Regularity and Approximability of Electronic Wave Functions

ISBN-13: 9783642122477 / Angielski / Miękka / 2010 / 188 str.

Harry Yserentant

Opis

The electronic Schrodi ] nger equation describes the motion of N electrons under Coulomb interaction forces in a eld of clamped nuclei. Solutions of this equation depend on 3N variables, three spatial dimensions for each electron. Approxim- ing the solutions is thus inordinately challenging, and it is conventionally believed that a reduction to simpli ed models, such as those of the Hartree-Fock method or density functional theory, is the only tenable approach. This book seeks to c- vince the reader that this conventional wisdom need not be ironclad: the regularity of the solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and the antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that for a system of one or two electrons. The present notes arose from lectures that I gave in Berlin during the academic year 2008/09 to introduce beginning graduate students of mathematics into this subject. They are kept on an intermediate level that should be accessible to an audience of this kind as well as to physicists and theoretical chemists with a c- responding mathematical training.

Recenzje

From the reviews:

"This monograph discusses the possible direct approximation of the wavefunction itself, in view of several nice properties of the functions at hand (spatial exponential decay, symmetries, regularity, etc.). ... the book is very pleasant to read, having been written very carefully, and with many proofs provided explicitly. The text should be understandable to any graduate student in (applied) mathematics, as well as students and researchers from chemistry and physics ... ." (Gabriel Stoltz, Mathematical Reviews, Issue 2011 d)

Spis treści

and Outline.- Fourier Analysis.- The Basics of Quantum Mechanics.- The Electronic Schrödinger Equation.- Spectrum and Exponential Decay.- Existence and Decay of Mixed Derivatives.- Eigenfunction Expansions.- Convergence Rates and Complexity Bounds.- The Radial-Angular Decomposition.

Blurb

The electronic Schrödinger equation describes the motion of N-electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions, depend on 3N variables, with three spatial dimensions for each electron. Approximating these solutions is thus inordinately challenging, and it is generally believed that a reduction to simplified models, such as those of the Hartree-Fock method or density functional theory, is the only tenable approach. This book seeks to show readers that this conventional wisdom need not be ironclad: the regularity of the solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and the antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that for a system of one or two electrons. The text is accessible to a mathematical audience at the beginning graduate level as well as to physicists and theoretical chemists with a comparable mathematical background and requires no deeper knowledge of the theory of partial differential equations, functional analysis, or quantum theory.