3.3 Toward Extended Applications
99
Neglected Differential Overlap) that sets equal to zero all overlap integrals but those
between valence orbitals. Moreover, also the integrals between valence integrals are
approximated either to an (arbitrary) reference value or to experimental data like the
ionization potential. A mitigation criterion that connects to the value of the exchange
integral K to the difference in energy of opposed and parallel spin integrals is the
one named INDO. In this case, integrals of the type (ml|ml) are not neglected if the
functions χ m and χ i are centered on the same atom. However, the role played by these
approaches has to be understood in terms of the possibility of covering large regions
of the molecular geometries to be considered in dynamical processes. In this case, in
fact, the approximate methods discussed here are mainly used to best fit locally the
potential energy values to a suitable functional representation by adjusting the value
of related parameters.
3.4 Full Range Process Potentials
3.4.1 The Three-Body Internuclear Coordinates
When tackling the problem of describing reactive processes at atomistic level one
needs to calculate the electronic structure of completely different molecular arrangements including those far from the equilibrium geometry. In order to better illustrate
this case, we concentrate here on the simplest reaction prototype that consists of three
atoms. Three-atom systems are, in fact, the ideal case study both because they include
the most investigated prototype atom–diatom reactive and nonreactive processes (that
are commonly used to the end of rationalizing chemical reaction mechanisms) and
because they are on the theoretical side the simplest reactive case to handle and
to understand. In order to describe chemical reactive processes, time t is the ideal
continuity variable. However, once that time (that is an ideal continuity variable for
describing chemical reactions) has been factored out, as it happens in time independent approaches, one faces the problem of devising a suitable alternative continuity
variable out of the position coordinates.
As already done for two-body systems, the first step is the reduction of the number
of position vectors by separating the centre-of-mass (CM) and the related motion.
From the vectors W A , W B , and W C (the position vectors of the nuclei in the chosen
axis frame omitted in the picture for the sake of clarity) we can build the three
internuclear vectors r (see Fig. 3.1) using the relationships (please notice hereinafter
the change of meaning of r AB , r BC and r CA from an electron-nucleus vector to a
nucleus–nucleus one)
12
r ν = r λμ = W λ − W μ
(3.43)
12 Single subscript notation is used mainly in the scheme of the separated atom formalism (in which
the label singles out the unbound atom) while the double subscript notation is mainly meant to
single out the bound atoms.
99
Neglected Differential Overlap) that sets equal to zero all overlap integrals but those
between valence orbitals. Moreover, also the integrals between valence integrals are
approximated either to an (arbitrary) reference value or to experimental data like the
ionization potential. A mitigation criterion that connects to the value of the exchange
integral K to the difference in energy of opposed and parallel spin integrals is the
one named INDO. In this case, integrals of the type (ml|ml) are not neglected if the
functions χ m and χ i are centered on the same atom. However, the role played by these
approaches has to be understood in terms of the possibility of covering large regions
of the molecular geometries to be considered in dynamical processes. In this case, in
fact, the approximate methods discussed here are mainly used to best fit locally the
potential energy values to a suitable functional representation by adjusting the value
of related parameters.
3.4 Full Range Process Potentials
3.4.1 The Three-Body Internuclear Coordinates
When tackling the problem of describing reactive processes at atomistic level one
needs to calculate the electronic structure of completely different molecular arrangements including those far from the equilibrium geometry. In order to better illustrate
this case, we concentrate here on the simplest reaction prototype that consists of three
atoms. Three-atom systems are, in fact, the ideal case study both because they include
the most investigated prototype atom–diatom reactive and nonreactive processes (that
are commonly used to the end of rationalizing chemical reaction mechanisms) and
because they are on the theoretical side the simplest reactive case to handle and
to understand. In order to describe chemical reactive processes, time t is the ideal
continuity variable. However, once that time (that is an ideal continuity variable for
describing chemical reactions) has been factored out, as it happens in time independent approaches, one faces the problem of devising a suitable alternative continuity
variable out of the position coordinates.
As already done for two-body systems, the first step is the reduction of the number
of position vectors by separating the centre-of-mass (CM) and the related motion.
From the vectors W A , W B , and W C (the position vectors of the nuclei in the chosen
axis frame omitted in the picture for the sake of clarity) we can build the three
internuclear vectors r (see Fig. 3.1) using the relationships (please notice hereinafter
the change of meaning of r AB , r BC and r CA from an electron-nucleus vector to a
nucleus–nucleus one)
12
r ν = r λμ = W λ − W μ
(3.43)
12 Single subscript notation is used mainly in the scheme of the separated atom formalism (in which
the label singles out the unbound atom) while the double subscript notation is mainly meant to
single out the bound atoms.