The general results (see previous Part II) on the structure of the discrete spectra of energy operators of neutral systems in a homogeneous magnetic field at a fixed pseudomomentum are proved to be applicable to Hamiltonians of arbitrary atoms. Asymptotic expressions for the discrete spectra of Hamiltonians in the presence of a homogeneous magnetic field are found for arbitrary atoms. This paper completes the investigation of the spectral properties of Hamiltonians of neutral systems in a homogeneous magnetic field at a fixed pseudomomentum. The essential and discrete parts of the spectrum for such systems were found previously; however, whether the theorems in Part II were valid for actual $n$-particle systems remained an open question for the case $n>3$.