Theor Chem Acc (2015) 134:138
1 3
strength scale of the interaction [ 6 – 10 ]. Nevertheless,
the inadequacy of this model has been noted, and a linear
dependence within the density functional theory (DFT)
and also for the state function approach was proposed
as an attempt to obtain the right energy dependence
for non-integral electron number N and its differentiability [ 4 , 5 , 11 ]. Recently, a general proof for that proposal going beyond the DFT and the pure state function
approaches has been presented [ 12 ] under the hypothesis of the ground state energy convexity for atomic
and molecular systems driven by Coulombic interactions [ 4 , 5 , 12 , 13 ]. To understand the mechanism of
charge transfer mentioned above, the fragments within
the molecular structure or even a whole molecular system may be interpreted as open systems that exchange
electrons and energy between them and/or with a reservoir [ 4 , 14 , 15 ]. Therefore, it follows that a non-integer
electron number may arise as a time average caused by
the fl uctuating number of particles and thus the open
system need to be described by a statistical mixture or
ensemble of states with different number of particles [ 4 ,
6 , 8 , 12 – 14 ]. Regarding the dependence of each magnitude, i.e., energy and/or the density, with the number of
particles, two kinds of descriptors arise from them. The
zero-order descriptors are those which are integrated
functions of the magnitudes itself such as those describing the electronic distribution from electron populations
as atomic charges, covalent bond orders, valencies, free
valencies among others [ 16 , 17 ] or local indicators as
those coming from the topological approach like critical points of the density, their ellipticity or the Laplacian functions of the density, among others [ 18 , 19 ].
The other type are the so-called higher-order descriptors depending on the successive derivatives of these
magnitudes as for instance, the chemical potential, the
hardness, Fukui functions, etc. [ 3 ]. The behavior of the
energy, density or other properties for ensemble [ 4 , 12 ,
13 ] or even for pure states distributions [ 20 , 21 ] is of
fundamental importance for the latter type because of
the discontinuities they undergo at integer numbers [ 4 ].
So that the system defi nition is supported on the AIM
notion giving rise to atomistic models for molecules [ 18 ,
22 , 23 ] which permits to determine the concept of net
charge on an atom as the key variable for determining
its energy [ 4 , 9 , 12 ]. The energy and the density matrices
(DM) are piecewise-continuous linear functions of the
number of particles N [ 4 , 12 ], and consequently, its fi rst
derivatives are N -staircase functions being undefi ned at
the integers and constant in between [ 24 ]. So that, second derivatives vanish in between and are not defi ned
at the integers. Hence, descriptors like hardness vanish
[ 3 ]. This dependence has contradictory consequences
as for instance, the violation of the electronegativity
equalization principle [ 3 , 6 ] closely related to reactivities and hardness [ 3 ].
In Ref. [ 24 ], it is clearly noted that the formal N piecewise-continuous linear dependence of the physical magnitudes with the number of particles contains the essence of
the model for non-integer electron systems [ 4 , 12 ]. Thus,
admitting the onset of a more accurate reactivity theory
going beyond the mentioned inconsistencies, it must be
recognized that reactivity descriptors are chemical environment dependent and may not be defi ned for isolated species without considering a fragment and/or reservoir interaction, i.e., generally, system–reservoir (S–R) interactions
from which the species exchanges or transfers electrons
[ 24 ]. A formal approach which addresses the problem and
proposed a formal solution at an ensemble level can found
in Ref. [ 25 ].
The objective of this work is to introduce some recent
rigorous developments about the structure of the density
matrices (DM), i.e., the state of the system as an ensemble of pure states of different number of fi xed particles M
commonly called grand-canonical ensemble (GC), the calculation of the energy under the hypothesis of its convexity for ground state isolated molecular systems and their
extensions to systems under the infl uence of an environment interaction, i.e., a fragment or a reservoir [ 14 ]. In this
way, we attempt to obtain a solution based on the interaction between the subsystems (S–R) inducing a coherent
DM distribution which overcomes the inconsistencies mentioned above. Hence, the solution lies within the formal
structure of reactivity theory, and the second-type chemical
descriptors (second derivatives) are obtained in the natural
scenario of the GC [ 12 , 13 ] and the chemical context in
which the species exchanges or transfers electrons.
Finally, the marginal distributions of the GC DMs, i.e.,
p-RDMs of the non-coherent (isolated systems) [ 13 ] distributions of the molecular open systems, are calculated
by means of the contraction mappings [ 13 , 26 ] in order to
evaluate the properties as averages of the associated quantum observable. As an example, an explicit derivation and
generalization of the Fukui functions are shown as a fi rstorder descriptor of the density from this formalism without
using the fi nite difference methods. The article is organized
as follows. Section 2 presents the theoretical aspects introducing the defi nition, characterization and features of the
systems, the energy determination, their states (DM) and
marginal distributions, i.e., the reduced density matrices
p -RDMs in the GC ensemble. Also in this section, some
important properties for the open systems are sketched. In
Sect. 3 , the chemical descriptors of interest and the solution for the quantum state of the system in the framework
of the S–R interaction are presented to show the machinery in action. A fi nal Section is dedicated to the concluding
remarks.
