= 2D IR center-line slope
{c}
A \b[2D IR center-line slope] is obtained by locating peak maxima in a sequence of slices through a <two-dimensional infrared spectroscopy> feature, then fitting the line traced by those maxima. Its definition includes which coordinate is held fixed, which coordinate is maximized and which coordinate is plotted vertically. Those choices constitute the <2D IR slope-axis convention>.
Let $x$ denote excitation-frequency offset and $y$ detection-frequency offset. For a positive bivariate Gaussian envelope with widths $\sigma_x,\sigma_y$ and <correlation coefficient> $\rho$, completing the square gives the slice maximum
$$
y_{\max}(x)=\rho\frac{\sigma_y}{\sigma_x}x,\qquad
m_{y|x}=\rho\frac{\sigma_y}{\sigma_x}.
$$
Reversing the slicing direction gives
$$
x_{\max}(y)=\rho\frac{\sigma_x}{\sigma_y}y,\qquad
m_{x|y}=\rho\frac{\sigma_x}{\sigma_y}.
$$
These conditional-maximum slopes are generally not reciprocals. For an original example with $\rho=0.6$, $\sigma_x=10$ and $\sigma_y=20$ in compatible frequency units, they are $1.2$ and $0.3$. Taking $1/1.2$ would not produce the reverse-slice value. Their product is $\rho^2=0.36$ for this particular Gaussian model.
A simple <linear regression> estimator from slice maxima $(x_i,y_i)$ is
$$
\widehat m=\frac{\sum_i(x_i-\overline x)(y_i-\overline y)}{\sum_i(x_i-\overline x)^2}.
$$
This fits the maxima, rather than calculating a spectral-intensity-weighted <covariance>. Weak slices, overlapping features or asymmetric peaks can move their maxima and change the fitted line. The fitting interval, baseline treatment and slice direction therefore belong with the reported value.
The primary 2007 method uses a specified inverse-slope convention and tests its relation to the <vibrational frequency-frequency correlation function> using calculated spectra.<Frequency-frequency correlation functions and apodization in two-dimensional infrared vibrational echo spectroscopy: A new approach by Kyungwon Kwak and others>[Abstract and Section II.B, center-line definition and inverse-slope convention]{ref} Its convention must be read before comparing its observable with a raw geometric slope or a software output.
For a validated response model, measuring the slope at successive <2D IR waiting times> can reveal changing <2D IR line-shape correlation>. A single fit does not by itself identify a molecular relaxation time, and finite time windows or signed spectral contributions require attention before treating a spectral feature as a probability envelope.
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