General framework for E3-equivariant neural network representation of density functional theory Hamiltonian Xiaoxun Gong1 2He Li1 5Nianlong Zou1Runzhang Xu1Wenhui Duan1 3 4 5 6and Yong Xu1 3 4 7y

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General framework for E(3)-equivariant neural network representation of density

functional theory Hamiltonian

Xiaoxun Gong,1, 2 He Li,1, 5 Nianlong Zou,1Runzhang Xu,1Wenhui Duan,1, 3, 4, 5, 6, ∗and Yong Xu1, 3, 4, 7, †

1State Key Laboratory of Low Dimensional Quantum Physics and

Department of Physics, Tsinghua University, Beijing, 100084, China

2School of Physics, Peking University, Beijing 100871, China

3Tencent Quantum Laboratory, Tencent, Shenzhen, Guangdong 518057, China

4Frontier Science Center for Quantum Information, Beijing, China

5Institute for Advanced Study, Tsinghua University, Beijing 100084, China

6Beijing Academy of Quantum Information Sciences, Beijing 100193, China

7RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan

Combination of deep learning and ab initio calculation has shown great promise in revolutionizing

future scientiﬁc research, but how to design neural network models incorporating a priori knowledge

and symmetry requirements is a key challenging subject. Here we propose an E(3)-equivariant

deep-learning framework to represent density functional theory (DFT) Hamiltonian as a function

of material structure, which can naturally preserve the Euclidean symmetry even in the presence

of spin-orbit coupling. Our DeepH-E3 method enables very eﬃcient electronic-structure calculation

at ab initio accuracy by learning from DFT data of small-sized structures, making routine study of

large-scale supercells (>104atoms) feasible. Remarkably, the method can reach sub-meV prediction

accuracy at high training eﬃciency, showing state-of-the-art performance in our experiments. The

work is not only of general signiﬁcance to deep-learning method development, but also creates new

opportunities for materials research, such as building Moir´e-twisted material database.

I. INTRODUCTION

It has been well recognized that deep learning methods

could oﬀer a potential solution to the accuracy-eﬃciency

dilemma of ab initio material calculations. Deep-learning

potential [1,2] and a series of other neural network

models [3–7] are capable of predicting the total energies

and atomic forces of given material structures, enabling

molecular dynamics simulation at large length and time

scales. The paradigm has been used for deep-learning

research of various kinds of physical and chemical prop-

erties [8–19]. Remarkably, a deep neural network rep-

resentation of density functional theory (DFT) Hamil-

tonian (named DeepH) was developed by employing the

locality of electronic matter, localized basis, and local

coordinate transformation [20]. By the DeepH approach

the computationally demanding self-consistent ﬁeld it-

erations could be bypassed and all the electron-related

physical quantities in the single-particle picture can in

principle be derived very eﬃciently. This opens oppor-

tunities for the electronic-structure calculation of large-

scale material systems.

Introducing physical insights and a priori knowledge

into neural networks is of crucial importance to the deep-

learning approaches. Speciﬁcally, the deep-learning po-

tential takes advantage of the invariance of the total en-

ergy under rotation, translation and spatial inversion as

well as permutation of atoms. For DeepH, the property

that the Hamiltonian matrix changes covariantly (i.e.

