The spin-restricted ensemble-referenced Kohn-Sham (REKS) method is based on an ensemble representation of the density and is capable of correctly describing the non-dynamic electron correlation stemming from (near-)degeneracy of several electronic configurations. The existing REKS methodology describes systems with two electrons in two fractionally occupied orbitals. In this work, the REKS methodology is extended to treat systems with four fractionally occupied orbitals accommodating four electrons and self-consistent implementation of the REKS(4,4) method with simultaneous optimization of the orbitals and their fractional occupation numbers is reported. The new method is applied to a number of molecular systems where simultaneous dissociation of several chemical bonds takes place, as well as to the singlet ground states of organic tetraradicals 2,4-didehydrometaxylylene and 1,4,6,9-spiro[4.4]nonatetrayl.

1.
D.
Cremer
,
M.
Filatov
,
V.
Polo
,
E.
Kraka
, and
S.
Shaik
,
Int. J. Mol. Sci.
3
,
604
(
2002
).
2.
W.
Kohn
and
L. J.
Sham
,
Phys. Rev.
140
,
A1133
(
1965
).
3.
P.
Hohenberg
and
W.
Kohn
,
Phys. Rev.
136
,
B864
(
1964
).
4.
L.
Noodleman
,
J. Chem. Phys.
74
,
5737
(
1981
).
5.
L.
Noodleman
and
E. R.
Davidson
,
Chem. Phys.
109
,
131
(
1986
).
6.
F.
Illas
,
I. d. P. R.
Moreira
,
J. M.
Bofill
, and
M.
Filatov
,
Phys. Rev. B
70
,
132414
(
2004
).
7.
J. L.
Sonnenberg
,
H. B.
Schlegel
, and
H. P.
Hratchian
, “
Spin contamination in inorganic chemistry calculations
,” in
Encyclopedia of Inorganic and Bioinorganic Chemistry
(
John Wiley & Sons, Ltd.
,
2011
).
8.
R.
Costa
,
R.
Valero
,
D. R.
Mañeru
,
I. de P. R.
Moreira
, and
F.
Illas
,
J. Chem. Theory Comput.
11
,
1006
(
2015
).
9.
S. M.
Valone
,
J. Chem. Phys.
73
,
4653
(
1980
).
10.
J. P.
Perdew
,
R. G.
Parr
,
M.
Levy
, and
J. L.
Balduz
, Jr.
,
Phys. Rev. Lett.
49
,
1691
(
1982
).
11.
E. H.
Lieb
,
Int. J. Quantum Chem.
24
,
243
(
1983
).
12.
H.
Englisch
and
R.
Englisch
,
Phys. Status Solidi B
123
,
711
(
1984
);
H.
Englisch
and
R.
Englisch
,
Phys. Status Solidi B
124
,
373
(
1984
).
13.
C. A.
Ullrich
and
W.
Kohn
,
Phys. Rev. Lett.
87
,
093001
(
2001
).
14.
R.
van Leeuwen
,
Adv. Quantum Chem.
43
,
25
(
2003
).
15.
M.
Filatov
and
S.
Shaik
,
Chem. Phys. Lett.
304
,
429
(
1999
).
16.
I. d. P. R.
Moreira
,
R.
Costa
,
M.
Filatov
, and
F.
Illas
,
J. Chem. Theory Comput.
3
,
764
(
2007
).
17.
M.
Filatov
,
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
5
,
146
(
2015
).
18.
M.
Filatov
,
T. J.
Martínez
, and
K. S.
Kim
,
Phys. Chem. Chem. Phys.
18
,
21040
(
2016
).
19.
R. C.
Ladner
and
W. A.
Goddard
 III
,
J. Chem. Phys.
51
,
1073
(
1969
).
20.
W. A.
Goddard
 III
,
T. H.
Dunning
, Jr.
,
W. J.
Hunt
, and
P. J.
Hay
,
Acc. Chem. Res.
6
,
368
(
1973
).
21.
F. W.
Bobrowicz
and
W. A.
Goddard
 III
, in
Methods of Electronic Structure Theory
, Modern Theoretical Chemistry Vol. 3, edited by
H. F.
Schaefer
(
Springer
,
New York
,
1977
), pp.
79
127
.
22.
T. H.
Dunning
, Jr.
,
L. T.
Xu
,
T. Y.
Takeshita
, and
B. A.
Lindquist
,
J. Phys. Chem. A
120
,
1763
(
2016
).
23.
F.
Faglioni
and
W. A.
Goddard
 III
,
Int. J. Quantum Chem.
73
,
1
(
1999
).
24.
A.
Voter
and
W. A.
Goddard
 III
,
J. Chem. Phys.
75
,
3638
(
1981
).
25.
M.
Filatov
,
Top. Curr. Chem.
368
,
97
124
(
2016
).
26.
M.
Filatov
,
M.
Huix-Rotllant
, and
I.
Burghardt
,
J. Chem. Phys.
142
,
184104
(
2015
).
27.

