A simple comparison between the exact and approximate correlation components U of the electron-electron repulsion energy of several states of few-electron harmonium atoms with varying confinement strengths provides a stringent validation tool for 1-matrix functionals. The robustness of this tool is clearly demonstrated in a survey of 14 known functionals, which reveals their substandard performance within different electron correlation regimes. Unlike spot-testing that employs dissociation curves of diatomic molecules or more extensive benchmarking against experimental atomization energies of molecules comprising some standard set, the present approach not only uncovers the flaws and patent failures of the functionals but, even more importantly, also allows for pinpointing their root causes. Since the approximate values of U are computed at exact 1-densities, the testing requires minimal programming and thus is particularly suitable for rapid screening of new functionals.

1.
T. L.
Gilbert
,
Phys. Rev. B
12
,
2111
(
1975
).
2.
R. A.
Donnelly
and
R. G.
Parr
,
J. Chem. Phys.
69
,
4431
(
1978
).
3.
M.
Levy
,
Proc. Natl. Acad. Sci.
76
,
6062
(
1979
).
4.
S. M.
Valone
,
J. Chem. Phys.
73
,
1344
(
1980
).
5.
G.
Zumbach
and
K.
Maschke
,
J. Chem. Phys.
82
,
5604
(
1985
).
6.
T. T.
Nguyen-Dang
,
E. V.
Ludeña
, and
Y.
Tal
,
J. Mol. Struct.: THEOCHEM
120
,
247
(
1985
).
7.
M.
Levy
, “
Correlation energy functionals of one-matrices and Hartree-Fock densities
,” in
Density Matrices and Density Functionals
, edited by
R.
Erdahl
and
V. H. J.
Smith
(
Reidel
,
Dordrecht
,
1987
), pp.
479
-
498
.
8.
K.
Yasuda
,
Phys. Rev. A
63
,
032517
(
2001
).
9.
J.
Cioslowski
,
K.
Pernal
, and
P.
Ziesche
,
J. Chem. Phys.
117
,
9560
(
2002
).
10.
J.
Cioslowski
,
J. Chem. Phys.
123
,
164106
(
2005
).
11.
M.
Taut
,
Phys. Rev. A
48
,
3561
(
1993
).
12.
J.
Cioslowski
and
K.
Pernal
,
J. Chem. Phys.
113
,
8434
(
2000
), and the references cited therein.
13.
M.
Taut
,
A.
Ernst
, and
H.
Eschrig
,
J. Phys. B
31
,
2689
(
1998
).
14.
Z.
Qian
and
V.
Sahni
,
Phys. Rev. A
57
,
2527
(
1998
).
15.
P.
Hessler
,
J.
Park
, and
K.
Burke
,
Phys. Rev. Lett.
82
,
378
(
1999
).
16.
S.
Ivanov
,
K.
Burke
, and
M.
Levy
,
J. Chem. Phys.
110
,
10262
(
1999
).
17.
V.
Sahni
,
Quantal Density Functional Theory II: Approximation Methods and Applications
(
Springer
,
2010
).
18.
P.
Gori-Giorgi
and
A.
Savin
,
Int. J. Quantum Chem.
109
,
2410
(
2009
).
19.
W. M.
Zhu
and
S. B.
Trickey
,
J. Chem. Phys.
125
,
094317
(
2006
).
20.
They are also potentially useful in benchmarking of DFT functionals; see, e.g.,
J.
Cioslowski
and
E.
Matito
,
J. Chem. Theory Comput.
7
,
915
(
2011
).
21.
J.
Cioslowski
,
J. Chem. Phys.
139
,
224108
(
2013
).
22.
J.
Cioslowski
and
M.
Buchowiecki
,
J. Chem. Phys.
125
,
064105
(
2006
).
23.
J.
Cioslowski
and
K.
Pernal
,
J. Chem. Phys.
125
,
064106
(
2006
).
24.
J.
Cioslowski
and
E.
Grzebielucha
,
Phys. Rev. A
77
,
032508
(
2008
).
25.
J.
Cioslowski
,
J. Chem. Phys.
142
,
114104
(
2015
).
26.
J.
Cioslowski
,
J. Chem. Phys.
136
,
044109
(
2012
).
27.
K.
Varga
,
P.
Navratil
,
J.
Usukura
, and
Y.
Suzuki
,
Phys. Rev. B
63
,
205308
(
2001
).
28.
J.
Cioslowski
,
K.
Strasburger
, and
E.
Matito
,
J. Chem. Phys.
136
,
194112
(
2012
).
29.
J.
Cioslowski
,
K.
Strasburger
, and
E.
Matito
,
J. Chem. Phys.
141
,
044128
(
2014
).
30.
J.
Cioslowski
and
E.
Matito
,
J. Chem. Phys.
134
,
116101
(
2011
).
31.
J.
Cioslowski
,
J. Chem. Phys.
142
,
114105
(
2015
).
32.
W.
Kutzelnigg
and
D.
Mukherjee
,
J. Chem. Phys.
110
,
2800
(
1999
).
33.
K. R.
Shamasundar
,
J. Chem. Phys.
131
,
174109
(
2009
).
34.

