Single-step perturbation

Fig. 1. A single step perturbation fails for the mutation of p-methylphenol to p-methoxyphenol because no configuration with a cavity large enough to accommodate the methoxy group is sampled in a simulation of p-methylphenol in water.

Single-step perturbation methods have also been applied to electrostatic processes. One study probed the dielectric properties of several proteins at a microscopic level [41,42], Test charges were inserted at many different positions within or around each protein, and a dielectric relaxation free energy was computed, which is related to a microscopic dielectric susceptibility (see Sect. 12.3). 

Oostenbrink C, van Gunsteren WF. Single-step perturbations to calculate free energy differences from unphysical reference states limits on size, flexibility, and character. J Comput Chem 2003 24 1730-1739. 

Nearly 10 years after Zwanzig published his perturbation method, Benjamin Widom [6] formulated the potential distribution theorem (PDF). He further suggested an elegant application of PDF to estimate the excess chemical potential -i.e., the chemical potential of a system in excess of that of an ideal, noninteracting system at the same density - on the basis of the random insertion of a test particle. In essence, the particle insertion method proposed by Widom may be viewed as a special case of the perturbative theory, in which the addition of a single particle is handled as a one-step perturbation of the liquid. 

This is the desired formula, which requires only statistical averages over the reference system at temperature T. If, instead, we start from (2.65) and perform identical steps, we obtain a similar, single-state perturbation formula for AS,, . As it turns out, that formula, however, is more cumbersome to use than (2.69) but does not seem to offer any benefits in terms of accuracy. 

First of all, the mathematical background will be developed for the case of a simple electrode reaction O + n e = R. In this treatment, contrasts like potential versus current perturbation, large amplitude versus small amplitude, and single step versus periodical perturbation are emphasized. While discussing these principles, the most common methods derived from them will be briefly mentioned. On the other hand, it will be shown that, by virtue of the method of Laplace transformation, these methods have much in common and contain, in principle, the same information if the detected cell response is of the same order. 

Also in this case, the occurrence of a single-step three-electron process indicated that the three ferrocenyl ligands, from which the electrons are concomitantly removed, are non-communicating. In addition, if one considers that, under the same experimental conditions, ferrocene undergoes oxidation at E° = 4-0.45 V, it is evident that the ferrocenyl ligands are only slightly electronically perturbed by complex formation with the CpTi fragment. 

In a single-step process that involves both isotopically sensitive processes (as primary isotope effects the bond to the isotope is broken), the isotope effects should be independent, according to the fundamental assumption of transition-state theory, because the bond to the atom (the isotopic atom in this case) in the transition state has no residual vibrational component (I). Therefore, the isotope effect is due exclusively to differences in ground-state vibrational levels. In a single step in which both isotopic bonds are involved for the two processes of interest, the isotope effects must function independently. In a sequential system, the first isotope effect perturbs the proportioning of the second. 

In the case of an anay of band electrodes or interdigitated electrode structures, the width of each single electrode element and the gap between the electrode elements must be considered carefully in the sensor design. Interactions between electrode elements and their effects on the transient response to a potential step perturbation will directly affect the overall sensor output. When an interdigitated electrode structure is used, chemical cross-talk among the reactants and products in both electrode elements (cathode and anode) may occur, which will then influence the sensor output [3], Therefore, the relative location of the sensing elements is also an essential consideration. 

The apparatus developed by Rees et al. can also be used for single-step FR measurements. In these experiments, the gas phase equilibrium volume, Ve, is only subjected to one half of a square wave perturbation of 1%. Up to 2,048 pressure readings can be recorded by the on-line computer during... 

To determine the state change at the end of I2, we repeat the steps of Section 5.4.1, which is concerned with a single pulse perturbation corresponding to the first subinterval Ii. 

---