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Analytic energy gradients and Hessians of exact two-component relativistic methods: Efficient implementation and extensive applications.

The algebraic exact two-component (X2C) relativistic Hamiltonian can be viewed as a matrix functional of the decoupling (X) and renormalization (R) matrices. It is precisely their responses to external perturbations that render X2C-based response theories different in form from the nonrelativistic counterparts. However, the situation is not really bad. Sticking to the energy gradients, it can be shown that the nuclear derivatives of X and R (Xμ and Rμ, respectively) can be transformed away to favor transformed, nucleus-independent density matrices, viz. the X2C energy gradients can be written in a form that does not depend explicitly on Xμ and Rμ. Further combined with the storage of quantities that are already available in the energy calculation, only 35 matrix multiplications are needed to construct the one-electron (relativistic) part of the X2C gradients, thereby rendering the gradient calculations very efficient. More efficiency can be gained by approximating the molecular X as the superposition of the atomic ones (denoted as X2C/AXR) and by further approximating the molecular R also as the superposition of the atomic ones (denoted as X2C/AU): The numbers of matrix multiplications required for constructing the one-electron (relativistic) parts of the X2C/AXR and X2C/AU gradients are reduced to 18 and 4, respectively. Similar approximations can also be applied to the X2C Hessian. It will be shown numerically that the X2C/AXR gradients and Hessians are extremely accurate (almost indistinguishable from the full X2C ones), whereas the X2C/AU ones do have discernible errors but which are tolerable in view of the dramatic gain in efficiency.