Virial relations for ultracold trapped Fermi gases with finite range interactions through the Bardeen-Cooper-Schrieffer Bose-Einstein-condensate crossover

We study the virial relations for ultracold trapped two-component Fermi gases in the case of short finite range interactions. Numerical verifications for such relations are reported through the Bardeen-Cooper-Schrieffer (BCS) Bose-Einstein-condensate (BEC) crossover. As an intermediate step, it is necessary to evaluate the partial derivatives of the many-body energy with respect to the inverse of the scattering length and with respect to the interaction range. Once the binding energy of the formed molecules in the BEC side is subtracted, the corresponding energy derivatives are found to have extreme values at the unitary limit. The value of the derivative with respect to the potential range in that limit is large enough to yield measurable differences between the total energy and twice the trapping energy unless the interacting system is described by extremely short potential ranges. The virial results are used to check the quality of the variational wavefunction involved in the calculations.