Internal torsion

The internal rotation of one group relative to another about a bond: a periodic potential \(V(\varphi)=\tfrac{V_n}{2}(1-\cos n\varphi)\) → the Mathieu equation. From the hindered rotor (ethane) to the nearly free one (nitromethane). Effective coordinate solved by finite differences with cyclic boundary conditions in the Python backend (via gw2py).

It is the third potential type of the suite: after the single well (harmonic) and the double well (inversion), the periodic well → Mathieu functions. The high-barrier limit is again a local harmonic oscillator; the zero-barrier limit, the free rotor.

Two bonded groups can rotate relative to each other: an internal rotor, hindered by a periodic barrier.

Periodic potential

Along the dihedral angle \(\varphi\) the potential is periodic with the group's symmetry (3-fold for CH₃):

\[ -F\,\frac{d^2\psi}{d\varphi^2} + \frac{V_n}{2}\big(1-\cos n\varphi\big)\,\psi = E\,\psi, \]

with \(F=\hbar^2/2I_{\rm red}\) the internal-rotation constant. It is the Mathieu equation: its solutions are the Mathieu functions.

From hindered to free

Ethane (V₃≈1024 cm⁻¹) is well hindered: torsional fundamental at ~287 cm⁻¹, organized in nearly degenerate triplets (the 3 equivalent minima, A+E symmetry). Nitromethane (V₆≈2 cm⁻¹) is almost a free rotor: levels tend to \(F\,m^2\). Tunnelling between the minima gives the A–E splitting.

The thread

After the single well (harmonic) and the double well (inversion), the periodic well completes the triad: the hindered limit is again a local harmonic oscillator.

Molecule

Select

The potential is periodic: the coordinate is the dihedral angle. High barrier → torsional oscillator; low barrier → nearly free rotor.

State to represent

State type Level \(n_1\)

Level \(n_2\)

The low levels come in A+E triplets (the three equivalent minima).

The special function: Mathieu

With a \(\cos n\varphi\) potential the equation is exactly Mathieu's; eigenvalues and eigenfunctions are the Mathieu functions \(\mathrm{ce}_r,\mathrm{se}_r\). The high-barrier limit reduces them to local harmonic oscillators (Hermite), the zero-barrier one to plane waves \(e^{im\varphi}\) (free rotor).

Symmetry and tunnelling

The 3-fold periodicity imposes the A+E structure of the levels: each torsional level is a multiplet whose components split by tunnelling between the three minima (A–E splitting), large for low barriers and for light hydrogen.

What is missing (honesty)

We consider a single \(\cos n\varphi\) term and constant \(F\); molecules like H₂O₂ have different cis and trans barriers (more terms), and the global torsion-rotation coupling matters in the fine spectrum. The accurate treatment is a rotor-torsion Hamiltonian (backend).

References

J. D. Lewis, T. B. Malloy, T. H. Chao, J. Laane, «Periodic potential functions…», J. Mol. Struct. 12, 427 (1972). doi.
W. Gordy, R. L. Cook, Microwave Molecular Spectra, Wiley, 1984 — internal rotors.

WebNIR · CNR-IFAC | demo interface — numerical work is provided by the Python backend (gw2py).

Keywords: torsion, internal rotor, periodic potential, Mathieu equation, ethane, nitromethane, A-E splitting, effective coordinate