= Two-dimensional infrared spectroscopy
<Two-dimensional infrared spectroscopy> measures a sample's vibrational response as a function of two frequency coordinates. In a coherent experiment, timed <infrared> pulses establish an excitation history and a later optical response; the spectrum correlates excitation and detection frequencies. The two dimensions are spectral coordinates, not two spatial directions through the sample.
In a time-domain implementation, the experiment scans a coherence delay $t_1$ and resolves a detection interval $t_3$ while holding a <2D IR waiting time> $T$ fixed. Schematically, a complex measured response $S(t_1,T,t_3)$ is transformed to
$$
\widetilde S(\omega_1,T,\omega_3)
=\int\!\int S(t_1,T,t_3)e^{i\omega_1t_1}e^{i\omega_3t_3}\,dt_1\,dt_3.
$$
The signs and combination of measured response pathways depend on the experimental convention. <Fourier analysis> converts the scanned delays into frequency information; it does not by itself establish which molecular mechanism produced a feature.
An original spectral example has modes near $1600$ and $1700\,\mathrm{cm^{-1}}$. Features at coordinates $(1600,1600)$ and $(1700,1700)$ lie on the diagonal. An off-diagonal feature at $(1600,1700)$ is a <2D IR cross peak>: excitation associated with one coordinate is correlated with response at another. Its physical interpretation depends on the pulse sequence and waiting time. A cross peak is not automatically evidence for chemical conversion between two species.
A primary2000 peptide experiment used phase-controlled femtosecond pulses, phase matching and <heterodyne detection> to measure the phase and amplitude of <vibrational photon echoes> as pulse delays varied. Cross peaks in acyl-proline-NH$_2$ spectra supplied evidence for coupling between amide-I modes, with solvent-sensitive structural differences.<Two-dimensional infrared spectroscopy of peptides by phase-controlled femtosecond vibrational photon echoes by M. C. Asplund and others>[Abstract, phase/amplitude detection, coupled amide-I modes and solvent dependence.]{ref}
Compared with a one-dimensional <infrared spectroscopy> trace, the extra coordinate can separate overlapping responses and expose relationships between vibrations. Varying the <2D IR waiting time> adds a controlled time dependence to test candidate mechanisms. Interpretation still requires a response model, sufficient time/frequency resolution and controls for solvent and instrument contributions; a richly structured spectrum does not uniquely determine a molecular structure by inspection.
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