Electronic states with fractional spins arise in systems with large static correlation (strongly correlated systems). Such fractional-spin states are shown to be ensembles of degenerate ground states with normal spins. It is proven here that the energy of the exact functional for fractional-spin states is a constant, equal to the energy of the comprising degenerate pure-spin states. Dramatic deviations from this exact constancy condition exist with all approximate functionals, leading to large static correlation errors for strongly correlated systems, such as chemical bond dissociation and band structure of Mott insulators. This is demonstrated with numerical calculations for several molecular systems. Approximating the constancy behavior for fractional spins should be a major aim in functional constructions and should open the frontier for density functional theory to describe strongly correlated systems. The key results are also shown to apply in reduced density-matrix functional theory.

1.
W.
Kohn
and
L.
Sham
,
Phys. Rev.
140
, A
1133
(
1965
).
2.
R. G.
Parr
and
W.
Yang
,
Density-Functional Theory of Atoms and Molecules
(
Oxford University Press
,
New York
,
1989
).
3.
A. J.
Cohen
,
P.
Mori-Sánchez
, and
W.
Yang
,
Science
321
,
792
(
2008
).
4.
A.
Savin
, in
Recent Developments and Applications of Modern Density Functional Theory
, edited by
J. M.
Seminario
(
Elsevier
,
Amsterdam
,
1996
), p.
327
.
5.
E. J.
Baerends
,
Phys. Rev. Lett.
87
,
133004
(
2001
).
6.
A. D.
Becke
,
J. Chem. Phys.
122
,
064101
(
2005
).
7.
M.
Fuchs
,
Y. M.
Niquet
,
X.
Gonze
, and
K.
Burke
,
J. Chem. Phys.
122
,
094116
(
2005
).
8.
M. J. G.
Peach
,
A. M.
Teale
, and
D. J.
Tozer
,
J. Chem. Phys.
126
,
244104
(
2007
).
9.
P.
Fazekas
,
Lecture Notes on Electron Correlation and Magnetism
,
Modern Condensed Matter Physics
, Vol.
5
(
World Scientific
,
Singapore
,
1999
).
10.
J. K.
Perry
,
J. Phys. Chem. A
104
,
2438
(
2000
).
11.
J. P.
Perdew
,
R. G.
Parr
,
M.
Levy
, and
J. L.
Balduz
, Jr.
,
Phys. Rev. Lett.
49
,
1691
(
1982
).
12.
W.
Yang
,
Y.
Zhang
, and
P. W.
Ayers
,
Phys. Rev. Lett.
84
,
5172
(
2000
).
13.
P.
Mori-Sánchez
,
A. J.
Cohen
, and
W.
Yang
,
Phys. Rev. Lett.
100
,
146401
(
2008
).
14.
A. D.
Becke
,
Phys. Rev. A
38
,
3098
(
1988
).
15.
C.
Lee
,
W.
Yang
, and
R. G.
Parr
,
Phys. Rev. B
37
,
785
(
1988
).
16.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
,
Phys. Rev. Lett.
77
,
3865
(
1996
).
17.
A. D.
Becke
,
J. Chem. Phys.
98
,
5648
(
1993
).
18.
A.
Ruzsinszky
,
J. P.
Perdew
,
G. I.
Csonka
,
O. A.
Vydrov
, and
G. E.
Scuseria
,
J. Chem. Phys.
125
,
194112
(
2006
).
19.
P.
Mori-Sánchez
,
A. J.
Cohen
, and
W.
Yang
,
J. Chem. Phys.
125
,
201102
(
2006
).
20.
A. J.
Cohen
,
P.
Mori-Sánchez
, and
W.
Yang
,
J. Chem. Phys.
126
,
191109
(
2007
).
21.
A. J.
Cohen
,
P.
Mori-Sánchez
, and
W.
Yang
,
Phys. Rev. B
77
,
115123
(
2008
).
22.
See EPAPS Document No. E-JCPSA6-129-008837 for details of the derivation of Eq. (5) and additional fractional-spin calculations on the H atom. For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html.
23.
J. P.
Perdew
,
A.
Ruzsinszky
,
G. I.
Csonka
,
O. A.
Vydrov
,
G. E.
Scuseria
,
V. N.
Staroverov
, and
J.
Tao
,
Phys. Rev. A
76
,
040501
R (
2007
).

Supplementary Material

You do not currently have access to this content.