Electric fields vibrational hyperpolarizabilities

The CCSD model gives for static and frequency-dependent hyperpolarizabilities usually results close to the experimental values, provided that the effects of vibrational averaging and the pure vibrational contributions have been accounted for. Zero point vibrational corrections for the static and the electric field induced second harmonic generation (ESHG) hyperpolarizability of methane have recently been calculated by Bishop and Sauer using SCF and MCSCF wavefunctions [51]. 

The vibrationally resonant hyperpolarizability of a molecule can be described with the following expression obtained utilizing perturbation theory [17] (assuming that the only interactions between the electric fields and the media are dipolar interactions). 

Apart from purely electronic effects, an asymmetric nuclear relaxation in the electric field can also contribute to the first hyperpolarizability in processes that are partly induced by a static field, such as the Pockels effect [55, 56], and much attention is currently devoted to the study of the vibrational hyperpolarizability, can be deduced from experimental data in two different ways [57, 58], and a review of the theoretical calculations of p, is given in Refs. [59] and [60]. The numerical value of the static P is often similar to that of static electronic hyperpolarizabilities, and this was rationalized with a two-state valence-bond charge transfer model. Recent ab-initio computational tests have shown, however, that this model is not always adequate and that a direct correlation between static electronic and vibrational hyperpolarizabilities does not exist [61]. 

One of the hurdles in this field is the plethora of definitions and abbreviations in the next section I will attempt to tackle this problem. There then follows a review of calculations of non-linear-optical properties on small systems (He, H2, D2), where quantum chemistry has had a considerable success and to the degree that the results can be used to calibrate experimental equipment. The next section deals with the increasing number of papers on ab initio calculations of frequency-dependent first and second hyperpolarizabilities. This is followed by a sketch of the effect that electric fields have on the nuclear, as opposed to the electronic, motions in a molecule and which leads, in turn, to the vibrational hyperpolarizabilities (a detailed review of this subject has already been published [2]). Section 3.3. is a brief look at the dispersion formulas which aid in the comparison of hyperpolarizabilities obtained from different processes. 

To obtain hyperpolarizabilities of calibrational quality, a number of standards must be met. The wavefunctions used must be of the highest quality and include electronic correlation. The frequency dependence of the property must be taken into account from the start and not be simply treated as an ad hoc add-on quantity. Zero-point vibrational averaging coupled with consideration of the Maxwell-Boltzmann distribution of populations amongst the rotational states must also be included. The effects of the electric fields (static and dynamic) on nuclear motion must likewise be brought into play (the results given in this section include these effects, but exactly how will be left until Section 3.2.). All this is obviously a tall order and can (and has) only been achieved for the simplest of species He, H2, and D2. Comparison with dilute gas-phase dc-SHG experiments on H2 and D2 (with the helium theoretical values as the standard) shows the challenge to have been met. 

The SFG technique probes the second-order nonhnear hyperpolarizability tensor this tensor includes the Raman and IR susceptibihty, which requires that the molecular vibrational modes are both Raman and IR active. Since Raman- and IR-dipole moment transition selection rules for molecules with a center of symmetry indicate that a vibrational mode is either Raman or IR active but not both, only molecules in a non-centrosymmetric environment on the surface interact with the electric fields molecules in the isotropic bulk phase show inversion symmetry where the third rank hyperpolarizability tensor goes to zero [25-27]. 

Perpete, Quinet and Champagne have shown that vibrational contributions to the y-hyperpolarizability of various symbioticaUy substituted quadrupolar JT-conjugated molecules are large. Using ab initio calculations at the HF level they calculate the response functions for the dc-Kerr effect, degenerate four wave mixing, hyper-Raman effect and electric field induced SHG. 

Bishop, D. M. and Sauer, S. P. A. (1997). Calculation, with the inclusion of vibrational corrections, of the DC-electric-field induced second-harmonic-generation hyperpolarizability of methane. J. Chem. Phys., 107, 8502-8509. 

The difference equation or numerical integration method for vibrational wavefunctions usually referred to as the Numerov-Cooley method [111] has been extended by Dykstra and Malik [116] to an open-ended method for the analytical differentiation of the vibrational Schrodinger equation of a diatomic. This is particularly important for high-order derivatives (i.e., hyperpolarizabilities) where numerical difficulties may limit the use of finite-field treatments. As in Numerov-Cooley, this is a procedure that invokes the Born-Oppenheimer approximation. The accuracy of the results are limited only by the quality of the electronic wavefunction s description of the stretching potential and of the electrical property functions and by the adequacy of the Born-Oppenheimer approximation. 

The experimental measures of these molecular electric properties involve oscillating fields. Thus, the frequency-dependence effects should be considered when comparing the experimental results . Currently, there are fewer calculations of the frequency-dependent polarizabilities and hyperpolarizabilities than those of the static properties. Recent advances have enabled one to study the frequency dispersion effects of polyatomic molecules by ab initio methods In particular, the frequency-dependent polarizability a and hyperpolarizability y of short polyenes have been computed by using the time-dependent coupled perturbed Hartree-Fock method. The results obtained show that the dispersion of a increases with the increase in the optical frequency. At a given frequency, a and its relative dispersion increase with the chain length. Also, like a, the hyperpolarizability y values increase with the chain length. While the electronic static polarizability is smaller than the dynamic one, the vibrational contribution is smaller at optical frequencies. ... 

---