My introduction to electron correlation is based on multideterminant methods. I introduce the electron-electron cusp condition, configuration interaction, complete active space self consistent field (CASSCF), and just a little information about perturbation theories. These slides were part of a workshop I organized in 2014 at the University of Pittsburgh and for a guest lecture in a Chemical Engineering course at Pitt.

1. Electron Correlation
Albert DeFusco
March 26, 2015

2. Hartree-Fock (SCF) methods
What it does well
Geometries
What it doesn’t do well
Energies
No electron correlation
The Fock operator assumes an average electron-electron repulsion
Each orbital energy is independent of another
Average interaction is optimized through the SCF procedure

3. Perturbation Theory
Corrections to the Hartree-Fock energy and wavefunction
Requires that the corrections be small
HF wavefunction and energy must be pretty good
Møller-Plesset perturbation (MP2, MP3, etc.)
Perturbations are not convergent
Coupled Cluster
CCSD, CCSD(t), CCSDT, CCSDTQ, etc.
Perturbations are guaranteed to converge

4. Correlated electron motion
He atom
ˆH = −
1
2
2
1 −
Z
r1Z
−
1
2
2
2 −
Z
r2Z
+
1
r1,2
This is not the Hartree-Fock Hamiltonian
There is no analytic solution
We must use hyrdrogen atom solutions as a guide
The Hamiltonian diverges at r1,2 = 0
r1,2 = r1 − r2
r1Z = r1 − RZ
r2Z = r2 − RZ
and at the nuclei ri = 0.
The kinetic energy must also diverge to achieve a finite sum.

5. A determinant is a wavefunction
Ψ(x1, x2, . . . , xn) =
1
√
n!
χ1(x1) χ2(x1) · · · χn(x1)
χ1(x2) χ2(x2) · · · χn(x2)
...
... · · ·
...
χ1(xn) χ2(xn) · · · χn(xn)
Rows are electron positions and spins
Columns are spin orbitals
Ψ(x1, x2, . . . ) = −Ψ(x2, x1, . . . )
Exchange either two rows or two columns: antisymmetric

6. Probability density
Ψ(x1, x2) = χ1(x1)χ2(x2)
P(r1, r2) = dω1dω2 |Ψ|2
dr1dr2
The two-electron density
Probability of finding electron 1 at r1 in a volume dr1 while
simultaneously finding electron 2 at r2 in a volume dr2.

7. The helium atom: ground state singlet wavefunction
Hartree-Fock single determinant for the 11
S state
Minimal 1-electron basis: 2 1s restricted spin orbitals
Ψ(x1, x2) =
1
√
2
ψ1s(r1)α(ω1) ψ1s(r1)β(ω1)
ψ1s(r2)α(ω2) ψ1s(r2)β(ω2)
= 2− 1
2 (ψ1s(r2)β(ω2)ψ1s(r1)α(ω1)
−ψ1s(r2)α(ω2)ψ1s(r1)β(ω1))
|Ψ|2
=
1
2
. . .
= ψ1s(r1)
2
ψ1s(r2)
2
The spatial orbital is identical for α and β spin
Spatial motion in uncorrelated

8. The helium atom: triplet wavefunction
Hartree-Fock single determinant for the 23
S state
Minimal 1-electron basis: 1s and 2s spin orbitals
Ψ(x1, x2) =
1
√
2
ψ1s(r1)α(ω1) ψ2s(r1)α(ω1)
ψ1s(r2)α(ω2) ψ2s(r2)α(ω2)
= 2− 1
2 (ψ2s(r2)α(ω2)ψ1s(r1)α(ω1)
−ψ1s(r2)α(ω2)ψ2s(r1)α(ω1))
|Ψ|2
=
1
2
. . .
= ψ1s(r1)
2
ψ2s(r2)
2
+ ψ1s(r2)
2
ψ2s(r1)
2
Parallel spin electrons have Fermi correlation

9. Cusp conditions
Figure: Hylleraas Hamiltonian for Helium with relative coordinates.(Figure from
Fred Manby)
electron-nucleas cusp
limri →0
∂Ψ
∂ri ave
= −ZΨ(ri = 0)
Exists in Slater orbitals, but approximated in Gaussian orbitals
More angular momentum means a sharper cusp
electron-electron cusp
limr1,2→0
∂Ψ
∂r1,2 ave
= 1
2 Ψ(r1,2 = 0)
Leads to a depletion in in the two-electron density at r1,2

10. Hylleraas wavefunction
Figure: Hartree-Fock wavefunction and
Hylleraas wavefunction. (Figure from Fred Manby)
One slater determinant is not
enough
To help electrons “avoid” each
other, we can distribute them
among more orbitals using
many determinants

11. Helium determinants: double zeta basis
Singlet Helium
ψ2s is a “virtual” or unoccupied spin orbital
Ψ2 is a “doubly excited” determinant
CI Singles and Double (CISD)
Why no singles?
Ψ0 = ψ1s
¯ψ1s
Ψ2 = ψ2s
¯ψ2s
Ψ(x1, x2) = K1 ψ1s
¯ψ1s + K2 ψ2s
¯ψ2s
P(r1, r2) = K1ψ1s(r1)ψ1s(r2) + K2ψ2s(r1)ψ2s(r2)
2
lim
r1→r2
P(r1, r2) = K1 ψ1s(r1)
2
+ K2 ψ2s(r1)
2 2

