Transverse-Coulomb Term

In the region where k a we may safely neglect the binding energies in the denominators of the second-order perturbation theory and thus simplify the integrand. After integration one obtains 

Let us emphasize once more that, as was discussed above, the high frequencies in this formula are effectively cut at the characteristic bound state 

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To order e2 the Coulomb interaction term contributes to the first-order term in the 5-matrix expansion, i.e., to fd4xJffin(x), and the term contributes to second-order. Diagrammatically, we have illustrated these contributions in Fig. 11-1 ( stands for Coulomb, / for transverse photons). To the order indicated, the part of the 5-matrix contributing to the process is... 

The calculated Rayleigh mode (SJ, the lowest lying phonon branch, is in good agreement with the experimental data of Harten et al. for all three metals. Due to symmetry selection rules the shear horizontal mode just below the transverse bulk band edge can not be observed by scattering methods. The mode denoted by Sg is the anomalous acoustic phonon branch discussed above. Jayanthi et al. ascribed this anomalous soft resonance to an increased Coulomb attraction at the surface, reducing the effective ion-ion repulsion of surface atoms. The Coulomb attraction term is similar for all three metals... 

Here, square brackets indicate functional dependence, and V,[P ] is the molecule-field interaction operator linear in the electronic charge e for a given transverse polarization field P (essentially equation (3)). V,[0] corresponds to the usual p.A terms in the Coulomb gauge theory (Div A = 0). It is related directly to the discussion presented here since, for example, the cross-section for absorption of a photon with wavevector k and polarization direction e (a = x,y) that causes an atomic transition n m is obtained from the Coulomb gauge theory as proportional to. 

If one neglects the transverse contribution, one arrives at what is termed the Dirac-Coulomb approximation (a standard in quantum chemistry). Inclusion of the transverse term, which describes retardation and magnetic effects, in perturbation theory (weakly relativistic hmit) leads to the Dirac-Coulomb-Breit Hamiltonian. 

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