Module 5
1. Electron correlation
Quantum Chemistry FK7009
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Updated May 20, 2016
Electron correlation
Electron correlation
Accuracy versus Speed
Limitations of the mean-field
approximations in HF
Definition of correlation energy
Speed
Force field
Semi-empirical
Ecorr = Eexact − EHF
Varies with basis set.
Complete basis set limit
Hartree–Fock
MP2
Size extensive
E(An) = n × E(A)
Chemical
accuracy
1 kcal/mol
Size consistent
CI
CCSD
E(A + B) = E(A) + E(B)
Accuracy
Quantum Chemistry FK7009
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Electron correlation
Electron correlation
Dynamic: Instantaneous electron–electron correlation
Static: Near degeneracy
Consider H2 dissociation:
At bonding distance: dynamic correlation
At infinite separation: static correlation
At intermediate distances: just correlation
Conditional probability in the density matrix:
Ψ∗ (r1 , r2 , . . . , rn−1 , rn )Ψ(r10 , r20 , . . . , rn0 −1 , rn0 )
Quantum Chemistry FK7009
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Electron correlation
Beyond Hartree–Fock theory
Wavefunction methods:
Density functional methods:
Post-HF Add determinants
DFT Introduces correlation in H
via the exchange–correlation
functional
+
+
+
+
+ ...
Correlation functional derived
from systems with dynamical
correlation.
Static correlation is still lacking.
Still one determinant with integer
occupation!
Static correlation (due to near
degeneracy) can be introduced
with fractional occupation.
Free energy functionals (metals)
or
Multi-configurational DFT
Configuration interaction:
excite from reference Φ
Optimize weight of each
determinant for
predetermined orbitals
Quantum Chemistry FK7009
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Electron correlation
Dynamical correlation
Adding one determinant helps only
little at short distances
But at large separation the static
correlation is fully captured
Conditional probability in the density matrix:
Ψ∗ (r1 , r2 , . . . , rn−1 rn )Ψ(r10 , r20 , . . . , rn0 −1 rn0 )
Quantum Chemistry FK7009
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Electron correlation
True or False?
True
False
Dynamic correlation is the most important when the atoms are far apart.
Static correlation is important for transition metal complexes.
The Hartree–Fock energy is the same as the true energy if the basis set is complete.
In general, more coefficients have to be calculated with post-HF methods than HF.
With configuration interaction, we can in principle calculate the true energy.
Quantum Chemistry FK7009
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Electron correlation
Correlation methods
I. Variational methods (e.g. Configuration interaction, CI)
Ia. Multiconfigurational SCF, e.g. CAS)
II. Perturbation theory (e.g. Møller–Plesset, MP2)
III. Coupled pair theories (e.g. Coupled cluster, CC)
II and III: Size consistent but not variational
I: Only size consistent when not truncated
Ia: Size consistent and variational, but mainly static correlation
a
Based on a configuration expansion, i.e. excitations out of one or several
reference configurations:
Ψ=
X
cn Φn +
ref
| {z }
Reference
X
a
a
ci Φi +
i,a
X
ab
ab
cij Φij +
ij,ab
| {z }
Single exc.
|
X
abc
abc
cijk Φijk + . . .
i
ijk ,abc
{z
Double exc.
}
| {z }
Triple exc.
Truncation!
Each determinant represents a particular electron configuration
Quantum Chemistry FK7009
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