1. New Variational Approaches to Excited and (Nearly) Degenerate States in Density Functional Theory 15:20 Wednesday June 24 2009 Tom Ziegler Department of Chemistry University of Calgary Calgary Alberta Canada Helsinki Finland June 22-27 2009 Helsinki Finland June 22-27 2009 Chair: Manuel Yanez

2. Excited States DFT-methods 1:  SCF (Slater, 1972) 2: TD-DFT (Gross,Cassida 1995) II. Discuss the the relation between  SCF and TD-DFT as special cass of CV-DFT II. Discuss the the relation between  SCF and TD-DFT as special cass of CV-DFT Ground state Excited States I.Demonstrate that the basic equation of TD-DFT can be derived from a time- independent constraint variational DFT (CV-DFT) procedure I.Demonstrate that the basic equation of TD-DFT can be derived from a time- independent constraint variational DFT (CV-DFT) procedure III. Application to charge transfer IV. Dissociation of molecules

3. A closed shell molecule is described in the Kohn-Sham formulation of density functional theory (KS-DFT) by a single Slater determinant: A closed shell molecule is described in the Kohn-Sham formulation of density functional theory (KS-DFT) by a single Slater determinant: with the corresponding IDEMPOTENT density matrix given by: The density matrix is optimized in such a way that it minimizes the energy expression Basic Ground State KS-theory Ziegler, et al. J. Chem. Phys. 2009, 130,154102

4. Here the kinetic- and nuclear attraction energy is given by The density matrix is optimized in such a way that it minimizes the energy expression The Hartree Coulomb interaction energy by Exchange correlation energy Ziegler, et al. J. Chem. Phys. 2009, 130,154102 Basic Ground State KS-theory

5. The set of spin-orbitals that optimize the ground state Slater determinant can be found as solutions to the one-electron KS-equation The set of spin-orbitals that optimize the ground state Slater determinant can be found as solutions to the one-electron KS-equation Where With the exchange correlation potential V XC given by Ziegler, et al. J. Chem. Phys. 2009, 130,154102 Basic Ground State KS-theory

6. Electronic Ground-State Hessian As a first step towards a variational theory for the determination of excitation energies by KS-DFT we consider a variation of each of the occupied spin-orbitals As a first step towards a variational theory for the determination of excitation energies by KS-DFT we consider a variation of each of the occupied spin-orbitals From this set we can generate a new determinantal wave function From the variations  i +  i we can form a set of orbitals orthonormal to second order in U Ziegler, et al. J. Chem. Phys. 2009, 130,154102 Here U ai are our variational parameters

7. Electronic Ground-State Hessian Corresponding to the determinantal wave-function We have to second order in U the density matrix

8. Electronic Ground-State Hessian Substituting  ’  ’  into the expression for E KS affords to second order Here Where Ziegler, et al. J. Chem. Phys. 2009, 130,154102

9. Stationary Points and Excitation Energies We now have to second order We shall next find stationary points U (I) on E KS [  0 +  ’ ], such that represents an excitation energy And such that

10. Stationary Points and Excitation Energies In the expression for the new density Only the last two terms give rise to a change in the energy if    ’  is optimized occ vir Ziegler, et al. J. Chem. Phys. 2009, 130,154102

11. Stationary Points and Excitation Energies Only the last two terms give rise to a change in the energy if    ’  is optimized occ vir We have for the total charge in  occ Whereas the total charge in  vir is given by We have that:

12. Stationary Points and Excitation Energies occ vir We have a transfer of charge In single excitations we shall require that a single charge is transferred This is a generalization of  SCF-DFT where an electron is promoted from an occupied to a virtual orbital This is a generalization of  SCF-DFT where an electron is promoted from an occupied to a virtual orbital Ziegler, et al. J. Chem. Phys. 2009, 130,154102

