We suggest new strict constraints that the two-particle cumulant matrix should fulfill. The constraints are obtained from the decomposition of

$\langle \hat{S}^{\,2}\rangle$
⟨Ŝ2⟩⁠, previously developed in our laboratory, and the vanishing number of electrons shared by two non-interacting fragments. The conditions impose stringent constraints into the cumulant structure without any need to perform an orbital optimization procedure thus carrying very small or no computational effort. These constraints are tested on the series of Piris natural orbital functionals (PNOF), which are among the most accurate ones available in the literature. Interestingly, even though all PNOF cumulants ensure correct overall
$\langle \hat{S}^{\,2}\rangle$
⟨Ŝ2⟩ values, none of them is consistent with the local spin structure of systems that dissociate more than one pair of electrons. A careful analysis of the local spin components reveals the most important missing contributions in the cumulant expression thus suggesting a means to improve PNOF5. The constraints provide an inexpensive tool for the construction and testing of cumulant structures that complement previously known conditions such as the N-representability or the square of the total spin angular momentum,
$\langle \hat{S}^{\,2}\rangle$
⟨Ŝ2⟩⁠.

1.
M.
 
Piris
,
Int. J. Quantum Chem.
 
113
,
620
(
2013
).
https://doi.org/10.1002/qua.24020
Crossref
Search ADS
 
2.
W.
 
Kutzelnigg
,
J. Chem. Phys.
 
125
,
171101
(
2006
).
https://doi.org/10.1063/1.2387955
Crossref
Search ADS
PubMed
 
3.
E. V.
 
Ludeña
,
F. J.
 
Torres
, and
C.
 
Costa
,
J. Mod. Phys.
 
4
,
391
(
2013
).
https://doi.org/10.4236/jmp.2013.43A055
Crossref
Search ADS
 
4.
D. A.
 
Mazziotti
,
Phys. Rev. Lett.
 
108
,
263002
(
2012
).
https://doi.org/10.1103/PhysRevLett.108.263002
Crossref
Search ADS
PubMed
 
5.
J.
 
Cioslowski
,
J. Chem. Phys.
 
123
,
164106
(
2005
).
https://doi.org/10.1063/1.2074527
Crossref
Search ADS
PubMed
 
6.
V. N.
 
Staroverov
 and
E. R.
 
Davidson
,
Int. J. Quantum Chem.
 
77
,
651
(
2000
).
https://doi.org/10.1002/(SICI)1097-461X(2000)77:3<651::AID-QUA6>3.0.CO;2-N
Crossref
Search ADS
 
7.
E.
 
Ramos-Cordoba
,
E.
 
Matito
,
I.
 
Mayer
, and
P.
 
Salvador
,
J. Chem. Theory Comput.
 
8
,
1270
(
2012
).
https://doi.org/10.1021/ct300050c
Crossref
Search ADS
PubMed
 
8.
M.
 
Piris
,
J. M.
 
Matxain
,
X.
 
Lopez
, and
J. M.
 
Ugalde
,
J. Chem. Phys.
 
131
,
021102
(
2009
).
https://doi.org/10.1063/1.3180958
Crossref
Search ADS
PubMed
 
9.
M.
 
Piris
,
Int. J. Quantum Chem.
 
106
,
1093
(
2006
).
https://doi.org/10.1002/qua.20858
Crossref
Search ADS
 
10.
M.
 
Piris
,
X.
 
Lopez
, and
J. M.
 
Ugalde
,
J. Chem. Phys.
 
126
,
214103
(
2007
).
https://doi.org/10.1063/1.2743019
Crossref
Search ADS
PubMed
 
11.
M.
 
Piris
,
J. M.
 
Matxain
,
X.
 
Lopez
, and
J. M.
 
Ugalde
,
J. Chem. Phys.
 
132
,
031103
(
2010
).
https://doi.org/10.1063/1.3298694
Crossref
Search ADS
PubMed
 
12.
M.
 
Piris
,
J. M.
 
Matxain
,
X.
 
Lopez
, and
J. M.
 
Ugalde
,
J. Chem. Phys.
 
133
,
111101
(
2010
).
https://doi.org/10.1063/1.3481578
Crossref
Search ADS
PubMed
 
13.
M.
 
Piris
,
X.
 
Lopez
,
F.
 
Ruipérez
,
J. M.
 
Matxain
, and
J. M.
 
Ugalde
,
J. Chem. Phys.
 
134
,
164102
(
2011
).
https://doi.org/10.1063/1.3582792
Crossref
Search ADS
PubMed
 
14.
Please note that on Piris' notation assumes Löwdin's normalization for 2-RDM,
$({{N} \primfrac{atop}{2}})$
(N2)⁠, while we use McWeeny's, N(N − 1).
15.
M.
 