∗duanw@tsinghua.edu.cn

†yongxu@mail.tsinghua.edu.cn

equivariantly) under rotation or gauge transformations

should be preserved by the neural network model for eﬃ-

cient learning and accurate prediction (Fig. 1). A strat-

egy is developed to apply local coordinate transformation

which changes the rotation covariant problem into an in-

variant one and thus the transformed Hamiltonian matri-

ces can be learned ﬂexibly via rotation-invariant neural

networks [20]. Nevertheless, the large amount of local

coordinate information seriously increases the computa-

tional load, and the model performance depends critically

on a proper selection of local coordinates, which relies

on human intuition and is not easy to optimize. Alter-

natively, one may get rid of the local coordinate trans-

formation by applying the equivariant neural network

(ENN) [21–24]. The key innovation of ENN is that all

the internal features transform under the same symmery

group with the input, thus the symmetry requirements

are explicitly treated and exactly satisﬁed, as shown by

a series of neural network models for various material

properties [6,7,13–15], including PhiSNet [25] for pre-

dicting the Hamiltonian of molecules with ﬁxed system

size. However, the key capability of DeepH that learns

from DFT results on small-sized material systems and

predicts the electronic structures of much larger ones has

not been demonstrated by ENN models. More critically,

the existing ENN models have neglected the equivari-

ance in the spin degrees of freedom, although the elec-

tronic spin and spin-orbit coupling (SOC) play a key role

in modern condensed matter physics and materials sci-

ence. With SOC, one should take care of the spin-orbital

Hamiltonian, whose spin and orbital degrees of freedom

are coupled and transform together under change of co-

ordinate system or basis set, as illustrated in Fig. 1. This

would raise critical diﬃculties in designing ENN models

arXiv:2210.13955v1 [physics.comp-ph] 25 Oct 2022

a1 a2 b1 b2

Electronic Structure Calculation

Atomic orbital basis

Nucleus

Spin-orbital wavefunction

Electron spin

Neglecting SOC Including SOC

FIG. 1. Equivarience in electronic structure calculations. Schematic wavefunctions and Hamiltonian matrices are shown for

the systems neglecting or including spin-orbit coupling (SOC). Structures a1 and a2 are related to each other by a 90◦rotation,

whose hopping parameters (i.e., Hamiltonian matrix elements) between pxorbitals are related by a unitary transformation.

This equivariant property of Hamiltonian must be preserved in all electronic structure calculations. When the SOC are taken

into account, the spin and orbital degrees of freedom are coupled and must transform together under global rotations, as shown

for structures b1 and b2.

due to a fundamental change of symmetry group. In this

context, the incorporation of ENN models into DeepH is

essential but remains elusive.

In this work, we propose DeepH-E3, a universal E(3)-

equivariant deep-learning framework to represent the

spin-orbital DFT Hamiltonian ˆ

HDFT as a function of

atomic structure {R} by neural networks, which enables

very eﬃcient electronic structure calculations of large-

scale materials at ab initio accuracy. A general theoreti-

cal basis is developed to explicitly incorporate covariance

transformation requirements of {R} 7→ ˆ

HDFT into neu-

ral network models that can properly take the electronic

spin and SOC into account, and a code implementation

of DeepH-E3 based on message passing neural network

is also presented. Since the principle of covariance is au-

tomatically satisﬁed, eﬃcient learning and accurate pre-

diction become feasible via the DeepH-E3 method. Our

systematic experiments demonstrate state-of-the-art per-

formance of DeepH-E3, which shows sub-meV accuracy

in predicting DFT Hamiltonian. The method works well

for various kinds of material systems, such as magic-angle

twisted bilayer graphene or twisted van der Waals ma-

terials in general, and the computational costs are re-

duced by several orders of magnitude compared to direct

DFT calculations. Beneﬁting from the high eﬃciency

and accuracy as well as the good transferability, there

could be promising applications of DeepH-E3 in elec-

tronic structure calculations. Also we expect that the

proposed neural-network framework can be generally ap-

plied to develop deep-learning ab initio methods and that

the interdisciplinary developments would eventually rev-

olutionize future materials research.

II. REALIZATION OF EQUIVARIANCE

It has long been established as one of the fundamen-

tal principles of physics that all physical quantities must

transform equivariantly between reference frames. For-

mally, a mapping f:X→Yis equivariant for vector

spaces Xand Ywith respect to group Gif DY(g)◦f=

f◦DX(g),∀g∈G, where DX, DYare representations

of group Gover vector spaces X, Y , respectively. The

problem considered in this work is the equivariance of

a mapping from the material structure {R} including

atom types and positions to the DFT Hamiltonian ˆ

HDFT

with respect to the E(3) group. The E(3) group is the

Euclidean group in three-dimensional (3D) space which

contains translations, rotations and inversion. Transla-

tion symmetry is manifest since we only work on the

relative positions between atoms, not their absolute po-

sitions. Rotations of coordinates introduce non-trivial

transformations which should be carefully investigated.

Suppose the same point in space is speciﬁed in two coor-

dinate systems by rand r0. If the coordinate systems are

related to each other by a rotation, the transformation

rule between the coordinates of the point is r0=Rr,

where Ris a 3 ×3 orthogonal matrix.

In order to take advantage of the nearsightedness of

electronic matter [26], the Hamiltonian operator is ex-

pressed in the picture of localized atomic orbital basis.

The basis is separated into radial and angular parts,

having the form φiα(r) = Ripl(r)Ylm(ˆr). Here iis the

site index, α≡(plm), where pis the multiplicity index,

Ylm is the spherical harmonics having angular momen-

tum quantum number land magnetic quantum number

m,r≡ |r−ri|and ˆr≡(r−ri)/|r−ri|. The transfor-

mation rule for the Hamiltonian matrix between the two

Elemental

embedding ...

