Complementary orbitals

It will be observed that the Coulson-Fisher functions satisfy the relations 

It will be recalled by examining Table 2.3 that there are 12 independent cr-AO-only VB functions in the MCVB. Our complementary orbital function has only five independent parameters, so it certainly cannot duplicate the MCVB energy, but it reproduces 96.8% of the binding energy of the latter calculation. 

We show a 3D altitude drawing of the amplitude of the A orbital in Fig. 3.1. It is easily seen to be extended over both nuclei, and it is this property that produces in the wave function the adjustment of the correlation and delocalization that is provided by the ionic function in the linear variation treatment with the same AO basis. 

We point out that these results are obtained without any ionic states in the wave function and such are not needed. As we argued in Chapter 2, the principal role of the ionic functions is to provide delocalization of the electrons when the molecule 

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Figure 3.1. Altitude drawing of the A optimal complementary orbital for values in thex-z plane. The H nuclei are on the z-axis. The two vertical lines point at the nuclei.

In an alternant hydrocarbon, the LCAO-coefficients of any pair of complementary orbitals (i.e. orbitals of energy a + kp and a — kfi) are identical apart from a change of sign in the coefficients of the atomic orbitals centred on the unstarred atoms . 

An illustration of this aspect of the Coulson-Rushbrooke Theorem is again provided by the alternant hydrocarbon, butadiene. The LCAO-coefficients of the two pairs of complementary orbitals in the molecule display this alternation of sign, as examination of equations (2-67) confirms (atoms 1 and 3 of Fig. 2-6 may be considered, for this purpose, as the starred atoms). 

We now prove the second part of the Coulson-Rushbrooke Theorem that in pairs of complementary orbitals the LCAO-coefficients of orbitals centred on the starred atoms are the same, while those of orbitals centred on the unstarred atoms are the same in magnitude but opposite in sign. Let us consider the 7th orbital of energy e, (let it be, for argument s sake, a bonding orbital) and its associated set of LCAO-coefficients c/rh r = 1, 2,. .., n. These will satisfy a series of secular equations, the rth one of which, on the simple Hiickel assumptions, is... 

Here, there are ten molecular orbitals which occur in five complementary pairs, as shown schematically in Fig. 6-7. In the ground state, all the five bonding-orbitals are filled, accommodating the 10 electrons in the jr-system and leaving all the anti-bonding orbitals empty. The lowest-energy excitation is approximately one from the highest-occupied MO (called Fj in Fig. 6-7) to the lowest-unoccupied ( F6)N2. These are complementary orbitals with... 

Hence, because of the result expressed in equation (6-37), the pairing of MO s and the consequent symmetry of LCAO-coefficients between pairs of complementary orbitals, the charge density on the rth carbon-atom of a neutral, even, alternant hydrocarbon in its ground state, is unity. This is the essence of the third part of the Coulson-Rushbrooke Theorem. Its proof depends on the fact that the square of a quantity is the same as the square of minus that quantity, and is thus seen to be a natural consequence of parts 1 and 2 of the Theorem. 

An interesting corollary to part 2 of the Coulson-Rushbrooke Theorem (which does not require assumption c), equation (6-4)) concerns the nodes in the various LCAO-MO s discussed in our sample calculation on the alternant hydrocarbon, butadiene ( 2.10). As an example, we consider just the highest-bonding and lowest-anti-bonding orbitals of butadiene which, in 2.7, we called 4 2 and 4V These are, of course, complementary orbitals their nodal behaviour has been redrawn in Fig. 6-6 which is a simplified... 

As a result of several complementary theoretical efforts, primarily the path integral centroid perspective [33, 34 and 35], the periodic orbit [36] or instanton [37] approach and the above crossover quantum activated rate theory [38], one possible candidate for a unifying perspective on QTST has emerged [39] from the ideas from [39, 40, 4T and 42]. In this theory, the QTST expression for the forward rate constant is expressed as [39]... 