Piris
,
J.
 
Matxain
, and
X.
 
Lopez
,
J. Chem. Phys.
 
139
,
234109
(
2013
).
https://doi.org/10.1063/1.4844075
Crossref
Search ADS
PubMed
 
16.
K.
 
Pernal
,
Comput. Theor. Chem.
 
1003
,
127
(
2013
).
https://doi.org/10.1016/j.comptc.2012.08.022
Crossref
Search ADS
 
17.
M.
 
Piris
,
J. Chem. Phys.
 
139
,
064111
(
2013
).
https://doi.org/10.1063/1.4817946
Crossref
Search ADS
PubMed
 
18.
P. R.
 
Surján
,
Á.
 
Szabados
,
P.
 
Jeszenszki
, and
T.
 
Zoboki
,
J. Math. Chem.
 
50
,
534
(
2012
).
https://doi.org/10.1007/s10910-011-9849-9
Crossref
Search ADS
 
19.
P.
 
Jeszenszki
,
V.
 
Rassolov
,
P. R.
 
Surján
, and
A.
 
Szabados
, “
Local spin from strongly orthogonal geminal wavefunctions
,”
Mol. Phys.
(published online).
https://doi.org/10.1080/00268976.2014.936919
20.
A. E.
 
Clark
 and
E. R.
 
Davidson
,
J. Chem. Phys.
 
115
,
7382
(
2001
).
https://doi.org/10.1063/1.1407276
Crossref
Search ADS
 
21.
I.
 
Mayer
,
Chem. Phys. Lett.
 
478
,
323
(
2009
).
https://doi.org/10.1016/j.cplett.2009.07.087
Crossref
Search ADS
 
22.
I.
 
Mayer
 and
E.
 
Matito
,
Phys. Chem. Chem. Phys.
 
12
,
11308
(
2010
).
https://doi.org/10.1039/c004427j
Crossref
Search ADS
PubMed
 
23.
D. R.
 
Alcoba
,
A.
 
Torre
,
L.
 
Lain
, and
R. C.
 
Bochicchio
,
J. Chem. Theory Comput.
 
7
,
3560
(
2011
).
https://doi.org/10.1021/ct200594f
Crossref
Search ADS
PubMed
 
24.
E.
 
Ramos-Cordoba
,
E.
 
Matito
,
P.
 
Salvador
, and
I.
 
Mayer
,
Phys. Chem. Chem. Phys.
 
14
,
15291
(
2012
).
https://doi.org/10.1039/c2cp42513k
Crossref
Search ADS
PubMed
 
25.
X.
 
Fradera
,
M. A.
 
Austen
, and
R. F. W.
 
Bader
,
J. Phys. Chem. A
 
103
,
304
(
1999
).
https://doi.org/10.1021/jp983362q
Crossref
Search ADS
 
26.
E.
 
Matito
,
M.
 
Solà
,
P.
 
Salvador
, and
M.
 
Duran
,
Faraday Discuss.
 
135
,
325
(
2007
).
https://doi.org/10.1039/b605086g
Crossref
Search ADS
PubMed
 
27.
R. F. W.
 
Bader
 and
M. E.
 
Stephens
,
Chem. Phys. Lett.
 
26
,
445
(
1974
).
https://doi.org/10.1016/0009-2614(74)89069-X
Crossref
Search ADS
 
28.
R. F. W.
 
Bader
 and
M. E.
 
Stephens
,
J. Am. Chem. Soc.
 
97
,
7391
(
1975
).
https://doi.org/10.1021/ja00859a001
Crossref
Search ADS
 
29.
M.
García-Revilla
 ,
E.
Francisco
 ,
A.
Costales
 , and
A.
Martín Pendás
 ,
J. Phys. Chem. A
 
116
,
1237
(
2012
);
https://doi.org/10.1021/jp204001n
Crossref
Search ADS
[PubMed]
M.
 
García-Revilla
,
E.
 
Francisco
,
A.
 
Costales
, and
A.
 
Martín Pendás
,
J. Phys. Chem. A
 
116
,
9216
(
2012
).
https://doi.org/10.1021/jp3074442
Crossref
Search ADS
30.
The inclusion of these terms is not as straightforward as it might seem because one should impose simultaneously additional cumulant constraints such as the antisymmetry. For instance, in the case of
$\Gamma _{i \tilde{j};j \tilde{i}}$
Γij̃;jĩ⁠, the same formulation does not automatically attain the antisymmetry condition.
© 2014 AIP Publishing LLC.
2014
AIP Publishing LLC
You do not currently have access to this content.