Perturbation series

MP3 has not been included in the above comparison. As aheady mentioned, MP3 results are often inferior to those at MP2. In fact MP2 often gives surprisingly good results, especially if large basis sets are used." Furthermore, it should be kept in mind that the MP perturbation series in many cases may actually be divergent. 

The improvement brought about by extending the perturbation series beyond second order is very small when a UHF wave function is used as the reference, i.e. the higher-order terms do very little to reduce the spin contamination. In the dissociation limit the spin contamination is inconsequential, and the MP2, MP3 and MP4 results are all in... 

The HF level as usual overestimates the polarity, in this case leading to an incorrect direction of the dipole moment. The MP perturbation series oscillates, and it is clear that the MP4 result is far from converged. The CCSD(T) method apparently recovers the most important part of the electron correlation, as compared to the full CCSDT result. However, even with the aug-cc-pV5Z basis sets, there is still a discrepancy of 0.01 D relative to the experimental value. 

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. 

Of the many quantum chemical approaches available, density-functional theory (DFT) has over the past decade become a key method, with applications ranging from interstellar space, to the atmosphere, the biosphere and the solid state. The strength of the method is that whereas conventional ah initio theory includes electron correlation by use of a perturbation series expansion, or increasing orders of excited state configurations added to zero-order Hartree-Fock solutions, DFT methods inherently contain a large fraction of the electron correlation already from the start, via the so-called exchange-correlation junctional. 

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]. 

Within the multireference BWPT [45], the exact wave functions for o = 1,..., d can be expanded in the Brillouin-Wigner (BW) perturbation series as follows... 

Certainly, we can took into account only a finite number of terms in the perturbation series (13). Let us assume that we perform calculations to the pth order of PT. If we use the M ller-Plesset PT then p < 4. The expression for AE2°" N) is easily obtained from Eqs. (19), (24), and (25)... 

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. 

The parameter is introduced to keep track of the order of the perturbation series, as will become clear. Indeed, one can perform a Taylor series expansion of the perturbed wave functions and perturbed energies using X to keep track of the order of the expansions. Since the set of eigenfunctions of the unperturbed SE form a complete and orthonormal set, the perturbed wave functions can be expanded in terms of them. Thus,... 

Figure 7.5 Slowly oscillatory behavior of MP /6-31G(d)//HF/6-31G(d) theory for the energy separation between carbonyl oxide and dioxirane. Accurate extrapolation from this perturbation series is an unlikely prospect...

Finally, it should be noted that, with minor modification, the standard Davidson procedure is able to produce systematically the terms of the Moller-Plesset perturbation series to any order, provided that a full Cl space is used. This has been used to study the convergence properties of the perturbation series. 

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