Electron Correlation
• Hartree-Fock results do not agree with experiment
• Heirarchy of methods to treat electron-electron interactions electron
correlation ie what approximation do we use for H?
-
repulsion
attraction
+
attraction
• Hartree-Fock theory – just consider 1 electron + “average” repulsion
– Need an initial guess of the average repulsion (ie the electron density)
– Iterate until self-consistent
What Tools Can We Use?
• Density Functional Theory
 quantum method
 in principle “exact”
 faster than traditional ab initio
 variable accuracy
 no systematic improvement
Walter Kohn, Nobel Prize 1998
Density Functional Theory
The energy and electronic properties of the ground state are
uniquely determined by the electron density:
E = E[]
• In principal this expression is exact!
• But we don’t know what the functional is
• Use model systems and fitting to derive expressions giving
different “functionals”
• The electron density is something we can “see”
• The electron density is a 3-dimensional property whereas
wavefunction-based methods are 3N dimensional
• Using the Kohn-Sham orbitals DFT is mathematically
equivalent to HF theory
Density Functional Theory
E[ ]  ET []  EV [ ]  E J []  E X [ ]  EC []
• ET The kinetic energy
• EV The Coulomb attraction of the electrons to the nucleus

• EJ The Coulomb energy of that the electrons would have in
their own field, assuming they moved independently and
if each electron repelled itself
• EX The Exchange energy
• EC The Correlation energy
EXC corrects for the false assumptions in EJ
First Generation DFT
E[ ]  ET []  EV [ ]  E J []  E X [ ]  EC []
•
•
•
•
Energy a functional of  alone
Analytic expressions derived from the uniform electron gas
Local Density Approximation
Local Spin Density Approximation
LDA functionals were originally developed for metals and assume the
electron density is constant, not a sensible assumption in a molecule.
LDA tends to underestimate exchange energies by up to 10%, to
overestimate correlation energies by up to a factor of two and to
“overbind” molecules.
This approximate cancellation of errors made initial LDA results look so
promising…
Second Generation DFT
E[ ]  ET []  EV [ ]  E J []  E X [ ]  EC []
• Used LDA/uniform electron gas expressions for ET, EV and EJ
• Invoked the generalised gradient approximation (GGA) for EXC
to attempt to correct for non-local interactions,
inhomogeneities in the electron gas, using the gradient of the
density:
E GGA
xc    f r,rdr
• Meta functionals incorporate the local kinetic energy density,
t (r), which is dependent on the Kohn-Sham orbitals:

metaGGA
E xc
   f r,r,t rdr
Second Generation DFT
E[ ]  ET []  EV [ ]  E J []  E X [ ]  EC []
• GGAs, and meta-GGAs are “local” functionals because the
electronic energy density at a single spatial point depends only
 on the behavior of the electronic density and kinetic energy at
and near that point.
• Examples of second generation functionals include Becke’s
1986 exchange functional, the LYP correlation functional, and
the PBE and the PW91 functionals
• These functionals are commonly used in plane wave DFT
calculations and in calculations on large systems
Third Generation DFT
E[ ]  ET []  EV [ ]  E J []  E X [ ]  EC []
• Functionals where the electronic energy is a functional of the
electron density, its gradient and its Laplacian, that is,
 E[; ; 2]
• Hybrid functionals where a proportion of the exact HF
exchange energy is included to introduces a degree of “nonlocal” behaviour
• The most popular hybrid functional is the B3LYP functional:
VW N
EXCr  0.2EXHF  0.8EXLDA  0.72EXB88  0.19EC
 0.81ECLYP
where the coefficients were found empirically
• Hybrid functionals generally perform better than GGA
functionals in chemical applications
Fourth Generation DFT
E[ ]  ET []  EV [ ]  E J []  E X [ ]  EC []
• Meta-hybrid functionals
• Double hybrid functionals

• Extensively parametrised functionals….
• These functionals attempt to correct for the “local” behaviour
of DFT and give much better results for systems with weak or
non-bonded interactions
DFT Performance
• LDA:
– Works well for anything where the uniform electron gas is a
sensible model (eg metals) and for bulk properties
– Not accurate for chemical applications
• GGA:
– Fast
– Binding energies to about 20 kcal/mol
• Hybrid functionals:
– Slower (HF exchange costs)
– 3-12 kcal/mol errors
• Fourth generation functionals:
– Relatively expensive…
– Claim to do a lot better
Density Functional Theory
E[ ]  ET []  EV [ ]  E J []  E X [ ]  EC []
• Density Functional Theory is only marginally more expensive
than HF theory.

