Hydrogen-like Atomic Wavefunctions: A First Sketch

This chapter reiterates the quantum numbers for atomic orbitals, known from general chemistry, and places them into the context developed so far. It is sketched how the Schrodinger equation (SE) for the hydrogen atom hydrogen-like systems (one electron plus a nucleus of charge Z) is set up. When the nucleus is treated as a fixed point charge, the SE is only for the electron. The solutions of the SE can be obtained by switching to spherical polar coordinates, such that the variables are separable in terms of the electron distance from the nucleus, r, and two angles. The kinetic energy of the electron then has a radial component, and an angular component. The latter is associated with the angular momentum quantum number, which is codified by the letters s, p, d, f, and so forth. A step by step solution of the SE is provided later, in chapter 19.

Relativistic compact effective potentials and efficient, shared-exponent basis sets for the third-, fourth-, and fifth-row atoms

Relativistic compact effective potentials (RCEP), which replace the atomic core electrons in molecular calculations, have been derived from numerical Dirac–Fock atomic wavefunctions using shape-consistent valence pseudo-orbitals and an optimizing procedure based on an energy-overlap functional. Potentials are presented for the third-, fourth-, and fifth-row atoms of the Periodic Table (excluding the lanthanide series). The efficiency of molecular calculations is enhanced by using compact Gaussian expansions (no more than three terms) to represent the radial components of the potentials, and energy-optimized, shared-exponent, contracted-Gaussian atomic orbital basis sets. Transferability of the potentials has been tested by comparing calculated atomic excitation energies and ionization potentials with values obtained from numerical relativistic Hartree–Fock calculations. For the alkali and alkaline earth atoms, core polarization potentials (CPP) have been derived which may be added to the RCEP to make possible accurate molecular calculations without explicitly including core-valence correlating configurations in the wavefunction. Keywords: model potentials, effective core potentials, transition metals, relativistic calculations.