Theor Chem Acc (2015) 134:138
1 3
where p D
i 1 ,i 2 ,...,i p
j 1 ,j 2 ,...,j p
(( M
k ) =
M
p
ˆ
L M
p { M D M
k
} stands for the
p -RDM associated with the | M
k kth accessible M-particle
pure state of the system. The physical meaning of this operation is nothing but an averaging process over the remaining M − p variables [ 45 ].
As stated above, any physical system featured by a
non-integer number of particles N cannot be described
by any other state than the GC. Therefore, a CM to take
into account properly the M-particle different states in D
defi ned by Eq. ( 2 ) in Fock space to calculate the p -RDMs
marginal distributions may be introduced. This expression
has been obtained recently [ 13 ] as
Equation ( 14 ) is the defi nition for the GC CM ˆ
L p and permits to note that it involves several pure states M D M
k
with
the condition that the number of particles was M ≥ p , i.e.,
the order of contraction p must be less than or equal to M
and all states in the mixture not lying in this interval, i.e.,
M < p , will not contribute to the GC distribution, while for
M = p , no action is needed [ 13 ]. These mathematical conditions are expressed by
and
with I and O , the identity and null superoperators, respectively [ 13 ]. These requirements complete the defi nition for
the CM in Fock space, and the p - RDM s may be expressed
by the expansion,
It is worthy to note that the trace operation calculated by
Tr( p D) =
{ M
k ,M≥p} ω M
k
M
p
= =
M
p
is the number of p-ons number in the system as an average which
is noted by the symbol . . . . In particular, for p = 1,
Tr
1 D
=
{ M
k ,M≥1} ω M
k
M = =M , is the number of particles expressed by the non-integer number
M = N + ν mentioned above.
To fi nish this section, let us mention some important
consequences coming from the marginal distributions in
the GC structure of the density matrices which we will not
treat in this work. As noted in this section, within the C
and MC states, any q D may be obtained from other matrix
p D with q < p by a contraction operation [ 26 ]. The same
(14)
p D = ˆ
L p {D} =
{ M
k ,M≥p}
ω M
k
M
p
ˆ
L
M
p {
M D M
k
}
ˆ
L
M
p {
M D M
k
} = O M < p
ˆ
L
p
p {
p D
p
k
} = I
p D
p
k
=
p D
p
k
(15)
p D =
{ M
k ,M≥p}
ω M
k
p D M
k
is not true within the GC distribution [cf. Eq. ( 2 )] without losing some information and hence any matrix may
be only obtained directly by contraction of D [ 13 ]. Nevertheless, for the case of our interest in which the energy
has a convex structure [cf. Eq. ( 6 )], no information is lost,
except for the case in which q = N and p = N + 1 [ 13 ].
The other consequence we want to mention is that for
a closed atomic and molecular systems, the energy is a
functional of the second-order reduced density matrix
as E
N
0 = Tr
2 D N
o
2 K N
, where 2 K N stands for the Coleman reduced Hamiltonian, 2 D N
o the ground state secondorder reduced density matrix with a supra-index N which
indicates that the 2-RDM comes from contraction of a
N-particle DM [ 26 , 38 ]. In contrast, for an open atomic or
molecular system, the energy cannot be expressed similarly as a functional of the corresponding 2 D N+ν
o
but as
E
N+ν
0
= νTr
2 D N+1
o
2 K N+1
+ (1 − ν) Tr
2 D N
o
2 K N
,
namely the energy is a functional F of 2 D N
o , 2 D N+1
o
and the fractional population number ν , i.e.,
E
N+ν
0
= F( 2 D N+1
o
, 2 D N
o , ν) [ 13 , 46 ].
3 Chemical descriptors: system–environment
interactions and derivative discontinuities
The higher chemical descriptors are derivatives of the
energy or of the electron density with respect of the number
of particles N [ 3 ]. They are related to the concept of reactivity interpreted as a response function to proper chemical interactions [ 24 ]. Joint together with the zero-order
descriptors, i.e., energy and functions of the density itself,
provides the complete and detailed description of a molecular system and its intra- (with a solvent, a reservoir, etc.)
and inner-interactions (between different fragments in the
molecule). So that it imposes the knowledge of these magnitudes, E N
0 and ρ N (or more generally the associated DM
from which ρ N is obtained), dependence with N [ 5 , 12 ,
13 ]. The common use of the method of fi nite differences
with respect to integer number of the particles of isolated
species to evaluate the derivatives [ 3 ] neglects their values
at non-integer numbers [ 24 ] and consequently the true electron exchange between molecular subsystems which constitutes the onset of chemical behavior.
For ground states, the dependence of the energy E N
0
and the DMs is a piecewise-continuous linear functions of
N and only the closed systems with integers N and N ± 1
enter in this ensemble as stated by Eqs. ( 6 ) and ( 7 ), respectively [ 12 ]. Hence, all ground state properties then have
similar dependence, and the fi rst derivatives of the energy
and the density are staircase functions of N , undefi ned at
the integers and constant in between [ 24 ] leading to the
second derivatives to vanish in between integers [ 4 , 24 ].