• Because it contains an estimate of the electron correlation
energy it should always be used in preference to HF
• HOWEVER, DFT calculations must be validated by comparison
against some higher level of theory (sometimes they fail
catastrophically…)
Model Chemistries
• Theoretical models are defined by specifying a correlation
procedure and a basis set
–
–
–
–
HF/STO-3G: A very simple theoretical model (level of theory)
MP2/6-31G(d): An intermediate level of theory
CCSD(T)/6-311+G(3d,2p): A high level of theory
B3LYP/6-31G(d): A cost effective level of theory
• Some properties (eg geometry) can be obtained reliably at
simple levels of theory
• Others (eg reaction energy, reaction barrier) require a high
level of theory
Beyond HF: Electron Correlation Methods
• For a given basis set, the difference between the exact energy
and the HF energy is the correlation energy, ~ 85 kJ/mol
correlation energy per electron pair
• Dynamic correlation: electrons repel each other and get out of
each other’s way; dynamical motions of electrons are
correlated, so electron repulsion is less than in an independent
electron model such as HF theory
• Static Electron Correlation/Non-Dynamic Electron
Correlation/Intrinsic Electron Correlation: arises when a single
configurational treatment (ie a single determinant) is not
adequate to describe the problem (eg the ground state of the
molecule)
Frozen Core Approximation
• assume that only the valence electrons are correlated
• the core orbitals are treated at the HF level of theory
• This assumption is normally good for systems involving first
and second row atoms
• For third row, or higher, the approximation should probably be
checked
• for example, if you are not careful in studying a molecule like
CaF2, you may find that in the Ca2+ species none of the
electrons have been correlated because the 3p orbitals are
considered core orbitals (and the F atoms have effectively
removed the valence 4s electrons)
Møller-Plesset Perturbation Theory
• Although correlation energy is large on a chemical scale it is
small compared to the total energy of an atom
• We can treat correlation as a perturbation to the HF
Hamiltonian
H  H 0  lH1
• Expand the perturbation in l:

  (0)  l(1)  l2(2)  l3(3)  ...
E  E(0)  lE(1)  l2E(2)  l3E(3)  ...
• Møller-Plesset perturbation theories, MP2, MP3, MP4… are
obtained by setting l=1 and truncating at the 2nd, 3rd, 4th…

order terms in l for the wavefunction (l+1 for the energy).
Møller-Plesset Perturbation Theory
• Not a variational method
• Overcorrection possible
• Not appropriate if compound is not well described by a simple
Lewis structure
• Does not do well in cases of spin contamination
• Computational effort nN4 (MP2) n3N4 (MP4) for n electrons and
N orbitals
• Does not converge smoothly (oscillates)
• Sometimes nonconvergent series (eg Ne)
• MP2 often gives better results than MP3, MP4…
Configuration Interaction
• Mathematically we want to allow electrons in the wavefunction
to be able to move together
• We can re-expand the wavefunction in terms of some
orthogobal basis that encapsulates this concerted movement
• We can use all possible HF-SCF determinants as this basis
• A determinant describes an electron configuration, “excited”
determinants excite one or more electrons into unoccupied
orbitals
• They are all orthogonal to each other
• This single, double, triple etc excitation correlates the electrons
Configuration Interaction
• The wavefunction is expanded as a linear combination of all
possible HF-SCF determinants
n
  b00   bs s
s 0
• The CI coefficients bs are determined variationally
• The size of the FCI calculation depends on the number of
 n, and the number of orbitals, N.
electrons,
• For N basis functions there are 2N spinorbitals and the total
number of determinants is (2N!)/[n!(2N-n)!] ~ eN
Configuration Interaction
O occupied
V virtual orbitals
•
•
•
•
CIS – include all possible single electron excitations:
simplest qualitative method for electronic excited states, but not for
correlation of the ground state
CISD – include all single and double excitations (yields ~O2V2 determinants)
most useful for correlating the ground state
CISDT – singles, doubles and triples (~O3V3 determinants)
Full CI (FCI) – (~((O+V)!/O!V!) determinants) exact for a given basis set
Coupled Cluster Theory
• The CI expansion converges slowly
• Some excitations are more important than others…
• Define a cluster operator:
T= 1 + T1 + T2 + T3 +…
• Write as
• Where

