Perturbation theory convergence properties

Since his appointment at the University of Waterloo, Paldus has fully devoted himself to theoretical and methodological aspects of atomic and molecular electronic structure, while keeping in close contact with actual applications of these methods in computational quantum chemistry. His contributions include the examination of stability conditions and symmetry breaking in the independent particle models,109 many-body perturbation theory and Green s function approaches to the many-electron correlation problem,110 the development of graphical methods for the time-independent many-fermion problem,111 and the development of various algebraic approaches and an exploration of convergence properties of perturbative methods. His most important... 

Using a numerical decomposition of the spectral density which describes the coupling of the system to the environment allows one to develop TL and TNL non-Markovian QMEs. Using the hierarchical approach the results can be extended from second-order perturbation theory to higher orders to be able to study the convergence properties of the different approaches. As shown in the example for bosonic baths, the TL formalism shows numerically almost converged results. Actually, this numerical finding has been analytically proven... 

Various symmetry forcing schemes, convergence properties, and asymptotic behavior are summarized in Table 1-3. An inspection of this Table leads to quite pessimistic conclusions. All perturbation theories that are asymptotically consistent... 

Table 1-3. Summary of the symmetry forcing operators, convergence properties, and asymptotic correctness of various symmetry-adapted perturbation theories...

Quasi-degeneracy Effects.—The convergence properties of the non-degenerate formulation of the many-body perturbation theory deteriorate when quasidegeneracy is present in the reference spectrum. In view of its simplicity, however, there is considerable interest in exploring the range of applicability of the nondegenerate formalism. 

Kirtman and Luis review some of the theoretical/computational methods which have been proposed over the past fifteen years for the calculation of vibrational contributions to the linear and NLO properties. They discuss (i) the time-dependent sum-over-states perturbation theory and the alternative nuclear relaxation/curvature approach, (ii) the static field-induced vibrational coordinates which reduce the number of n -order derivatives to be evaluated, (hi) tire convergence behavior of the perturbation series, (iv) an approach to treat large amplitude (low frequency) vibrations, (v) the effect of the basis set and electron correlation on the vibrational properties, and (vi) techniques to compute the linear and NLO properties of infinite polymers. 

Taylor series at second order is a questionable approximation. This approximation is justified if the third- and higher-order direct correlation functions of the liquid phase are negligibly small, but this does not appear to be the case. In particular, Curtin [130] and Cerjan et al. [132] have studied the effect of including third-order terms in the perturbation series, and have found that the agreement with computer simulations is significantly worse than for the second-order theory. This clearly shows that third-order (and perhaps higher-order) terms are important, and that the convergence properties of the perturbation series are poor. 

Since the symmetry-adapted perturbation theory provides the basis for the understanding of weak intermolecular interactions, it is useful to discuss the convergence properties of the sapt expansion. High-order calculations performed for model one-electron (Hj) (30), two-electron (H2) (14, 15), and four-electron (He and He-Hz) (31) systems show that the sapt series converges rapidly. In fact, already the second-order calculation reproduces the exact variational interaction energies with errors smaller than 4%. Several recent applications strongly indicate that this optimistic result holds for larger systems as well. 

Perturbation theory approach appears to be the most natural tool for theoretical investigations of weak intermolecular interactions (1). It provides the basis for present understanding of interactions between atoms and molecules, and defines the asymptotic constraints (via the multipole expansion (2, 3)) on the interaction potential. It is not surprising, then, that since the early 1970 s the convergence properties of various perturbation expansions for the intermolecular interaction energies are subject of extensive theoretical studies (4)-(23). 

The SRS perturbation theory is the simplest of all SAPT expansions proposed thus far, and in view of numerous recent applications to complexes of direct experimental interest (see Refs. (1, 31) for reviews) it is important to study the convergence properties of this expansion, and its applicability in low orders to interactions of many-electron systems. When... 

In the present paper we investigate the convergence properties of the RS, SRS, and HS perturbation series for He2 and HeH2 molecules, i.e. for the interaction of two ground-state two-electron systems. These perturbation formalisms correspond to none, weak, and strong symmetry forcing, respectively. In Sec. II the perturbation equations of the RS, SRS, and HS theories are briefly summarized. In Sec. Ill the computational details of the calculations are presented. The numerical results are presented and discussed in Sec. IV. 

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