Electronic state properties, vibrationally averaged

Vibrational motion is affected by the presence of an external field, field gradient, and so on. This introduces a response to an applied potential beyond that of the vibrationally averaged electronic state properties. Physically, it arises from the change in the stretching potential experienced when the molecule is in a field. 

Property Vibrationally Averaged Value Complete value for Vibrational-Electronic State... 

XXXVI compares properties for hydrogen fluoride. Most of the contribution to the complete value for the vibrational-electronic ground state comes from the change in the potential, and this reinforces the validity of doubly harmonic analyses. The difference between the vibrationally averaged and complete values is small for a but quite significant for p, as Adamowicz and Bartlett first discovered [70]. 

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. 

If high accuracy is required, vibrational effects must be taken into account. In a proper adiabatic Born-Oppenheimer treatment, the groimd state wave function would be written as a product of an electronic and a vibrational wave function. The response of this wave function should then be computed and subsequently used to construct vibronic response functions. The sum-over-states expressions would include contributions from vibrational states in the electronic groimd and excited states. Since each set of vibrational wave functions is tied to a specific electronic state within the adiabatic Bom-Oppenheimer approximation, this approach is not feasible in practice. Hence, the electronic properties are considered as electronic ground state properties and therefore, averaged in a vibrational state of the electronic ground state. 

The physical and spectroscopic properties of a spin-equilibrium complex can appear to be either the average or the superposition of the properties of the separate spin states. Which occurs is dependent on the time scale of the observation relative to the relaxation time of the equilibrium. Thus the electronic and vibrational spectra always appear as a superposition of the two isomers because each spin state possesses a distinctive potential energy surface with its characteristic electronic and vibrational properties. On the other hand, the NMR spectra appear as the average of the spectra of the two spin states, for all but the slowest interconversions, because the frequency of the interconversion is high compared with the frequency differences of the chemical shifts or the inverse of the spin relaxation times of the two isomers. 

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