Molecular rotational levels

In summary, molecular rotational levels refer to the different energy states a molecule can occupy due to its rotation, determined by its moment of inertia. These levels help explain the rotational spectra of molecules and can be manipulated using radiation to gain information about their structure and properties. With increasing temperature, molecules can occupy higher rotational levels and the spacing between levels decreases.
  • #1
DanAbnormal
23
0
Im doing a study on a polyatomic molecule which is assigned by six quantum numbers, namely the three vibrational modes, and the Rotational quantum number J, and its projections on the A and C axes, (Ka and Kc):

J, Ka, Kc, v1, v2, v3

Each combination of these numbers has an energy in wavenumbers, ie. 0 0 0 0 0 0 has energy 0 cm^-1 (ground level)

I have to use fortran to manipulate a few lists of these numbers (big lists!) and was wondering if there was any sort of method of combining the numbers into a compound, unique number because its the only way I can perform the manipulation in my program.

My project supervisor says there is a formula which can be used to combine the rotational quantum numbers (J, Ka, Kc) as such:

0 0 0 0

1 0 1 1
1 1 0 2
1 1 1 3

2 0 2 4
2 1 1 5
2 1 2 6
2 2 0 7
2 2 1 8

3 0 3 9
3 1 2 10
3 1 3 11
3 2 1 12
3 2 2 13
3 3 0 14
3 3 1 15

and so on...

The list is ordered according to the value of J, in the first column. The groupings arise as a result of the 2J+1 degeneracy. So apparently there is a formula which fits this, I can't figure it out!
Ive tried by noting that the first "number assignment" for a new grouping is the square of the J value, can't get anything else out of that.
Also, even though its not my first priority, it would be nice if there was a similar situation for the vibrational quantum numbers, such that I could ultimately comine all six numbers into a unique identifier, but for now the rotational numbers is what I am concerned about.

Cheers
Dan
 
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  • #2
The formula for combining the rotational quantum numbers into a unique number is: Number = (J * (J + 1) / 2) + Ka + Kc + 1 For example, the first group in your list, with J=0, Ka=0, and Kc=0, would be Number = (0 * (0+1) / 2) + 0 + 0 + 1 = 1. The second group, with J=1, Ka=0, and Kc=1, would be Number = (1 * (1+1) / 2) + 0 + 1 + 1 = 3, and so on. As far as combining all six quantum numbers into one unique identifier, you could use a similar formula to combine the three vibrational modes and the rotational quantum numbers together. For example, you could use the following formula: Number = ((v1 * (v1 + 1) / 2) + v2 + v3 + 1) * (J * (J + 1) / 2) + Ka + Kc + 1 Using this formula, the first group in your list, with J=0, Ka=0, Kc=0, v1=0, v2=0, and v3=0, would be Number = ((0 * (0+1) / 2) + 0 + 0 + 1) * (0 * (0 + 1) / 2) + 0 + 0 + 1 = 1. The second group, with J=1, Ka=0, Kc=1, v1=0, v2=1, and v3=1, would be Number = ((0 * (0+1) / 2) + 1 + 1 + 1) * (1 * (1 + 1) / 2) + 0 + 1 + 1 = 7. Hope this helps!
 

What are molecular rotational levels?

Molecular rotational levels refer to the different possible energy states that a molecule can occupy due to its rotation. These levels are quantized, meaning they can only exist at specific energy values.

How are molecular rotational levels determined?

The rotational levels of a molecule are determined by its moment of inertia, which is a measure of how the mass of the molecule is distributed around its axis of rotation. The moment of inertia is dependent on the molecular structure and can be calculated using quantum mechanical principles.

What is the significance of molecular rotational levels?

The existence of molecular rotational levels helps to explain the rotational spectra of molecules. By studying the transitions between these levels, scientists can gain information about the molecular structure and properties, such as bond lengths and dipole moments.

How do molecular rotational levels relate to temperature?

As temperature increases, molecules have a higher kinetic energy and are able to occupy higher rotational levels. This results in more frequent transitions between levels, leading to a broader rotational spectrum. Additionally, the spacing between rotational levels decreases with increasing temperature.

Can molecular rotational levels be manipulated?

Yes, molecular rotational levels can be manipulated using various techniques such as microwave or infrared radiation. By applying a specific frequency of radiation, molecules can be forced to transition between rotational levels, providing valuable information about their structure and behavior.

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