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arXiv:2603.14515

Excited Pfaffians: Generalized Neural Wave Functions Across Structure and State

Nicholas Gao, Till Grutschus, Frank Noé, Stephan Günnemann

cs.LGphysics.chem-phphysics.comp-phquant-phSubmitted Mar 15, 2026• Announced Mar 17, 2026• 0 citations
v1 · Mar 15, 2026v2 · May 27, 2026
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Abstract

Neural-network wave functions in Variational Monte Carlo (VMC) have achieved great success in accurately representing both ground and excited states. However, achieving sufficient numerical accuracy in state overlaps requires increasing the number of Monte Carlo samples, and consequently the computational cost, with the number of states. We present a nearly constant sample-size approach, Multi-State Importance Sampling (MSIS), that leverages samples from all states to estimate pairwise overlap. To efficiently evaluate all states for all samples, we introduce Excited Pfaffians. Inspired by Hartree-Fock, this architecture represents many states within a single neural network. Excited Pfaffians also serve as generalized wave functions, allowing a single model to represent multi-state potential energy surfaces. On the carbon dimer, we match the $O(N_s^4)$-scaling natural excited states while training $>200\times$ faster and modeling 50\% more states. Our favorable scaling enables us to be the first to use neural networks to find all distinct energy levels of the beryllium atom. Finally, we demonstrate that a single wave function can represent excited states across various molecules.

