In this exercise, you will calculate the rovibrational spectrum of a diatomic molecule of interest (e.g.HCl) using different levels of theory: a normal mode analysis and ab initio methods.
Normal Mode Analysis:
As a first approximation, you will use the harmonic oscillator / rigid rotor approximation to calculate the fundamental vibrational frequency, and rotational energies. In this approximation, the total rovibrational energy is given by Eq. (1):
(1)
where n = 0, 1, 2, ... is the vibrational quantum number, l = 0, 1, 2, ... is the rotational quantum number, is the equilibrium bond distance, and μ is the reduced mass of the diatomic.
Using the Bohr correspondence principle, one can calculate a given transition between any two rovibrational states. However, selection rules for a rotating/vibrating heteronuclear diatomic are Δl = ±1 and Δn = ±1. Therefore, the (0,0) --> (1,0) fundamental transition would not be observed. What would be observed are the P-branch (corresponding to all transitions with Δl = -1) and the R-branch (all transitions corresponding to Δl = +1):
P-branch: = (2)
R-branch: = (3)
Transition energies are typically expressed in wavenumbers ():
. (4)
You will use the GeometryOptimization function to calculate for the diatomic of interest and then use the VibrationalModes function to calculate the fundamental frequency, You will then calculate the rotational energies given by the second term in Eq. (1) for l = 0, 1, ..., . You will then calculate P- and R-branches according to Eqs. (2) and (3) and compare to experimental values.
Ab Initio Calculations
While the above approximation is simple, more sophisticated methods can be used to capture the inherent anharmonicity in the system. In the second portion of this activity, you will calculate the potential energy surface (PES) for the diatomic molecule of interest using different levels of electronic structure theory. To assess the accuracy of each PES, you will calculate the corresponding rovibrational motion and compare with experimental observations. In doing so, you will make use of the Born Oppenheimer approximation to separate the electronic and rovibrational degrees of freedom. To this end, you will first calculate electronic energies by solving the electronic Schrodinger equation for a range of nuclear separations (R) to construct a 'potential energy surface' (PES), V(R). You will explore different levels of theory, including Hartree-Fock (HF), many-body perturbation theory (MP2), coupled cluster (CC), and density functional theory (DFT) as well as different atomic basis sets, including STO-3G, 6-31G, cc-pVDZ, and cc-pVTZ.
Once V(R) has been constructed, the nuclear Schrodinger equation is given by
(R) = (R). (5)
where is the corresponding rovibrational wavefunction, and the Hamiltonian, , is given by
= -(6)
While the details of how one solves Eq. (5) are beyond the scope of this exercise, the variational principle is used: represent Eq. (5) in an underlying basis representation for Ψ and diagonalize the resulting matrix to get energy eigenvalues, . (For more on matrix representations and the variational principle, see Variational Theorem exercise). Once values have been calculated, the transition frequencies can again be calculated using the appropriate selection rules and Eqs. (2) and (3), and the results can be compared to experiment.