 2D IR response-factor correction

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 $r=C/(C+D)$ obeysThe 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 $4/7\simeq 0.571$, while the corrected fraction is $8/(12+8)=0.4$.

The effect is especially transparent on an odds scale: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 $q=8/21.2\simeq 0.377$. A response factor that is uncertain should contribute to uncertainty propagation, rather than being treated as an exact correction.^[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.^[ref] Signed or complex spectral components do not satisfy the positive-component population model merely because they have been divided by a sum.