5 Introduction to Quantum Vibrational Spectroscopy
85
origin in the formalism of the harmonic approximation and will be outlined in detail
in Sect. 5.4 of this chapter.
In polyatomic systems, symmetric and antisymmetric modes occur due to
symmetry factors [1, 2]. In addition, deformation modes appear as well; they involve
change of valence and dihedral angles between the atoms in the system. As a rule,
force constants associated with stretching modes are typically higher, and thus, the
wave numbers of these vibrations are higher compared to associated deformation
modes. For instance, CO 2 is a linear molecule and thus has 3N − 5 = 4 modes of
vibration (Fig. 5.1). Among them are the two stretching modes, being symmetric and
antisymmetric. The CO 2 symmetric stretch (v 1 ) is IR inactive because there is no
change in the dipole moment of the molecule along the associated vibrational coordinate. In contrast, the antisymmetric stretching vibration (v 3 ) generates a significant
net change of the dipole moment giving rise to a strong IR band observed at ca.
2345 cm
−1 in gas phase. The two deformation modes of CO 2 involve the bending
of the OCO angle in the molecule. These two modes differ only from the point
of view of an external coordinate system; the vibrations occur along perpendicular
planes. However, from the molecule’s point of view, they are indistinguishable, and
their energies (and thus wavenumbers) are degenerate giving rise only to a single IR
absorption band v 2 located at 667 cm
−1 in gas phase. Therefore, despite possessing
four vibrational degrees of freedom, only two fundamental bands of CO 2 are observed
in the respective IR spectrum. However, IR spectra of gaseous molecules are further
complicated because of rotational–vibrational coupling. Water serves as an archetypical nonlinear molecule; it has 3N − 6 = 3 vibrational degrees of freedom, with only
a single deformation mode v 2 (Fig. 5.1).
Since the center of mass of the vibrating molecule may not change its position
in space, the atomic displacements associated to these normal modes often involve
displacements of all atoms in the molecule. These are not necessarily large amplitude
motions, however. Water serves a good example, as large amplitude motions of the
light-weighted hydrogen atoms are accompanied by a low-amplitude motion of the
heavy oxygen atom; this is reflected in an exaggerated way in Fig. 5.1. These complex
Fig. 5.1 Equilibrium geometries and normal modes of a carbon dioxide (top) and a water (bottom)
molecule, respectively
85
origin in the formalism of the harmonic approximation and will be outlined in detail
in Sect. 5.4 of this chapter.
In polyatomic systems, symmetric and antisymmetric modes occur due to
symmetry factors [1, 2]. In addition, deformation modes appear as well; they involve
change of valence and dihedral angles between the atoms in the system. As a rule,
force constants associated with stretching modes are typically higher, and thus, the
wave numbers of these vibrations are higher compared to associated deformation
modes. For instance, CO 2 is a linear molecule and thus has 3N − 5 = 4 modes of
vibration (Fig. 5.1). Among them are the two stretching modes, being symmetric and
antisymmetric. The CO 2 symmetric stretch (v 1 ) is IR inactive because there is no
change in the dipole moment of the molecule along the associated vibrational coordinate. In contrast, the antisymmetric stretching vibration (v 3 ) generates a significant
net change of the dipole moment giving rise to a strong IR band observed at ca.
2345 cm
−1 in gas phase. The two deformation modes of CO 2 involve the bending
of the OCO angle in the molecule. These two modes differ only from the point
of view of an external coordinate system; the vibrations occur along perpendicular
planes. However, from the molecule’s point of view, they are indistinguishable, and
their energies (and thus wavenumbers) are degenerate giving rise only to a single IR
absorption band v 2 located at 667 cm
−1 in gas phase. Therefore, despite possessing
four vibrational degrees of freedom, only two fundamental bands of CO 2 are observed
in the respective IR spectrum. However, IR spectra of gaseous molecules are further
complicated because of rotational–vibrational coupling. Water serves as an archetypical nonlinear molecule; it has 3N − 6 = 3 vibrational degrees of freedom, with only
a single deformation mode v 2 (Fig. 5.1).
Since the center of mass of the vibrating molecule may not change its position
in space, the atomic displacements associated to these normal modes often involve
displacements of all atoms in the molecule. These are not necessarily large amplitude
motions, however. Water serves a good example, as large amplitude motions of the
light-weighted hydrogen atoms are accompanied by a low-amplitude motion of the
heavy oxygen atom; this is reflected in an exaggerated way in Fig. 5.1. These complex
Fig. 5.1 Equilibrium geometries and normal modes of a carbon dioxide (top) and a water (bottom)
molecule, respectively