PHYCS 498CQM: Homework 3

1. Problem constructed by Markus Holzmann on the self-consistent Gross-Pitaevski equation for Bose condensation.
2. In a Hartree-Fock or DFT program one generally needs a starting guess for the wavefunctions or the density. One choice are the variational hydrogenic-like orbitals described in Thijssen Ch 3. More explicitly, Koonin (Project 3) and the description of the method, p. 76-77, describes trial orbitals that have the form of hydrogenic orbitals with an effective charge. In the program a way to find the effective charge ZSTAR is given at the bottom of page 435. Work out analytically why this method to find the optimal hydrogenic starting orbital is the one that satisfies the virial theorem and simultaneously is the gives the minimum total energy for this class of orbitals.
3. Using the class atomic codes carry out HF calculations for H, He, C, and Ne. For H and C treat both spin-restricted and spin-unrestricted, and compare the results for the total energies.
4. Using the class DFT atomic codes (presently limited to spin-independent DFT) carry out calculations for H, He, C, and Ne. Compare with HF and the NIST site that lists DFT results (see link on class notes.) (The output of the class code at present lists "exchange energy" that really is "exchange correlation energy". Compare this number specifically with the NIST site for each element.

EXTRA - NOT Required
5. Using the GAUSSIAN code carry out calculations for these elements, comparing A) different basis sets, and B) HF, LDA, and The BLYP gradient approximation (GGA) functional. Compare with the class codes that solve the problem more accurately on a radial grid.