Relativistic effects in ADF Erik van Lenthe, SCM
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.
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
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
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
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
June 2007ADF overview
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
June 2007ADF overview Uranium atom
June 2007ADF overview Uranium atom
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
June 2007ADF overview Scalar Relativistic effects
June 2007ADF overview
June 2007ADF overview Spin-orbit effects
June 2007ADF overview
June 2007ADF overview Spin-orbit splitting Bi atomBi 2
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
June 2007ADF overview
June 2007ADF overview
June 2007ADF overview (non-)collinear approximation XC-functional depends on spin-density collinearnoncollinear
June 2007ADF overview Excitation energies PbO SR SO
June 2007ADF overview 1 H NMR V.G.Malkin, O.L.Malkina, and D.R.Salahub, Chem.Phys.Lett. 261 (1996) 335.
June 2007ADF overview 13 C NMR S.K.Wolff et all, J.Chem.Phys. 110 (1999) 7689.
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
June 2007ADF overview CO on (111) Pt surface Exp.NRSRSO top hollow Adsorption energy (eV) of CO Pt (111) surface one-third coverage ( 3 3)R30 -CO on Pt (111) surface
June 2007ADF overview TDDFT calculation on bulk InSb Imaginary part of the dielectric function of InSb (eV) Im[ ( )]
June 2007ADF overview EPR Experiment Effective Spin Hamiltonian Theory Electron Paramagnetic Resonance Open shell molecule Degenerate levels are split by magnetic field
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
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
June 2007ADF overview g-tensor [(P)Fe(ImH) 2 ] +
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
June 2007ADF overview Nuclear electric quadrupole hyperfine interaction: I N Q N I N Electric field gradient (EFG) at nucleus hyperfine interactions (Q-tensor)
June 2007ADF overview 127 I quadrupole coupling constants
June 2007ADF overview g || gg a || (Ni)a (Ni)Q || (Ni)Q (Ni)a iso (H) unrestricted experiment Ni(CO) 3 H A-values, Q-values (MHz) Experiment, krypton matrix Experiment Morton, Preston, JCP 81 (1984) 5775
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
June 2007ADF overview Basis Sets effects valence
June 2007ADF overview Basis Sets effects valence
June 2007ADF overview Basis Sets effects valence
June 2007ADF overview
June 2007ADF overview
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
June 2007ADF overview Thank you for your attention