Localization criteria

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The following presentation is limited to closed-shell molecular orbital wave-functions. The first section discusses the unique ability of molecular orbital theory to make chemical comparisons. The second section contains a discussion of the underlying basic concepts. The next two sections describe characteristics of canonical and localized orbitals. The fifth section examines illustrative examples from the field of diatomic molecules, and the last section demonstrates how the approach can be valuable even for the delocalized electrons in aromatic ir-systems. All localized orbitals considered here are based on the self-energy criterion, since only for these do the authors possess detailed information of the type illustrated. We plan to give elsewhere a survey of work involving other types of localization criteria. 

Since then, a variety of other reasonable localization criteria have been proposed and investigated. A survey and discussion of this aspect of localization theory is given by Weinstein, H., Pauncz, R., Cohen, M. Advances in Atomic and Molecular Physics, Vol. 7. New York Academic Press 1971. 

All these statements, although correct in principle, are not precise from the technical point of view. For example, the zero approximate wave function in the PCILO method is a one-electron approximate function constructed from the bond wave functions determined by an a posteriori localization procedure from an HFR function. Thus the bond orbitals appear after a unitary transformation of the canonical MOs, which correspond to some more or less arbitrary localization criteria [123-125]. 

Among the non-local criteria, the possible role of the long range electrostatic interactions was recognized quite early on. On the basis of a simple energy decomposition scheme, where the OH bond energy is considered as the sum... 

These two criteria appear to be generalizations of the localization criteria named after Edmiston-Ruedenberg and Boys respectively [1, 2, 3]. Nevertheless the pair functions ipp, constructed by either of these criteria, are generally symmetry-adapted (in spite of having localized features as well) unlike for the one-electron case there is no strict alternative symmetry-adapted vs. localized [5],... 

The direct method of Gilbert leads to a special eigenvalue problem by the introduction of a so-called localization potential [19]. The solutions of the equation corresponding to the above problem provide directly the localized orbitals. This method is of theoretical interest, as using the formalism of the localization potential all of localization methods can be interpreted in a common way. The various localization criteria correspond to functionals of extreme value problems. These latters can be solved by using the variation method, and the solutions are the best localized orbitals (corresponding to a given localization criterion). 

The adoption of different localization criteria does not lead to equivalent orbitals, as the given procedure usually applies iterative method. The iteration is repeated until the criterion chosen is fulfilled. In spite of this, LMOs obtained by different algorithms are quite similar. This similarity increases the hope, that a) the LMOs imply a serious, general (even interpretative) significance in the study of molecular electronic structure and b) the use of LMOs (especially their transferability property) might be helpful in studying extended/weakly interacting systems. 

The same technique of successive 2x2 rotations applies to other localization criteria. 

As noted earlier, a necessary and sufficient condition for TST to be exact at a dividing surface is that any classical trajectory crossing the surface will never recross it. It is of practical interest to determine when TST must fail. Usually, the complete potential energy surface is not available so that one would want to develop local criteria for the failure of TST. Here we will shoxv that the stability properties of periodic orbits can be used to answer this question. Loosely speaking, a periodic orbit is defined as stable in the sense of Lyapunov if every trajectory originating at t=0 close enough to the periodic orbit remains close to the orbit for all time t. Obviously, TST cannot be exact if the pods is stable, since there are, by definition, trajectories crossing the pods that stay in its vicinity forever. For these trajectories and so TST is not exact. 

Pancif [17] and Basilevsky/Shamov [18] were the first who formulated local criteria for describing a valley floor line. Pancir determined two conditions which he assumed to be obviously given ... 

The localized orbitals generation from the set of canonical MO allowed by symmetry requires additional choice of the localization criteria. The different orbital localization methods are implemented in molecular computer codes [35]. All of them are connected with the search for the coefficients connecting LMO with the canonical MO to satisfy the localization criteria. 

The localized molecular orbitals (LMO) are extensively used not only for the chemical-bonding analysis in molecules but also in the local correlation methods [41] (we consider the problem of electron correlation in molecules and crystals in Chap. 5). The LMO are generated from the canonical MO occupied by electrons and found in the Hartree-Fock or DFT calculations. This generation is based on one or other localization criteria [42]. 

Localized crystalUne orbitals (LCO) are generated from a canonical delocahzed Bloch functions (CO). As in the case of the molecules one or other localization criteria is used. The orthonormaUzed LCO in crystals are known as Wannier functions. Wannier functions (WFs) have attracted much attention in solid-state physics since their first introduction in 1937 [43] and up to now. The analytical behavior of Bloch... 

