Second-order MP theory

BH is actually atypical. For most other molecules, the MP series characteristically exhibits an oscillatory behavior [92], as found, for example, in a previous study of the NNO molecule [52]. Unfortunately, however, no other FCI shieldings (apart from H2) are available and comparisons with experimental data are necessary for further validation of the methods. Such a comparison is shown in Figure 6.9 for N2. Again, this is a case with large correlation effects of about 55 ppm. Second-order MP theory overestimates the correlation contribution by nearly 20 ppm (36%), while MP3 underestimates it by 15 ppm (27%). Fourth-order MP again overestimates the correlation corrections, in particular when triple excitations are included. On the other hand, the CC results are very accurate, thus lending support to the general belief that near quantitative accuracy is achieved with these methods. 

Electron correlation does not appear to be of crucial importance to evaluation of the optimal angular characteristic of these complexes. The a angle changes by about 10° or less when second-order MP theory is employed. On the other hand, if one wishes to calculate the height of the energy barrier for inversion separating the two equivalent geometries, correlation does become important. For example, the SCF value of this barrier is only 45 cm for H2O HF, as compared to an MP2 value of 144 cm" [35]. 

The quality of these basis sets can be demonstrated by the example of the H2 molecule. The HF energy extrapolated to the complete basis set is hf = -2976.26 kJ mol at the optimal interatomic distance of / = 0.1386 nm. Suppose, for instance, that the fifth basis set in Table 1 is used, then we obtain hf = -2973.07 kJ mol and R = 0.1398 nm. Application of second-order MP theory yields for hydrogen in this basis set a correlation energy of -72.94 kJ mol as compared with the exact value of -107.09 kJ mol . ... 

A very recent thorough set of calculations by Del Bene [36] has applied a large doubly polarized basis set, in tandem with full fourth-order MP theory, to binary complexes including HOH. Her results suggest that going beyond the second order has very little influence upon the calculated energetics of these systems, regardless of the basis set employed. 

MR-MP second-order perturbation theory are close to the experimental values, and the terms arising from corrections to the photon frequency Siv = in both zero- and first-order transition amplitudes... 

The Moller-Plesset second-order perturbation theory (MP/2) is comparatively simple because only matrix elements of the form Po v 0 ) need to be calculated, while in third order there are already matrix elements of type V 0 (as in a Cl calculation). There are many more of these matrix elements (even for an atomic basis of middle quality there are many more virtual than occupied bands) than those corresponding to excitations between the ground state and the different doubly excited states. 

It was seen in Section 5.3 that to determine the QP band structures of a polymeric chain one must use a size-consistent method to determine the major part of the correlation [many-body perturbation theory (MBPT) in the Moller-Plesset partitioning, coupled-cluster theory, etc.]. Suhai, in his QP band-structure calculation on polyacetylenes and polydiace-tylenes, used second-order (MP/2) Moller-Plesset MBPT. For polydiacetylenes he obtained 5.7 eV as first ionization potential (using the generalized Koopmans theorem) for the PTS structure (see Figure 8.1), in reasonable agreement with experiment (A = 5.5 0.1 while the HF value (the simple Koopmans theorem) is 6.8 eV.< > For the TCDU diacetylene structure the theoretical value is 5.0 eV (HF value, 6.2 eV). Unfortunately, there is no reliable experimental ionization-potential value available for the TCDU structure of polydiacetylene. 

Frequency-dependent response functions can only be computed within approximate electronic structure models that allow definition of the time-dependent expectation value. Hence, frequency-dependent response functions are not defined for approximate methods that provide an energy but no wave function. Such methods include MoUer-Plesset (MP) perturbation theory, multiconfigurational second-order perturbation theory (CASPT2), and coupled cluster singles and doubles with non-iterative perturbative triples [CCSD(T)]. As we shall see later, it is possible to derive static response functions for such methods. 

This article is organized in the following way. In Section 2 a brief summary (without any derivations) of the ab initio HF CO theory will be given. In Section 3 the MP second-order perturbation theory applied for polymers will be presented. In this way one can correct for correlation the total energy per unit cell of a periodic polymer. 

CC) methods, which have largely superseded Cl methods, in the limit can also be used to give exact solutions but again with same prohibitive cost as full Cl. As with Cl, CC methods are often truncated, most commonly to CCSD (N cost), but as before these can still only be applied to systems of modest size. Finally, Moller-Plesset (MP) perturbation theory, which is usually used to second order (MP2 has a cost), is more computationally accessible but does not provide as robust results. 

The original ab initio approach to calculating electronic properties of molecules was the Hartree-Fock method [31,32,33,34]. Its appeal is that it preserves the concept of atomic orbitals, one-electron functions, describing the movement of the electron in the mean field of all other electrons. Although there are some inherent deficiencies in the method, especially those referred to the absence of correlation effects. Improvements have included the introduction of many-body perturbation theory by Mollet and Plesset (MP) [35] (MP2 to second-order MP4 to fourth order). The computer power required for Hartree-Fock methods makes their use prohibitive for molecules containing more than very few atoms. 

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