1. Ĥ  = E  Quantum Mechanics and Atomic Orbitals Bohr and Einstein
particle nature of light
DeBroglie
wave nature of particles
Schrödinger
theoretical descriptions of atoms
Heisenberg
quantum or wave mechanics
Dirac
wave function
= 
every allowed e- state
has unique 
to calculate energy
use Ĥ
Ĥ 
= E 

2. Ĥ  = E  solved for hydrogen wave functions  energies E 2 =
probability distribution
probability of finding an e- in H
at a particular distance from the nucleus
orbital

3. orbital
requires
3 quantum numbers
“address” n l ml
magnetic
-l, …, l
orientation
angular momentum
0, 1, 2, …, (n - 1)
shape
principal
1, 2, 3, …
size and energy

4. principal quantum number size energy as n increases
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

5. 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

6. n = 1 l = 0 designated by letters n = 2 l = 0, 1 l = 0 s orbital n = 3
p orbital
n = 4
l = 0, 1, 2, 3
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

7. orbital
requires 3
quantum numbers n l ml
magnetic quantum number
-l,…, l s
row s
n = 3
l = 0
m = 0 1
n = 1
l = 0
m = 0 1 p s
l = 1
m = -1
n = 2
l = 0
m = 0 1 3 p
m = 0
l = 1
m = -1
m = 1 d
m = 0 3
l = 2
m = -2
m = 1
m = -1
1 s orbital 5
m = 0
3 p orbitals
m = 1
5 d orbitals
m = 2

8. 1 s orbital 3 p orbitals 5 d orbitals each orbital holds 2e-
4th quantum number ms
spin  7
f orbitals 
n = 1
n = 2
n = 3
n = 4
n = 5
n = 6
n = 7 p d f s

9. 2
1s orbital
spherical 2
2s and 3s

10. 1p orbital
dumbbell shape
2p orbitals 3
3p, 4p, 5p etc.
similar shapes
larger

11. 3
d orbitals 5
cloverleaf
larger n
same shapes
larger

12. 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  2s He
2e-
1s2   2p
which orbital fills next? 3s 3p 4s
where is 3d?

13. H He Li Be no! Hund’s rule parallel spins B C N O F Ne Na [Ne]
1s 2s 2px 2py 2pz 3s 3px 3py 3pz 4s 3d 3d 3d 3d 3d H He Li Be
no!
Hund’s rule
parallel spins B C N O F Ne Na
[Ne]

14. K [Ar] Ca [Ar] Sc [Ar] Ti [Ar] half full shell stable V [Ar] Cr [Ar]
3dxz
3dyz
3dxy
3dx2-z2
3dz2
4px K
[Ar] Ca
[Ar] Sc
[Ar] Ti
[Ar]
half full shell
stable V
[Ar] Cr
[Ar] no Mn
[Ar]
full shell
stable Cu
[Ar] no