CHEM 938: Density Functional Theory

1. CHEM 938: Density Functional Theory Electron Correlation January 19, 2010

2. Electron Correlation based on the variation principle: Hartree-Fock energy real ground state energy why is the Hartree-Fock energy too high? Hartree-Fock method uses a single Slater determinant wavefunction • electron-electron interactions are treated in an average sense • each electron ‘feels’ the average Coulomb repulsion arising from the static distrubution of the other N-1 electrons • instantaneous electron-electron interactions are neglected • causes Coulomb energy to be too high • remedied partially by exchange interactions, which correlate positions of electrons with the same spin (this is kind of correlation)

3. Electron Correlation the correlation energy is: • the correlation energy is everything that is not included in a Hartree-Fock calculation • Ecorr is similar to bond energies  important if calculating reaction energies there are two types of electron correlation 1. Dynamic correlation: • accounts for fact that electrons move such that they avoid other electrons • this is the instantaneous electron-electron interactions missing in Hartree-Fock 2. Static correlation: • in systems with multiple resonance states, electrons can avoid each other by occupying different ‘resonance states’ • a single Slater determinant only represents 1 resonant state, so Hartree-Fock cannot describe this type of correlation • also called non-dynamic correlation

4. Electron Correlation the correlation energy is: • the correlation energy is everything that is not included in a Hartree-Fock calculation • Ecorr is similar to bond energies  important if calculating reaction energies Hartree-Fock also capture exchange, which is a type of electron correlation 3. Exchange: • electrons of the same spin avoid each other to satisfy the Pauli exclusion principle

5. Dynamic Correlation • electrons are charged particles moving in space • their motions are correlated so that they avoid each other • Hartree-Fock calculations do not capture this • consequence of considering ‘static’ electron distributions • direct consequence of using a single Slater determinant wavefunction • only correlation is from exchange interactions Solution: change the form of the wavefunction 1. Multi-determinant Wavefunction • called post-Hartree-Fock methods 2. Perturbation Theory

6. Multideterminant Wavefunctions Slater determinants in a multideterminant wavefunction are derived from the Hartree-Fock Slater determinant wavefunction in Hartree-Fock calculations: • molecular orbitals are expanded as a linear combination of K basis functions • taking linear combinations of K basis functions yields 2K molecular orbitals (considering spin) • only the lowest energy N orbitals are used to build the Hartree-Fock Slater determinant • the remaining 2K – N orbitals are called ‘unoccupied’ or ‘virtual’ orbitals • we can ‘switch’ occupied and unoccupied orbitals to make new Slater determinants

7. Multideterminant Wavefunctions Slater determinants in a multideterminant wavefunction are derived from the Hartree-Fock Slater determinant wavefunction • ‘switching’ orbitals corresponds to exciting electrons into higher energy states virtual energy occupied

8. Multideterminant Wavefunctions Slater determinants in a multideterminant wavefunction are derived from the Hartree-Fock Slater determinant wavefunction • ‘switching’ orbitals corresponds to exciting electrons into higher energy states • we can use single excitations virtual labeled r, s, … energy occupied labeled i, j,…

9. Multideterminant Wavefunctions Slater determinants in a multideterminant wavefunction are derived from the Hartree-Fock Slater determinant wavefunction • ‘switching’ orbitals corresponds to exciting electrons into higher energy states • we can use single excitations virtual labeled r, s, … • we can use double excitations energy • we can excitations from up to all N occupied orbitals into any of the 2K – N virtual orbitals occupied • this gives a whole series of new Slater determinant wavefunctions labeled i, j,… • note that the Hartree-Fock molecular orbitals, , are not re-optimized in this procedure

10. Multideterminant Wavefunctions Slater determinants in a multideterminant wavefunction are derived from the Hartree-Fock Slater determinant wavefunction how does including ‘excited’ Slater determinants improve the wavefunction? 1. we get more variational parameters • more parameters to variationally optimize in guess wavefunction  lower energy solution 2. including extra orbitals gives electrons more space to avoid each other 3 2 consider allyl anion: • excitation allows one of the ‘original’ 2 electrons to ‘spread out’ into 3 1

11. Multideterminant Wavefunctions how are multideterminant wavefunctions used in practice? Full configuration interaction: • also called ‘Full CI’ • includes entire set of excitations from N occupied orbitals into all possible combinations of 2K – N virtual orbitals • this will give the ‘exact’ wavefunction for a specific basis set • the number of Slater determinants is tremendous: • computational effort is prohibitive, and full CI calculations are only used on very small molecules

12. Multideterminant Wavefunctions how are multideterminant wavefunctions used in practice? Truncated configuration interaction: • only consider certain levels of excitations • most common include all single and double excitations (CISD) • reduces significantly the number of determinants in the wavefunction, making the calculations more computationally tractable • double excitations are most important for capturing correlation energy • still recovers ~95 % of the correlation energy • has problems related to size-consistency

