Chemistry Reference
In-Depth Information
E
/
|
E
0
|
1
1
2
3
/
a
-1
-1
Figure A.1
Variational estimate of the electron ground state energy in a hydrogen
atom as a function of the arbitrary parameter
γ
in the trial wavefunction,
e
−
γ
r
. In this case, the lowest variational estimate equals the true ground
state energy. The energy scale (vertical axis) is in units of
|
E
0
|
, the hydro-
gen ground state binding energy, with the horizontal axis in units of
a
−
1
0
,
(inverse Bohr radius).
The ground state variational energy can be calculated using eq. (1.37) as
∞
f
∗
(
r
)
[
Hf
(
r
)
]
d
V
=
∞
E
(A.3)
f
∗
(
)
(
)
r
f
r
d
V
As the integrands in both the numerator and denominator are spherically
symmetric, we can solve eq. (A.3) by integrating outwards over spherical
shells of radius
r
and width d
r
, for which d
V
r
2
d
r
. The denominator
=
4
π
in eq. (A.3) is given by
∞
π
γ
e
−
2
γ
r
4
r
2
d
r
π
=
(A.4)
3
r
=
0
where we use the standard integral
0
e
−
ar
r
n
d
r
a
n
+
1
while the
=
n
!
/
numerator is given by
∞
e
−
γ
r
πε
0
r
e
−
γ
r
4
r
2
d
2
e
2
d
d
r
e
−
γ
r
r
2
d
r
−
d
r
(
)
−
π
(A.5)
2
mr
2
4
=
r
0
which it can be shown is equal to
2
e
2
π
2
m
−
(A.6)
γ
ε
(
2
γ)
2
0
with the estimated ground state energy
E
then obtained by dividing
eq. (A.6) by (A.4) to give
2
2
m
−
2
e
2
γ
γ
=
E
(A.7)
4
πε
0
