Molecular orbitals localized orthogonal

Step 2. The set of CMOs orthogonal molecular orbitals (LMOs) Xj using, e.g., Ruedenberg s localization criterion205. This is achieved by multiplying up with an appropriate unitary transformation matrix L ... 

In order to construct localized orbitals for molecules, it is necessary to define a measure for the degree of localization of an arbitrary set of molecular orbitals. The localized orbitals are then defined as that set of orthogonal molecular orbitals obtained by a transformation of the type given in Eq. (5), for which the measure of localization has the maximum value. It is clear that the resulting localized orbitals will depend, at least to some degree, upon the choice of the localization measure. In the present work the localized molecular orbitals are defined as those self-consistent-field orbitals which maximize the localization sum 14)... 

All lone pair orbitals have a node between the two atoms and, hence, have a slightly antibonding character. This destabilizing effect of the lone pair localized molecular orbitals corresponds to the nonbonded repulsions between lone pair atomic orbitals in the valence bond theory. In the MO theory all bonding and antibonding resonance effects can be described as sums of contributions from orthogonal molecular orbitals. Hence, the nonbonded repulsions appear here as intra-orbital antibonding effects in contrast to the valence-bond description. 

Aromatic systems play a central role in organic chemistry, and a great deal of this has been fruitfully interpreted in terms of molecular orbital theory that is, in terms of electrons moving more-or-less independently of one another in delocalized orbitals. The spin-coupled model provides a clear and simple picture of the motion of correlated electrons in such systems. The spin-coupled and classical VB descriptions of benzene are very similar, except for the small but crucial distortions of the orbitals. The localized character of the orbitals allows the electrons to avoid one another. Nonetheless, the electrons are still able to influence one another directly because of the non-orthogonality of the orbitals. 

We have proposed two types of CASVB method. The first one is a method where the valence bond structures are constructed from orthogonal localized molecular orbitals (LMOs) [1], and the second is one from nonorthogonal localized molecular orbitals [2]. 

Fig. 2. CASSCF, orthogonal localized, non-orthogonal localized, and generalized valence bond molecular orbitals for the hydrogen molecule.

Eq. (5) where the are localized molecular orbitals (LMOs) which are related to the familiar CMOs by a unitary transformation. Because of the orthogonality of molecular orbitals the autocorrelation function for the MO approximation is simply the sum of the time-dependent probability amplitudes for the hole to be in the various LMOs,... 

It is not necessary to choose the orthogonal orbitals q> and

simple model we consider all configurations that can be constructed from the two nonbonding orbitals by any permissible electron occupancy, the energies and wave functions of the resulting states are invariant to any mixing of these orbitals. Thus, we may equally well choose them to be the most delocalized orthogonal molecular orbitals, related to the most localized ones by... 

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