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Physical Review Letters

Exactness of Two-Body Cluster Expansions in Many-Body Quantum Theory

Piotr Piecuch B, Karol Kowalski, Peng‐Dong Fan, Krzysztof Jedziniak

Published Mar 20, 2003• 45 citations
Publisher

Abstract

The Horn-Weinstein formula and the variational principle, combined with numerical results for a few many-electron systems, are used to provide support for a conjecture that the exact ground-state wave function for a Hamiltonian system containing up to two-body terms may be represented by an exponential cluster expansion employing a finite two-body operator.

DOI
10.1103/physrevlett.90.113001

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The Horn-Weinstein formula and the variational principle, combined with numerical results for a few many-electron systems, are used to provide support for a conjecture that the exact ground-state wave function for a Hamiltonian system containing up to two-body terms may be represented by an exponential cluster expansion employing a finite two-body operator.
It is proved that a recent conjecture that the exact ground-state wave function of an arbitrary many-fermion system with one- and two-body interactions may be represented by an exponential cluster expansion employing finite two-body operators, starting from any reference function sufficiently close to the exact eigenfunction, is not valid. We show that the space of initial reference functions which lead to the exact ground state is of dimension equal to the number of two-body operators. If the dimension of the multiparticle space is greater than the number of two-body operators, then the space of good reference functions is of measure zero in it.
Progress toward the solution of the strongly correlated electron problem has been stymied by the exponential complexity of the wave function. Previous work established an exact two-body exponential product expansion for the ground-state wave function. By developing a reduced density-matrix analog of Dalgarno-Lewis perturbation theory, we prove here that (i) the two-body exponential product expansion is rapidly and globally convergent with each operator representing an order of a renormalized perturbation theory, (ii) the energy of the expansion converges quadratically near the solution, and (iii) the expansion is exact for both ground and excited states. The two-body expansion offers a reduced parametrization of the many-particle wave function as well as the two-particle reduced density matrix with potential applications on both conventional and quantum computers for the study of strongly correlated quantum systems. We demonstrate the result with the exact solution of the contracted Schr\"odinger equation for the molecular chains H4{\mathrm{H}}_{4}H4​ and H5{\mathrm{H}}_{5}H5​.
Recent studies have considered the possibility that the exact ground-state wavefunction from any Hamiltonian with two-particle interactions may be generated from a single finite two-body exponential transformation acting on an arbitrary Slater determinant [Piecuch et al., Phys. Rev. Lett. 90, 113001 (2003)]. Using the Campbell-Baker-Hausdorff relation, we show that it is difficult for the variational minimum of this trial wave function to satisfy the contracted Schr\"odinger equation which is a necessary and sufficient condition for the wave function to satisfy the Schr\"odinger equation. A counterexample is presented through the Lipkin quasispin model with 4--50 fermions. When the number of fermions exceeds four, the wave function from a finite two-body exponential transformation is shown to be inexact. If the trial wave function ansatz is extended to include products of finite two-body exponential transformations acting on an arbitrary Slater-determinant reference, then we show that the ansatz includes the exact ground-state wave function from any Hamiltonian with only two-particle interactions. Connections between the two-body exponential transformation of the wave function and recent research on two-body exponential similarity transformations of the Hamiltonian [S.R. White, J. Chem. Phys. 117, 7472 (2002)] are discussed.
Using a general two-body exponential parametrization for the wave function, the Nakatsuji two-particle density equation [Phys. Rev. A 14, 41 (1976)] is transformed into a set of nonlinear algebraic equations in which the number of unknowns precisely equals the number of equations. Since the Nakatsuji two-particle density equation is equivalent to the time-independent Schrodinger equation for Hamiltonians containing up to two-body interactions, the answer to the title question is affirmative, provided the equations have solutions. Practical implications of the parametrization and possible approximation schemes are briefly discussed.