The third-order nonlinear response functions for infrared vibrational spectroscopy are often applied to a weakly anharmonic vibration. For high frequency vibrations in which only the $$\nu = 0$$ state is initially populated, when the incident fields are resonant with the fundamental vibrational transition, we generally consider diagrams involving the system eigenstates $$\nu = 0, 1$$ and 2, and which include v=0-1 and v=1-2 resonances. Then, there are three distinct signal contributions:
Note that for the $$S_I$$ and $$S_{II}$$ signals there are two types of contributions: two diagrams in which all interactions are with the v=0-1 transition (fundamental) and one diagram in which there are two interactions with v=0-1 and two with v=1-2 (the overtone). These two types of contributions have opposite signs, which can be seen by counting the number of bra side interactions, and have emission frequencies of $$\omega_{10}$$ or $$\omega_{21}$$. Therefore, for harmonic oscillators, which have $$\omega_{10} = \omega_{21}$$ and $$\sqrt{2}\mu_{10}=\mu_{21}$$, we can see that the signal contributions should destructively interfere and vanish. This is a manifestation of the finding that harmonic systems display no nonlinear response. Some deviation from harmonic behavior is required to observe a signal, such as vibrational anharmonicity $$\omega_{10} \ne \omega_{21}$$, electrical anharmonicity (\sqrt{2}\mu_{10}\ne\mu_{21}\), or level-dependent damping $$\Gamma_{10}\ne\Gamma_{21}$$ or \(\Gamma_{00}\ne\Gamma_{11}.