Illustrating the universal relations for a trapped four-fermion system with arbitrary s-wave scattering length

A two-component four-fermion system with equal masses, interspecies s-wave scattering length $a_s$, and vanishing intraspecies interactions under external spherically symmetric harmonic confinement is considered. For all calculations we employ a correlated Gaussian basis set expansion approach. We determine the energies and various structural properties of the energetically lowest-lying gaslike state throughout the crossover for various ranges of the underlying two-body potential. Extrapolating to the zero-range limit, our numerical results show explicitly [1] that the total energy, the trap energy, as well as certain aspects of the pair distribution function and of the momentum distribution are related through the so-called integrated contact intensity I($a_s$) [2]. Furthermore, it is shown explicitly that the total energy and the trap energy are related through a generalized virial theorem [3] that accounts for a nonzero range.\\[4pt] [1] D. Blume, K.M. Daily, Phys. Rev. A \textbf{80}, 053626 (2009).\\[0pt] [2] S. Tan, Ann. Phys. \textbf{323}, 2952 (2008); 2971 (2008); 2987 (2008).\\[0pt] [3] F. Werner, Phys. Rev. A \textbf{78}, 025601 (2008).