Localized representation

In the full quantum mechanical picture, the evolving wavepackets are delocalized functions, representing the probability of finding the nuclei at a particular point in space. This representation is unsuitable for direct dynamics as it is necessary to know the potential surface over a region of space at each point in time. Fortunately, there are approximate formulations based on trajectories in phase space, which will be discussed below. These local representations, so-called as only a portion of the FES is examined at each point in time, have a classical flavor. The delocalized and nonlocal nature of the full solution of the Schtddinger equation should, however, be kept in mind. 

Crl, Grr, A r and A r are the potential parameters of the constituents A and B of the alloy, S r r r, is the structure matrix in the most localized representation, tir are local site-occupation variables which randomly takes value 1 or 0 according to whether the site is occupied by an atom of type A or not, with probabilities proportional to the concentrations of the constituents. According to the prescription of the augmented space formalism, the effective non-random Hamiltonian H in augmented space is then... 

While the 4220 structure (3.255) is the uniquely best localized representation, strong delocalization is again evident in the strength of intramolecular donor-acceptor interactions. Most important by far are the four interactions involving donation from the Bi—B2 bond into each pair of adjacent Tbbb(a) and Tbhb(a) antibonds,... 

The specific realization (3.259) of the favored 0.10.3.0 styx pattern is only one of many equivalent localized representations. The central 1—4—5 triangle of Fig. 3.110 could equivalently be chosen as any of the 20 possible triangular facets in Fig. 3.111, leading to 20 equivalent three-center Lewis structures that contribute to the overall resonance delocalization of Bi2Hi22. The non-Lewis density of any such structure is found to be... 

Since rigorous theoretical treatments of molecular structure have become more and more common in recent years, there exists a definite need for simple connections between such treatments and traditional chemical concepts. One approach to this problem which has proved useful is the method of localized orbitals. It yields a clear picture of a molecule in terms of bonds and lone pairs and is particularly well suited for comparing the electronic structures of different molecules. So far, it has been applied mainly within the closed-shell Hartree-Fock approximation, but it is our feeling that, in the future, localized representations will find more and more widespread use, including applications to wavefunctions other than the closed-shell Hartree-Fock functions. 

The development of localized-orbital aspects of molecular orbital theory can be regarded as a successful attempt to deal with the two kinds of comparisons from a unified theoretical standpoint. It is based on a characteristic flexibility of the molecular orbital wavefunction as regards the choice of the molecular orbitals themselves the same many-electron Slater determinant can be expressed in terms of various sets of molecular orbitals. In the classical spectroscopic approach one particular set, the canonical set, is used. On the other hand, for the same wavefunction an alternative set can be found which is especially suited for comparing corresponding states of structurally related molecules. This is the set of localized molecular orbitals. Thus, it is possible to cast one many-electron molecular-orbital wavefunction into several forms, which are adapted for use in different comparisons fora comparison of the ground state of a molecule with its excited states the canonical representation is most effective for a comparison of a particular state of a molecule with corresponding states in related molecules, the localized representation is most effective. In this way the molecular orbital theory provides a unified approach to both types of problems. 

The scope of this work is to deal with the possible treatments of electron correlation in a localized representation. Several methods will be discussed in detail elaborated by present authors. Special attention will be payed to the analysis of the transferability of certain correlation energy contributions. The use of their transferability will be discussed for extended systems series of hydrocarbons and polyenes will be investigated. The transferable properties of the contributions to the correlation energy, furthermore, turned out to be useful in the study of weakly interacting intermolecular systems. A detailed description of this procedure will be given in the present work. 

Due to the localization terms entering the localized representation, an extra computational work is necessary. It represents only a small fraction of the total computing time, in a given order, because the number of indices to be summed up are always less in the localization terms than in the canonical ones. 

In the canonical representation the total energy correction of a given order n is denoted by he localized representation consists of two terms The... 

The actual calculation were performed through fourth order, the perturbation corrections in both the canonical and the localized representation for the interval -2 > P > - 10. The results for C H(, and CioT/jo were compared to those obtained by full CL It was shown (Kapuy et al., 1984 Kapuy et al, 1988) that the perturbation theory recovers a fraction of the total correlation correction. The localization correction are relatively small (compared to the canonical ones) in the localized representation. 

The zero differential overlap approximation can be applied in the localized representation. This was demonstrated by calculating for C H, CioTfio and C14//14, respectively the total energy corrections and the pair correlation energies through second and third order in different approximations. When the strongly local contributions were only... 

The following important conclusion can be drown, that the diagrammatic many-body perturbation theory can be used in a localized representation. 

The main advantage suggested by the use of the localized many-body perturbation theory (LMBPT) is that the local effects can be separated from the non-local ones. The summations in the corrections at a given order can be truncated. As to the practical applicability of the localized representation, a localization (separation) method, satisfying a double requirement is highly desired. Well-localized (separated) orbitals with small off-diagonal Lagrangianmultipliers are required (Kapuy etal., 1983). 

The cyclic polyenes (in PPP approximation) have been examined in (Kapuy et ah, 1984) too. The correlation energy contributions obtained for the zW.-trans polyenes further were analyzed both in canonical and localized representation, respectively (Kapuy et ah, 1994). The results are in agreement with those found for the smaller cyclic polyenes in the series. 

Another field, were the transferability property of certain correlation energy contributions can be demonstrated is in the study of weakly interacting systems in localized representation. Although the interaction energy between the systems will be discussed... 

Some other systems - beside the series of relative ones - have also been studied in localized representation, using the LMBPT scheme. Some of these systems have importance in biology. Two kinds of molecular systems have been investigated, each of them is related to the CH2O molecule which is now believed to be one of the most important small molecular species in human life. 

Table 4 Total and interaction energies obtained at the correlated MP2 level for some studied dimers in canonical and localized representation...

Several studies demonstrated that the canonical contributions at the correlated level (MP2, MP3, MP4) can be approximated to a very good extent by their corresponding values obtained in localized representation. The transferability of the intra-parts can be further used this makes it also possible to discuss conveniently the interaction energy of weakly bounded-systems in terms of SMOs. 

Treatment of Electron Correlation in Localized Representation Table 7 Correlated energy components, the E(intra/total) values calculated for some studied dimers... 

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