Atomic Orbitals: n, ℓ and mℓ Counting Orbitals I Atomic Orbitals: n, ℓ and mℓ Joseph F. Lomax U.S. Naval Academy Copy permission with this footer

Counting up Orbitals: In subshells Number of orbitals in a subshell Seven f-orbitals Five d-orbitals Three p-orbitals One s-orbital Joseph F. Lomax U.S. Naval Academy Copy permission with this footer

Counting up Orbitals: in shells Number of orbitals in a subshell Number of orbitals in a shell Seven f-orbitals + n = 4, 16 orbitals Five d-orbitals + n = 3, 9 orbitals Three p-orbitals + n = 2, 4 orbitals One s-orbital n =1, 1 orbital 1= 12 Because it’s a square 1+3 = 22 Because it’s a square 1+3+5 = 32 Because it’s a square 1+3+5+7 = 42 Because it’s a square Joseph F. Lomax U.S. Naval Academy Copy permission with this footer

Orbitals in shells: Stacking them up 1+3+5+7 = 42 16 Seven f-orbitals Five d-orbitals Three p-orbitals One s-orbital n = 4 1+3+5 = 32 9 Five d-orbitals Three p-orbitals One s-orbital n = 3 1+3 = 22 4 Three p-orbitals One s-orbital n = 2 In Hydrogen-like Orbitals All Orbitals in a shell are Degenerate, i.e. same energy n = 1 1= 12 One s-orbital 1 Joseph F. Lomax U.S. Naval Academy Copy permission with this footer

This works because of the rules of waves. Electrons follow these. Labeling ℓ and mℓ Identify mℓ labels Label ℓ with number or letter ℓ 3 2 1 +3 +2 +1 Seven f-orbitals It is symmetrical. beautiful. +2 +1 -1 Five d-orbitals +1 -1 -2 Three p-orbitals -1 -2 -3 One s-orbital This works because of the rules of waves. Electrons follow these. (“This is bowling. There are rules.” – John Goodman as Walter Sobchak in The Big Lebowski, 1998. Joseph F. Lomax U.S. Naval Academy Copy permission with this footer

Many electron energies 5d 4f 6s Orbitals in shells: Many electron energies 5p 4d 5s One electron with each spin: ms = +1/2 and ms = -1/2 for each orbital Average distance of electrons in same shell is the same. Energy increases (stability decreases) with increasing ℓ, i.e. s, p, d, f If all else fails follow the Periodic Table. 4p 4s 3d 1s 1s 3p 2s 6s 6p 5p 4p 3p 2p 6s 6p 5p 4p 3p 2p 6s 6p 5p 4p 3p 2p 3s 3s 4s 3d 2p 5s 4d 2s 6s 5d 1s 4f 5f Joseph F. Lomax U.S. Naval Academy Copy permission with this footer