Answer to Question #82730 in Physical Chemistry for Szu-An Lin

Question

When using an independent particle model, an MO ψ is expressed in LCAO form, i.e. as a linear combination of atomic orbitals {ϕ_1, ϕ_2, . . . , ϕ_K}, (1)
and the expansion coefficients are determined from secular equation: the Hamiltonian is the one electron operator h and the “orbital energy” ε. Find the MOs and orbital energies for a linear
polyene (Cn Hn+2), taking ϕ_µ to be 2p_z AOs on C and including only nearest-neighbor matrix
element, with
(ϕ_µ|h|ϕ_µ) = α, (ϕ_µ|h|ϕ_µ±1) = β (2)
and with the neglect of the overlap.
Give explicitly the MOs and orbital energies for N = 3 and N = 4.