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Neural-network wave functions in Variational Monte Carlo (VMC) have achieved great success in accurately representing both ground and excited states. However, achieving sufficient numerical accuracy in state overlaps requires increasing the number of Monte Carlo samples, and consequently the computational cost, with the number of states. We present a nearly constant sample-size approach, Multi-State Importance Sampling (MSIS), that leverages samples from all states to estimate pairwise overlap. To efficiently evaluate all states for all samples, we introduce Excited Pfaffians. Inspired by Hartree-Fock, this architecture represents many states within a single neural network. Excited Pfaffians also serve as generalized wave functions, allowing a single model to represent multi-state potential energy surfaces. On the carbon dimer, we match the O(Ns4)O(N_s^4)O(Ns4​)-scaling natural excited states while training >200×>200\times>200× faster and modeling 50% more states. Our favorable scaling enables us to be the first to use neural networks to find all distinct energy levels of the beryllium atom. Finally, we demonstrate that a single wave function can represent excited states across various molecules.
Describing the electronic structure of molecules using computational methods is of central importance in many areas of chemical research and industry. To this day, the variational quantum Monte Carlo (VMC) algorithm, which utilizes the variational principle of quantum mechanics to optimize an ansatz towards the true electronic wave function, is one of the lesser employed of such computational methods. While it is generally applicable, and provides direct access to the wave function, its accuracy has been limited by insufficiently expressive ansatzes. This barrier was recently lifted with the introduction of deep-learning-based ansatzes, providing unprecedented flexibility and corresponding accuracy, at the cost of increased computational requirements. In this thesis, our implementation of the VMC algorithm with deep-learning ansatzes is described, along with a number of developments facilitating the computation of electronic excited states, extraction of various observables, and characterization of potential energy surfaces, at a reduced computational cost. The method is implemented as open-source software, employing a common framework to encompass most ansatzes present in the literature. Utilizing the direct wave function access granted by VMC, an approach is developed to extract one-electron reduced densities, that are free from Gaussian basis set effects. Significant advances in the methodology of computing electronic excited states via the ladder-of-states approach are made, including an improved form of the overlap penalty term with an automatically determined prefactor, and a new loss term with which the total spin of the solution can be selected. The efficiency of computing excited potential energy surfaces are enhanced by optimizing a single ansatz for a range of molecular configurations. To enable such transferable training in the presence of state crossings, a dynamic, geometry-wise ordering in the ladder of states is introduced. Geometry-optimization-based tasks are identified as natural fits for such transferable ansatzes, and a transferability-native approach for extracting the necessary information from VMC data is devised. Finally, transferable training is employed to simultaneously model different chemical species, resulting in a foundation model predicting bond breaking at a cost-accuracy ratio competitive with mature multi-reference methods. The work presented here contributes significantly to the development of a practically useful deep-learning-based VMC method. The described approaches have the potential to become tools for a wide community of computational chemists, as a source of accurate data, describing strong correlation in both ground and excited states.
Obtaining accurate ground and low-lying excited states of electronic systems is crucial in a multitude of important applications. One ab initio method for solving the Schrödinger equation that scales favorably for large systems is variational quantum Monte Carlo (QMC). The recently introduced deep QMC approach uses ansatzes represented by deep neural networks and generates nearly exact ground-state solutions for molecules containing up to a few dozen electrons, with the potential to scale to much larger systems where other highly accurate methods are not feasible. In this paper, we extend one such ansatz (PauliNet) to compute electronic excited states. We demonstrate our method on various small atoms and molecules and consistently achieve high accuracy for low-lying states. To highlight the method's potential, we compute the first excited state of the much larger benzene molecule, as well as the conical intersection of ethylene, with PauliNet matching results of more expensive high-level methods.
We propose a general framework for finding the ground state of many-body fermionic systems by using feed-forward neural networks. The anticommutation relation for fermions is usually implemented to a variational wave function by the Slater determinant (or Pfaffian), which is a computational bottleneck because of the numerical cost of O(N3)O({N}^{3})O(N3) for NNN particles. We bypass this bottleneck by explicitly calculating the sign changes associated with particle exchanges in real space and using fully connected neural networks for optimizing the rest parts of the wave function. This reduces the computational cost from O(N3)O({N}^{3})O(N3) to O(N2)O({N}^{2})O(N2) or less, while the overall cost depends on the number of Monte Carlo samples and the number of parameters in neural networks. We show that the accuracy of the approximation can be improved by optimizing the ``variance'' of the energy simultaneously with the energy itself. We also find that a reweighting method in Monte Carlo sampling can stabilize the calculation. These improvements can be applied to other approaches based on variational Monte Carlo methods. Moreover, we show that the accuracy can be further improved by using the symmetry of the system, the representative states, data preprocessing, and an additional neural network implementing a generalized Gutzwiller-Jastrow factor. We demonstrate the efficiency of the method by applying it to a two-dimensional Hubbard model.
We propose a general framework for finding the ground state of many-body fermionic systems by using feed-forward neural networks. The anticommutation relation for fermions is usually implemented to a variational wave function by the Slater determinant (or Pfaffian), which is a computational bottleneck because of the numerical cost of O(N3)O(N^3)O(N3) for NNN particles. We bypass this bottleneck by explicitly calculating the sign changes associated with particle exchanges in real space and using fully connected neural networks for optimizing the rest parts of the wave function. This reduces the computational cost to O(N2)O(N^2)O(N2) or less. We show that the accuracy of the approximation can be improved by optimizing the variance of the energy simultaneously with the energy itself. We also find that a reweighting method in Monte Carlo sampling can stabilize the calculation. These improvements can be applied to other approaches based on variational Monte Carlo methods. Moreover, we show that the accuracy can be further improved by using the symmetry of the system, the representative states, and an additional neural network implementing a generalized Gutzwiller-Jastrow factor. We demonstrate the efficiency of the method by applying it to a two-dimensional Hubbard model.
Solving the Schrödinger equation is key to many quantum mechanical properties. However, an analytical solution is only tractable for single-electron systems. Recently, neural networks succeeded at modeling wave functions of many-electron systems. Together with the variational Monte-Carlo (VMC) framework, this led to solutions on par with the best known classical methods. Still, these neural methods require tremendous amounts of computational resources as one has to train a separate model for each molecular geometry. In this work, we combine a Graph Neural Network (GNN) with a neural wave function to simultaneously solve the Schrödinger equation for multiple geometries via VMC. This enables us to model continuous subsets of the potential energy surface with a single training pass. Compared to existing state-of-the-art networks, our Potential Energy Surface Network PESNet speeds up training for multiple geometries by up to 40 times while matching or surpassing their accuracy. This may open the path to accurate and orders of magnitude cheaper quantum mechanical calculations.