  Motivated by the prospect of optical lattice experiments with two-component
Fermi gases consisting of different atomic species such as Li and K, we
calculate the energies for N fermions under harmonic confinement as a function
of the mass- and trap-imbalance, i.e., as a function of the ratio between the
masses and frequencies of species one and two, using microscopic approaches.
Our energies for N=2 through 6 can be used to determine the energetically most
favorable configuration for a given number of atoms per species of a deep
lattice in which each lattice site is approximately harmonic and in which
tunneling between neighboring sites can be neglected. Furthermore, our energies
determine one of the input parameters, namely the onsite interaction strength,
of the corresponding lattice Hamiltonian. We also determine and interpret the
excitation gap for unequal-mass systems with up to N=13 atoms for equal
oscillator lengths.
