Rotational fine structure refers to the splitting of vibrational energy levels into sub-levels due to the rotational motion of molecules. This phenomenon occurs in electronic-vibrational transitions within molecules and is crucial for understanding the detailed structure of their absorption or emission spectra. Here’s an explanation of rotational fine structure in electronic-vibrational transitions:

Basics of Rotational Fine Structure:

Molecular Energy Levels: Molecules can absorb or emit photons corresponding to transitions between different electronic, vibrational, and rotational energy levels. Electronic transitions involve changes in the electronic configuration (from one electronic state to another), vibrational transitions involve changes in vibrational energy levels (within the same electronic state), and rotational transitions involve changes in rotational energy levels (within the same vibrational and electronic state).
Selection Rules: The selection rules for transitions between rotational levels depend on the change in the rotational quantum number J.
Energy Levels and Transitions: When a molecule undergoes an electronic-vibrational transition, the resulting vibrational energy levels can have different rotational energy levels associated with them due to the molecule's rotation. This results in a series of closely spaced spectral lines corresponding to transitions between different rotational levels of the same vibrational state.

Interpretation and Significance:

Line Spacing: The spacing between rotational lines within a vibrational band depends on the rotational constant BBB of the molecule. The rotational energy levels are quantized and typically follow the formula:
E_r = BJ(J+1)hc ...1
where J =0,1,2,..... is rotational Quantum number and B = h/ 8𝜋²IC rotational constant.

Let us consider electronic, vibrational and rotational energies of molecules are independent of each other (called Born - oppenheimer approximation). Then total energy of molecule will be

E_T = E_e + E_𝛾+ E_r

= E_e + E_𝛾+ BJ(J+1)hc....2

And the change in total energy will be

ΔE_T = ΔE_e + ΔE_𝛾+ ΔBJ(J+1)hc ....3

Then, the frequency of rotational corresponding to such change in total energy is

𝛾 = ΔE_T/ ch = ΔE_e + ΔE_𝛾 / ch + Δ BJ(J+1)]

= 𝛾_o + Δ BJ(J+1)] ...4

𝛾_o is wave number of electronic vibrational transition and it is the frequency of origin of particular band

Let us consider the transition between upper electron state with B', J' and lower electronic state with B'', J''. Then the frequency of radiation in the unit of wave number will be

𝛾 = 𝛾_o + B'J'(J'+1)- B''J''(J''+1) ........5

The transition for which

i. ΔJ = -1 i.e J'' = J' + 1 gives the lines which constitute a fine structure called P branch.

ii. ΔJ = +1 i.e J'' = J' + 1 gives the lines which constitute a fine structure called Q branch

The frequency of P branch lines in wave number term is

𝛾 = 𝛾_o + B'J'(J'+1)- B''J''(J'+1)(J'+2)

= 𝛾_o - (B'+ B'')(J'+ 1) + (B'- B'')(J'+1)² ....6

Similarly frequency of Q branch lines in wave number term is

𝛾 (R) = 𝛾_o + B'(J''+ 1)(J'' +2) - B''J''(J''+1)

= 𝛾_o + (B' + B'')(J''+1) + (B'- B'')(J'+1)² ...7

Combining 6 and 7

𝛾 (PR) = 𝛾_o + (B' + B'')m + (B'- B'')m² ...8

where m = - (J' +1) for P branch J' = 0,1,2

= (J'' +1) for R branch J'' = 0,1,2

Putting J'= J'' in equation 5, we get the frequency in wave number for Q branch

𝛾 (Q) = 𝛾_o + B'J''(J''+ 1)(J'' +2) - B''J''(J''+1) ; J'' = 1,2,3

= 𝛾_o + (B'- B'')J''(J''+1)

= 𝛾_o + (B'- B'')J''² + (B'- B'')J'' ...9

Case I: (B'- B'') is negative i.e B'< B''

In this case the separation of successive R branch lines decreases with increasing value of m(1,2,3), the separation of successive P branch lines increases with decreasing value of m(-1,-2,-3..) and separation of Q branch lines increases with increasing value of m(1,2,3).

From figure, it is seen that band head appears on the high wave number side i.e R branch and band tail appears on the low wave number side i.e P branch. Thus the intensity band falls off towards res end of spectrum and hence the band is degraded towards red end.

Case II (B' - B'') is positive i.e B'> B''

Separation of R branch line increase with increasing value of m, the separation of P branch line decrease with decreasing value of m and the separation of Q branch line decrease with increasing value of m.

This note is a part of the Physics Repository

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