Answer Happy

Suppose that the diagonal elements of the Hamiltonian in the
on-site basis, i.e. the energies associated with each site, are i
for i = 1, 2, 3, 4, and the nearest-neighbour matrix elements, i.e.
the hopping parameters, are t1, t2, t3, as denoted in the figure
below. Explain why we must have 4 = 1, 3 = 2 and t3 = t1. Hence
write down the Hamiltonian matrix in the on-site basis, in terms of
the parameters 1, 2, t1 and t2. [You may assume all these
parameters are real.] State, and briefly explain, the connection
between the space-inversion symmetry of the molecule and the form
of the Hamiltonian in the basis of the eigenvectors of P. Construct
the Hamiltonian matrix in the basis of eigenvectors of P where P is
the space inversion operator.
4 tz=tı 2 tz 3 ti 1