Amount of information present in the one-particle density matrix and the charge density
Juan Carlos Ramı
´
rez, Julio Manuel Herna
´
ndez Pe
´
rez, Robin P. Sagar, and Rodolfo O. Esquivel
Departamento de Quı ´mica, Universidad Auto ´noma Metropolitana, Apartado Postal 55-534, Iztapalapa,
09340 Me ´xico Distrito Federal, Mexico
Minhhuy Ho
ˆ
and Vedene H. Smith, Jr.
Department of Chemistry, Queen’s University, Kingston, Ontario, Canada K7L 3N6
Received 5 September 1997; revised manuscript received 20 February 1998
The amount of statistical information present in the one-particle density matrix and its diagonal, the charge
density, is probed by comparison of the Jaynes and Shannon entropies for molecular systems. Numerical
results show that these information entropies are not equal but that there exists a correspondence between them.
The entropies are also compared among members of isoelectronic groups, and it is shown that these quantities
are sensitive to the chemical environment and may be used as measures of the degree of electron delocalization
or localization within a chemical system. S1050-29479801210-4
PACS numbers: 31.15.Ew, 31.30.Jv
I. INTRODUCTION
The one-particle reduced density matrix ODM of an
N -particle system may be defined in terms of the N -particle
wave function as
1;1 ' =N
1,2, . . . , N * 1 ' ,2, . . . , N d 2 ••• dN ,
1.1
where the index denotes a combined spin and spatial coordi-
nate of a particular electron. This quantity is of fundamental
importance in quantum chemistry. Through it, all one-
electron properties of the system may be calculated.
In order to calculate the energy and other two electron
properties, one needs the second-order reduced density ma-
trix or 2-matrix,
1,2;1 ' ,2 ' =
N
2
1,2, . . . , N
* 1 ' ,2' ,..., N d 3 ••• dN . 1.2
Clearly is related to by a simple reduction of variables;
that is, one can integrate over d 2 in to obtain . Also,
within the Hartree-Fock HF approximation, the energy as
well as other two electron properties may be formulated in
terms of the ODM.
The ODM may be expanded in a set of orthonormal spin
orbitals
i
1;1 ' =
ij
c
ij
i
1
j
* 1 ' , 1.3
and then diagonalized to yield
1;1 ' =
j
v
j
j
1
j
* 1 ' , 1.4
where the eigenfunctions
j
are the natural spin orbitals and
the eigenvalues v
j
are the occupation numbers. In order to be
N representable, the ODM should have occupation numbers
that sum to N and lie in the range 0,11. Within the
independent-particle model, these occupation numbers are
either 0 or 1, which identifies (1;1 ' ) as idempotent.
One may also consider an integration over the spin vari-
ables of the ODM to obtain a spin-free charge-density matrix
CDM
r ; r ' =
1;1 '
s
1
=s
1
'
ds
1
, 1.5
where r is the spatial coordinate and s
1
is a coordinate that
represents the spin variable, both of electron 1. There is a
similar expansion of ( r ; r ' ) to Eq. 1.4, namely,
r ; r ' =
i
n
i
i
r
i
* r ' . 1.6
It is the diagonal of ( r ; r ' ) that corresponds to the charge
density ( r ),
r =
i
n
i
|
i
r |
2
, 1.7
where n
i
are the occupation numbers that lie in the range
0,2, of the natural orbitals
i
.
Furthermore, the ODM is also important because it pro-
vides a connection to other representations, among them the
momentum space one. The ODM in position space is con-
nected to the ODM in momentum space
˜
(1;1 ' ) by a Dirac-
Fourier transform 2
˜
1;1 ' = 2
-3
1;1 ' exp -ı p • r - p ' • r ' dr dr ' .
1.8
Although these two ODM’s are connected by this Fourier
transformation, there is no rigorous mathematical relation-
ship between ( r ) and its counterpart in momentum space
( p ) 3, since both densities overlook the nonlocal behav-
ior in the respective spaces. The Dirac-Fourier relationship
PHYSICAL REVIEW A NOVEMBER 1998 VOLUME 58, NUMBER 5
PRA 58 1050-2947/98/585/35079/$15.00 3507 ©1998 The American Physical Society