Perturbation theory, regular

Instead of estimating -/ 1 ln(exp(-/W(f)) directly using (5.44), one can use cumulant expansion approaches, as in regular free energy perturbation theory (see e.g., [20, 39] for combining cumulant expansions about the initial and final states). Unbiased estimators for cumulants can be used. Probably the most useful relations involve averages and variances of the work ... 

The free propagator in (1.8) determines other building blocks and the form of a two-body equation equivalent to the BS equation, and the regular perturbation theory formulae in this case were obtained in [9, 10]. 

Even though the total kernel in (1.23) is unambiguously defined, we still have freedom to choose the zero-order kernel Kq at our convenience, in order to obtain a solvable lowest order approximation. It is not difficult to obtain a regular perturbation theory series for the corrections to the zero-order ap>-proximation corresponding to the difference between the zero-order kernel Kq and the exact kernel Kq + 6K... 

The only apparent difference of the EDE (1.23) from the regular Dirac equation is connected with the dependence of the interaction kernels on energy. Respectively the perturbation theory series in (1.25) contain, unlike the regular nonrelativistic perturbation series, derivatives of the interaction kernels over energy. The presence of these derivatives is crucial for cancellation of the ultraviolet divergences in the expressions for the energy eigenvalues. 

Table 3 Optimized regular interatomic distance (in A) and gain in energy per C2H2 unit (in eV) of regular polyacetylene by report to the alternating ground state, with the Neel state based higher order and the RVB ansatze. Results from Local-density-functional approach of Mintmire and White [48], Hartree-Fock and Mpller-Plesset perturbation Theory of Suhai [50], or Ashkenazi [49] are included.

Patkowski K, Jeziorski B, Szalewicz K (2001) Symmetry-adapted perturbation theory with regularized Coulomb potential. J Mol Struct (Theochem) 547 293-307... 

Here, we explain the origin of the term singular. It is used in contrast with the expression regular. In general, regular perturbation theory presupposes that the solution obtained by setting s = 0 resembles the one for a small and positive E. However, for Eq. (4), there exist no solutions at all for arbitrary initial conditions when we set s = 0. 

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