Solving the vibrational Schrödinger equation with artificial neural networks

Artificial neural networks are universal function approximators and have shown great ability in computing the ground-state energy of the electronic Schrödinger equation, yet have not established themselves as a practical and accurate approach for solving the vibrational Schrödinger equation for realistic polyatomic molecules. Here, we propose an efficient neural-network approach for solving the vibrational Schrödinger equation and provide a detailed illustration using the methane molecule. To demonstrate the power of the proposed method, we then apply it to propane, an 11-atom molecule with 27 vibrational degrees of freedom. Using a neural network with fewer than 15,000 parameters, we obtain the ground-state energy within 1 cm^-1 of the reference value obtained from a diffusion Monte Carlo calculation, as well as vibrational energies for three excited states involving C-C-C stretching/bending modes that agree with the corresponding experimental values within the experimental uncertainties. The proposed method is expected to provide highly accurate vibrational energies and wavefunctions for molecules with more than 20 atoms.