2D IR center-line slope

A 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 maximumReversing the slicing direction givesThese 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 $(x_i,y_i)$ isThis 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.^[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.

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.