One can partition the molecular density into its atomic contributions by minimizing the divergence of the atom-in-molecule densities from their corresponding reference pro-atomic densities, subject to the constraint that the sum of the atom-in-molecule densities is the total molecular density. We expose conditions on the divergence measure that are necessary, and sufficient, to recover the popular Hirshfeld partitioning. Specifically, among all local measures of the divergence between two probability distribution functions, the Hirshfeld partitioning is obtained only for f-divergences.
REFERENCES
1.
J.
Rychlewski
and R. G.
Parr
, J. Chem. Phys.
84
, 1696
(1986
). 2.
P.
Hohenberg
and W.
Kohn
, Phys. Rev.
136
, B864
(1964
). 3.
M. H.
Cohen
and A.
Wasserman
, J. Phys. Chem. A
111
, 2229
(2007
). 4.
P.
Elliott
, M. H.
Cohen
, A.
Wasserman
, and K.
Burke
, J. Chem. Theory Comput.
5
, 827
(2009
). 5.
R. G.
Tang
, J.
Nafziger
, and A.
Wasserman
, Phys. Chem. Chem. Phys.
14
, 7780
(2012
). 6.
Y.
Zhang
and A.
Wasserman
, J. Chem. Theory Comput.
6
, 3312
(2010
). 7.
8.
S.
Kullback
and R. A.
Leibler
, Ann. Math. Stat.
22
, 79
(1951
). 9.
R. F.
Nalewajski
and R. G.
Parr
, Proc. Natl. Acad. Sci. U.S.A.
97
, 8879
(2000
). 10.
R. F.
Nalewajski
and R. G.
Parr
, J. Phys. Chem. A
105
, 7391
(2001
). 11.
F. L.
Hirshfeld
, Theor. Chim. Acta
44
, 129
(1977
). 12.
F.
Heidar-Zadeh
, P. W.
Ayers
, and P.
Bultinck
, J. Chem. Phys.
141
, 094103
(2014
). 13.
P.
Bultinck
, C.
Van Alsenoy
, P. W.
Ayers
, and R.
Carbó-Dorca
, J. Chem. Phys.
126
, 144111
(2007
). 14.
D.
Ghillemijn
, P.
Bultinck
, D.
Van Neck
, and P. W.
Ayers
, J. Comput. Chem.
32
, 1561
(2011
). 15.
T.
Verstraelen
, P. W.
Ayers
, V.
Van Speybroeck
, and M.
Waroquier
, J. Chem. Theory Comput.
9
, 2221
(2013
). 16.
T. C.
Lillestolen
and R. J.
Wheatley
, Chem. Commun.
(45
), 5909
(2008
). 17.
T.
Verstraelen
, P. W.
Ayers
, V.
Van Speybroeck
, and M.
Waroquier
, Chem. Phys. Lett.
545
, 138
(2012
). 18.
T. A.
Manz
and D. S.
Sholl
, J. Chem. Theory Comput.
8
, 2844
(2012
). 19.
T. A.
Manz
and D. S.
Sholl
, J. Chem. Theory Comput.
6
, 2455
(2010
). 20.
21.
R. G.
Parr
, P. W.
Ayers
, and R. F.
Nalewajski
, J. Phys. Chem. A
109
, 3957
(2005
). 22.
P. W.
Ayers
, Theor. Chem. Acc.
115
, 370
(2006
). 23.
S. M.
Ali
and S. D.
Silvey
, J. R. Stat. Soc. Ser. B
28
, 131
(1966
), http://www.jstor.org/stable/2984279.24.
25.
I.
Csiszár
, Magyar. Tud. Akad. Mat. Kutato Int. Kozl
8
, 85
(1963
).26.
Not every partitioning satisfies this requirement. For example, the Rychlewski-Parr partitioning (a.k.a. partition DFT) does not.
27.
F.
Liese
and I.
Vajda
, IEEE Trans. Inf. Theory
52
, 4394
(2006
). 28.
F.
Osterreicher
and I.
Vajda
, Ann. Inst. Stat. Math.
55
, 639
(2003
). 29.
M. M.
Deza
and E.
Deza
, Encyclopedia of Distances
, 2nd ed. (Springer-Verlag
, Berlin
, 2013
).© 2015 AIP Publishing LLC.
2015
AIP Publishing LLC
You do not currently have access to this content.