Molecular orbital symmetry rotations

Bicycloheptene I was found to rearrange to II at 307° with greater than 95% stereoselectivity to the predicted exo-l-d II. This rearrangement, of course, requires rotation about the C-6—C-7 bond as carbon-7 moves across the face of the cyclopentene ring to carbon-3. The stereo-chemically more comfortable path (no rotation about the C-6—C-7 bond) is symmetry-forbidden. That the rearrangement proceeds with a methylene rotation in preference to the smooth, unhindered 1,3-migration illustrates the depth of control that molecular orbital symmetry conservation holds on transforming molecules. 

The effects of molecular structure on the rates of energy transfer are manifested in many ways. Additional pathways are available in collisions with molecules, such as the influence of ionic potential surfaces [35], the availability of near resonant electronic-to-vibrational and rotational energy transfer pathways [36], and the introduction of nonadiabatic transitions due to the breaking of the molecular orbital symmetry [37]. For the studies considered here, we might also add the competition between reaction and the desired energy transfer process, the possibility of energy transfer processes in the entrance or exit channels, selective changes in... 

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... 

Symmetry of this type will be present if rotation of the molecule around the axis by 180° gives rise to a molecular orbital identical with the original. 

A Straightforward answer to the above question would be a consideration of molecular symmetry. According to group theory, doubly degenerate molecular orbitals, denoted by the symbol e, can arise if a given molecule has one three-fold or higher axis of rotation, or if it has Djj symmetry. It is well... 

Hyperspherical harmonics are now explicitly considered as expansion basis sets for atomic and molecular orbitals. In this treatment the key role is played by a generalization of the famous Fock projection [5] for hydrogen atom in momentum space, leading to the connection between hydrogenic orbitals and four-dimensional harmonics, as we have seen in the previous section. It is well known that the hyperspherical harmonics are a basis for the irreducible representations of the rotational group on the four-dimensional hypersphere from this viewpoint hydrogenoid orbitals can be looked at as representations of the four-dimensional hyperspherical symmetry [14]. 

It is assumed that the reader has previously learned, in undergraduate inorganic or physical chemistry classes, how symmetry arises in molecular shapes and structures and what symmetry elements are (e.g., planes, axes of rotation, centers of inversion, etc.). For the reader who feels, after reading this appendix, that additional background is needed, the texts by Cotton and EWK, as well as most physical chemistry texts can be consulted. We review and teach here only that material that is of direct application to symmetry analysis of molecular orbitals and vibrations and rotations of molecules. We use a specific example, the ammonia molecule, to introduce and illustrate the important aspects of point group symmetry. 

In using symmetry to help simplify molecular orbital or vibration/rotation energy level calculations, the following strategy is followed ... 

In a cyclic (CH)n molecule with rotational symmetry Cn, there will always be n n molecular orbitals, one belonging to each irreducible representation of the group Cn. 

Now the molecular orbitals in equation (3) also have interesting transformation properties. Consider the effect of a rotation through 2w/5 radians about the fivefold axis of symmetry, perpendicular to the plane of the ring. For example, in place of 0+i take that function which is obtained from it by replacing atom 5 (and thus 0b) by atom 4 (and thus 04), 4 by 3, and so on, namely... 

Figure 1.9 shows the three-dimensional shape of the electron distributions Pmo2px and symmetry element is called a C axis. An orbital with this kind of symmetry is called a 77 orbital. Atomic orbitals of the s type can form only a molecular orbitals atomic... 

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. 

Fukui, 1965a). Depending on k, the highest occupied molecular orbital (HOMO) will be symmetric (a) or antisymmetric (6), with respect to a symmetry plane (or) perpendicular to the plane of the carbon chain of 1. Thus, la is a (in + 2)v and lb is a 4nir electron system (Fig. 6). Now, in order for the ring to close and form a cr-bond, the ends of the 7r-system of la must rotate in opposite directions or in a disrotatory mode, while the ends of lb must rotate in the same direction, or in a conrotatory mode. In either case, rotation occurs so as to maximize orbital overlap or bonding as measured by P in (14). 

A second 7r interaction between that it orbital of the organic group and a molecular orbital of a symmetry centered on the metal fragment (Fig. 11), although of less energy than the former tt interaction, results in a calculated bond order between M and Ca of between 2 and 3 and, more importantly, indicates that rotation of the vinylidene unit about the M-Ca bond should be facile. This prediction is easily deduced from the lone pair... 

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