Neural Network Representation of Three-State Quasidiabatic Hamiltonians Based on the Transformation Properties from a Valence Bond Model: Three Singlet States of H₃⁺

Abstract

A neural network (NN) approach was recently developed to construct accurate quasidiabatic Hamiltonians for twostate 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 H 3 + , 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 (D 3h ). 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.

Diabatic States of Molecules

“…Employing machine learning techniques for fitting potential energy surfaces is a growing field, − but we limit our discussion here to diabatic potentials, in particular to the DDNN method. We shall see that the DDNN method (unlike most previous applications of neural networks in chemistry) uses the DNN as more than just an interpolation tool; it is used to discover the diabatic representation as well as to (simultaneously) fit it.…”

Data Quality, Data Sampling and Data Fitting: A Tutorial Guide for Constructing Full-dimensional Accurate Potential Energy Surfaces (PESs) of Small Molecular Systems

“…Relevant excellent reviews and work are available. [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] Then various properties can be determined by running either classical or quantum dynamic calculations on the PES of the system. Consequently, although PES can't be measured directly by experiments, its performance significantly affects the dynamic outcome.…”

Unified Hamiltonian Formalism of Jahn–Teller and Pseudo-Jahn–Teller Problems in Axial Symmetries

“…Usually, one such Hamiltonian operator is resolved in a set of electronic states within which the JT/pJT interactions occur, and the set of states are selected to be diabatic states. [20][21][22][23][24][25] The diabatic states vary smoothly along nuclear structural change, so that their Hamiltonian matrix elements are differentiable functions of vibrational coordinates. The matrix elements can hence be expanded as polynomial series of vibrational coordinates.…”

“…The JT and pJT interactions belong to the broader category of vibronic interactions, since they involve interactions between electronic and vibrational degrees of freedom. , Naturally, accurate vibronic Hamiltonians for JT/pJT problems are of critical importance to simulate and understand the associated JT/pJT effects. Usually, one such Hamiltonian operator is resolved in a set of electronic states within which the JT/pJT interactions occur, and the set of states are selected to be diabatic states. − The diabatic states vary smoothly along nuclear structural change, so that their Hamiltonian matrix elements are differentiable functions of vibrational coordinates. The matrix elements can hence be expanded as a polynomial series of vibrational coordinates.…”