Perturbation technique

Nayfeh, A.H., 1993. Introduction to Perturbation Techniques, Wiley, New York. 

Matched-Asymptotic Expansions Sometimes the coefficient in front of the highest derivative is a small number. Special perturbation techniques can then be used, provided the proper scaling laws are found. See Refs. 32, 170, and 180. 

Electrochemical Techniques Although the linear polarisation resistance technique has moved beyond the infancy status attributed to it in the original material, its inherent limitations remain, i.e. it is a perturbation technique, sensitive to environmental conductivity and insensitive to localised corrosion. Two developments have occurred ... 

The idea of coupling variational and perturbational methods is nowadays gaining wider and wider acceptance in the quantum chemistry community. The background philosophy is to realize the best blend of a well-defined theoretical plateau provided by the application of the variational principle coupled to the computational efficiency of the perturbation techniques. [29-34]. In that sense, the aim of these approaches is to improve a limited Configuration Interaction (Cl) wavefunction by a perturbation treatment. 

We have first been concerned with the computational point of view. Through the calculation of the dynamic polarizability of CO, we have developed a method based on the conventional SCF-Cl method, using the variational- perturbation techniques the first-order wavefunction includes two parts (i) the traditional one, developed over the excited states and (ii) additional terms obtained by multiplying the zeroth—order function by a polynomial of first-order in the electronic coordinates. This dipolar... 

These compounds have been the subject of several theoretical [7,11,13,20)] and experimental[21] studies. Ward and Elliott [20] measured the dynamic y hyperpolarizability of butadiene and hexatriene in the vapour phase by means of the dc-SHG technique. Waite and Papadopoulos[7,ll] computed static y values, using a Mac Weeny type Coupled Hartree-Fock Perturbation Theory (CHFPT) in the CNDO approximation, and an extended basis set. Kurtz [15] evaluated by means of a finite perturbation technique at the MNDO level [17] and using the AMI [22] and PM3[23] parametrizations, the mean y values of a series of polyenes containing from 2 to 11 unit cells. At the ab initio level, Hurst et al. [13] and Chopra et al. [20] studied basis sets effects on and y. It appeared that diffuse orbitals must be included in the basis set in order to describe correctly the external part of the molecules which is the most sensitive to the electrical perturbation and to ensure the obtention of accurate values of the calculated properties. 

In reality, this term is not small in comparison with the other terms so we should not expect the perturbation technique to give accurate results. With this choice for the perturbation, the Schrodinger equation for the unperturbed Hamiltonian operator may be solved exactly. 

Serrano-Andres L, Merchan M, Lindh R (2005) Computation of conical intersections by using perturbation techniques. J Chem Phys 122 104107... 

Schmidt MW, Baldridge KK, Boatz JA, Elbert ST, Gordon MS, Jensen JH, Koseki S, Matsunaga N, Nguyen KA, Su S, Windus TL, Dupuis M, Montgomery JA (1993) Computation of conical intersections by using perturbation techniques. J Comput Chem 14 1347... 

Exact analytical solutions of the coupled equations for simultaneous mass transfer, heat transfer, and chemical reaction cannot be obtained. However, various authors have employed linear approximations (56-57), perturbation techniques (58), or asymptotic approaches (59) to obtain approximate analytical solutions to these equations. Numerical solutions have also been obtained (60-61). Once the solution for the concentration profile has been determined, equation 12.3.98 may be used to determine the temperature profile. The effectiveness factor may also be determined from the concentration profile, using the approach we have... 

Matched-Asymptotic Expansions Sometimes the coefficient in front of the highest derivative is a small number. Special perturbation techniques can then be used, provided the proper scaling laws are found. See Kevorkian, J., and J. D. Cole, Perturbation Methods in Applied Mathematics, Springer-Verlag, New York (1981) and Lager-strom, P. A., Matched Asymptotic Expansions Ideas and Techniques, Springer-Verlag, New York (1988). 

Perturbation theory is one of the oldest and most useful, general techniques in applied mathematics. Its initial applications to physics were in celestial mechanics, and its goal was to explain how the presence of bodies other than the sun perturbed the elliptical orbits of planets. Today, there is hardly a field of theoretical physics and chemistry in which perturbation theory is not used. Many beautiful, fundamental results have been obtained using this approach. Perturbation techniques are also used with great success in other fields of science, such as mathematics, engineering, and economics. 

The finite-difference method can be combined with the perturbation technique that was previously used to derive the basic formulas in Sect. 2.2. This yields another single-state perturbation formula [49, 50]. Starting from (2.66), we get... 

Quantum mechanical models at different levels of approximation have been successfully applied to compute molecular hyperpolarizabilities. Some authors have attempted a complete determination of the U.V. molecular spectrum to fill in the expression of p (15, 16). Another approach is the finite-field perturbative technique (17) demanding the sole computation of the ground state level of a perturbated molecule, the hyperpolarizabilities being derivatives at a suitable order of the perturbed ground state molecule by application of the Hellman-Feynman theorem. 