Due to the real-valuedness of the natural orbitals in question, no distinction is made in this paper between the exchange and the so-called L integrals.

35.
A. M. K.
Müller
,
Phys. Lett. A
105
,
446
(
1984
).
36.
M. A.
Buijse
and
E. J.
Baerends
,
Mol. Phys.
100
,
401
(
2002
).
37.
S.
Goedecker
and
C. J.
Umrigar
,
Phys. Rev. Lett.
81
,
866
(
1998
).
38.

Strictly speaking, UGU is not a genuine 1-matrix functional as it is not invariant to unitary transformations among NOs with degenerate occupation numbers.

39.
G.
Csányi
and
T. A.
Arias
,
Phys. Rev. B
61
,
7348
(
2000
).
40.
G.
Csányi
,
S.
Goedecker
, and
T. A.
Arias
,
Phys. Rev. A
65
,
032510
(
2002
).
41.
O.
Gritsenko
,
K.
Pernal
, and
E.
Baerends
,
J. Chem. Phys.
122
,
204102
(
2005
).
42.

Like UGU, these expressions are not invariant to unitary transformations among NOs with degenerate occupation numbers.

43.
M. A. L.
Marques
and
N. N.
Lathiotakis
,
Phys. Rev. A
77
,
032509
(
2008
).
44.

This expression shares the aforementioned invariance problem with UGU, UBBC1, and UBBC2.

45.
J.
Cioslowski
and
K.
Pernal
,
J. Chem. Phys.
111
,
3396
(
1999
).
46.
J.
Cioslowski
and
K.
Pernal
,
Phys. Rev. A
61
,
034503
(
2000
).
47.
S.
Sharma
,
J. K.
Dewhurst
,
N. N.
Lathiotakis
, and
E. K. U.
Gross
,
Phys. Rev. B
78
,
201103
(
2008
).
48.
E. N.
Zarkadoula
,
S.
Sharma
,
J. K.
Dewhurst
,
E. K. U.
Gross
, and
N. N.
Lathiotakis
,
Phys. Rev. A
85
,
032504
(
2012
).
49.
N.
Helbig
,
G.
Theodorakopoulos
, and
N. N.
Lathiotakis
,
J. Chem. Phys.
135
,
054109
(
2011
).
50.
N. N.
Lathiotakis
and
M. A. L.
Marques
,
J. Chem. Phys.
128
,
184103
(
2008
).
51.
N. N.
Lathiotakis
,
N.
Helbig
, and
E. K. U.
Gross
,
Phys. Rev. A
72
,
030501
(
2005
).
52.
Such functionals are generalizations of their “JK-only” counterparts defined in
J.
Cioslowski
,
K.
Pernal
, and
M.
Buchowiecki
,
J. Chem. Phys.
119
,
6443
(
2003
).
53.
M.
Piris
,
Int. J. Quantum Chem.
106
,
1093
-
1104
(
2006
).
54.
M.
Piris
,
J. M.
Matxain
,
X.
Lopez
, and
J. M.
Ugalde
,
J. Chem. Phys.
131
,
021102
(
2009
).
55.
M.
Piris
,
X.
Lopez
, and
J. M.
Ugalde
,
J. Chem. Phys.
126
,
214103
(
2007
).
56.
M.
Piris
,
J. M.
Matxain
,
X.
Lopez
, and
J. M.
Ugalde
,
J. Chem. Phys.
132
,
031103
(
2010
).
57.
M.
Piris
,
J. M.
Matxain
,
X.
Lopez
, and
J. M.
Ugalde
,
J. Chem. Phys.
133
,
111101
(
2010
).
58.
M.
Piris
,
J. Chem. Phys.
141
,
044107
(
2014
).
59.
M.
Piris
,
X.
Lopez
,
F.
Ruiperez
,
J. M.
Matxain
, and
J. M.
Ugalde
,
J. Chem. Phys.
134
,
164102
(
2011
).
60.
J.
Cioslowski
,
Theor. Chem. Acc.
134
,
113
(
2015
).
61.

This is a particular case of a general property of the “JKL-only” functionals.62 

62.
J.
Cioslowski
(unpublished).
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