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arXiv:1604.05514

New method of solving the many-body Schrödinger equation

Vladimir M. Tapilin

Published Apr 19, 2016• 0 citations
arXivHTMLPDF

Abstract

A method of solving the Schrödinger equation based on the use of constant particle-particle interaction potential surfaces (IPS) is proposed. The many-body wave function is presented in a configuration interaction form, with coefficients depending on the total interaction potential. The corresponding set of linear ordinary differential equations for the coefficients was developed. To reduce the computational work, a hierarchy of approximations based on interaction potential surfaces of a part of the particle system was worked out. The solution of a simple exactly solvable model and He-like ions proves that this method is more accurate than the conventional configuration interaction method and demonstrates a better convergence with a basis set increase.

DOI
10.48550/arxiv.1604.05514

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Over time, many different theories and approaches have been developed to tackle the many-body problem in quantum chemistry, condensed-matter physics, and nuclear physics. Here we use the helium atom, a real system rather than a model, and we use the exact solution of its Schrödinger equation as a benchmark for comparison between methods. We present new results beyond the random-phase approximation (RPA) from a renormalized RPA (r-RPA) in the framework of the self-consistent RPA (SCRPA) originally developed in nuclear physics, and compare them with various other approaches like configuration interaction (CI), quantum Monte Carlo (QMC), time-dependent density-functional theory (TDDFT), and the Bethe-Salpeter equation on top of the GW approximation. Most of the calculations are consistently done on the same footing, e.g. using the same basis set, in an effort for a most faithful comparison between methods.
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Over time, many different theories and approaches have been developed to tackle the many-body problem in quantum chemistry, condensed-matter physics, and nuclear physics. Here we use the helium atom, a real system rather than a model, and we use the exact solution of its Schrödinger equation as a benchmark for comparison between methods. We present new results beyond the random-phase approximation (RPA) from a renormalized RPA (r-RPA) in the framework of the self-consistent RPA (SCRPA) originally developed in nuclear physics, and compare them with various other approaches like configuration interaction (CI), quantum Monte Carlo (QMC), time-dependent density-functional theory (TDDFT), and the Bethe-Salpeter equation on top of the \boldsymbol{GW} 𝐆 𝐖 approximation. Most of the calculations are consistently done on the same footing, e.g. using the same basis set, in an effort for a most faithful comparison between methods.
We propose a method of solution of the many-body Schr\"odinger equation that involves an expansion of the wave function in terms of a finite, time-dependent basis of a relevant subspace. The equations of motion for the expansion coefficients generalize previous proposals of approximate dynamics. The method is illustrated in the case of an N-particle system with an SU(2) Hamiltonian, and it is shown that it improves the approximation that disregards off-diagonal elements of the dynamical matrices.