Polarization expansion

This tutorial deals with nonlinear optical effects associated with the first nonlinear term in expression for the polarization expansion described in the next section. The first nonlinear term is the origin of several interesting and important effects including second-harmonic generation, the linear electrooptic or Pockels effect,... 

The approaches to this problem follow along two general lines. In the first approach, one computes derivatives of the dipole moment with respect to the applied field and relates them to the terms in the polarization expansion of equation 8. Inspection of equation 8 suggests that the second derivative of the dipole moment with respect to the field gives p. The choice of the exact form of the Hamiltonian, which incorporates the optical field and the atomic basis set, determines the accuracy of this procedure. In one popular version of this approach, the finite field method, the time dependence of the Hamiltonian is ignored for purposes of simplification and the effects of dispersion on p, therefore, cannot be accounted for. 

R is I l, v4 + 1 s,. Obviously, < >, = 1. v, is not a good approximation to The component lsB due to the perturbation V is as large as the unperturbed function itself. Hence, the operator V cannot be considered as a small perturbation. Since the polarization wave functions for this system are all localized at the nucleus A, i.e. they decay exponentially with the distance from the nucleus A, the polarization expansion can possibly recover the component of P, localized on the nucleus B, only in very high orders. 

Although the polarization expansion cannot be used to compute the interaction energies of weakly bound complexes, it does provide the correct asymptotic expansion of the interaction energy in the following sense84... 

Table 1-1. Convergence of the polarization expansion for the interaction of two ground-state helium atoms at R = 1 and 5.6 bohr, and of the lithium and hydrogen atoms in their ground states at R = 10 and 12 bohr. The Coulomb energies represent 53.50% (He2) R = 5.6bohr), 73.4% (LiH, R = 10 bohr), and 85.53% (LiH, R= 12 bohr) of the energies of the fully symmetric (Pauli forbidden) states. The quantity S(n) represents the percent error of the perturbation series through the nth-order with respect to the variational interaction energy of the Pauli forbidden state...

Extensive numerical calculations for small systems (H2 (Ref. 79,87), H2 (Ref. 89,90) and He2 Ref. (81) show that the convergence properties of the SRS theory are excellent. Since in most cases the polarization expansion is divergent, one can expect that for many-electron monomers the SRS expansion will not be strictly convergent. However, the experience gained thus far for large many-electron systems suggests that a second-order SRS calculation correctly accounts for all major polarization... 

Cwiok T, Jeziorski B, Kolos W, Moszynski R, Rychlewski J, Szalewicz K (1992) Convergence properties and large-order behavior of the polarization expansion for the interaction energy of hydrogen atoms. Chem Phys Lett 195 67-76... 

The manifestation of NLO behavior can easily be seen by substituting a sinusoidal field equation into the polarization expansion equation [Eq. (22)]. This gives... 

The starting wave function (Po = used in the polarization expansion is not anti-... 

Chipman,D. M., ScHirschfelder, J. O. (1980). On symmetry in the polarization expansion for intermolecular forces. Journal of Chemical PhysicSy 73y 5164. 

When one of the monomers has three or more electrons, the situation is quite different. Group theoretical analysis and noncrossing rule-type arguments show that the polarization expansion cannot converge in such a case, at least for sufficiently large distances R. These theoretical arguments were recently confirmed by Adams in a numerical study of the... 

Table 3 Convergence of the regularized polarization expansion for J = 11.5 bohr and both regularizations (Gaussian and SNC). The numbers listed are percent errors of the sum of the first n R-RS corrections with respect to 6 p(/j) (gp( )). The values of gp (and of the Coulomb energy Q for = oo), given in microhartrees, are displayed in the row marked gp, whereas the last row lists convergence radii of the regularized polarization series ...

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