Schrödinger Equation in Many-Electron System: Helium, Cluster

Abstract

In hydrogenic atom, Schrödinger equation can be analytically solved. Quantum wave function is concretely given. However, the situation dramatically changes in many-electron system. As analytical solution cannot be given, we need to solve Schrödinger equation numerically. In this chapter, necessary contents for numerical approach are explained. First, we consider representative two-electron system: helium atom. The Hamiltonian includes operators of kinetic energy of electron, kinetic energy of atomic nucleus, Coulomb interaction between electron and atomic nucleus, and Coulomb interaction between two electrons. In comparison with hydrogenic atom, Coulomb interaction between two electrons is added. In many-electron cluster or molecule, we need to incorporate the operator of Coulomb interaction between atomic nuclei. In Born–Oppenheimer approximation, the Coulomb interaction can be negligible. When considering more than two electrons, electron spin must be taken into account, by the introduction of spin function into quantum wave function. The quantum wave function in many-electron system is represented by the product of spatial orbital and spin function. Spin function satisfies orthonormality. As electron is categorised as fermion, inverse principle must be satisfied for total quantum wave function. It implies that total quantum wave function changes the sign, when the labels of any two identical fermions are exchanged. Hartree product does not satisfy the condition. In quantum chemistry, Slater determinant is introduced for the purpose. Finally, Slater determinant for two-electron system is shown.