# Molecular Geometry - The Influence of Thermal Excitation

The Influence of Thermal Excitation

Since the motions of the atoms in a molecule are determined by quantum mechanics, one must define “motion” in a quantum mechanical way. The overall (external) quantum mechanical motions translation and rotation hardly change the geometry of the molecule. (To some extent rotation influences the geometry via Coriolis forces and centrifugal distortion, but this is negligible for the present discussion.) A third type of motion is vibration, which is the internal motion of the atoms in a molecule. The molecular vibrations are harmonic (at least to good approximation), which means that the atoms oscillate about their equilibrium, even at the absolute zero of temperature. At absolute zero all atoms are in their vibrational ground state and show zero point quantum mechanical motion, that is, the wavefunction of a single vibrational mode is not a sharp peak, but an exponential of finite width. At higher temperatures the vibrational modes may be thermally excited (in a classical interpretation one expresses this by stating that “the molecules will vibrate faster”), but they oscillate still around the recognizable geometry of the molecule.

To get a feeling for the probability that the vibration of molecule may be thermally excited, we inspect the Boltzmann factor, where is the excitation energy of the vibrational mode, the Boltzmann constant and the absolute temperature. At 298K (25 °C), typical values for the Boltzmann factor are: 0.089 for ΔE = 500 cm−1 ; ΔE = 0.008 for 1000 cm−1 ; 7 10−4 for ΔE = 1500 cm−1. That is, if the excitation energy is 500 cm−1, then about 9 percent of the molecules are thermally excited at room temperature. The lowest excitation vibrational energy in water is the bending mode (about 1600 cm−1). Thus, at room temperature less than 0.07 percent of all the molecules of a given amount of water will vibrate faster than at absolute zero.

As stated above, rotation hardly influences the molecular geometry. But, as a quantum mechanical motion, it is thermally excited at relatively (as compared to vibration) low temperatures. From a classical point of view it can be stated that more molecules rotate faster at higher temperatures, i.e., they have larger angular velocity and angular momentum. In quantum mechanically language: more eigenstates of higher angular momentum become thermally populated with rising temperatures. Typical rotational excitation energies are on the order of a few cm−1.

The results of many spectroscopic experiments are broadened because they involve an averaging over rotational states. It is often difficult to extract geometries from spectra at high temperatures, because the number of rotational states probed in the experimental averaging increases with increasing temperature. Thus, many spectroscopic observations can only be expected to yield reliable molecular geometries at temperatures close to absolute zero, because at higher temperatures too many higher rotational states are thermally populated.