Lecture 6. Many-Electron Atoms. Pt.4. Physical significance of Hartree-Fock solutions: Electron correlation, Aufbau principle, Koopmans

1
Lecture 6. Many-Electron Atoms. Pt.4.Physical
significance of Hartree-Fock solutionsElectron
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

2
Helium Atom First (1 nucleus 2 electrons)
(Review)

• Electron-electron repulsion
• Indistinguishability

newly introduced
1. Electron-electron repulsion (correlation)
H atom electron at r1
H atom electron at r2
Correlated, coupled
The 1/r12 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.
3
Hartree-Fock equation (One-electron equation)
spherically symmetric
Veff includes

• - Two-electron repulsion operator (1/rij) is
  replaced by one-electron operator VHF(i), which
  takes it into account in an average way.
• - Any one electron sees only the spatially
  averaged
• position of all other electrons.
• - VHF(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
4
Electron Correlation (P.-O. Löwdin, 1955)
Ref) F. Jensen, Introduction to Computational
Chemistry, 2nd ed., Ch. 4

• A single Slater determinant never corresponds to
  the exact wave function.
• EHF gt E0 (the exact ground state energy)
• Correlation energy a measure of error introduced
  through the HF scheme
• EC E0 - EHF (lt 0)
• Dynamical correlation
• Non-dynamical (static) correlation
• Post-Hartree-Fock method (Well see later.)
• Møller-Plesset perturbation MP2, MP4,
• Configuration interaction CISD, QCISD, CCSD,
  QCISD(T),
• Multi-configuration self-consistent-field method
  MCSCF, CAFSCF,

5
Solution of HF-SCF equation gives
6
Solution of HF-SCF equation Z-? (measure of
shielding)
0
0.31
1.72
2.09
2.42
2.78
3.15
3.51
3.87
4.24
2.58
2.86
3.17
3.55
3.90
4.24
8.49
8.88
9.10
9.36
9.73
9.93
10.24
8.69
8.93
9.71
10.11
10.52
10.88
11.24
less shielded
more shielded
7
Solution of HF-SCF equation Effective nuclear
charge (Z-? is a measure of shielding.)
higher energy, bigger radius
lower energy, smaller radius
8
smaller
larger
Source www.chemix-chemistry-software.com/school/p
eriodic_table/atomic-radius-elements.html
www.periodictable.com/Properties/A/AtomicRadius.v.
wt.html
9
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.
Ostlund/Szabo Ch.3.3
Orbital energy Approximate ionization energy
10
Atomic orbital energy levels Ionization
energy of H-like atoms
Total energy eigenvalues are negative by
convention. (Bound states)
?
?
IE (1 Ry for H)
length
Minimum energy required to remove an electron
from the ground state
atomic units
?
energy
1
Ry
depend only on the principal quantum number.
11
Koopmans theorem Validation from experiments
12
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
?
?
?


13
Aufbau (Building-up) principle for transition
metals
14
Aufbau (Building-up) principle for transition
metals
15
Electronegativity ( IE EA)
Na? Cl ? ? ? NaCl ? ? ? Na Cl?
Lowest Unoccupied AO/MO (LUMO)
large
small
high
large
small
high
low or deep
large
low or deep
Highest Occupied AO/MO (HOMO)
small
16
Periodic trends of many-electron atoms
17
Periodic trends of many-electron atoms
Electronegativity
http//www.periodictable.com/Properties/A/Electron
egativity.bt.wt.html
18
Periodic trends of many-electron atoms 1st
ionization energy
http//www.periodictable.com/Properties/A/Ionizati
onEnergies.bt.wt.html
19
Periodic trends of many-electron atoms Electron
affinity
http//www.periodictable.com/Properties/A/Electron
Affinity.bt.wt.html
20
Periodic trends of many-electron atoms Atomic
radius
http//www.periodictable.com/Properties/A/AtomicRa
dius.bt.wt.html