Quantum dynamics at conical intersections in solution. I. Multiplicative neural networks and thermofields

Environmental effects on the vibronic dynamics at a conical intersection can be captured by collective modes, which affect both the topology of the nonadiabatically coupled potential surfaces and the transient dynamics. Here, we show how neural network (NN) potentials can be adapted to a combination of intramolecular coordinates and collective environmental modes. Specifically, we use multiplicative NN (m-NN) potentials, which are fitted to a diabatic representation of regularized diabatic states type. These potentials are readily combined with multiconfigurational wave functions for high-dimensional quantum dynamics. The thermofield dynamics (TFD) approach is employed to include thermal averaging at the wave function level, and we formulate a thermal NN/TFD Hamiltonian that accommodates initial vibronic correlations and collective/residual-mode coupling in the environmental subspace. For a model system describing the isomerization of protonated Schiff bases in solution, the solvent is represented by an overdamped Brownian-oscillator spectral density. In a companion paper [B. Błasiak et al., J. Chem. Phys. 163, 124109 (2025)], the resulting m-NN/TFD Hamiltonian is employed in real-time quantum dynamical simulations using the multi-layer multiconfiguration time-dependent Hartree method.