Nuclear motions Rotation and vibration

From the spectrum of a molecule we can obtain experimental information about the geometry of the molecule (bond lengths), and the energy states from which bond strengths are ultimately obtained. The molecular spectrum depends on the characteristics of the nuclear motions as well as on the electronic motions. In Section 23.1, by invoking the Born-Oppenheimer approximation, we discussed the electronic motion that produces the bonding between the atoms as a problem separate from that of the nuclear motions. We begin the discussion of molecular spectroscopy with a brief recapitulation of the description of the nuclear motions. 

The motions of the nuclei are of three kinds the translational motion of the molecule as a whole, which we discard as uninteresting the rotation of the molecule and the vibrations of the nuclei within the molecule. To a good approximation these motions are independent and can be discussed separately. 

A molecule containing N atoms has 3N nuclear coordinates and 3N nuclear momenta therefore there are 3N independent modes of motion or 3N degrees of freedom. Discarding three coordinates and three momenta that pertain to the translation of the whole molecule, there remain 3N — 3 degrees of freedom. If the molecule is linear and the axis of the molecule is the z-axis, then two independent modes of rotation, about the x- and y-axis, are possible. For linear molecules the number of coordinates and momenta remaining to describe the vibrations is 3N — 3 — 2 = 3N — 5. Nonlinear molecules have three independent modes of rotation about three mutually perpendicular axes, so the number of coordinates and momenta remaining to describe the vibrations is 3N — 3 — 3 = 3N — 6. The number of modes of each type of motion is shown in Table 25.1. 

Total number of degrees of freedom Number of translational degrees of freedom Number of rotational degrees of freedom Number of vibrational degrees of freedom 

In addition to the selection rules restricting the changes in the quantum numbers, the presence or absence of a dipole moment in the molecule imposes a restriction on the appearance of lines and bands in the spectrum. If the transition between one vibratio.nal or rotational state to another is to produce emission or absorption of radiation the vibration or rotation must be accompanied by an oscillation in the magnitude of the dipole moment of the molecule. 

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