Perturbation parameter

If A = 0, then H = Hq, 4/ = o md W = Eq. As the perturbation is increased from zero to a finite value, the new energy and wave function must also change continuously, and they can be written as a Taylor expansion in powers of the perturbation parameter A. 

Although applications of perturbation theory vary widely, the main idea remains the same. One starts with an initial problem, called the unperturbed or reference problem. It is often required that this problem be sufficiently simple to be solved exactly. Then, the problem of interest, called the target problem, is represented in terms of a perturbation to the reference problem. The effect of the perturbation is expressed as an expansion in a series with respect to a small quantity, called the perturbation parameter. It is expected that the series converges quickly, and, therefore, can be truncated after the first few terms. It is further expected that these terms are markedly easier to evaluate than the exact solution. 

According to the perturbation method, we introduce the perturbation parameter X that labels the order of interaction,... 

In complete analogy with this procedure, the energy functional can be expanded in powers of the differential perturbation parameter A ... 

This chapter is intended to provide basic understanding and application of the effect of electric field on the reactivity descriptors. Section 25.2 will focus on the definitions of reactivity descriptors used to understand the chemical reactivity, along with the local hard-soft acid-base (HSAB) semiquantitative model for calculating interaction energy. In Section 25.3, we will discuss specifically the theory behind the effects of external electric field on reactivity descriptors. Some numerical results will be presented in Section 25.4. Along with that in Section 25.5, we would like to discuss the work describing the effect of other perturbation parameters. In Section 25.6, we would present our conclusions and prospects. 

In addition to these external electric or magnetic field as a perturbation parameter, solvents can be another option. Solvents having different dielectric constants would mimic different field strengths. In the recent past, several solvent models have been used to understand the reactivity of chemical species [55,56]. The well-acclaimed review article on solvent effects can be exploited in this regard [57]. Different solvent models such as conductor-like screening model (COSMO), polarizable continuum model (PCM), effective fragment potential (EFP) model with mostly water as a solvent have been used in the above studies. 

In order to calculate the free energy of mixing for a mixture consisting of the two cations A+ and B+ and the anion X-, three separate perturbation calculations are carried out. If AX is salt 1 and BX salt 2, then there are two size parameters (or characteristic sums of cation-anion radii) dx and d3. Dimensionless perturbation parameters are defined by... 

The second type of interaction is dependent on the ion sizes and for the cation-anion pairs on the perturbation parameter, g. The core repulsive potential for a cation-anion pair has the form... 

To summarize exactly the same self-consistent polarizabilities may be used in discussing the effect of heteroatoms as in discussing framework perturbations, but the perturbation parameter 8a , is replaced by given in (120). The changes in charges and bond orders are given by... 

Assuming 1/c as the perturbation parameter the first-order energy can be written as... 

We expand the Hamiltonian and the IBCj in terms of a perturbation parameter p. in the spirit of M0ller-Plesset perturbation theory [34]. Details are found in Ref. [25]. We need not worry about the particle rank to which we have to go, since this is fully controlled by the perturbation expansion. We limit ourselves to a closed-shell state, such that the zeroth order is simply closed-shell Hartree-Fock. 

In the early days of quanttrm mechanics Bom and Oppenheimer[18] showed that the energy and motion of the nuclei and electrons could be separated approximately. This was accomplished using a perturbation treatment in which the perturbation parameter is In actuality, the term Bom-Oppenheimer approximation ... 

Second, Eq. 1.1 corresponds to electrons moving in the field of fixed nuclei — the clamped-nucleus Bom-Oppenheimer approximation. This approximation derives from a perturbation theory analysis in which the perturbation parameter is M , where M is the nuclear mass in atomic units (electron mass unity). Thus even for the worst case of a hydrogen nucleus this quantity is about 0.15, and for heavier atoms it is smaller. The Bom-Oppenheimer approximation is thus generally a very good... 

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