Chemistry Reference
In-Depth Information
function of the orbital exponent c
s
. The too attractive h
ss
has its bad
minimum at c
s
¼
1
5 (see Table 4.2) and is contrasted by the repulsive
E
pn
so as to give the best variational minimum of
:
«
w
¼
0
:
1234E
h
occurring for c
s
¼
0
:
4222. This result is within 98.7%of the exact result,
E
2s
¼
0
:
125E
h
. Even without doing orbital exponent optimization
ð
c
s
¼
0
1192E
h
would be obtained
(about 95% of the exact value), showing the crucial importance of
Schmidt's orthogonalization in assessing a reasonable variational approx-
imation to the 2s excited state.
:
5
Þ
, the remarkable result of
«
w
¼
0
:
4.2.3 The First Excited 2p State of the Hydrogenic System
We can take as an appropriate variational function for this case the
normalized 2p
z
STO:
!
1
=
2
c
p
p
w ¼
exp
ð
c
p
r
Þ
r cos
u
ð
4
:
20
Þ
orthogonal to
c
0
by symmetry. The orthogonality constraint is now
satisfied from the outset, and simple calculation gives the upper bound
(Equation 3.66):
1
2
r
Z
r
1
2
ð
c
p
Zc
p
Þ
E
2p
2
«
w
¼
2p
z
2p
z
¼
ð
4
:
21
Þ
giving upon c
p
optimization the exact value for the excited 2p energy level
of the H-like system:
«
w
dc
p
¼
c
p
Z
2
8
d
Z
2
¼
0
)
c
p
ð
best
Þ¼
Z
2
) «
w
ð
best
Þ¼
ð
4
:
22
Þ
4.2.4 The Ground State of the He-like System
With reference to Figure 4.4, the two-electron Hamiltonian (in atomic
units) for the He-like system is
1
r
12
;
1
2
r
Z
r
H
¼
h
1
þ
h
2
þ
h
¼
2
ð
4
:
23
Þ