Disadvantages of the method an efficient procedure for the case when the lattice has no the center of inversion it is not developed the WF maximal localization criteria is not formulated the method is complex enough in realization. 

Advantages of the method the algorithm is efficient and universal, it allows one to construct quickly well localized WF the procedure is completely automated and included in the computer code CRYSTAL. The applications of this method to actual crystals are discussed in Chap. 9. Disadvantages of the method in this method it is possible to use only Bloch functions calculated in LCAO approximation (PW-based Bloch functions can not be used) there is no possibility to apply different localization criteria in WF generation the Muthken atomic populations used in the iteration procedure are essentially dependent on the AO basis chosen and for diffuse AO are sometimes unrealistic. 

An attempt to overcome the disadvantages mentioned was made in [42] where a variational method of Wannier-type function generation was suggested. This method is applicable with the different localization criteria, the Bloch functions can be calculated both in LCAO and in PW basis, the full s3Tnmetry is taken into account. In the next section we consider it in more detail. 

The essential progress in the correlation effects inclusion was achieved in the so-called local correlation methods [109,110] taking into account the short-range nature of the correlation. In these methods, the localized MOs are generated from the occupied canonical MOs using different localization criteria, see Sect. 3.3.1. For the virtual space the atomic-orbital basis is projected out of the occupied MO space. 

The LCAO approximation allows comparison of the electron-charge distribution in the bulk crystal and on the surface. The application of Wannier functions for this comparison can be found in [781], where the Boys localization criteria was appUed and only the valence-band states were included for WFs generation (for bulk crystals such an approach is discussed in Chap. 9). As was seen in Chap. 9 for bulk crystals, the population analysis using Wannier-type atomic orbitals (WTAO) gives more adequate... 

In addition to the exclusionary and discretionary criteria listed above, the SSA has identified other criteria which will need to be considered at the site specific planning/design stage for any sites identified in the Nuclear NPS that are subsequently taken forward towards development. These (local) criteria are ... 

The transformational freedom of the occupied MOs gives the possibility to localize them into certain regions of space leading to MOs with a shape corresponding to classical chemical expectations (bonds, lone pairs, etc.). Several localization criteria have been proposed which define particular unitary transformations. We do not discuss this issue here since it is not directly related to second quantization. The interested reader is referred to the book of Chalvet et al. (1975) and the review of Surjan (1989). 

Our approach relies on the use of localized occupied molecular orbitals (LMOs) that can be obtained relatively easily by a unitary transformation in the subspace of occupied orbitals, according to either an external or an internal localization criterion [19]. The localization of virtual orbitals is much more difficult, since the usual localization criteria for the occupied orbitals lead often to divergent results. It is to be noted that recently a significant progress has been reported [20] for the efficient localization of virtual orbitals. However, we follow another strategy here and build... 

The a posteriori localization of the subspace of the occupied orbitals is a relatively standard procedure, which can be achieved following a large variety of localization criteria (for a succinct overview, see Ref. [53]). In the context of correlation energy calculations, i.e., in various local correlation approaches , the most widely used localization methods are based either on the criterion of Foster and Boys [24] or that proposed by Pipek and Mezey [54]. For reasons which become clearer below, in the present work we will use the Foster-Boys localization criterion, which can be expressed in various equivalent forms [26]. In its the most suggestive formulation, the Foster-Boys localization procedure consists in the maximization of the squared distance between the centroids of the orbitals ... 

While most of the focus on and use of localized orbitals has been with regard to closed shell restricted Hartree-Fock (RHF) wavefunctions. the applications of localization criteria are considerably more general. Of particular importance is the fact that complex multi-conligurational wavefunctions can also be localized, as long as the full active space of electrons and orbitals, according to the FORS or CASSCF prescription, is included. MCSCF localized orbitals are particularly useful for the interpretation of bonding. Consider, for example, a typical double bond between a transition metal (M) and a main group element (E). A simple RHF description of such a bond is provided by the wavefunction... 

The local lubricant film breakdown, promotes an increase in the local friction coefficient caused by metal/metal contact, which is extended to the global friction coefficient. Being so, the probability of lubricant film breakdown is strong for high loads and speeds. It s clear that local criteria for lubricant film breakdown must be developed and introduced into the mixed film lubrication model. 

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