Chemistry Reference
In-Depth Information
The general solution of the radial equation is not straightforward. We are interested in
solutions for which the electron is in a stationary state bound to the nucleus. This gives
a boundary condition that
R
nl
(
r
)
0 at large
r
values. The set of functions that satisfy
Equation (A9.46) is then a product of an associated Laguerre polynomial,
L
2
l
+
1
→
n
+
1
(
ρ
r
), and a
decaying exponential that ensures this boundary condition is met:
2
Z
eff
n
3
(
n
1) !]
3
L
2
l
+
1
r
)exp
−
2
r
−
l
−
1) !
2
Z
na
0
(A9.47)
R
nl
(
r
)
=−
1
(
ρ
with
ρ
=
n
+
2
n
[(
n
+
From this general expression the factorial terms tell us that
n
must be a positive integer
and that
l
1.
n
is the principal quantum number, which determines which electronic
shell the orbital belongs to.
The first few radial functions are listed in Table A9.2; for completeness, the length scale
is shown in these functions by including the Bohr radius
a
0
explicitly.
≤
n
−
Table A9.2
The solutions for the radial equation, Equa-
tion (A9.7), for principle quantum number n from 1 to 3. Note:
Z
eff
is the effective nuclear charge, a
0
is the Bohr radius (0.529
177 Å), r is the radial coordinate, and
ρ
=
(
2Z
eff
/
na
0
)
.
n
l
R
nl
(r)
Z
eff
a
0
3
/
2
1
0(1s)
2e
−(
ρ/
2)
r
Z
eff
a
0
3
/
2
2
0(2s)
1
2
√
2
(2
−
ρ
r
) e
−(
ρ/
2)
r
Z
eff
a
0
3
/
2
1
2
√
6
1 (2p)
ρ
r
e
−(
ρ/
2)
r
Z
eff
a
0
3
/
2
3
0(3s)
1
9
√
3
(6
−
6
ρ
r
+
ρ
2
r
2
) e
−(
ρ/
2)
r
Z
eff
a
0
3
/
2
1
9
√
6
(4
1 (3p)
−
ρ
r
)
ρ
r
e
−(
ρ/
2)
r
Z
eff
a
0
3
/
2
2 (3d)
1
9
√
30
ρ
2
r
2
e
−(
ρ/
2)
r
To test out these solutions we will substitute some examples back into Equation (A9.46),
remembering that this equation is in atomic units and so takes
a
0
=1.
To obtain the kinetic energy term for
R
10
we will need
r
2
∂
∂
exp
−
2
r
A
∂
∂
2
r
R
10
=
∇
r
r
r
2
exp
−
2
r
A
2
∂
∂
=−
r
