Enabling ab initio geometry optimization of strongly correlated systems with transferable deep quantum Monte Carlo

Abstract

A faithful description of chemical processes requires exploring extended regions of the molecular potential energy surface (PES), which remains challenging for strongly correlated systems. Transferable deep-learning variational Monte Carlo (VMC) offers a promising route by efficiently solving the electronic Schrödinger equation jointly across molecular geometries at consistently high accuracy, yet its stochastic nature renders direct exploration of molecular configuration space nontrivial. Here, we present a framework for highly accurate ab initio exploration of PESs that combines transferable deep-learning VMC with a cost-effective estimation of energies, forces, and Hessians. By continuously sampling nuclear configurations during VMC optimization of electronic wave functions, we obtain transferable descriptions that achieve zero-shot chemical accuracy within chemically relevant distributions of molecular geometries. Throughout the subsequent characterization of molecular configuration space, the PES is evaluated only sparsely, with local approximations constructed by estimating VMC energies and forces at sampled geometries and aggregating the resulting noisy data using Gaussian process regression. Our method enables accurate and efficient exploration of complex PES landscapes, including structure relaxation, transition-state searches, and minimum-energy pathways, for both ground and excited states. This opens the door to studying bond breaking, formation, and large structural rearrangements in systems with pronounced multi-reference character.

Figures (8)

• Figure 1: Sketch of the DeepQMC/GPR ab initio geometry optimization method. The flowchart depicts the process of optimizing the minimum energy pathway (MEP) of the formaldehyde isomerization. a) A transferable wave function is optimized on continuously sampled nuclear configurations of a two dimensional cut of the formaldehyde configuration space. Since the training loss consists of expectation values over small electron batches at random nuclear configurations, during transferable training the full PES is never materialized beyond noisy estimates. b) To extract energy and gradient information from the optimized transferable wave function, Monte Carlo estimators of the molecular energy and Hellmann-Feynman force are evaluated. The predictions are further improved by evaluating these expectation values at a range of molecular configurations in the vicinity of the geometry under investigation and fitting a local approximation of the PES with Gaussian process regression (GPR). From the Gaussian process energy, force and Hessian are computed analytically, which suffice to compute second order geometry updates. c) By a consecutive application of optimization steps, molecular geometries can be relaxed, transition states can be found and intermediates can be obtained when employing additional constrains. d) Combining multiple geometry optimizations, the full MEP of a chemical process can be characterized. Notably, for the MEP optimization the PES was only approximated sparsely via the samples depicted in gray.
• Figure 2: Optimizing equilibrium bond lengths for diatomic molecules. DeepQMC/GPR results are compared to experimental data huber1979 and to MP2@CBS and CCSD(T)@cc-pV6Z calculations pawlowski2003.
• Figure 3: Minimum energy path search on two dimensional cuts of the ammonia and formaldehyde potential energy surfaces. On panel a) the minimum energy pathway for the inversion of ammonia is shown, while panel b) depicts the isomerization of formaldehyde. Reference and baseline results for the ammonia inversion are taken from Ref. iyerForceFreeIdentificationMinimumEnergy2024, CCSD results for formaldehyde are computed in house. The heatmaps visualize PESs extracted from the transferable deep-learning VMC solution and serve as a guide to the eye.
• Figure 4: Adiabatic excitation energies for the ethylene molecule. On panels a)-d) the internal coordinates of the relaxation trajectory of the triplet state after a vertical excitation from the singlet ground state equilibrium geometry are depicted. The VMC barboriniStructuralOptimizationQuantum2012 reference equilibrium geometries of the singlet and triplet states are well reproduced. Corresponding estimates of the relative energy with respect to the triplet equilibrium geometry are displayed on panel e). The inserts show the optimized equilibrium geometry of the singlet and triplet state. Panel f) and panel g) compare the adiabatic and the vertical excitation energies to reference calculations barboriniStructuralOptimizationQuantum2012, where the shaded area indicates chemical accuracy.
• Figure 5: Potential energy surfaces of the H2 + NH -> H* + *NH2 reaction. The reaction might proceed either on the $a^1A"$ or the $b^1A'$ PES. Inset plots show the relevant reactant, transition state, and product structures highlighting the differences between the two electronic states. The FCI results taken from Wu et al.wu2020 were computed on geometries optimized at the CASPT2 level of theory, with the relatively small basis set of 6-311G*. The energies of these geometries were also evaluated with DeepQMC, shown with solid lines.