Gaussian

basis ...

Spherical

harmonics ... Equivariant neural network

Including

SOC

Neglecting

SOC

Wigner-

Eckart

layer

FIG. 2. Method of constructing an equivariant mapping {R} 7→ ˆ

HDFT. Take the Hamiltonian matrix between l= 1 and

l= 2 orbitals for example. The atomic numbers Ziand interatomic distances |rij |are used to construct the l= 0 vectors,

and the unit vectors of relative positions ˆrij are used to construct vectors of l= 1,2, . . . . These vectors are passed to the

equivariant neural network. If neglecting spin-orbit coupling (SOC), the output vectors of the neural network are converted to

the Hamiltonian using the rule 1 ⊕2⊕3 = 1 ⊗2 via the Wigner-Eckart layer. If including SOC, the output consists of two

sets of real vectors which are combined to form complex-valued vectors. These vectors are converted to the spin-orbital DFT

Hamiltonian according to a diﬀerent rule (1 ⊕2⊕3) ⊕(0 ⊕1⊕2) ⊕(1 ⊕2⊕3) ⊕(2 ⊕3⊕4) = 1⊗1

2⊗2⊗1

∗.

coordinate systems described above is

H0

ip1,jp2l1l2

m1m2=

1=−l1

2=−l2

Dl1

m1m0

1(R)Dl2

m2m0

2(R)∗(Hip1,jp2)l1l2

1m0

2,(1)

where Dl

mm0(R) is the Wigner D-matrix. The equivari-

ance of the mapping {R} 7→ ˆ

HDFT requires that, if the

change of coordinates causes the positions of the atoms to

transform, the corresponding Hamiltonian matrix must

transform covariantly according to Eq. (1). If we further

consider the spin degrees of freedom, the transformation

rule for the Hamiltonian becomes

H0

ip1,jp2l11

2l21

m1σ1m2σ2=

1=−l1

2=−l2X

σ0

1=↑,↓X

σ0

2=↑,↓

Dl1

m1m0

1(R)D1

σ1σ0

1(R)

Dl2

m2m0

2(R)∗D1

σ2σ0

2(R)∗(Hip1,jp2)l11

2l21

1σ0

1m0

2σ0

2,(2)

where σ1, σ2are the spin indices (spin up or down).

ENN is applied to construct the mapping {R} 7→

HDFT in order to preserve equivariance. The input, out-

put and internal features of ENNs all belong to a special

set of vectors which have the form xl= (xl,l, . . . , xl,−l)

and transform according to the following rule:

lm =

m0=−l

mm0(R)xlm.(3)

This vector is said to carry the irreducible representation

of the SO(3) group of dimension 2l+ 1. If the input vec-

tors are transformed according to Eq. (3), then all the

internal features and the output vectors of the ENN will

also be transformed accordingly. Under this constraint,

the ENN incorporates learnable parameters in order to

model equivariant relationships between inputs and out-

puts.

The method of constructing the equivariant mapping

{R} 7→ ˆ

HDFT is illustrated in Fig. 2. The atomic num-

bers Ziand interatomic distances |rij |are used to con-

struct the l= 0 input vectors (scalars). Spherical har-

monics acting on the unit vectors of relative positions

ˆrij constitute input vectors of l= 1,2, . . . . The out-

put vectors of the ENN are passed through the Wigner-

Eckart layer before representing the ﬁnal Hamiltonian.

This layer exploits the essential concept of the Wigner-

Eckart theorem:

l1⊗l2=|l1−l2| ⊕ · · · ⊕ (l1+l2).(4)

“⊕” and “⊗” signs stand for direct sum and tensor prod-

uct of representations, respectively. “=” denotes equiva-

lence of representations, i.e., they diﬀer from each other

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GeneralframeworkforE(3)-equivariantneuralnetworkrepresentationofdensityfunctionaltheoryHamiltonianXiaoxunGong,1,2HeLi,1,5NianlongZou,1RunzhangXu,1WenhuiDuan,1,3,4,5,6,andYongXu1,3,4,7,y1StateKeyLaboratoryofLowDimensionalQuantumPhysicsandDepartmentofPhysics,TsinghuaUniversity,Beijing,100084,China2Sc...

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General framework for E3-equivariant neural network representation of density functional theory Hamiltonian Xiaoxun Gong1 2He Li1 5Nianlong Zou1Runzhang Xu1Wenhui Duan1 3 4 5 6and Yong Xu1 3 4 7y

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