Small perturbation theory

In discussing many problems which cannot be directly solved, a solution can be obtained of a wave equation which differs from the true one only in the omission of certain terms whose effect on the system is small. Perturbation theory provides a method of treating such problems, whereby the approximate equation is first solved and then the small additional terms are introduced as corrections ... 

Mirels,H. and Mullen,J.F. Small Perturbation Theory for Shock Tube Attenuation and Nonuniformity, P.Fluids, Vol.7,... 

Hfcc have been computed using essentially all available correlated ab initio methods, from small perturbation theory expansions to large-scale multireference configuration... 

If //j is small compared with EI we may treat EI by perturbation theory. The first-order perturbation theory fomuila takes the fonn [18, 19, 20 and 21] ... 

First-principles models of solid surfaces and adsorption and reaction of atoms and molecules on those surfaces range from ab initio quantum chemistry (HF configuration interaction (Cl), perturbation theory (PT), etc for details see chapter B3.1 ) on small, finite clusters of atoms to HF or DFT on two-dimensionally infinite slabs. In between these... 

In his classical paper, Renner [7] first explained the physical background of the vibronic coupling in triatomic molecules. He concluded that the splitting of the bending potential curves at small distortions of linearity has to depend on p, being thus mostly pronounced in H electronic state. Renner developed the system of two coupled Schrbdinger equations and solved it for H states in the harmonic approximation by means of the perturbation theory. 

A Moeller-Plesset Cl correction to v / is based on perturbation theory, by which the Hamiltonian is expressed as a Hartree-Fock Hamiltonian perturbed by a small perturbation operator P through a minimization constant X... 

CSFs have equal or nearly equal amplitudes beeause it obtains the amplitudes of all but the dominant CSF from perturbation theory formulas that assume the perturbation is small. ... 

The golden rule is a reasonable prediction of state-crossing transition rates when those rates are slow. Crossings with fast rates are predicted poorly due to the breakdown of the perturbation theory assumption of a small interaction. 

It has been postulated that jet breakup is the result of aerodynamic interaction between the Hquid and the ambient gas. Such theory considers a column of Hquid emerging from a circular orifice into a surrounding gas. The instabiHty on the Hquid surface is examined by using first-order linear theory. A small perturbation is imposed on the initially steady Hquid motion to simulate the growth of waves. The displacement of the surface waves can be obtained by the real component of a Fourier expression ... 

Substitution r = 0 gives a value of 1422.7 MHz (50.765 mT) and the small difference between theory and experiment is essentially due to the use of first-order perturbation theory in the derivation above. 

The idea in perturbation methods is that the problem at hand only differs slightly from a problem which has already been solved (exactly or approximately). The solution to the given problem should therefore in some sense be close to the solution of the already known system. This is described mathematically by defining a Hamilton operator which consists of two part, a reference (Hq) and a perturbation (H )- The premise of perturbation methods is that the H operator in some sense is small compared to Hq. In quantum mechanics, perturbational methods can be used for adding corrections to solutions which employ an independent particle approximation, and the theoretical framework is then called Many-Body Perturbation Theory (MBPT). 

These concepts play an important role in the Hard and Soft Acid and Base (HSAB) principle, which states that hard acids prefer to react with hard bases, and vice versa. By means of Koopmann s theorem (Section 3.4) the hardness is related to the HOMO-LUMO energy difference, i.e. a small gap indicates a soft molecule. From second-order perturbation theory it also follows that a small gap between occupied and unoccupied orbitals will give a large contribution to the polarizability (Section 10.6), i.e. softness is a measure of how easily the electron density can be distorted by external fields, for example those generated by another molecule. In terms of the perturbation equation (15.1), a hard-hard interaction is primarily charge controlled, while a soft-soft interaction is orbital controlled. Both FMO and HSAB theories may be considered as being limiting cases of chemical reactivity described by the Fukui ftinction. 

The perturbation theory used by Holstein in his small-polaron model confines its validity to an upper limit for J of around hto0, which corresponds to a non-adiabatic process. The adiabatic process, for which J > has been studied less extensively. In the high temperature limit, Emin and Holstein [46] arrive at the result that... 

The key element in London s approach is the expansion of the electrical potential energy in multipole series. Since neutral molecules or portions of molecules are involved, the leading term is that for dipole-dipole interaction. While attention has been given to higher-order terms, these are usually small, and the greater need seems to be for improved treatment of the dipole-dipole terms. London used second order perturbation theory in his treatment, but Slater and Kirkwood38,21 soon followed with a variation method treatment which yielded similar results. Other individual papers will be mentioned later, but the excellent review of Mar-genau26 should not be overlooked. 

The perturbation theory presented in Chapter 2 implies that orientational relaxation is slower than rotational relaxation and considers the angular displacement during a free rotation to be a small parameter. Considering J(t) as a random time-dependent perturbation, it describes the orientational relaxation as a molecular response to it. Frequent and small chaotic turns constitute the rotational diffusion which is shown to be an equivalent representation of the process. The turns may proceed via free paths or via sudden jumps from one orientation to another. The phenomenological picture of rotational diffusion is compatible with both... 

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