Asymptotic Methods in Quantum Mechanics: Application to Atoms, Molecules and Nuclei

Quantum mechanics and the Schrodinger equation are the basis for the de­ scription of the properties of atoms, molecules, and nuclei. The development of reliable, meaningful solutions for the energy eigenfunctions of these many­ is a formidable problem. The usual approach for obtaining particle systems the eigenfunctions is based on their variational extremum property of the expectation values of the energy. However the complexity of these variational solutions does not allow a transparent, compact description of the physical structure. There are some properties of the wave functions in some specific, spatial domains, which depend on the general structure of the Schrodinger equation and the electromagnetic potential. These properties provide very useful guidelines in developing simple and accurate solutions for the wave functions of these systems, and provide significant insight into their physical structure. This point, though of considerable importance, has not received adequate attention. Here we present a description of the local properties of the wave functions of a collection of particles, in particular the asymptotic properties when one of the particles is far away from the others. The asymptotic behaviour of this wave function depends primarily on the separation energy of the outmost particle. The universal significance of the asymptotic behaviour of the wave functions should be appreciated at both research and pedagogic levels. This is the main aim of our presentation here.

Table of Contents:
1. Introduction.- 2. General Properties of Wave Functions.- 2.1 Asymptotic Form of Wave Functions.- 2.2 Asymptotic Perturbed Wave Function.- 2.3 Wave Function for rij ? 0.- 2.4 Wave Function for rij and rik ? 0.- 2.5 Local Satisfaction of Schrödinger Equation.- 2.6 Variational Stationary Property.- 2.7 Variational Approach to Perturbations.- 2.8 Generalised Virial Theorem.- 2.9 A Simple Example.- 3. Two- and Three-Electron Atoms and Ions.- 3.1 A Simple Wave Function.- 3.2 Wave Functions Satisfying Cusp, Coalescence and Asymptotic Conditions.- 3.3 Three-Electron Wave Functions.- 4. Polarizabilities and Dispersion Coefficients.- 4.1 Polarizabilities.- 4.2 Dispersion Coefficients.- 4.3 Alkali Isoelectronic Sequences.- 4.4 Asymptotic Polarizabilities and Dispersion Coefficients.- 5. Asymptotically Correct Thomas-Fermi Model Density.- 5.1 Thomas-Fermi Model.- 5.2 Solution for the Thomas-Fermi Density.- 5.3 Asymptotic Density.- 5.4 Modified Density.- 5.5 Applications.- 6. Molecules and Molecular Ions with One and Two Electrons.- 6.1 Wave Functions for One-Electron Molecular Ions.- 6.2 Energies for One-Electron Molecular Ions.- 6.3 Wave Function for H2 and He2++.- 6.4 Results for the Ground State.- 7. Interaction of an Electron with Ions, Atoms, and Molecules.- 7.1 Atomic Rydberg States.- 7.2 Electron-Atom and Electron-Molecule Scattering at High Energies.- 8. Exchange Energy of Diatomic Systems.- 8.1 Exchange Energy of Dimer Ions.- 8.2 Exchange Energy of Diatomic Molecules.- 9. Inter-atomic and Inter-ionic Potentials.- 9.1 Exchange Energy and Exchange Integral in the Heitler-London Theory.- 9.2 Generalized Heitler-London Theory.- 9.3 Inter-atomic and Inter-ionic Potentials.- 10. Proton and Neutron Densities in Nuclei.- 10.1 Semi-phenomenological Density.- 10.2 Determination of the Parameters.- 10.3 Results.- References.