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Molecular Spectra

Friday, March 18, 2022

Molecular Transitions

A molecule can transition between electronic states when excited by a photon. However, the energy required to transition between electronic states is on the order of electron-volts. Within an electronic state, there are vibrational states (on the order of 0.11 eV0.1-1~\text{eV}) and rotational states (on the order of 0.010.1 eV0.01-0.1~\text{eV}).

Most molecules have both vibrational and rotational states, represented by vibrational quantum number NN and rotational quantum number LL. These molecules must follow both selection rules:

ΔN=1     and     ΔL=1\left|\Delta N\right|=1~~~~~\text{and}~~~~~\left|\Delta L\right|=1

Since all transitions must satisfy both these requirements, transitions to the next lowest rotational state, for instance, are not allowed (i.e. N=1,L=4N=1,L=4 to N=1,L=3N=1,L=3). Instead, the molecule would need to transition to N=0,L=3N=0,L=3 or N=0,L=5N=0,L=5 for example.


The energy of one of these states is found as



N=0,1,2,...     L=0,1,2,...N=0,1,2,...~~~~~L=0,1,2,...

Since the vibrational term is usually larger than the rotational term, the emission wavelengths typically correspond to NN1N\rightarrow N-1 with LL±1L\rightarrow L\pm 1. The absorption wavelengths will be one where NN increases by one unit.

For an absorption from N,LN,L to N+1,L±1N+1,L\pm 1, the possible energies are

ΔE=EN+1,L±1ENLΔE=[(N+32)hf+B(L±1)(L±1+1)][(N+12)hf+BL(L+1)]ΔE=hf+2B(L+1)     for LL+1ΔE=hf2BL     for LL1\Delta E=E_{N+1,L\pm 1}-E_{NL}\newline\Delta E=\left[\left(N+\frac{3}{2}\right)hf+B\left(L\pm 1\right)\left(L\pm 1+1\right)\right]-\left[\left(N+\frac{1}{2}\right)hf+BL\left(L+1\right)\right]\newline\Delta E=hf+2B\left(L+1\right)~~~~~\text{for }L\rightarrow L+1\newline\Delta E=hf-2BL~~~~~\text{for }L\rightarrow L-1

According to the energies above, absorption photons can have energies hf+2Bhf+2B, hf+4Bhf+4B, hf+6Bhf+6B, ...... , but also hf2Bhf-2B, hf4Bhf-4B, hf6Bhf-6B, ...... . Note: hfhf is missing since it would represent a pure vibrational transition, which is not allowed.

Example: HCl\text{HCl}

Below is the absorption spectrum of HCl\text{HCl}:

Absorption spectrum of HCl

Notice a few qualities where this differs from the ideal version:

  1. Transitions are not exactly an energy of 2B2B apart. In fact, the energies decreased moving away from the center. This is because the bond can stretch slightly when rotating at such speeds, increasing the value of ReqR_{\text{eq}}.
  2. The heights of the peaks are different. This is because the heights represent the intensity of the transitions, which depends on the population of the particular origin level. Since these decrease according to the Maxwell-Boltzmann distribution factor eE/kTe^{-E/kT}, they are not exactly equal. Additionally, the degeneracy of each level has an effect on the population, according to


Note: the level with maximum intensity can be found by locating when dp/dL=0dp/dL=0:


  1. Each peak looks like two very close peaks. This is because chlorine has two isotopes, one with a mass of 35 u35~\text{u} and another with a mass of 37 u37~\text{u}. The heavier one has slightly smaller energies since it takes more energy to vibrate and rotate it. The intensity has to do with the relative abundance of each isotope.


Just like how atomic spectroscopy helps identify atoms from their characteristic emission and absorption spectra, molecular spectroscopy helps identify molecules by which wavelengths of radiation they can absorb and emit. This gives each molecule a unique fingerprint which can be used to determine the composition of a material.