CTC2 (NWI-MOL175): Computer assignment 3: Ar+CO with \(J>0\) and in the rigid rotor approximation

This assignment consists of two parts. The first part continues from assignment 2 by computing higher rotational levels, and the second part uses the rigid rotor approximation to calculate those same rotational levels.

Higher rotational levels using the variational method

• Adapt the code of assignment 2 to compute \(J>0\) energy levels of Ar+CO.
  • Note that the special case of Clebsch-Gordans for \(J=0\) cannot be used anymore.
  • Note also that for the basis functions \(|(jl)JM\rangle\) we have \(|j-l| \leq J \leq j+l\).
  • The Hamiltonian is diagonal in \(J\) and \(M\). A rigorous proof of this requires some angular momentum theory that has not been shown in this course.
• To speed up the computations, approximate the bending potential with only two terms: \[ V_\mathrm{bend}(z) = c_1 P_1(z) + c_2 P_2(z) \] Choose \(c_1\) and \(c_2\) such that the minimum of the potential is at the same angle as the full bending potential in assignment 2. Also, make sure that the second derivative in the minimum remains the same. A simple way to find the minimum is to evaluate the potential on a grid. Also, with three points around the minimum you can find the second derivative - high precision is not very important here.
• Compute the energy levels for \(J=0, 1, 2\) and make a plot of the lowest 8 energy levels as a function of \(J\). Make sure your energy levels are converged. Since the potential is a polynomial of lower order, converging the results will be easier than in assignment 2.
• Determine the parity of your basis functions. Repeat the calculation keeping only the basis functions of even parity. This should again speed up the calculation.
• Do the same for odd parity functions.
• Make an energy level plot using different colors for odd and even parity states.
• If your parity adapted code reproduces the results of the original code, extend your calculations and plots to \(J_\mathrm{max}=4\). This calculation can be expected to run for at least a few minutes.

Rigid rotor approximation for Ar+CO

• Fix the Ar-CO angle and determine Cartesian coordinates of the atoms - put them in the xy-plane.
• Compute the center-of-mass and translate them such that the c.o.m. is in the origin of your coordinate system.
• Compute the inertia tensor.
• Diagonalize the inertia tensor to compute the principle moments of inertia.
• Is the molecule prolate-top like, or oblate-top like?
• Approximate the molecule as a (prolate or oblate) symmetric top, and compute the energy levels.
• Write a program to setup and diagonalize the asymmetric rigid-rotor Hamiltonian.
• Compare the results of the symmetric top, asymmetric rotor and the variational method.

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