Analytical Gradient Theory for Density-Fitted Exact Two-Component Hartree–Fock, State-Specific Complete Active Space Self-Consistent Field, and Second-Order Møller–Plesset Perturbation Theories

The exact two-component (X2C) relativistic quantum chemistry calculations can be used to describe scalar relativistic effects and spin-orbit couplings at reasonable computational cost. However, they have limited applicability to wave function-based quantum chemistry methods, particularly geometric optimizations and dynamics simulations, owing to the high computational demands of these methods in sizable molecular systems. In this work, we report our implementation of an analytical gradient algorithm with a density-fitting approximation for Hartree-Fock, state-specific complete active space self-consistent field (CASSCF), and second-order Møller-Plesset perturbation theory (MP2) calculations with the X2C one-electron Hamiltonian. This implementation uses a second-order orbital optimization scheme to facilitate convergence in X2C-CASSCF calculations, as well as a response (<i>Z</i>-vector) equation for evaluation of the X2C-MP2 nuclear gradient. We demonstrate the applicability of the algorithm for optimization of the geometry of Ir(ppy)₂(bpy)⁺ and evaluate its computational cost and parallelization (multithreading) efficiency.