Non-Adiabiatic Excited States

hippynn has features for training to excited-state energies, transition dipoles, and the non-adiabatic coupling vectors (NACR). These features can be found in excited.

For a more detailed description, please see the paper [Li2023]

Multi-targets nodes are recommended over the usage of one node per target.

For energies, the node can be constructed just like the ground-state counterpart:

energy = targets.HEnergyNode("E", network, module_kwargs={"n_target": n_states + 1})
mol_energy = energy.mol_energy
mol_energy.db_name = "E"

Note that a multi-target node is used here, defined by the keyword module_kwargs={"n_target": n_states + 1}. Here, n_states is the number of excited states in consideration. The extra state is for the ground state, which is often useful. The database name is simply E with a shape of (n_molecules, n_states+1).

Predicting the transition dipoles is also similar to the ground-state permanent dipole:

charge = targets.HChargeNode("Q", network, module_kwargs={"n_target": n_states})
dipole = physics.DipoleNode("D", (charge, positions), db_name="D")

The database name is D with a shape of (n_molecules, n_states, 3).

For NACR, to avoid singularity problems, we enforcing the training of NACR*ΔE instead:

nacr = excited.NACRMultiStateNode(
    "ScaledNACR",
    (charge, positions, energy),
    db_name="ScaledNACR",
    module_kwargs={"n_target": n_states},
)

For NACR between state i and j, \(\boldsymbol{d}_{ij}\), it is expressed in the following way

\[\boldsymbol{d}_{ij}\Delta E_{ij} = \Delta E_{ij}\boldsymbol{q}_i \frac{\partial\boldsymbol{q}_j}{\partial\boldsymbol{R}}\]

\(E_{ij}\) is energy difference between state i and j, which is calculated internally in the NACR node based on the input of the energy node. \(\boldsymbol{R}\) corresponding the positions node in the code. \(\boldsymbol{q}_{i}\) and \(\boldsymbol{q}_{j}\) are the transition atomic charges for state i and j contained in the charge node. This charge node can be constructed from scratch or reused from the dipole predictions. The database name is ScaledNACR with a shape of (n_molecules, n_states*(n_states-1)/2, 3*n_atoms).

Due to the phase problem, when the loss function is constructed, the phase-less version of MAE or RMSE should be used:

energy_mae = loss.MAELoss.of_node(energy)
dipole_mae = excited.MAEPhaseLoss.of_node(dipole)
nacr_mae = excited.MAEPhaseLoss.of_node(nacr)

MAEPhaseLoss and MSEPhaseLoss are the phase-less version MAE and MSE, which take the minimum error over the possible signs of the output.

For a complete script, please take a look at examples/excited_states_azomethane.py.

[Li2023]

Machine Learning Framework for Modeling Exciton-Polaritons in Molecular Materials.

Li et. al, 2023. https://arxiv.org/abs/2306.02523