Electron Correlation

Albert DeFusco

March 26, 2015

Hartree-Fock (SCF) methods

I What it does wellI Geometries

I What it doesn’t do wellI Energies

I No electron correlationI The Fock operator assumes an average electron-electron repulsionI Each orbital energy is independent of anotherI Average interaction is optimized through the SCF procedure

Perturbation Theory

I Corrections to the Hartree-Fock energy and wavefunctionI Requires that the corrections be smallI HF wavefunction and energy must be pretty good

I Møller-Plesset perturbation (MP2, MP3, etc.)I Perturbations are not convergent

I Coupled ClusterI CCSD, CCSD(t), CCSDT, CCSDTQ, etc.I Perturbations are guaranteed to converge

Correlated electron motion

He atom

H = −12∇

21 −

Zr1Z−

12∇

22 −

Zr2Z

+1

r1,2

I This is not the Hartree-Fock HamiltonianI There is no analytic solutionI We must use hyrdrogen atom solutions as a guide

I The Hamiltonian diverges at r1,2 = 0I r1,2 = r1 − r2I r1Z = r1 − RZI r2Z = r2 − RZ

I and at the nuclei ri = 0.I The kinetic energy must also diverge to achieve a finite sum.

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)

∣∣∣∣∣∣∣∣∣∣∣∣I Rows are electron positions and spinsI Columns are spin orbitalsI Ψ(x1,x2, . . . ) = −Ψ(x2,x1, . . . )

I Exchange either two rows or two columns: antisymmetric

Probability density

Ψ(x1,x2) =∣∣∣χ1(x1)χ2(x2)

⟩P(r1, r2) =

∫dω1dω2 |Ψ|

2 dr1dr2

I The two-electron densityI Probability of finding electron 1 at r1 in a volume dr1 while

simultaneously finding electron 2 at r2 in a volume dr2.

The helium atom: ground state singlet wavefunctionI Hartree-Fock single determinant for the 11S stateI 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−

12 (ψ1s(r2)β(ω2)ψ1s(r1)α(ω1)

−ψ1s(r2)α(ω2)ψ1s(r1)β(ω1))

|Ψ|2 =12. . .

=∣∣∣ψ1s(r1)

∣∣∣2 ∣∣∣ψ1s(r2)∣∣∣2

I The spatial orbital is identical for α and β spinI Spatial motion in uncorrelated

The helium atom: triplet wavefunction

I Hartree-Fock single determinant for the 23S stateI 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−

12 (ψ2s(r2)α(ω2)ψ1s(r1)α(ω1)

−ψ1s(r2)α(ω2)ψ2s(r1)α(ω1))

|Ψ|2 =12. . .

=∣∣∣ψ1s(r1)

∣∣∣2 ∣∣∣ψ2s(r2)∣∣∣2 +

∣∣∣ψ1s(r2)∣∣∣2 ∣∣∣ψ2s(r1)

∣∣∣2I Parallel spin electrons have Fermi correlation

Cusp conditions

Figure: Hylleraas Hamiltonian for Helium with relative coordinates.(Figure from

Fred Manby)

I electron-nucleas cusplimri→0

(∂Ψ∂ri

)ave

= −ZΨ(ri = 0)

I Exists in Slater orbitals, but approximated in Gaussian orbitalsI More angular momentum means a sharper cusp

I electron-electron cusplimr1,2→0

(∂Ψ∂r1,2

)ave

= 12 Ψ(r1,2 = 0)

I Leads to a depletion in in the two-electron density at r1,2

Hylleraas wavefunction

Figure: Hartree-Fock wavefunction andHylleraas wavefunction. (Figure from Fred Manby)

I One slater determinant is notenough

I To help electrons “avoid” eachother, we can distribute themamong more orbitals usingmany determinants

Helium determinants: double zeta basis

I Singlet HeliumI ψ2s is a “virtual” or unoccupied spin orbitalI Ψ2 is a “doubly excited” determinantI CI Singles and Double (CISD)

I Why no singles?

Ψ0 =∣∣∣ψ1sψ1s

⟩Ψ2 =

∣∣∣ψ2sψ2s

⟩Ψ(x1,x2) = K1

∣∣∣ψ1sψ1s

⟩+ K2

∣∣∣ψ2sψ2s

⟩P(r1, r2) =

∣∣∣ K1ψ1s(r1)ψ1s(r2) + K2ψ2s(r1)ψ2s(r2)∣∣∣2

limr1→r2

P(r1, r2) =(K1

∣∣∣ψ1s(r1)∣∣∣2 + K2

∣∣∣ψ2s(r1)∣∣∣2)2

Configuration Interaction

I Solving for the determinant coefficients

Ψ = K0Ψ0 + K2Ψ2

H =

( ⟨Ψ0|H|Ψ0

⟩ ⟨Ψ0|H|Ψ2

⟩⟨Ψ2|H|Ψ0

⟩ ⟨Ψ2|H|Ψ2

⟩ )⟨Ψ2|H|Ψ0

⟩=

⟨11

∣∣∣22⟩−

⟨11

∣∣∣22⟩

= K12

H(

K0K2

)= ECISD

(K0K2

)

Helium convergence

ERHF = −2.85516 a.u.

