Exchange Potential for Nearly Free Electrons

The approximation of the exchange potential which varies as the cube root of the density is examined. The optimum value of the coefficient in this expression, i.e., the value which minimizes the total energy of the system, is determined for a sinusoidal perturbation upon a system of free electrons as a function of the wavelength of the perturbation. For very long waves the coefficient assumes the value proposed by Gaspar. However, over a range of wavelengths which, in relation to the Fermi energy, are more appropriate for conduction electrons in a metal, the coefficient increases, well beyond Slater's original value, and then experiences a fairly sharp cutoff. Beyond this range the coefficient tends to vanish for decreasing wavelengths. A higher-order approximation incorporating this characteristic is also presented.