The consequences of this dependence are nonphysical,
93
Reprinted from the journal
1 3
where p D
i 1 ,i 2 ,...,i p
j 1 ,j 2 ,...,j p
(( M
k ) =
M
p
ˆ
L M
p { M D M
k
} stands for the
p -RDM associated with the | M
k kth accessible M-particle
pure state of the system. The physical meaning of this operation is nothing but an averaging process over the remaining M − p variables [ 45 ].
As stated above, any physical system featured by a
non-integer number of particles N cannot be described
by any other state than the GC. Therefore, a CM to take
into account properly the M-particle different states in D
defi ned by Eq. ( 2 ) in Fock space to calculate the p -RDMs
marginal distributions may be introduced. This expression
has been obtained recently [ 13 ] as
Equation ( 14 ) is the defi nition for the GC CM ˆ
L p and permits to note that it involves several pure states M D M
k
with
the condition that the number of particles was M ≥ p , i.e.,
the order of contraction p must be less than or equal to M
and all states in the mixture not lying in this interval, i.e.,
M < p , will not contribute to the GC distribution, while for
M = p , no action is needed [ 13 ]. These mathematical conditions are expressed by
and
with I and O , the identity and null superoperators, respectively [ 13 ]. These requirements complete the defi nition for
the CM in Fock space, and the p - RDM s may be expressed
by the expansion,
It is worthy to note that the trace operation calculated by
Tr( p D) =
{ M
k ,M≥p} ω M
k
M
p
= =
M
p
is the number of p-ons number in the system as an average which
is noted by the symbol . . . . In particular, for p = 1,
Tr
1 D
=
{ M
k ,M≥1} ω M
k
M = =M , is the number of particles expressed by the non-integer number
M = N + ν mentioned above.
To fi nish this section, let us mention some important
consequences coming from the marginal distributions in
the GC structure of the density matrices which we will not
treat in this work. As noted in this section, within the C
and MC states, any q D may be obtained from other matrix
p D with q < p by a contraction operation [ 26 ]. The same
(14)
p D = ˆ
L p {D} =
{ M
k ,M≥p}
ω M
k
M
p
ˆ
L
M
p {
M D M
k
}
ˆ
L
M
p {
M D M
k
} = O M < p
ˆ
L
p
p {
p D
p
k
} = I
p D
p
k
=
p D
p
k
(15)
p D =
{ M
k ,M≥p}
ω M
k
p D M
k
is not true within the GC distribution [cf. Eq. ( 2 )] without losing some information and hence any matrix may
be only obtained directly by contraction of D [ 13 ]. Nevertheless, for the case of our interest in which the energy
has a convex structure [cf. Eq. ( 6 )], no information is lost,
except for the case in which q = N and p = N + 1 [ 13 ].
The other consequence we want to mention is that for
a closed atomic and molecular systems, the energy is a
functional of the second-order reduced density matrix
as E
N
0 = Tr
2 D N
o
2 K N
, where 2 K N stands for the Coleman reduced Hamiltonian, 2 D N
o the ground state secondorder reduced density matrix with a supra-index N which
indicates that the 2-RDM comes from contraction of a
N-particle DM [ 26 , 38 ]. In contrast, for an open atomic or
molecular system, the energy cannot be expressed similarly as a functional of the corresponding 2 D N+ν
o
but as
E
N+ν
0
= νTr
2 D N+1
o
2 K N+1
+ (1 − ν) Tr
2 D N
o
2 K N
,
namely the energy is a functional F of 2 D N
o , 2 D N+1
o
and the fractional population number ν , i.e.,
E
N+ν
0
= F( 2 D N+1
o
, 2 D N
o , ν) [ 13 , 46 ].
3 Chemical descriptors: system–environment
interactions and derivative discontinuities
The higher chemical descriptors are derivatives of the
energy or of the electron density with respect of the number
of particles N [ 3 ]. They are related to the concept of reactivity interpreted as a response function to proper chemical interactions [ 24 ]. Joint together with the zero-order
descriptors, i.e., energy and functions of the density itself,
provides the complete and detailed description of a molecular system and its intra- (with a solvent, a reservoir, etc.)
and inner-interactions (between different fragments in the
molecule). So that it imposes the knowledge of these magnitudes, E N
0 and ρ N (or more generally the associated DM
from which ρ N is obtained), dependence with N [ 5 , 12 ,
13 ]. The common use of the method of fi nite differences
with respect to integer number of the particles of isolated
species to evaluate the derivatives [ 3 ] neglects their values
at non-integer numbers [ 24 ] and consequently the true electron exchange between molecular subsystems which constitutes the onset of chemical behavior.
For ground states, the dependence of the energy E N
0
and the DMs is a piecewise-continuous linear functions of
N and only the closed systems with integers N and N ± 1
enter in this ensemble as stated by Eqs. ( 6 ) and ( 7 ), respectively [ 12 ]. Hence, all ground state properties then have
similar dependence, and the fi rst derivatives of the energy
and the density are staircase functions of N , undefi ned at
the integers and constant in between [ 24 ] leading to the
second derivatives to vanish in between integers [ 4 , 24 ].
The consequences of this dependence are nonphysical,
93
Reprinted from the journal