The (2n+1)th-order off-resonant spectroscopy from the (n+1)th-order anharmonicities of molecular vibrational modes in the condensed phase

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Abstract
Assuming that the polarizability is a linear function of the nuclear coordinate, i.e., α(q)=α0+α1q, we obtain analytical expressions of the (2n+1)th-order signals and show that the leading order of the signals (n>1) is proportional to gn+1, where gn+1 is the coefficient of the anharmonic potential V(q)=g3q3/3!+g4q4/4!+⋅⋅⋅. In other words, detection of the (2n+1)th-order signal implies the direct observation of the (n+1)th-order anharmonicity within the approximation. Based on this fact we discuss a possibility to detect the (n+1)th-order anharmonicity directly from the (2n+1)th-order experiment. Calculations are made by using novel Feynman rules for the nonequilibrium multitime correlation functions relevant to the higher-order off-resonant spectroscopy. The rules have been developed by the authors and are presented compactly in this paper. With the help of a conventional double-sided Feynman diagram, we draw physical pictures of higher-order off-resonant optical processes. Representative calculations for CHCl3 of the fifth-, seventh-, and ninth-order optical processes are presented and discussed.