Biomedical Engineering Reference
In-Depth Information
A
B
Hartree Fock
Exact
Figure 14.4
Hartree-Fock energy
Table 14.6
Properties output from an atomic HF study
ε
Orbital energy
A
Initial slope
P
nl
(
r
) /
r
l
+1
,
r
→
0
s
Screening parameter
1
/
R
3
Expectation value of 1/
R
3
1
/
R
Expectation value of 1/
R
R
Expectation value of
R
R
2
Expectation value of
R
2
Potential
/
Virial ratio
Kinetic
Spin-orbit coupling
ζ
nl
Orbit-orbit coupling
M
k
(
nl
,
nl
)
3
n
independent variables in the Schrödinger equation (plus spin). Hartree (1957) expressed
the need most dramatically as follows:
One way of representing a solution quantitatively would be by a table of its numerical values,
but an example will illustrate that such a table would be far too large ever to evaluate, or to
use if it were evaluated. Consider, for example, the tabulation of a solution for one stationary
state of Fe. Tabulation has to be at discrete values of the variables, and 10 values of each
variable would provide only a very coarse tabulation; but even this would require 10
78
entries
to cover the whole field; and even though this might be reduced to, say, 5
78
=
10
53
by use of the
symmetry properties of the solution, the whole solar system does not contain enough matter
to print such a table. And, even if it could be printed, such a table would be far too bulky to
use. And all this is for a single stage of ionization of a single atom.
14.11.1 Zener's Wavefunctions
In Section 14.5, we addressed ways of improving the analytical 1s orbital for helium,
especially by treating the effective nuclear charge
Z
as a variational parameter. In his
keynote paper Zener (1930) extended this simple treatment to first-row atoms. As usual,
I will let the author tell the story in his own words through the abstract.
