Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Textbook errors: Hund’s rule 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 Also applies to restroom urinals 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 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, Vee, 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 (Eh) 1 Eh = 4.35974410-18 J Units of length in bohrs (a0) 1 a0 = 53.9177211 pm In atomic units, most atomic/molecular quantities have magnitudes ~0.1 – 100 Hydrogen atom ground state Ionization energy = 0.5 Eh Average radius = 1 a0 Hund’s rule 6 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 Calculation method Hylleraas variational perturbation theory* 70-term correlated wavefunction s r1 r2 , t r1 r2 , u r12 s,t, u e s s t li mi u ni i li , mi , ni are integers, li mi ni 7 Energy corrections through 25th order *H. E. Montgomery, Jr. European Journal of Physics 32 (2011) 1275-1284 8 Useful expectation values p12 p22 2me 2me 2 2 r1 r2 1 r12 T electron kinetic energy Ven electron-nucleus attraction Vee interelectronic repulsion r12 average interelectronic distance r average electron-nucleus distance 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 mi even 11S (ground state singlet) Hund’s rule 10 He 1 1S energy components E = T + Ven + Vee = -2.903724 Eh T = +2.903724 Eh Ven = -6.753267 Eh Vee = +0.945818 Eh Virial theorem: 2T = -(Ven + Vee) Hund’s rule 11 Excited L = 0 states Promote one electron to the 2s state Two possible spin states Singlet - (1)(2) 21S - mi even Triplet - (1)(2) 13S - mi odd Energies 21S -2.145974 Eh 13S -2.175229 Eh = 0.029 Eh Hund’s rule 12 Comparison of average values E T Ven Vee <r12> <r> 21S -2.146 2.146 -4.542 0.250 5.270 2.937 13S -2.175 2.175 -4.619 0.268 4.448 2.550 Hund’s rule 21S-13S 0.029 -0.029 0.077 -0.018 0.822 0.387 13 What’s going on Vee in the 13S state is greater than Vee in the 21S state Ven in the 13S state is more negative than Ven in the 21S state Electrons in the 13S state are closer together than electrons in the 21S state Electrons in the 13S state are closer to the nucleus than electrons in the 21S state Hund’s rule 14