Artificial Neural Networks as Mappings between Proton Potentials, Wave Functions, Densities, and Energy Levels

Artificial neural networks (ANNs) have become important in quantum chemistry. Herein, applications to nuclear quantum effects, such as zero-point energy, vibrationally excited states, and hydrogen tunneling, are explored. ANNs are used to solve the time-independent Schrodinger equation for single- and double-well potentials representing hydrogen-bonded molecular systems capable of proton transfer. ANN mappings are trained to predict the lowest five proton vibrational energies, wave functions, and densities from the proton potentials and to predict the excited state proton vibrational energies and densities from the proton ground state density. For the inverse problem, ANN mappings are trained to predict the proton potential from the proton vibrational energy levels or the proton ground state density. This latter mapping is theoretically justified by the first Hohenberg-Kohn theorem establishing a one-to-one correspondence between the external potential and the ground state density. ANNs for twoand three-dimensional systems are also presented to illustrate the straightforward extension to higher dimensions.

Automated summary

Artificial neural networks play a crucial role in quantum chemistry research involving nuclear quantum effects, particularly in solving problems related to proton transfer in hydrogen-bonded molecular systems. The use of ANN mappings to predict proton vibrational energies, wave functions, densities, and proton potentials demonstrates the potential for accurately modeling quantum phenomena in higher-dimensional systems.