^{1,a)}and Yoshitaka Tanimura

^{1}

### Abstract

We present an analytical expression for the linear and nonlinear infrared spectra of interacting molecular vibrational motions. Each of the molecular modes is explicitly represented by a classical damped oscillator on an anharmonic multidimensional potential-energy surface. The two essential interactions, the dipole-dipole (DD) and the dipole-induced-dipole (DID) interactions, are taken into account, and each dipole moment and polarizability are expanded to nonlinear order with respect to the nuclear vibrational coordinate. Our analytical treatment leads to expressions for the contributions of anharmonicity, DD and DID interactions, and the nonlinearity of dipole moments and polarizability elements to the one-, two-, and three-dimensional spectra as separated terms, which allows us to discuss the relative importance of these respective contributions. We can calculate multidimensional signals for various configurations of molecules interacting through DD and DID interactions for different material parameters over the whole range of frequencies. We demonstrate that contributions from the DD and DID interactions and anharmonicity are separately detectable through the third-order three-dimensional IR spectroscopy, whereas they cannot be distinguished from each other in either the linear or the second-order IR spectroscopies. The possibility of obtaining the intra- or intermolecular structural information from multidimensional spectra is also discussed.

We would like to express our sincere thanks to K. Okumura, S. Saito, H. Torii, and R. J. Dwayne Miller for their fruitful discussions, and M. Maroncelli and Y. Ueno for reviewing our manuscripts. This research is partially supported by Grant-in-Aids for Scientific Research from Japan Society for the Promotion of Science, Grants No. 17740278 and No. A15205005.

I. INTRODUCTION

II. EQUATIONS OF MOTION FOR INTERACTING MOLECULAR VIBRATION MODES: A FORCED OSCILLATOR PICTURE

III. MOLECULAR VIBRATIONAL COORDINATES

A. The first-order solution

B. The second-order solution

C. The third-order solution

IV. LINEAR IR SPECTROSCOPY

V. NONLINEAR IR SPECTROSCOPY

A. The second-order spectroscopy

B. The third-order spectroscopy

VI. REPRESENTATIVE CALCULATIONS

A. Linear absorption spectroscopy

B. The second-order IR spectroscopy

C. The third-order IR spectroscopy

VII. SUMMARY AND DISCUSSION

### Key Topics

- Infrared spectroscopy
- 26.0
- Electric dipole moments
- 21.0
- Infrared spectra
- 14.0
- Vibrational spectroscopy
- 10.0
- Polarizability
- 9.0

## Figures

A pair of dipoles and which interact with each other under an external electric field along the axis.

A pair of dipoles and which interact with each other under an external electric field along the axis.

A schematic description of nonlinear IR spectroscopy. A sample interacts with the first electric field at , then with the second one at , and finally with the third one at . The sample radiates an IR field at .

A schematic description of nonlinear IR spectroscopy. A sample interacts with the first electric field at , then with the second one at , and finally with the third one at . The sample radiates an IR field at .

The linear IR spectral density in Eq. (B1) . The unit of vertical axis is with the speed of light . The upper, middle, and lower figures correspond to the contributions from the linear dipole coefficient, the DID interaction, and the DD interaction in , respectively. The left three figures show real parts of the linear spectral density, , while the right ones show imaginary parts of it, .

The linear IR spectral density in Eq. (B1) . The unit of vertical axis is with the speed of light . The upper, middle, and lower figures correspond to the contributions from the linear dipole coefficient, the DID interaction, and the DD interaction in , respectively. The left three figures show real parts of the linear spectral density, , while the right ones show imaginary parts of it, .

(Color) Contour plots of the second-order IR spectra . The left four figures show , while the right ones show . The upper, upper-middle, lower-middle, and lower figures show the contributions from the nonlinear dipole coefficient, the DID interaction, the DD interaction, and the anharmonicity in , respectively.

(Color) Contour plots of the second-order IR spectra . The left four figures show , while the right ones show . The upper, upper-middle, lower-middle, and lower figures show the contributions from the nonlinear dipole coefficient, the DID interaction, the DD interaction, and the anharmonicity in , respectively.

(Color) Contour plots of the third-order IR spectra at . The left four figures show , while the right ones show . The upper, upper-middle, lower-middle, and lower figures show the contributions from the nonlinearity of the dipole moment, the DID interaction, the DD interaction, and the anharmonicity, respectively.

(Color) Contour plots of the third-order IR spectra at . The left four figures show , while the right ones show . The upper, upper-middle, lower-middle, and lower figures show the contributions from the nonlinearity of the dipole moment, the DID interaction, the DD interaction, and the anharmonicity, respectively.

(Color) The same as Fig. 5 but at , and the horizontal axis is .

(Color) Magnified views of the real part of Fig. 5 (the upper two figures) and Fig. 6 (the lower two figures). The left figures show sums of the contributions from the nonlinear dipole coefficient, the DD interaction, and the anharmonicity, while the right ones show those from the DID interaction.

(Color) Magnified views of the real part of Fig. 5 (the upper two figures) and Fig. 6 (the lower two figures). The left figures show sums of the contributions from the nonlinear dipole coefficient, the DD interaction, and the anharmonicity, while the right ones show those from the DID interaction.

(Color) The same as Fig. 5 but at , and the horizontal and the vertical axes are and , respectively.

(Color) The same as Fig. 5 but at , and the horizontal and the vertical axes are and , respectively.

(Color) The same as Fig. 8 but at .

(Color) The third-order response functions from the nonlinear dipole coefficient (upper), the DID interaction (middle-upper), the DD interaction (middle-lower), and the anharmonicity (lower). The unit of both axes and is . The unit of vertical axis is . We set in the left figures and in the right ones.

(Color) The third-order response functions from the nonlinear dipole coefficient (upper), the DID interaction (middle-upper), the DD interaction (middle-lower), and the anharmonicity (lower). The unit of both axes and is . The unit of vertical axis is . We set in the left figures and in the right ones.

(Color) Magnified views of the real part of Fig. 9 . The upper-left figure represents the effects from the nonlinear dipole coefficient, the DID interaction, and the anharmonicity, while the upper-right one represents the effect of the DD interaction. The lower-left figure shows a sum of the contributions from the nonlinear dipole coefficient, the DID interaction, and the DD interaction, while the lower-right one shows the contribution from the anharmonicity.

(Color) Magnified views of the real part of Fig. 9 . The upper-left figure represents the effects from the nonlinear dipole coefficient, the DID interaction, and the anharmonicity, while the upper-right one represents the effect of the DD interaction. The lower-left figure shows a sum of the contributions from the nonlinear dipole coefficient, the DID interaction, and the DD interaction, while the lower-right one shows the contribution from the anharmonicity.

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