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

Despite the rich literature on Born–Oppenheimer molecular dynamics (BOMD) with grand-canonical density-functional theory (GC-DFT), the nuclear derivatives of GC-DFT have rarely been worked out systemically. In this work, we derive the analytical nuclear gradient and Hessian of GC-DFT. It is shown that the nuclear gradient of GC-DFT has a form similar to that of microcanonical (μC) DFT, which confirms the validity of the GC-BOMD results in literature. 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 nonidempotent (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.