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The basis set that is normally used for H2 consists
of the two 1s orbitals: 1sA and 1sB. A basis
transformation replaces these orbitals by two others that are linear
combinations of 1sA an 1sB and are linear
independent. Sometimes such a basis transformation can make the solution
of the secular equation easier.
The H2 molecule has a high symmetry. In particular it has a mirror plane perpendicular to and through the middle of the molecular axis. The basis set 1sA+1sB and 1sA-1sB is adapted to this symmetry: i.e., these basis functions either do not change or only change sign when they are reflected in the mirror plane.
a) Write down the secular equation with 1s A+1sB and 1sA-1sB as basis: i.e., write the solutions as c+(1sA+1sB)+c-(1sA-1sB) and write the secular equation to determine the orbital energies and the coefficients c+ and c-. For example, the overlap matrix is
Use
| a = á1sA|F|1sAñ = á1sB|F|1sBñ, |
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| b = á1sA|F|1sBñ = á1sB|F|1sAñ, |
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and
| S = á1sA|1sBñ = á1sB|1sAñ. |
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b) Solve the secular equation of a).
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