Perturbation theory convergence

Claverie [M] has argued that if perturbation theory converges at all, it will converge to one of these... 

Electron correlation (how well does MoUer-Plesset perturbation theory converge for these problems )... 

As you can see, the computations necessary for these corrections become increasingly complicated as we go to higher-ordei When perturbation theory works well, the relatively little effort required for the first- and second-order corrections accounts for most of the perturbation. We can see that perturbation theory converges fairly quickly for the two electrons in ground state helium because the second-order correction to the energy is much smaller than the first-order correction. Similarly, Fig. 4.10 shows that the first-order, two-electron wavefunction is about 67% Is and about 30% ls 2s with much smaller contributions from other states. 

Things have moved on since the early papers given above. The development of Mpller-Plesset perturbation theory (Chapter 11) marked a turning point in treatments of electron correlation, and made such calculations feasible for molecules of moderate size. The Mpller-Plesset method is usually implemented up to MP4 but the convergence of the MPn series is sometimes unsatisfactory. The effect... 

It is seen that the result obtained is sensitive to both the molecular symmetry and the strength of collision y, which is a quantitative measure of the degree of correlation. However, the latter affects only correction to the Hubbard relation which appears in the second order in (t))2 linear dependence of product on (L/)2 for any y, but the slope of the lines differs by a factor of two, being minimal for y=l and maximal for y=0. In principle, it is possible to calculate corrections of the higher orders in (t))2 and introduce them into (2.91). In practice, however, this does not extend the application range of the results due to a poor convergence of the perturbation theory series. 

As we have already pointed out, the theoretical basis of free energy calculations were laid a long time ago [1,4,5], but, quite understandably, had to wait for sufficient computational capabilities to be applied to molecular systems of interest to the chemist, the physicist, and the biologist. In the meantime, these calculations were the domain of analytical theories. The most useful in practice were perturbation theories of dense liquids. In the Barker-Henderson theory [13], the reference state was chosen to be a hard-sphere fluid. The subsequent Weeks-Chandler-Andersen theory [14] differed from the Barker-Henderson approach by dividing the intermolecular potential such that its unperturbed and perturbed parts were associated with repulsive and attractive forces, respectively. This division yields slower variation of the perturbation term with intermolecular separation and, consequently, faster convergence of the perturbation series than the division employed by Barker and Henderson. 

Although applications of perturbation theory vary widely, the main idea remains the same. One starts with an initial problem, called the unperturbed or reference problem. It is often required that this problem be sufficiently simple to be solved exactly. Then, the problem of interest, called the target problem, is represented in terms of a perturbation to the reference problem. The effect of the perturbation is expressed as an expansion in a series with respect to a small quantity, called the perturbation parameter. It is expected that the series converges quickly, and, therefore, can be truncated after the first few terms. It is further expected that these terms are markedly easier to evaluate than the exact solution. 

Among various model H(0) s that could be considered, the best such model is evidently that for which the perturbative corrections are most rapidly convergent, i.e., for which /7(perl) is in some sense smallest and the model Em> and < 0) are closest to the true E and T. Perturbation theory can therefore be used to guide selection of the best possible H(0) within a class of competing models, as well as to evaluate systematic corrections to this model. 

Although any of these decompositions might be employed in the formal machinery of perturbation theory, one can expect that choices of F(0) for which the perturbation elements in (Per9 are small will lead to more rapid convergence, and thus serve as better models. 

It has been well known for some time (e.g. [36]) that the next component in importance is that of connected triple excitations. By far the most cost-effective way of estimating them has been the quasiper-turbative approach known as CCSD(T) introduced by Raghavachari et al. [37], in which the fourth-order and fifth-order perturbation theory expressions for the most important terms are used with the converged CCSD amplitudes for the first-order wavefunction. This account for substantial fractions of the higher-order contributions a very recent detailed analysis by Cremer and He [38] suggests that 87, 80, and 72 %, respectively, of the sixth-, seventh-, and eighth-order terms appearing in the much more expensive CCSDT-la method are included implicitly in CCSD(T). 

Many-body perturbation theory (MBPT) for periodic electron systems produces many terms. All but the first-order term (the exchange term) diverges for the electron gas and metallic systems. This behavior holds for both the total and self-energy. Partial summations of these MBPT terms must be made to obtain finite results. It is a well-known fact that the sum of the most divergent terms in a perturbation series, when convergent, leads often to remarkably accurate results [9-11]. 

This approximation can be justified from a perturbation theory viewpoint that assumes the smallness of ff and is analogous to the treatment of connected triples in CCSDT-1 [64]. The simplification in the equations allowed the CCSD(R12) and CCSD(T)(R12) methods to be implemented by a modest extension of the computational elements developed in the MP2-R12 implementations. Since they do not rely on the SA, they need an auxiliary basis set for the RI, but the rapid basis-set convergence can be obtained. 

On the convergence of the many-body perturbation theory second-order energy component for negative ions using systematically constructed basis sets of primitive Gaussian-type functions... 

Using the F ion as a prototype, the convergence of the many-body perturbation theory second-order energy component for negative ions is studied when a systematic procedure for the construction of even-tempered btisis sets of primitive Gaussian type functions is employed. Calculations are reported for sequences of even-tempered basis sets originally developed for neutral atoms and for basis sets containing supplementary diffuse functions. 

---