Biomedical Engineering Reference
In-Depth Information
c
0
n
n
X
i
Ψ
0
dτ
X
i
Ψ
XY
X
=
Ψ
0
+
Ψ
XY
AB
AB
dτ
(17.27)
i
=
1
A,B,X,Y
i
=
1
At first sight, one-electron properties calculated fromHF wavefunctions ought to be very
accurate because the lowest 'correction' terms are certain doubly excited configurations.
The argument is correct as far as it goes, but if we include
both
the singly and doubly
excited states together
c
A
Ψ
A
c
XY
AB
Ψ
XY
Ψ
=
c
0
Ψ
0
+
+
(17.28)
AB
A,X
A,B,X,Y
the singly excited states couple with the doubly excited ones, which couple with the ground
state and so the singly excited states do make a contribution to the ground state.
In any case, the key choice when performing HF-LCAO calculations is the choice of
atomic orbital basis set, as can be seen from my representative HF-LCAO calculations on
pyridine in Table 17.3. In each case I optimized the geometry.
Table 17.3
Electric dipole moment for
pyridine; HF-LCAO model
Level of theory
p
e
/10
−30
Cm
HF/STO-3G
6.870
HF/6-31G
8.811
HF/6-31G*
7.720
HF/6-311G**
7.553
HF/cc-pvQz
10.846
Experiment
7.31
±
2%
17.10 Electric Field Gradients
I have made mention at several points in the text of electron spin, the internal angu-
lar momentum possessed by electrons just because they are electrons. The total angular
momentum of an atomic electron is found by summing the electron spin and the electron
orbital angular momentum.
Many nuclei have a corresponding internal angular momentum, which is given the sym-
bol
I
and the vector is characterized by two quantumnumbers
I
and
m
I
. The vector obeys the
laws of quantummechanics appropriate to angular momenta; for each value of
I
,
m
I
=−
I
,
−
I
+
1,...
+
I
, the size of the vector is
I
(
I
1)
h
2π
and we can only measure simultaneously the size of the vector and one component that we
conventionally take as the
z
component. The
z
component has size
m
I
h
2π
The nuclear spin quantum number is characteristic of a given nucleus and can have
different values for different isotopic species. A sample is shown in Table 17.4.
+
