  We employ \textit{ab initio} methods of quantum chemistry to investigate
spin-1/2 fermions interacting via a two-body contact potential in a
one-dimensional harmonic trap. The convergence of the total energy with the
size of the one-particle basis set is analytically investigated for the
two-body problem and the same form of the convergence formula is numerically
confirmed to be valid for the many-body case. Benchmark calculations for two to
six fermions with the full configuration interaction method equivalent to the
exact diagonalization approach, and the coupled cluster method including
single, double, triple, and quadruple excitations are presented. The
convergence of the correlation energy with the level of excitations included in
the coupled cluster model is analyzed. The range of the interaction strength
for which single-reference coupled cluster methods work is examined. Next, the
coupled cluster method restricted to single, double, and noniterative triple
excitations, CCSD(T), is employed to study a two-component Fermi gas composed
of 6 to 80 atoms in a one-dimensional harmonic trap. The density profiles of
trapped atomic clouds are also reported. Finally, a comparison with
experimental results for few-fermion systems is presented. Upcoming possible
applications and extensions of the presented approach are discussed.

Quantum Gases;Atomic Physics;Quantum Physics