Ĥ  = E  Quantum Mechanics and Atomic Orbitals Bohr and Einstein particle nature of light DeBroglie wave nature of particles Schrödinger theoretical descriptions of atoms Heisenberg quantum or wave mechanics Dirac wave function =  every allowed e- state has unique  to calculate energy use Ĥ Ĥ  = E 

Ĥ  = E  solved for hydrogen wave functions  energies E 2 = probability distribution probability of finding an e- in H at a particular distance from the nucleus orbital

orbital requires 3 quantum numbers “address” n l ml magnetic -l, …, l orientation angular momentum 0, 1, 2, …, (n - 1) shape principal 1, 2, 3, … size and energy

principal quantum number size energy as n increases orbital requires 3 quantum numbers n l ml principal quantum number size energy as n increases orbitals become larger e- is further from the nucleus n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7

orbital requires 3 quantum numbers n l ml angular momentum shape 0  n - 1 n = 1 l = 0 designated by letters n = 2 l = 0, 1 l = 0 s orbital n = 3 l = 0, 1, 2 l = 1 p orbital n = 4 l = 0, 1, 2, 3 l = 2 d orbital l = 3 f orbital

n = 1 l = 0 designated by letters n = 2 l = 0, 1 l = 0 s orbital n = 3 p orbital n = 4 l = 0, 1, 2, 3 l = 2 d orbital l = 3 f orbital s p n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 d f

orbital requires 3 quantum numbers n l ml magnetic quantum number -l,…, l s row s n = 3 l = 0 m = 0 1 n = 1 l = 0 m = 0 1 p s l = 1 m = -1 n = 2 l = 0 m = 0 1 3 p m = 0 l = 1 m = -1 m = 1 d m = 0 3 l = 2 m = -2 m = 1 m = -1 1 s orbital 5 m = 0 3 p orbitals m = 1 5 d orbitals m = 2

1 s orbital 3 p orbitals 5 d orbitals each orbital holds 2e- 4th quantum number ms spin  7 f orbitals  n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 p d f s

2 1s orbital spherical 2 2s and 3s

1p orbital dumbbell shape 2p orbitals 3 3p, 4p, 5p etc. similar shapes larger

3 d orbitals 5 cloverleaf larger n same shapes larger

Polyelectronic Atoms Pauli exclusion principle no 2 electrons same 4 quantum numbers lowest energy orbitals fill first 1s orbital is lowest energy H 1e- 1s1  2s He 2e- 1s2   2p which orbital fills next? 3s 3p 4s where is 3d?

H He Li Be no! Hund’s rule parallel spins B C N O F Ne Na [Ne] 1s 2s 2px 2py 2pz 3s 3px 3py 3pz 4s 3d 3d 3d 3d 3d H He Li Be no! Hund’s rule parallel spins B C N O F Ne Na [Ne]

K [Ar] Ca [Ar] Sc [Ar] Ti [Ar] half full shell stable V [Ar] Cr [Ar] 3dxz 3dyz 3dxy 3dx2-z2 3dz2 4px K [Ar] Ca [Ar] Sc [Ar] Ti [Ar] half full shell stable V [Ar] Cr [Ar] no Mn [Ar] full shell stable Cu [Ar] no