13. Stationary Points and Excitation Energies We now have that the stationary points for E’ that fulfills the condition: We demand Introducing and is a Lagrange multiplier Ziegler, et al. J. Chem. Phys. 2009, 130,154102

14. Stationary Points and Excitation Energies Thus the requirement that  L/  U ai leads to We must thus have in order for  L/  U =0 to hold for any variation  U

16. Stationary Points and Excitation Energies For non-relativistic cases with A KS and B KS real By making use of the Tamm-Dankoff approximation B KS =0 This equation is identical to that obtained from TD-DFT/TD within the adiabatic approximation This equation is identical to that obtained from TD-DFT/TD within the adiabatic approximation

17. Derivation of transition moments In wave function mechanics we have from Raleigh-Schrödinger perturbation theory for the polarizability tensor In wave function mechanics we have from Raleigh-Schrödinger perturbation theory for the polarizability tensor Here Electric dipole operator Electric transition dipole moment component Excitation energies Ziegler, et al. J. Chem. Phys. 2009, 130,154102

18. Derivation of transition moments Ziegler, et al J.Chem Phys. 2007,126,174103 In DFT the static polarizability tensor component  rs is given by : Here Writing next the real matrix [A KS +B KS ] -1 in its spectral resolution form affords or by comparison to the Raleigh-Schrödinger expression Ziegler, et al. J. Chem. Phys. 2009, 130,154102

19. Derivation of transition moments For a component of the electronic magnetic susceptibility tensor  rs, Raleigh-Schrödinger perturbation theory affords the following SOS expression For a component of the electronic magnetic susceptibility tensor  rs, Raleigh-Schrödinger perturbation theory affords the following SOS expression Here : In DFT we find : Ziegler, et al J.Chem Phys. 2007,126,174103 Here :

20. Derivation of transition moments Making use of the fact Ziegler, et al J.Chem Phys. 2007,126,174103 In DFT we find : We get : or by comparison to the Raleigh-Schrödinger expression Ziegler, et al. J. Chem. Phys. 2009, 130,154102

21. 21 Time-dependent density functional theory (TD-DFT) at the generalized gradient (GGA) level of approximation has shown systematic errors in the calculated excitation energies. This is especially the case for energies representing electron transitions between two separated regions of space or between orbitals of different spatial extent. Charge Transfer Ground state Triplet excited state Ziegler, et al. J. Chem. Phys. 2008, 129, 184114

22. 22 HF-Case  E-HF TD-HF Same results R Charge Transfer Ground state Triplet excited state Ziegler, et al. J. Chem. Phys. 2008, 129, 184114 Ziegler et al. Theochem. 2009

23. 23 R Charge Transfer Ground state Triplet excited state KS-Case  E-KS Different result TD-DFT Ziegler, et al. J. Chem. Phys. 2008, 129, 184114 Ziegler et al. Theochem. 2009

24. 24 Charge Transfer in steps Ground state Triplet excited state Frozen orbitals I. Separation step KS-Case HF-Case Ziegler et al. Theochem. 2009 M. J. G. Peach, P. Bernfield, T. Helgaker, and D. J. Tozer, J. Chem. Phys. 128, 044118 2008. A.Dreuw and M. Head-Gordon, J. Am. Chem. Soc. 126, 4007 2004. J. Neugebauer, O. Gritsenko, and E. J. Baerends, J. Chem. Phys. 124, 214102 2006.

25. 25 Charge Transfer in steps Frozen orbitals II. Ionization HF-Case KS-Case Ziegler et al. Theochem. 2009

26. 26 Charge Transfer in steps Frozen orbitals III. Electron attachment HF-Case KS-Case Ziegler et al. Theochem. 2009

27. 27 R Charge Transfer Ground state Triplet excited state IV. Recombining KS-Case HF-Case Ziegler et al. Theochem. 2009

28. 28 R Charge Transfer Ground state Triplet excited state HF-case TD-HF Same result Combining all terms Ziegler et al. Theochem. 2009 KS-Case Different result TD-DFT

