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Lecture 6. Many-Electron Atoms. Pt.4.
Physical significance of 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
~H atom electron
at r1
: Correlated, coupled
at r2
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.
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
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 > E0 (the exact ground state energy)
• Correlation energy: a measure of error introduced through the HF scheme
EC = E0 - EHF (< 0)
– Dynamical correlation
– Non-dynamical (static) correlation
• 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
Solution of HF-SCF equation:
Z- (measure of shielding)
0
0.31
1.72
2.09
2.42
2.58
2.78
2.86
3.15
3.17
3.51
3.55
3.87
3.90
4.24
4.24
8.49
8.69
8.88
8.93
9.10
9.71
9.36
10.11
9.73
10.52
9.93
10.88
10.24
11.24
less shielded
more shielded
Solution of HF-SCF equation:
Effective nuclear charge
(Z- is a measure of shielding.)
higher energy, bigger radius
lower energy, smaller radius
smaller
larger
Source: www.chemix-chemistry-software.com/school/periodic_table/atomic-radius-elements.html
www.periodictable.com/Properties/A/AtomicRadius.v.wt.html
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)
Z 2 e 4
En  32 2e02 2 n 2
with n  1,2,3...
40  2
a0 
 ee2
m
length

atomic units
energy
1
Ry
depend only on the
principal quantum number.
IE (1 Ry for H)
Minimum energy
required to remove
an electron from
the ground state
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)
large
small
high
large
small
high
large
small
~Highest Occupied
AO/MO (HOMO)
low or deep
low or deep
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:
1st 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