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K. B. Be´ c et al.
5.1 Introduction
The aim of this chapter is to present the essential information required to obtain
a fundamental understanding of quantum vibrational absorption spectroscopy, in
particular near-infrared (NIR) spectroscopy. The discussion highlights the critical
aspects to provide an in-depth and accessible overview aimed at a spectroscopic
audience. The necessary basics include the commonly used coordinate frame for
the description of molecular vibrations, an overview of the role of the vibrational
potential and methods for its determination, as well as the critical factor in all
applications of quantum chemistry being the computational complexity of a given
approach. Considerable attention is focused toward the harmonic approximation,
the fundamental framework underlying most applications of theoretical vibrational
spectroscopy. The harmonic approximation is in general not sufficiently accurate for
the needs of NIR spectroscopy. However, it is an essential foundation for advanced
anharmonic treatments. The majority of these methods are either built on the basis
of a harmonic Hamiltonian (VPT2), adopt a harmonic Hessian as the reference state
(VSCF) or use the harmonic analysis (i.e., harmonic normal modes) in the process
of probing the true vibrational potential (grid-based methods). The details of various
anharmonic approaches are discussed, and their specific merits and shortcomings
examined from the point of view of applications in NIR spectroscopy. This outline
is based on several examples selected from recent literature.
5.2 Normal Modes of Vibration
Commonly, literature introducing the principles of vibrational spectroscopy mainly
employs the example of a simple diatomic molecule. However, this kind of twobody system is limited to a single mode of vibration resulting from a one-dimensional
potential and is not suitable for a complete presentation of the main concepts in vibrational analysis [1–4]. The total number of degrees of freedom (DOF) in a chemical
system is 3N, where N is the number of atoms. Translational and rotational motion
can only be defined in an external coordinate system; thus, the translational and rotational DOF are invariant in the molecule’s frame of reference. This sets them apart
from the internal DOF (vibrational DOF; vibrational modes). The number of vibrational DOF equals to 3N − N inv · N inv is generally partitioned into three translational
and three rotational DOF (along the x, y, z directions); however, no change in the
potential energy is associated to the rotation over the main rotational axis of linear
molecules (including diatomic ones). Additionally, N inv of periodic systems only
considers uniform translational DOF (x, y, z) of the entire lattice. This effectively
leaves 3N − 6 modes for nonlinear molecules, 3N − 5 modes for linear molecules,
and 3N − 3 modes for periodic systems. Note that these different DOFs need to be
separated, e.g., no translation of the molecule’s center of mass may occur along the
vibrational mode. The concept of normal modes in computational chemistry has its
K. B. Be´ c et al.
5.1 Introduction
The aim of this chapter is to present the essential information required to obtain
a fundamental understanding of quantum vibrational absorption spectroscopy, in
particular near-infrared (NIR) spectroscopy. The discussion highlights the critical
aspects to provide an in-depth and accessible overview aimed at a spectroscopic
audience. The necessary basics include the commonly used coordinate frame for
the description of molecular vibrations, an overview of the role of the vibrational
potential and methods for its determination, as well as the critical factor in all
applications of quantum chemistry being the computational complexity of a given
approach. Considerable attention is focused toward the harmonic approximation,
the fundamental framework underlying most applications of theoretical vibrational
spectroscopy. The harmonic approximation is in general not sufficiently accurate for
the needs of NIR spectroscopy. However, it is an essential foundation for advanced
anharmonic treatments. The majority of these methods are either built on the basis
of a harmonic Hamiltonian (VPT2), adopt a harmonic Hessian as the reference state
(VSCF) or use the harmonic analysis (i.e., harmonic normal modes) in the process
of probing the true vibrational potential (grid-based methods). The details of various
anharmonic approaches are discussed, and their specific merits and shortcomings
examined from the point of view of applications in NIR spectroscopy. This outline
is based on several examples selected from recent literature.
5.2 Normal Modes of Vibration
Commonly, literature introducing the principles of vibrational spectroscopy mainly
employs the example of a simple diatomic molecule. However, this kind of twobody system is limited to a single mode of vibration resulting from a one-dimensional
potential and is not suitable for a complete presentation of the main concepts in vibrational analysis [1–4]. The total number of degrees of freedom (DOF) in a chemical
system is 3N, where N is the number of atoms. Translational and rotational motion
can only be defined in an external coordinate system; thus, the translational and rotational DOF are invariant in the molecule’s frame of reference. This sets them apart
from the internal DOF (vibrational DOF; vibrational modes). The number of vibrational DOF equals to 3N − N inv · N inv is generally partitioned into three translational
and three rotational DOF (along the x, y, z directions); however, no change in the
potential energy is associated to the rotation over the main rotational axis of linear
molecules (including diatomic ones). Additionally, N inv of periodic systems only
considers uniform translational DOF (x, y, z) of the entire lattice. This effectively
leaves 3N − 6 modes for nonlinear molecules, 3N − 5 modes for linear molecules,
and 3N − 3 modes for periodic systems. Note that these different DOFs need to be
separated, e.g., no translation of the molecule’s center of mass may occur along the
vibrational mode. The concept of normal modes in computational chemistry has its