A new scheme combining the finite element method and the basis set exp
ansion method in the framework of the finite element approach has been
developed. The factors which affect the numerical accuracy of the sol
ution of Hartree-Fock-Slater equation have been investigated, and the
total energies of H-2(+), H-2, LiH, Li-2, BH, N-2 and CO molecules hav
e been calculated. In comparison with the results obtained with the tr
aditional finite element method, it can be found that the new approach
produces more accurate results with less computational efforts.