Simplified procedure for calculation and parametrisation of tight binding integrals

The two-center integrals ∫ψ*_l(r - d) V(r) ψ_m(r) d³r for vanadium atoms are estimated by expanding ψ_l(r - d) around r = 0. For typical metallic densities, the lowest order term approximates the two-center integrals to about 15 per cent. Thus, within this accuracy, the two-center integrals can be expressed as a product of a one-center integral ∫u(r) V(r) r⁴dr where u(r) is the radial wave function, and other simple numbers. Moreover, the one-center integral is an atomic property, which does not depend strongly on the lattice, nor on the distance between the atoms for which the two-center integrals are evaluated. The various two-center integrals are evaluated for the intermetallic compound V₃Ga, and compared with the effective integrals determined from Mattheiss' APW calculation. Most large two-center integrals approximate the effective integrals to within about 20 per cent and for the few cases where the discrepancy is greater, it is found to be due to three-center integrals. These can also be evaluated to a good accuracy by a similar expansion procedure. By treating the exponent Q of the asymptotic form ( 1 r)e^-Qr of the radial part of the wavefunction and the one-center integrals as adjustable or partly adjustable parameters, four parameters fit the main effective integrals to within 10-20 per cent.