Suppose that two identical, neighboring atoms have substantial overlap of one s-orbital and one p-orbital (we can

Question:

Suppose that two identical, neighboring atoms have substantial overlap of one s-orbital and one p-orbital (we can assume, for example, that the p-orbitals in the x-direction overlap, and that the other two p-orbitals of the two atoms, pointing in the y- and z-directions, have very little overlap).

(a) In a mathematical program like Mathematica, construct a 4 × 4 matrix of the LCAO matrix elements:

Note that the atomic s- and p-orbitals of a single atom are orthogonal. Assume that the coupling integrals are all real.

(b) Let E_s = 1, E_p = 1.3, U_pp = 3U_ss, and U_sp = U_ss. Plot the eigenvalues of the matrix as a function of Uss for Uss in the range 0–1. Will this system allow bound states?

(c) Solve for the eigenvectors at U_ss = 0.6. What is the character of the states – which are mostly p-like? Which are mostly s-like? Which are bonding? Which are antibonding?

(d) What happens to the character of the bonds if you change the relative weights of the coupling terms U_ss, U_sp, and U_pp? Try plotting the eigenvalues vs U_ss for some different values of the ratios.