Physical Chemistry for Engineers

CHEM 4521

Homework: Molecular Structure

(1) Consider the cation, HeH+.(a) Write the Hamiltonian for this system (there should be 10 terms). Indicate

the physical meaning of each term.(b) Using your result for the above, make the Born-Oppenheimer approximation

for the cation and write the electronic Hamiltonian. Explain the assumptionsmade to obtain this smaller Hamiltonian.

(2) Consider the molecule HF. Suppose that we wish to construct the molecular orbitalsof HF.(a) Write a linear combination of atomic orbitals (LCAO) that we would use to

represent our molecular orbitals (for example, φ1sH

would represent a 1s orbital

centered on the H atom and φ2pxF

would represent the 2px orbital centeredon the F atom). There should be (a minimum of) 6 terms in you linearcombination.

(b) The highest occupied molecular orbital is primarily constructed with the 1sorbital of H and the 2pz of the F. Write the approximate LCAO form of thehighest occupied molecular orbital of HF.

(c) HF is a very polar molecule, so the electrons in the highest occupied orbital(a σ bonding orbital) are primarily localized around the F. What can you sayabout the magnitude of the coefficients of your LCAO in (b)?

(3) In this problem, you will perform a variational calculation to approximate theground state energy and wave function for the electron in the H atom. You willneed to make the following approximations:(a) Take of the origin of the coordinate system to be the nucleus of the H atom

and the reduced mass of the system to be the electron’s mass, µ = me.(b) Perform all calculations in spherical coordinates (r,θ,φ). The wave function

we are approximating depends only on r and not θ or φ. You can make thefollowing simplifications for (i) the volume differential for the integrals and (ii)the Laplacian operator:

(i) dx dy dz = r2sinθ dr dθ dφ =⇒ 4πr2dr

(ii) ∇2ψ(r, θ,φ) = 1

r2∂

∂r(r2 ∂ψ(r,θ,φ)

∂r)+ 1

r2sinθ

∂

∂θ(sinθ ∂ψ(r,θ,φ)

∂θ)+ 1

r2sin2θ

∂2ψ(r,θ,φ)∂φ2 =⇒

1r2

∂

∂r(r2 ∂ψ(r)

∂r)

(c) You may find the following integral relations helpful,�∞0 e

−γ2r2dr =

√π

2γ ,�∞0 re

−γ2r2dr = 1

2γ2 ,�∞0 r

2e−γ

2r2dr =

√π

4γ3 ,�∞0 r

4e−γ

2r2dr = 3

√π

8γ5 .Here is how to do the problem:

(a) Use the trial wave function ψ(r) = e−α2

2 r. The parameter α will be yourvariational parameter (what you will change to perform the minimization).

(b) Setup and perform the integral for �ψ|H|ψ�. It should be a function of α (andthe constants a0, me, and �0).

(c) Setup and perform the integral for �ψ|ψ�. It should be a function of α (andthe constants a0, me, and �0).

(d) Divide the quantity found in (a) by the quantity found in (b). This is the

expectation energy �ψ|H|ψ��ψ|ψ� = �E� you are going to minimize.

(e) Take the derivative of the quantity you found in (c) with respect to α.(f) To find the minimum of the expectation energy, set the quantity you found

for part (d) equal to zero and solve for α. You should express your answer interms of the fundamental constants �0, me, e, and �.

(g) Now use α to determine the energy of your variational approximation. Youshould express your answer in terms of the fundamental constants �0, me, e,and �.

(h) The energy of the ground state of a hydrogen atom is − mee4

32π2�2�20. Compare

your answer for the energy of your variational approximation to this number.Is your answer consistent with the variational theorem? Is the wave functionyou’ve found equal to the actual ground state function?

(i) The approximate radius of the ground state hydrogen atom is given by the

Bohr radius, 4π�0�2mee

2 . The radius of your variational approximation is approx-

imately 1α. Does your wave function have a larger or smaller radius than the

actual hydrogen atom. Is this consistent with the variational theorem?

(4) This problem demonstrates an elementary application of Huckel theory. Considerthe ethene molecule, H2CH=CH2. The carbons are bonded together through twotypes of bonds. The first is a σ bond resulting from the LCAO of the 2px, 2py, and2s orbitals of the C atoms. We’re not going to worry about this one. The secondis called a π bond and is the result of a linear combination of the 2pz orbitals,the p orbitals on the carbon atoms perpendicular to the molecular plane. Use thevariational procedure performed on H+

2 in class to determine the energy differencebetween the π bonding and antibonding orbitals. The matrix elements for theHamiltonian matrix are H11=H22=α = -80 kcal mol−1 and H12=H21=β≈ -15 kcalmol−1. The overlap matrix has off diagonal terms S12=S21=0.20 (you should knowthe other two S matrix elements by the definition of the S-matrix given in class).