Semiclassical Mode-Coupling Factorizations of Coherent Nonlinear Optical Response

The identification of relevant collective coordinates is crucial for the interpretation of coherent nonlinear spectroscopies of complex molecules and liquids. Using an $\hbar$ expansion of Liouville space generating functions, we show how to factorize multitime nonlinear response functions into products of lower-order correlation functions of collective coordinates, and derive closed expressions for linear, second and third order response functions. In addition to providing systematic quantum corrections, $\hbar$ offers a convenient bookkeeping device even for the purely classical response, since including quantum fluctuations allows to circumvent the expensive computation of stability matrices which is a major bottleneck in Molecular Dynamics (MD) simulations. The existing classical simulation strategies, including Mode-Coupling in $\mathbf{k}$ space and in real-space, Langevin equations, and Instantaneous Normal Modes are compared from a unified viewpoint.