H-atom p.9

Aufbau: Building Up Multi-Electron Atoms

A consequence of the Spin-Statistics Theorem is the Pauli Exclusion Principle which states that at most two electrons can be in the same spatial wavefunction. (If two electrons share a spatial wavefunction, they must be in the spin "paired", i.e., singlet, state.)

The lessons learned from the Helium atom are that

as a result of the electron-electron repulsion the n² states (ns, np, up to l=n-1) that were degenerate in the H-atom are instead "l-tilted" so the ns level lies below the np level, etc.
other things being equal, the spatially antisymmetric state (and hence the state with "all the electron spinning the same way") has lowest energy.

Follow along as we build up the atoms of the period table. (Denoted below is the electron configuration followed by [^2S+1L_J] which describes the ground state term.

H: 1s (one electron in the 1s state) [²S_1/2]
He: 1s² (two electrons in the 1s state, that state is now full) [¹S₀]
That completes the first row
Li: 1s²2s (two electrons filling the 1s state, because of l-tilting the next available state is 2s) [²S_1/2]
Be: 1s²2s² (two electrons filling the 1s state; two electrons filling the 2s state) [¹S₀]
B: 1s²2s²2p (two electrons filling the 1s state; two electrons filling the 2s state; the last electron in the lowest available state: 2p) [²P_1/2]
C: 1s²2s²2p² [³P₀]
N: 1s²2s²2p³ [⁴S_3/2]
O: 1s²2s²2p⁴ [³P₂]
F: 1s²2s²2p⁵ [²P_3/2]
Ne: 1s²2s²2p⁶ [¹S₀]
That completes the second row
Na: 1s²2s²2p⁶3s [²S_1/2]
Mg: 1s²2s²2p⁶3s² [¹S₀]
Al: 1s²2s²2p⁶3s²3p [²P_1/2]
Si: 1s²2s²2p⁶3s²3p² [³P₀]
P: 1s²2s²2p⁶3s²3p³ [⁴S_3/2]
S: 1s²2s²2p⁶3s²3p⁴ [³P₂]
Cl: 1s²2s²2p⁶3s²3p⁵ [²P_3/2]
Ar: 1s²2s²2p⁶3s²3p⁶ [¹S₀]
Surprisingly that completes the third row. l-tilting is so extreme that the 4s has slightly lower energy than 3d. For larger Z the 3d will go below 4s, but currently 4s fills.
K: 1s²2s²2p⁶3s²3p⁶4s [²S_1/2]
Ca: 1s²2s²2p⁶3s²3p⁶4s² [¹S₀]
Now we start to fill 3d in row 4! The result is the "transition elements". There are two oddities it the filling sequence...can you find them? I'm tired of repeating all the lower levels...I'll just use: (Ar) to mean the filled shells of Argon: 1s²2s²2p⁶3s²3p⁶
Sc: (Ar) 3d4s² [²D_3/2]
Ti: (Ar) 3d²4s² [³F₂]
V: (Ar) 3d³4s² [⁴F_3/2]
Cr:(Ar) 3d⁵4s [⁷S₃]
Mn: (Ar) 3d⁵4s² [⁶S_5/2]
Fe: (Ar) 3d⁶4s² [⁵D₄]
Co: (Ar) 3d⁷4s² [⁹F_9/2]
Ni: (Ar) 3d⁸4s² [³F₄]
Cu: (Ar) 3d¹⁰4s [²S_1/2]
Zn: (Ar) 3d¹⁰4s² [¹S₀]

The below is a physicist's view of the periodic table. Helium has been moved over from its proper place with the other inert gases on the far right, to above the "alkaline earth" elements like Magnesium. Thats to make the normal filling sequence more systematic: you just move across the row. Note the insertion of the "rare earth elements" the "lanthanides" and "actinides"; in rows n=6,7 we start filling the (n-2)f orbitals.

As you might expect for larger n the levels are more closely spaced (just as in the H-atom), and unexpected fillings become more common. Here are some "odd" configurations:

Pd: (Kr)4d¹⁰ [¹S₀]
La: (Xe)5d6s² [²D_3/2]
Ce: (Xe)4f5d6s² [¹G₄]
Gd: (Xe)4f⁷5d6s² [⁹D₂]
Pt: (Xe)4f¹⁴5d⁹6s [³D₃]
Th: (Rn)6d²7s² [³F₂]
U: (Rn)5f³6d7s² [⁵L₆]
Lr: (Rn)5f¹⁴7s²7p [²P_1/2]