Perturbation Theory Methods

All discussions of transport processes currently available in the literature are based on perturbation theory methods applied to kinetic pictures of micro-scattering processes within the macrosystem of interest. These methods do involve time-dependent hamiltonians in the sense that the interaction operates only during collisions, while the wave functions are known only before and after the collision. However these interactions are purely internal, and their time-dependence is essentially implicit the over-all hamiltonian of the entire system, such as the interaction term in Eq. (8-159) is not time-dependent, and such micro-scattering processes cannot lead to irreversible changes of thermodynamic (ensemble average) properties. 

The GIAO-MP2/TZP calculated 13C NMR chemical shifts of the cyclopropylidene substituted dienyl cation 27 show for almost all carbon positions larger deviations from the experimental shifts than the other cations 22-26. The GIAO-MP2/TZP method overestimates the influence of cr-delocalization of the positive charge into the cyclopropane subunit on the chemical shifts. Electron correlation corrections for cyclopropylidenemethyl cations such as 27 and 28 are too large to be adequately described by the GIAO-MP2 perturbation theory method and higher hierarchies of approximations such as coupled cluster models are required to rectify the problem. 

One area in which Cl methods appear to be thoroughly superior to perturbation theory is in the treatment of multireference problems problems with substantial nondynamical correlation effects. Even UHF-based single reference perturbation theory methods may not cope with some such situations, and multireference perturbation theory, despite many efforts over the years, still appears to be far from developing a general flexible approach that is competitive with MRCI. Transition-metal chemistry, in particular, is a graveyard for UHF-based MP methods. 

Second, where nondynamical correlation is unimportant (that is, mainly closed-shell molecules near their equilibrium geometries), size-extensive (or nearly size-extensive) treatments like CPF, CCSD, and CISD+Q perform well. However, as bonds are stretched, or in species where nondynamical correlation is important because of near-degeneracies or open-shell effects, the performance of these methods deteriorates fairly rapidly. As might be expected, the CCSD method generally deteriorates slowest. In general, low-order perturbation theory methods do not agree well... 

In table 2 our result is compared with the UV spectroscopic result of Klein et al. [26], Also shown are the theoretical results of Zhang et al. [2], Plante et al. [27], and Chen et al. [28], The first of these uses perturbation theory, with matrix elements of effective operators derived from the Bethe-Salpeter equation, evaluated with high precision solutions of the non-relativistic Schrodinger equation. This yields a power series in a and In a. The calculations of Zhang et al. include terms up to O(o5 hi a) but omit terms of 0(ary) a.u. The calculations of Plante et al. use an all orders relativistic perturbation theory method, while those of Chen et al. use relativistic configuration interaction theory. These both obtain all structure terms, up to (Za)4 a.u., and use explicit QED corrections from Drake [29],... 

Tel. 904-392-1597, fax 904-392-8722, e-mail bartlett qtp.ufl.edu Ab initio molecular orbital calculations using coupled-cluster and many-body perturbation theory methods. 

The contributions of Komatsuzaki and Berry, and of Uzer s group, discuss these manifolds, and they present their calculations using Lie perturbation theory methods. The contribution of Wiesenfeld discusses these manifolds in reaction processes involving angular momenta, and the contribution by Joyeux et al. shows applications of the perturbation theory method to reactions involving Fermi resonance. The contribution of Sano discusses invariant manifolds in the Coulomb three-body problem. 

Amos, R.D. (1982) Multipole moments and polarizabilities of hydrogen fluoride. A comparison of configuration interaction and perturbation theory methods. Chem. Phys. Lett., 88, 89-94. 

As seen in equation (26), the quasi-relativistic Hamiltonian and the operators describing the difference between the exact Dirac Hamiltonian and the quasi-relativistic one are now explicitly separated and the direct perturbation theory method can be applied. In the direct perturbation theory approach, the metric is also affected by the perturbation [12]. Note that the interaction matrix is block diagonal at the lORA level of theory, whereas the coupling between the upper and the lower components still appears in the metric. 

To see the overall picture of the benchmark test, the mean absolute deviations A are given for several methods and basis sets in Table II. These methods include the previously mentioned ACPF and CCSD methods, but also the MCPF (modified coupled pair functional) method [11], the MP2 (Mpller-Plesset second-order perturbation theory) method and the CCSD(T) method [12], where a perturbational estimate of the triple excitations has been added. The basis sets include the DZP basis discussed above and the nearly equivalent VDZ basis set, a DZ basis set... 

In this section we have tried to indicate the usefulness of sensitivity functions for the large number of applications. Considerable development is still required, however, before full benefit can be derived from perturbation theory methods for all these applications. Cross-section sensitivity studies, for example, will be more useful and reliable when cross-section error files supplement the cross-section files in present use. New computer code systems that can process these error files and perform sensitivity studies, allowing... 

Perturbation theory methods, the development of which will be useful. 

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