The development of quantum chemistry, that is, the solution of the Schrödinger equation for molecules, is almost exclusively founded on the expansion of the molecular electronic wave function as a linear combination of atom-centered functions, or atomic orbitals—the LCAO approximation. These orbitals are usually built up out of some set of basis functions. The properties of the atomic functions at large and small distances from the nucleus determines to a large extent what characteristics the basis functions must have, and for this purpose it is sufficient to examine the properties of the hydrogenic solutions to the Schrödinger equation. If we are to do the same for relativistic quantum chemistry, we should first examine the properties of the atomic solutions to determine what kind of basis functions would be appropriate. However, the atomic solutions of the Dirac equation provide more than merely a guide to the choice of basis functions. The atoms in a molecule retain their atomic identities to a very large extent, and the modifications caused by the molecular field are quite small for most properties. In order to arrive at a satisfactory description of the relativistic effects in molecules, we must first of all be able to treat these effects at the atomic level. The insight gained into the effects of relativity on atomic structure is therefore a necessary and useful starting point for relativistic quantum chemistry. As in the nonrelativistic case, most of the salient features of the atomic systems are exposed in the treatment of the simplest of these, the hydrogen-like one-electron atoms. In Hartree atomic units the time-independent Dirac equation yields the coupled equations where we have shifted the energy by −mc2 (with m = 1), as discussed in section 4.6. We will use this shifted energy scale for the rest of the book unless otherwise explicitly indicated. V is here a scalar, central potential.
Inverse Multiquadratic Functions as the Basis for the Rectangular Collocation Method to Solve the Vibrational Schrödinger Equation
