= Vibrational spectral diffusion
\b[Vibrational spectral diffusion] is the evolution of a molecular vibrational transition frequency as its surroundings or configuration fluctuate. It concerns frequency memory within a vibrational response, whereas <vibrational relaxation> concerns loss or redistribution of vibrational excitation. Their characteristic times need not coincide.
In <two-dimensional infrared spectroscopy>, an initial frequency label is followed by an independently varied <2D IR waiting time> and a later frequency-sensitive readout. The <vibrational frequency-frequency correlation function> models the relation between those earlier and later frequencies, while a <2D IR center-line slope> summarizes part of the measured spectral geometry under specified response assumptions.<Frequency-frequency correlation functions and apodization in two-dimensional infrared vibrational echo spectroscopy: A new approach by Kyungwon Kwak and others>[Introduction and §II.A, frequency labeling, population interval and correlation-function model]{ref}
An original <two-state vibrational frequency memory> model has equally populated environments with spectroscopic frequencies $\widetilde\nu_0-\Delta$ and $\widetilde\nu_0+\Delta$. Suppose switching in either direction follows a <continuous-time Markov chain> with rate $k$. With a sign variable $s=\pm1$, the probability of a flip during $dt$ is $k\,dt$; each flip reverses its frequency deviation. Hence the normalized sign memory satisfies
$$
\frac{dc}{dT}=-2kc,\qquad c(0)=1,
\qquad C_{\widetilde\nu}(T)=\Delta^2e^{-2kT}.
$$
The memory time is $1/(2k)$, not the mean residence time $1/k$ in either state. Returning to the initial state contributes to the measured memory, so these two time scales describe different events.
For stipulated transitions at 1000 and $1010\,\mathrm{cm^{-1}}$ with $k=0.25\,\mathrm{ps^{-1}}$, the mean is $1005\,\mathrm{cm^{-1}}$, the variance is $25\,\mathrm{cm^{-2}}$ and the memory time is 2 ps. At 2 ps the correlation is $25e^{-1}\approx9.20\,\mathrm{cm^{-2}}$. The probability of occupying the initial environment is $(1+e^{-1})/2\approx0.684$, including trajectories with an even number of switches.
If an independent vibrational population in this illustrative model decays with lifetime 10 ps, its surviving fraction at 2 ps is $e^{-0.2}\approx0.819$, while normalized frequency memory is only 0.368. This explicitly separates population survival from frequency decorrelation. These are model values, not measured solvent or vibrational constants.
The model uses <spectroscopic wavenumber>, not angular frequency. Converting to an angular-frequency correlation multiplies by $(2\pi c_0)^2$ when the speed of light $c_0$ uses compatible length units. Mixing these conventions changes the fluctuation amplitude without changing the switching time.
To test the interpretation, acquire spectra at several waiting times, characterize pulse timing and phase, and assess overlap, population decay and instrument response. A decaying spectral statistic can constrain frequency fluctuations, but the same observed decay may fit more than one microscopic environment model. The two-state calculation is a reusable mechanism example, not a universal explanation of <spectral diffusion>.
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