Answer all questions in this section.

1.

(a)

Describe the electron correlation problem in the context of SCF calculations. [5 marks]

(b)

Briefly describe two computational methods by which electron correlation can be treated. [4 marks]

(c)

When calculating $\Delta E$'s, in what kinds of reactions are correlation effects likely to be important? In what kinds of reactions is correlation unimportant? [4 marks]

(d)

Provide a specific example of a reaction in which correlation is unimportant. Support your example with a HyperChem calculation. [10 marks]

(e)

Provide a specific example of a reaction in which correlation has a material effect on the computed $\Delta E$. Support your example with a HyperChem calculation. [10 marks]

 

2.

Calculate the expectation value of r² for an electron in a hydrogen atom in the 2p₀ orbital. The wavefunction is

$$\psi_\mathrm{2p_0}(r,\theta,\phi) = \frac{1}{8}\sqrt{\frac{2}{\pi a_0^3}}\,\rho\,e^{-\rho/2}\cos\theta$$

where $\rho = r/a_0$. [7 marks]

Maple hints: $\pi$ is Pi in Maple, $\sqrt{u}$ is sqrt(u), and e^u is exp(u). Maple will need to know that a₀>0.

3.

We have not explicitly considered the effects of relativity in this course. At intermediate values of the momentum (large enough that relativistic effects are measurable, but not so large that they dominate the behavior), relativistic effects can be treated by adding the following (small) term to the Hamiltonian:

$$-\frac{\hat{p}^4}{8m^3c^2}.$$

Obtain an equation for the energy levels of a particle in a box which includes the relativistic correction. Reduce all instances of $\hbar$ to h in your final answer. [6 marks]

Hints: This question can be answered without explicitly evaluating any integrals. However, if you do apply the formal procedure, you will need to tell Maple that the quantum number is an integer. Recall also that $\pi$ is spelled Pi in Maple and that $\sqrt{u}$ is sqrt(u).

4.

Write down the complete Hamiltonian for H₂⁺. (Avoid using summation notation if you can.) Explain how the Born-Oppenheimer approximation can be used to separate the associated quantum mechanical problem into two smaller problems and outline the solution procedure. Include in your answer the Hamiltonians which arise from the separation. [10 marks]

5.

(a)

A diatomic molecule with a reduced mass of $1.3\times 10^{-26}\,\mathrm{kg}$ has an effective potential energy curve approximately given by

$$V_\mathrm{eff}(R) = A\left[\left(\frac{B}{R}\right)^2 - \frac{B}{R}\right]$$

where $A = 3\times 10^{-18}\,\mathrm{J}$ and $B = 6\times 10^{-11}\,\mathrm{m}$. Find the equilibrium bond length and potential energy at equilibrium. [5 marks]

(b)

Using the trial wavefunction

$$\phi(R) = R^2e^{-\alpha R^2},$$

estimate the ground-state energy. [10 marks]

Maple hints: Maple will need to know that $\alpha$ is a positive parameter. Enter numeric values for all constants and parameters before carrying out any mathematical operations. Don't forget that e^u is exp(u) in Maple. Use assign() to set the value of the variational parameter at the end of the computation. Syntax:

assign(u=a);

sets the value of variable u to a.

(c)

Find the location of the maximum of the wavefunction. Comment on the quality of the variational wavefunction. [5 marks]

(d)

Estimate the dissociation energy. [2 marks]

Hint: What is $$\lim_{R\rightarrow\infty}V_\mathrm{eff}(R)$$?

(e)

What could you do to improve the accuracy of these calculations? [2 marks]