UC Irvine

It has been known for decades that Thomas-Fermi theory becomes exact in the semiclassical limit. The corrections to Thomas-Fermi theory in this limit are much more difficult to derive. This thesis describes the successful derivation of these post-TF corrections for two systems which admit WKB series approximations. These corrections are kinetic and total energy potential functionals which can be used in orbital-free calculations. I also describe the history of this research, which goes back decades, and earlier failed attempts to find these corrections. These potential functionals are derived from an asymptotic semiclassical series which I work out to all orders for the pure quartic oscillator. I use the quartic oscillator to show how subdominant corrections to this asymptotic series can yield more accurate approximations. I also give the exact eigenvalues, to 40 digits, for potentials with quartic and harmonic pieces which should be useful for testing one dimensional models.