# 2.8: Appendix- Third-order diagrams for a vibration

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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}.