7/29/2014. Electronic Structure. Electrons in Momentum Space. Electron Density Matrices FKF FKF. Ulrich Wedig

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1 Electron Density Matrices Density matrices Γ, an alternative to the wavefunction Ψ, for the description of a quantum system Electronic Structure The N-particle density matrix Electrons in Momentum Space can be reduced to the 2-particle- Ulrich Wedig or to the 1-particle-level Max Planck Institute for Solid State Research - Stuttgart Dept. Quantum Materials / Takagi * 1-particle properties 1

2 Electron Density Matrices Electron Density in Position Space Interrelationships between various representations Electron position density ρ(r) and form factor F(χ) diagonal of the 1-particle density matrix momentum position momentum position X-ray diffraction Wolf Weyrich Lecture Notes in Chem. 67 FD 6D Fourier-Dirac transformation FD 3D Fourier-Dirac transformation P Projection into a sub S Selection of a sub 2

3 Electron Density in Momentum Space Electron Density in Momentum Space Electron momentum density ϖ(r) and reciprocal form factor B(s) Electron momentum density ϖ(r) and reciprocal form factor B(s) Compton scattering diagonal of the 1-particle density matrix Compton scattering diagonal of the 1-particle density matrix momentum position momentum position X-ray diffraction projections parallel to the diagonal r into the sub {s} orthogonal to r FD 3D Fourier-Dirac transformation P Projection into a sub S Selection of a sub X-ray diffraction X-ray diffraction and Compton scattering are complementary methods to get information about the 1-particle density matrix projections parallel to the diagonal r into the sub {s} orthogonal to r FD 3D Fourier-Dirac transformation P Projection into a sub S Selection of a sub 3

4 Compton Scattering - Principles Compton Profiles k 1 p 1 k = k 2 k q 1 p 2 p 1 k 2 p 2 k 1 wave vector of photon before collision k 2 wave vector of photon after collision k = k 2 k 1 scattering vector p 1 electron momentum before collision p 2 electron momentum after collision q projection of p onto k inelastic scattering of photons (X-ray, γ) momentum transfer from photon to electron (energy and momentum conservation) electrons at rest Compton line h 1 cos m c moving electrons Compton profile (Doppler broadening) e The formula is valid within the impulse approximation Projection of the electron s initial momentum onto the scattering vector Probing several directions yields information on the Electron Momentum Density - Energy transfer is much larger than the binding energy of the electron - Corrections for multiple scattering processes P. Eisenberger, P. M. Platzman, Phys. Rev. A 2, 415 (1970) 4

5 Compton Profiles Computed Compton Profiles Projection of the electron s initial momentum onto the scattering vector Directional Compton Profiles can be computed in two ways: 1) 2D-integration of the Electron Momentum Density π(p) Probing several directions yields information on the Electron Momentum Density The formula is valid within the impulse approximation synchrotron-, γ-radiation 2) Via the reciprocal form factor B(r) In the AO basis set Beamline ID15B at ESRF, Grenoble 5

6 Computed Compton Profiles Computed Compton Profiles Directional Compton Profiles can be computed in two ways: 1) 2D-integration of the Electron Momentum Density π(p) Directional Compton Profiles can be computed in two ways: BRG, interfaced to CRYSTAL98: A. Saenz, T. Asthalter, W. Weyrich, Int. J. Quant. Chem. 65, 213 (1997) 1) 2D-integration of the Electron Momentum Density π(p) Since CRYSTAL09: Keyword BIDIERD A. Erba, C. Pisani, S. Casassa. L. Maschio, M. Schütz, D. Usvyat, Phys. Rev. B 81, (2010) (use of corrected density matrices (MP2)) 2) Via the reciprocal form factor B(r) CRYSTAL: Keyword PROF In the AO basis set - analytical integration valence contribution only for molecules and non conducting polymers 2) Via the reciprocal form factor B(r) In the AO basis set - numerical integration dense k-mesh required 6

