= Spectral diffusion
\b[Spectral diffusion] is the time-dependent wandering of a transition frequency as its local environment changes. A collection of molecules can retain the same instantaneous frequency distribution while individual members lose memory of their earlier frequencies. In the <infrared spectroscopy> setting, <vibrational spectral diffusion> concerns vibrational transitions. The name describes frequency evolution, not necessarily spatial <diffusion> of the molecules.
A useful original stationary model writes $\omega(t)=\overline\omega+\delta\omega(t)$ and defines the <autocovariance function>
$$
C(T)=\mathbb E[\delta\omega(0)\delta\omega(T)].
$$
For a real <stationary process>, its frequency-increment variance is
$$
\mathbb E[(\omega(T)-\omega(0))^2]=2[C(0)-C(T)].
$$
This follows by expanding the square: stationarity makes the two squared-frequency terms equal, while the cross term is $2C(T)$. The relation supplies a <spectral-diffusion frequency-increment variance> without assuming that the mean transition frequency shifts.
Take the stipulated model $C(T)=\Delta^2e^{-|T|/\tau}$, with $\Delta=1\,\mathrm{rad\,ps^{-1}}$ and $\tau=2\,\mathrm{ps}$. After 2 ps, normalized memory is $e^{-1}\approx0.368$ and the root-mean-square frequency change is $\sqrt{2(1-e^{-1})}\approx1.124\,\mathrm{rad\,ps^{-1}}$. The long-time increment spread approaches $\sqrt2\Delta$, while the single-time spread remains $\Delta$. Thus loss of temporal correlation need not broaden the stationary ensemble distribution over time.
A static distribution offers a contrasting physical limit. Molecules may have different, but individually fixed, frequencies. Then $C(T)=C(0)$ and the increment variance is zero despite a broad ensemble spectrum. A single <spectral line> width cannot determine how much of that diversity is static or dynamically sampled.
In <two-dimensional infrared spectroscopy>, controlled infrared pulses first label frequency and later interrogate the response after a <2D IR waiting time>. Repeating over waiting times tests the frequency relation rather than merely measuring one ensemble average.<Frequency-frequency correlation functions and apodization in two-dimensional infrared vibrational echo spectroscopy: A new approach by Kyungwon Kwak and others>[Introduction, pulse timing, initial-frequency labeling and later frequency readout]{ref} Its calculated spectra illustrate the associated evolution of spectral geometry.<Frequency-frequency correlation functions and apodization in two-dimensional infrared vibrational echo spectroscopy: A new approach by Kyungwon Kwak and others>[Fig. 1, calculated spectra and center lines at different waiting times]{ref}
Environmental fluctuations, population loss and exchange between resolved species affect spectra in different ways. A changing <2D IR line-shape correlation> is useful evidence only with an appropriate response model and controls. A fitted time scale is not automatically a molecular transport coefficient or the lifetime of every environmental structure.
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