Cross terms molecular properties

Mixed derivatives refer to cross terms if the energy is expanded in more than one perturbation. There are many such mixed derivatives which translate into molecular properties, below are a few examples. 

Zhao and Rice then developed an approximation to locate the intramolecular bottlenecks and calculate the associated flux. There are two principal motivations for the development of such an approximation. These are, first, the need to simplify the very complicated mapping-based calculation of the flux crossing a cantorus so as to make the calculation practical in systems with many DOFs and, second, the desirability of having a simple representation of the intramolecular energy transfer barrier in terms of molecular properties. 

A central issue is the number of different atom types that are used in a particular force field. There is always a compromise between increasing the number to allow for the inclusion of more environmental effects (i.e., local electronic interactions) vs. the increase in the number of parameters to be determined to adequately represent a new atom type. In general, the more subtypes of atoms (how many different kinds of nitrogen, for example), the less likely that the parameters for a particular application will be available in the force field. The extreme, of course, would be a special atom type for each kind of atomic environment in which the parameters were chosen, so that the calculated properties of each molecule would simply reproduce the experimental observations. One major assumption, therefore, is that the force constants (parameters) and equilibrium values of the equations are functions of a limited number of atom types and can be transferred from one molecular environment to another. This assumption holds reasonably well where one may be primarily interested in geometric issues, but is not so valid in molecular spectroscopy. This had led to the introduction of additional equations, the so-called "cross-terms" which allow additional parameters to account for correlations between bond lengths and bond angles... 

We have seen how the molecular properties in nonlinear optics are defined by the expansion of the molecular polarization in orders of the external electric field, see Eq. (5) beyond the linear polarization this definition introduces the so-called nonlinear hyperpolarizabilities as coupling coefficients between the two quantities. The same equation also expresses an expansion in terms of the number of photons involved in simultaneous quantum-mechanical processes a, j3, y, and so on involve emission or absorption of two, three, four, etc. photons. The cross section for multiphoton absorption or emission, which takes place in nonlinear optical processes, is in typical cases relatively small and a high density of photons is required for these to occur. 

The preceding development illustrates the assumptions that are necessary to develop the cross relationship. The more detailed theory that follows provides further understanding in terms of the molecular properties of the reactants and solvent. 

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