= 2D IR response-factor correction
{c}
\b[2D IR response-factor correction] adjusts a measured signal ratio when the spectral components being compared do not produce equal response per unit population. It is part of <2D IR population normalization>, rather than a rule that every <2D IR cross peak> directly counts transferred molecules.
For positive fitted components, suppose $C=\beta Pq$ and $D=\alpha P(1-q)$, where $q$ is the fraction of interest and $P$ is a shared population-survival factor. Define the <2D IR response-factor ratio> $\kappa=\beta/\alpha>0$. The measured fraction $r=C/(C+D)$ obeys
$$
r=\frac{\kappa q}{\kappa q+1-q},\qquad
q=\frac{r}{\kappa(1-r)+r}=\frac{C}{\kappa D+C}.
$$
The second expression follows by solving the first; $q=r$ requires $\kappa=1$ in this model. An original example uses $C=8$, $D=6$ and $\kappa=2$. The raw ratio is $4/7\simeq0.571$, while the corrected fraction is $8/(12+8)=0.4$.
The effect is especially transparent on an odds scale:
$$
\ln\frac{q}{1-q}=\ln\frac{C}{D}-\ln\kappa,\qquad
\frac{\partial q}{\partial\ln\kappa}=-q(1-q).
$$
At $q=0.4$, a small relative error of $10\%$ in $\kappa$ changes the inferred fraction by about $0.024$ in magnitude to first order. Using $\kappa=2.2$ instead of $2$ in the example gives $q=8/21.2\simeq0.377$. A response factor that is uncertain should contribute to <uncertainty propagation>, rather than being treated as an exact correction.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[Sections 5.1–5.2, sensitivity coefficients and correlated inputs]{ref}
A useful experimental check measures a suitable reference under the same pulse, detection and fitting conditions, and asks whether the inferred response ratio transfers to the unknown sample. Common multiplicative gain cancels; additive background, differing response pathways or a changing ratio need additional modeling. Primary phase-sensitive peptide measurements recorded both phase and amplitude of <vibrational photon echoes>, illustrating why the fitted response component must be specified.<Two-dimensional infrared spectroscopy of peptides by phase-controlled femtosecond vibrational photon echoes by M. C. Asplund and others>[Abstract, phase matching and heterodyne measurement of echo phase and amplitude]{ref} Signed or complex spectral components do not satisfy the positive-component population model merely because they have been divided by a sum.
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