Angular Orbitals: p and d
Unlike s orbitals, orbitals with l > 0 have an angular component – the electron density varies with orientation in space. z
The 2p z
orbital
The angular component is Y( p z
)
∝
cos
θ
θ cos
θ
•
•
•
•
•
Two lobes, separated by angular node p z
oriented along z-axis: node is xy plane p x
and p
For
Ψ
, two lobes have different phase
For
Ψ 2 y
oriented along x- and y-axes
, lobes must always be (+)
The 3d xy
orbital
The angular component is Y( d xy
)
∝
sin2 ϕ x,y y
•
•
•
How many nodes? total nodes = n – 1 sin2 ϕ
Four lobes, two angular nodes
Other d orbitals have different orientations
Nodes are xz/yz planes, lobes between x/y axes n l orbital radial nodes angular nodes
1 0 1s 0 0 angular nodes = l radial nodes = n – 1 – l
2 0 2s 1 0
2 1 2p 0 1
3 0 3s 2 0
3 1 3p 1 1
3 2 3d 0 2
4 0 4s 3 0
4 1 4p 2 1
4 2 4d 1 2
4 3 4f 0 3 ϕ x

Aufbau (“Building Up”)
As Z increases in multielectron atoms, electrons are added one by one from the lowest energy to progressively higher energy orbitals: spdf notation orbital diagram
1s
H: 1s
1 ↑
Lowest energy orbital:
1s (only orbital with n = 1)
1s
He: 1s
2 ↑↓
Pauli Principle : each orbital occupied by two electrons with opposite spin .
Li: [He]2s
Be: [He]2s
B: [He]2s
C: [He]2s
N: [He]2s
O: [He]2s
F: [He]2s
2
1
2
2
2
2
2
2p
2p
2p
2p
2p
1
2
3
5
4
↑↓
2s
↑ next-lowest energy orbital. Of n = 2, s is lowest energy.
↑↓ ↑↓
4th electron also in the 2s orbital, opposite spin.
↑↓ ↑↓ ↑
2s is full. 2p is next lowest energy
↑↓
↑↓
↑↓
↑↓
↑↓
↑↓
↑
↑
↑
↑
↑↓ ↑
↑
↑
2nd p electron occupies different p orbital, with same spin as first.
Hund’s Rule : in degenerate orbitals, electrons occupy different orbitals and adopt parallel spins .
3rd p electron occupies 3rd p orbital
4th p electron must pair spins
↑↓ ↑↓ ↑↓ ↑↓ ↑
5th p electron pairs in 2nd p orbital
Ne: [He]2s
2
2p
6 ↑↓ ↑↓ ↑↓ ↑↓ ↑↓
Ne has filled all orbitals of n = 2
Na: [Ne]3s
1 ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑
Na begins the new n = 3 shell