= 2D IR diagonal-peak overlap
{c}
<2D IR diagonal-peak overlap> is the contribution of extended diagonal spectral features to regions where an off-diagonal signal might be assigned in <two-dimensional infrared spectroscopy>. A nonzero value at an off-diagonal coordinate is not sufficient evidence for a separate <2D IR cross peak>.
Peak orientation matters as well as width. An original positive-envelope model for a diagonal feature centered at $(c,c)$ is
$$
G_c(x,y)=\exp\!\left[-\frac{(x-c)^2-2\rho(x-c)(y-c)+(y-c)^2}{2\sigma^2(1-\rho^2)}\right],\qquad |\rho|<1.
$$
Here $\rho$ specifies <2D IR line-shape correlation>. It is a shape parameter in this illustrative model, not automatically a measured population correlation. Let $S=G_a+G_b$ for two diagonal features, with $\Delta=b-a$. At the off-diagonal coordinate $(a,b)$,
$$
S(a,b)=2\exp\!\left[-\frac{\Delta^2}{2\sigma^2(1-\rho^2)}\right].
$$
For original values $a=1600$, $b=1650$ and $\sigma=20$, all in $\mathrm{cm^{-1}}$, the off-diagonal amplitude is approximately 0.0879 at $\rho=0$, but only 0.000340 at $\rho=0.8$. Both models have the same individual marginal width. An overlap estimate based only on that width can therefore miss a substantial shape dependence.
\Image[/ir-diagonal-overlap-correlation.png]
{title=Original two-diagonal-feature model: changing <2D IR line-shape correlation> changes off-diagonal tails without adding a <2D IR cross peak>}
The toy envelope is positive. Actual coherent spectra can contain signed features, so fitting and subtraction must respect the measured response and phase convention. The primary phase-controlled peptide experiment used phase and amplitude information to interpret mode-coupling cross peaks.<Two-dimensional infrared spectroscopy of peptides by phase-controlled femtosecond vibrational photon echoes by M. C. Asplund and others>[Abstract, phase-sensitive measurement and amide-I cross peaks.]{ref}
Useful tests vary the fit region, include plausible neighboring features and compare the proposed cross signal across <2D IR waiting times> and other controlled conditions. A residual can indicate a missing feature, an unsuitable diagonal line shape or an unmodeled background. Its persistence alone does not uniquely select one molecular explanation.
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