Chemistry Reference
In-Depth Information
Table 4.1
Variational approximations to the ground state of the
hydrogenic system
w
«(c)
c
min
«
min
c
2
2
Zc
Z
2
2
w
1
¼
exp
ð
cr
Þ
Z
c
2
6
Zc
2
3
2
Z
3
8
Z
2
w
2
¼
exp
ð
cr
Þ
r
1
=
2
3
2
c
Z
8c
p
8
9
4
3
w
3
¼
exp
ð
cr
2
Z
2
Z
2
Þ
p
p
Table 4.2
Numerical results of different variational
approximations for the ground-state H atom
w
c
min
«(c
min
)/E
h
w
1
¼
exp
ð
cr
Þ
1
0.5
w
2
¼
exp
ð
cr
Þ
r
0.375
1.5
w
3
¼
exp
ð
cr
2
Þ
0.4244
0.2829
energy result, with no more than 75%of the truth. The third function
w
3
is the prototype of the 1s Gaussian function. Even if it has the wrong
behaviour at the origin (it has a zero derivative here) and decreases
too quickly far from the nucleus, it nevertheless gives a fair result for the
energy, namely about 85% of the true value, which is 10% better
than
w
2
. However, increasing the number of 1s GTOs improves the
energy, but does not improve the wavefunction sufficiently; even taking
N
¼
10 optimized 1s GTOs, it is still very different from the correct
one.
4
It is interesting to note fromTable 4.2 that the variational theorem
tries to do its best to correct the inappropriate form of functions
w
2
and
w
3
by strongly increasing (c
min
¼
1
:
5, the function contracts)or
decreasing (c
min
¼
0
2829, the function expands) respectively the best
value for their orbital exponents. Taking
:
c
¼
1in
w
2
would give
«
w
¼
0
333 333E
h
, which is about 67% of the true value. Hence,
optimization of c improves energy by 8%.
:
4
The incorrect behaviour at the origin of 1s GTOs and the way of correcting for it are fully
discussed by Magnasco (2007).
