Environmental Engineering Reference
In-Depth Information
Table 3.1
One-electron wavefunctions in real form [33].
Equation for real form of wavefunction
a
,
where r¼Zr
Wavefunction
designation
Wavefunction name,
real form
=
p
3=2
/
a
o
and
C
1
¼ Z
C
1
e
r
Y
100
1
s
C
2
(2
r
)e
r
/2
Y
200
2
s
e
r
/2
Y
21,cos
j
2
p
x
C
2
r
sin
cos
j
e
r
/2
Y
21,sin
j
2
p
y
C
2
r
sin
sin
j
e
r
/2
Y
210
2
p
z
C
2
r
cos
C
3
(27
18
r þ
2
r
2
)e
r
/3
Y
300
3
s
C
3
(6
rr
2
) sin
e
r
/3
Y
31,cos
j
3
p
x
cos
j
C
3
(6
rr
2
) sin
e
r
/3
Y
31,sin
j
3
p
y
sin
j
C
3
(6
rr
2
) cos
e
r
/3
Y
310
3
p
z
3d
z
2
1) e
r
/3
C
4
r
2
(3cos
2
Y
320
C
5
r
2
sin
e
r
/3
Y
32,cos
j
3
d
xz
cos
cos
j
C
5
r
2
sin
e
r
/3
Y
32,sin
j
3
d
yz
cos
sin
j
3d
x
2
y
2
C
6
r
2
sin
2
e
r
/3
Y
32,cos2
j
cos2
j
C
6
r
2
sin
2
e
r
/3
Y
32,sin2
j
3
d
xy
sin2
j
a) C
2
¼C
1
/4
p
2, C
3
¼2C
1
/81
p
3, C
4
¼C
3
/2, C
5
¼
p
6C
4
, C
6
¼C
5
/2.
l
exp
ðr
R
n;l
ðrÞ¼ðr
=
a
o
Þ
=
na
o
ÞL
n;l
ðr
=
a
o
Þ:
ð
3
:
23
Þ
a
o
Þ
is a Laguerre polynomial in
r¼ r
/
a
o
, and the radial function has
n l
1 nodes. The parameter
a
o
is identical to its value in the Bohr model, but it no
longer signi
es the exact radius of an orbit. The energies of the electron states of the
one electron atom,
E
n
¼Z
2
E
o
/
n
2
(where
E
o
¼
13.6 eV, and
Z
is the charge on
nucleus) are unchanged from the Bohr model. The energy can still be expressed as
E
n
¼kZe
2
/2
r
n
, where
r
n
¼n
2
a
o
/
Z
, and
a
o
¼
0.0529 nm is the Bohr radius, as we
found in Equations 3.1
-
3.4 above.
The lowest energy wavefunctions
Here,
L
n;l
ðr
=
Y
n
,
l
,
m
of the one electron atom are listed in
Table 3.1 [33]. As before, to represent the hydrogen atom, set
Z¼
1. In common
chemical usage, the letters s, p, d, f correspond to angularmomentumvalues 0, 1, 2, 3.
An s-state is spherically symmetric and has no angular momentum.
=
p
Y
100
¼ðZ
3
=
2
a
3
=
2
Þ
exp
ðZr
=
a
o
Þ
ð
3
:
24
Þ
represents the lowest energy ground state that we earlier found had energy
13.6 eV.
The probability
P
(
r
)ofnding the electron at a radius
r
is
PðrÞ¼
4
pr
2
2
100
Y
;
ð
3
:
25
Þ
which is a smooth function with amaximumat
r ¼a
o
/
Z
. This is not an orbit of radius
a
o
, but a spherical probability cloud inwhich the electronsmost probable radius is
a
o
.
There is no angular momentum in this wavefunction.
This new solution represents a correction in concept, and in some numerical
values, from the results of the Bohr model. Note that
Y
100
is real, as opposed to
