Chemistry 6440 / 7440
Electron Correlation Effects
Resources
• Cramer, Chapter 7
• Jensen, Chapter 4
• Foresman and Frisch, Exploring Chemistry
with Electronic Structure Methods,
Chapter 6
Electron Correlation Energy
• in the Hartree-Fock approximation, each electron
sees the average density of all of the other
electrons
• two electrons cannot be in the same place at the
same time
• electrons must move two avoid each other,
i.e. their motion must be correlated
• for a given basis set, the difference between the
exact energy and the Hartree-Fock energy is the
correlation energy
• ca 20 kcal/mol correlation energy per electron pair
General Approaches
• include r12 in the wavefunction
– suitable for very small systems
– too many difficult integrals
– Hylleras wavefunction for helium
• expand the wavefunction in a more convenient
set of many electron functions
– Hartree-Fock determinant and excited determinants
– very many excited determinants, slow to converge
– configuration interaction (CI)
Goals for Correlated Methods
• well defined
– applicable to all molecules with no ad-hoc choices
– can be used to construct model chemistries
• efficient
– not restricted to very small systems
• variational
– upper limit to the exact energy
• size extensive
– E(A+B) = E(A) + E(B)
– needed for proper description of thermochemistry
• hierarchy of cost vs. accuracy
– so that calculations can be systematically improved
Configuration Interaction
  0   tia ia   tijab ijab 
ia
ijab
abc
abc
t

 ijk ijk  
ijkabc
0 | 1 n | reference determinan t
(Hartree - Fock wavef unction)
ia | 1 i 1ai 1 n | singly excited determinan t
(excite occupied orbital i to unoccupied orbital a )
ijab | 1 i 1ai 1  j 1b j 1 n | doubly excited determinan t
(i  a ,  j  b )
etc.
Configuration Interaction
• determine CI coefficients using the variational principle
  0   tia ia   tijab ijab 
ia
ijab
abc
abc
t

 ijk ijk  
ijkabc
ˆ d /  *d with respect to t
minimize E    *H

• CIS – include all single excitations
– useful for excited states, but on for correlation of the ground state
• CISD – include all single and double excitations
– most useful for correlating the ground state
– O2V2 determinants (O=number of occ. orb., V=number of unocc. orb.)
• CISDT – singles, doubles and triples
– limited to small molecules, ca O3V3 determinants
• Full CI – all possible excitations
– ((O+V)!/O!V!)2 determinants
– exact for a given basis set
– limited to ca. 14 electrons in 14 orbitals
Configuration Interaction
Ht  Et
• very large eigenvalue problem, can be
solved iteratively
• only linear terms in the CI coefficients
• upper bound to the exact energy
(variational)
• appliciable to excited states
• gradients simpler than for non-variational
methods
Size Extensivity
• E(A+B) =E(A)+E(B) for two systems at long
distance
• CID for helium with 2 basis functions
  0  t A12A1A2A A
• two helium’s calculated separately
– E=E(A)+E(B), = AB
  (0 A  t A 12A1A2A A )( 0 B  t B 12BB1B2 B )
 0  t A 
 tB 
 t At B 
• CID for two helium’s at long distance
(no quadruply excited determinant)
2 A2 A
2B 2B
  0  t A1A1A  t B 1B1B
2 A2 A
1A1A
2B2B
1B1B
2 A2 A2 B 2 B
1A1A1B1B
Perturbation Theory
• divide the Hamiltonian into an exactly solvable
part and a perturbation
ˆ   E H
ˆ 0  E 00 H
ˆ H
ˆ V
ˆ
H
0 i
i
i
0
• expand the Hamiltonian, energy and
wavefunction in terms of l
ˆ H
ˆ  lV
ˆ E  E  lE  l2 E      l  l2  
H
0
0
1
2
0
1
2
• collect terms with the same power of l
ˆ  E
H
0 0
0 0
ˆ  V
ˆ  E  E  E   V
ˆ  d
H
0
1
0
0
1
1
0
1

ˆ  V
ˆ  E  E  E 
H
0 2
1
0 2
1 1
2 0
0
0
ˆ  d
E2   0 V
1
Møller-Plesset Perturbation Theory
• choose H0 such that its eigenfunctions are
determinants of molecular orbitals
ˆ   Fˆ
H
0
i
• expand perturbed wavefunctions in terms of
the Hartree-Fock determinant and singly,
doubly and higher excited determinants
1   aia ia   aijab ijab 
ia
ijab
abc
abc
a

 ijk ijk  
ijkabc
• perturbational corrections to the energy
ˆ  d   V
ˆ  d
EHF  E0  E1   0 H
0 0
0
0

ˆ  d  E 
EMP 2  EHF  E2  EHF   0 V
1
HF

i  j , a b
ˆ  ab d ]2
[  0 V
ij
a  a  i   j
Møller-Plesset Perturbation Theory
• size extensive at every order
• MP2 - second order relatively cheap (requires only
double excitations)
• 2nd order recovers a large fraction of the correlation
energy when Hartree-Fock is a good starting point
• practical up to fourth order (single, doubly, triple
and quadruple excitations)
• MP4 order recovers most of the rest of the
correlation energy
• series tends to oscillate (even orders lower)
• convergence poor if serious spin contamination or
if Hartree-Fock not a good starting point
Coupled Cluster Theory
• CISD can be written as
ˆ T
ˆ )
CISD  (1  T
1
2
0
• T1 and T2 generate all possible single and
double excitations with the appropriate
coefficients
T̂2 0 
ab
ab
t

 ij ij
i  j , a b
• coupled cluster theory wavefunction
ˆ T
ˆ )
CCSD  exp(1  T
1
2
0
Coupled Cluster Theory
• CCSD energy and amplitudes can be
obtained by solving
ˆ  E ) d  0

(
H
CC
CC
 0
ˆ  E ) d  0

(
H
CC
CC

a
i
ˆ  E ) d  0

(
H
CC
CC

ab
ij
Coupled Cluster Theory
•
•
•
•
•
CC equations quadratic in the ampitudes
must be solve iteratively
cost of each iteration similar to MP4SDQ
CC is size extensive
CCSD - single and double excitations correct to
infinite order
• CCSD - quadruple excitations correct to 4th order
• CCSD(T) - add triple excitations by pert. theory
• QCISD – configuration interaction with enough
quadratic terms added to make it size extensive
(equivalent to a truncated form of CCSD)