Deep neural network solution of the electronic Schrodinger equation¨ · 2020. 12. 18. · Deep neural network solution of the electronic Schrodinger equation¨ Jan Hermann, 1 ;2

Deep neural network solution of the electronicSchrodinger equation

Jan Hermann,1,2 Zeno Schatzle,1 and Frank Noe1,3,41FU Berlin, Dept. of Mathematics and Computer Science, Arnimallee 6, 141 95 Berlin, Germany

2TU Berlin, Machine Learning Group, Marchstr. 23, 105 87 Berlin, Germany3FU Berlin, Dept. of Physics, Arnimallee 14, 141 95 Berlin, Germany

4Rice University, Dept. of Chemistry, Houston, TX 77005, USA{jan.hermann,frank.noe}@fu-berlin.de

Abstract

The electronic Schrodinger equation describes fundamental properties ofmolecules and materials, but cannot be solved exactly for larger systems than thehydrogen atom. Quantum Monte Carlo is a high-accuracy approximation method,in principle only limited by the flexibility of the used wave-function ansatz. Herewe develop the deep-learning real-space wave-function ansatz PauliNet. PauliNethas the Hartree–Fock solution built in as a baseline, incorporates the physicsof valid wave functions, and is trained using variational quantum Monte Carlo(VMC). PauliNet achieves higher accuracy than comparable state-of-the-art VMCfor atoms, diatomic molecules, and a strongly-correlated hydrogen chain.

1 Introduction

A cornerstone of predicting the physical and chemical properties of a given molecule or materialdefined by Hamiltonian operator, H is the solution of the corresponding electronic Schrodingerequation for the unknown electronic wave function, ψ(r1, . . . , rN ), and electronic energy, E,

Hψ(r1, . . . , rN ) = Eψ(r1, . . . , rN ) (1)

H :=∑i

(− 1

2∇2ri −

∑I

ZI|ri −RI |

)+∑i<j

1

|ri − rj |(2)

Here, (r1, . . . , rN ) ≡ r ∈ R3N are the spatial coordinates of N electrons, and (ZI ,RI) are thecharges and coordinates of M atomic nuclei. In addition to spatial coordinates, each electrons has asi ∈ {↑, ↓}, and the solution of Eq. (1-2) must obey antisymmetry with respect to exchanging equal-spin electrons. The antisymmetry constraint makes solving the electronic Schrodinger equation toa fixed accuracy level NP-hard, i.e. it scales exponentially with N [Troyer and Wiese, 2005]. Anexact solution of Eq. (1-2) is only available for N = M = 1, the hydrogen atom, but quantumchemistry has developed numerous approximation methods with varying degrees of accuracy andcomputational efficiency [Piela, 2014].

Many quantum chemistry methods build on the variational principle, which states that the solutionof (1) with the lowest energy, ψ0, E0, can be obtained by minimizing the energy expectation valueover all antisymmetric wave functions. Thus, minimizing over a trial subset of all wave functions,called an ansatz, parametrized by θ, gives an upper bound to the true ground-state energy E0:

E0 = minψE[ψ] ≤ min

θE[ψθ], E[ψ] =

∫drψ(r)Hψ(r) (3)

Second Workshop on Machine Learning and the Physical Sciences (NeurIPS 2019), Vancouver, Canada.

Number of electronsNumber of spin-up electronsNumber of spin-down electronsNumber of nucleiEmbedding dimension, electronsEmbedding dimension, nuclei

xn

Equivariant InteractionBlock

HartreeFock

Backflow Jastrow ElectronicCusps

xn+1

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Fixed function

Neural Network (trainable function)

expexp

detdet

J γ


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Wave function

r↑ r↓

r↑

r↓R

XY

InputArray

HiddenArray

trainablearray

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Figure 1: Schematic of the PauliNet neural network structure and information flow.

In variational quantum Monte Carlo (VMC) [Foulkes et al., 2001, Needs et al., 2010, Austinet al., 2012], the energy integral is written as an expected value of the local energy, Eloc[ψ](r) =

Hψ(r)/ψ(r), over the probability distribution |ψ2(r)|,E[ψ] = Er∼|ψ|2

[Eloc[ψ](r)

](4)

The primary motivation to improve existing variational wave-function ansatzes is that although thereare several extensions of VMC that increase its accuracy, such as the diffusion quantum Monte Carlo(DMC), improving the actual ansatz is the only way to improve the nodal surface of a wave function,which is the hyperplane of electronic coordinates on which the wave function changes sign, andwhose accuracy sets fundamental limitation to the accuracy of all quantum Monte Carlo (QMC)techniques. The existing wave-function ansatzes are partially motivated by the physics of electrons,partially by trial-and-error experience. Here, we demonstrate that ansatzes that incorporate the rightphysics while using DNNs in place of the existing ad-hoc functional forms can outperform state-of-the-art variational ansatzes of comparable types.

