  Two-dimensional infrared (2DIR) spectroscopy is widely used to study
molecular dynamics but it is typically restricted to solid and liquid phase
samples and modest spectral resolution. Only recently, its potential to study
gas-phase dynamics is beginning to be realized. Moreover, the recently proposed
technique of cavity-enhanced 2D spectroscopy using frequency combs and
developments in multi-comb spectroscopy are expected to dramatically advance
capabilities for acquisition of rotationally-resolved 2DIR spectra. This
demonstrates the need for rigorous and quantitative treatment of
rotationally-resolved, polarization-dependent third-order response of gas-phase
samples. In this article, we provide a rigorous and quantitative description of
rotationally-resolved 2DIR spectroscopy using density-matrix, time-dependent
perturbation theory and angular momentum algebra techniques. We describe the
band and branch structure of 2D spectra, decompose the molecular response into
polarization-dependence classes, use this decomposition to derive and explain
special polarization conditions and relate the liquid-phase polarization
conditions to gas-phase ones. Furthermore, we discuss the rotational coherence
dynamics during waiting time.
