The Coupled Pair Functional Method

We shall now demonstrate the second approach towards variational pair theory. A convenient starting point is the classiflcation of the problems arising from the use of CI(SD). Again we shall rely on PT as a tool for investigations. 

In order to motivate the following modiflcations of the Cl(SD) energy expression, we reconsider the statement concerning Eq. (48). The CI(SD) correlation energy may be written 

This fact is exploited by Davidson s correction which recommends omitting the second term and the sixth-order terms in (62). This prescription, however, suffers from two basic shortcomings. First, the formula tends to overshoot the contributions of higher excitations, which is obvious for the two-electron case where no correction should be made. Secondly, the Davidson corrected energy expression is evaluated with the CI(SD) wavefunction. One thereby underestimates the cluster corrections, as is easily verified for simple models such as He , where both the CI(SD) and the corrected correlation energies increase like n. 

The discussion so far reveals the CI(SD) denominator to be too large— which causes the size-consistency problem—which is rectified to some extent by the Davidson correction. A reasonable procedure to cure these shortcomings from the very beginning would be to modify the denominator in the CI(SD) energy expression in order to get rid of the size-consistency problem and then to treat the modified energy expression by means of a variational procedure. To be more specific, one would attempt to fix the denominator by forcing the whole procedure to give exact results for some representative but simple systems, for example He,. 

The approach just sketched was used in the derivation of the CPF method. We shall not go into all details but only give a short outline of our reasoning. The expression (61) for the CI(SD) correlating energy is rewritten in the form 

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Sagarik, K. R, Ahlrichs, R., and Erode, S., Intermolecular potentials for ammonia based on the test particle model and the coupled pair functional method. Mol. Phys. 57, 1247-1264 (1986). 

K. P. Sagarik, R. Ahlrichs, and S. Brode, Mol. Phys., 57, 1247 (1986). Intermolecular Potentials for Ammonia Based on the Test Particle Model and the Coupled Pair Functional Method. 

Kofraneck and coworkers24 have used the geometries and harmonic force constants calculated for tram- and gauche-butadiene and for traws-hexatriene, using the ACPF (Average Coupled Pair Functional) method to include electron correlation, to compute scaled force fields and vibrational frequencies for trans-polyenes up to 18 carbon atoms and for the infinite chain. 

Very similar in spirit to CEPA, but formulated as a functional to be made stationary, is the coupled-pair functional (CPF) approach of Ahlrichs and co-workers [28]. CPF can be viewed as modifying the CISD energy functional to obtain size-extensivity for the special case of noninteracting two-electron systems. One disadvantage of some of the CEPA methods is that, unlike CISD or CCSD, the results are not invariant to a unitary transformation that mixes occupied orbitals with one another. CPF... 

Theoretical chemistry at UBC was further strengthened with the arrival of Delano Chong and Keith Mitchell in 1965 and 1966, respectively. Chong s interests in quantum chemistry have spanned the full range from semiempirical to ab initio molecular orbital methods. His long-standing interests in perturbation methods and constrained variations have figured prominently in his publications. He is probably best known for his attempts to calculate the X-ray and UV photoelectron spectra of molecules, often by means of perturbation corrections to Koopmans theorem.40 More recently he has shifted his focus to coupled pair functional methods and density functional methods, with a special interest in polarizabilities and hyperpolarizabilities.41... 

Table 4 shows the results for the best LSD potential of Vosco, Wilk and Nusair [12] (DFT/L) compared with experiment [21] as well as with other theoretical extensive configuration interaction and coupled-pair functional method [23], where available. Four quintets, one doublet and two sextet states are included in the table. The overall impression which arises from inspection of numbers is that the LCGTO-LSD method works for vanadium oxide even better than could be expected for such a difficult example. Equilibrium bond distances approach the experimental values very closely, similarly to other spectroscopic constatnts even though they have been obtained from a very simple polynomial fit to an approximate, few-point curve. As a rule, the... 

Closely related to CEPA are the method of self-consistency electron pairs (SCEP) [138] and also the coupled-pair functional approach (CPF) [139]. 

To see the overall picture of the benchmark test, the mean absolute deviations A are given for several methods and basis sets in Table II. These methods include the previously mentioned ACPF and CCSD methods, but also the MCPF (modified coupled pair functional) method [11], the MP2 (Mpller-Plesset second-order perturbation theory) method and the CCSD(T) method [12], where a perturbational estimate of the triple excitations has been added. The basis sets include the DZP basis discussed above and the nearly equivalent VDZ basis set, a DZ basis set... 

Figure4.7 Relativistic bond contractions A re for Au2 calculated in the years from 1989 to 2001 using different quantum chemical methods. Electron correlation effects Acte = te(corn) — /"e(HF) at the relativistic level are shown on the right hand side of each bar if available. From the left to the right in chronological order Hartree-Fock-Slater results from Ziegler et al. [147] AIMP coupled pair functional results from Stbmberg and Wahlgren [148] EC-ARPP results from Schwerdtfeger [5] EDA results from Haberlen and Rdsch [149] Dirac-Fock-Slater...

Another point worth making is that since the SD-CI method is exact within the chosen basis set for a two-electron system, it must be size-consistent in this particular case. Nevertheless, when Davidson s correction is applied to an SD-CI wave-function for a two-electron system it will give a non-zero contribution, which is thus an artefact of this correction. (The same error appears also when the functional (10.1) is used with g=0.) This artefact can be simply removed and this is done in the Averaged Coupled Pair Functional (ACPF) method. In this method the factor g is considered to be a function of the number of electrons N, g=g(N), and one considers the special case of n separated He atoms. If the denominator De in (10.1) for one He atom is... 

The properties of oligomers of thiophene at the ground and excited states have been investigated by semi-empirical and ab initio methods. Semi-empirical calculations for the ground state of thiophene involve the QCFF/PI -f CISD method, while ab initio calculations are carried out at the HF/6-31G and the CASSF/3-21G" levels and also at the SCF and the averaged coupled-pair functional (ACPF) levels, with (basis set I) and without (basis set II) the sulfur 3d-orbitals. 

Two recent ab-initio studies are devoted to the determination of the ground-state structure of Cg. Parasuk and AlmlOf concluded that the cyclic polyacetylenic structure ( Ag) and the linear cumulenic g state are essentially isoenergetic [260]. However, the choice of basis sets and methods (MRCI modified coupled-pair functional, MCPF) heavily affects the difference in energy between the linear and cyclic forms of Cg, and so the results are quite uncertain. On the other hand, Slanina et al. proposed a nonplanar Z)2d symmetrical cyclic structure as the minimum-energy structure of Cg [261]. According to their MP2/6-31G calculations, this form is 13 kcal/mol lower in energy than the planar Qa structure. Unexpectedly, all the C-C bond lengths in the species are equivalent (1.339 A). DFT calculations favor the linear structure as compared with a planar cyclic one, but nonplanar species seem to have not been considered in this study [236]. 

One can further try to formulate a generalization of CEPA or rather CPF to a multiconfiguration reference function. On these lines the average coupled pair functional (ACPF) has been derived by Gdanitz and Ahlrichs [157]. A straightforward MC-CEPA method has been proposed by Fink and Staeramler [158]. This is extensive, while ACPF is only approximatively so. 

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