Why are 4f orbitals filled when they are?
Thomas-Fermi explanation

• Large angular momentum for an f-orbital (l = 3) Þ large centrifugal potential

  tends to keep the electron away from the nucleus Æ Aufbau order

• Increased Z increases Coulombic attraction to a larger extent for smaller n

  due to a proportionately greater change in Z_eff Æ reasserts Hydrogenic order

  This can be viewed empirically as due to differing penetration effects

 Radial Wavefunctions P_n,l² for 4f, 5d, 6s in Ce

• 4f orbitals (and the atoms in general) steadily contract across the lanthanide series
• Effective electron potential for the excited states of Ba {[Xe] 6s 4f} & La {[Xe] 6s 5d 4f}

  show a `sudden' change in the broadness & depth of the 4f "inner well"

Þ for Ba (Z = 56) 4f is an outer orbital with ·4fÒ close to its value for the H atom

Þ for La (Z = 57) 4f is an inner orbital with ·4fÒ ca. 0.7a_o

though only for the next atom, Ce (Z = 58) is the 4f electron of sufficiently high binding energy to appear in the ground state configuration

Bibliography