Theoretical and computational chemistry

Vibrational, rotational and electronic spectra

Connect molecular energy-level structure to rotational, vibrational–rotational and vibronic spectra, and learn how line positions and intensities encode molecular properties.

1 Lower secondary 2 Upper secondary 3 Undergraduate 4 Advanced 5 Research

IntuitionThree energy scales, three spectral regions

A molecule can rotate, vibrate, and rearrange its electrons. Rotational spacings are usually smallest, vibrational spacings larger, and electronic excitation energies largest. Absorption occurs when photon energy matches an allowed level difference.

Switch between rotational CO lines, the HCl vibration–rotation band with P/R branches, and a Franck–Condon vibronic progression.

SchoolReading line positions

Definition: Wavenumber

Spectroscopists commonly report photon energy as wavenumber $ν̃=1/λ$ in cm $^{-1}$ . The energy is $E=hcν̃$ .

E_{J}=hcB J(J+1),\qquad \tilde\nu_{J\to J+1}=2B(J+1)

For a rigid diatomic rotor, neighboring rotational absorption lines are separated by approximately $2B$ . Since $B=h/(8π^2Ic)$ , larger moment of inertia means closer lines.

Example: Estimate a rotational spacing

A molecule has $B=1.93$ cm $^{-1}$ . What is the spacing between adjacent rigid-rotor lines?

Solution

Use $2B$ : the spacing is $3.86$ cm $^{-1}$ .

UndergraduateVibration–rotation structure

G(v)=\omega_e(v+\tfrac12)-\omega_ex_e(v+\tfrac12)^2;\quad F_v(J)=B_vJ(J+1)

The harmonic oscillator gives equally spaced vibrational levels; real bonds are anharmonic, so spacings shrink toward dissociation. A rovibrational transition changes both vibrational and rotational quantum numbers. For a diatomic electric-dipole band, $\Delta J=\pm1$ produces P and R branches; the absent Q branch leaves a gap near the band origin.

Motion	Typical region	Useful information
Rotation	Microwave / far IR	Moment of inertia and geometry

Electronic absorption often carries vibrational structure because nuclei do not move appreciably during the fast electronic transition (Franck–Condon principle). The intensity envelope depends on overlaps between vibrational wavefunctions on the two electronic surfaces.

I_{v'v''}\propto|\langle\chi_{v'}^{(e')}|\chi_{v''}^{(e'')}\rangle|^2

References

Molecular Spectra and Molecular Structure. II. Infrared and Raman Spectra of Polyatomic Molecules · Gerhard Herzberg, 1945
Molecular Spectra and Molecular Structure. III. Electronic Spectra and Electronic Structure of Polyatomic Molecules · Gerhard Herzberg, 1966