= Vibrational frequency-frequency correlation function
A \b[vibrational frequency-frequency correlation function] describes how deviations of a vibrational transition frequency at two times vary together. For a real <stationary process> $\omega(t)$ with mean $\overline\omega$, define
$$
\delta\omega(t)=\omega(t)-\overline\omega,\qquad
C(T)=\langle\delta\omega(0)\delta\omega(T)\rangle.
$$
This is an <autocovariance function>. It has squared-frequency units, and $C(0)$ is the <variance> of the frequency fluctuations. If $C(0)>0$, the normalized quantity $c(T)=C(T)/C(0)$ starts at one. Normalizing removes the fluctuation amplitude, so two systems can have the same $c(T)$ but different $C(0)$.
<Vibrational spectral diffusion> is the associated evolution of transition frequencies in their changing molecular surroundings. Decay of their correlation need not mean loss of vibrational population.
An original exponential example is
$$
C(T)=\Delta^2e^{-|T|/\tau}.
$$
With $\Delta=0.5\,\mathrm{rad\,ps^{-1}}$ and $\tau=2\,\mathrm{ps}$, $C(0)=0.25\,\mathrm{rad^2\,ps^{-2}}$ and $C(2\,\mathrm{ps})\simeq0.0920\,\mathrm{rad^2\,ps^{-2}}$, while $c(2\,\mathrm{ps})=e^{-1}$. The mean transition frequency has not been specified by these fluctuation parameters.
The function also connects a frequency model with coherence decay. For a zero-mean classical <Gaussian process>, neglect population loss and other broadening, and let accumulated phase be $\phi(t)=\int_0^t\delta\omega(s)\,ds$. Its Gaussian average gives
$$
\langle e^{-i\phi(t)}\rangle=e^{-g(t)},\qquad
g(t)=\int_0^t(t-u)C(u)\,du.
$$
For the exponential model,
$$
g(t)=\Delta^2\left[\tau t-\tau^2(1-e^{-t/\tau})\right].
$$
Thus the chosen example has $g(2\,\mathrm{ps})=e^{-1}$ and coherence factor about $0.692$. These are stipulated model values, not an experimental fit.
The 2007 primary analysis relates <vibrational spectral diffusion> to changing <2D IR line-shape correlation>. Three controlled infrared pulses label an initial frequency and later read out the response, with the intervening <2D IR waiting time> varied.<Frequency-frequency correlation functions and apodization in two-dimensional infrared vibrational echo spectroscopy: A new approach by Kyungwon Kwak and others>[Introduction, pulse delays, frequency labeling and frequency-frequency correlation function]{ref} A measured <2D IR center-line slope> can constrain such dynamics under appropriate response approximations; it is not a direct time trace of each molecule's frequency. <Vibrational relaxation> and instrumental effects need separate consideration.
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