= 2D IR line-shape correlation
{c}
\b[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> $\rho$:
$$
G(x,y)\propto\exp\!\left[-\frac{x^2/\sigma_x^2-2\rho xy/(\sigma_x\sigma_y)+y^2/\sigma_y^2}{2(1-\rho^2)}\right],\qquad |\rho|<1.
$$
The associated <covariance matrix> is
$$
\Sigma=\begin{pmatrix}\sigma_x^2&\rho\sigma_x\sigma_y\\\rho\sigma_x\sigma_y&\sigma_y^2\end{pmatrix}.
$$
At fixed $x$, maximizing $G$ over $y$ gives the line
$$
y_{\max}(x)=\rho\frac{\sigma_y}{\sigma_x}x.
$$
Thus 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 $\mathrm{cm^{-1}}$ 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 $u=(x+y)/\sqrt2$ and $v=(y-x)/\sqrt2$ have <variances> $\sigma^2(1+\rho)$ and $\sigma^2(1-\rho)$. At $\rho=0.75$, the diagonal-to-antidiagonal width ratio is $\sqrt7$. 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.<Frequency-frequency correlation functions and apodization in two-dimensional infrared vibrational echo spectroscopy: A new approach by Kyungwon Kwak and others>[Abstract, Introduction and Section II.B, center-line definition and calculated-spectrum tests]{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 $C(T)=\langle\delta\omega(0)\delta\omega(T)\rangle$. 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.
Back to article page
