Download CHEMISTRY 131d Atomic and Molecular Structure

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
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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
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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
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Einstein’s razor
“Everything should be made as
simple as possible, but no
simpler.”
A. Einstein (sort of)
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Atomic units
 Units of energy in hartrees (Eh)
1 Eh = 4.35974410-18 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
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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
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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
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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   
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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)
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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)
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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
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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
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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
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