29. 29 Coupling of a Single Occupied/Virtual Orbital Pair to all Orders. We consider the variation The corresponding variation in energy is T=HF,KS Ziegler, et al. J. Chem. Phys. 2008, 129, 184114

31. 31 KS-Case We have We can not only retain second order terms as it is done in TD-DFT Since

32. 32 A Revised Electronic Hessian for Approximate CV-DFT We now consider With Keeping terms in the energy to Ziegler, et al. J. Chem. Phys. 2008, 129, 184114

33. 33 A Revised Electronic Hessian for CV-DFT where We now obtain the revised Hessian Ziegler, et al. J. Chem. Phys. 2008, 129, 184114

34. 34 Revised CV-KS  E-KS Same result Revised KS-Case Ground state Triplet excited state A Revised Electronic Hessian for CV-DFT

35. TD-DFT CV-DFT I+A-1/R Exp 35 Numerical example He Li R Orbital optimization

36. 36 Numerical example CV-DFT TD-DFT

37. 37 Numerical example CV-DFT TD-DFT

38. 92nd Canadian Chemistry Conference and Exhibition in Hamilton, ON, May 30 - June 3, 2009 16:00 Tuesday June 2 2009 Heritage-Sheridan Excited States Ground state Excited States DFT-methods 1:  SCF (Slater, 1972) 2: TD-DFT (Gross,Cassida 1995)

39. 92nd Canadian Chemistry Conference and Exhibition in Hamilton, ON, May 30 - June 3, 2009 Excited States Degeneracies Ground state Excited States DFT-methods 1:  SCF (Slater, 1972) 2: TD-DFT (Gross,Cassida 1995) Slater Sum rules Ziegler 1976 Spin-Restricted Open-Shell-KS ROKS: Filatov 2000

40. 92nd Canadian Chemistry Conference and Exhibition in Hamilton, ON, May 30 - June 3, 2009 Excited States Ground state Excited States DFT-methods 1:  SCF (Slater, 1972) 2: TD-DFT (Gross,Cassida 1995) Near Degeneracies

41. 92nd Canadian Chemistry Conference and Exhibition in Hamilton, ON, May 30 - June 3, 2009 Excited States Ground state Excited States DFT-methods 1:  SCF (Slater, 1972) Near Degeneracies I.Ensamble theory Ziegler 1979 Filatov REKS (2002) II. Broken Symmetry Fukotome (1973) III. Density matrix functional theory Baerends (2009)

43. Exact 12 34 H-H Bond Length (Å) 0 H 2 Dissociation Missing orbital optimization

44. Dr. Mike SethDr. Mykhylo Krykunov Prof. Fan Wang Prof. Jochen Autschbach

46. 46 Concluding Remarks

47. 47 Basic Time Dependent Density Functionl Theory Basic Equation : Definition of A and B Matrices : M.E.Casida Gross,E.K.; Kohn W. Where : TD-approximation

48. 48 The Ground State Hessian Energy change due to change in density away from ground state Change in density away from ground state Ground state density

49. 49 The Ground State Hessian for HF HF Change in density away from ground state

50. 50 The Ground State Hessian for HF KS Change in density away from ground state

51. Normal Restrictions Spin Symmetry Restrictions: Space Symmetry Restrictions: All KS-orbitals belongs to a symmetry representation characteristic for the point group of the molecule Near degenerate systems Solutions subject to space and spin symmetry breaking: Problems:Problems: Problems:Problems: Not all systems can be described by a single determinant: Degenerate systems Basic Ground State KS-theory

52. Similar Constraint in Wave Function Mechanics In wave function mechanics we can write. where by expanding in terms of determinants to second order

53. Similar Constraint in Wave Function Mechanics To second order The corresponding energy to second order is For E’ has no diagonal contribution from the ground state. It can be considered as the energy of an excited state E’ has no diagonal contribution from the ground state. It can be considered as the energy of an excited state