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Solving Many-Electron Schrödinger Equation Using Deep Neural Networks
Jiequn Han, Linfeng Zhang, Weinan E
TL;DR
The paper addresses the difficulty of solving many-electron Schrödinger equations with a DNN trial-wave-function that explicitly enforces the Pauli exclusion principle and is optimized through VMC. DeepWF gives consistent ground-state energies and reproduces electronic structures across the tested atoms, molecules, and H10 chain, while its antisymmetric ansatz may not fully capture Pauli-induced electronic correlations.
Problem
Accurately solving many-electron Schrödinger equations is difficult because of high dimensionality, entanglement, electron interactions, and the Pauli exclusion principle.
Method
DeepWF represents many-electron trial wave-functions with deep neural networks, explicitly enforcing antisymmetry and optimizing parameters through variational Monte Carlo.
Results
DeepWF shows consistent ground-state energies across H2, He, LiH, Be, B, and H10, and reproduces tested electronic structures without prior atomic-basis knowledge.
Takeaways & Limitations
DeepWF provides a simple DNN-based route for representing tested many-electron ground states and electronic structures from scratch.
Takeaways & Limitations
The antisymmetric ansatz may be insufficient to represent electronic correlations caused by the Pauli exclusion rule, despite its quadratic scaling.
Abstract
from arXiv · showhide
We introduce a new family of trial wave-functions based on deep neural networks to solve the many-electron Schrödinger equation. The Pauli exclusion principle is dealt with explicitly to ensure that the trial wave-functions are physical. The optimal trial wave-function is obtained through variational Monte Carlo and the computational cost scales quadratically with the number of electrons. The algorithm does not make use of any prior knowledge such as atomic orbitals. Yet it is able to represent accurately the ground-states of the tested systems, including He, H2, Be, B, LiH, and a chain of 10 hydrogen atoms. This opens up new possibilities for solving large-scale many-electron Schrödinger equation.
1. Introduction
Many-electron Schrödinger equations are difficult because of high dimensionality, entanglement, and the Pauli exclusion principle. The paper introduces DeepWF, a DNN trial-wave-function representation optimized with VMC and tested on several systems.
- 1. Introduction: Many-electron Schrödinger equations are difficult because of Hilbert-space dimensionality, electron interactions, entanglement, and antisymmetry requirements.The wave-function must change sign when identical electrons exchange places.
- 1. Introduction: Existing approaches include Hartree-Fock, configuration interaction, coupled cluster, Monte Carlo, DMRG, and DMET methods.These methodologies have achieved substantial progress in treating many-electron systems.
- 1. Introduction: VMC uses variational optimization and Monte Carlo sampling to obtain an optimized parametrized trial wave-function.Common trial wave-functions combine an antisymmetric Slater determinant with a symmetric Jastrow factor.
- 1. Introduction: DeepWF develops a DNN representation of the many-electron wave-function that explicitly satisfies the Pauli exclusion principle and integrates with VMC.The work targets the limited use of DNNs for many-electron Schrödinger equations.
- 1. Introduction: DeepWF produces reasonably accurate ground-state energies and reproduces electronic structures for tested systems including Be, B, LiH, and H10.The model learns from scratch without prior atomic-orbital or atomic-basis information.
2. Method
The method represents many-electron trial wave-functions with deep neural networks while enforcing required symmetry properties, then optimizes them using variational Monte Carlo.
- System and objective: The system contains N electrons and M ions under the Born-Oppenheimer approximation, with spin-up and spin-down electrons treated as separate particle types.The Hamiltonian is spin-independent, allowing the electrons to be ordered by spin.
- Wave-function representation: The wave-function is decomposed into a symmetric function and separate anti-symmetric functions for spin-up and spin-down electrons.The symmetric component is analogous to a Jastrow factor, while the anti-symmetric components are Slater determinant-like.
- Anti-symmetric components: The anti-symmetric components are constructed from DNN-based pair functions and inherit anti-symmetry through the network architecture and sign construction.The same construction is applied to spin-down electrons.
- Symmetric component: The symmetric component extends DeepPot-SE by embedding extended relative coordinates into permutation-invariant feature matrices and fitting particle contributions with neural networks.This construction is designed to preserve symmetry and capture electron-ion and many-body correlations.
- Asymptotic and cusp properties: An additional asymptotic term promotes exponential decay and incorporates electron-nuclear and unlike-spin cusp behavior into the trial wave-function.The authors report that adding this term makes training much more stable, although the cusp conditions are not strictly satisfied after contributions from other components are included.
- Optimization: Variational Monte Carlo optimizes randomly initialized parameters by minimizing the second moment of local energy, using walker reweighting, correlated sampling, backpropagation, and stochastic-gradient updates.The local energy is defined as Eloc(r) := HΨθ(r)/Ψθ(r), and correlated sampling reduces variance when evaluating the objective.
3. Results and Discussion
DeepWF produces ground-state energies and electronic structures that generally agree with benchmarks, while accuracy declines as electron count increases and the antisymmetric ansatz remains a limitation.
- Ground-state energies: DeepWF ground-state energies show good consistency with benchmarks across H2, He, LiH, Be, B, and H10, but accuracy deteriorates as electron number increases.The benchmark energies use CISD for H2, He, LiH, Be, and B, and MRCI+Q for H10, with results extrapolated to the CBS limit.
- Potential-energy curves: H2’s potential-energy curve is very close to the benchmark, whereas H10 has good relative energy and the correct local minimum but remains above the benchmark.Figure 2 compares potential energy per atom against bond length for H2 and H10.
- Electronic structures: DeepWF learns the shell structure of Be electrons from scratch and agrees closely with the ccpv5z result, while sto6g does not adequately describe the electronic dispersion.For LiH, DeepWF and the comparison methods show excellent agreement in the axial electron distribution.
- Limitations and future improvements: The antisymmetric ansatz has quadratic scaling but might not sufficiently represent electronic correlations caused by the Pauli exclusion rule.The authors identify this representation issue as a major area for improvement and also suggest more efficient VMC sampling and alternative optimization methods.
Appendix A. Details of the Training Procedure for Each System
The appendix specifies system-dependent network and sampling settings, including activation functions, parameter sharing, optimization schedules, and common VMC sampling hyperparameters.
- Network structure: The network uses hyperbolic-tangent activation functions, and Netanti is omitted for the two-electron systems H2 and He.The network is represented by the number of nodes in each hidden layer from input to output.
- Network structure: Net↑anti parameters are shared except for B, where the spin-up and spin-down electron counts differ by one.
- Network structure: Net↑odd is implemented as an identity function because network approximation of this term showed no significant effect in the tested systems.
- Optimization: Adam stochastic gradient descent optimizes all parameters using exponentially decaying learning rates, batch size 256, and 5000 epochs.Each epoch comprises 24 batch stochastic-gradient-descent iterations, and samples may be reused within an epoch.
- VMC sampling: VMC sampling uses 2048 walkers and 80 Metropolis-Hastings steps after each parameter epoch, with Gaussian proposals adjusted to maintain 15%–75% acceptance.The sampling hyperparameters are shared across systems except for the reference energy Eref.