Preface Acknowledgements Notation Part I. Overview and Background Topics: 1. Introduction 2. Overview 3. Theoretical background 4. Periodic solids and electron bands 5. Uniform electron gas and simple metals Part II. Density Functional Theory: 6. Density functional theory: foundations 7. The Kohn-Sham ansatz 8. Functionals for exchange and correlation 9. Solving the Kohn-Sham equations Part III. Important Preliminaries on Atoms: 10. Electronic structure of atoms 11. Pseudopotentials Part IV. Determination of Electronic Structure, The Three Basic Methods: 12. Plane waves and grids: basics 13. Plane waves and grids: full calculations 14. Localized orbitals: tight binding 15. Localized orbitals: full calculations 16. Augmented functions: APW, KKR, MTO 17. Augmented functions: linear methods Part V. Predicting Properties of Matter from Electronic Structure - Recent Developments: 18. Quantum molecular dynamics (QMD) 19. Response functions: photons, magnons ... 20. Excitation spectra and optical properties 21. Wannier functions 22. Polarization, localization and Berry's phases 23. Locality and linear scaling O (N) methods 24. Where to find more Appendixes References Index.

Electronic Structure: Basic Theory and Practical Methods

The x‐ray form factors for a bonded hydrogen in the hydrogen molecule have been calculated for a spherical approximation to the bonded atom. These factors may be better suited for the least‐squares refinement of x‐ray diffraction data from organic molecular crystals than those for the isolated hydrogen atom. It has been shown that within the spherical approximation for the bonded hydrogens in H2, a least‐squares refinement of the atomic positions will result in a bond length (Re value) short of neutron diffraction or spectroscopic values. The spherical atoms are optimally positioned 0.07 A off each proton into the bond. A nonspherical density for the bonded hydrogen atom in the hydrogen molecule has also been defined and the corresponding complex scattering factors have been calculated. The electronic density for the hydrogen molecule in these calculations was based on a modified form of the Kolos—Roothaan wavefunction for H2. Scattering calculations were made tractable by expansion of a plane wave in spheroidal wavefunctions.

Coherent X‐Ray Scattering for the Hydrogen Atom in the Hydrogen Molecule

In the following chapters the electronic structure of molecules will be discussed and the techniques of electronic structure calculations presented. Without exception the molecular electronic wave functions will be expanded in some convenient, but physically motivated, set of one-electron functions. Since the computational effort strongly depends on the number of expansion functions (see, e.g., the following chapters), the set of functions must be limited as far as possible without adversely affecting the accuracy of the wave functions. This chapter will discuss the choice of such functions for molecular calculations.

Gaussian Basis Sets for Molecular Calculations

An analytic semiempirical expression is developed for the ground‐state potential‐energy surface of the H3 system. Use is made of the valence‐bond formulation with the inclusion of overlap and three‐center terms. The diatomic Coulomb and exchange integrals are estimated from accurate values for the H2 molecule by a modified London—Eyring—Sato procedure, while the three‐center integrals are approximated by simple formulas. A linear, symmetric saddle‐point configuration of minimum energy is found that has properties in approximate agreement, though not identical, with other estimates from theory and experimental rate data. The details of the surface are presented in terms of contour maps for a variety of distances and angles. Because of the inclusion of overlap and three‐center terms, a more realistic energy is expected for the nonlinear configurations than could be obtained from previous potential energy formulas. The utility of the present method for dynamical studies of reaction cross sections is indicated.

Potential Energy Surface for H3

The current state of the theory and its application to low-energy electron-molecule collisions is reviewed. The emphasis is on elastic scattering and vibrational and rotational excitation of small diatomic and polyatomic molecules. New and traditional theoretical approaches are described, and the results of calculations are compared with existing experimental measurements.

The theory of electron-molecule collisions

The solution of Schroedinger's equation for the normal hydrogen molecule is approximated by the function $C[{e}^{\ensuremath{-}\frac{z({r}_{1}+{p}_{2})}{a}}+{e}^{\ensuremath{-}\frac{z({r}_{2}+{p}_{1})}{a}}]$ where $a=\frac{{h}^{2}}{4{\ensuremath{\pi}}^{2}m{e}^{2}}$, ${r}_{1}$ and ${p}_{1}$ are the distances of one of the electrons to the two nuclei, and ${r}_{2}$ and ${p}_{2}$ those for the other electron. The value of $Z$ is so determined as to give a minimum value to the variational integral which generates Schroedinger's wave equation. This minimum value of the integral gives the approximate energy $E$. For every nuclear separation $D$, there is a $Z$ which gives the best approximation and a corresponding $E$. We thus obtain an approximate energy curve as a function of the separation. The minimum of this curve gives the following data for the configuration corresponding to the normal hydrogen molecule: the heat of dissociation = 3.76 volts, the moment of inertia ${J}_{0}=4.59\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}41}$ gr. ${\mathrm{cm}}^{2}$, the nuclear vibrational frequency ${\ensuremath{\nu}}_{0}=4900$ ${\mathrm{cm}}^{\ensuremath{-}1}$.

S. C. Wang

Papers

The Problem of the Normal Hydrogen Molecule in the New Quantum Mechanics