Chapter 3
Microscopic Expressions of Nonlinear
Polarization
Abstract In the SFG processes described in the preceding Chap. 2, material
properties were treated as given parameters. This chapter formulates the material properties relevant to the SFG spectroscopy from a microscopic viewpoint.
The most important property in the SFG spectra is the frequency-dependent
second-order nonlinear susceptibility tensor, χ (2) ((, ω 1 , ω 2 ). We derive χ (2) by
the quantum mechanical perturbation theory of light-matter interactions. In the
vibrational SFG spectroscopy, vibrational resonance plays a critical role in the
nonlinear response of polarization. We further discuss some basic features of χ (2) ,
including the relation of its tensor elements to the light polarizations and molecular
orientation.
Keywords χ (2) · Perturbation theory · Vibrational resonance · Molecular
orientation
3.1 Density Matrix
In the present discussion on the light-matter interactions, we employ the semiclassical description that the material is treated by quantum mechanics while the light
is treated as classical electromagnetic waves which perturb the material states. 1 The
quantum states of materials in condensed phase are expressed preferably with the
density matrix rather than the wavefunction, for the reasons described below. Before
discussing the nonlinear susceptibility, we briefly introduce the density matrix and
summarize its merits [1, 13].
1 The semiclassical theory suffices for properly describing susceptibilities of materials, including
the nonlinear ones, as the light fields provide perturbation on the materials states [12].
© Springer Nature Singapore Pte Ltd. 2018
A. Morita, Theory of Sum Frequency Generation Spectroscopy,
Lecture Notes in Chemistry 97, https://doi.org/10.1007/978-981-13-1607-4_3
47
Microscopic Expressions of Nonlinear
Polarization
Abstract In the SFG processes described in the preceding Chap. 2, material
properties were treated as given parameters. This chapter formulates the material properties relevant to the SFG spectroscopy from a microscopic viewpoint.
The most important property in the SFG spectra is the frequency-dependent
second-order nonlinear susceptibility tensor, χ (2) ((, ω 1 , ω 2 ). We derive χ (2) by
the quantum mechanical perturbation theory of light-matter interactions. In the
vibrational SFG spectroscopy, vibrational resonance plays a critical role in the
nonlinear response of polarization. We further discuss some basic features of χ (2) ,
including the relation of its tensor elements to the light polarizations and molecular
orientation.
Keywords χ (2) · Perturbation theory · Vibrational resonance · Molecular
orientation
3.1 Density Matrix
In the present discussion on the light-matter interactions, we employ the semiclassical description that the material is treated by quantum mechanics while the light
is treated as classical electromagnetic waves which perturb the material states. 1 The
quantum states of materials in condensed phase are expressed preferably with the
density matrix rather than the wavefunction, for the reasons described below. Before
discussing the nonlinear susceptibility, we briefly introduce the density matrix and
summarize its merits [1, 13].
1 The semiclassical theory suffices for properly describing susceptibilities of materials, including
the nonlinear ones, as the light fields provide perturbation on the materials states [12].
© Springer Nature Singapore Pte Ltd. 2018
A. Morita, Theory of Sum Frequency Generation Spectroscopy,
Lecture Notes in Chemistry 97, https://doi.org/10.1007/978-981-13-1607-4_3
47