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International Journal of Quantum Chemistry

Wavelets?something for quantum chemistry?

Jean‐Louis Calais

Published Jan 1, 1996• 13 citations
Publisher

Abstract

We propose to use the relatively new mathematical and numerical tool wavelets in quantum chemical studies. A short survey is given of the most elementary aspects of wavelet theory. The connection between wavelets and coherent states is discussed together with Zak's so-called kq-representation. © 1996 John Wiley & Sons, Inc.
DOI
10.1002/(sici)1097-461x(1996)58:6<541::aid-qua1>3.0.co;2-y

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We present a survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.
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