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A 2D IR cross peak is an off-diagonal spectral feature correlating excitation and detection frequencies that differ in a two-dimensional infrared spectroscopy experiment. Its location is a relation between frequency coordinates, not a second physical location in the sample. Such features can reveal interactions between vibrations, but their interpretation depends on the measured response and timing.

For a concrete primary example, phase-controlled peptide spectra showed cross peaks interpreted as coupling between amide-I modes. The experiment measured the phase as well as amplitude of vibrational photon echoes, supplying information beyond an unsigned intensity map.^[ref]

An original positive-envelope model illustrates 2D IR diagonal-peak overlap. Let $g(x)=e^{-x^2/(2{\sigma }^2)}$ and put two purely diagonal features at frequencies $a$ and $b$:At the off-diagonal coordinate $(a,b)$, this model givesFor $a=1600$, $b=1650$ and $\sigma =20$, all in $\mathrm{c}{\mathrm{m}}^{-1}$, the value is approximately0.0879. The diagonal value $S(a,a)$ is approximately1.0019. Thus a nonzero off-diagonal pixel about8.8% of the diagonal height can arise solely from broad diagonal tails, with no separate cross feature in the stipulated model.

A real response can have positive and negative components, so this envelope is not a universal line-shape formula. It demonstrates why a peak assignment needs a fitted background/overlap model and appropriate spectral resolution, rather than the observation of any off-diagonal intensity.

Changing 2D IR waiting time, polarization or sample conditions can test proposed mechanisms. Growth or decay of a 2D IR cross peak alone does not uniquely establish chemical conversion: coupling, relaxation and changes in response factors can contribute. Interpret the feature jointly with the rest of the spectrum and the experimental pulse sequence.

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 $(c,c)$ isHere $\rho$ 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 $(a,b)$,For original values $a=1600$, $b=1650$ and $\sigma =20$, all in $\mathrm{c}{\mathrm{m}}^{-1}$, 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.

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.^[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.

2D IR line-shape correlation describes how excitation- and detection-frequency coordinates vary together within a spectral feature in two-dimensional infrared spectroscopy. An elongated feature can retain a relation between the initially selected frequency and the frequency read out later. Its shape can change with 2D IR waiting time, but a fitted geometric parameter needs a response model before it is assigned a molecular dynamical meaning.

An original Gaussian-envelope example makes the geometry precise. Let centered frequency offsets be $x,y$, with widths ${\sigma }_x,{\sigma }_y$ and correlation coefficient $\rho$:The associated covariance matrix isAt fixed $x$, maximizing $G$ over $y$ gives the lineThus a 2D IR center-line slope equals $\rho$ only for this orientation with equal widths in compatible frequency coordinates. For stipulated widths ${\sigma }_x=12$ and ${\sigma }_y=18$ in $\mathrm{c}{\mathrm{m}}^{-1}$ and $\rho =0.75$, the slope is $1.125$. A slope greater than one is not an impossible correlation coefficient; the two quantities have different width dependence.

For equal widths $\sigma$, rotated coordinates $u=(x+y)/\sqrt{2}$ and $v=(y-x)/\sqrt{2}$ have variances ${\sigma }^2(1+\rho )$ and ${\sigma }^2(1-\rho )$. At $\rho =0.75$, the diagonal-to-antidiagonal width ratio is $\sqrt{7}$. This geometric identity also helps explain why 2D IR diagonal-peak overlap depends on orientation as well as marginal width.

A 2007 primary analysis by Kyungwon Kwak, Sungnam Park, Ilya J. Finkelstein and M. D. Fayer extracted a waiting-time-dependent observable from maxima of spectral slices and tested it against calculated spectra with known frequency dynamics. Its center-line convention explicitly depends on the direction of the slices.^[ref]

A vibrational frequency-frequency correlation function instead concerns frequency fluctuations at two times. Under an appropriate stationary model, its normalized form is $C(T)/C(0)$ with $C(T)=⟨\delta \omega (0)\delta \omega (T)⟩$. Relating it to spectral geometry requires the pulse sequence, broadening model and slope convention. Multiple overlapping peaks, signed responses and finite acquisition windows must be addressed before interpreting a change of shape as a unique molecular time scale.