Biomedical Engineering Reference
In-Depth Information
where the operators refer to each of the
n
electrons. Usually it is necessary to add a corres-
ponding nuclear contribution, since we workwithin the Born-Oppenheimer approximation.
For example, the electric dipole moment operator is
N
n
p
e
=
ˆ
e
Z
α
R
α
−
e
r
i
α
=
1
i
=
1
where the first sum runs over the
N
nuclei and the second sum over the
n
electrons. All
electrons enter the sum on an equal footing, as they should, and the expectation value can
be written in terms of the charge density
P
(
r
)as
e
N
ˆ
p
e
=
e
Z
α
R
α
−
r
P
(
r
) dτ
(17.2)
α
=
1
Such electric moments are often reported in non-SI units; the old-fashioned unit of
length is the angstrom and the debye, itself a relic from the days of electrostatic units, is
the dipole moment corresponding to a pair of equal and opposite charges of magnitude
10
−
10
electrostatic units (esu
´
Å(
=
g
−
1/2
cm
3/2
s
−
1
) separated by 1
=
10
−
10
m). There are
2.9989
×
10
9
esu per coulomb, and so 1 D
=
10
−
10
esu
×
10
−
10
m or 3.336
×
10
−
30
Cm.
The atomic unit of electric dipole is
ea
0
=
8.4784
×
10
−
30
Cm, which is 2.5418 D.
17.3.1 Electrostatic Potential
One of the fundamental objectives of chemistry is to predict and rationalize chemical react-
ivity. In principle, this involves searching a potential energy surface for saddle points (i.e.
transition states) andminima (reactants and products), and this kind of detailed investigation
has only become possible in the last decade. Most of the traditional theories of chemical
reactivity have concerned themselves with organic molecules, and the simplest theories
have attempted to extract useful information from the electronic properties of reactants.
We can distinguish
static
theories, which in essence make use of the electronic wavefunc-
tion and/or electronic properties appropriate to an isolated molecule in the gas phase, and
dynamic
theories. Dynamic theories aim (for example) to predict the likely reaction sites
for the approach of a charged reagent, usually modelled as a point charge.
The electrostatic potential gives an index that has been widely used since the 1970s for
just this purpose (e.g. Scrocco and Tomasi 1978).
Figure 17.4 shows phenylanine; within the Born-Oppenheimer approximation,
molecules are thought of as point positive charges (the nuclei) surrounded by continuous
distributions of electron charge. A small portion of the molecular electron charge resides
in the volume element dτ . I can now calculate the electrostatic potential at points in space
around the molecule using the methods of classical electromagnetism. I have placed a point
charge
Q
at the origin; the electrostatic potential φ at this point will contain contributions
from the nuclei such as N:
1
4πε
0
eZ
N
R
N
