Coupling parameters

We conclude this section by discussing an expression for the excess chemical potential in temrs of the pair correlation fimction and a parameter X, which couples the interactions of one particle with the rest. The idea of a coupling parameter was mtrodiiced by Onsager [20] and Kirkwood [Hj. The choice of X depends on the system considered. In an electrolyte solution it could be the charge, but in general it is some variable that characterizes the pair potential. The potential energy of the system... 

This is Kirkwood s expression for the chemical potential. To use it, one needs the pair correlation fimction as a fimction of the coupling parameter A as well as its spatial dependence. For instance, if A is the charge on a selected ion in an electrolyte, the excess chemical potential follows from a theory that provides the dependence of g(i 2, A) on the charge and the distance r 2- This method of calculating the chemical potential is known as the Gimtelburg charging process, after Guntelburg who applied it to electrolytes. 

The first step is to divide the total potential into two parts a reference part and the remainder treated as a perturbation. A coupling parameter X is introduced to serve as a switch which turns the perturbation on or off. 

Series expansion Smith and van Gunsteren [4] investigated the first approach expanding the free energy as a function of the coupling parameter A into a T ylor series around a given reference state, A = 0,... 

The relationship between the initial, final and intermediate states is usefully describer terms of a coupling parameter, A. As A is changed from 0 to 1, the Hamiltonian varies fi Isfx to Y- Each of the terms in the force field for an intermediate state A can be wri as a linear combination of the values for X and Y ... 

Other attempts to predict free energies from a single simulation have explored the relati ship between the coupling parameter. A, and the free energy. Specifically, the free energ ... 

If the coupling parameter (the Bath relaxation constant in HyperChem), t, is too tight (<0.1 ps), an isokinetic energy ensemble results rather than an isothermal (microcanonical) ensemble. The trajectory is then neither canonical or microcanon-ical. You cannot calculate true time-dependent properties or ensemble averages for this trajectory. You can use small values of T for these simulations ... 

Another conventional simplification is replacing the whole vibration spectrum by a single harmonic vibration with an effective frequency co. In doing so one has to leave the reversibility problem out of consideration. It is again the model of an active oscillator mentioned in section 2.2 and, in fact, it is friction in the active mode that renders the transition irreversible. Such an approach leads to the well known Kubo-Toyozawa problem [Kubo and Toyozava 1955], in which the Franck-Condon factor FC depends on two parameters, the order of multiphonon process N and the coupling parameter S... 

Naturally, neither of these approximations is valid near the border between the two regions. Physically sensible are only such parameters, for which b < 1. Note that even for a low vibration frequency Q, the adiabatic limit may hold for large enough coupling parameter C (see the bill of the adiabatic approximation domain in fig. 30). This situation is referred to as strong-fiuctuation limit by [Benderskii et al. 1991a-c], and it actually takes place for heavy particle transfer, as described in the experimental section of this review. In the section 5 we shall describe how both the sudden and adiabatic limits may be viewed from a unique perspective. 

In addition, the frequency cooo, as well as the tunneUng distance can also be extracted from the same empirical data. Thus all the information needed to construct a PES is available. Of course, this PES is a rather crude approximation, since all the skeleton vibrations are replaced by a single mode with effective frequency cooo and coupling parameter C. From the experimental data it is known that the strong hydrogen bond (roo < 2.6 A) is usually typical of intramolecular hydrogen transfer. 

There is a nice point as to what we mean by the experimental energy. All the calculations so far have been based on non-relativistic quantum mechanics. A measure of the importance of relativistic effects for a given atom is afforded by its spin-orbit coupling parameter. This parameter can be easily determined from spectroscopic studies, and it is certainly not zero for first-row atoms. We should strictly compare the HF limit to an experimental energy that refers to a non-relativistic molecule. This is a moot point we can neither calculate molecular energies at the HF limit, nor can we easily make measurements that allow for these relativistic effects. 

As can be seen from Table 2, the agreement between measured and calculated isotropic hyperfine coupling parameters is good, confirming previous interpretations [9] of ESR data. 

Since the only angle dependence conies from 0 , and the actions /, L are constant. From this point onwards we concentrate on motion under the reduced Hamiltonian which depends, apart from the scaling parameter y, only on the values of scaled coupling parameter p and the scaled detuning term p. In other words, we investigate the monodromy only in a fixed J (or polyad number N = 2J) section of the three-dimensional quanmm number space. 

---