13. Size Consistency the energy of two well-separated molecules should be the sum of their energies • truncated CI methods are not size-consistent • consider the double excitations for an A B system • all excitations are from 2 occupied orbitals on A into 2 virtual orbitals on A A • all excitations are from 2 occupied orbitals on B into 2 virtual orbitals on B B • some excitations are from occupied orbitals on A into virtual orbitals on A 50 Å B A • some excitations are from occupied orbitals on B into virtual orbitals on B • some excitations will be from 1 occupied orbital on A into 1 virtual orbital on A and 1 occupied orbital on B into 1 virtual orbital on B more excitations are included in the supersystem than in the sum of the individual molecules with truncated CI

14. Coupled-Cluster Calculations if we include the right excitations, we can use a truncated CI expansion and retain size-consistency coupled-cluster operator: generates all Nth excitations generates all double excitations generates all single excitations coupled-cluster wavefunction:

15. Coupled-Cluster Calculations truncated coupled-cluster wavefunction: • includes all possible single and double excitations • product of operating with the singles operator twice • gives excitations the prevent CISD from being size consistent currently, the CCSD(T) method is the ‘gold standard’ quantum chemical method • includes all possible single, double and some triple excitations • wavefunction involves as huge number of Slater determinants • computational effort prevents application of CCSD(T) to systems with more than ~20 atoms

16. Perturbation Theory basic idea: start with a problem we can solve and add in parts we cannot solve as small perturbations part we cannot solve (perturbation) total Hamiltonian part we can solve dimensionless parameter • varies from 0 to 1 • turns perturbation on or off part we can solve gives: zeroth order ground state energy zeroth order ground state wavefunction

17. Perturbation Theory we are interested in ground state eigenfunctions and eigenvalues for the whole Hamiltonian expand  and E as Taylor series about (0) and E(0): first order correction to the wavefunction first order correction to the energy

18. Perturbation Theory inserting the Taylor expansions into the eigenvalue expression collecting terms with the same power of : set of equations to get the corrections to the energy and wavefunction

19. Perturbation Theory let’s enforce orthonormality: then multiply each equation on the left by 0(0) and integrate: this gives use the zeroth order energy:

20. Perturbation Theory let’s continue: 1 0 0 zeroth order wavefunctions first order correction to the energy in general, the Nth order correction to the energy depends on the N-1, N-2, … corrections to the wavefunction

21. Perturbation Theory how do we get the wavefunction corrections? using 4th postulate of quantum mechanics: eigenfunctions of H(0) linear expansion coefficients first order correction to the wavefunction

22. Perturbation Theory setting: yields (after some algebra): zeroth order ground state wavefunction zeroth order wavefunction for the ith excited state zeroth order ground state energy zeroth order energy of the ith excited state

23. Perturbation Theory first order correction to the wavefunction we also have: so we can solve for E0(2) from: giving (we won’t work it out):

24. Perturbation Theory in summary: we split the Hamiltonian into parts we can solve and parts we cannot solve starting from a wavefunction that we can solve, we get corrections to the energy and wavefunction that account of the effect of V

25. Perturbation Theory how do we apply perturbation theory to chemical problems? we split H into a part we can solve, and a part we cannot solve we can solve the Fock operator, so: the difference between H and H(0)gives:

26. Perturbation Theory we also take the zeroth order wavefunction to be a single Slater determinant the zeroth order energy is then: energy of molecular orbital i • the zeroth order energy is just the sum of the orbital energies • recall, this does not equal the Hartree-Fock energy • it double counts the electron-electron interactions

27. Perturbation Theory now we can solve for the energy corrected to first order Hartree-Fock wavefunction full electronic Hamiltonian in perturbation theory, the energy corrected to first order is the Hartree-Fock energy!!! 1st order corrections get a single Slater determinant result

28. Perturbation Theory consider the second order correction to the energy and first order correction to the wavefunction ground state Slater determinant excited Slater determinant these corrections ‘mix in’ extra Slater determinants!!!

29. Perturbation Theory let’s look at the second order correction to the energy if you work this out:

30. Møller-Plesset Perturbation Theory if we put this all together, we get what’s called second order Møller-Plesset Perturbation Theory basically, if we do a Hartree-Fock calculation we have all of the basic quantities needed to calculate the corrections to the wavefunction and energy • second order Moller-Plesset theory is called MP2 • Nth order Moller-Plesset theory is called MPn • calculations up to MP4 are not unusual • for larger values of n, the corrections to the wavefunction and energy become very complex

31. Møller-Plesset Perturbation Theory MP2 calculations were the standard way of capturing electron correlation prior to the advent of density functional theory methods Advantages of MP methods: • MP2 captures ~ 90% of electron correlation • necessary integrals in corrections can be evaluated relatively rapidly Disadvantages of MP methods: • MP methods are not variational • we can’t be sure the real energy is lower than the MPn energy • electron correlation is not a minor perturbation • Taylor series only converges if perturbation is small • going from MP2  MP3  MP4  … does not guarantee an improvement in the wavefunction and energy

32. Static Correlation the electronic structure of some systems are best described as a combination of several states (think resonance structures) • allowing electrons to ‘spread out’ among more that one state decreases electron-electron repulsion • called static correlation • has nothing to do with the correlated motions of electrons Example:  system of singlet trimethylenemethane 4 • 4  electrons • 2 degenerate frontier orbitals energy 2 3 • 4  orbitals in total 1 how do we distribute the  electrons?

