Neural Network Representation of Three-State Quasidiabatic Hamiltonians Based on the Transformation Properties from a...

A neural network (NN) approach was recently developed to construct accurate quasidiabatic Hamiltonians for two-state systems with conical intersections. Here, we derive the transformation properties of elements of 3 × 3 quasidiabatic Hamiltonians based on a valence bond (VB) model and extend the NN-based method to accurately diabatize the three lowest electronic singlet states of H₃⁺, which exhibit the avoided crossing between the ground and first excited states and the conical intersection between the first and second excited states for equilateral triangle configurations (D₃). The current NN framework uses fundamental invariants (FIs) as the input vector and appropriate symmetry-adapted functions called covariant basis to account for the special symmetry of complete nuclear permutational inversion (CNPI). The resulting diabatic potential energy matrix (DPEM) can reproduce the ab initio adiabatic energies, energy gradients, and derivative couplings between adjacent states as well as the particular symmetry. The accuracy of DPEM is further validated by full-dimensional quantum dynamics calculations. The flexibility of the FI-NN approach based on the VB model shows great potential to resolve diabatization problems for many extended and multistate systems.