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Quantum Mechanics and Atomic Orbitals
Bohr and Einstein particle nature of light
DeBroglie
wave nature of particles
theoretical descriptions of atoms
Schrödinger
Heisenberg
quantum or wave mechanics
Dirac
wave function = 
every allowed e- state has unique 
to calculate energy use Ĥ
Ĥ=E
Ĥ=E
wave functions 
energies E
2 = probability distribution
probability of finding an e- in H
at a particular distance from the nucleus
solved for hydrogen
orbital
orbital requires 3 quantum numbers
n l ml
magnetic
-l, …, l
orientation
angular momentum
0, 1, 2, …, (n - 1)
shape
principal
1, 2, 3, …
size and energy
“address”
orbital requires 3 quantum numbers
n l ml
principal quantum number size
energy
as n increases orbitals become larger
e- is further from the nucleus
n=1
n=2
n=3
n=4
n=5
n=6
n=7
orbital requires 3 quantum numbers
n l ml
angular momentum shape 0  n - 1
n=1 l=0
designated by letters
n = 2 l = 0, 1
l = 0 s orbital
n = 3 l = 0, 1, 2
l = 1 p orbital
n = 4 l = 0, 1, 2, 3
l = 2 d orbital
l = 3 f orbital
n=1
n=2
n=3
n=4
l=0
l = 0, 1
l = 0, 1, 2
l = 0, 1, 2, 3
designated by letters
l = 0 s orbital
l = 1 p orbital
l = 2 d orbital
l = 3 f orbital
s
p
n=1
n=2
n=3
n=4
n=5
n=6
n=7
d
f
orbital requires 3 quantum numbers
n l ml
magnetic quantum number -l,…, l
s
row
s
n=3 l=0
n=1 l=0 m=0 1
p
s
l=1
n=2 l=0 m=0 1
p
l = 1 m = -1
d
m=0 3
l=2
m=1
1 s orbital
3 p orbitals
5 d orbitals
m=0 1
m = -1
m=0 3
m=1
m = -2
m = -1
5
m=0
m=1
m=2
1 s orbital
3 p orbitals
5 d orbitals
each orbital holds 2e- 4th quantum number ms spin

7 f orbitals
s
p
n=1
n=2
n=3
n=4
n=5
n=6
n=7
d
f
2

1s orbital
spherical
2

2s and 3s
dumbbell shape
3p, 4p, 5p etc.
1p orbital
2p orbitals
3
similar shapes larger
3 d orbitals 5
cloverleaf
larger n same shapes larger
Polyelectronic Atoms
Pauli exclusion principle
no 2 electrons same 4 quantum numbers
lowest energy orbitals fill first
1s orbital is lowest energy
H 1e- 1s1 
He 2e- 1s2  
which orbital fills next?
2s
2p
3s
3p
4s
where is 3d?
1s 2s 2px 2py 2pz 3s 3px 3py 3pz 4s 3d 3d 3d 3d 3d
H
He
Li
Be
B
C
N
O
F
Ne
Na [Ne]
no! Hund’s rule parallel spins
4s 3dxz 3dyz 3dxy 3dx2-z2 3dz2 4px
K [Ar]
Ca [Ar]
Sc [Ar]
Ti [Ar]
V [Ar]
Cr [Ar]
Mn [Ar]
Cu [Ar]
half full shell stable
no
full shell stable
no