1 3
Theor Chem Acc (2015) 134:138
DOI 10.1007/s00214-015-1743-2
REGULAR ARTICLE
On the non-integer number of particles in molecular system
domains: treatment and description
Roberto C. Bochicchio
1
Received: 26 May 2015 / Accepted: 30 September 2015 / Published online: 23 October 2015
© Springer-Verlag Berlin Heidelberg 2015
1 Introduction
Generally, quantum chemical calculations of electronic
structure take the number of electrons in the molecule as
a fi xed parameter which states for a closed system [ 1 , 2 ],
i.e., for its neutral state of N electrons or for any of their
ionic confi gurations. This is usually the correct approach
for a molecule in gas phase but not within the framework
of a surrounding environment which may donate or accept
electrons as for instance in the treatment of solvation phenomena, surface chemistry, or enzyme mechanisms, among
others. Therefore, this scenario induces to describe these
problems by means of fragments or physical domains
as moieties like individual or group of atoms within the
molecular structure. Thus, to associate non-integer charges
to them is the main key to the understanding at atomic scale
of complex processes of electron distributions undergoing charge fl ux transfer among subsystems of atoms and
molecules under the infl uence of reactive interactions and/
or external perturbations, conformational changes or interactions, related to chemical reactivity [ 3 – 5 ]. For such a
goal, the fundamental magnitudes to be described are the
energy, the electron density and their derivatives [ 3 , 4 , 6 ].
Therefore, an accurate quantum treatment which attempts
to reach a complete and rigorous description of the electron distribution and of ulterior way the determination of
the physicochemical properties, needs a precise defi nition
of the system, its energy and state. These problems merit
the introduction of the Atoms in Molecules (AIM) concept
and the energy and state dependence with the number of
particles in the system.
It has been a common trend in the literature to assume
under certain success that the quadratic electrostatic
interactions constitute a suitable approach of the energy
dependence on fractional charge and independent of the
Abstract The energy of an atomic or molecular system undergoing Coulomb interactions is well known at
the integer numbers of its neutral or ionic confi gurations.
Nevertheless, the physical domains (atoms in molecules)
inside the whole molecular system possess a non-integer
number of particles due to the electron exchange with its
surrounding. Hence, the dependence of the energy, the
density matrix and their marginal distributions (reduced
density matrices) with the number of particles become
a problem of fundamental importance in the description
of the electron distribution, its properties and transformations. In this work, we present a rigorous mathematical and
physical basis for the treatment of this problem within the
grand-canonical statistical distribution of few particles.
In this context, the derivatives of the energy and the density referred as chemical descriptors (especially chemical
potential and hardness) are analyzed in both cases, when
the system is isolated and when it is subject to the interaction with an environment. The ground state energy convexity dependence with the number of particles of these systems simplifi es the description.
Keywords Grand-canonical ensemble · Electronic
structure · Density matrix · Descriptors
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Roberto C. Bochicchio
rboc@df.uba.ar
1
Departamento de Física, Facultad de Ciencias Exactas
y Naturales , Universidad de Buenos Aires and IFIBA
(CONICET), Ciudad Universitaria , 1428 Buenos Aires ,
Argentina
89
Reprinted from the journal
Theor Chem Acc (2015) 134:138
DOI 10.1007/s00214-015-1743-2
REGULAR ARTICLE
On the non-integer number of particles in molecular system
domains: treatment and description
Roberto C. Bochicchio
1
Received: 26 May 2015 / Accepted: 30 September 2015 / Published online: 23 October 2015
© Springer-Verlag Berlin Heidelberg 2015
1 Introduction
Generally, quantum chemical calculations of electronic
structure take the number of electrons in the molecule as
a fi xed parameter which states for a closed system [ 1 , 2 ],
i.e., for its neutral state of N electrons or for any of their
ionic confi gurations. This is usually the correct approach
for a molecule in gas phase but not within the framework
of a surrounding environment which may donate or accept
electrons as for instance in the treatment of solvation phenomena, surface chemistry, or enzyme mechanisms, among
others. Therefore, this scenario induces to describe these
problems by means of fragments or physical domains
as moieties like individual or group of atoms within the
molecular structure. Thus, to associate non-integer charges
to them is the main key to the understanding at atomic scale
of complex processes of electron distributions undergoing charge fl ux transfer among subsystems of atoms and
molecules under the infl uence of reactive interactions and/
or external perturbations, conformational changes or interactions, related to chemical reactivity [ 3 – 5 ]. For such a
goal, the fundamental magnitudes to be described are the
energy, the electron density and their derivatives [ 3 , 4 , 6 ].
Therefore, an accurate quantum treatment which attempts
to reach a complete and rigorous description of the electron distribution and of ulterior way the determination of
the physicochemical properties, needs a precise defi nition
of the system, its energy and state. These problems merit
the introduction of the Atoms in Molecules (AIM) concept
and the energy and state dependence with the number of
particles in the system.
It has been a common trend in the literature to assume
under certain success that the quadratic electrostatic
interactions constitute a suitable approach of the energy
dependence on fractional charge and independent of the
Abstract The energy of an atomic or molecular system undergoing Coulomb interactions is well known at
the integer numbers of its neutral or ionic confi gurations.
Nevertheless, the physical domains (atoms in molecules)
inside the whole molecular system possess a non-integer
number of particles due to the electron exchange with its
surrounding. Hence, the dependence of the energy, the
density matrix and their marginal distributions (reduced
density matrices) with the number of particles become
a problem of fundamental importance in the description
of the electron distribution, its properties and transformations. In this work, we present a rigorous mathematical and
physical basis for the treatment of this problem within the
grand-canonical statistical distribution of few particles.
In this context, the derivatives of the energy and the density referred as chemical descriptors (especially chemical
potential and hardness) are analyzed in both cases, when
the system is isolated and when it is subject to the interaction with an environment. The ground state energy convexity dependence with the number of particles of these systems simplifi es the description.
Keywords Grand-canonical ensemble · Electronic
structure · Density matrix · Descriptors
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Roberto C. Bochicchio
rboc@df.uba.ar
1
Departamento de Física, Facultad de Ciencias Exactas
y Naturales , Universidad de Buenos Aires and IFIBA
(CONICET), Ciudad Universitaria , 1428 Buenos Aires ,
Argentina
89
Reprinted from the journal