Ab-initio simulation of excited-state potential energy surfaces with transferable deep quantum Monte Carlo

TL;DR

Addressing the expensive ab-initio prediction of excited-state PESs, the paper proposes a transferable deep-learning QMC framework that optimizes a neural-network wave function across multiple geometries and electronic states. It combines weight sharing and dynamic state ordering within variational Monte Carlo to achieve near two orders of magnitude cost reduction while maintaining high accuracy. The method is demonstrated on ethylene, the carbon dimer, and the methylenimmonium cation, delivering accurate relative energies and smooth behavior across conical intersections. This approach enables efficient, accurate ab-initio excited-state PESs with potential applications in nonadiabatic dynamics and excited-state force-field development.

Abstract

The accurate quantum chemical calculation of excited states is a challenging task, often requiring computationally demanding methods. When entire ground and excited potential energy surfaces (PESs) are desired, e.g., to predict the interaction of light excitation and structural changes, one is often forced to use cheaper computational methods at the cost of reduced accuracy. Here we introduce a novel method for the geometrically transferable optimization of neural network wave functions that leverages weight sharing and dynamical ordering of electronic states. Our method enables the efficient prediction of ground and excited-state PESs and their intersections at the highest accuracy, demonstrating up to two orders of magnitude cost reduction compared to single-point calculations. We validate our approach on three challenging excited-state PESs, including ethylene, the carbon dimer, and the methylenimmonium cation, indicating that transferable deep-learning QMC can pave the way towards highly accurate simulation of excited-state dynamics.

Figures (4)

• Figure 1: Neural network architecture for the transferable excited-state wave function with orthogonality constraints. Diagram of the transferable wave function. Nuclear positions inform the electron representations in the CausalSelfAttn block, introducing an explicit dependency on the molecular configuration. If parameters are shared between states, the same Electron Encoder, Nuclei Encoder and CausalSelfAttn block are applied for each batch of electron coordinates, respectively. The orthogonality of the states is implemented by ordering wave functions based on energy and applying a directional overlap penalty \ref{['eqn:loss_function']}.
• Figure 2: Lowest-lying singlet PESs for ethylene relaxation. Simulation of the ground and first singlet excited state of ethylene along the torsion around the C--C bond and pyramidalitzation of a CH$_2$ group. a): Relative energy with respect to the equilibrium geometry. DeepQMC results for transferable optimization with parameter sharing and dynamic state ordering are presented alongside highly accurate single-point NES-VMC simulations pfau2024. Sampling errors are smaller then the marker size. b): Convergence of the DeepQMC MAEs to NES-VMC results in relative energy. Different variants of transferable DeepQMC optimization are compared to single-point calculations performed in earlier work szabo2024. The horizontal axis indicates the total cumulative iterations across all geometries of the torsion and pyramidalization PES, respectively. The dashed horizontal line gives the accuracy of evaluating the final single-point wave functions. c): Relative energies with respect to $\phi=90^\circ$ around the conical intersection (top two panels) and intermolecular overlaps (Eq.\ref{['eq:intermolecular_overlap']}, bottom two panels) for transferable DeepQMC runs with parameter sharing and dynamic state ordering and without parameter sharing and fixed ordering.
• Figure 3: Carbon dimer dissociation curves along the lowest-lying eight electronic states. Transferable DeepQMC ansatzes were trained jointly across ten geometries, with bond lengths ranging from 1.0 to 1.9 Å. Semistochastic heat-bath configuration interaction calculations in quintuple-$\zeta$ basis, extrapolated to the full configuration interaction limit are used as reference holmes2017. a): The final relative energies obtained with transferable DeepQMC, for the lowest-lying four singlet and four triplet states of the carbon dimer. Both DeepQMC and reference curves were smoothed by cubic interpolation. b): Convergence of the mean absolute error of relative energies between the eight lowest-lying states of the carbon dimer. Geometrically transferable optimization is compared with single-point training performed in earlier work szabo2024. The horizontal axis indicates the total cumulative iterations across all geometries. The dashed horizontal line marks the evaluated final accuracy of the single-point simulations.
• Figure 4: 2D singlet PESs of the methylenimmonium cation. Transferable DeepQMC simulations on CH2NH2+ for varying C-N bond lengths (1.2--1.54 Å) and torsion angles $\tau$ (0--90$^\circ$). a): Relative energies of the three lowest-lying singlet PESs of the CH2NH2+ on a uniform 10$\times$10 grid. b): 1D slice of the 2D PES at $\tau=0$, comparing transferable DeepQMC results with MR-CISD+Q/SA-9-CAS(6,4)/d-aug-cc-pVDZ data from Ref. barbatti2006. c): Convergence of the mean absolute error of relative energies between the three lowest-lying singlet states evaluated at ten configurations with $\tau=0$ (C-N bond length between 1.2--1.54 Å), using MR-CISD barbatti2006 as a reference. Single-point and two transferable neural network VMC optimizations are compared. The first transferable training was carried out on the 10$\times$10 grid (varying C-N and $\tau$), while the second only on the 10-point subset at $\tau=0$. The horizontal axis indicates the total cumulative iterations across all geometries considered, re-normalized to take into account differences in the electron batch sizes. The dashed horizontal line marks the final accuracy of the single-point simulations.