 2D IR population normalization

2D IR population normalization relates selected amplitudes in two-dimensional infrared spectroscopy to a population fraction while accounting for factors shared between the signals. It is a measurement-model operation: a normalized ratio is not automatically a molecular probability.

Suppose a positive signal component obeys $C=\beta Pq$ for a transferred population and $D=\alpha P(1-q)$ for the remaining population. Here $P$ is a common survival factor, $q$ the fraction of interest, and $\alpha ,\beta$ the respective response factors. The raw ratio isIt equals $q$ only when the response factors match. Solving instead gives the 2D IR response-factor correctionFor an original example, $q=0.5$, $\alpha =1$ and $\beta =2$ yield $r=2/3$. Substituting those response factors into the inverse relation recovers 0.5. Reading the uncorrected ratio as a population would overestimate it.

Common multiplicative gain cancels from the ratio, but common additive background generally does not. Stipulated signals $C=6,D=4$ give $r=0.6$; doubling both preserves 0.6, while adding 1 to each gives $7/12\approx 0.583$.

Uncertainty also depends on shared inputs. For $r=C/(C+D)$,This follows from correlated-input uncertainty propagation.^[ref] Small denominators and uncertain backgrounds can make the ratio unstable.

Real 2D IR cross peaks can have signed or complex response contributions. Normalizing such values requires defining the fitted signal component and demonstrating its connection to populations. A waiting-time-dependent cross peak is an observation to model, rather than a population count supplied directly by the spectrometer.