33. Static Correlation energy energy possible electron configurations energy energy

34. Static Correlation 4 ’4 3 ’3 2 ’2 energy energy 1 ’1 • the molecular orbitals with these two electron configurations are different • putting electrons in 2 will affect 1 in a different way than putting electrons in 3 • each configuration has its own Slater determinant

35. Multi-Reference Methods a single Slater determinant represents 1 electronic state • to describe systems with many accessible states we need to consider multiple determinants • molecular orbitals used to form determinants are optimized variationally • linear expansion coefficients {ai} are optimized variationally to minimize the energy Questions: 1. how do we get the extra Slater determinants? 2. which extra Slater determinants do we include in the expansion?

36. Multi-Reference Methods generating additional Slater determinants • additional Slater determinants are generate by ‘excitations’ out of Hartree-Fock Slater determinant as in CI, coupled-cluster, etc. • orbitals 3 and 4 are still present as virtual states 4 • so, we can still ‘excite’ electrons into these virtual orbitals to generate new electronic configurations energy 2 3 1

37. Multi-Reference Methods which Slater determinants do you use • we want to include all Slater determinants corresponding to relevant ‘resonance states’ in the multi-determinant wavefunction Complete Active Space Self-Consistent Field • pick a number of orbitals that may be occupied in various resonance states 4 • pick how many electrons can be distributed through those states • take all possible combinations energy 2 3 e.g., trimethylenemethane: 1 4  electrons 20 possible configurations 4  orbitals

38. Multi-Reference Methods Complete Active Space Self-Consistent Field (CASSCF) is the most common method for treating static correlation • involves generating Slater determinants by considering all possible excitations that can be generated by distributing m electrons in n molecular orbitals • set of orbitals is called the ‘active’ space • standard notation is CASSCF calculation with (m,n) active space number of ‘active’ electrons number of ‘active’ orbitals • large active spaces yield a large number of determinants • selection of active space is ‘black art’, and probably not accessible to novices (and many experienced computational chemists) CASSCF calculations are necessary when static correlation is significant • singlet diradicals • reactions at some transition metal centers • excited states/electron transfer • describing bond dissociation

39. Multi-Reference Methods Complete Active Space Self-Consistent Field (CASSCF) is a generalization of Hatree-Fock calculations • one Slater determinant will be the Hartree-Fock wavefunction • if static correlation is negligible, aHF 1, and it will be a Hartree-Fock calculation CASSCF can be the basis for CI, MPn and coupled-cluster calculations • generate ‘excited’ states using CASSCF wavefunction instead of Hartree-Fock wavefunction • called MRCI, MRMP2, CASMP2, CASPT2, etc. • very accurate, but very intensive computationally

40. Accuracy and Cost so how costly are these methods? Scaling Method K3 Density functional theory K4 Hartree-Fock K5 MP2 K6 MP3, CISD, MP4SDQ, CCSD, QCISD K7 MP4, CCSD(T), QCISD(T) K8 MP5, CISDT, CCSDT K9 MP6 K10 MP7, CISDTQ, CCSDTQ K! Full CI • this gives an idea of how each method scales with the number of basis functions • other factors determine the computational effort required per basis function

41. Accuracy and Cost how does scaling affect calculations in practice? Hartree-Fock: • suitable for systems with up to a few hundred atoms • geometry optimizations, frequency calculations, etc. are feasible MP2: • suitable for systems with up to ~50 atoms • geometry optimizations, frequency calculations, etc. are feasible CCSD(T): • suitable for systems containing up to ~20 atoms • usually only single point calculations Full CI: • suitable for systems containing up to ~3 or 4 atoms • usually only single point calculations

42. Energies and Geometries Average errors (kcal/mol) on atomization energies Average errors (Å) on bond lengths method error method error HF/6-31G(d) HF/6-31G(d,p) 85.9 0.021 HF/6-311G(2df,p) HF/6-311G(d,p) 82.0 0.022 MP2/6-31G(d) MP2/6-31G(d,p) 22.4 0.014 MP2/6-31G(d,p) MP2/6-311G(d,p) 23.7 0.014 QCISD/6-31G(d) QCISD/6-311G(d,p) 28.8 0.013 CCSD(T)/6-311G(2df,p) CCSD(T)/6-311G(d,p) 11.5 0.013 CCSD(T) is usually better than indicated here, and can often yield results with errors as low as 2.0 kcal/mol on reaction energies