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International Journal of Quantum Chemistry

Utilization of deformations in molecular quantum chemistry and application to density functional theory

Olivier Bokanowski, Beno�t Grebert

Published Jan 1, 1998• 18 citations
Publisher

Abstract

The aim of this article is to present in a way accessible to most quantum chemists a general mathematical method which consists in deforming wave functions and density functions (in the spirit of the local scaling transformation). This deformation method allows us to obtain several new results, including a characterization of the set of wave functions that have the same given density function (which gives a new insight on a result of G. Zumbach and K. Maschke, Phys. Rev. A 28, 544 (1983)) and an N-representability result where symmetry is taken into account. We also propose new theoretical ways to generate approximations of the exact density functional and give a numerical example. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 221–231, 1998

DOI
10.1002/(sici)1097-461x(1998)68:4<221::aid-qua1>3.0.co;2-x

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The aim of this article is to present in a way accessible to most quantum chemists a general mathematical method which consists in deforming wave functions and density functions (in the spirit of the local scaling transformation). This deformation method allows us to obtain several new results, including a characterization of the set of wave functions that have the same given density function (which gives a new insight on a result of G. Zumbach and K. Maschke, Phys. Rev. A 28, 544 (1983)) and an N-representability result where symmetry is taken into account. We also propose new theoretical ways to generate approximations of the exact density functional and give a numerical example. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 221–231, 1998
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.
Current experiments on semiconductor devices using femtosecond lasers provide new theoretical challenges for the description of charge carrier dynamics. Among the new features of such experiments are states driven very far from equilibrium and probes on time scales short compared to scattering and other characteristic material relaxation times. Standard many-body methods must be modified and extended to accommodate these features. We propose that the quantum hierarchy for reduced density operators is an ideal formulation of such initial value problems and describe how the dominant effects of exchange and charge correlations can be accounted for in a simple and physically transparent closure of the hierarchy of equations. The transformations, approximations, and interpretation can be accomplished independent of any particular matrix representation. Decomposition into kinetic equations for band occupation densities and polarization densities follows in a straightforward way after the many-body problem has been brought under control. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 929–940, 1997
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.