Localized molecular orbitals amplitudes

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,... 

Sometimes several different types of localized group orbitals contribute significantly to the molecular orbital. The molecular orbital is then labeled by listing all the different types of group orbitals which have a large amplitude in the overall molecular orbital. Thus... 

For Eq. (9.35) to be useful the density matrix employed must be accurate. In particular, localization of excess spin must be well predicted. ROHF methods leave something to be desired in this regard. Since all doubly occupied orbitals at the ROHF level are spatially identical, they make no contribution to P only singly occupied orbitals contribute. As discussed in Section 6.3.3, this can lead to the incorrect prediction of a zero h.f.s. for all atoms in the nodal plane(s) of the singly occupied orbital(s), since their interaction with the unpaired spin(s) arises from spin polarization. In metal complexes as well, the importance of spin polarization compared to tire simple analysis of orbital amplitude for singly occupied molecular orbitals (SOMOs) has been emphasized (Braden and Tyler 1998). 

A theoretical study of the effects of structure and substituents on reactivity in allylboration has recently been completed [116]. Electron delocalization from the oxygen of an attacking aldehyde to the boron p-type atomic orbital is crucial in the allylboration reaction. The ab initio molecular orbital study indicates that the complex between the allylic borane and the aldehyde is weak. The relative elec-trophilicity of the boron atom in allylboron reagents is estimated by projecting out the unoccupied reactive orbital having the maximum amplitude onto the boron p-type atomic orbital. Two factors which are considered to be of importance in the reaction include the electron accepting level of the reactive unoccupied orbital and the efficiency of localization of the orbital in the unoccupied MO space. 

Finally, according to Equation (1.39) the transition moment also vanishes if the differential overlap is zero everywhere in space. This is not strictly possible (except within the ZDO approximation), but the product lvery small values if the amplitude of MO only is large in those regions of space where the amplitude of tpi is very small, and vice versa. Consequently the transition dipole moment will also be very small in such a case. This is true for n- r transitions, where an electron is excited from a lone-pair orbital in the molecular plane into a n orbital of an unsaturated system, for which the molecular plane is also a nodal plane. Similar reasoning applies to so-called charge transfer transitions, that is, those in which an electron is transferred from one subsystem to another, the orbitals different regions of space. Such transitions are overlap forbidden. This is not, however, a very strict selection rule, since it is not based on a vanishing but only on a small value of the transition moment. The differential overlap is never exactly zero in practice. 

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