A new variant of the nonempirical valence bond method, which uses a set of numerical Hartree-Fock atomlike orbitals as its basis, is proposed for calculating the electronic structure of diatomic molecules. The method is based on using the procedure of configuration interaction for various covalent and ionic configurations of the molecule. As a many-electron basis, a set of Slater determinants constructed from nonorthogonal orbitals of the atomic type localized at different centers is employed. This approach is designed for calculations of molecules containing heavy atoms and is an all-electron method, which does not require the introduction of a pseudopotential or an effective core potential to eliminate core states. The use of the numerical Hartree-Fock basis is especially efficient in calculations of such physical characteristics as densities at a nucleus, hyperfine splitting parameters, energies of X-ray transitions, and chemical shifts. The results of calculations of spectral characteristics for the AgH molecule and AgH⁺ ion are presented.