  eT0
HeT0  EeT0
• This is particularly clever because


 1
eX  
Xk
k0 k!
Coupled Cluster Theory
• If we truncate T
T= 1 + T1 + T2
• Then eT will contain products of T1 and T2 that are equivalent
to higher order excitations
– T12 represents all double excitations arising from “disconnected”
single excitations,
– T22 represents all quadruple excitations arising from disconnected
double excitations
These disconnected excitations turn out to be important (than
the generic n-electron excitations) so the coupled cluster
wavefunction converges much more rapidly than the CI
expansion
CCSD(T)
• Truncate T
T= 1 + T1 + T2
• Include T3 as a perturbation
• Simpler and faster and almost as accurate as CCSDT
The CCSD(T) method is the highest level theory available for
routine use. With a large basis set CCSD(T) is considered the
“Gold Standard” for dynamic electron correlation:
CCSD(T)/aug-ccpVTZ
Static Correlation
• If the wavefunction is not well described as a single
determinant
–
–
–
–
–
–
Species with significant diradical character
Transition States (frequently)
Bond breaking processes
Often for excited electronic states
Unsaturated transition metal complexes
molecules containing atoms with low-lying excited states (Li, Be,
transition metals, etc)
– along reaction paths in many chemical and photochemical
reactions
– Generally any species with near degeneracies
The T1 diagnostic in CC methods is an indicator of the validity of a single
reference approach.
T1 > 0.01 casts suspicion on the applicability of single reference methods.
Multi-Configuration Methods
• Similar to the CI expansion
• Optimise the one-electron orbitals rather than leave them at
their HF values
• Eg MR-CI(SD), CASSCF, CASPT2 …
• Starting to get into some serious computational expense…
Cyclobutadiene
• 4 p electrons and 4 p molecular orbitals
• Each diagram represents a determinant (a configuration state
function)
• The overall wavefunction is a combination of the possible
determinants
• The coefficients of the orbitals change with their occupancy
(consider square vs rectangular cyclobutadiene)
Cyclobutadiene
Active space: 4 electrons in 4 orbitals
Core inactive space: remaining 16 electrons in 8 orbitals
Virtual/Unoccupied orbitals
Assessment of Correlated Methods
Method
Approx. Time Factor
HF
Av. Error (kcal/mol)
vs FCI
5-30
DFT
2-10
ON3
MP2
17.4
ON4
MP3
14.4
MP4
3.7
MP5
3.2
CISD
13.8
CCSD
4.4
CCSD(T)
0.7
O2N7
CCSDT
0.5
O2N>7
CCSDTQ
0.0
O2N>>7
ON2-3
ON5
ON6
Model Chemistries
Choosing a method (theoretical model) in ab initio calculations
involves striking a compromise between accuracy and
computational expense – the more reliable the calculations
generally the more computationally demanding
The method chosen depends on
–
–
–
–
The
The
The
The
size of the molecule being examined
property being calculated
accuracy that is required
computing resources that are available
Pople Diagram
•
John Pople
Nobel Prize 1998
A specific level of theory (theoretical model)
corresponds to a combination of correlation procedure
and basis set
Improvement of
Correlation Treatment
Improvement
of Basis Set
HF MP2 MP4 CCSD(T)
Full
Configuration
Interaction
STO-3G
3-21G
6-31G(d)
6-311+G(2df,p)
Completely
Flexible Basis
Set
Exact Soln of
Schrödinger
Equation
Composite Methods
• Extrapolate to the bottom right corner of the Pople diagram
• Aim is better than 1 cal/mol accuracy
•
• Gaussian “n” methods
– G1, G2, G2MP2, G3…
• Complete Basis Set Limit (CBS) methods
• Weizmann Wn methods
• HEAT method…
Summary
• Computational chemistry can be used to predict molecular
properties, such as:
1. Equilibrium geometries
2. Transition structures
3. Reaction potential energy surfaces
• Many tools are available. In general, the more accurate the
method the more costly it is to use
• Before using a particular approach and methodology, you need
to make sure it is accurate enough for your particular problem
Some Observations
• Chemists like simple systems
• Chemists are interested in electrons so they tend to use
the most accurate methods they can
• Big problems need to be distilled into small enough bits to
provide sensible results
• Everything kicks up more
questions, nothing is ever
as simple as it seems