what are the 2 angular nodes for d_xy

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