Analytic second derivatives for the spin-free exact two-component theory

The formulation and implementation of the spin-free (SF) exact two-component (X2c) theory at the one-electron level (SFX2c-1e) is extended in the present work to the analytic evaluation of second derivatives of the energy. In the X2c-1e scheme, the four-component one-electron Dirac Hamiltonian is block diagonalized in its matrix representation and the resulting "electrons-only" two-component Hamiltonian is then used together with untransformed two-electron interactions. The derivatives of the two-component Hamiltonian can thus be obtained by means of simple manipulations of the parent four-component Hamiltonian integrals and derivative integrals. The SF version of X2c-1e can furthermore exploit available nonrelativistic quantum-chemical codes in a straightforward manner. As a first application of analytic SFX2c-1e second derivatives, we report a systematic study of the equilibrium geometry and vibrational frequencies for the bent ground state of the copper hydroxide (CuOH) molecule. Scalar-relativistic, electron-correlation, and basis-set effects on these properties are carefully assessed.