Scott’s Guide to Electron Correlation
Scott Allen, Kozlowski Lab, April 10, 2012
The overall energy of a molecule is a function of the electrons, the nuclei, and their interactions.
It would be impossible to calculate this without making approximations.
E = f (Nuclei, Electrons)
•
Born-Oppenheimer Approximation – Compared to the movement of the electrons, the
nuclei are, for all intents and purposes, motionless (think flies on a cow). Thus, the nuclei
can be held stationary, and they can be removed from the equation.
•
Hartree Approximation – Rather than calculate each electron against each other electron,
we calculate each electron against a field of all other electrons and nuclei
E = f (A0 ) f (A1 ) f (A2 )... f (An ) " f (electrons)
!
E = f (A0 )... f (An "1 ) f (An ) # f (e0 ) f (e1 ) f (e2 )... f (en "1 ) f (en )
BUT
No extra consideration is given to the electron with which it shares an orbital. The
energy that comes from the direct interactions of two electrons in the same orbital is
! called correlation energy (or Coulomb correlation because it is a coulombic
repulsion).
o Thus:
!
o
e1
e1
d1
e2
d1'
d2
e2
By the Hartree Approximation,
these two configurations have
the same energy
d2 '
=
If d1 = d1'
and
d2 = d2 '
Normally this is fine, because electrons don’t want to get close (like charges repel)
BUT
Dispersion Interactions: The weak interaction between two orbitals caused by rearrangement of
the electrons, creating a weak dipole
Textbook definition:
•
±
±
≠
+– +–
∴ HF (Hartree-Fock) theory usually reports dispersion interactions as repulsions between two
orbitals.
NOTE: HF includes a bit of correlation, because each electron is calculated against an
average field of all other electrons, and its neighbor is part of that field. Also, HF includes terms to
prevent two electrons of parallel spin from occupying the same space (called Fermi correlation)
An Example:
•
H
H
H
H
H
H
H
H
Interaction Energy:
= Ecomplex – (Eethylene + Ebenzene)
H
H
HF/6-311G(d,p) +1.14 kcal/mol
MP2/6-311G(d,p) –1.27 kcal/mol
•
This CH/π interaction does have an
electrostatic component, but it’s not
strong enough to overcome the
repulsion in HF theory.
The MP2 level of theory includes a
calculation of electron correlation, so it
calculates a favorable interaction.
1/2
Amount of Correlation
Number of
basis sets
100%
Density Functional
Theory (DFT)
Older
methods
(B3LYP, etc)
MP2
Newer
methods
vdW-DFT,
M06
CI
∞
Hartree-Fock Limit
CCSD
The exact solution of
Schrödinger equation
Some notes on above chart:
• Chart shows the accuracy of the energy calculated by the given method.
o Relative energies can still be calculated using lower levels of theory.
o The diagonal of the chart goes with increasing computational cost. Large organic
systems (especially transition states) are far too expensive for CCSD and CI, this is
why DFT or HF is usually used.
• HF is variational: The energy calculated is always higher than the actual energy of the
system, and the more basis sets, the more exact the energy gets, eventually converging on
the Hartree-Fock Limit. The HF Limit
is NOT the solution to the Schrödinger
The dependence of accuracy on number
Energy
equation because it includes no
of basis sets for variational (red) and
non-variational (blue) methods
correlation.
All other methods are non-variational
MP2 tends to overestimate electron
correlation
Hartree-Fock Limit
• DFT correlation is parameterized
correct energy (soln
to Schrödinger Eqn)
(based on empirical results)
• CCSD (coupled-cluster) and CI
(configuration interaction) very
expensive for all but the smallest
Number of Basis Sets
systems (handful of atoms) but are
accepted as the most accurate.
Further Reading (all available at UPenn Library):
• Hehre, Warren J. A Guide to Molecular Mechanics and Quantum Chemical Calculations.
• Bachrach, Steven M. Computational Organic Chemistry
• Young, David Computational Chemistry
• Online Resources:
o http://comporgchem.com/blog - Companion blog to Bachrach’s Book
o http://www.ch.imperial.ac.uk/rzepa/blog - Henry Rzepa of Imperial College
•
•
2/2
!