., Sixth International Congress of the International Society for Theoretical Chemical Physics, 2008; Alcoba et al., Int J Quantum Chem, in press) reported the form of the G‐particle‐hole hypervirial equation, which can be identified with the anti‐Hermitian part of the correlation contracted Schrödinger equation (Alcoba, Phys Rev A, 2002, 65, 032519), as a tool to obtain the second‐order reduced density matrix of an N‐electron system without previous knowledge of the wave‐function. The results which have been obtained when solving the G‐particle‐hole hypervirial equation with an iterative method also described in (Valdemoro et al., Sixth International Congress of the International Society for Theoretical Chemical Physics, 2008; Alcoba et al., Int J Quantum Chem, in press) have been highly accurate. The convergence of these test calculations has been very smooth, though rather slow. One of the factors which determines the performance of the method is the accuracy with which the 3‐order correlation matrices (3‐CM) involved in the calculations are approximated. It is, therefore, necessary to optimize to the utmost the construction algorithms of these 3‐order matrices in terms of the 2‐CM. In this article, the main theoretical features of the p‐CM are described. Also, some aspects of the correlation contracted Schrödinger equation and of the G‐particle‐hole hypervirial equation are revisited. A new theorem, concerning the sufficiency of the hypervirial of the 3‐order correlation operator to guarantee a correspondence between its solution and that of the Schrödinger equation, and some preliminary results concerning the constructing algorithms of the 3‐CM in terms of the 2‐CM, are reported in the second part of this article. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

International Journal of Quantum Chemistry•Jan 1, 2009•31 citations
Abstract In this article a new procedure aimed at controlling the N‐representability defect of an approximated second‐order reduced density matrix is reported. This procedure should prove to be effective in enhancing the convergence of the iterative solution of the contracted Schrödinger equation besides being of direct applicability in other methods that are reduced density matrices oriented. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 93: 212–222, 2003
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
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
A a procedure is proposed by which, in the course of an iterative solution of the second-order contracted Schr\"odinger equation, The N representability of the second-order reduced density matrix can be tested with increasing stringency. This procedure was suggested by an extended study of the G conditions and from the contraction into the two-electron space of an N-fermion relation, expressing in matrix form the antisymmetry and normalization properties of the N-electron wave function. Several relations are reported.
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
A purification procedure that simultaneously corrects the NNN- and SSS-representability main defects of a second-order reduced density matrix (2RDM), second-order hole reduced density matrix (2HRDM), and second-order G\mathcal{G}G matrix is presented here. In this purifying procedure, the generalized unitarily invariant second-order matrix decomposition [D. R. Alcoba, Int. J. Quantum Chem. 97, 776 (2004)] for the 2RDM and the 2HRDM as well as for the G\mathcal{G}G matrix is combined with the SSS-representability conditions. In particular, here we will focus our attention on the RDMs corresponding to doublet states. We will thus explicitly give the SSS-representability conditions that a two-body correlation matrix has to satisfy when an NNN-electron system is in a doublet spin-state in the spin-component of maximum projection. Furthermore, as a consequence of the G\mathcal{G}G-matrix spin properties (which directly affect the SSS-representability of the 2RDM), we show that a different contracting form for the 2RDM is possible. The numerical results presented in this work confirm the efficiency of our purifying procedure.
Abstract We have recently (Valdemoro et al., Sixth International Congress of the International Society for Theoretical Chemical Physics, 2008; Alcoba et al., Int J Quantum Chem, in press) reported the form of the G‐particle‐hole hypervirial equation, which can be identified with the anti‐Hermitian part of the correlation contracted Schrödinger equation (Alcoba, Phys Rev A, 2002, 65, 032519), as a tool to obtain the second‐order reduced density matrix of an N‐electron system without previous knowledge of the wave‐function. The results which have been obtained when solving the G‐particle‐hole hypervirial equation with an iterative method also described in (Valdemoro et al., Sixth International Congress of the International Society for Theoretical Chemical Physics, 2008; Alcoba et al., Int J Quantum Chem, in press) have been highly accurate. The convergence of these test calculations has been very smooth, though rather slow. One of the factors which determines the performance of the method is the accuracy with which the 3‐order correlation matrices (3‐CM) involved in the calculations are approximated. It is, therefore, necessary to optimize to the utmost the construction algorithms of these 3‐order matrices in terms of the 2‐CM. In this article, the main theoretical features of the p‐CM are described. Also, some aspects of the correlation contracted Schrödinger equation and of the G‐particle‐hole hypervirial equation are revisited. A new theorem, concerning the sufficiency of the hypervirial of the 3‐order correlation operator to guarantee a correspondence between its solution and that of the Schrödinger equation, and some preliminary results concerning the constructing algorithms of the 3‐CM in terms of the 2‐CM, are reported in the second part of this article. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009