Lecture 3: Radial wavefunctions & energy levels of hydrogen

Radial Node

- occur where the radial component of the wave function passes through zero

- total number of radial nodes for any s orbital (ns) equal to n-1

np has n - 2 radial nodes

nd has n - 3 radial nodes

nf has n - 4 radial nodes

2s Wavefunction

- n=2 ,l=0, ml =0, wavefunction known as 2s

- ψ is non zero at nucleus

passes through zero, 1 node at r = 2

Radial Distribution for 2s

- Two local maxima in electron

density / probability

- Most probable radius for the electron in a H atom when it is in the 2s orbital is at about 5 a₀

3s Wavefunction

- n=3, l=0, ml =0, wave function known as 3s

- ψ is non zero at the nucleus, 2 nodes

Radial Distribution for 3s (compared to 1s and 2s)

- 3s has 3 local maxima

2p Wavefunction

- n=2, l=1, ml = -1, 0, +1

- Maximum for radial distribution is at 4 a₀

Comparison between 2p wavefunction and 2s

- 2p is on average closer to the nucleus, it maximum is nearer

- The 2s has a high probability of being very close to the nucleus - due to the small inner maximum

- Penetration is the potential for the presence of an electron inside shells of other electrons

- A 2p electron does not penetrate so effectively through the core (filled inner shells of electrons) because its wave function goes to zero at the nucleus. As a result, it is more fully shielded from the nucleus by core electrons

- This means in a many-electron atom, a 2s has a lower energy and will be occupied before the 2p

3p Wavefunction

- n=3, l=1, ml = -1, 0, +1

- 1 radial node

- Two maxima for radial distribution curve

3d Wavefunction

- n=3, l=2, m= -2, -1, 0, +1, +2

- no radial nodes (nd has n-3 radial nodes)

- Maxima in radial distribution curve is at 9 a₀

All relevant wavefunction graphs