1. Textbook errors: Hund’s rule

2. Hund’s rules (empirical - 1925) 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

3. 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

4. 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

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

6. Atomic units Units of energy in hartrees (E h ) 1 E h = 4.359744  10 -18 J Units of length in bohrs (a 0 ) 1 a 0 = 53.9177211 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

7. Helium atom G.W.F. Drake and Z-C. Yan, Advances in Quantum Chemistry 53 (2008) 37–56. DOI:10.1016/S0065-3276(07)53004-2 Hund’s rule 7

8. 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) 1275-1284 8

9. Useful expectation values Hund’s rule 9

10. 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

11. He 1 1 S energy components E = T + V en + V ee = -2.903724 E h  T = +2.903724 E h  V en = -6.753267 E h  V ee = +0.945818 E h Virial theorem: 2T = -(V en + V ee ) Hund’s rule 11

12. 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 -2.145974 E h  1 3 S -2.175229 E h Hund’s rule 12  = 0.029 E h

13. Comparison of average values 21S21S13S13S2 1 S-1 3 S E-2.146-2.175 0.029 T 2.146 2.175-0.029 V en -4.542-4.619 0.077 V ee 0.250 0.268-0.018 5.270 4.448 0.822 <r><r> 2.937 2.550 0.387 Hund’s rule 13

14. 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