Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Equivalence theorems between the solutions of the fourth-order modified contracted Schrödinger equation and those of the Schrödinger equation

Diego R. Alcoba

Published Mar 1, 2002• 20 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.65.032519

Related papers

Family of modified-contracted Schrödinger equations

Diego R. Alcoba, Carmela Valdemoro

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.

Physical Review A•Nov 13, 2001•32 citations

Comment on “Family of modified contracted Schrödinger equations”

Koji Yasuda

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.

Physical Review A•Jan 27, 2009•1 citation

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.

Physical Review A•Jan 27, 2009•2 citations

Comparison of contracted Schrödinger and coupled-cluster theories

David A. Mazziotti

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, ${\mathrm{BeH}}_{2},$ ${\mathrm{BH}}_{3},$ and ${\mathrm{H}}_{2}\mathrm{O}$ as well as for systems with more general two-particle interactions. The TCSE offers a unified approach to electronic structure.

Physical Review A•Dec 1, 1999•99 citations

Uniqueness of the solution of the contracted Schrödinger equation

Koji Yasuda

In this paper two fundamental questions in the contracted Schr\"odinger equation (CSE) approach are considered by using Lipkin's quasispin model: 1-1 mapping between the second-order reduced density matrix (2-RDM) and the wave function of an excited state, and the uniqueness of the solution of CSE under incomplete N-representability conditions. We present some examples of the wave functions that give the same 2-RDM as the excited state. Thus 2-RDM of an excited state does not determine the wave function uniquely, and it alone cannot be used as basic variable for excited states of the density-matrix theory. Under the incomplete representability constraints the solution of the second-order CSE contains all the exact 4-RDMs together with the spurious ones. We examined the distribution of the solutions as a function of energy, and found that the solutions are well separated from each other under the P- and G-representability conditions of 4-RDM in the low-energy region, but with moderate interaction, or in the higher-energy region, there exist spurious solutions for almost all energies. Thus the G condition of 4-RDM is not sufficient to solve the excited states, although it gives accurate results for the ground state of Lipkin's model.

Physical Review A•May 9, 2002•20 citations

Equation for the direct determination of the density matrix

Hiroshi Nakatsuji

A nonvariational equation, called the density equation, is proposed for the direct determination of the density matrix without using a wave function. It is connected with the Schr\"odinger equation by a necessary and sufficient theorem. The equation for the lowest order depends explicitly only on the fourth-order density matrix ${\ensuremath{\Gamma}}^{(4)}$ (or ${\ensuremath{\Gamma}}^{(3)}$ in a special case) and not on the higher-order density matrices. The equation always gives the density matrix and the associated energy which coincide with those obtained indirectly from the Schr\"odinger equation. This is true even if we solve the equation only with the known and tractable $N$-representability conditions, although in such a case some unphysical solution may also occur in the non-$N$-representable space. The equation is applicable to both fermion and boson systems, and to both ground and excited states. In contrast to the Schr\"odinger equation, the labor of solving the equation does not increase when the number of particles of the system is increased. When we have the Hartree-Fock solution, the equation is transformed such that the correlated density matrix and the correlation energy are the direct solution. The correlated density equation thus obtained is suitable for the study of electron correlations.

Physical Review A•Jul 1, 1976•257 citations
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.
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.
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.
In this paper two fundamental questions in the contracted Schr\"odinger equation (CSE) approach are considered by using Lipkin's quasispin model: 1-1 mapping between the second-order reduced density matrix (2-RDM) and the wave function of an excited state, and the uniqueness of the solution of CSE under incomplete N-representability conditions. We present some examples of the wave functions that give the same 2-RDM as the excited state. Thus 2-RDM of an excited state does not determine the wave function uniquely, and it alone cannot be used as basic variable for excited states of the density-matrix theory. Under the incomplete representability constraints the solution of the second-order CSE contains all the exact 4-RDMs together with the spurious ones. We examined the distribution of the solutions as a function of energy, and found that the solutions are well separated from each other under the P- and G-representability conditions of 4-RDM in the low-energy region, but with moderate interaction, or in the higher-energy region, there exist spurious solutions for almost all energies. Thus the G condition of 4-RDM is not sufficient to solve the excited states, although it gives accurate results for the ground state of Lipkin's model.
A nonvariational equation, called the density equation, is proposed for the direct determination of the density matrix without using a wave function. It is connected with the Schr\"odinger equation by a necessary and sufficient theorem. The equation for the lowest order depends explicitly only on the fourth-order density matrix ${\ensuremath{\Gamma}}^{(4)}$ (or ${\ensuremath{\Gamma}}^{(3)}$ in a special case) and not on the higher-order density matrices. The equation always gives the density matrix and the associated energy which coincide with those obtained indirectly from the Schr\"odinger equation. This is true even if we solve the equation only with the known and tractable NNN-representability conditions, although in such a case some unphysical solution may also occur in the non-NNN-representable space. The equation is applicable to both fermion and boson systems, and to both ground and excited states. In contrast to the Schr\"odinger equation, the labor of solving the equation does not increase when the number of particles of the system is increased. When we have the Hartree-Fock solution, the equation is transformed such that the correlated density matrix and the correlation energy are the direct solution. The correlated density equation thus obtained is suitable for the study of electron correlations.