Symmetry-correct molecular orbital

Symmetric and antisymmetric orbitals The first point is that each symmetry correct molecular orbital may be either unchanged or transformed to... 

One important aspect of MO theory— the concept of symmetry-correct molecular orbitals— not only makes it possible to explain this experimental result but, in fact, requires it. A fimdamental property of molecular orbitals is that they have the full symmetry of the basis set of atomic orbitals used to generate the molecular orbitals. This means that the molecular orbitals must be either symmetric or antisymmetric with respect to the symmetry operations provided for by the symmetry group of the atomic orbitals.If we consider each C-H bond formed by overlap of an sp orbital on a carbon atom with a Is orbital on a hydrogen atom to be an MO, then clearly each of these MOs lacks the full symmetry of the basis set of s and p atomic orbitals. 

As a prelude to further analysis, it is useful to review one important property of molecular orbitals. As noted in Chapter 1, symmetry-correct molecular orbitals must be either symmetric or antisymmetric with respect to the full symmetry of the basis set of atomic orbitals that are used to construct the molecular orbitals. In the analysis of orbital symmetries, we will need to consider only the number of molecular S5nnmetry elements that are sufficient to distinguish between allowed and forbidden pathways. Also, it is not necessary to consider here the minor perturbation of molecular orbital symmetry that results from isotopic or alkyl substitution. In other words, to a first approximation the basis set orbitals of any conjugated diene are considered to be the same as those for 1,3-butadiene. Figtue 11.13 shows the... 

What do molecular orbitals and their nodes have to do with pericyclic reactions The answer is, everything. According to a series of rules formulated in the mid-1960s by JR. B. Woodward and Roald Hoffmann, a pericyclic reaction can take place only if the symmetries of the reactant MOs are the same as the symmetries of the product MOs. In other words, the lobes of reactant MOs must be of the correct algebraic sign for bonding to occur in the transition state leading to product. 

The nature of the electronic states for fullerene molecules depends sensitively on the number of 7r-electrons in the fullerene. The number of 7r-electrons on the Cgo molecule is 60 (i.e., one w electron per carbon atom), which is exactly the correct number to fully occupy the highest occupied molecular orbital (HOMO) level with hu icosahedral symmetry. In relating the levels of an icosahedral molecule to those of a free electron on a thin spherical shell (full rotational symmetry), 50 electrons fully occupy the angular momentum states of the shell through l = 4, and the remaining 10 electrons are available... 

The implied implication that even 7r-electrons are independent of each other is grossly unjustified. What saves the day is that all wave functions and the overlap function that is being ignored, have the same symmetry. The shape of the molecular orbital remains qualitatively correct. 

Each reaction species must have molecular orbitals available and with the correct symmetry to allow bonding. These will be called frontier orbitals composed of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). In addition to their involvement in bonding between species, these orbitals are of considerable interest in that they are largely responsible for many of the chemical and spectroscopic characteristics of molecules and species and are thus important in analytical procedures and spectroscopic methods of analysis [5-7],... 

The NMRD profiles of V0(H20)5 at different temperatures are shown in Fig. 35 (58). As already seen in Section I.C.6, the first dispersion is ascribed to the contact relaxation, and is in accordance with an electron relaxation time of about 5 x 10 ° s, and the second to the dipolar relaxation, in accordance with a reorientational correlation time of about 5 x 10 s. A significant contribution for contact relaxation is actually expected because the unpaired electron occupies a orbital, which has the correct symmetry for directly overlapping the fully occupied water molecular orbitals of a type (87). The analysis was performed considering that the four water molecules in the equatorial plane are strongly coordinated, whereas the fifth axial water is weakly coordinated and exchanges much faster than the former. The fit indicates a distance of 2.6 A from the paramagnetic center for the protons in the equatorial plane, and of 2.9 A for those of the axial water, and a constant of contact interaction for the equatorial water molecules equal to 2.1 MHz. With increasing temperature, the measurements indicate that the electron relaxation time increases, whereas the reorientational time decreases. 

Section treats the spatial, angular momentum, and spin symmetries of the many-electron wavefunctions that are formed as antisymmetrized products of atomic or molecular orbitals. Proper coupling of angular momenta (orbital and spin) is covered here, and atomic and molecular term symbols are treated. The need to include Configuration Interaction to achieve qualitatively correct descriptions of certain species electronic structures is treated here. The role of the resultant Configuration Correlation Diagrams in the Woodward-Hoffmann theory of chemical reactivity is also developed. 

From simple symmetry arguments concerning the electron density, we can deduce that cA = cB and we label the two molecular orbitals by symmetry lcrg = lsA + lsB and lo-u = lsA — lsB. Neither is a solution of the electronic Schrodinger equation, but each has the correct boundary conditions and so they are possible approximate solutions. 

In order to take advantage of the symmetry criterion for orbital interaction, it is necessary to have orbitals that are symmetry correct. The molecular orbitals obtained from atomic orbitals by the methods described in Sections 1.2 and 10.1 will sometimes be symmetry correct and sometimes not. 

It is evident from a comparison of 5 and 4 that the reflection transforms >pi into i/j2. The reader should verify that the a reflection and the C2 rotation also transform symmetry correct. The result we have found will generally hold when molecular orbitals constructed by the LCAO method from hybrid atomic orbitals are subjected to symmetry operations. Each of those orbitals in the set of MO s that is not already symmetry correct will be transformed by a symmetry operation into another orbital of the set. 

Examine each molecular orbital to see whether it is already symmetry correct. 

Figure 11.3 The it orbitals for 1-methylbutadiene. Now the correct molecular symmetry consists of only the plane a, but the rr orbitals are still constructed according to their approximate local symmetry, which remains as shown in Figure 11.2.

There is danger of error only when such elements are used exclusively.) The atomic orbital basis consists of a p function on each of the four carbon atoms Figure 11.15 illustrates these orbitals and the derived symmetry correct orbitals. As a result of the reflection plane that carries one ethylene into the other, these molecular orbitals are delocalized over both molecules. Figure 11.15 also shows the orbitals of the product we consider only the two C—C bonds formed and ignore the other two, which were present from the beginning and did not undergo any change. 

They are strong a-donors, but have no energetically accessible molecular orbitals available of the correct ji-symmetry for retrodative combi-... 

The bonding in monometal alkyne complexes is usually interpreted in terms of the Dewar-Chatt-Duncanson model (293), since the alkyne molecule has a pair of n and n molecular orbitals which lie in the plane of the metal and the two carbon atoms. These two orbitals are denoted n and n, and are analogous to those in jr-bonded alkene complexes (394). There is also a pair of n and n molecular orbitals which lie perpendicular to the metal-carbon plane, denoted nL and n . These orbitals are illustrated in Fig. 14. Both sets of n and n orbitals have the correct symmetry to interact with metal d orbitals. The interaction... 

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