 2D IR line-shape correlation

2D IR line-shape correlation describes how excitation- and detection-frequency coordinates vary together within a spectral feature in two-dimensional infrared spectroscopy. An elongated feature can retain a relation between the initially selected frequency and the frequency read out later. Its shape can change with 2D IR waiting time, but a fitted geometric parameter needs a response model before it is assigned a molecular dynamical meaning.

An original Gaussian-envelope example makes the geometry precise. Let centered frequency offsets be $x,y$, with widths ${\sigma }_x,{\sigma }_y$ and correlation coefficient $\rho$:The associated covariance matrix isAt fixed $x$, maximizing $G$ over $y$ gives the lineThus a 2D IR center-line slope equals $\rho$ only for this orientation with equal widths in compatible frequency coordinates. For stipulated widths ${\sigma }_x=12$ and ${\sigma }_y=18$ in $\mathrm{c}{\mathrm{m}}^{-1}$ and $\rho =0.75$, the slope is $1.125$. A slope greater than one is not an impossible correlation coefficient; the two quantities have different width dependence.

For equal widths $\sigma$, rotated coordinates $u=(x+y)/\sqrt{2}$ and $v=(y-x)/\sqrt{2}$ have variances ${\sigma }^2(1+\rho )$ and ${\sigma }^2(1-\rho )$. At $\rho =0.75$, the diagonal-to-antidiagonal width ratio is $\sqrt{7}$. This geometric identity also helps explain why 2D IR diagonal-peak overlap depends on orientation as well as marginal width.

A 2007 primary analysis by Kyungwon Kwak, Sungnam Park, Ilya J. Finkelstein and M. D. Fayer extracted a waiting-time-dependent observable from maxima of spectral slices and tested it against calculated spectra with known frequency dynamics. Its center-line convention explicitly depends on the direction of the slices.^[ref]

A vibrational frequency-frequency correlation function instead concerns frequency fluctuations at two times. Under an appropriate stationary model, its normalized form is $C(T)/C(0)$ with $C(T)=⟨\delta \omega (0)\delta \omega (T)⟩$. Relating it to spectral geometry requires the pulse sequence, broadening model and slope convention. Multiple overlapping peaks, signed responses and finite acquisition windows must be addressed before interpreting a change of shape as a unique molecular time scale.