The Perturbation Method

Although no new numerical information regarding the hydrogen molecule-ion can be obtained by treating the wave equation by perturbation methods, nevertheless it is of value to do this. For perturbation methods can be applied to many systems for which the wave equation can not be accurately solved, and it is desirable to have some idea of the accuracy of the treatment. This can be gained from a comparison of the results of the perturbation method of the hydrogen molecule-ion and of Bureau s accurate numerical solution. The perturbation treatment assists, more-... 

By allowing the direct calculation of the successive derivatives (thus without resorting to any effective value of the field), the perturbation methods offers an elegant... 

The perturbation method presented in Section 9.3 applies only to non-degenerate eigenvalues eX of the unperturbed system. When E is degenerate, the denominators vanish for those terms in equations (9.40) and (9.41) in which Ef is equal to E making the perturbations to E and ipn indeterminate. In... 

In this section we will discuss perturbation methods suitable for high-energy electron diffraction. For simplicity, in this section we will be concerned with only periodic structures and a transmission diffraction geometry. In the context of electron diffraction theory, the perturbation method has been extensively used and developed. Applications have been made to take into account the effects of weak beams [44, 45] inelastic scattering [46] higher-order Laue zone diffraction [47] crystal structure determination [48] and crystal structure factors refinement [38, 49]. A formal mathematical expression for the first order partial derivatives of the scattering matrix has been derived by Speer et al. [50], and a formal second order perturbation theory has been developed by Peng [22,34],... 

A different approach to obtaining approximation solutions to quantum mechanical problems is provided by the variation method. It is particularly useful when there is no closely related problem that yields exact solutions. The perturbation method is not applicable in such a case. 

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

Fig. 8.34 Surface sensing sensitivity of different radial order modes are simulated by using the perturbation method with the same azimuthal number m 700.The adsorbed polymer layer is assumed to have a refractive index of 1.46...

The perturbation method is a unique method to determine the correlation energy of the system. Here the Hamiltonian operator consists of two parts, //0 and H, where //0 is the unperturbed Hamiltonian and // is the perturbation term. The perturbation method always gives corrections to the solutions to various orders. The Hamiltonian for the perturbed system is... 

The RF interference associated with an infinite number of effective RF fields is a typical problem addressed by the Floqute theory.28 32 To solve the problem, one may, however, face the diagonalization of a matrix with infinite dimensions, which is often unlikely to be done analytically. Certain approximations, such as the perturbation method, may be used.32... 

The protocol described in Section 7.1.2 involves isotopic competition, but with the different isotopomers held in separate containers. Equations 7.10 to 7.13 apply equally well to a type of competition experiment known in biochemistry as the perturbation method for determining KIE s of reversible enzyme catalyzed reactions. The perturbation method differs from simultaneous non-competitive measurements in several important ways. One begins by mixing equilibrium concentrations of substrate and product but with one component (substrate or product) at a different isotopic composition than the other. Thus, the mixture is in chemical, but not isotopic equilibrium. At this stage no enzyme is present and the interconversion is... 

In Appendix A2, we have formally applied the perturbation method to find the energy levels of a d ion in an octahedral environment, considering the ligand ions as point charges. However, in order to understand the effect of the crystalline field over d ions, it is very illustrative to consider another set of basis functions, the d orbitals displayed in Figure 5.2. These orbitals are real functions that are derived from the following linear combinations of the spherical harmonics ... 

A reexamination of the chiral discrimination of PPA herbicides was done on a TE CSP using the perturbation method to calculate the solute distribution isotherms [86]. The effect of temperature was well described by the bi-Langmuir model and enabled confirmation of the previous results by the same authors [85]. 

There have been a few recent studies of the corrections due to nuclear motion to the electronic diagonal polarizability (a ) of LiH. Bishop et al. [92] calculated vibrational and rotational contributions to the polarizability. They found for the ground state (v = 0, the state studied here) that the vibrational contribution is 0.923 a.u. Papadopoulos et al. [88] use the perturbation method to find a corrected value of 28.93 a.u. including a vibrational component of 1.7 a.u. Jonsson et al. [91] used cubic response functions to find a corrected value for of 28.26 a.u., including a vibrational contribution of 1.37 a.u. In all cases, the vibrational contribution is approximately 3% of the total polarizability. 

Equations 10.47 through 10.49 constitute the basis for the method of isotherm determination by the perturbation method. The retention time of a small injection of a solute in a column equilibrated... 

Fig. 7.6. Perturbation theory of the attractive atomic force in STM. (a) The geometry of the system. A separation surface is drawn between the tip and the sample, (b) The potential of the coupled system, (c) The potential surface of the unperturbed Hamiltonian of the sample, Us, which may be different from the potential surface of the free sample, Uso, (d) The potential surface of the unperturbed Hamiltonian of the tip, Ut, which may be different from the potential surface of the free tip, U-m- The effect of the difference between the "free" tip (sample) potential and the "distorted" tip (sample) potential can be evaluated using the perturbation method see Chapter 2. (Reproduced from Chen, 1991b, with permission.)...

The perturbation method (abr. p.m.) is one of the most important methods of approximation in quantum mechanics as well as in some fields of classical mechanics. It is usually presented in the following form. Let H0 be the operator representing some physical quantity of the unperturbed system and let i c0 be the perturbation, where k is a parameter assumed to be small. Then p.m. consists in solving problems concerning the perturbed operator H% = H0 + by expanding the results into power series of k, assuming that they are already solved for the unperturbed operator H0. 

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