Molecular inversion

The umbrella motion of NH₃ and related molecules, reduced to a 1D double well: the central atom crosses the plane of the three hydrogens through the planar form (the barrier). Each level splits into a tunnelling doublet — the transition of the ammonia maser (0.79 cm⁻¹). Effective coordinate solved by finite differences in the Python backend (via gw2py).

It is literally the abstract double well rendered in a real molecule: tunnelling through the planar configuration opens the parity doublet. Same physics as the doublet of pear-shaped nuclei.

A pyramidal molecule can flip like an umbrella in the wind, passing through the planar form.

A single coordinate

The inversion motion of NH₃ is dominated by one large-amplitude coordinate: the central atom crossing the plane of the three hydrogens. Along it the potential is a symmetric double well: two minima (the two mirror pyramids) separated by a barrier (the planar configuration).

\[ -B\,\frac{d^2\psi}{dq^2} + V(q)\,\psi = E\,\psi,\qquad V(q)=V_b\Big[(q/q_0)^2-1\Big]^2. \]

The tunnelling doublet

Each level splits into a pair (symmetric / antisymmetric) by tunnelling through the barrier. The ground-state doublet of NH₃ is at 0.79 cm⁻¹ (23.87 GHz): it is the transition of the ammonia maser, the first maser (1954).

With a 2020 cm⁻¹ barrier and μ≈3 u the FD solver gives a ground-state doublet ≈0.68 cm⁻¹; increasing the mass (ND₃) the doublet collapses, raising the barrier (PH₃) the molecule essentially no longer inverts.

The thread

It is the double well rendered in a real molecule — and the same physics as the parity doublet of pear-shaped nuclei.

Molecule

Select

The barrier is the planarity of the molecule; the doublet opens by tunnelling through it. Parameters (barrier, reduced mass) adopted from spectroscopy.

State to represent

State type Level \(n_1\)

Level \(n_2\)

The \((0,1)\) doublet is the state that «oscillates» between the two pyramids (the inversion motion).

Why it works

Inversion is a soft, large-amplitude mode, well separated in frequency from the rigid stretches/bends: the reduction to 1D is therefore legitimate. Barrier and reduced mass are the only two ingredients; the FD solver is the same as the rest of the suite.

Isotope effect and barrier

Tunnelling depends exponentially on \(\sqrt{\mu V_b}\): ND₃ (larger mass) has a much smaller doublet than NH₃; PH₃ and AsH₃, with much higher barriers, do not invert on observable timescales. The same formula explains why tetrahedral carbon does not racemize spontaneously.

What is missing (honesty)

We neglect the coupling with the other modes and the coordinate dependence of the reduced mass (\(G(q)\)); the parameters are adopted. An accurate computation uses an ab initio potential energy surface and a multidimensional vibrational Hamiltonian (backend).

References

C. H. Townes, A. L. Schawlow, Microwave Spectroscopy, McGraw-Hill, 1955.
J. D. Swalen, J. A. Ibers, «Potential Function for the Inversion of Ammonia», J. Chem. Phys. 36, 1914 (1962). doi.

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

Keywords: inversion, ammonia, NH3, maser, double well, tunnelling, effective coordinate, isotope effect, spectroscopy