Ring-puckering

The out-of-plane puckering mode of small rings, described by Laane's quartic potential \(V(q)=a\,q^4+b\,q^2\): from the double well (sharply bent ring) to the nearly harmonic single well (nearly planar ring). Effective coordinate solved by finite differences in the Python backend (via gw2py).

Four- and five-membered rings do not stay still: they oscillate out of the plane along a single soft mode. The dominant quartic term links this page to the quartic anharmonic oscillator; for a negative quadratic term it is the double well of inversion.

Four- and five-membered rings oscillate out of the plane along a single soft mode.

Laane's quartic potential

Puckering is described by an out-of-plane coordinate \(q\) with a four-plus-two potential:

\[ V(q)=a\,q^4 + b\,q^2. \]

If \(b<0\) two minima appear (ring bent both ways) separated by a barrier at planarity; if \(b\ge0\) a single well remains (planar or nearly planar ring).

From double to single well

Cyclopentene has a ~232 cm⁻¹ barrier → doublet below the barrier; cyclobutane ~500 cm⁻¹ → nearly degenerate doublet (sharply bent ring); oxetane is nearly planar (~15 cm⁻¹ barrier) → a nearly harmonic level ladder. The dominant \(a\,q^4\) term links this page to the quartic anharmonic oscillator.

Why it matters

Ring-puckering levels fall in the far infrared (10–200 cm⁻¹) and are a historical benchmark of large-amplitude vibrational spectroscopy (Laane, 1970s).

Molecule

Select

Strained rings → double well; nearly planar rings → nearly harmonic single well. Parameters (barrier, mass) adopted from spectroscopy.

State to represent

State type Level \(n_1\)

Level \(n_2\)

One coordinate, one polynomial

Ring-puckering is the lowest-frequency mode of the ring, well separated from the C–C stretches: the reduction to a \(a q^4+b q^2\) potential is Laane's historical choice and reproduces the observed far-IR sequences with just two parameters.

The effective masses are calibrated on data. Puckering moves the heavy ring atoms, so in ångström coordinates the effective mass is ~10 u: cyclopentene 10.75, cyclobutane 9.15, oxetane 11.08. Each one is derived from one observed transition; for cyclopentene the second follows by itself — calibrating on 1→2 = 127 cm⁻¹ the doublet comes out at 0.861 against 0.91 observed (5% discrepancy), which is the check that the one-degree-of-freedom coordinate holds.

Link with the quartic

For \(b\to0\) it is the purely quartic well: levels scale as \(a^{1/3}(v+\tfrac12)^{4/3}\), the same physics as the anharmonic-oscillator page. For \(b<0\) it is the double well of inversion.

What is missing (honesty)

The puckering reduced mass depends on the coordinate (\(G(q)\)) and there is coupling with torsions/twisting in larger rings; here \(G\) is constant and the parameters adopted. The accurate treatment is Laane's Hamiltonian with \(G(q)\) computed from the geometry (backend).

References

J. Laane, «Origin of the Ring-Puckering Potential Energy Function…», J. Phys. Chem. 95, 9246 (1991). doi.
T. B. Malloy, «Far-infrared spectra of ring compounds», J. Mol. Spectrosc. 44, 504 (1972). doi.

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

Keywords: ring-puckering, puckering, ring, quartic potential, Laane, cyclopentene, oxetane, effective coordinate, far infrared