Biomedical Engineering Reference
In-Depth Information
whilst the
Hartree-Fock (HF)
model uses a fully antisymmetrized wavefunction such as
ψ
A
(
r
1
) α (
s
1
)
ψ
A
(
r
1
) β (
s
1
)
ψ
B
(
r
1
) α (
s
1
)
ψ
B
(
r
1
) β (
s
1
)
ψ
A
(
r
2
) α (
s
2
)
ψ
A
(
r
2
) β (
s
2
)
ψ
B
(
r
2
) α (
s
2
)
ψ
B
(
r
2
) β (
s
2
)
Ψ
Hartree
−
Fock
=
(14.29)
ψ
A
(
r
3
) α (
s
3
)
ψ
A
(
r
3
) β (
s
3
)
ψ
B
(
r
3
) α (
s
3
)
ψ
B
(
r
3
) β (
s
3
)
ψ
A
(
r
4
) α (
s
4
)
ψ
A
(
r
4
) β (
s
4
)
ψ
B
(
r
4
) α (
s
4
)
ψ
B
(
r
4
) β (
s
4
)
In the simplest version, HF theory concentrates on electronic states that can be represented
as a single Slater determinant. We find an extra term in the energy expression called the
exchange energy
, discussed earlier in Chapter 12. Evaluation of the energy needs a more
complicated numerical procedure then the simpler Hartree theory. Inclusion of electron
exchange by the numerical methods used in their day proved more and more difficult for
atoms towards the bottom right-hand corner of the Periodic Table. The Hartrees were able
to study a wide range of atoms in different electronic states, together with their ions. To
give a flavour of systems studied, I have reproduced in Table 14.5 the systems that Hartree
(1957) reports in his topic, which cover the period 1948 through 1957.
Table 14.5
Selection of atoms treated in Hartree (1957)
Atom
Atom
Atom
Atom
H
−
Ne
2+
Cl
−
Fe
16+
He
+
Na
Cl
Zn
2+
Li
+
Na
+
Ca
Zr
4+
Be
Mg
Ti
2+
Mo
+
B
Al
2+
V
+
In
3+
C
Al
3+
Mn
Sb
3+
O
6+
S
−
Mn
+
Au
+
F
−
S
Fe
13+
Tl
+
Tl
2+
The author mentions that all up to Mn
2+
and also Fe
16+
and Zn
2+
have
exchange terms included in the calculations.
14.11 Atomic Shielding Constants
Hartree-Fock wavefunctions are the best wavefunctions that are possible within the orbital
model, as shown in Figure 14.4. Wavefunctions A and B are simple orbital wavefunctions,
perhaps hydrogenic or improved hydrogenic. The difference between the Hartree-Fock
energy and experiment is called the
correlation energy
.
Output from an atomic HF program consists of the radial function, together with data
for each shell such as those shown in Table 14.6.
It is usual to work with normalized radial functions, and this determines the functions
apart from their sign (for if
P
nl
(
r
) is a solution, so is
P
nl
(
r
)). Hartree used a convention
that the radial function should be positive near the nucleus, whilst other authors use a
convention that it should be positive for large r.
Several authors pointed out that it was often desirable to have simple approximations to
the wavefunctions and energy levels of atoms and ions. For an atom of
n
electrons, there are
−
