We propose a grid-based local representation of electronic quantities that can be used in machine learning applications for molecules, which is compact, fixed in size, and able to distinguish different chemical environments. We apply the proposed approach to represent the external potential in density functional theory with modified pseudopotentials and demonstrate its proof of concept by predicting the Perdew-Burke-Ernzerhof and local density approximation electronic density and exchange-correlation potentials by kernel ridge regression. For 16 small molecules consisting of C, H, N, and O, the mean absolute error of exchange-correlation energy was 0.78 kcal/mol when trained for individual molecules. Furthermore, the model is shown to predict the exchange-correlation energy with an accuracy of 3.68 kcal/mol when the model is trained with a small fraction (4%) of all 16 molecules of the present dataset, suggesting a promising possibility that the current machine-learned model may predict the exchange-correlation energies of an arbitrary molecule with reasonable accuracy when trained with a sufficient amount of data covering an extensive variety of chemical environments.

1.
D.
Weininger
,
J. Chem. Inf. Comput. Sci.
28
,
31
(
1988
).
2.
R. W.
Homer
,
J.
Swanson
,
R. J.
Jilek
,
T.
Hurst
, and
R. D.
Clark
,
J. Chem. Inf. Comput. Sci.
48
,
2294
(
2008
).
3.
M.
Rupp
,
A.
Tkatchenko
,
K.-R.
Müller
, and
O. A.
von Lilienfeld
,
Phys. Rev. Lett.
108
,
058301
(
2012
).
4.
F. A.
Faber
,
L.
Hutchison
,
B.
Huang
,
J.
Gilmer
,
S. S.
Schoenholz
,
G. E.
Dahl
,
O.
Vinyals
,
S.
Kearnes
,
P. F.
Riley
, and
O. A.
von Lilienfeld
,
J. Chem. Theory Comput.
13
,
5255
(
2017
).
5.
K.
Hansen
,
F.
Biegler
,
R.
Ramakrishnan
,
W.
Pronobis
,
O. A.
von Lilienfeld
,
K.-R.
Müller
, and
A.
Tkatchenko
,
J. Phys. Chem. Lett.
6
,
2326
(
2015
).
6.
A. P.
Bartók
,
R.
Kondor
, and
G.
Csányi
,
Phys. Rev. B
87
,
219902
(
2013
).
7.
J. C.
Snyder
,
M.
Rupp
,
K.
Hansen
,
K.-R.
Müller
, and
K.
Burke
,
Phys. Rev. Lett.
108
,
253002
(
2012
).
8.
J. C.
Snyder
,
M.
Rupp
,
K.
Hansen
,
L.
Blooston
,
K.-R.
Müller
, and
K.
Burke
,
J. Chem. Phys.
139
,
224104
(
2013
).
9.
L.
Li
,
J. C.
Snyder
,
I. M.
Pelaschier
,
J.
Huang
,
U.-N.
Niranjan
,
P.
Duncan
,
M.
Rupp
,
K.-R.
Müller
, and
K.
Burke
,
Int. J. Quantum Chem.
116
,
819
(
2016
).
10.
L.
Li
,
T. E.
Baker
,
S. R.
White
, and
K.
Burke
,
Phys. Rev. B
94
,
245129
(
2016
).
11.
F.
Brockherde
,
L.
Vogt
,
L.
Li
,
M. E.
Tuckerman
,
K.
Burke
, and
K.-R.
Müller
,
Nat. Commun.
8
,
872
(
2017
).
12.
K.
Yao
and
J.
Parkhill
,
J. Chem. Theory Comput.
12
,
1139
(
2016
).
13.
J.
Seino
,
R.
Kageyama
,
M.
Fujinami
,
Y.
Ikabata
, and
H.
Nakai
,
J. Chem. Theory Comput.
148
,
241705
(
2018
).
14.
B.
Kolb
,
L. C.
Lentz
, and
A. M.
Kolpak
,
Sci. Rep.
7
,
1192
(
2017
).
15.
J.
Behler
and
M.
Parrinello
,
Phys. Rev. Lett.
98
,
146401
(
2007
).
16.
J.
Behler
,
J. Phys.: Condens. Matter
26
,
183001
(
2014
).
17.
K.
Schütt
,
P.-J.
Kindermans
,
H. E.
Sauceda Felix
,
S.
Chmiela
,
A.
Tkatchenko
, and
K.-R.
Müller
, “
Schnet: A continuous-filter convolutional neural network for modeling quantum interactions
,”
Adv. Neural Inf. Process. Syst.
30
,
991
(
2017
).
18.
K. T.
Schütt
,
H. E.
Sauceda
,
P.-J.
Kindermans
,
A.
Tkatchenko
, and
K.-R.
Müller
,
J. Chem. Phys.
148
,
241722
(
2018
).
19.
L.
Ruddigkeit
,
R.
van Deursen
,
L. C.
Blum
, and
J.-L.
Reymond
,
J. Chem. Inf. Model.
52
,
2864
(
2012
).
20.
R.
Ramakrishnan
,
P. O.
Dral
,
M.
Rupp
, and
O. A.
von Lilienfeld
,
Sci. Data
1
,
140022
(
2014
).
21.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
,
Phys. Rev. Lett.
77
,
3865
(
1996
).
22.
M.
Krack
and
A. M.
Köster
,
J. Chem. Phys.
108
,
3226
(
1998
).
23.
Q.
Sun
,
T. C.
Berkelbach
,
N. S.
Blunt
,
G. H.
Booth
,
S.
Guo
,
Z.
Li
,
J.
Liu
,
J. D.
McClain
,
E. R.
Sayfutyarova
,
S.
Sharma
,
S.
Wouters
, and
G. K.-L.
Chan
,
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
8
,
e1340
(
2017
).
24.
K. F.
Garrity
,
J. W.
Bennett
,
K. M.
Rabe
, and
D.
Vanderbilt
,
Comput. Mater. Sci.
81
,
446
(
2014
).
25.
C.
Musco
and
C.
Musco
, “
Recursive sampling for the Nyström method
,” e-print arXiv:1605.07583 (
2016
).
26.
A.
Paszke
,
S.
Gross
,
S.
Chintala
,
G.
Chanan
,
E.
Yang
,
Z.
DeVito
,
Z.
Lin
,
A.
Desmaison
,
L.
Antiga
, and
A.
Lerer
, PyTorch, http://pytorch.org.
27.
S. G.
Johnson
, The NLopt nonlinear-optimization package, http://ab-initio.mit.edu/nlopt.
28.
R. A.
Friesner
,
Chem. Phys. Lett.
116
,
39
(
1985
).
29.
R. B.
Murphy
,
M. D.
Beachy
,
R. A.
Friesner
, and
M. N.
Ringnalda
,
J. Chem. Phys.
103
,
1481
(
1995
).
30.
Y.
Imamura
,
A.
Takahashi
, and
H.
Nakai
,
J. Chem. Phys.
126
,
034103
(
2007
).

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