In this work, we use Lebedev grids to perform the numerical integration. The second option consists in computing the σ, χ and τ tensors analytically as derivatives of the Helmholtz free energy and using Eq. ( ) to obtain the shift. For σ and χ, which are second derivatives of the free energy, the analytical sum-over-states equations (Van Vleck, Van den Heuvel-Soncini) are well-known. In order to calculate the τ tensor, we generalized our previous rederivation of the Van den Heuvel-Soncini equation (appendix of Reference 27) to obtain sum-over-states expressions for analytical third and fourth derivatives of the free energy.