Orbital orthogonality

Most radicals located on double bonds (e.g. 4, 5) or aromatic systems (e.g. 6) are a-radicals. The free spin is located in an orbital orthogonal to the it-bond system and it is not delocalized. The orbital of the vinyl radical (4) containing the free spin can be cis- or trans- with respect to substituents on the double bond. The barrier for isomerization of vinyl radicals can be significant with respect to the rate of reaction. 

The K conjugate molecules usually have planar geometries and no difference between the two faces above and below the molecular plane. When substitutions break the symmetry with respect to the plane, n orbitals mix a orbitals orthogonal prior to the substitution. Rehybridization occurs and the unsaturated bonds have... 

Some small molecules have a filled and vacant orbital available on the same atom for bonding to other atoms. Sulphur dioxide, carbon monoxide and singlet carbenes are examples. In each case one atom (sulphur or carbon) has a lone pair electrons in the plane of the molecule and a vacant p orbital orthogonal to it. 

Heteroatoms (such as O, S, N) that are double-bonded to other atoms in a ring can t donate lone-pair electrons to the pi system because their p-electrons are already involved in the double bond. Single-bonded heteroatoms can donate a single lone-pair to the pi system but not two, because one lone-pair must be in an unhybridized p orbital orthogonal (at 90 degrees) to the sp ring plane. 

In order to explain the stabilization of the CT excited state even in a nonpolar solvent, the orthogonal intramolecular charge-transfer (OICT) mechanism, which is a conceptually similar term to the TICT mechanism but is used to stress the orbital orthogonality rather than the twisting of the molecular frame, has been proposed. The OICT mechanism for the CT fluorescence of aryldisilanes in nonpolar solvents was supported by the remarkable dependence of the fluorescence spectra on the dihedral angle between a benzene pjr... 

Since a CASSCF calculation is faster than a direct SC calculation, owing to the advantages associated with orbital orthogonality in CASSCF, it is practical to extract an approximate SC wave function (or another type of VB function, e.g., a multiconfigurational one) from a CASSCF wave function. The conversion from one wave function to the other relies on the fact that a CASSCF wave function is invariant under linear transformations of the active orbitals. Based on this invariance principle, two different procedures were developed and both share the same name CASVB . Thus, CASVB is not a straightforward VB method, but rather a projection method that bridges between CASSCF and VB wave functions. 

Hartree s original idea of the self-consistent field involved only the direct Coulomb interaction between electrons. This is not inconsistent with variational theory [163], but requires an essential modification in order to correspond to the true physics of electrons. In neglecting electronic exchange, the pure Coulombic Hartree mean field inherently allowed an electron to interact with itself, one of the most unsatisfactory aspects of pre-quantum theories. Hartree simply removed the self-interaction by fiat, at the cost of making the mean field different for each electron. Orbital orthogonality, necessary to the concept of independent electrons, could only be imposed by an artificial variational constraint. The need for an ad hoc self-interaction correction (SIC) persists in recent theories based on approximate local exchange potentials. 

So far we have considered the more usual orthogonal orbital type wave-functions in which the constraints are those of orbital orthogonality. However, for wavefunctions in which orbital orthogonality is not required (or for more general wavefunctions) the above discussion need not apply since in these cases it is possible to use an unconstrained minimization method directly on the functional... 

Please note that the alternative conjugation1 shown in the structure below Is wrong. The structure with two adjacent doubte bonds in a six-membered ring is Impossible and, in any case, as you saw in Chapter 8, the lone pair electrons on nitrogen are in an sp2 orbital orthogonal to the p orbitals in the ring. There is no interaction between orthogonal orbitals. 

Sophisticated theoretical pictures of carbonyl compounds all point out that the M-orbital and the C—-C a orbitals are mixed. Therefore, an n,n excitation intrinsically weakens the carbonyl-a-carbon bond and predisposes that bond to rupture. However, the actual cleavage process is certainly distinct from the electronic transition and apparently follows potential surfaces analogous to those followed by alkoxy radicals in their -cleavage reactions. Mixing of the n-orbital with the C—C a orbital is presumably very similar in the transition states for both kinds of cleavage and involves three electrons. The n electron in the excited state reaction is in an orbital orthogonal to the reaction coordinate and is not apparently involved in the reaction. 

At a later date, however, it was the MO approach that carried the day. The main reason for its success was not only its conceptual simplicity, but also the mathematical convenience of the orbited orthogonality that drastically simplified any quantitative computations. Only relatively recently, and thanks to our better understanding of suitable AO-like basis sets [30], has the VB approach, with its appeal to a chemist s intuition and its direct link with Lewis electron pairing ideas started to enjoy a definite renaissance... 

In the case of the group V jt-excessive heterocycles, the single lone pair of electrons is associated with a p orbital orthogonal to the plane of the ring and overlap occurs with the p orbitals of the carbon framework to give a complement of six n electrons, the aromatic sextet. This involvement of the pair of electrons from the heteroatom with the closed aromatic shell effectively renders the group V 7t-excessive heterocycles nonbasic. 

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