Hartree-Fock solutions:
Electron correlation, Aufbau principle,
Koopmans’ theorem & Periodic trends
References
• Ratner Ch. 9.5-, Engel Ch. 10.5-, Pilar Ch. 10
• Modern Quantum Chemistry, Ostlund & Szabo (1982) Ch. 3.3
• Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch.7
• Computational Chemistry, Lewars (2003), Ch. 5
• A Brief Review of Elementary Quantum Chemistry http://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.html
http://vergil.chemistry.gatech.edu/notes/hf-intro/hf-intro.html

Helium Atom First (1 nucleus + 2 electrons) (Review)
• Electron-electron repulsion
• Indistinguishability newly introduced
1. Electron-electron repulsion (correlation)
~H atom electron at r
1
~H atom electron at r
2
: Correlated, coupled
The 1/r
12 term removes the spherical symmetry in He.
We cannot solve this Schrödinger equation analytically.
(Two electrons are not separable nor independent any more.)
 A series of approximations will be introduced.

Hartree-Fock equation (One-electron equation)
V eff includes spherically symmetric
&
- Two-electron repulsion operator (1/r ij
) is replaced by one-electron operator V which takes it into account in an “average” way.
HF
(i),
- Any one electron sees only the spatially averaged position of all other electrons.
- V
HF
(i) is spherically symmetric.
(Instantaneous, dynamic) electron correlation is ignored.
- Spherical harmonics (s, p, d, …) are valid angular-part eigenfunctions (as for H-like atoms).
- Radial-part eigenfunctions of H-like atoms are not valid any more.
optimized

Electron Correlation (P.-O. Löwdin, 1955)
Ref) F. Jensen, Introduction to Computational Chemistry, 2 nd ed., Ch. 4
• A single Slater determinant never corresponds to the exact wave function.
E
HF
> E
0
(the exact ground state energy)
• Correlation energy: a measure of error introduced through the HF scheme
E
C
= E
0
– Dynamical correlation
– Non-dynamical (static) correlation
E
HF
(< 0)
• Post-Hartree-Fock method (We’ll see later.)
– Møller-Plesset perturbation: MP2, MP4, …
– Configuration interaction: CISD, QCISD, CCSD, QCISD(T), …
– Multi-configuration self-consistent-field method: MCSCF, CAFSCF, …

Solution of HF-SCF equation gives

0
Solution of HF-SCF equation:
Z (measure of shielding) 0.31
1.72
2.09
2.42
2.58
2.78
2.86
8.49
8.69
less shielded
8.88
8.93
more shielded
9.10
9.71
3.15
3.17
9.36
10.11
3.51
3.55
9.73
10.52
3.87
3.90
9.93
10.88
4.24
4.24
10.24
11.24

Solution of HF-SCF equation:
Effective nuclear charge
(Z is a measure of shielding.) higher energy, bigger radius lower energy, smaller radius

larger
Source: www.chemix-chemistry-software.com/school/periodic_table/atomic-radius-elements.html
www.periodictable.com/Properties/A/AtomicRadius.v.wt.html
smaller

Physical significance of orbital energies (  i
):
Koopmans’ theorem (T. C. Koopmans, 1934) Physica,
1, 104
As well as the total energy, one also obtains a set of orbital energies.
Remove an electron from occupied orbital a .
Orbital energy = Approximate ionization energy
Ostlund/Szabo
Ch.3.3

Atomic orbital energy levels & Ionization energy of H-like atoms
Total energy eigenvalues are negative by convention. (Bound states)
E n
 -
32

Z
2
 e
2
 e
2
0

4
2 n
2
with n

1 , 2 , 3 ...
a
0

4

0

 e e
2
2 length
 atomic units
IE (1 Ry for H)
Minimum energy required to remove an electron from the ground state energy
1
Ry depend only on the principal quantum number.

Koopmans’ theorem: Validation from experiments

Hartree-Fock orbital energies  i
& Aufbau principle
Hartree-Fock orbital energies  i depend on both the principal quantum number (n) and the angular quantum number (l).
Within a shell of principal quantum number n,
 ns
  np
  nd
  nf
 …

 
For H-like atoms degenerate
”
”

Aufbau (Building-up) principle for transition metals
10.3

Aufbau (Building-up) principle for transition metals

Electronegativity (~ IE + EA)
Na + Cl +   NaCl -   Na + + Cl -
~Lowest Unoccupied
AO/MO (LUMO) small small large high small high large
~Highest Occupied
AO/MO (HOMO) low or deep low or deep large

Periodic trends of many-electron atoms

Periodic trends of many-electron atoms:
Electronegativity http://www.periodictable.com/Properties/A/Electronegativity.bt.wt.html

Periodic trends of many-electron atoms:
1 st ionization energy http://www.periodictable.com/Properties/A/IonizationEnergies.bt.wt.html

Periodic trends of many-electron atoms:
Electron affinity http://www.periodictable.com/Properties/A/ElectronAffinity.bt.wt.html

Periodic trends of many-electron atoms:
“Atomic” radius http://www.periodictable.com/Properties/A/AtomicRadius.bt.wt.html