1. Relativistic effects in ADF Erik van Lenthe, SCM

2. June 2007ADF overview Why relativistic effects in Quantum Chemistry ? Is the velocity nuclei and valence electrons not small compared to the speed of light? Yes, in chemistry normally the velocity of nuclei and valence electrons in the valence region small. However, the velocity of valence electrons in the core region is large.

3. June 2007ADF overview Outline Introduction relativistic methods The Dirac equation  the ZORA equation Relativistic effects in ADF –scalar relativistic effects, spin-orbit coupling –diatomics: bond distance, energy –excitation energies, NMR, TDDFT in BAND EPR –Zeeman interaction, hyperfine interactions, EFG ADF: basis set effects Summary

4. June 2007ADF overview Relativistic Hamiltonians Quantum Electrodynamics –quantized electromagnetic field –interaction electrons with electromagnetic field Dirac-Coulomb-Breit Hamiltonian –4-component equation –Breit interaction: modified electron-electron repulsion Spin-orbit coupled Hamiltonian –2-component equation Scalar relativistic Hamiltonian –1-component equation, including part of relativistic effects Non-relativistic Hamiltonian –Schrödinger equation

5. June 2007ADF overview Relativistic approximations Starting point 4-component Dirac equation Approximate Foldy-Wouthuysen transformation to 2- or 1-component Expansion in 1/2c 2 - non-relativistic Hamiltonian, Pauli Hamiltonian Expansion in ([p,V])/2c 2 - Douglas-Kroll-Hess Hamiltonian Expansion in E/(2c 2 -V) - ZORA, scaled ZORA Relativistic effective core potentials Relativistic extended Hückel theory

6. June 2007ADF overview (non-)relativistic equations Kohn-Sham equation: [V + T]  i =  i  i Dirac equation for the large component  T DIRAC = 1/2  p K  pK=2c 2 /(2c 2 +  i -V) ZORA T ZORA = 1/2  p K  pK=2c 2 /(2c 2 -V) Non-relativistic T NR = 1/2 p 2 = 1/2 p  KpK=2c 2 /2c 2 = 1

7. June 2007ADF overview

8. June 2007ADF overview One-electron atom Hydrogen-like atom, V =  Z/r Exact relation  i ZORA = 2c 2  i DIRAC /(2c 2 +  i DIRAC )  i scaled ZORA =  i DIRAC (bound states) DIRAC ZORA Scaled ZORA

9. June 2007ADF overview Uranium atom

10. June 2007ADF overview Uranium atom

11. June 2007ADF overview Relativistic effects Scalar relativistic and spin-orbit coupling Atoms: s and p 1/2 orbitals contract p 3/2 orbitals intermediate d and f orbitals expand Molecules: often bond contraction depends on character of the bonding orbitals and the Pauli repulsion

12. June 2007ADF overview Scalar Relativistic effects

13. June 2007ADF overview

14. June 2007ADF overview Spin-orbit effects

15. June 2007ADF overview

16. June 2007ADF overview Spin-orbit splitting Bi atomBi 2

17. June 2007ADF overview Visualization spinors  = ( ) spinor complex and 2-component:   R + i   I   R + i   I  =  †  m =  †   spinor is now defined up to a constant e i   are the Pauli spin matrices

18. June 2007ADF overview

19. June 2007ADF overview

20. June 2007ADF overview (non-)collinear approximation XC-functional depends on spin-density collinearnoncollinear

21. June 2007ADF overview Excitation energies PbO SR SO

22. June 2007ADF overview 1 H NMR V.G.Malkin, O.L.Malkina, and D.R.Salahub, Chem.Phys.Lett. 261 (1996) 335.

23. June 2007ADF overview 13 C NMR S.K.Wolff et all, J.Chem.Phys. 110 (1999) 7689.

24. June 2007ADF overview BAND Periodic structure program: bulk, slabs, polymers Properties –DOS, Time-dependent DFT –Potential energy surfaces Relativistic effects (ZORA, same as ADF) –Scalar relativistic effects, spin-orbit coupling –Transition and heavy metal compounds Uses numerical functions and Slater-type functions

25. June 2007ADF overview CO on (111) Pt surface Exp.NRSRSO top1.3-1.50.831.411.29 hollow1.021.171.05 Adsorption energy (eV) of CO Pt (111) surface one-third coverage (  3  3)R30  -CO on Pt (111) surface

26. June 2007ADF overview TDDFT calculation on bulk InSb Imaginary part of the dielectric function of InSb  (eV) Im[  (  )]

27. June 2007ADF overview EPR Experiment Effective Spin Hamiltonian Theory Electron Paramagnetic Resonance Open shell molecule Degenerate levels are split by magnetic field

28. June 2007ADF overview Effective Spin Hamiltonian H = S  g  B +  N S  A N  I N +  N I N  Q N  I N + S  D  S + … S  g  BZeeman interaction S  A N  I N Nuclear magnetic dipole hyperfine interaction I N  Q N  I N Nuclear electric quadrupole hyperfine interaction S  D  SZero-field splitting

29. June 2007ADF overview Theoretical Hamiltonian H = T + V + H so + H zeeman + H hfs +  N I N  Q N  I N + … T+VKinetic energy + Electric potential H so Spin-orbit coupling H zeeman Zeeman interaction H hfs Nuclear magnetic dipole hyperfine interaction I N  Q N  I N Nuclear electric quadrupole hyperfine interaction

30. June 2007ADF overview g-tensor [(P)Fe(ImH) 2 ] +

31. June 2007ADF overview hyperfine interactions (A-tensor) Nuclear magnetic dipole hyperfine interaction: S  A N  I N Most important Fermi contact term (~S  I N  (r N ) ) Relativistic effect factor 3 for the gold atom! 197 AuA-value (MHz) non-relativistic978 relativistic3109 experiment3053

32. June 2007ADF overview Nuclear electric quadrupole hyperfine interaction: I N  Q N  I N Electric field gradient (EFG) at nucleus hyperfine interactions (Q-tensor)

33. June 2007ADF overview 127 I quadrupole coupling constants

34. June 2007ADF overview g || gg a || (Ni)a  (Ni)Q || (Ni)Q  (Ni)a iso (H) unrestricted2.00252.0469-81558.6-4.3312 experiment2.00422.0674-79538.4-4.2293 Ni(CO) 3 H A-values, Q-values (MHz) Experiment, krypton matrix Experiment Morton, Preston, JCP 81 (1984) 5775

35. June 2007ADF overview Basis Sets in ADF: Slater Orbitals for the whole periodic system –all electron, frozen core –DZ, TZP, TZ2P, QZ4P for H-Kr –SZ, DZP, even tempered, diffuse, augmented

36. June 2007ADF overview Basis Sets effects valence

37. June 2007ADF overview Basis Sets effects valence

38. June 2007ADF overview Basis Sets effects valence

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40. June 2007ADF overview

41. June 2007ADF overview Summary ZORA accurate relativistic approximation Relativistic effects –scalar relativistic –spin-orbit coupling BAND has same relativistic options as ADF DFT can give reasonable EPR parameters ADF STO basis sets available for Z = 1-118

42. June 2007ADF overview Thank you for your attention