Comparison of Screened Exchange with the Slater Approximation for Silicon

The exchange self-energy in silicon is calculated using the dynamic Penn dielectric function and pseudopotential wave functions. Comparison is made with Slater's ${\rho }^{\frac{1}{3}}$ approximation, and it is concluded that errors of the order of 0.5 to 2 eV result. The static dielectric function gives 15% errors in the average self-energy but only 2-3% errors in the exchange-gap enhancements.