90
Reprinted from the journal
1 3
strength scale of the interaction [ 6 – 10 ]. Nevertheless,
the inadequacy of this model has been noted, and a linear
dependence within the density functional theory (DFT)
and also for the state function approach was proposed
as an attempt to obtain the right energy dependence
for non-integral electron number N and its differentiability [ 4 , 5 , 11 ]. Recently, a general proof for that proposal going beyond the DFT and the pure state function
approaches has been presented [ 12 ] under the hypothesis of the ground state energy convexity for atomic
and molecular systems driven by Coulombic interactions [ 4 , 5 , 12 , 13 ]. To understand the mechanism of
charge transfer mentioned above, the fragments within
the molecular structure or even a whole molecular system may be interpreted as open systems that exchange
electrons and energy between them and/or with a reservoir [ 4 , 14 , 15 ]. Therefore, it follows that a non-integer
electron number may arise as a time average caused by
the fl uctuating number of particles and thus the open
system need to be described by a statistical mixture or
ensemble of states with different number of particles [ 4 ,
6 , 8 , 12 – 14 ]. Regarding the dependence of each magnitude, i.e., energy and/or the density, with the number of
particles, two kinds of descriptors arise from them. The
zero-order descriptors are those which are integrated
functions of the magnitudes itself such as those describing the electronic distribution from electron populations
as atomic charges, covalent bond orders, valencies, free
valencies among others [ 16 , 17 ] or local indicators as
those coming from the topological approach like critical points of the density, their ellipticity or the Laplacian functions of the density, among others [ 18 , 19 ].
The other type are the so-called higher-order descriptors depending on the successive derivatives of these
magnitudes as for instance, the chemical potential, the
hardness, Fukui functions, etc. [ 3 ]. The behavior of the
energy, density or other properties for ensemble [ 4 , 12 ,
13 ] or even for pure states distributions [ 20 , 21 ] is of
fundamental importance for the latter type because of
the discontinuities they undergo at integer numbers [ 4 ].
So that the system defi nition is supported on the AIM
notion giving rise to atomistic models for molecules [ 18 ,
22 , 23 ] which permits to determine the concept of net
charge on an atom as the key variable for determining
its energy [ 4 , 9 , 12 ]. The energy and the density matrices
(DM) are piecewise-continuous linear functions of the
number of particles N [ 4 , 12 ], and consequently, its fi rst
derivatives are N -staircase functions being undefi ned at
the integers and constant in between [ 24 ]. So that, second derivatives vanish in between and are not defi ned
at the integers. Hence, descriptors like hardness vanish
[ 3 ]. This dependence has contradictory consequences
as for instance, the violation of the electronegativity
equalization principle [ 3 , 6 ] closely related to reactivities and hardness [ 3 ].
In Ref. [ 24 ], it is clearly noted that the formal N piecewise-continuous linear dependence of the physical magnitudes with the number of particles contains the essence of
the model for non-integer electron systems [ 4 , 12 ]. Thus,
admitting the onset of a more accurate reactivity theory
going beyond the mentioned inconsistencies, it must be
recognized that reactivity descriptors are chemical environment dependent and may not be defi ned for isolated species without considering a fragment and/or reservoir interaction, i.e., generally, system–reservoir (S–R) interactions
from which the species exchanges or transfers electrons
[ 24 ]. A formal approach which addresses the problem and
proposed a formal solution at an ensemble level can found
in Ref. [ 25 ].
The objective of this work is to introduce some recent
rigorous developments about the structure of the density
matrices (DM), i.e., the state of the system as an ensemble of pure states of different number of fi xed particles M
commonly called grand-canonical ensemble (GC), the calculation of the energy under the hypothesis of its convexity for ground state isolated molecular systems and their
extensions to systems under the infl uence of an environment interaction, i.e., a fragment or a reservoir [ 14 ]. In this
way, we attempt to obtain a solution based on the interaction between the subsystems (S–R) inducing a coherent
DM distribution which overcomes the inconsistencies mentioned above. Hence, the solution lies within the formal
structure of reactivity theory, and the second-type chemical
descriptors (second derivatives) are obtained in the natural
scenario of the GC [ 12 , 13 ] and the chemical context in
which the species exchanges or transfers electrons.
Finally, the marginal distributions of the GC DMs, i.e.,
p-RDMs of the non-coherent (isolated systems) [ 13 ] distributions of the molecular open systems, are calculated
by means of the contraction mappings [ 13 , 26 ] in order to
evaluate the properties as averages of the associated quantum observable. As an example, an explicit derivation and
generalization of the Fukui functions are shown as a fi rstorder descriptor of the density from this formalism without
using the fi nite difference methods. The article is organized
as follows. Section 2 presents the theoretical aspects introducing the defi nition, characterization and features of the
systems, the energy determination, their states (DM) and
marginal distributions, i.e., the reduced density matrices
p -RDMs in the GC ensemble. Also in this section, some
important properties for the open systems are sketched. In
Sect. 3 , the chemical descriptors of interest and the solution for the quantum state of the system in the framework
of the S–R interaction are presented to show the machinery in action. A fi nal Section is dedicated to the concluding
remarks.
90
Reprinted from the journal