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Introduction to Electron Correlation
The document discusses electron correlation and methods for calculating electronic structures in quantum chemistry, including Hartree-Fock, perturbation theory, and configuration interaction approaches. Key topics include the divergence of Hamiltonians, the limitations of Hartree-Fock methods regarding electron correlation, and the advantages of configuration interaction and coupled-cluster methods for achieving accurate energies. It also delves into the specific cases of the helium atom and ozone, emphasizing the importance of multi-configurational techniques for optimizing molecular orbitals.
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Introduction to Electron Correlation
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- 2. Hartree-Fock (SCF) methods Whatit 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 tothe 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 Heatom 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 isa 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: HylleraasHamiltonian 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-Fockwavefunction 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: doublezeta 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 forthe 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: Heliumwavefunction 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 whenall 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 twoelectrons 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 withexcitation 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 stillmissing? Bond breaking Including transition states Diradicals Degenerate orbital pictures
- 19. H2 dissociation UHF Not smoothdue to spin contamination RHF MP2 likely to fail
- 20. H2 dissociation determinants Asingle 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 FullCI minimum basis
- 22. Separate correlation Dynamic correlation Shortrange 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 theLewis structure of ozone?
- 24. Ozone What are themolecular 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.
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- 26. Ozone Distribute 4 electronsin 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 ConsistentField (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 forCASSCF optimized orbitals 0.864 -0.240 -0.240 -0.295 -0.159 0.159
- 29. What about theorbitals? 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 innatural orbitals for the active space 1.993 1.598 0.409
- 31. Dynamic correlation CASSCF hasno 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: FromChem. Soc. Rev., 2012,41, 6259-6293