= 2D IR population normalization
{c}
<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 is
$$
r=\frac{C}{C+D}=\frac{\beta q}{\beta q+\alpha(1-q)}.
$$
It equals $q$ only when the response factors match. Solving instead gives the <2D IR response-factor correction>
$$
q=\frac{\alpha r}{\beta(1-r)+\alpha r}.
$$
For 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 $7/12\approx0.583$.
Uncertainty also depends on shared inputs. For $r=C/(C+D)$,
$$
u_r^2\approx\frac{D^2u_C^2+C^2u_D^2-2CD\operatorname{Cov}(C,D)}{(C+D)^4}.
$$
This follows from correlated-input <uncertainty propagation>.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[§5.2, covariance terms and sensitivity coefficients.]{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.
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