Single-Point Grand Potential and Nuclear Derivatives of Finite-Temperature Kohn-Sham Density-Functional Theory

We present an implementation of single-point grand potential, nuclear gradient and nuclear hessian for finite-temperature Kohn-Sham density-functional theory (FT-DFT) in the grand canonical ensemble. It is shown that evaluation of single-point grand potential and nuclear gradient of FT-DFT is much similar to that of single-point energy and nuclear gradient of zero-temperature DFT, with no need for nested-loop optimization or approximation. The nuclear hessian is divided into two parts, the fixed-occupation-number component and the variable-occupation-number one, as a result of fractional occupation of molecular orbitals. We have developed two techniques, namely the non-idempotent CPSCF and the occupation-gradient CPSCF for those two components, respectively. The convergence of single-point grand potential is discussed, while the analytical nuclear derivatives are verified via comparison with numerical results.