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. 

Shortly after the above was written, a paper by Jaszuhski et al. [47] on FH appeared. They used the polarization propagator technique (a Danish tradition ) and calculated the QRF using a multi-configurational-self-consistent-field reference state (the previously unimplemented formalism had been developed by Olsen and Jprgensen [48]). They also took into account zero-point-vibrational averaging. For p(SHG) at X = 6943 A, and using CAS 4220 functions (more than 125000 determinants), their final value was (in the Ward convention) -7.8 a.u. or 71% of the experimental value they considered that Sekino and Bartlett s [38] estimate to be an exaggeration of "how close to experiment theory can get". However, their own value is, in fact, within the limits that Sekino and Bartlett proposed. 

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]. 

Pure vibrational and zero-point vibrational average contributions... 

Santiago et a/. ° have calculated at the Time-Dependent Hartree-Fock level the vibrational contributions to the dynamic (hyper)polarizabilities of H2O2 and have demonstrated that, though smaller than their electronic counterparts, the zero-point vibrational average contributions increase faster with the frequency. 

Kongsted, J., Christiansen, O. (2006). Automatic generation of force fields and property surfaces for use in variational vibrational calculations of anharmonic vibrational energies and zero-point vibrational averaged properties. Journal of Chemical Physics, 125,124108. 

The remaining (higher-order) vibrational contributions can, in principle be computed as well using a related formulation [36]. However, that treatment requires computation of the field-dependent zero-point vibrationally averaged properties, which was not feasible for the systems studied here because of their large size and complicated potential energy surface (PES). Indeed, of the dynamic properties mentioned above, we were only able to obtain the dc-Pockels Effect due to instabilities for high fields that wiU be described later. 

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