This chapter focuses on analytical CL methodologies, with emphasis on the kinetic connotations of typical approaches such as the stopped-flow, the continuous-addition-of-reagent (a new kinetic methodology) and the pulse perturbation technique developed for oscillating reactions, among others. Recent contributions to kinetic simultaneous determinations of organic substances using CL detection (kinetometric approaches included) are also preferentially considered here. 

In this study, identification of the critical atomic and molecular determinants pertaining to the mechanism of dihydrofolate to tetrahydrofolate reduction was achieved by (i) ab initio quantum mechanics, (ii) molecular mechanics, and (iii) free energy perturbation techniques. For the first time, the complete free energy profile was calculated for the proton transfer from Asp27 of the enzyme E. Coli DHFR to the N5 position of the dihydropterin moiety of the substrate dihydrofolate. In addition, the free... 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

To date the structure and reactivity of numerous complexes derived from aromatic compounds and nitrosonium cation have been studied (5, 56-63). However, relatively few studies are available on the nitrosonium complexes of cyclophanes (5, 57, 59, 61, 62), cf ref. (63). The interaction of [2.2]paracyclophane with nitrosonium tetrachloroaluminate was studied by H and 13C NMR spectroscopy using deuterium isotope perturbation technique (64). It was found that the resulting nitrosonium complexes containing one (25) or two NO groups (26) are involved in fast interconversion (on the NMR time scale) (Scheme 17). 

A widely-used alternative is the free energy perturbation technique (FEP),9"11,72"74 which will be discussed for the Gibbs free energy G. It involves a thermodynamic cycle such as the following, in which X and Y are two different molecules that are undergoing solvation in a particular solvent ... 

Fig. 11.8 Equilibrium perturbation trace for deuterated L-malate reaction catalyzed by malic enzyme. The equilibrium perturbation technique is discussed in Section 7.1.4. The label on the ordinate, p,M has been recalculated from the change in absorbance of TPNH at 340 nm (Schimerlik, M. I., Rife, J. E. and Clcland, W. W. Biochemistry 14, 5347 (1975))...

Loukili, B. etal.. Study of tryptophan enantiomer binding to a teicoplanin-based stationary phase using the perturbation technique. Investigation of the role of sodium perchlorate in solute retention and enantioselectivity, J. Chromatogr. A, 986, 45,... 

With the availability of perturbation techniques for measuring the rates of rapid reactions (Sec. 3.4), the subject of relaxation kinetics — rates of reaction near to chemical equilibrium — has become important in the study of chemical reactions.Briefly, a chemical system at equilibrium is perturbed, for example, by a change in the temperature of the solution. The rate at which the new equilibrium position is attained is a measure of the values of the rate constants linking the equilibrium (or equilibria in a multistep process) and is controlled by these values. 

Static charge-density susceptibilities have been computed ab initio by Li et al (38). The frequency-dependent susceptibility x(r, r cd) can be calculated within density functional theory, using methods developed by Ando (39 Zang-will and Soven (40 Gross and Kohn (4I and van Gisbergen, Snijders, and Baerends (42). In ab initio work, x(r, r co) can be determined by use of time-dependent perturbation techniques, pseudo-state methods (43-49), quantum Monte Carlo calculations (50-52), or by explicit construction of the linear response function in coupled cluster theory (53). Then the imaginary-frequency susceptibility can be obtained by analytic continuation from the susceptibility at real frequencies, or by a direct replacement co ico, where possible (for example, in pseudo-state expressions). 

In contrast to metal electrodes, for a semiconductor-electrolyte interface most of the potential drop is located in the semiconductor making it difficult to study interfacial processes using potential perturbation techniques [11,20,55,58,60-65,75-78]. H. Gerischer [76] proposed a model in which electrons and holes are considered as individual interfacial reactants. Distinct and preferential electron transfer reactions involve either the conduction band or valence band as dependent on the nature of the redox reactants of the electrolyte, with specific properties dependent upon the energy state location. 

The similarity of the shape of the voltammetric curves of platinized system to those reported for electrodes treated by the fast repetitive potential perturbation technique or obtained by annealing electrodeposited platinum and the appearance of signs characteristic for stepped surfaces allows us to assume that these phenomena should be ascribed to the similarities in the surface structure. 

Detonation, Nonlinear Theory of Unstable One-Dimensional. J.J. Erpenbeck describes in PhysFluids 10(2), 274-89(1969) CA 66, 8180-R(1967) a method for calcg the behavior of 1-dimensional detonations whose steady solns are hydrodynamic ally unstable. This method is based on a perturbation technique that treats the nonlinear terms in the hydro-dynamic equations as perturbations to the linear equations of hydrodynamic-stability theory. Detailed calcns are presented for several ideal-gas unimol-reaction cases for which the predicted oscillations agree reasonably well with those obtd by numerical integration of the hydrodynamic equations, as reported by W. Fickett W.W. Wood, PhysFluids 9(5), 903-16(1966) CA 65,... 

---