Two-dimensional (2D) infrared (IR) spectra are commonly interpreted using a quantum diagrammatic expansion that describes the changes to the density matrix of quantum systems in response to light–matter interactions. Although classical response functions (based on Newtonian dynamics) have shown promise in computational 2D IR modeling studies, a simple diagrammatic description has so far been lacking. Recently, we introduced a diagrammatic representation for the 2D IR response functions of a single, weakly anharmonic oscillator and showed that the classical and quantum 2D IR response functions for this system are identical. Here, we extend this result to systems with an arbitrary number of bilinearly coupled, weakly anharmonic oscillators. As in the single-oscillator case, quantum and classical response functions are found to be identical in the weakly anharmonic limit or, in experimental terms, when the anharmonicity is small relative to the optical linewidth. The final form of the weakly anharmonic response function is surprisingly simple and offers potential computational advantages for application to large, multi-oscillator systems.
The frequencies needed for the g_spec calculation are the local transition energies of each oscillator for the 0 → 1 and 1 → 2 transitions. Under the full Hamiltonian ĤS(0) + ĤS(1), the ground state for each individual oscillator has energy ; the first excited state energy is ; and the energy of the second excited state is . The 0 → 1 transition frequency is thus ωn + 2ℏΔ, while the 1 → 2 frequency is ωn + 4ℏΔ.