Textbook errors: Hund’s rule

Hund’s rules (empirical ) For a given electron configuration, the state with maximum multiplicity has the lowest energy For a given multiplicity, the state with the largest value of the orbital angular momentum number has the lowest energy In an atom with outermost subshell half-filled or less, the state with the lowest value of the total angular momentum quantum number has the lowest energy Hund’s rule 2 Also applies to restroom urinals

Friedrich Hermann Hund 4 February 1896 – 31 March 1997 German physicist Doctoral advisor – Max Born Contributions  Hund’s rules  Hund-Mulliken MO theory  Quantum tunneling Hund’s rule 3

The myth In almost all textbooks, Hund’s rule has long been interpreted as a reduction in the electron-electron repulsion energy, V ee, in the higher multiplicity state Electrons with the same spin are kept apart due to Pauli’s exclusion principle Electron-electron repulsion is smaller in the higher multiplicity state Hund’s rule 4

Einstein’s razor “Everything should be made as simple as possible, but no simpler.” A. Einstein (sort of) Hund’s rules 5

Atomic units Units of energy in hartrees (E h ) 1 E h =  J Units of length in bohrs (a 0 ) 1 a 0 = pm In atomic units, most atomic/molecular quantities have magnitudes ~0.1 – 100 Hydrogen atom ground state  Ionization energy = 0.5 E h  Average radius = 1 a 0 Hund’s rule 6

Helium atom G.W.F. Drake and Z-C. Yan, Advances in Quantum Chemistry 53 (2008) 37–56. DOI: /S (07) Hund’s rule 7

Calculation method Hylleraas variational perturbation theory * 70-term correlated wavefunction Energy corrections through 25 th order * H. E. Montgomery, Jr. European Journal of Physics 32 (2011)

Useful expectation values Hund’s rule 9

Helium ground state 1s(1)  (1)1s(2)  (2)  = spin  = +½  = spin  = -½ Multiplicity = S + 1 (+½) + (-½) + 1 = 1 = singlet m i even 1 1 S (ground state singlet) Hund’s rule 10

He 1 1 S energy components E = T + V en + V ee = E h  T = E h  V en = E h  V ee = E h Virial theorem: 2T = -(V en + V ee ) Hund’s rule 11

Excited L = 0 states Promote one electron to the 2s state Two possible spin states  Singlet -  (1)  (2) 2 1 S - m i even  Triplet -  (1)  (2) 1 3 S - m i odd Energies  2 1 S E h  1 3 S E h Hund’s rule 12  = E h

Comparison of average values 21S21S13S13S2 1 S-1 3 S E T V en V ee <r><r> Hund’s rule 13

What’s going on V ee in the 1 3 S state is greater than V ee in the 2 1 S state V en in the 1 3 S state is more negative than V en in the 2 1 S state Electrons in the 1 3 S state are closer together than electrons in the 2 1 S state Electrons in the 1 3 S state are closer to the nucleus than electrons in the 2 1 S state Hund’s rule 14