Geometries and vibrational frequencies with Kohn-Sham methods using $\sigma$-functionals for the correlation energy

Recently, Kohn-Sham (KS) methods with new correlation functionals, called $\sigma$-functionals, have been introduced. Technically, $\sigma$-functionals are closely related to the well-known random phase approximation (RPA), formally, $\sigma$-functionals are rooted in perturbation theory along the adiabatic connection. If employed in a post-SCF (post self-consistent field) manner in a Gaussian basis set framework, then $\sigma$-functional methods are computationally very efficient. Moreover, for main group chemistry $\sigma$-functionals are highly accurate and can compete with high-level wave-function methods. For reaction and transition state energies, e.g., chemical accuracy of 1 kcal/mol is reached. Here, we show how to calculate first derivatives of the total energy with respect to nuclear coordinates for methods using $\sigma$-functionals and then carry out geometry optimizations for test sets of main group molecules, transition metal compounds, and non covalently bonded systems. For main group molecules, we additionally calculate vibrational frequencies. $\sigma$-Functional methods are found to yield very accurate geometries and vibrational frequencies for main group molecules superior not only to those from conventional KS methods but also to those from RPA methods. For geometries of transition metal compounds, not surprisingly, best geometries are found for RPA methods, while $\sigma$-functional methods yield somewhat less good results. This is attributed to the fact that in the optimization of $\sigma$-functionals transition metal compounds could not be represented well due to the lack of reliable reference data. For non-covalently bonded systems $\sigma$-functionals yield geometries of the same quality as the RPA or as conventional KS schemes combined with dispersion corrections.