We will consider four scalar and two vector fields that we are starting to explore
as potentially useful candidates in QCT studies. The exchange-correlation potential
V xc r
ð Þ, which determines the covalent interaction energy density among electrons at
point r and the rest of the electrons of the system, the potential acting on an electron
in a molecule as defined by Zhao and Yang [7, 8] (PAEM), which determines the
interaction of an electron belonging to a molecule and the remaining electrons and
nuclei, and the additive and effective energy densities E
add r
ð Þ; E
eff r
ð Þ, that provide
additive (counting half the interaction of an electron with the rest of the system’s
particles) and effective (counting them all) energetic contributions of a volume
element located at r to the total energy. All these scalar fields may be subjected to
the standard QCT procedure, so their gradients will induce topological partitions of
the physical space, providing new sets of critical points, inter-basin surfaces, etc.
The vector fields explored are the PAEM force, defined as minus the PAEM gradient, and the Ehrenfest force field, i.e. the force density acting on electrons at r due
to the remaining particles comprising the molecule. Vector fields can also be used
to directly determine partitions of the space, as shown in the case of the Ehrenfest
field [27].
Since our aim here is presenting the overall features of these quantities, we will
focus on their basic properties, showing their basic topological portraits in a small
set of archetypal molecules. We will stress the similarities and differences with
other known fields, as well as their potential usefulness.
The rest of the Chapter is organized as follows: first we will briefly define all the
quantities we will discuss, leaving some technical details for an appendix. Then we
will present the behaviour of the scalar and vector fields in a couple of systems,
computed at several levels of theory, discussing their similarities and differences.
We will end with some conclusions and prospects.
6.2 On QCT Fields Depending on the Pair Density
As shown in the introduction, we have decided to focus in this contribution on some
fields that depend on the second order density, q 2 r 1 ; r 2
ð
Þ. This is an essential
ingredient of the molecular energy, entering the electron-electron repulsion. Any
total energy index that does not use it relies either on (i) some approximate density
functional, or (ii) on a local theorem that allows for a mapping of the two body
complexity of the electron-electron interactions onto one body contributions (like the
local virial theorem). Since the quantities we will manipulate are related to total
energy (or force) components, we will easily recognize in their definitions a set of
well known components of the energy: electron-nucleus attraction, electron-electron
repulsion, nucleus-nucleus repulsion, and kinetic energy (densities).
In order to ease the comparison among the different quantities that we will
introduce, it is instructive to recall two important auxiliary fields that will appear in
several of our discussions. The first is the molecular electrostatic potential (MEP),
6 Emergent Scalar and Vector Fields in Quantum Chemical Topology
133
as potentially useful candidates in QCT studies. The exchange-correlation potential
V xc r
ð Þ, which determines the covalent interaction energy density among electrons at
point r and the rest of the electrons of the system, the potential acting on an electron
in a molecule as defined by Zhao and Yang [7, 8] (PAEM), which determines the
interaction of an electron belonging to a molecule and the remaining electrons and
nuclei, and the additive and effective energy densities E
add r
ð Þ; E
eff r
ð Þ, that provide
additive (counting half the interaction of an electron with the rest of the system’s
particles) and effective (counting them all) energetic contributions of a volume
element located at r to the total energy. All these scalar fields may be subjected to
the standard QCT procedure, so their gradients will induce topological partitions of
the physical space, providing new sets of critical points, inter-basin surfaces, etc.
The vector fields explored are the PAEM force, defined as minus the PAEM gradient, and the Ehrenfest force field, i.e. the force density acting on electrons at r due
to the remaining particles comprising the molecule. Vector fields can also be used
to directly determine partitions of the space, as shown in the case of the Ehrenfest
field [27].
Since our aim here is presenting the overall features of these quantities, we will
focus on their basic properties, showing their basic topological portraits in a small
set of archetypal molecules. We will stress the similarities and differences with
other known fields, as well as their potential usefulness.
The rest of the Chapter is organized as follows: first we will briefly define all the
quantities we will discuss, leaving some technical details for an appendix. Then we
will present the behaviour of the scalar and vector fields in a couple of systems,
computed at several levels of theory, discussing their similarities and differences.
We will end with some conclusions and prospects.
6.2 On QCT Fields Depending on the Pair Density
As shown in the introduction, we have decided to focus in this contribution on some
fields that depend on the second order density, q 2 r 1 ; r 2
ð
Þ. This is an essential
ingredient of the molecular energy, entering the electron-electron repulsion. Any
total energy index that does not use it relies either on (i) some approximate density
functional, or (ii) on a local theorem that allows for a mapping of the two body
complexity of the electron-electron interactions onto one body contributions (like the
local virial theorem). Since the quantities we will manipulate are related to total
energy (or force) components, we will easily recognize in their definitions a set of
well known components of the energy: electron-nucleus attraction, electron-electron
repulsion, nucleus-nucleus repulsion, and kinetic energy (densities).
In order to ease the comparison among the different quantities that we will
introduce, it is instructive to recall two important auxiliary fields that will appear in
several of our discussions. The first is the molecular electrostatic potential (MEP),
6 Emergent Scalar and Vector Fields in Quantum Chemical Topology
133