Density of States: IR Transitions in CO2 Molecule

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Discussion Overview

The discussion revolves around determining the density of states for the CO2 molecule, specifically focusing on vibrational transitions at infrared (IR) wavelengths. Participants explore methods for calculating these states and their implications for statistical distributions, particularly the Maxwell-Boltzmann distribution.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about an easy method to determine the density of states for CO2, focusing on vibrational modes relevant to IR transitions.
  • Another participant questions the concept of density of states for an isolated molecule and notes that CO2 has four vibrational eigenmodes likely in the IR range, while electronic transitions occur in the UV-vis range.
  • A suggestion is made to calculate the IR spectra through normal mode analysis or by using Monte Carlo or molecular dynamics methods to autocorrelate the dipole moment.
  • A participant expresses a desire to plot the Maxwell-Boltzmann distribution for vibrational modes, similar to existing work on rotational levels, and questions whether the degeneracy G(J) is 4 for vibrational modes.
  • Another participant confirms that a similar analysis can be done for vibrational modes but notes that excluding rotational fine structure is generally unphysical. They also clarify that there is no G(J) for vibrational modes, as degeneracy is typically 1 unless in rare cases of commensurate eigenvalues.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of density of states for isolated molecules and the relevance of rotational modes in the context of vibrational analysis. There is no consensus on the specific degeneracy of vibrational modes.

Contextual Notes

Some limitations include the dependence on specific definitions of density of states and the assumptions regarding the physical conditions under which the analysis is conducted. The discussion also highlights the complexity of integrating vibrational and rotational modes in statistical mechanics.

Steleo
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Hi All
Im just wondering if there is an easy way to determine the density of states for a molecule such as CO2. I am interested in transitions at IR wavelengths so I'm wondering if there is an 'easy' way to get at the vibrational modes only to chuck into something like the Maxwell-Boltzmann distribution.
 
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Density of states for an isolated molecule? I don't follow.

The CO2 molecule has 4 vibrational eigenmodes - these are probably in the IR range. It has 12 valence molecular orbitals (from overlap of the one 2s and three 2p atomic orbitals of the 3 atoms). Transitions between the electronic states are in the UV-vis range. The rotational eigenmodes are in the microwave regime.
 
the IR spectra of the gas phase molecule can be calculated through normal mode analysis.

another way would be to calculate a set of states either by monte carlo or molecular dynamics, and then autocorrelate the dipole moment. formally this arises by being able to rewrite the fermi golden rule expression in terms of a TCF (interestly enough, a classical approach yields the same expression, see Frenkel or McQuarrie for details).

look into time correlation theory and normal mode analysis. a suitable text such as herzberg will tell you all you'll want to know. there are public domain codes that will do these things or you can implement your own.
 
Hi thanks for the replies,
Sorry for being a bit unclear in the original question, basically I want to do something similar to http://webphysics.davidson.edu/alumni/jimn/Final/Pages/FinalMolecular.htm
Where they plot the Maxwell-Boltzmann distribution for the rotational levels, however I want to do it for the vibrational modes. I'm really wondering whether G(J) is 4 for the vibrational modes of a molecule such as CO2?
Thanks
 
1. Yes, you can do a similar thing with only the vibrational modes (but leaving out the fine structure from the rotational modes makes the system specific to ones where rotation is inhibited - this is rare, and in general, unphysical).

2. There isn't a G(J) for the vibrational modes (there's no angular momentum involvement). The degeneracy of the vibration modes is 1 except in the rare case of commensurate eigenvalues.
 

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