Linear chains and inversion symmetry

We saw in Section 3.3.2 that the molecular-orbital states of linear chains are eigenstates of the inversion operator, i. The ground state is constructed by occupying each of the valence molecular-orbital states with two electrons. Thus, the overall inversion symmetry of the ground state must be even (or Ag for a many-body state). Now, because the molecular orbital states alternate in symmetry, the highest occupied molecular orbital (HOMO) state will be either even or odd, while the lowest unoccupied molecular orbital (LUMO) state will be either odd or even. In fact, using eqns (3.19) and (3.20), with [3 = 7r/2a replacing k, the HOMO is 

a particle-hole excitation from the HOMO to the LUMO must have overall odd symmetry. This is the state. The first Ag excitation (the 2Ag state) will be HOMO—1 to LUMO (or, equivalently HOMO to LUMO+1). Such an excitation will lie higher in energy than the 1B state.These transitions are shown in Fig. 3.8. 

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