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Physical Review A

Comment on “Family of modified contracted Schrödinger equations”

Koji Yasuda

Published Jan 27, 2009• 1 citation
Publisher

Abstract

It is shown that the modified contracted Schr\"odinger equation (MCSE) proposed by Alcoba and Valdemoro [Phys. Rev. A 64, 062105 (2001)] is equivalent to the original CSE of lower-order [Phys. Rev. A 14, 41 (1976)]. In particular, MCSEs of the second, third, and fourth orders are reduced to CSEs of the first, first, and second orders, respectively. Thus these MCSEs are indeterminate as the corresponding CSEs.

DOI
10.1103/physreva.79.016101

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Reply to “Comment on ‘Family of modified contracted Schrödinger equations’ ”

Diego R. Alcoba, Carmela Valdemoro

The preceding Comment by Yasuda [Phys. Rev. A 79, 016101 (2009)] on our earlier paper [Phys. Rev. A 64, 062105 (2001)] raises several objections about the modified contracted Schr\"odinger equations (MCSEs) to which we reply here. In his comment, Yasuda also questions the possibility of obtaining the exact solution of the fourth-order MCSE due to its indeterminacy. We maintain the opposite, at least from a theoretical point of view.

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Four basic theorems involving the correlation matrices are demonstrated here. One of these theorems establishes, as a necessary and sufficient condition, that the solutions of the fourth-order modified contracted Schr\"odinger equation correspond uniquely to those of the Schr\"odinger equation. The complete equivalence between these two equations is demonstrated. Another equation equivalent to the Schr\"odinger equation but involving only the correlation matrices is also obtained as a consequence of the second theorem.

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Antoine Royer

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Physical Review Letters•Apr 22, 1996•0 citations
It is shown that the modified contracted Schr"odinger equation (MCSE) proposed by Alcoba and Valdemoro [Phys. Rev. A 64, 062105 (2001)] is equivalent to the original CSE of lower-order [Phys. Rev. A 14, 41 (1976)]. In particular, MCSEs of the second, third, and fourth orders are reduced to CSEs of the first, first, and second orders, respectively. Thus these MCSEs are indeterminate as the corresponding CSEs.
A family of equations that combines contracted Schr\"odinger equations of different orders is reported here. Attention is focussed on the resulting second order, third order, and fourth order of these modified-contracted Schr\"odinger equations. Some of these equations are self-contained and have as fixed points those corresponding to the full-configuration interaction eigenstates. The indeterminacy, which hindered initially the use of the contracted Schr\"odinger equations, does not formally exist in these equations. Relations linking the lower-order reduced density matrices with the higher-order matrices are exactly incorporated into the modified-contracted Schr\"odinger-equations structure. The cancellation of high-order correlation terms, which is hidden in the contracted Schr\"odinger equations, now takes an explicit form.
Four basic theorems involving the correlation matrices are demonstrated here. One of these theorems establishes, as a necessary and sufficient condition, that the solutions of the fourth-order modified contracted Schr\"odinger equation correspond uniquely to those of the Schr\"odinger equation. The complete equivalence between these two equations is demonstrated. Another equation equivalent to the Schr\"odinger equation but involving only the correlation matrices is also obtained as a consequence of the second theorem.
The contracted Schr\"odinger equation (CSE) technique through its direct determination of the two-particle reduced density matrix (2RDM) without the wave function may offer a fresh alternative to traditional many-body quantum calculations. Without additional information the CSE, also known as the density equation, cannot be solved for the 2RDM because it also requires a knowledge of the 4RDM. We provide theoretical foundations through a reconstruction theorem for recent attempts at generating higher RDMs from the 2RDM to remove the indeterminacy of the CSE. With Grassmann algebra a more concise representation for Valdemoro's reconstruction functionals [F. Colmenero, C. Perez del Valle, and C. Valdemoro, Phys. Rev. A 47, 971 (1993)] is presented. From the perspective of the particle-hole equivalence we obtain Nakatsuji and Yasuda's correction for the 4RDM formula [H. Nakatsuji and K. Yasuda, Phys. Rev. Lett. 76, 1039 (1996)] as well as a corrective approach for the 3RDM functional. A different reconstruction strategy, the ensemble representability method (ERM), is introduced to build the 3- and 4-RDMs by enforcing four-ensemble representability and contraction conditions. We derive the CSE in second quantization without Valdemoro's matrix contraction mapping and offer the first proof of Nakatsuji's theorem for the second-quantized CSE. Both the functional and ERM reconstruction strategies are employed with the CSE to solve for the energies and the 2RDMs of a quasispin model without wave functions. We elucidate the iterative solution of the CSE through an analogy with the power method for eigenvalue equations. Resulting energies of the CSE methods are comparable to single-double configuration-interaction (SDCI) energies, and the 2RDMs are more accurate by an order of magnitude than those from SDCI. While the CSE has been applied to systems with 14 electrons, we present results for as many as 40 particles. Results indicate that the 2RDM remains accurate as the number of particles increases. We also report a direct determination of excited-state 2RDMs through the CSE. By circumventing the wave function, the CSE presents new possibilities for treating electron correlation.
The theory of the contracted Schr\"odinger equation (CSE) [D. A. Mazziotti, Phys. Rev. A 57, 4219 (1998)] is connected with traditional methods of electronic structure including configuration-interaction (CI) and coupled-cluster (CC) theory. We derive a transition contracted Schr\"odinger equation (TCSE) which depends on the wave function $\ensuremath{\psi}$ as well as another N-particle function $\ensuremath{\chi}$ through the two-, three-, and four-particle reduced transition matrices (RTMs). By reconstructing the 3 and 4 RTMs approximately from the 2-RTM, the indeterminacy of the equation may be removed. The choice of the reconstruction and the function $\ensuremath{\chi}$ determines whether one obtains the CI, CC, or CSE theory. Through cumulant theory and Grassmann algebra we derive reconstruction formulas for the 3- and 4-RTMs which generalize both the reduced density matrix (RDM) cumulant expansions as well as the exponential ansatz for the CC wave function. This produces a fresh approach to CC theory through RTMs. Two theoretical differences between the CC and the CSE theories are established for energetically nondegenerate states: (i) while the CSE has a single exact solution when the 3- and 4-RDMs are N-representable, the CC equations with N-representable 3- and 4-RTMs have a family of solutions. Thus, N-representability conditions offer a medium for improving the CSE solution but not the CC solution, and (ii) while the 2-RDM for an electronic Hamiltonian reconstructs to unique N-representable 3- and 4-RDMs, the 2-RTM builds to a family of N-representable 3- and 4-RTMs. Hence, renormalized reconstructions beyond the cumulant expansion may be developed for the 2-RDM but not for the 2-RTM without explicit use of the Hamiltonian. In the applications we implement our recently developed reconstruction formula for the 3-RDM which extends beyond the cumulant approximation. Calculations compare the 3-RDM and 3-RTM reconstructions for the molecules LiH, BeH2,{\mathrm{BeH}}_{2},BeH2​, BH3,{\mathrm{BH}}_{3},BH3​, and H2O{\mathrm{H}}_{2}\mathrm{O}H2​O as well as for systems with more general two-particle interactions. The TCSE offers a unified approach to electronic structure.