Navigating the Electronic Grand Potential Surface with Analytical Nuclear Derivatives of Grand-Canonical Density-Functional Theory

We present an implementation of analytical nuclear gradient and hessian for grand-canonical density-functional theory (GC-DFT) in the Kohn-Sham formulation. Despite the fact that the average occupation numbers of orbitals change with the geometric structures of molecules, it is shown that evaluation of nuclear gradient of GC-DFT can be formulated in a way similar to that of nuclear gradient of micro-canonical (μC) DFT, without the need for the knowledge of the changes in the occupation numbers. On the other hand, in contrast to that of μC-DFT, the nuclear hessian of GC-DFT encompasses two components concerning both the fixed and variable occupation numbers, as a result of the chain rule of differentiation. We have developed three techniques, namely the non-idempotent (NI) coupled-perturbed self-consistent field (CPSCF), the occupation-gradient (OG) CPSCF and the occupation-fluctuation (OF) CPSCF for those two components. The analytical nuclear derivatives are verified via comparison with numerical results given by the finite-difference method. The application of the nuclear derivatives of GC-DFT is demonstrated with an example.