Deviations, perturbation theory

The results of the Debye theory reproduced in the lowest order of perturbation theory are universal. Only higher order corrections are peculiar to the specific models of molecular motion. We have shown in conclusion how to discriminate the models by comparing deviations from Debye theory with available experimental data. 

As can be seen from Table I, the C-C bond distance as described by LDF is closer to experiment than the corresponding HF value obtained with a 6-3IG basis. Including correlation via second and third order Moller-Plesset perturbation theory and via Cl leads to very close agreement with experiment. The C-H bond length is significantly overestimated in the LDF calculations by almost 2%. The HCH bond angle is reasonably well described and lies close to all the HF and post-HF calculations. Still, all the theoretical values are too small by more than one degree compared with experiment the deviation from experiment is particularly pronounced for the semi-empirical MNDO calculation. 

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. 

The G3SX method based on the third-order perturbation theory, G3SX(MP3), is especially noteworthy in that it has a mean absolute deviation of 1.04 kcal/mol for the 376 energies in the G3/99 test set and 0.90 kcal/mol for the 222 enthalpies of formation. In this respect, it is as accurate as G3 theory and much less expensive. All of the G3SX methods have the advantage of being suitable for studies of potential energy surfaces. 

P. Gaspard To answer the question by Prof. Marcus, let me say that we have observed, in particular in Hgl2, that higher order perturbation theory around the saddle equilibrium point of the transition state may indeed be used to predict with a good accuracy the resonances just above the saddle. However, deviations appear for higher resonances and periodic-orbit quantization then turns out to be in better agreement than equilibrium point quantization. 

The autoionizing two electron states we have considered so far are those which can be represented sensibly by an independent electron picture. For example, an autoionizing Ba 6pnd state is predominantly 6pnd with only small admixtures of other states, and the departures from the independent electron picture can usually be described using perturbation theory or with a small number of interacting channels. In all these cases one of the electrons spends most of its time far from the core, in a coulomb potential, and the deviation of the potential from a coulomb potential occurs only within a small zone around the origin. 

Let us now embed the renormalization group, Constructed in Chap. 8, iftto this general framework. As mentioned above, relation (8.5) shows that the RG we are searching for must be a nonlinear representation of the group of dilatations in the space of parameters. , n,/ e). These are the microscopic parameters of the model, and the representation shall leave macroscopic observables invariant. Furthermore we want the representation to show a nontrivial fixed point. In Sect. 8.2 we have constructed such a representation based on first order perturbation theory. The invariance constraint is obeyed within deviations of order 1+e 2, no = n(A = 1). Equations (8.38), (8.42) give the parameter flow under this nonlinear representation in the standard form (10.28),... 

Comparison of the experimental data to the CASPT2 values shows that for the large majority of states the calculated value does not deviate more than 0.2 eV from the experimentally determined one. Exceptions are the 4A2g state in Co-O (1.79 vs. 2.14) and the lower d-d transitions in MnO, for which the calculated transition energies are too high. In none of these cases are there indications from the perturbation theory that intruder states artificially affect the calculated transition energies. Hence, it is unlikely that the discrepancy between calculated and experimental values can be attributed to an incorrect or incomplete treatment of the electron correlation effects. 

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