Variational grand-canonical electronic structure of Li+Li at ~10⁴ K with second-order perturbation theory corrections

An ab initio variational grand-canonical electronic structure mean-field method, based on the Gibbs-Peierls-Bogoliubov minimum principle for the Gibbs free energy, is applied to the di-lithium (Li+Li) system at temperatures around T ≈ 10⁴ K and electronic chemical potential of μ ≈ -0.1E_h. The method is an extension of the Hartree-Fock approach to finite temperatures. We first study the Li₂ molecule at a frozen inter-nuclear distance of R = 3 Å as a function of temperature. The mean-field electronic structure changes smoothly as temperature increases, up to 10⁴ K, where a sharp spontaneous spin-polarization emerges as the variational mean-field solution. Further increase in the temperature extinguishes this polarization. We analyze the mean-field behavior using a correlated single-site Hubbard model and show it arises from an attempt of the mean-field to mimic the polarization of the spin-spin correlation function of the exact solution. Next, we keep constant the temperature at 10⁴ K and examine the electronic structure as a function of inter-nuclear distance R. At R = 3.7 Å, a crossing between two free energy states occurs: One state is "spin-unpolarized" (becomes lower in energy when R > 3.7 Å), while the other is "spin polarized". This crossing causes near-discontinuous jumps in calculated properties of the system and is associated with using the noninteracting electron character of our mean-field approach. Such problems will likely plague FT-DFT calculations as well. We use second-order perturbation theory (PT2) to study effects of electron correlation on the potential of mean force between the two colliding Li atoms. We find that PT2 correlation free energy at ~10⁴ K is larger than at 0 K and tends to restore the spin-polarized state as the lowest free energy solution.