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Incoming links: Trihydrogen cation
Past exam of the physics B course of the University of Cambridge 2010 Paper 2A Section A 5 Solution 2026-10-20
The trihydrogen cation is a nonlinear three-nucleus molecule. Removing three translations and three rigid rotations leaves vibrational degrees of freedom. One normal mode is the symmetric in-plane breathing motion: every nucleus moves radially toward or away from the centre. The remaining two normal modes form a degenerate in-plane distortion pair, stretching one direction while compressing another.
For an explicit sketch, let be the equilibrium vertices relative to their centre. Displacements proportional togive the breathing mode and a basis of the degenerate pair. Linear combinations within the latter pair are equally valid. Their actual frequencies require molecular force constants, which are not supplied. An apparent out-of-plane motion of three point nuclei is a translation or infinitesimal rigid rotation, rather than an additional independent vibration.
Breathing and degenerate distortion modes of an equilateral molecule
. Original displacement vectors in a simple symmetric bond-spring illustration. The degeneracy is fixed by symmetry; no measured molecular frequencies are asserted. Details.
Past exam of the physics B course of the University of Cambridge 2010 Paper 2B Section A 5 Solution 2026-10-20
The trihydrogen cation has one breathing normal mode and two degenerate in-plane distortion normal modes. Rigid translations and rotations do not count as vibrations.
Full derivation: solution.