Molecular Orbitals for HJ Excited States

FIGURE 13.10 Coordinate systems for a homonuclear diatomic molecule. 

One final point. The approximate wave functions in this chapter are written in atomic units. When rewriting these functions in ordinary units, we must remember that wave functions are not dimensionless. A one-particle wave function tj/ has units of length / (Section 3.5). The AOs Is and ISj, that occur in the functions (13.57) and (13.58) are given by (13.44) in atomic units. In ordinary units, li = kj 

In the preceding section, we used the approximate functions (13.57) and (13.58) for the two lowest electronic states. Now we construct approximate functions for further excited states so as to build up a supply of H2 -like molecular orbitals. We shall then use these MOs to discuss many-electron diatomic molecules qualitatively, just as we used hydrogenlike AOs to discuss many-electron atoms. 

To get approximations to higher MOs, we can use the linear-variation-function method. We saw that it was natural to take variation functions for Hj as linear combinations of hydrogenlike atomic-orbital functions, giving LCAO-MOs. To get approximate MOs for higher states, we add in more AOs to the linear combination. Thus, to get approximate wave functions for the six lowest r states, we use a linear combination of the three lowest m = 0 hydrogenlike functions on each atom  

As found in the preceding section for the function (13.43), the symmetry of the homo-nuclear diatomic molecule makes the coefficients of the atom-fc orbitals equal to 1 times the corresponding atom-a orbital coefficients  

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