Why a spectral line broadens

Motion can persist after phases spread

A short pulse prepares related phases across many molecules. Represent each vibration by a rotating arrow. Their vector sum gives the macroscopic polarization, the collective electric-dipole response that radiates from the sample. Aligned arrows add strongly; dispersed arrows can nearly cancel.

A coherence describes an amplitude with a definite phase relation between quantum states. A population describes how much occupation an energy level has. Let $T_1$ denote population relaxation time and $T_2$ coherence decay time. Population loss contributes half its decay rate to the coherence rate.

\frac{1}{T_2}=\frac{1}{2T_1}+\frac{1}{T_2^*}

This is a simple two-level Markov model. The starred time denotes pure dephasing; no separate orientational relaxation is included.

Reading: Andrei Tokmakoff · Fenn and Fayer

The same broad peak can have different origins

If every molecule loses coherence at the same exponential rate, the broadening is homogeneous. A distribution of long-lived resonance frequencies instead causes inhomogeneous broadening: individual arrows keep rotating, but their sum decays. In liquids, those frequencies can change as environments evolve. We will call that spectral diffusion.

For a static Gaussian frequency distribution, write the angular-frequency offset as $\delta\omega$ and its standard deviation as $\sigma_\omega$ . Averaging the phase arrows gives a Gaussian envelope. An independent homogeneous decay multiplies that envelope.

\left\langle e^{-i\delta\omega t}\right\rangle=e^{-\sigma_\omega^2t^2/2},\qquad P(t)\propto e^{-t/T_2}e^{-\sigma_\omega^2t^2/2}

Offsets in radians per picosecond require times in picoseconds. The two factors describe different mechanisms.

Transform a decay into a line

Take the causal response $e^{-t/T_2}e^{-i\omega_0t}$ and integrate over positive time. At resonance, successive contributions add in phase. Away from resonance, their phases spread. The real part of the following integral gives an absorptive line.

\operatorname{Re}\int_0^\infty e^{-t/T_2}e^{i(\omega-\omega_0)t}\,dt=\frac{1/T_2}{(1/T_2)^2+(\omega-\omega_0)^2}

This Lorentzian has angular-frequency half width equal to the inverse coherence time. Converting its full width to wavenumber introduces the speed of light and a full-turn factor.

Reading: Andrei Tokmakoff · Fenn and Fayer

Worked example for one picosecond of coherence

Set $T_2=1\,\mathrm{ps}$ . The angular-frequency full width is $2\,\mathrm{rad\,ps}^{-1}$ , or about $10.6\,\mathrm{cm}^{-1}$ . If $T_1=2\,\mathrm{ps}$ , population loss contributes $0.25\,\mathrm{ps}^{-1}$ to the coherence decay rate. The remaining $0.75\,\mathrm{ps}^{-1}$ is pure dephasing.

Adding a static frequency distribution broadens the line further. The inverse total linewidth can no longer be read directly as $T_2$ . An echo tests which part of the phase spread can return.

\Delta\widetilde\nu_{\mathrm{FWHM}}=\frac{1}{\pi cT_2}=10.6\,\mathrm{cm}^{-1}

This conversion applies to the exponential coherence decay used here. A Gaussian distribution follows a different linewidth formula.

Exercises and explanations

For a purely Lorentzian line, how does doubling coherence time change the wavenumber full width?

Read the explanation

It halves the width because the coherence time enters the denominator. The conclusion assumes exponential decay and does not directly describe static Gaussian broadening.

Can the macroscopic polarization nearly vanish while individual molecules still oscillate?

Read the explanation

Yes. Polarization is a phase-sensitive vector sum. Opposing contributions can cancel without each molecular vibration losing its energy.

Sources and calculation scope

Andrei Tokmakoff, Nonlinear and Two-Dimensional Spectroscopy (June 2011, v1)

Author lecture notes, nonlinear-spectroscopy section: density-matrix perturbation theory; printed pp. 23–25, Eqs. (51)–(55), for rephasing and nonrephasing; pp. 29–30 for the vibrational harmonic limit. The course examples are new teaching calculations.

Fenn and Fayer, Extracting 2D IR frequency-frequency correlation functions from two component systems (2011)

Journal of Chemical Physics 135, 074502, publisher version, DOI 10.1063/1.3625278. Read Figs. 1–3 and Eqs. (1)–(4), (12), (14)–(15). The bounded numerical benchmark is the single-component Gaussian slope relation in Eq. (12), with original synthetic parameters; it is not a fit to the paper’s water measurements.