2.3 Electronic Structures
33
On the other hand, the total electron density in the open-shell molecule is obtained
by
ρ(r) ≡ ρ α (r) + ρ β (r) =
occ
i
n α,i ψ
∗
α,i (r)ψ α,i (r) +
occ
j
n β, j ψ
∗
β, j (r)ψ β, j (r) (2.16)
which corresponds to Eq. (2.12) for the ordinary closed-shell molecule. As a matter
of course, the numbers of α and β spins become the same in the singlet closed-shell
molecule.
2.3.4 Atomic Net Charge
This quantity gives the measure to understand whether an atom involved in a molecule
is positively or negatively charged, which provides convenient chemical intuition
with experimental chemists. Atomic net charge N A (A signifies an atom) is based on
the information of electron population N r from a certain AO χ r (r) centered on atom
A, where N r is given by, what is called, population analysis. The atomic net charge
N A of the atom A can be formally expressed as
N A = Z A −
on A
r
N r
(2.17)
where Z A designates the nuclear charge on the atom A. It is noted that there is an
essential arbitrariness for defining the electron population N r ascribed to particular
AO χ r (r) since one has to cut off the electron density at a certain point although
this quantity extends to the whole space. Hence this arbitrariness is also brought into
atomic net charge N A .
The atomic net charge, nonetheless, is a convenient and indispensable concept
for experimentalists, in particular, to grasp broad idea of electronically positive or
negative characteristics of each atom as well as the charge distribution over the whole
molecule. This quantity is also useful for considering Lewis structure, degree of the
orbital hybridizations, and so on, corresponding to the “chemical intuition” often
quoted. There have been developed several ways to represent the electron population
and the associated atomic net charge. In Table 2.8 are listed atomic net charges of a
formaldehyde molecule, for instance, with respect to all the atoms in three ways of
estimations of N r using several kinds of basis sets as described in the following:
(1) Mulliken population analysis
Name of this analysis is after the proposer (Mulliken 1952), who is one of the founders
of the MO theory in early days. This analysis is the simplest one which defines the
electron population at the AO χ r (r) as
33
On the other hand, the total electron density in the open-shell molecule is obtained
by
ρ(r) ≡ ρ α (r) + ρ β (r) =
occ
i
n α,i ψ
∗
α,i (r)ψ α,i (r) +
occ
j
n β, j ψ
∗
β, j (r)ψ β, j (r) (2.16)
which corresponds to Eq. (2.12) for the ordinary closed-shell molecule. As a matter
of course, the numbers of α and β spins become the same in the singlet closed-shell
molecule.
2.3.4 Atomic Net Charge
This quantity gives the measure to understand whether an atom involved in a molecule
is positively or negatively charged, which provides convenient chemical intuition
with experimental chemists. Atomic net charge N A (A signifies an atom) is based on
the information of electron population N r from a certain AO χ r (r) centered on atom
A, where N r is given by, what is called, population analysis. The atomic net charge
N A of the atom A can be formally expressed as
N A = Z A −
on A
r
N r
(2.17)
where Z A designates the nuclear charge on the atom A. It is noted that there is an
essential arbitrariness for defining the electron population N r ascribed to particular
AO χ r (r) since one has to cut off the electron density at a certain point although
this quantity extends to the whole space. Hence this arbitrariness is also brought into
atomic net charge N A .
The atomic net charge, nonetheless, is a convenient and indispensable concept
for experimentalists, in particular, to grasp broad idea of electronically positive or
negative characteristics of each atom as well as the charge distribution over the whole
molecule. This quantity is also useful for considering Lewis structure, degree of the
orbital hybridizations, and so on, corresponding to the “chemical intuition” often
quoted. There have been developed several ways to represent the electron population
and the associated atomic net charge. In Table 2.8 are listed atomic net charges of a
formaldehyde molecule, for instance, with respect to all the atoms in three ways of
estimations of N r using several kinds of basis sets as described in the following:
(1) Mulliken population analysis
Name of this analysis is after the proposer (Mulliken 1952), who is one of the founders
of the MO theory in early days. This analysis is the simplest one which defines the
electron population at the AO χ r (r) as