Electronic correlation is a fundamental topic in many-electron systems. To characterize this correlation, one may introduce the concept of exchange-correlation hole. In this work, we first briefly revisit its definition and relation to electron and geminal densities, followed by their intimate relations to copula functions in probability theory and statistics. We then propose a copula-based approach to estimate the exchange-correlation hole from the electron density. It is anticipated that the proposed scheme will become a promising ingredient towards the future development of strongly correlated electronic structure calculations.

[1]
A.
Szabo
,
Modern Quantum Chemistry
,
New York
:
Dover Publications Inc
., (
1996
).
[2]
R. M.
Martin
,
Electronic Structure: Basic Theory and Practical Methods
,
Cambridge
:
Cambridge University Press
, (
2004
).
[3]
G.
Xu
,
L.
Li
, and
D.
Wang
,
Quantum Chemistry (in Chinese)
,
Beijing
:
China Science Publishing and Media Ltd
., (
2007
).
[4]
I. N.
Levine
,
Quantum Chemistry
, 7th Edn.,
Boston
:
Pearson
, (
2018
).
[5]
P.
Hohenberg
and
W.
Kohn
,
Phys. Rev.
136
,
B864
(
1964
).
[6]
W.
Kohn
and
L. J.
Sham
,
Phys. Rev.
140
,
A1133
(
1965
).
[7]
R. G.
Parr
and
W. T.
Yang
,
Density-Functional Theory of Atoms and Molecules
,
New York
:
Oxford University Press
, (
1995
).
[8]
[9]
N.
Mardirossian
and
M.
Head-Gordon
,
Mol. Phys.
115
,
2315
(
2017
).
[10]
A. D.
Becke
,
J. Chem. Phys.
140
,
18A301
(
2014
).
[11]
[12]
D. C.
Langreth
and
J. P.
Perdew
,
Phys. Rev. B
21
,
5469
(
1980
).
[13]
D. C.
Langreth
and
M. J.
Mehl
,
Phys. Rev. B
28
,
1809
(
1983
).
[14]
[15]
C.
Lee
and
R. G.
Parr
,
Phys. Rev. A
35
,
2377
(
1987
).
[16]
J. P.
Perdew
,
J. A.
Chevary
,
S. H.
Vosko
,
K. A.
Jackson
,
M. R.
Pederson
,
D. J.
Singh
, and
C.
Fiolhais
,
Phys. Rev. B
46
,
6671
(
1992
).
[17]
J. P.
Perdew
and
Y.
Wang
,
Phys. Rev. B
45
,
13244
(
1992
).
[18]
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
,
Phys. Rev. Lett.
77
,
3865
(
1996
).
[19]
J. P.
Perdew
and
K.
Schmidt
,
AIP Conf. Proc.
577
,
1
(
2001
).
[20]
J. P.
Perdew
,
A.
Ruzsinszky
,
G. I.
Csonka
,
O. A.
Vydrov
,
G. E.
Scuseria
,
L. A.
Constantin
,
X.
Zhou
, and
K.
Burke
,
Phys. Rev. Lett.
100
,
136406
(
2008
).
[21]
C.
Noce
,
J. Phys. Condens. Matter.
3
,
7819
(
1991
).
[22]
A. J.
Cohen
,
P.
Mori-Sánchez
, and
W. T.
Yang
,
Chem. Rev.
112
,
289
(
2012
).
[23]
R.
Schade
,
E.
Kamil
, and
P.
Blöchl
,
Eur. Phys. J. Spec. Top.
226
,
2677
(
2017
).
[24]
M.
Sklar
,
Publ. Inst. Statist. Univ. Paris
8
,
229
(
1959
).
[25]
R. B.
Nelsen
,
An Introduction to Copulas
,
New York
:
Springer Science & Business Media
, (
2007
).
[26]
P. K.
Trivedi
and
D. M.
Zimmer
,
Found. Trends Econ.
1
,
1
(
2007
).
[27]
J. C.
Rodriguez
,
J. Empir. Finance
14
,
401
(
2007
).
[28]
[29]
A. K.
Nikoloulopoulos
and
D.
Karlis
,
Environmetrics
19
,
251
(
2008
).
[30]
H. N.
Li
,
X. W.
Zheng
, and
C.
Li
,
Int. J. Struct. Stab. Dyn.
19
,
1950046
(
2019
).
[31]
M.
Sato
,
K.
Ichiki
, and
T. T.
Takeuchi
,
Phys. Rev. D
83
,
023501
(
2011
).
[32]
E. R.
Davidson
,
Reduced Density Matrices in Quantum Chemistry
,
New York, San Francisco, London
:
Academic Press
, (
1976
).
[33]
D. M.
Bishop
,
Group Theory and Chemistry
,
North Chelmsford
:
Courier Corporation
, (
1993
).
[34]
H.
Joe
and
D.
Kurowicka
,
Dependence Modeling: Vine Copula Handbook
,
Singapore
:
World Scientific
, (
2011
).
[35]
D.
Kraus
and
C.
Czado
,
Comput. Stat. Data Anal.
110
,
1
(
2017
).
[36]
E.
Runge
and
E. K.
Gross
,
Phys. Rev. Lett.
52
,
997
(
1984
).
[37]
M. A.
Marques
and
E. K.
Gross
,
Annu. Rev. Phys. Chem.
55
,
427
(
2004
).
[38]
M. E.
Casida
and
M.
Huix-Rotllant
,
Annu. Rev. Phys. Chem.
63
,
287
(
2012
).
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