Rigorous Perturbation Theories

Here we look briefly at what might be called proper perturbation methods those which may, at least in principle, be made reasonably rigorous, as distinct from those of the previous section, which generally involve some element of handwaving. 

Rigorous perturbational treatments of the interaction between two molecules belong to the field of intermolecular forces, and I shall not attempt a comprehensive review, since the topic has been reviewed by Stamper129 in the previous volume in this series. However, several authors have devised perturbation schemes with a view to their application in problems of reactivity, which is a departure from conventional theory of intermolecular forces, where the possibility of making and breaking of bonds is usually excluded, on the reasonable grounds that the problem is quite hard enough anyway. 

A major difficulty is the problem of electron indistinguishability. The natural choice of the unperturbed Hamiltonian is the sum of the Hamiltonians for the separated molecules, but this is not symmetric with respect to permutations of electrons on one molecule with electrons on the other. The order of a term in the perturbation expansion then becomes undefined,129 and although this difficulty can be overcome,130 the application to large systems is probably not in sight. 

Stamper, in Theoretical Chemistry , ed. R. N. Dixon and C. Thomson, (Specialist Periodical Reports), The Chemical Society, London, 1975, Vol. 2, p. 66. 

Sustmann and Binsch132 described a method which started from the same point, but invoked the zero-differential-overlap approximation on the other hand, it was not confined to jr-electrons and the perturbation energy was refined iteratively. Using a MINDO parameterization they then applied the method to Diels-Alder reactions, and were able to account for effects which cannot be explained in terms of simple jr-electron theory, such as the preference for endo addition of cyclopropene to cyclo-pentadiene. 

One approach of this type is due originally to Bader, who used perturbation theory to show that unimolecular reaction is particularly favoured along a normal co-ordinate whose symmetry is the same as that of the transition density between the ground state and a low-lying excited state, because then second-order Jahn-Teller interaction depresses the energy of the ground state. The idea has been taken up by Salem and Pearson and used, in conjunction with a semilocalized MO model, to determine the allowed routes in the pyrolysis of cyclobutane and cyclohexene. Fukui et a/. have described a similar method. 

Rigorous perturbational treatments of the interaction between two molecules belong to the field of intermolecular forces, and I shall not attempt a comprehensive review, since the topic has been reviewed by Stamper in the previous volume in this 

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For one-dimensional rotation (r = 1), orientational correlation functions were rigorously calculated in the impact theory for both strong and weak collisions [98, 99]. It turns out in the case of weak collisions that the exact solution, which holds for any happens to coincide with what is obtained in Eq. (2.50). Consequently, the accuracy of the perturbation theory is characterized by the difference between Eq. (2.49) and Eq. (2.50), at least in this particular case. The degree of agreement between approximate and exact solutions is readily determined by representing them as a time expansion... 

To illustrate the accuracy of the perturbation theory these results are worth comparing with the well-known values of h and I4 for t = 1 rigorously found from first principles in [8]. It turns out that the second moment in Eq. (2.65a) is exact. The evaluation of I4, however, is inaccurate its first component is half as large as the true one. The cause of this discrepancy is easily revealed. Since M = / and (/) = J/xj, the second component in Ux) is linear in e. Hence, it is as exact in this order as perturbation theory itself. In contrast, the first component in IqXj is quadratic in A and its value in the lowest order of perturbation theory is not guaranteed. Generally speaking... 

In this section we consider how to express the response of a system to noise employing a method of cumulant expansions [38], The averaging of the dynamical equation (2.19) performed by this technique is a rigorous continuation of the iteration procedure (2.20)-(2.22). It enables one to get the higher order corrections to what was found with the simplest perturbation theory. Following Zatsepin [108], let us expound the above technique for a density of the conditional probability which is the average... 

Starting from the normal mode approximation, one can introduce anharmonicity in different ways. Anharmonic perturbation theory [206] and local mode models [204] may be useful in some cases, where anharmonic effects are small or mostly diagonal. Vibrational self-consistent-field and configuration-interaction treatments [207, 208] can also be powerful and offer a hierarchy of approximation levels. Even more rigorous multidimensional treatments include variational calculations [209], diffusion quantum Monte Carlo, and time-dependent Hartree approaches [210]. 

Complete formal analysis of the recoil corrections in the framework of the relativistic two-particle equations, with derivation of all relevant kernels, perturbation theory contributions, and necessary subtraction terms may be performed along the same lines as was done for hyperfine splitting in [3]. However, these results may also be understood without a cumbersome formalism by starting with the simple scattering approximation. We will discuss recoil corrections below using this less rigorous but more physically transparent approach. 

Using first-order, time-dependent perturbation theory it can be shown15 that one can rigorously speak of the excited system, eqs. (12-1)— (12-7), as being in the initial nonstationary state, 5>. With this preparation of our molecule assured, then, we shall, for the time being, neglect the... 

There have been attempts to generalize the many-body perturbation theory to cover the relativistic regime in a rigorous and systematic manner [238-241]. Unfortunately, practical applications, so far, are only to simple atoms or ions. 

CIS(D) can be rigorously derived by applying the Lowdin-type (as opposed to Rayleigh-Schrodinger) perturbation theory [70] to CIS, according to Meissner [71]. Additional off-diagonal second-order corrections to CIS have been considered by Head-Gordon et al. [72],... 

In this and the next sections, we will present the basic ideas of singular perturbation theory without going into mathematical rigor. 

The main trend in the experimental ranges listed in Table 9-3 is an increase with increasing mctallicity. Sokel and Harrison (1976) explained the nature of these forces and the trend by noting that the interaction between atoms can be calculated rigorously in perturbation theory, by Eq. (1-14). Let k,> be an eigenstate... 

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