Angular Orbitals: p and d
Unlike s orbitals, orbitals with l > 0 have an angular component – the electron density
varies with orientation in space.
The 2pz orbital
The angular component is Y(pz) ∝ cosθ
• Two lobes, separated by angular node
• pz oriented along z-axis: node is xy plane
• px and py oriented along x- and y-axes
• For Ψ, two lobes have different phase
• For Ψ2, lobes must always be (+)
The 3dxy orbital
The angular component is Y(dxy) ∝ sin2ϕ
• Four lobes, two angular nodes
• Nodes are xz/yz planes, lobes between x/y axes
• Other d orbitals have different orientations
How many nodes?
total nodes = n – 1
angular nodes = l
radial nodes = n – 1 – l
n l orbital radial
nodes
angular
nodes
1 0 1s 0 0
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
0
45
90
135
180
225
270
315
360
cos θ
0
45
90
135
180
225
270
315
360
sin2ϕ
ϕ
y
x
z
x,y
θ
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: 1s1 ↑
1s
He: 1s2 ↑↓
1s 2s
Li: [He]2s1 ↑↓ ↑
1s 2s
Be: [He]2s2 ↑↓ ↑↓
1s 2s 2p
B: [He]2s22p1 ↑↓ ↑↓ ↑
1s 2s 2p
C: [He]2s22p2 ↑↓ ↑↓ ↑ ↑
1s 2s 2p
N: [He]2s22p3 ↑↓ ↑↓ ↑ ↑ ↑
1s 2s 2p
O: [He]2s22p4 ↑↓ ↑↓ ↑↓ ↑ ↑
1s 2s 2p
F: [He]2s22p5 ↑↓ ↑↓ ↑↓ ↑↓ ↑
1s 2s 2p
Ne: [He]2s22p6 ↑↓ ↑↓ ↑↓ ↑↓ ↑↓
Lowest energy orbital:
1s (only orbital with n = 1)
Pauli Principle: each orbital
occupied by two electrons with
opposite spin.
1s is full, so 3rd electron goes to
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 has filled all orbitals of n = 2
1s 2s 2p 3s
Na: [Ne]3s1 ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑ Na begins the new n = 3 shell
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