Hyperpolarizability tensor perturbation

The result is a perturbation expression for the components of the hyperpolarizability tensor which can be written as... 

One should point out, however, that the described procedures only pertain to an early stage in the theory of polarizabilities and hyperpolarizabilities of polymers. Further theoretical research is required to solve the problem of the unbounded perturbation operator in equation (10.35) which blocks direct CO calculations. Moreover, the Rombeig algorithm could be employed not only to test the accuracy of the numerical differentiation (10.41), but also to extrapolate the a (and other polarizability and hyperpolarizability tensor element) values obtained for a few terms in a series of similar molecules for large n(n- oo). Our first results for the band structure of a (-H-H H-H-) chain have been very promising in this respect. ... 

A computer program for the theoretical determination of electric polarizabilities and hyperpolarizabilitieshas been implemented at the ab initio level using a computational scheme based on CHF perturbation theory [7-11]. Zero-order SCF, and first-and second-order CHF equations are solved to obtain the corresponding perturbed wavefunctions and density matrices, exploiting the entire molecular symmetry to reduce the number of matrix element which are to be stored in, and processed by, computer. Then a /j, and iap-iS tensors are evaluated. This method has been applied to evaluate the second hyperpolarizability of benzene using extended basis sets of Gaussian functions, see Sec. VI. 

Table 2 lists the results of calculations of the static first hyperpolarizability. As the perturbation theory expansion shows, for w = 0, Kleinman symmetry holds exactly and the components of the tensor are invariant under permutations of the three co-ordinate indices so that Pz as defined in eqn (4.12) reduces to,... 

In the perturbation theory for degenerate states the resonant hyperpolarizability is determined by the tensor part of polarizability [9] and may be extracted out of the fourth-order terms self-consistently in the case of nondegenerate perturbation theory the resonant part appears for separate sublevels of an atomic multiplet. The numerical results are listed in Table 2. 

TDHF [52, 53] is one of the most widely-employed ab initio techniques to evaluate nonlinear-optical response tensors. The fDHF approach is size consistent but cannot account for the finite lifetime of the excited states. The matrices of the TDHF equation are expanded in a Taylor series of the perturbation due to the static and/or dynamic electric fields and are solved for each order [52, 53], The so-obtained successive field-derivatives of the density matrix are then inserted into the expressions for the hyperpolarizability,... 

Here, pa and ma are the electric and magnetic dipole moment functions, aap, Papy> YapyS 81 the polarizability and first and second hyperpolarizabilities, is the magnetizability, and, of the other terms, only the hypermagnetizability will be of interest (it relates to the Cotton-Mouton effect). The Greek subscripts a, p,... denote vector or tensor quantities and can be equal to the Cartesian coordinates x, y, or z. Einstein summation over these subscripts is implied both here and elsewhere. Differentiation of this expression with respect to F (or B) leads to an expression for the dipole moment (or polarization) of the species in the presence of the perturbing fields and it is clear that P, y, t), etc. will govern the non-linear terms in the induced electric (or magnetic) dipole moment - hence, non-linear optics. 

Bartkowiak and Misiaszek used the eoupled perturbed Hartree-Fock method and the sum-over-modes formalism to ealeulate the eleetronie and vibrational /i-tensors for 4-nitroaniline, 4-nitro-4 -aminostilbene, 4-amino-4 -nitrobiphenyl and 4-amino-4 -nitrophenylacetylene, all typieal push-pull eonjugated molecules of the kind that have been associated with seeond order optieal non-linearities derived from their large P values. Their ealeulations refer to the gas phase and to chloroform and aqueous solutions, the solvent effects being included through a continuum self-consistent reaction field model. They demonstrate that the solvent effects are much greater for the vibrational hyperpolarizability than for the electronic contribution. 

---