12. Configuration Interaction
Solving for the determinant coefficients
Ψ = K0Ψ0 + K2Ψ2
H =
Ψ0|H|Ψ0 Ψ0|H|Ψ2
Ψ2|H|Ψ0 Ψ2|H|Ψ2
Ψ2|H|Ψ0 = 1¯1 2¯2 − 1¯1 ¯22
= K12
H
K0
K2
= ECISD
K0
K2

13. Helium convergence
ERHF = −2.85516 a.u.
Table: FCI/cc-pVQZ
Spatial Orbital CI coefficient
1s 0.995974
2s -0.039200
2p -0.028429
3d -0.005164
3f -0.001472
Table: Helium Full CI total energy J.
Chem. Phys. 127 , 224104 (2007)
Basis Set Energy (a.u.)
cc-pVDZ -2.8875948
cc-pVTZ -2.9002322
cc-pVQZ -2.9024109
cc-pV5Z -2.9031519
Exact Energy -2.9037225
The difference between Full CI and Hartree-Fock energies is
called the correlation energy

14. Helium cusp
Figure: Helium wavefunction cusps for cc-pVDZ,
cc-pVTZ, cc-pVQZ and cc-pV5Z basis sets from
Martin Sch¨utz
Slater determinant
expansions have even
powers of r1,2
(Helgaker)

15. Configuration Interaction
Exact when all possible determinants used with a complete basis
set
Full CI
Basis sets
HF has a factorizable two-electron density and requires fewer basis
functions to converge
Correlation methods seek to improve the two-electron density and
require much more functions to converge
Orbitals are frozen at the HF solution

16. More than two electrons
Excited determinants and matrix elements (Slater Rules)
Single excitations are zero by Brilloun’s theorem
Determinants differening by one spin orbital
Ψ|H|Ψp
m = m ˆh p +
N
n
( mn|pn − mn|np )
Determinants differing by two spin orbitals
Ψ|H|Ψpq
mn = mn|pq − mn|qp
determinants differing by more than two spin orbitals contributes 0
H =


Ψ0 H Ψ0 0 Ψ0 H Ψ2 0 0 · · ·
Ψ1 H Ψ1 Ψ1 H Ψ2 Ψ1 H Ψ3 0 · · ·
Ψ2 H Ψ2 Ψ2 H Ψ3 Ψ2 H Ψ4 · · ·
Ψ3 H Ψ3 Ψ3 H Ψ4 · · ·
Ψ4 H Ψ4 · · ·
...



17. Expansions
Slow convergence with excitation
Scaling problem
Full CI grows factorially
Many billion determinant Full CI is possible, but only small
molecules
Truncation CI expansions
CISD: CI Singles and Doubles
CISDT: With triple excitations
CISDTQ: With quadruple excitations
QCISD is closely related to coupled-cluster

18. What is still missing?
Bond breaking
Including transition states
Diradicals
Degenerate orbital pictures

19. H2 dissociation
UHF
Not smooth due to spin
contamination
RHF
MP2 likely to fail

20. H2 dissociation determinants
A single determinant has a spurious J integral at long range
ψσ = φA + φB
ψσ∗ = φA − φB
Eeq = 2hσσ + Jσσ
E∞ = 2hσσ + Jσσ = 2hσ∗σ∗ + Jσ∗σ∗

21. H2 dissociation determinants
Full CI minimum basis

22. Separate correlation
Dynamic correlation
Short range cusp conditions
Perturbation theories are very efficient
Requires large basis sets
Static correlation
Degeneracies in orbitals or determinants
“chemical intuition”
Full CI does both

23. Diradicals
What is the Lewis structure of ozone?

24. Ozone
What are the molecular orbitals?
Each oxygen supplies 1s, 2s and 3 2p orbitals
Hartree-Fock will only fill the first 12 orbitals
The HOMO energy is -0.4810 a.u.
The LUMO energy is -0.0299 a.u.

25. Ozone

26. Ozone
Distribute 4 electrons in the 3 π orbitals
Full CI = CISD
0.929
-0.224
-0.224
-0.166
-0.066
0.066
Determinant weights for Hartree-Fock orbitals

27. Ozone
Multi-configurational Self Consistent Field (MCSCF)
Improve the energy by minimizing the orbitals
New definition of Fock matrix
Fully Optimized Reaction Space (FORS)
Complete Active Space Self Consistent Field (CAS)
Full CI within a “chosen” orbital space
MCSCF does not imply CASSCF

28. Ozone
Determinant weights for CASSCF optimized orbitals
0.864
-0.240
-0.240
-0.295
-0.159
0.159

29. What about the orbitals?
Hartree-Fock orbitals have little physical meaning
DFT orbitals even less
Orbital energies may be helpful to pick active space
Determinants define electron configurations
Natural orbitals have populations
Diagonalize the density matrix
U are orbitals and n are electron occupations
Ψ =
k
Ak Ψk
P = Ψ∗
Ψ
PU = nU

30. Ozone
Electron populations in natural orbitals for the active space
1.993 1.598 0.409

31. Dynamic correlation
CASSCF has no dynamic correlation
Similar to Hartree-Fock
Multi-reference CI (MR-CISD)
Very accurate for excitation energies
Still not size-consistent
Alternatives to MR-CI expansions, perturbation theory
MRPT (MRMP), CASPT, MCQDPT
MRCC: Multiple types
Good for potential energy surfaces

32. Pople Diagram
Figure: From Chem. Soc. Rev., 2012,41, 6259-6293