Magnetic circular dicliroism (MCD) is independent of, and thus complementary to, the natural CD associated with chirality of nuclear stmcture or solvation. Closely related to the Zeeman effect, MCD is most often associated with orbital and spin degeneracies in cliromophores. Chemical applications are thus typically found in systems where a chromophore of high symmetry is present metal complexes, poriihyrins and other aromatics, and haem proteins are... 

Generally speaking the three models offer complementary information Organic chemists use all three emphasizing whichever one best suits a particular feature of struc ture or reactivity Until recently the Lewis and orbital hybridization models were used far more than the molecular orbital model But that is changing... 

When both the 1,3-dipoIe and the dipolarophile are unsymmetrical, there are two possible orientations for addition. Both steric and electronic factors play a role in determining the regioselectivity of the addition. The most generally satisfactory interpretation of the regiochemistry of dipolar cycloadditions is based on frontier orbital concepts. As with the Diels-Alder reaction, the most favorable orientation is that which involves complementary interaction between the frontier orbitals of the 1,3-dipole and the dipolarophile. Although most dipolar cycloadditions are of the type in which the LUMO of the dipolarophile interacts with the HOMO of the 1,3-dipole, there are a significant number of systems in which the relationship is reversed. There are also some in which the two possible HOMO-LUMO interactions are of comparable magnitude. 

The complementary relationship between thermal and photochemical reactions can be illustrated by considering some of the same reaction types discussed in Chapter 11 and applying orbital symmetry considerations to the photochemical mode of reaction. The case of [2ti + 2ti] cycloaddition of two alkenes can serve as an example. This reaction was classified as a forbidden thermal reaction (Section 11.3) The correlation diagram for cycloaddition of two ethylene molecules (Fig. 13.2) shows that the ground-state molecules would lead to an excited state of cyclobutane and that the cycloaddition would therefore involve a prohibitive thermal activation energy. 

Based on experimental results and complementary calculations, an out-of-plane n-delocalization is suggested for thiirene dioxides39. As far as the thiirene oxide is concerned, theoretical calculations predict possible spiroconjugative-type53 interaction between the n c—c orbital of the ring and the jr-orbitals of the SO (which leads to aromatic stabilization and a n charge transfer backward from the SO to the C=C). There exists, however, a rather strong destabilization effect, due to the jr so(d )-orbital. 

There are two complementary lines of approach to examining the part played by 3d orbitals in molecular orbital theory and to appreciating current doubts as to their role. On the one hand, there is the question of 3d orbitals in relation to the basic formulation of molecular orbitals by overlapping atomic orbitals on the other hand, there is the question of the effect of including or excluding 3d functions in molecular orbital calculations, particularly of the ab initio type. We shall consider each of these briefly in turn. 

It was pointed out in my 1949 paper (5) that resonance of electron-pair bonds among the bond positions gives energy bands similar to those obtained in the usual band theory by formation of Bloch functions of the atomic orbitals. There is no incompatibility between the two descriptions, which may be described as complementary. It is accordingly to be expected that the 0.72 metallic orbital per atom would make itself clearly visible in the band-theory calculations for the metals from Co to Ge, Rh to Sn, and Pt to Pb for example, the decrease in the number of bonding electrons from 4 for gray tin to 2.56 for white tin should result from these calculations. So far as I know, however, no such interpretation of the band-theory calculations has been reported. 

The electron distributions in term wavefunctions and orbitals may be the same or complementary, as shown below... 

Photocycloaddition of Alkenes and Dienes. Photochemical cycloadditions provide a method that is often complementary to thermal cycloadditions with regard to the types of compounds that can be prepared. The theoretical basis for this complementary relationship between thermal and photochemical modes of reaction lies in orbital symmetry relationships, as discussed in Chapter 10 of Part A. The reaction types permitted by photochemical excitation that are particularly useful for synthesis are [2 + 2] additions between two carbon-carbon double bonds and [2+2] additions of alkenes and carbonyl groups to form oxetanes. Photochemical cycloadditions are often not concerted processes because in many cases the reactive excited state is a triplet. The initial adduct is a triplet 1,4-diradical that must undergo spin inversion before product formation is complete. Stereospecificity is lost if the intermediate 1,4-diradical undergoes bond rotation faster than ring closure. 

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