Three-orbital perturbation

For the moment, suffice it to say that the two-orbital perturbation schemes give the orbital energies and the sign of the MO coefficients in the supermolecule (A B) with a reasonable degree of precision. However, three-orbital perturbations are needed to determine the relative sizes of the coefficients. 

A theory of three-orbital interactions [17-20] is helpful to understand and design molecules and reactions. The orthogonal atomic, bond, or molecular orbitals and are both assumed to interact with a perturbing orbital (j). The orbitals and cannot interact directly but do so indirectly or mix with each other through (j). Orbital... 

CIS does not improve on the Hartree-Fock energy, as there is no interaction between the fundamental state and the singly excited states (Brillouin theorem). In CISD, the singly excited states can interact with the fundamental state through the intermediacy of the doubly excited states. We have met a similar situation in three orbital interactions (p. 27) two MOs belonging to the same molecule can interact if perturbed by a third MO belonging to another molecule. 

On the other hand we can also see that the degeneracy of the orbitals, yz>, zx>, and xy>, depends on the relative orientation of the water molecules. If, for example, the water molecule no. 1 is rotated around its central ion to ligator bond 45° so as to have its molecular plane lying in X — Y, and at the same time water molecule no. 2 is rotated 90° around its bond to the central ion, the above three orbitals obtain the Ti-pertur-bation energies e , en and 2en, respectively. We have a pseudotetra-gonal perturbation. 

Figure 10. Bond diagrammatic representation of Ax", A3", Ax" - A3 and Ax - A3 perturbations for the four-electron-three-orbital system.

The reader should not despair at the complexity introduced by equations 1.29 and 1.30. Symmetry and perturbation theory will allow us to treat any problem as an example of two or three orbitals interacting with each other. The former will be explicitly treated in Chapter 2 using equations 1.29 and 1.30. 

We have used examples where two and three orbitals interact with each other. As a final exercise, consider a case for a generic M-L a bond where now there are four orbitals to be considered. Draw out the resultant interaction diagram and orbital shapes using perturbation theory. 

After planetary accretion was complete there remained two groups of surviving planetesimals, the comets and asteroids. These populations still exist and play an important role in the Earth s history. Asteroids from the belt between Mars and Jupiter and comets from reservoirs beyond the outer planets are stochastically perturbed into Earth-crossing orbits and they have collided with Earth throughout its entire history. The impact rate for 1 km diameter bodies is approximately three per million years and impacts of 10 km size bodies occur on a... 

The analysis of the regioselective reactivity of olefins in identical topochemical environments by three computational methods concludes that both steric factors (cavity and potential energy) and electronic factors (perturbation energy from orbital interactions) play important cooperative roles in determining which C—C double bond in a molecule reacts first in [2-1-2] photodimerization. The steric factor is considered to be effective in the movement of olefins at an early stage of the reaction, whereas the electronic factors are effective in the adduction of olefins at a later stage of the reaction. 

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