On the Exchange Interaction in the Collective Electron Approximation

A brief discussion of the previous work on the subject is given. A general expression for the exchange integral in the collective electron approximation is obtained in terms of the Fourier coefficients of the exchange charge density. These coefficients are evaluated for a particular case of tightly bound electrons and it is shown that the exchange integral is a strong function of the difference of the wave vectors characterizing the two states between which exchange is considered. The significance of this result is discussed and the inadequacies of the collective electron approximation in dealing with problems involving exchange are pointed out.