Neural Canonical Transformation for the Spectra of Fluxional Molecule CH5+

TL;DR

The protonated methane $CH_5^+$ is a highly fluxional system whose vibrational spectrum is strongly affected by large-amplitude hydrogen motion and delocalized nuclear wavefunctions. The authors extend the neural canonical transformation (NCT) to Cartesian coordinates and apply it to a full-dimensional PES fitted to $36{,}173$ CCSD(T)/aug-cc-pVTZ data, using a normalizing-flow to map an orthonormal HO basis to anharmonic, orthogonal eigenfunctions; energies are obtained via Boltzmann-weighted variational training with MCMC evaluation. The study computes the ground state and 31 excited vibrational states, reproduces a ground-state zero-point energy consistent with the PES, and reveals that both low- and high-energy excited states preferentially sample three stationary points on the PES, indicating a highly delocalized, fluxional character. The results show pronounced anharmonicity and numerous low-energy excitations not captured by harmonic models, and demonstrate that NCT can extend to fluxional molecules with no fixed geometry, offering a scalable ab initio tool for interpreting complex spectra in CH$_5^+$ and similar systems.

Abstract

Protonated methane, CH5+, is a highly fluxional molecule with large spatial motions of the hydrogen atoms. The molecule's anharmonic effects and the delocalized wavefunction of the hydrogen atoms significantly affect the excitation spectrum of the molecule. The neural canonical transformation (NCT) approach, which we previously developed to solve the vibrational spectra of molecules and solids, is a powerful method that effectively treats nuclear quantum effects and anharmonicities. Using NCT with wavefunctions in atomic coordinates rather than normal coordinates, we successfully calculate the ground and excited states of CH5+. We found that the wavefunctions for the ground state, as well as for low- and high-energy excited states, show preferences for the three stationary points on the potential energy surface. This work extends the applicability of the NCT approach for calculating excited states to fluxional molecules without fixed geometry.

Figures (5)

• Figure 1: The structures of the three stationary points on the $\text{CH}_5^+$ PES with energies relative to the global minimum at the CCSD(T) level from Ref. huang_quantum_2006. In the $C_s\text{(I)}$ configuration, the hydrogens A, B, and C lie in the same plane as the central carbon. D and E are positioned symmetrically with respect to this plane. The $\text{H}_2$ moiety is formed by A and B. In the $C_s\text{(II)}$ configuration, the $\text{H}_2$ moiety (also formed by A and B) is rotated by 30$^\circ$ relative to the $C_s\text{(I)}$ configuration, and another 30$^\circ$ rotation of the H$_2$ moiety would lead back to another $C_s\text{(I)}$ configuration. In the $C_{2v}$ configuration, the hydrogens labeled A and C are symmetric with respect to the plane formed by D, B, E, and the central carbon.
• Figure 2: Neural Canonical Transformation: a sketch on a one-dimensional example. The $\Psi_n(x)$ in the left panel denotes the true wavefunction and the $\Phi_n(f_\theta(x))$ on the right refers to the wavefunction basis. $f_\theta(x)$ in the middle is the bijection parameterized by neural network that connects two sets of wavefunctions.
• Figure 3: The radial distribution functions for C-H distances (in angstroms, left axis) and H-H distances (in angstrom, right axis) of the converged NCT wavefunction of ground state CH$_5^+$.
• Figure 4: The Lorentzian spectra of the energy levels (convolved with FWHM of 30 $\text{cm}^{-1}$) for (a) $\text{CH}_5^+$ and (b) $\text{CH}_4$ from the NCT calculation and their harmonic approximations. The harmonic approximation result of $\text{CH}_5^+$ is from the normal mode frequencies in Ref. jin_ab_2006. The harmonic results for $\text{CH}_4$ are from the PES in Ref. lee_accurate_1995.
• Figure 5: Ternary diagram of the relative similarity of samples from each state to the three types of stationary point configurations on PES. The values are percentages and are normalized for each sample.