Hyperpolarizability zero-point vibrational average

Quinet, O., Kirtman, B., Champagne, B. Analytical time-dependent Hartree-Fock evaluation of the dynamic zero-point vibrationally averaged (ZPVA) first hyperpolarizability. J. Chem. Phys. 118, 505-513 (2003)... 

In the second part of this section (3.1.2.), with one exception, I will limit my survey to other calculations which have used ab initio techniques to determine frequency-dependent hyperpolarizabilities. It is unfortunate that, again with one exception, none of these calculations takes account of nuclear vibrations, not even to the extent of zero-point vibrational averaging (i.e. a fixed nuclear geometry is assumed). Any close agreement with experiment, which doesn t happen often, must therefore be considered coincidental. To redress (somewhat) the balance of this section, I will also report on an excellent paper dealing with a set of non-ab-initio calculations. 

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. 

There are two ways in which molecular vibrations affect non-linear optical properties. The first, which is well understood, is zero-point-vibrational averaging of the calculated electronic properties. This need not delay us long. The second comes about from the effect that the electromagnetic radiation has on the vibrational motions themselves and this leads to the vibrational polarizabilities and hyperpolarizabilities which are the exact counterparts of the electronic ones which stem from the effect that the radiation has on the electronic motions. This phenomenon is now receiving long overdue attention and will be the main subject of this section. A more extensive review is available elsewhere [2]. 

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]. 

---