Wave Function for the Free Electron. I. The Coulomb Potential

The wave function for a free electron in the presence of an atom is considered for the case of the Coulomb potential, a potential which ignores both the effects of electron exchange and core polarization. A sample calculation involving the neutral oxygen atom is carried through. From the single-determinant wave function for the ${}_{}{}^3{}_{}{}^{}P$ ground state of this atom the Coulomb potential is developed, and the important parts of the potential for the $s$- and $d$-wave portions of the free-electron wave function are considered. The relevant Schrödinger equations are solved by means of an IBM 704 program which is written so as to be readily adaptable to electrons in the presence of other atoms or ions having electrons through $2p$. In our oxygen example the wave function calculations were carried out for values of the free-electron linear momentum ranging from 0.01 through 0.80 atomic unit.