Perturbational molecular orbital theory

In principle, we can perform some sort of molecular orbital calculation on molecules of almost any complexity. It is, however, often extremely profitable to relate the properties of a complex system to those of a simpler one. Take, for example, the hydrogen atom in an electric field. It is much more instructive to see how the unperturbed levels of the atom are altered as a field is applied, than to solve the Schrodinger wave equation for the more complex case of the molecule with the field on. Analogously, to appreciate the orbital structure of complex systems it is much more insightful to start off with the levels of a simpler one and switch on a perturbation. 3.1-3.3 show three examples of different types of perturbations 

Thomas A. Albright, Jeremy K. Burdett, and Myung-Hv n Whangbo. 

Consider a set of unperturbed (zeroth order in the language of perturbation theory) orbitals with energy e corresponding to the left-hand side of 3.1-3.3. In general, these orbitals are given in terms of atomic orbitals as 

importantly, within the framework of perturbation theory, the new wave-functions that result after the perturbation have been switched on may be written as a linear combination of the unperturbed orbitals, that is, rj/i is given by 

This series represents a set of corrections to the unperturbed wavefunction, Subscripts are problematic in this chapter. Recall the convention that Greek characters are reserved for atomic orbitals and the Roman alphabet in italics is used for molecular orbitals (i.e., some combination of atomic orbitals whether they are for a molecule or a fragment in a molecule). Thus, stands for the /.tth atomic orbital in molecular orbital /. The order of the subscripts in equations 3.3 and 3.5 are important 4 stands for how much molecular orbital j mixes into molecular orbital i to the qth order of perturbation. Before the perturbation is switched on, t = I and all tji = 0. Later, the weight of j/rf (i.e., the mixing coefficient tg) has to be smaller than one since is normalized to unity just like r/rf itself. 

There is another extension of molecular orbital theory that organic chemists find quite useful. Perturbational molecular orbital (PMO) theory provides an estimate of the change in electronic energy levels and molecular orbitals that result from the interaction of one structure or molecular fragment having 

Weinhold, F, Landis, C. R. Valency and Bonding A Natural Bond Orbital Donor-Acceptor Perspective Cambridge University Press Cambridge, UK, 2005. 

Valence NBOs of ethene showing rcH (top), o-cc (middle), and Jicc (bottom) bonds as electron density surface (left) and contour (right) views. (Reproduced from reference 187.) 

4 APPLICATIONS OF MOLECULAR ORBITAL THEORY AND VALENCE BOND THEORY 

Since EJ, the second term on the right in equation 4.74 is a negative quantity. Therefore, E i E, meaning that the perturbation has lowered the energy of the fth level. We also have 

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This theory proves to be remarkably useful in rationalizing the whole set of general rules and mechanistic aspects described in the previous section as characteristic features of the Diels-Alder reaction. The application of perturbation molecular orbital theory as an approximate quantum mechanical method forms the theoretical basis of Fukui s FMO theory. Perturbation theory predicts a net stabilization for the intermolecular interaction between a diene and a dienophile as a consequence of the interaction of an occupied molecular orbital of one reaction partner with an unoccupied molecular orbital of the other reaction partner. 

Correlation of the effect of substituents on the rates of reactions with early transition states often is best accomplished in terms of perturbational molecular orbital theory, and polar effects can play a major role for such reactions [100, 101]. Essentially this theory states that energy differences between the highest occupied molecular orbital (HOMO) of one reactant and the lowest unoccupied molecular orbital (LUMO) of the other reactant are decisive in determining the reaction rate the smaller the difference in energy, the faster the predicted rate of reaction [102,103]. Since the HOMO of a free radical is the SOMO, the energy difference between the SOMO and the alkene HOMO and/or LUMO is of considerable importance in determining the rates of radical additions to alkenes [84],... 

Up to a few years ago chemical reactivity was discussed in term of reactivity indexes. These approaches, although valuable, will not be discussed here, since they have been frequently reviewed in the past40-44). Nor will we discuss the perturbation molecular orbital theory for reactants, which has been the subject of extensive reviews 45—47) Extensions of this method can be found in papers by Klopman 48 5°) and Dougherty 51). I shall now mention some methods which have not yet found wide popularity but seem very promising. I mean the criterion of maxi-... 

Hard and soft acid and base theory gives access to an early part of the slope in a reaction profile like that in Fig. 3.3, just as perturbation molecular orbital theory does. Using the definitions of absolute electronegativity and absolute hardness derived in Equations 3.5 and 3.6, the (fractional) number of electrons AN transferred is given by Equation 3.14. 

B. Qualitative Model Perturbation Molecular Orbital Theory... 

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