 2D IR slope-axis convention

A 2D IR slope-axis convention specifies which frequency coordinate is held fixed when taking spectral slices, which coordinate supplies each slice maximum, and which coordinate is plotted vertically. It also specifies whether the reported 2D IR center-line slope is that geometric slope or its inverse. These choices are needed before comparing slopes from two-dimensional infrared spectroscopy experiments or software.

Let $x$ be an excitation-frequency offset and $y$ a detection-frequency offset. An original positive bivariate normal distribution envelope with widths ${\sigma }_x,{\sigma }_y$ and correlation coefficient $\rho$ has conditional maximaThe first relation maximizes over $y$ at fixed $x$; the second maximizes over $x$ at fixed $y$. They trace different lines unless the distribution is degenerate. Changing the slicing direction is not the same operation as taking the reciprocal of an already fitted line's slope.

For $\rho =0.6$, ${\sigma }_x=10$ and ${\sigma }_y=20$ in matching frequency units, the first slope is 1.2. The second conditional slope $dx/dy$ is 0.3. Plotting that second set of maxima with $y$ vertical gives geometric slope $dy/dx=1/0.3=10/3$; its inverse is 0.3, not 1.2. Both sets came from the same stipulated band.

The 2007 primary method takes cuts parallel to the initial-frequency axis and defines its reported quantity through an inverse-slope convention. Its relationship with a vibrational frequency-frequency correlation function is established within the paper's response approximations.^[ref] A generic graphing routine's slope label does not identify that observable.

Units can change a geometric slope without changing the underlying correlation coefficient. Under coordinate scalings $x^{\prime }=ax$ and $y^{\prime }=by$, a fitted slope becomes $m^{\prime }=(b/a)m$. Expressing only the vertical axis in GHz instead of THz multiplies its numerical slope by 1000. Using the same conversion on both axes leaves the slope unchanged. Reciprocal wavelength coordinates require their own transformation rather than a constant rescaling over an arbitrary bandwidth.

Report axis labels, units, slice direction, fit range and inversion convention with the result. At $\rho =0$ in the ideal envelope, the maxima at fixed $y$ lie on $x=0$: that line is vertical even though its inverse slope tends to zero. A failed finite geometric fit near this limit need not mean molecular frequency memory increased. Noise, overlapping signed bands and finite fitting ranges require separate checks.