2 Method

2.1 Neural wave function PauliNet incorporating the physics of electrons

Our proposed PauliNet wave function is of the Slater–Jastrow–backflow type, where both the Jastrowfactor J and the backflow f are represented by DNNs with trainable parameters θ (Figure 1),

ψθ(r) = det[ϕµ(r↑i )fθ,µi(r)] det[ϕµ(r↓i )fθ,µi(r)]eγ(r)+Jθ(r) (5)Jθ and fθ(r) are invariant and equivariant DNNs, respectively, with respect to exchanging same-spin electrons, and ϕµ and γ are fixed functions. This ansatz ensures basic physics of electronicwave functions as follows.

Antisymmetry We fix the first N↑ (N↓) electrons to be spin-up (spin-down), The wave functionthen must be antisymmetric in the first N↑ and in the last N↓ = N −N↑ coordinates.

ψ(r↑1, . . . , r

↑N↑, r↓N↑+1, . . . , r

↓N

)≡ ψ(r↑, r↓) ≡ ψ(r) (6)

ψ(. . . , ri, . . . , rj , . . .) = −ψ(. . . , rj , . . . , ri, . . .), (7)As common in quantum chemistry, our ansatz enforces antisymmetry among same-spin electronsvia the matrix determinants in Eq. (5).

2

Hartree–Fock baseline To ensure a good starting point for the optimization problem, we buildthe standard Hartree–Fock (HF) wave function as a baseline into PauliNet. The HF wave functionuses one-electron functions, ϕµ(r), called molecular orbitals, and achieved antisymmetry by theso-called Slater determinant.

ψHF(r) := det[ϕµ(r↑i )] det[ϕµ(r↓i )] (8)HF is straightforward, computationally efficient, captures much of the basic physics of atoms andmolecules, and makes qualitative reasonable predictions. We take the molecular orbitals directlyfrom the HF method and keep them fixed in our ansatz (5).

Beyond Hartree–Fock with trainable Jastrow factor and backflow The main limitation to ac-curacy in HF is that it does not depend on electron-electron distances, and therefore captures noelectron-electron correlation besides the antisymmetric constraint. We combine two establishedstrategies to improve the HF baseline. First, the HF wave function is multiplied by a nonnegativetotally symmetric function eJθ(r), with the so-called Jastrow factor J [Drummond et al., 2004]. Thismodification can build complex electron correlations into the wave function, but cannot modify thenodal surface inherited from HF. Second, we can generalize the molecular orbitals to all-electronfunctions as long as we retain equivariance (and thus the determinant antisymmetry) with respect toexchanging same-spin electrons,

ϕµ(ri)→ ϕµi(r), Pϕµi(r) = ϕµi(Pr) (9)where P is the exchange operator for same-spin electrons. This approach is called the backflow, andunlike the Jastrow factor it can modify the nodal surface of the HF wave function [Lopez Rıos et al.,2006]. We achieve backflow by multiplying the orbitals ϕµ with trainable DNN functions fθ,µ.

Cusp conditions Any ground-state electronic wave function obeys exact asymptotic behavior de-fined by the cusp conditions as electrons approach each other and the nuclei [Kato, 1957],

1

ψ0

∂ψ0

∂|ri −RI |

∣∣∣∣ri=RI

= −ZI ,1

ψ0

∂ψ0

∂|ri − rj |

∣∣∣∣ri=rj

=

{14 si = sj12 si 6= sj

(10)

PauliNet ensures the nuclear cusp conditions via the molecular orbitals ϕµ(ri) as described in Maet al. [2005]. The electronic cusp conditions are enforced by a fixed function γ(r). To preserve thecusp conditions built into ϕµ and γ, the Jastrow factor and backflow DNNs must be cuspless, whichwe ensured by construction of the DNN architecture (see below).

∇riJ(r)∣∣ri={rk,Ra} = 0, ∇rifµi(r)

∣∣ri={rk,Ra} = 0 (11)

Neural network architecture The requirements of invariance and equivariance with respect topermutation of particles, and the fact that particle interactions are a function of their distances areclosely related with the aim of constructing DNNs that learn potential energy functions. Therefore,we adapt the graph-convolutional neural network SchNet [Schutt et al., 2018], as follows: (i) Theiterated feature vectors x(n)

i of the graph represent electrons, not atoms as in SchNet, and we onlydistinguish two initial embeddings: spin-up and spin-down. (ii) The graph messages received bythe electron feature vectors at each iteration are split into three channels: same-spin electrons, op-posite-spin electrons, and nuclei. (iii) Each nucleus is represented by a trainable embedding Yθ,I ,shared across all iterations of the graph network. (iv) Two new trainable functions ηθ and κθ of thefinal electron feature vectors return the values of the Jastrow factor and the backflow, respectively.As feature vectors x(n)

i are equivariant with respect to exchange at each iteration, so are the backflowvectors fi. (v) The distance features e are constructed to be cuspless to maintain the cusp behaviorencoded in ϕµ and γ.

