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Acta Physico-Chimica Sinica

Theoretical Analysis of Two Reconstruction Schemes of Reduced Density Matrices

X GAO, Feiwu Chen

Published Apr 1, 2007• 0 citations
Publisher

Abstract

Abstract The effectiveness of the approach for systematical reconstruction of higher order reduced density matrices with lower order ones, which was developed by Chen ( Science in China B , 2006, 49 : 402), was compared theoretically with that of Mazziotti's method through Harris model, by omitting the cumulant terms 3 Δ M , 4Δ M in the latter method and the normal product terms 3 Δ, 4 Δ in the former method, it was found that the errors from both approaches were comparable. As a better approximation, if only fourth-order terms 4Δ M and 4 Δ in both methods are neglected whereas the third-order terms 3Δ M and 3 Δ are computed from their cor-responding fourth-order terms 4Δ M and 4 Δ, respectively, through contractions, the results calculated with Chen's approach not only have the correct signs but also are very close to the exact normal products 3 Δ whereas some of the results calculated with Mazziotti's method do not even have the correct signs with respect to the exact cumulants 3Δ M .

DOI
10.1016/s1872-1508(07)60036-9

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Abstract The effectiveness of the approach for systematical reconstruction of higher order reduced density matrices with lower order ones, which was developed by Chen ( Science in China B , 2006, 49 : 402), was compared theoretically with that of Mazziotti's method through Harris model, by omitting the cumulant terms 3 Δ M , 4Δ M in the latter method and the normal product terms 3 Δ, 4 Δ in the former method, it was found that the errors from both approaches were comparable. As a better approximation, if only fourth-order terms 4Δ M and 4 Δ in both methods are neglected whereas the third-order terms 3Δ M and 3 Δ are computed from their cor-responding fourth-order terms 4Δ M and 4 Δ, respectively, through contractions, the results calculated with Chen's approach not only have the correct signs but also are very close to the exact normal products 3 Δ whereas some of the results calculated with Mazziotti's method do not even have the correct signs with respect to the exact cumulants 3Δ M .
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
Preface. I: Properties of Reduced Density Matrices. 1. RDMs: How Did We Get Here? A.J. Coleman. 2. Some Theorems on Uniqueness and Reconstruction of Higher-Order Density Matrices M. Rosina. 3. Cumulant Expansions of Reduced Densities, Reduced Density Matrices, and Green's Functions P. Ziesche. 4. On Calculating Approximate and Exact Density Matrices R. Erdahl, B. Jin. II: The Contracted Schroedinger Equation. 5. Density Equation Theory in Chemical Physics H. Nakatsuji. 6. Critical Questions Concerning Iterative Solution of the Contracted Schroedinger Equation C. Valdemoro, L.M. Tel, E. Perez-Romero 7. Cumulants and the Contracted Schroedinger Equation D.A. Mazziotti.. III: Density Matrix Functional Theory. 8. Natural Orbital Functional Theory S. Goedecker, C.J. Umrigar. 9. The Pair Density in Approximate Density Functional Theory: The Hidden Agent N.T. Maitra, K. Burke. 10. Functional N-representability in Density Matrix and Density Functional Theory: An Illustration for Hooke's Atom E.V. Ludena, V. Karasiev, A. Artemiev, D. Gomez.. IV: Electron Intracule and Extracule Densities. 11. Intracule and Extracule Densities: Historical Perspectives and Future Prospects E. Valderrama, J.M. Ugalde, R.J. Boyd. 12. Topology of Electron Correlation J. Cioslowski. 13. Electron-Pair Densities of Atoms T. Koga. Index.
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.
With the use of the density equation [H. Nakatsuji, Phys. Rev. A 14, 41 (1976)], the second-order density matrices are directly determined without any use of the wave functions. The third- and fourth-order reduced density matrices (RDM's) are decoupled into lower-order ones using the Green's function technique. This method is applied to Be, Ne, H2{\mathrm{H}}_{2}H2​O, H3O+{H}_{3}{O}^{+}H3​O+, NH3{\mathrm{H}}_{3}H3​, CH4{\mathrm{H}}_{4}H4​, ${\mathrm{BH}}_{4}^{\phantom{\rule{0ex}{0ex}}\ensuremath{-}}$, NH4+{\mathrm{NH}}_{4}^{\phantom{\rule{0ex}{0ex}}+}NH4+​, and CH3{\mathrm{H}}_{3}H3​F, and the results are successfully compared with the full configuration interaction results. The convergence is fairly good, and the calculated second-order RDM's almost satisfy the necessary conditions of the N representability, the P, Q, and G conditions, and the first-order RDM's are exactly N representable. These results show that the present method is very promising.
We consider, for systems of indistinguishable fermions, approximate reconstruction of the three- and four-particle reduced density matrices (RDMs) from the one- and two-particle RDMs, $\mathit{\ensuremath{\gamma}}$ and $\mathit{\ensuremath{\Gamma}}.$ Our ansatz for reconstructing the four-particle RDM is the linear combination $a(\mathit{\ensuremath{\Gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\Gamma}})+b(\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\Gamma}})+c(\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}),$ where ``\ensuremath{\wedge}'' denotes the antisymmetrized (Grassmann) product. This is a generalization of reconstruction functionals employed recently to perform direct RDM calculations without wave functions via the contracted Schr\"odinger equation. Here we consider relationships between the parameters a, b, and c that are required in order for the reconstruction functionals to respect the hierarchy of contraction relations between RDMs. To this end we establish several general theorems concerning contractions of antisymmetrized tensor products of $\mathit{\ensuremath{\gamma}},$ $\mathit{\ensuremath{\Gamma}},$ and various products thereof. The accuracy of proposed reconstruction functionals is evaluated using accurate density matrices for the ground state of Be.
The solution of N-representability for the two-particle reduced density matrix (2-RDM) is intimately connected through the contracted Schr\"odinger equation with the problem of reconstructing the 3- and 4-RDM's from the 2-RDM [D. A. Mazziotti, Phys. Rev. A 57, 4219 (1998)]. We derive an explicit formula for building the 3-RDM from the 2-RDM which significantly improves the accuracy of the 3-RDM cumulant expansion. Calculations indicate that the improvement is better than that achieved by the correction of Nakatsuji and Yasuda (NY) [Phys. Rev. Lett. 76, 1039 (1996); Phys. Rev. A 56, 2648 (1997)]. We also present a more efficient formulation of the NY correction. Both the present and NY reconstructions are shown to satisfy the particle-hole duality. Comparison of the reconstruction methods is made through several molecules LiH, BeH2,{\mathrm{BeH}}_{2},BeH2​, BH3,{\mathrm{BH}}_{3},BH3​, H2O,{\mathrm{H}}_{2}\mathrm{O},H2​O, and CH4,{\mathrm{CH}}_{4},CH4​, as well as systems with more general pairwise correlation.