Biomedical Engineering Reference
In-Depth Information
with wavefunctions that were explicit functions of the interelectron coordinate
r
12
. His first
attempt was to write
cr
12
)
(14.19)
where
A
is the normalization constant and
Z
and
c
are adjustable parameters. In later work
he made use of expansions such as
A
exp
−
r
2
)
(1
=
Z
(
r
1
+
+
Φ
A
exp
−
r
2
)
(polynomial in
r
1
,
r
2
,
r
12
)
Z
(
r
1
+
Φ
=
(14.20)
and was able to demonstrate impressive agreement with experiment. We refer to such
wavefunctions as
correlated
wavefunctions; as we will soon see, the most sophisticated
orbital models average over the electron interactions whilst correlated wavefunctions allow
for the 'instantaneous' interaction between electrons.
Some of Hylleraas's results are shown in Table 14.3; the most accurate theoretical value
known is that of Frankowski and Pekeris (1966).
Table 14.3
Hylleraas's variational results for helium
Approximate wavefunction
ε/
E
h
exp
−
r
2
)
k
(
r
1
+
−
2.8478
exp
−
r
2
)
exp
(
−
k
(
r
1
+
c
1
r
12
)
−
2.8896
exp
−
r
2
)
exp
(
−
k
(
r
1
+
c
1
r
12
)
cosh
(
c
(
r
1
−
r
2
))
−
2.8994
exp
r
1
+
c
0
+
r
2
r
2
)
2
−
c
1
r
2
+
c
2
(
r
1
−
+
c
3
(
r
1
+
r
2
)
−
2.9032
2
c
5
r
12
r
2
)
2
+
c
4
(
r
1
+
+
'Exact'
−
2.90372437703
Hylleraas's approach whereby we write the interelectron distances explicitly in the wave-
function (and so abandon the orbital model) gives by far the most accurate treatment of
atomic systems, but like most attractive propositions there is a catch. I glossed over calcu-
lation of the two-electron integral in the discussion above, but this causes a major problem
for molecular systems. Any attempt to use hydrogenic orbitals for a molecular system
leads to two-electron integrals that are impossibly hard to evaluate. Despite many attempts,
Hylleraas's method has never been successfully applied to a large polyatomic molecule.
14.6 Linear Variation Method
The variation principle as stated above applies only to the lowest energy solution of any
given symmetry (spatial and spin). A special kind of variation function widely used in
molecular structure theory is the
linear variation function
. For the sake of argument, suppose
we try to improve the helium atom ground-state wavefunction Ψ
1
by adding Ψ
2
through Ψ
6
(given in Table 14.1) and so write
Φ
=
c
1
Ψ
1
+
c
2
Ψ
2
+···+
c
6
Ψ
6
