Describe the born-oppenheimer approximation

Question

Assignment Task: Theoretical Chemistry

Answer as completely as possible. Show your work.

Problem 1: True or false? If false, briefly explain why.

a. No classical analog of the exchange operator exists.

b. An orbital rotation of the occupied canonical Hartree-Fock orbitals yields new orbitals with the same ground state energy.

c. At all points on the potential energy surface of a molecule, - = 2.

d. Given a two-electron operator O^₂, 2|D'> =0 if D and D' differ by two or more spin-orbitals.

e. The pseudoeigen value equations F^aC^a = SC^aε^a;F^βC^β = SC^βε^β can be solved independently.

f. There is no single, unique method to determine partial charges on atoms in molecules.

Problem 2: Is the following two electron spatial function symmetric or antisymmetric with exchange of the electrons? What spin function(s) is/are associated with this spatial function? What is the multiplicity of a state with this spatial function?

|Φ> = 1/√2 [ψ_1,s (1)ψ_2s (2)-ψ_2s (1)ψ_1s (2)]

Problem 3: Find the terms that arise from a (σ)²(δ₁)¹(δ₂)¹ configuration. Which term is expected to be the lowest in energy?

Problem 4: What is the MS value associated with the following function? (Show work for full credit)

1/√2 [ψ₁ (1)ψ₂(2) + ψ2(1)ψ₁(2)] 1/√2 [α(1)β(2) - β(1)α(2)]

Problem 5: Describe the Born-Oppenheimer Approximation and show how the Hamiltonian reduces with this approximation.

Problem 6: What are the advantages of Slater basis functions over Gaussian functions? What are the advantages of Gaussians over Slaters?

Problem 7:  What is a polarization function? Why is it useful to include polarization functions in calculations? What is a diffuse function? What are diffuse functions good at representing?

Problem 8: The restricted Hartree Fock STO-3G orbital energies (in a.u.) for an optimized water molecule are:

ε₁: -20.2516

ε₂: -1.2575

ε₃: -0.5939

ε_4: -0.4597

ε₅: -0.3926

ε_6: 0.5818

ε₇: 0.6927

Water has five occupied orbitals and ten electrons. Estimate the ionization potential, electron affinity and first excitation energy for water using Koopman's Theorem. What is one approximation that Koopman's Theorem makes?

Problem 9: Why do the Hartree-Fock equations need to be solved iteratively?

Problem 10: Explain what spin-orbit coupling is and briefly discuss the difference between Russell- Saunders coupling and j-j coupling.

Problem 11: Convert 1/2 ∑_a^N∑_b^N ( - ) from spin orbitals to spatial orbitals.

Problem 12: Simplify the following integral: <χkχlχmχn|> ∑_i=1^N∑_j>i^N 1/rij|χkχlχmχp> The bra and ket represent Slater determinants. Show your work.