State-Specific Analytical Nuclear Gradients for the Density-Fitted Exact Two-Component State-Averaged Complete Active Space Self-Consistent Field (X2C-SA-CASSCF) Theory

Abstract When a chemical system of interest contains heavy elements, it is important to obtain a balanced description of relativistic effects and electron correlation. The exact two-component (X2C) Hamiltonian includes both scalar-relativistic and spin–orbit effects with a reasonable computational cost. By using the spin–orbit X2C Hamiltonian in state-averaged complete active space self-consistent field (SA-CASSCF) calculations, it is possible to obtain equivalent descriptions of the states with different spin characters. For molecular geometry optimizations within this framework, the nuclear gradients should be efficiently evaluated. In this work, we describe a state-specific analytical gradient algorithm for X2C-SA-CASSCF theory with the density-fitting approximation. To realize such calculations, we formulate and implement the response (Z-vector) equation for X2C-SA-CASSCF theory. Using the obtained state-specific nuclear gradients, we optimize the equilibrium geometries of metal porphyrins and the singlet–singlet and singlet–triplet minimum energy crossing points or minimum energy conical intersections of 2-thiouracil.