would represent a 1s orbital centered on the H atom and φ 2px )+ 1 r 2 sinθ

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1 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 assumptions made to obtain this smaller Hamiltonian. (2) Consider the molecule HF. Suppose that we wish to construct the molecular orbitals of HF. (a) Write a linear combination of atomic orbitals (LCAO) that we would use to represent our molecular orbitals (for example, φ 1s H would represent a 1s orbital centered on the H atom and φ 2px F would represent the 2p x orbital centered on the F atom). There should be (a minimum of) 6 terms in you linear combination. (b) The highest occupied molecular orbital is primarily constructed with the 1s orbital of H and the 2p z of the F. Write the approximate LCAO form of the highest 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 say about the magnitude of the coefficients of your LCAO in (b)? (3) In this problem, you will perform a variational calculation to approximate the ground state energy and wave function for the electron in the H atom. You will need 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 = m e. (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 the following simplifications for (i) the volume differential for the integrals and (ii) the Laplacian operator: (i) dx dy dz = r 2 sinθ dr dθ dφ = 4πr 2 dr (ii) 2 ψ(r, θ, φ) = 1 r 2 r (r2 ψ(r,θ,φ) r )+ 1 r 2 sinθ 1 r 2 r (r2 ψ(r) r ) (c) You may find the following integral relations helpful, 0 re γ2 r 2 dr = 1, 2γ 2 0 r 2 e γ2 r 2 dr = Here is how to do the problem: ψ(r,θ,φ) θ (sinθ θ )+ 1 r 2 sin 2 θ 0 e γ2 r 2 dr = π 4γ 3, 0 r 4 e γ2 r 2 dr = 3 π 8γ 5. 2 ψ(r,θ,φ) φ 2 = π 2γ,

2 (a) Use the trial wave function ψ(r) =e α2 2 r. The parameter α will be your variational parameter (what you will change to perform the minimization). (b) Setup and perform the integral for ψ Ĥ ψ. It should be a function of α (and the constants a 0, m e, and 0 ). (c) Setup and perform the integral for ψ ψ. It should be a function of α (and the constants a 0, m e, and 0 ). (d) Divide the quantity found in (a) by the quantity found in (b). This is the expectation energy ψ Ĥ 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 in terms of the fundamental constants 0, m e, e, and. (g) Now use α to determine the energy of your variational approximation. You should express your answer in terms of the fundamental constants 0, m e, e, and. (h) The energy of the ground state of a hydrogen atom is mee4. Compare 32π your answer for the energy of your variational approximation to this number. Is your answer consistent with the variational theorem? Is the wave function you 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 2 m ee. The radius of your variational approximation is approximately 1 α 2. 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 Hückel theory. Consider the ethene molecule, H 2 CH=CH 2. The carbons are bonded together through two types of bonds. The first is a σ bond resulting from the LCAO of the 2p x, 2p y, and 2s orbitals of the C atoms. We re not going to worry about this one. The second is called a π bond and is the result of a linear combination of the 2p z orbitals, the p orbitals on the carbon atoms perpendicular to the molecular plane. Use the variational procedure performed on H + 2 in class to determine the energy difference between the π bonding and antibonding orbitals. The matrix elements for the Hamiltonian matrix are H 11 =H 22 =α = -80 kcal mol 1 and H 12 =H 21 =β -15 kcal mol 1. The overlap matrix has off diagonal terms S 12 =S 21 =0.20 (you should know the other two S matrix elements by the definition of the S-matrix given in class).

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