Atomic Orbitals
1
2
Atomic Orbitals
• What is an orbital?
Miessler and Tarr, Chapter 2
– An electron wave, a matter wave
• How is an orbital described?
– Three quantum numbers
Number
Value
Property
n
1, 2, 3, …
Orbital size, E of e- in H atom
l (“ell”)
0, 1, 2 …(n-1) Orbital shape
ml
- l ..0..+ l
Orbital orientation
3
Where do QN Come From?
• Schrodinger eqn. For e- wave in 3
dimensions
!2 "
!x2
+
! 2"
!y 2
+
! 2"
!z2
+
8π 2m
h2
4
! usually obtained
in polar coordinates
z
(E − V)" = 0
r
!
Where E = total energy and V = potential energy
m = electron mass and h = Planck’s constant
y
Solution to equation gives only certain values:
1.
For E = eigenvalues
2.
For ! = eigenfunctions or wave functions
This introduces the quantum condition naturally
Wave Functions
! (r, ", #) = [Rn,l (r) ] ["l,m #m]
Radial
Spherical harmonic
"
x
5
Wave Functions
6
! (r, ", #) = [Rn,l (r) ] ["l,m #m]
Radial
Spherical harmonic
= Y l,m (",#)
Wave functions are normalized. The wave
function must lead to a probability of finding
an electron somewhere in space of 1.
" = Zr/ao where ao = 52.9 pm
1
Wave Functions
7
! (r, ", #) = [Rn,l (r) ] ["l,m #m]
8
! (r, ", #) = [Rn,l (r) ] ["l,m #m]
Spherical harmonic
Radial
Wave Functions
Radial
Spherical harmonic
Consider radial portion—
• n cannot be 0 because R would then be 0
• n - l - 1 restricts values of l to n-1
– n = 1, then l = 0
– n = 2, then l = 0, 1
– n = 3, then l = 0, 1, 2
Wave Functions
9
! (r, ", #) = [Rn,l (r) ] ["l,m #m]
Radial
= Y l,m (",#)
= Y l,m (",#)
l - |m| sets up the condition that values of m l
are - l … 0 …+ l
! 1$
R1,0 = 2 # &
" a0 %
•
•
•
•
3/2
Spherical harmonic
$m portion is imaginary
Therefore, ! is itself meaningless, but !2
does have a physical interpretation
#l,m portion:
2 l + 1 = number of values of m l
For n = 1 and l = 0
1s atomic orbital
10
! (r, ", #) = [Rn,l (r) ] ["l,m #m]
Spherical harmonic
Radial
Wave Functions
!2 proportional to WAVE INTENSITY or
PROBABILITY
When l = 0 there is no dependence on angle.
11
Radial Wave Functions
12
e-r/a
R has units of (1/pm)3/2
R2 proportional to (1/pm)3
Therefore, R2 = density function
R2 proportional to probability of finding
electron in a given volume with radius r.
2
13
For n = 1 and l = 0
1s atomic orbital
! 1$
R1,0 = 2 # &
" a0 %
•
•
•
•
•
3/2
! 1$
R1,0 = 2 # &
" a0 %
e-r/a
R21,0 = (4/ao3)e-2r/a
When r = 0, R21,0 = 4/ao3 = 2.7 x 10-5 /pm3
Because Y0,0 = (√2/2)(1/√2 π) = 1/2√π
R2Y2 for 1s = 6.8 x 10-7 /pm3
This is the probability density for a 1s electron
essentially at the nucleus.
• A NODE is a plane or point of 0 electron
probability.
• s orbitals all have finite probability density
“at” the nucleus.
• p, d, and f orbitals have 0 density “at” the
nucleus. That is, there is a node passing
thru the nucleus.
• All orbitals can have nodes in the electron
wave beyond the nucleus.
orbital
1
0.6
Series1
0.4
0.2
0
2
3
Distance
from
4
5
16
Nodes in Electron Waves
0.8
1
e-r/a
• It is important to recognize that s
orbitals (l = 0) have probability
density greater than 0 “at” the nucleus.
1.2
0
3/2
15
1s Atomic Orbital
! (= e-r) vs. r
1s
14
For n = 1 and l = 0
1s atomic orbital
6
nucleus
17
18
2s Atomic Orbital
! [= (2-r)e-r/2] vs. r
2s Atomic Orbital
!2 vs. r
2s Orbital
2s Wave Function Squared
2.5
0.4
0.35
2
0.3
1.5
0.25
Series1
1
0.2
Series1
0.15
0.5
0.1
0.05
0
0
2
4
6
8
10
12
0
0
-0.5
Distance
from
nucleus
2
4
Distance
6
from
8
10
12
nucleus
3
19
2p Atomic Orbital
! ( = r·e-r/2) vs. r
20
Radial Wave Functions
A Summary
R = (const) (eqn in !) !l e-!/n
2p Atomic Orbital
0.8
0.7
0.5
0.4
Series1
Larger n
makes
R larger.
If l not 0
there is a
node at
nucleus.
No. of radial nodes
= n-l-1
0.6
0.3
0.2
0.1
l = number of nodal planes
0
0
2
4
6
Distance
from
8
the
10
12
nucleus
Number of radial nodes = n - l - 1
Angular Functions/Orbital Shapes
Consider pz orbital:
Y =
3
4π
" z$ =
# r%
z
Note that
the number
of planar
nodes = l
Wave
function
is + in +z
direction
21
3
cos &
4π
Consider dxy orbital:
cos ! = cos (0˚) = 1
a) When either x or y is
0, then Y = 0 and there
is a node.
b) When either x or y
is negative, then Y has
negative value.
c) When both x and y
are negative, Y has a
positive value.
cos (90˚) = 0
x
Planar node
Wave
function
is - in -z
direction
Angular Functions/Orbital Shapes
Y=
60 " x y%
$ '
16π # r 2 &
nodal plane
-x, -y
-x, +y
+
–
–
+
nodal plane
cos (180˚) = -1
y
+x, -y
x
+x, +y
23
Polyelectronic Atoms
22
24
Effective Nuclear Charge, Z*
• In polyelectronic atoms, orbital energies
increase in the order (n+ l)
• For two orbitals of same (n+l), the one with
smaller n lies lower in energy.
(n+l) 1 2 3
1s 2s 2p,3s
4
3p, 4s
5
3d, 4p, 5s
6
4d, 5p, 6s
Why should 2s lie lower in energy than
2p in a polyelectronic atom?
Shielding ---> effective nuclear charge, Z*.
Electron cloud
for 1s electrons
4
Slater’s Rules
Calculating Z*
Z* = Z - % where % is the shielding constant
Electrons of higher n have no effect
Electrons of same n, % = 0.35/electron
Electrons of n - 1, % = 0.85/electron
Electrons of lower than n - 1, % = 1.00
For Li 2s electron: % = 2 x 0.85 = 1.70
Z* = 3.00 - 1.70 = 1.30
25
Calculated Effective Nuclear
Charges, Z*
B
2.60
Al
3.50
Ga
5.00
In
5.00
Tl
5.00
C
3.25
Si
4.15
Ge
5.65
Sn
5.65
Pb
5.65
N
3.90
P
4.80
As
6.30
Sb
6.30
Bi
6.30
O
4.55
S
5.45
Se
6.95
Te
6.95
Po
6.95
F
5.20
Cl
6.10
Br
7.60
I
7.60
At
7.60
26
Ne
5.85
Ar
6.75
Kr
8.25
Xe
8.25
Rn
8.25
5