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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 )

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Page 1: CHEM 4642 Physical Chemistry II - d.umn.edu · CHEM 4642 Physical Chemistry II Spring 2018 Problem Set 3 1. (3 points) Recall states ψa and ψb from problem set 2. They represent

CHEM 4642 Physical Chemistry IISpring 2018Problem Set 3

1. (3 points) Recall states ψa and ψb 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 ψa and ψb 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 ψa and what is nx for ψb ?

b) Calculate N so that ψc is normalized.

c) Calculate the energy of ψc.

d) Write a normalized superposition of ψa and ψb that has the same energy as

the average of the energies of ψa and ψb.

e) For an electron represented by ψa, 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.

ψa =1

√4 Asin(

π x4 A ) , ψb =

1

√4Asin(

π x2 A )

ψc = N (2ψa−ψb )