CHEM 4642 Physical Chemistry II - d.umn.edu .CHEM 4642 Physical Chemistry II Spring 2018 Problem
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CHEM 4642 Physical Chemistry IISpring 2018Problem Set 3
1. (3 points) Recall states a and b from problem set 2. They represent an electron confined to a one-dimension box between x=0 and x=8 . Potential energy V=0.
Also consider this superposition state, , where N is a normalization constant.
a) Functions a and 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 nx for a and what is nx for b ?b) Calculate N so that c is normalized.c) Calculate the energy of c.d) Write a normalized superposition of a and b that has the same energy as the average of the energies of a and b.e) For an electron represented by a, what is the probability that 3x5 Angstroms?
2. (3 points) Two-dimensional molecular box.Consider an electron in a rectangular two-dimensional box. Dimensions are 6 by 4 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.
4 Asin( x4 A ) , b = 14A sin(
x2 A )
c = N (2ab )