Writing the Secular Equations and Determinant for Any Molecule

In the two-orbital mixing problem, we showed that when molecular orbitals are defined as linear combinations of atomic orbitals and are put into the Schrbdinger equation, followed by differentiation to minimize E, a series of simultaneous equations in the c s and E results. When mixing two orbitals, only two energies result along with two molecular orbitals. It is not much of a stretch to realize that there will be as many molecular orbitals with distinct energies as the number of atomic orbitals (or basis functions) we use to create the molecular orbitals. Moreover, there will be the same number of secular equations as the number of starting atomic orbitals or basis functions ( ), and hence the secular determinant will be n by n. 

It is actually easier to write the secular determinant first, and then create the secular equations (although we saw in the Two-Orbital Mixing Problem that they are derived in the opposite order). All secular determinants have the following form  

The diagonal elements represent the energies of the atomic orbitals in the molecule minus . The off-diagonal elements consist of the resonance integrals between the various atomic orbitals minus their respective overlaps times . 

The secular equations can be written by inspection of the secular determinant. Each row in the determinant corresponds to a separate secular equation. To generate all the secular equations, we take the first term in each row and multiply it by Ci, the second term is multiplied by C2, the third term by C3, and so on. Equation 14.51 shows a generalized form of the equations that result from the first row of the secular determinant. The result is ti equations in n C/t s and a single e,. 

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