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Mathematical Models and Methods in Applied Sciences

A DECOMPOSITION THEOREM FOR WAVE FUNCTIONS IN MOLECULAR QUANTUM CHEMISTRY

Olivier Bokanowski, Benoît Grébert

Published Jun 1, 1996• 18 citations
Publisher

Abstract

A general way to decompose an antisymmetric wave function into its density function and a wave function of a given density is proposed. Its usefulness for molecular quantum chemistry is discussed, in particular in the context of density functional theory.
DOI
10.1142/s021820259600016x

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A theorem is proved which gives the necessary and sufficient conditions on the ppp-particle reduced-density operator restricted to a functional subspace that ensures the separability of the wave function of a fermion system. This theorem is the theoretical basis of a general method of analyzing atomic and molecular wave functions, a method which is able to reveal the existence of somehow chemically discernible subsystems of electrons.
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The field of quantum chemistry is concerned with the modelling and simulation of the behaviour of molecular systems on the basis of the fundamental equations of quantum mechanics. Since these equations exhibit an extreme case of the curse of dimensionality (the Schrödinger equation for N electrons being a partial differential equation on \mathbb R^{3N} ), the quantum-chemical simulation of even moderate-size molecules already requires highly sophisticated model-reduction, approximation, and simulation techniques. The workshop brought together selected quantum chemists and physicists, and the growing community of mathematicians working in the area, to report and discuss recent advances on topics such as coupled-cluster theory, direct approximation schemes in full configuration-interaction (FCI) theory, interacting Green’s functions, foundations and computational aspects of density-functional theory (DFT), low-rank tensor methods, quantum chemistry in the presence of a strong magnetic field, and multiscale coupling of quantum simulations.
Quantum chemistry focuses on modelling and simulating the behaviour of molecular systems using the fundamental principles of quantum mechanics. However, the underlying equations, such as the Schrödinger equation for computing the ground state of N electrons, which is a partial differential equation defined on \mathbb{R}^{3N} , suffer from the curse of dimensionality. As a result, simulating even moderately sized molecules demands advanced techniques in analysis, approximation and reduced-order modelling for overcoming the naive computational complexity. In this workshop, the rapidly growing mathematical community working in the field together with several quantum chemists and physicists were gathered to discuss recent advances in areas such as (1) the mathematical and numerical analysis of standard models used in the field, including Density Functional Theory, Coupled Cluster, and the Density Matrix Renormalization Group (2) the development of efficient numerical methods, relying e.g. on the development of error bounds and (3) the opportunities of recent data-driven methods towards the approximation of wavefunctions or density matrices.
Quantum chemistry focuses on modelling and simulating the behaviour of molecular systems using the fundamental principles of quantum mechanics. However, the underlying equations, such as the Schrödinger equation for computing the ground state of NNN electrons, which is a partial differential equation defined on R3N\mathbb{R}^{3N}R3N, suffer from the curse of dimensionality. As a result, simulating even moderately sized molecules demands advanced techniques in analysis, approximation and reduced-order modelling for overcoming the naive computational complexity. In this workshop, the rapidly growing mathematical community working in the field together with several quantum chemists and physicists were gathered to discuss recent advances in areas such as (1) the mathematical and numerical analysis of standard models used in the field, including Density Functional Theory, Coupled Cluster, and the Density Matrix Renormalization Group (2) the development of efficient numerical methods, relying e.g. on the development of error bounds and (3) the opportunities of recent data-driven methods towards the approximation of wavefunctions or density matrices.