Electron-Nuclear Wave Functions in Multiphonon Processes

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Abstract
For most purposes, the wave functions of a crystal with an electron trapped at an impurity are approximated by the product of two functions, one involving the lattice coordinates and the other the position of the electron. The electronic function may be determined for the equilibrium position of the lattice (static approximation) or one may assume that the electronic wave function continuously adjusts itself to the instantaneous position of the lattice (adiabatic approximation). The author relates the wave functions to Hamiltonian operators which do not have some of the most common operational properties. A comparison of the two approximations is made using the variational principle. The static underestimates the kinetic energy and overestimates the potential energy, while the second does just the reverse. Although the formal treatment is quite general and includes all the effects for harmonic vibrations, the actual terms were evaluated for a simple model. The calculations show that for extremely shallow traps the static approximation may be slightly superior, while for deep traps the adiabatic approach should be used. For traps of depths less than 0.1 ev the methods are essentially equivalent; in any actual calculations, other approximations must be made and they are of greater importance than the slight difference between either of these approaches.

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