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philhellenephysicist
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- TL;DR
- How does one go from choosing an effective Hamiltonian for a single vibrational level to obtaining algebraic expressions for the rotational term values?
Hello, I am fairly new to the world of molecular spectroscopy, so I apologize for any ignorance on my part. For the last few months, I've been working on a diatomic spectral simulation tool and have reached a point where I want to incorporate more advanced theory to model complex interactions in the molecule without needing to resort to a bunch of different hardcoded formulae. So far, I've been using the Born-Oppenheimer approximation to find the transition energies, that is ##T = T_e + G + F##. This has worked well for a simulation of molecular oxygen, for which I used rotational term values tabulated in Herzberg's Spectra of Diatomic Molecules for ##^3\Sigma## states. However, adding more molecules involves different molecular term symbols, different coupling cases, etc., each of which require their own rotational terms. Now, Herzberg does have an excellent number of different formulae for all kinds of coupling and uncoupling cases, but I don't want to have a bunch of hardcoded cases in my program and would much rather be able to derive the energies through more fundamental means.
When I started working on my tool, I read about PGOPHER (a spectral tool which predicts transitions, does line fitting, etc.). Their website includes a small short course, which outlines the methods used to find transitions between two states on this page using matrix elements of an effective Hamiltonian. I essentially want to replicate this method of obtaining the line transitions by understanding the underlying theory.
From what I understand, the general method for analyzing molecular spectra is to choose an appropriate effective Hamiltonian, choose an appropriate basis set, work out the matrix elements (simplify where needed using symmetry arguments), and finally calculate the eigenvalues (energy levels). The effective Hamiltonian for a single vibrational level (what I'm interested in) can be written as the sum of individual perturbations, that is:
$$\mathbf{H}_{eff} = T_e + G_v + \mathbf{H}_{rot} + \mathbf{H}_{cd} + \dotsb$$
My question is, how can one go from an effective Hamiltonian to algebraic formulae representing the rotational energy levels? I'm most confused on how to get from the Hamiltonian to the matrix elements, forming the matrix, and then obtaining the eigenvalues.
This is my (very rough) interpretation: The chosen Hamiltonian operates on a chosen basis set in the form of ##|J, \Lambda, \Omega>## or something similar to get an expectation value ##<J, \Lambda, \Omega|\mathbf{H}|J, \Lambda, \Omega>##. Since there are multiple expectation values based on different ##\Omega, \Lambda## combinations, these can then be arranged into diagonal and off-diagonal elements of the representative matrix. This matrix is then somehow diagonalized with its corresponding eigenvectors (##V##) to get the eigenvalues from ##V^{-1}AV = \Lambda##. These eigenvalues are the allowed energy transitions, and the multiplicity of the eigenvalues is directly related to the spin multiplicity of the molecule (a doublet state with ##S = \frac{1}{2}## would have two rotational energy terms).
I've started reading Rotational Spectroscopy of Diatomic Molecules by Brown & Carrington, along with a text by Lefebvre-Brion & Field and various papers introducing forms of the effective hamiltonian for different coupling (and uncoupling) cases. This thread has some interesting discussion on effective molecular Hamiltonians, but my fundamental question is a bit different.
I think having a roadmap of what things I should be reading, and in what order, would be immensely helpful. Thank you!
When I started working on my tool, I read about PGOPHER (a spectral tool which predicts transitions, does line fitting, etc.). Their website includes a small short course, which outlines the methods used to find transitions between two states on this page using matrix elements of an effective Hamiltonian. I essentially want to replicate this method of obtaining the line transitions by understanding the underlying theory.
From what I understand, the general method for analyzing molecular spectra is to choose an appropriate effective Hamiltonian, choose an appropriate basis set, work out the matrix elements (simplify where needed using symmetry arguments), and finally calculate the eigenvalues (energy levels). The effective Hamiltonian for a single vibrational level (what I'm interested in) can be written as the sum of individual perturbations, that is:
$$\mathbf{H}_{eff} = T_e + G_v + \mathbf{H}_{rot} + \mathbf{H}_{cd} + \dotsb$$
My question is, how can one go from an effective Hamiltonian to algebraic formulae representing the rotational energy levels? I'm most confused on how to get from the Hamiltonian to the matrix elements, forming the matrix, and then obtaining the eigenvalues.
This is my (very rough) interpretation: The chosen Hamiltonian operates on a chosen basis set in the form of ##|J, \Lambda, \Omega>## or something similar to get an expectation value ##<J, \Lambda, \Omega|\mathbf{H}|J, \Lambda, \Omega>##. Since there are multiple expectation values based on different ##\Omega, \Lambda## combinations, these can then be arranged into diagonal and off-diagonal elements of the representative matrix. This matrix is then somehow diagonalized with its corresponding eigenvectors (##V##) to get the eigenvalues from ##V^{-1}AV = \Lambda##. These eigenvalues are the allowed energy transitions, and the multiplicity of the eigenvalues is directly related to the spin multiplicity of the molecule (a doublet state with ##S = \frac{1}{2}## would have two rotational energy terms).
I've started reading Rotational Spectroscopy of Diatomic Molecules by Brown & Carrington, along with a text by Lefebvre-Brion & Field and various papers introducing forms of the effective hamiltonian for different coupling (and uncoupling) cases. This thread has some interesting discussion on effective molecular Hamiltonians, but my fundamental question is a bit different.
I think having a roadmap of what things I should be reading, and in what order, would be immensely helpful. Thank you!