where E 0 c is the cohesive energy. This relation was used [45] to understand the near-constancy of N (0) < I 2 > in transition metals and their alloys. A modification of this calculation [18] to include the first correction due to the non- orthogonality S of the nearest neighbor tight binding orbitals gives that where the ( ± ) sign holds if the fermi-level falls in the antibonding/bonding part of the band. This reflects merely that the velocity in the antibonding/bonding parts of the band is larger/smaller approximately by (1 ± S ). We wish to derive [15] the observed relation shown in Fig.( ?? between < I 2 > and the average lattice stiffness M < ω 2 > . The phonon frequencies ω$_{q}$ are renormalized from their rigid-ion values Ω$_{q}$ due to the creation of particle-hole pairs created by the lattice deformation: where Π( q, 0) is the self-energy and χ ( k, k + q ) is the electronic polarizability at zero frequency. The average rigid-ion stiffness is given by the second-derivative of the cohesive energy of the lattice: Contributions to E ( R ) come from (1) The sum of the one-electron (band-structure energy) discussed above and (2) core-core repulsion energy minus the electron-electron repulsion energy, the latter having been counted twice in a self-consistent band-structure energy. For any long-wavelength property (2) sums to zero and it is negligible for consideration of the average stiffness compared to that of (1) because it is slowly varying with inter-site separation while (1) varies on the scale of the variation of the overlap integrals. In fact chemists calcualting the structure and vibration frequency of metals have found that good results are obtained by neglecting it altogether. For transition metals and compounds, we may note further the justification that about 60 − 80% of the cohesive energy is provided 57 M < Ω 2 > = ∑ α ∂ $^{2}$E ∂R 2 $_{α}$| R = R$_{0}$ (56) Π( q, 0) = ∑ k | g$_{k,k}$$_{+}$$_{q}$ | $^{2}$χ ( k, k + q ) (55) ω 2 q ≈ Ω 2 q + 2 ω$_{q}$ Π( q, 0) (54) N (0) < I 2 > ≈ (1 ± S ) E 0 $_{c}$/r 2 $_{0}$, (53) 2. Relation of average electron-phonon scattering to average lattice stiffness