Semi-rigid molecule

A semi-rigid molecule is a molecule which has a potential energy surface with a well-defined minimum corresponding to a stable structure of the molecule. The only (quantum mechanical) motions that a semi-rigid molecule makes are (small) internal vibrations around its equilibrium geometry and overall translations and rotations.

Potential energy surface

A molecule consists of atoms held together by chemical bonding forces. The potential, derived from these forces, is a function of the Cartesian nuclear coordinates R₁, ..., R_N. These coordinates are expressed with respect to a frame attached to the molecule. The potential function is known as force field or potential energy surface written as V(R₁, ..., R_N).

Often a more accurate representation of the potential V is obtained by the use of internal curvilinear coordinates, so-called valence coordinates. We mention bond stretch, valence angle bending, out-of-plane-rotation angles, and dihedral (torsion) angles. Although the curvilinear internal coordinates can give a good description of the molecular potential, it is difficult to express the kinetic energy of nuclear vibrations in these coordinates.

Identical nuclei

When a molecule contains identical nuclei—which is commonly the case—there are a number of minima related by the permutations of the identical nuclei. The minima, distinguished by different numberings of identical nuclei, can be partitioned in equivalent classes. Two minima are equivalent if they can be transformed into one other by rotating the molecule, that is, without surmounting any energy barrier (bond breaking or bond twisting). The molecules with minima in different equivalent classes are called versions. To transform one version into another version an energy barrier must be overcome.

Example

Take for instance the pyramidal ammonia (NH₃) molecule. There are 3!=6 permutations of the hydrogen atoms. If we count the hydrogens looking down from the nitrogen onto the plane of the hydrogens, then we see that

{H₁-H₂-H₃, H₃-H₁-H₂, H₂-H₃-H₁}

forms one equivalence class, (class I), because the members can be transformed into each other by simply rotating around the 3-fold axis without overcoming an energy barrier. The other equivalence class (class II) consists of

{H₂-H₁-H₃, H₃-H₂-H₁, H₁-H₃-H₂}.

To transform a member (version) of class I to class II, an energy barrier has to be overcome. (The lowest path on the potential energy surface is actually via the flipping of the ammonia "umbrella". The umbrella up and the umbrella down are separated by an energy barrier of height of ca. 1000 cm⁻¹).

In a semi-rigid molecule all the barriers between different versions are so high that the tunneling though the barriers may be neglected. Under these conditions identical nuclei may be seen as distinguishable particles to which the Pauli principle does not apply. This is a very common point of view in chemistry.

Floppy molecule

In a non-rigid (floppy) molecule (some of) the potential barriers between the different versions are so low that tunneling through the barrier is appreciable, or, in other words, that splittings due to tunneling are spectroscopically observable. Under these conditions one must take care that the identical nuclei obey the Pauli principle (are described by either a symmetric or antisymmetric wavefunction).

References

• H. W. Kroto, Molecular Rotation Spectra, Wiley, New York, 1975 (Reprinted by Dover 1992).
• P. R. Bunker and P. Jensen, Molecular Symmetry and Spectroscopy, 2nd edition, NRC Research Press, Ottawa, 1998.
• D. Papoušek and M. R. Aliev, Molecular Vibrational-Rotational Spectra Elsevier, Amsterdam, 1982.
• E. B. Wilson, J. C. Decius, and P. C. Cross, Molecular Vibrations, McGraw-Hill, New York, 1955 (Reprinted by Dover 1980).