CHEM 4642 Physical Chemistry II Spring 2018 Problem Set 3 1. (3 points) Recall states ψ a and ψ b from problem set 2. They represent an electron confined to a one- dimension box between x=0 and x=8 Å. Potential energy V=0. Also consider this superposition state, , where N is a normalization constant. a) Functions ψ a and ψ b are in fact particle-in-box eigenfunctions, as in equation 4.13 (or 15.13). What are their quantum numbers? I.e., what is n x for ψ a and what is n x for ψ b ? b) Calculate N so that ψ c is normalized. c) Calculate the energy of ψ c . d) Write a normalized superposition of ψ a and ψ b that has the same energy as the average of the energies of ψ a and ψ b . e) For an electron represented by ψ a , what is the probability that 3≤x≤5 Angstroms? 2. (3 points) Two-dimensional molecular box. Consider an electron in a rectangular two-dimensional box. Dimensions are 6 Å by 4 Å in the x and y directions. a) Calculate the first (i.e., lowest) five energies. Shift the energies so the lowest energy is zero. with the "minimal" STO-3G basis set to calculate the five lowest-energy pi orbitals of naphthalene. List those energies. Shift them so the lowest energy is zero. c) Produce images of the five lowest-energy pi molecular orbitals. (MacMolPlt will do this.) Compare them qualitatively to the wave functions of the two-dimensional particle-in-a-box. Are the numbers and locations of nodes the same? d) Convert RHF energy units so that your shifted particle-in-a-box energies and your shifted RHF energies are in aJ (1 attojoule equals 10 -18 J). Compare them. ψ a = 1 √ 4 ˚ A sin ( π x 4 ˚ A ) , ψ b = 1 √ 4 ˚ A sin ( π x 2 ˚ A ) ψ c = N ( 2 ψ a −ψ b )