As a number of symmetry relations, e.g., ρ5α=ρ6β holds for the microstate densities in Eq. (2), only 8 microstates out of 12 need to be calculated explicitly.

28.
K. J. H.
Giesbertz
and
E.-J.
Baerends
,
J. Chem. Phys.
132
,
194108
(
2010
).
29.
P.
Noziéres
and
D.
Pines
,
The Theory of Quantum Liquids
(
Perseus Books Publishing LLC
,
Cambridge, MA
,
1966
), pp.
296
298
.
30.
J.
Harris
and
R. O.
Jones
,
J. Phys. F: Metal Phys.
4
,
1170
(
1974
).
31.
A. D.
Becke
,
J. Chem. Phys.
88
,
1053
(
1988
).
32.

In the limit of non-interacting particles, i.e., λ=0, Eq. (1) becomes equivalent to the exact ensemble energy as first given by Perdew et al.10 and Lieb.11 As demonstrated in Ref. 18 (see also Refs. 25 and 26), the energy expression obtained for λ0 differs from Eq. (1) only in that there occur single determinant (sd) energies Esd[ρLα,ρLβ] of the usual wavefunction theory with the electron-electron interaction scaled down by the factor λ. As the density matrices ρLσ, σ=α,β are idempotent, integrating these energies with respect to λ, while constraining the density by external potential,25,26 results in the single determinant density functional energies EDFT[ρLα,ρLβ] occurring in the standard KS theory. In the practical application of Eq. (1), the energies EDFT[ρLα,ρLβ] are calculated by suitable density functional approximations.

33.
K.
Hirao
and
H.
Nakatsuji
,
J. Chem. Phys.
59
,
1457
(
1973
).
34.
H.
Hellmann
,
Einführung in die Quantenchemie
(
Franz Deuticke
,
Leipzig
,
1937
), p.
285
;
R. P.
Feynman
,
Phys. Rev.
56
,
340
(
1939
).
35.
I. S.
Ufimtsev
and
T. J.
Martínez
,
J. Chem. Theory Comput.
5
,
2619
(
2009
).
36.
A. V.
Titov
,
I. S.
Ufimtsev
,
N.
Luehr
, and
T. J.
Martínez
,
J. Chem. Theory Comput.
9
,
213
(
2013
).
37.
C.
Song
,
L.-P.
Wang
, and
T. J.
Martínez
,
J. Chem. Theory Comput.
12
,
92
(
2016
).
38.
T. H.
Dunning
, Jr.
,
J. Chem. Phys.
90
,
1007
(
1989
).
39.
R.
Krishnan
,
J. S.
Binkley
,
R.
Seeger
, and
J. A.
Pople
,
J. Chem. Phys.
72
,
650
(
1980
).
40.
A. D.
Becke
,
Phys. Rev. A
38
,
3098
(
1988
);
A. D.
Becke
,
J. Chem. Phys.
98
,
1372
(
1993
);
C.
Lee
,
W.
Yang
, and
R. G.
Parr
,
Phys. Rev. B
37
,
785
(
1988
).
41.
N.
Luehr
,
I. S.
Ufimtsev
, and
T. J.
Martínez
,
J. Chem. Theory Comput.
7
,
949
(
2011
).
42.
M.
Lundberg
and
P. E.
Siegbahn
,
J. Chem. Phys.
122
,
224103
(
2005
).
43.
E. A.
Carter
and
W. A.
Goddard
 III
,
J. Chem. Phys.
88
,
3132
(
1987
).
44.
H.
Kollmar
and
V.
Staemmler
,
J. Am. Chem. Soc.
99
,
3583
(
1977
).
45.
P.
Karafiloglou
,
J. Chem. Educ.
66
,
816
(
1989
).
46.
D.
Casanova
,
L. V.
Slipchenko
,
A. I.
Krylov
, and
M.
Head-Gordon
,
J. Chem. Phys.
130
,
044103
(
2009
).
47.
W. C.
Lineberger
and
W. T.
Borden
,
Phys. Chem. Chem. Phys.
13
,
11792
(
2011
).
48.
L.
McElwee-White
and
D. A.
Dougherty
,
J. Am. Chem. Soc.
106
,
3466
(
1984
).
49.
L.
McElwee-White
,
W. A.
Goddard
 III
, and
D. A.
Dougherty
,
J. Am. Chem. Soc.
106
,
3461
(
1984
).
50.
S. O.
Odoh
,
G. L.
Manni
,
R. K.
Carlson
,
D. G.
Truhlar
, and
L.
Gagliardi
,
Chem. Sci.
7
,
2399
(
2016
).
51.
J.
Nafziger
and
A.
Wasserman
,
J. Chem. Phys.
143
,
234105
(
2015
).
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