Training To generate samples of r from the wave-function distribution |ψ2(r)|, needed to evaluatethe expected value in (4), we use a standard Langevin Monte Carlo approach [Umrigar et al., 1993].To optimize the parameters θ in the Jastrow and backflow neural networks, we use Adam [Kingmaand Ba, 2014] with the variational energy (4) as the loss function. The stochastic loss gradient overa batch of samples is computed with a formula that exploits the Hermitian nature of the Hamiltonianin (3) [Ceperley et al., 1977]. This formulation only requires second derivatives of the wave function(for the Laplace operator), whereas direct differentiation would require third derivatives.L(θ) = Er∼|ψ′2|

[Eloc(r;θ)

], ∇θL(θ) = 2Er∼|ψ′2|

[(Eloc(r;θ)− L(θ)

)∇θ ln|ψθ|

](12)

3

Table 1: Ground-state energies of five test systems obtained by four different methods.Ea

0/Eh (% Ecorr)system VMC DMC DeepWF PauliNet

H2 −1.1738 98.4% −1.174 37(6) 99.7%LiH −8.0635f 91.5% −8.0703f 99.7% −7.8732 b −8.0690(3) 98.1%Be −14.6311d 61.6% −14.6572d 89.2% −14.6141 43.6% −14.6546(7) 86.5%B −24.6056d 60.0% −24.6398d 88.3% −24.2124 b −24.634(2) 83.5%He

10 −5.5685 63.8% −5.655(2) 96.0%aReference exact ground-state and HF energies are taken from Kołos and Wolniewicz [1965], Cencek andRychlewski [2000], Brown et al. [2007], Motta et al. [2017], Casalegno et al. [2003]. bFor LiH and B, DeepWFdoes not reach the accuracy of the HF method.

1 2 3dHH/a0

0.55

0.50

E 0/(N

HE h

)

(a)

1 2 3dHH/a0

(b)Hartree FockPauliNetDeepWFexact

Figure 2: Electronic energy as a function of distance between atoms in H2 (a) and H10 (b).

3 Results and Conclusions

In this work, we test only the Jastrow part of our architecture and turn the backflow part off, whichenables us to directly compare against fixed-node DMC calculations. We use the same set of atomsand small molecules that were used to test DeepWF in [Han et al., 2018] (Table 1).

PauliNet outperforms state-of-the-art Jastrow factors A common way to compare the accuracyof ab-initio methods in quantum chemistry is via the fraction of correlation energy captured, de-fined as the difference between the exact ground-state energy and that of the HF method. WhereasDeepWF does not even reach the accuracy of the HF method for LiH and B, our PauliNet givesvariational energies for atoms and diatomics that are better than those from state-of-the-art VMCcalculations, and are only outperformed by the fixed-node limits of DMC calculations by severalpercents of the correlation energy. It took several hundreds of iterations to converge the energy tothe reported values, corresponding to tens of minutes on a single GTX1080Ti GPU.

PauliNet captures strong correlation Unlike the atoms and diatomics, the linear hydrogen chainH10 exhibits strong correlation, which describes a situation where the single-determinant descriptionof the HF method is qualitatively insufficient [Motta et al., 2017]. This is a hard test for the ansatz,because the Jastrow factor needs to be a complicated many-body function under such circumstances[Goetz and Neuscamman, 2017]. Already H2 exhibits strong correlation when the two atoms aredissociated, but our Jastrow factor is able to cover the transition from dynamic to strong correlationsmoothly (Figure 2a). For H10, we recover 96% (80%) of the correlation energy in the equilibrium(stretched) geometry (Figure 2b). The supposedly harder problem of learning the Jastrow factor fora strongly-correlated system leads to a longer optimization time, here on the order of hours.

Conclusions DNN representations of real-space electronic wave functions can outperform state-of-the-art variational ansatzes for simple and strongly-correlated systems, as a result of the increasedflexibility in the functional form. In parallel, [Pfau et al., 2019] developed a deep-learning archi-tecture that exhibits higher accuracy, but also much greater computational costs. Combining thephysical constraints built into PauliNet with the improved optimizer of [Pfau et al., 2019] has thepotential to combine their advantages. In future work, we plan to explore numerical performance ofthe neural backflow in PauliNet, which in principle removes any inherent limitations of the flexibil-ity of our ansatz. We hope that introducing DNNs to quantum Monte Carlo opens the possibility toexploit the striking advances in deep learning for scientific discovery.

4

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Deep neural network solution of the electronic Schrodinger equation¨ · 2020. 12. 18. · Deep neural network solution of the electronic Schrodinger equation¨ Jan Hermann, 1 ;2

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