Bond Dipoles and Higher-Order Susceptibilities

Polarizations of higher order in the applied field have become important recently because of their use in generating harmonic signals with lasers and because lasers 

For a complete description of the theory and a detailing of its complexities, see Flytzanis (1975). 

The procedure here will be to compute the dipole to be associated with a bond, proportional to h x 6 , multiply it by eS to obtain a correction to the bond energy, and again minimize the energy to obtain the equilibrium bond dipole in the presence of a field. This will give an alternative derivation of Eq. (4-28), in fact, the way the formula was first derived (Harrison, 1973). It will then also directly give terms of higher order in the electric field. Note that here the field is independent of time, so that x = Zi(0)isreal wecandrop the subscript and parentheses and write X- 

It is desirable here to change to vector notation, writing the vector distance from the nucleus of atom 1 to that of atom 2 as d, so that this result becomes 

This gives the center of gravity of the bond electron density relative to the bond center. The corresponding shift in electron energy due to an electric field S is 

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This perturbation-theoretic formulation of the dielectric susceptibility is most appropriate and will be used in the treatment of dielectric properties based upon pscudopotcntials in Chapter 18. However, for the calculation of higher-order susceptibilities, a more direct approach in terms of bond dipoles is more convenient. Because the two derivations are equivalent, the bond-dipole approach will also enable us to establish parameters for the bond dipoles and effective charges in terms of the parameters listed in Table 4-1. 

Since the field is not assumed to be small, the result may be summed over bonds and used to calculate higher-order susceptibilities. Note that if the field is taken equal to zero, the dipole eyd p summed over the four bonds surrounding each atom is zero. The value to first order in the field is... 

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