Study of a Two-Orbital Interaction

Equations (3.3) define the essence of the Huckel molecular orbital (HMO) theory. Notice that the total energy is just the sum of the energies of the individual electrons. Simple Huckel molecular orbital (SHMO) theory requires further approximations that we will discuss in due course. 

Let us consider the simplest possible case of a system which consists of two orbitals, XaW and /6b(1)j with energies a and g which can interact. It should be emphasized here that these may be orbitals of any kind, atomic orbitals, group orbitals, or complicated MOs. We wish to investigate the results of the interaction between them, that is, what new wave functions are created and what their energies are. Let us also be clear about what the subscripts A and B represent. The subscripts denote orbitals belonging to two physically distinct systems the systems, and therefore the orbitals, are in separate positions in space. The two systems may in fact be identical, for example two water molecules or two sp3 hybrid orbitals on the same atom or on different but identical atoms (say, both C atoms). In this case, sa — cb- Or the two systems may be different in 

The energy of this orbital is given by the expectation value [see equation (A.8) for the definition of expectation value] 

In proceeding from equation (3.5) to (3.6), we notice that the integrals 

Notice that the overlap integral, SAB, will depend on the position and orientation of the orbitals at the sites A and B, as does the intrinsic interaction integral, h. The minimum-energy solution is found by the variational method, which we use twice in Appendix A. Equation (3.6) is differentiated with respect to cA and with respect to cB, resulting in two linear equations which can be solved. Thus, 

The operator h[ ) describes the kinetic energy of an electron and its attractive potential energy for all of the nuclei. The operators 7j(l) and 7Tj(l) together account for the repulsive interactions of the electron with a second electron whose distribution is given by MO b, The sum is only over half the number of electrons since each MO is doubly 

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