A Closer Look at ψ
1
• Contains Information about the Probability
of finding the Quantum Mechanical Entity
in a Certain State
– For atom, know energy so ψ is related to
probability of finding electron at a certain point
in space
• The Probability is not ψ, rather ψ2
– Actually this is ψ*ψ
2
More on Orbitals
• Wavefunctions for Atomic Orbitals can be
divided into Two Parts
– Radial (depends on distance from nucleus)
– Angular (depends on angles φ and θ)
• For Chemistry Angular Part is (most) Important
– Molecular shape
– Bonding
1
3
Nodes
• Places where the Probability of finding the
Electron is Zero (ψ = 0 so ψ2 = 0 )
• When ψradial is zero, called a radial (or
spherical) node
– There are n - l - 1 radial nodes
• When ψangular is zero, called an angular
node (or a nodal plane)
– There are l angular nodes
4
1s Radial Wavefunction
1.2
ψ radial
ψ (arbitrary units)
1.0
0.8
0.6
⎛Z⎞
= 2⎜⎜ ⎟⎟
⎝ a0 ⎠
3/ 2
e − Zr / a0
There are n - l - 1 = 1 - 0 - 1 = 0 radial nodes.
Note that ψ ≠ 0 at x = 0.
0.4
0.2
0.0
2
4
6
8
10
12
14
16
-0.2
Distance from Nucleus (arbitrary units)
2
5
1s Orbital
6
2s Radial Wavefunction
1.2
ψ radial
ψ (arbitrary units)
1.0
0.8
0.6
1 ⎛Z⎞
⎜ ⎟
=
2 2 ⎜⎝ a0 ⎟⎠
3/ 2
⎛
Z
⎜⎜ 2 −
a0
⎝
⎞
r ⎟⎟e − Zr / 2 a0
⎠
There are n - l - 1 = 2 - 0 - 1 = 1 radial node.
Note that ψ ≠ 0 at x = 0.
0.4
0.2
0.0
2
4
6
8
10
12
14
16
-0.2
Distance from Nucleus (arbitrary units)
3
7
2s Orbital
8
3s Radial Wavefunction
1.2
ψ (arbitrary units)
1.0
ψ radial
1 ⎛Z⎞
⎜ ⎟
=
9 3 ⎜⎝ a0 ⎟⎠
0.8
0.6
3/ 2
⎛ ⎛ 4Z ⎞ ⎛ 2Z ⎞ 2 ⎞
⎜6 − ⎜
⎟⎟r + ⎜⎜
⎟⎟ r 2 ⎟e − Zr / 3a0
⎜
⎜ ⎝ a0 ⎠ ⎝ 3a0 ⎠ ⎟
⎝
⎠
There are n - l - 1 = 3 - 0 - 1 = 2 radial nodes.
Note that ψ ≠ 0 at x = 0.
0.4
0.2
0.0
2
4
6
8
10
12
14
16
-0.2
Distance from Nucleus (arbitrary units)
4
9
3s Orbital
10
Probability of Finding an Electron
• Remember ψ2, not ψ, is Probability
ψ2 (arbitrary units)
1.2
ψ2 for 1s, 2s and 3s orbitals
1.0
Note s orbitals have a non-zero
probability of being found at the
nucleus.
0.8
0.6
0.4
0.2
0.0
0
2
4
6
8
10
12
14
16
Distance from the Nucleus (arbitrary units)
5
11
Probability of Finding an Electron
0.10
Ψ2 (arbitrary units)
1s
0.08
2s
0.06
Note wherever there was a node
ψ2 = 0, there is no probability that
the electron can be found there.
3s If the electron can’t be in a certain
place, how does it get across?
0.04
0.02
0.00
0
2
4
6
8
10
12
14
16
Distance from the Nucleus (arbitrary units)
12
Radial Distribution Function
• Problem with ψ2, it over estimates Probability
Close to Nucleus and under estimates it
Further Out
• Correct by multiplying ψ2 by 4πr2
– Takes into account that a wedge is smaller toward
the center than ends
– This correction only works for s orbitals
6
13
Radial Distribution Function
Electrons in orbitals with higher n are
usually found further from nucleus.
6
4 π r2 ψ2 (arbitrary units)
5
2s
Note that 2s and
3s electrons have
probability of being
closer to nucleus
than 1s!