Table: FCI/cc-pVQZ

Spatial Orbital CI coefficient1s 0.9959742s -0.0392002p -0.0284293d -0.0051643f -0.001472

Table: Helium Full CI total energy J.

Chem. Phys. 127 , 224104 (2007)

Basis Set Energy (a.u.)cc-pVDZ -2.8875948cc-pVTZ -2.9002322cc-pVQZ -2.9024109cc-pV5Z -2.9031519

Exact Energy -2.9037225

I The difference between Full CI and Hartree-Fock energies iscalled the correlation energy

Helium cusp

Figure: Helium wavefunction cusps for cc-pVDZ,cc-pVTZ, cc-pVQZ and cc-pV5Z basis sets fromMartin Schutz

I Slater determinantexpansions have evenpowers of r1,2(Helgaker)

Configuration Interaction

I Exact when all possible determinants used with a complete basisset

I Full CI

I Basis setsI HF has a factorizable two-electron density and requires fewer basis

functions to convergeI Correlation methods seek to improve the two-electron density and

require much more functions to converge

I Orbitals are frozen at the HF solution

More than two electronsI Excited determinants and matrix elements (Slater Rules)

I Single excitations are zero by Brilloun’s theoremI Determinants differening by one spin orbital⟨

Ψ|H|Ψpm

⟩=

⟨m

∣∣∣h∣∣∣p⟩+

N∑n

(〈mn|pn〉 − 〈mn|np〉)

I Determinants differing by two spin orbitals⟨Ψ|H|Ψpq

mn

⟩= 〈mn|pq〉 − 〈mn|qp〉

I 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⟩· · ·

...

Expansions

I Slow convergence with excitation

I Scaling problemI Full CI grows factoriallyI Many billion determinant Full CI is possible, but only small

molecules

I Truncation CI expansionsI CISD: CI Singles and DoublesI CISDT: With triple excitationsI CISDTQ: With quadruple excitationsI QCISD is closely related to coupled-cluster

What is still missing?

I Bond breakingI Including transition states

I DiradicalsI Degenerate orbital pictures

H2 dissociation

I UHFI Not smooth due to spin

contaminationI RHF

I MP2 likely to fail

H2 dissociation determinantsI A single determinant has a spurious J integral at long range

ψσ = φA + φB

ψσ∗ = φA − φB

Eeq = 2hσσ + JσσE∞ = 2hσσ + Jσσ = 2hσ∗σ∗ + Jσ∗σ∗

H2 dissociation determinants

I Full CI minimum basis

Separate correlation

I Dynamic correlationI Short range cusp conditionsI Perturbation theories are very efficientI Requires large basis sets

I Static correlationI Degeneracies in orbitals or determinantsI “chemical intuition”

I Full CI does both

Diradicals

I What is the Lewis structure of ozone?

Ozone

I What are the molecular orbitals?I Each oxygen supplies 1s, 2s and 3 2p orbitals

I Hartree-Fock will only fill the first 12 orbitals

I The HOMO energy is -0.4810 a.u.

I The LUMO energy is -0.0299 a.u.

Ozone

Ozone

I Distribute 4 electrons in the 3 π orbitalsI Full CI = CISD

0.929

-0.224

-0.224

-0.166

-0.066

0.066

Determinant weights for Hartree-Fock orbitals

Ozone

I Multi-configurational Self Consistent Field (MCSCF)I Improve the energy by minimizing the orbitalsI New definition of Fock matrix

I Fully Optimized Reaction Space (FORS)I Complete Active Space Self Consistent Field (CAS)I Full CI within a “chosen” orbital space

I MCSCF does not imply CASSCF

Ozone

I Determinant weights for CASSCF optimized orbitals

0.864

-0.240

-0.240

-0.295

-0.159

0.159

What about the orbitals?

I Hartree-Fock orbitals have little physical meaningI DFT orbitals even lessI Orbital energies may be helpful to pick active space

I Determinants define electron configurations

I Natural orbitals have populationsI Diagonalize the density matrixI U are orbitals and n are electron occupations

Ψ =∑

k

Ak Ψk

P = Ψ∗Ψ

PU = nU

OzoneElectron populations in natural orbitals for the active space

1.993 1.598 0.409

Dynamic correlation

I CASSCF has no dynamic correlationI Similar to Hartree-Fock

I Multi-reference CI (MR-CISD)I Very accurate for excitation energiesI Still not size-consistent

I Alternatives to MR-CI expansions, perturbation theoryI MRPT (MRMP), CASPT, MCQDPTI MRCC: Multiple typesI Good for potential energy surfaces

Pople Diagram

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