Chapter 5
The Vibrations of Polyatomic Molecules
Abstract The vibrational spectroscopy of the semirigid (i.e., without largeamplitude motions) polyatomic molecule is reviewed. The normal modes are defined.
The importance of molecular symmetry is pointed out. The Coriolis interaction and
the anharmonic resonances are discussed.
5.1 Introduction
This chapter is limited to the introduction of the notions permitting to understand how
the vibrations affect the structure determination. Reference books are: Bunker and
Jensen (1998), Califano (1976), Herman et al. (1999), Papousek and Aliev (1982),
Perrin et al. (2011), Quack and Merkt (2011), and Wilson et al. (1955).
We will consider a molecule of N atoms assumed to be point masses. A real
molecule is not rigid, its atoms vibrate around their equilibrium position. One may
distinguish three different kinds of motion: translation (t), rotation (r), and vibrations
(v). Note that to vibrations belong internal rotation and ring puckering. The total
kinetic energy T may be written
T = T t + T r + T v + T tr + T tv + T rv
(5.1)
The subscripts have the following meaning: tr = interaction between translation
and rotation, tv = interaction between translation and vibration, and rv = interaction
between rotation and vibration.
One may choose the molecule fixed axis system (see Chap. 4) so that the coupling
terms in (5.1) are minimal. This is done with the help of six equations called Eckart
conditions, see (5.17). The T tr and T tv terms are strictly zero because the molecule
is isolated. But the last term, T rv , is only minimized. It is called Coriolis energy and
may be important in some cases. It will be discussed in Sect. 5.5. First, we will only
consider the vibrations.
The position of the atoms may be described by 3N Cartesian coordinates. One
needs three coordinates to describe the motion of the center of mass of the molecule
© Springer Nature Switzerland AG 2020
J. Demaison and N. Vogt, Accurate Structure Determination of Free
Molecules, Lecture Notes in Chemistry 105,
https://doi.org/10.1007/978-3-030-60492-9_5
105
The Vibrations of Polyatomic Molecules
Abstract The vibrational spectroscopy of the semirigid (i.e., without largeamplitude motions) polyatomic molecule is reviewed. The normal modes are defined.
The importance of molecular symmetry is pointed out. The Coriolis interaction and
the anharmonic resonances are discussed.
5.1 Introduction
This chapter is limited to the introduction of the notions permitting to understand how
the vibrations affect the structure determination. Reference books are: Bunker and
Jensen (1998), Califano (1976), Herman et al. (1999), Papousek and Aliev (1982),
Perrin et al. (2011), Quack and Merkt (2011), and Wilson et al. (1955).
We will consider a molecule of N atoms assumed to be point masses. A real
molecule is not rigid, its atoms vibrate around their equilibrium position. One may
distinguish three different kinds of motion: translation (t), rotation (r), and vibrations
(v). Note that to vibrations belong internal rotation and ring puckering. The total
kinetic energy T may be written
T = T t + T r + T v + T tr + T tv + T rv
(5.1)
The subscripts have the following meaning: tr = interaction between translation
and rotation, tv = interaction between translation and vibration, and rv = interaction
between rotation and vibration.
One may choose the molecule fixed axis system (see Chap. 4) so that the coupling
terms in (5.1) are minimal. This is done with the help of six equations called Eckart
conditions, see (5.17). The T tr and T tv terms are strictly zero because the molecule
is isolated. But the last term, T rv , is only minimized. It is called Coriolis energy and
may be important in some cases. It will be discussed in Sect. 5.5. First, we will only
consider the vibrations.
The position of the atoms may be described by 3N Cartesian coordinates. One
needs three coordinates to describe the motion of the center of mass of the molecule
© Springer Nature Switzerland AG 2020
J. Demaison and N. Vogt, Accurate Structure Determination of Free
Molecules, Lecture Notes in Chemistry 105,
https://doi.org/10.1007/978-3-030-60492-9_5
105