3s
4
3
1s
2
1
0
0
2
4
6
8
10
12
14
16
Distance from Nucleus (arbitrary units)
14
1s, 2s, and 3s orbitals
7
15
Angular Part of Wavefunction
• Every Time ψ goes through a node Sign of
Wavefunction changes
– s orbital has same angular sign throughout
– p orbital lobes have different signs
– Lobes alternate signs in a d orbital
• Difference in Phase
p orbital
d orbital
dz2 orbital
16
p Orbitals
Typical
Typicalpporbital
orbital
When
When nn == 2,
2, then
then ll == 00 and
and 11
Therefore,
Therefore, in
in nn == 22 shell
shell there
there
are
2
types
of
orbitals
(2
are 2 types of orbitals (2
subshells
)
subshells)
For
For ll == 00 m
ml l == 00
this
this is
is aa ss subshell
subshell
For
1, 0,
For ll == 11 m
ml l == --1,
0, +1
+1
this
this is
is aa pp subshell
subshell
with
with 33 orbitals
orbitals
planar
planarnode
node
When l = 1, there is a
PLANAR NODE thru
the nucleus.
8
17
p Orbitals
The three p
orbitals lie 90o
apart in space
A p orbital
18
2p Radial Wavefunction
ψ radial
1.2
1 ⎛Z⎞
⎜ ⎟
=
4 6 ⎜⎝ a0 ⎟⎠
3/ 2
⎛ 2Z
⎜⎜
⎝ a0
⎞
r ⎟⎟e − Zr / 2 a0
⎠
ψ (arbitrary units)
1.0
0.8
There are n - l - 1 = 2 - 1 - 1 = 0 radial nodes.
Note that ψ = 0 at x = 0. Only s
orbitals have any probability
density at the nucleus.
0.6
0.4
0.2
0.0
2
4
6
8
10
12
14
16
-0.2
Distance from Nucleus (arbitrary units)
9
19
2px Orbital
20
2py Orbital
10
21
2pz Orbital
22
Degenerate 2p Orbitals
• All 3 orbitals have the same energy (n and l),
but differ in orientation (ml)
11
23
3p Radial Wavefunction
0.10
ψ radial
0.08
Ψradial ((Z/a0)3/2)
0.06
0.04
1 ⎛Z⎞
⎜⎜ ⎟⎟
=
27 6 ⎝ a0 ⎠
3/ 2
⎛ ⎛ 2 Z ⎞ ⎞⎛ 2 Z
⎜4 −⎜
⎟ ⎟⎜
⎜ ⎜ 3a ⎟r ⎟⎜ a
⎝ ⎝ 0 ⎠ ⎠⎝ 0
⎞
r ⎟⎟e − Zr / a0
⎠
Number of radial nodes = n - l - 1
Number of radial nodes = 3 - 1 - 1 = 1
0.02
Distance from Nucleus (Zr/a0)
0.00
5
10
15
20
25
30
-0.02
-0.04
24
3px Orbital
12
25
3py Orbital
26
3pz Orbital
13
27
d Orbitals
When n = 3, what are the values of l?
l = 0, 1, 2
so there are 3 subshells in the shell.
For l = 0, ml = 0
---> s subshell with single orbital
For l = 1, ml = -1, 0, +1
---> p subshell with 3 orbitals
For l = 2, ml = -2, -1, 0, +1, +2
---> d subshell with 5 orbitals
28
d Orbitals
typical d orbital
planar node
planar node
s orbitals have no planar nodes (l = 0) and are
spherical.
p orbitals have l = 1, have 1 planar node, and
are “dumbbell” shaped.
This means d orbitals (l = 2) have 2 planar nodes
14
29
3d Radial Wavefunction
0.05
ψ radial
Ψradial ((Z/a0)3/2)
0.04
0.03
1 ⎛Z⎞
⎜⎜ ⎟⎟
=
81 30 ⎝ a0 ⎠
3/ 2
2
⎛ 2 Zr ⎞ − Zr / 3a0
⎜⎜
⎟⎟ e
a
⎝ 0 ⎠
Number of radial nodes = 0
0.02
0.01
0.00
0
5
10
15
20
25
30
35
Distance from Nucleus (Zr/a0)
30
3dxy Orbital
15
31
3dxz Orbital
32
3dyz Orbital
16
33
3dz2 Orbital
34
3dx2-y2 Orbital
17
35
d Orbitals
36
f orbitals
When n = 4, l = 0, 1, 2, 3 so there are 4
subshells in the shell.
For l = 0, ml = 0
→ s subshell with single orbital
For l = 1, ml = -1, 0, +1
→ p subshell with 3 orbitals
For l = 2, ml = -2, -1, 0, +1, +2
→ d subshell with 5 orbitals
For l = 3, ml = -3, -2, -1, 0, +1, +2, +3
→
f subshell with 7 orbitals
18
37
f orbitals
19