7 Computed Compton Profiles Computed Compton Profiles Hartree-Fock vs. Correlated wavefunction vs. Kohn-Sham Hartree-Fock vs. Correlated wavefunction vs. Kohn-Sham Hartree-Fock: Electron correlation is missing. 1-particle density matrix is idempotent. eigenvalues (occupation numbers) of the spin-orbitals either 1 or 0 Correlated wavefunction: 1-particle density matrix is non-idempotent. eigenvectors (natural orbitals) and non-integer eigenvalues 0 < n < 1 Kohn-Sham: DFT electron position density minimizing the energy for a given functional Kohn-Sham: Total density is the sum of orbital densities (noninteracting electrons). 1-particle density matrix constructed from KS-orbitals is idempotent. Error in the kinetic energy term, compensated in the exchange-correlation functional E xc Corrections: L. Lam, P. M. Platzman, Phys. Rev. B 9, 5122 (1974) Hartree-Fock: Electron correlation is missing. 1-particle density matrix is idempotent. eigenvalues (occupation numbers) of the spin-orbitals either 1 or 0 Correlated wavefunction: 1-particle density matrix is non-idempotent. eigenvectors (natural orbitals) and non-integer eigenvalues 0 < n < 1 Kohn-Sham: The Kohn-Sham approach is incorrect for momentum related properties correlated density from uncorrelated density matrix DFT electron position density minimizing the energy for a given functional Kohn-Sham: Total density is the sum of orbital densities (noninteracting electrons). 1-particle density matrix constructed from KS-orbitals is idempotent. Error in the kinetic energy term, compensated in the exchange-correlation functional E xc Corrections: L. Lam, P. M. Platzman, Phys. Rev. B 9, 5122 (1974) 7

8 Theory vs. Experiment Theory vs. Experiment We have to consider: We have to consider: - Spectrometer specific corrections - Corrections for multiple scattering - Deviations from the impulse approximation (photon energy) - Limited resolution expt. data - Spectrometer specific corrections - Corrections for multiple scattering expt. data - Deviations from the impulse approximation (photon energy) Most effects are less relevant - Limited resolution if differences of directional CP (anisotropies in the EMD) are examined. theor. data theor. data - 0K vs. finite temperature statistically averaged density matrix (Alessandro Erba) - 0K vs. finite temperature statistically averaged density matrix (Alessandro Erba) 8

9 An Example CP of Zn and Mg An Example CP of Zn and Mg Results from a project within the DFG priority program 1178: Deviations from Ideal Structures in Metallic Elements and Simple Intermetallics: Combined Experimental and Theoretical Studies of Electron Distributions Results from a project within the DFG priority program 1178: Deviations from Ideal Structures in Metallic Elements and Simple Intermetallics: Combined Experimental and Theoretical Studies of Electron Distributions Stuttgart M. Jansen, U. Wedig Electron density J. Nuss, B. Boldrini, Th. Buslaps Electron density J. Nuss, H. Nuss, D. Fischer, H. Schlenz, K. Friese, W. Morgenroth, Ch. Busch, Ch. Hauf, W. Scherer Momentum Position W. Weyrich Experiments and Theory A. Kirfel Konstanz Momentum Experiments and Theory Position Berlin B. Paulus N. Gaston, D. Andrae, E. Voloshina P. Sony, K. Rosciszewski U. Wedig, H. Nuss, J. Nuss, M. Jansen, D. Andrae, B. Paulus, A. Kirfel, W. Weyrich, Z. Anorg. Allg. Chem, 639, 2036 (2013) 9

10 An Example CP of Zn and Mg Computed CP of Zn and Mg Hexagonal close packed elements unusual structures Anisotropies in the directional CP in the range of Zn Cd 1.8 c / a % BIDIERD: TOLINTEG vs BRG: TOLINTEG vs BIDIERD vs. BRG: TOLINTEG Mg Crucial is the first parameter (ITOL1) in keyword TOLINTEG overlap threshold for coulomb integrals but also governs the truncation of the lattice sums with the construction of the density matrix v o l u m e / Å 3 anisotropic in-plane and out-of-plane bonding tiny effects in the electron density Truncation errors may have the same magnitude as the physical effects Thanks to Alessandro Erba for valuable hints 10

11 Directional CP of Zn and Mg Anisotropy of the Momentum Density Zn 7 x 3 x 0.7 mm Mg 7 x 3 x 1 mm (Z3M [001]) (M1M [423]) Samples: Single crystals, one for each direction sheets perpendicular to the scattering vector Dispersion-compensating scanning spectrometer Beamline ID15B, ESRF, Grenoble [423] direction 1 a.u. = 1 DuMond = ħ / Bohr Directions considered: [100] intraplanar bond [423] interplanar bond [001] nonbonding direction as reference Experimental resolution: Zinc 0,07 0,09 a.u.; Magnesium a.u. 11

12 Anisotropy of the Momentum Density Anisotropy of the Momentum Density Comparing the bonding directions [100] [423] (B3PW) Comparing the bonding directions [100] [423] (B3PW) [423] [100] Zinc in momentum : 4s 2 -valence shell: Reduction of the anisotropy (change of the Fermi surface) 3d 10 -shell: Enhancement of the anisotropy (dynamical correlation) 12

13 Anisotropy of the Momentum Density The Method of Increments (MOI) for Metals Comparing the bonding directions [100] [423] (B3PW) The method of increments for metals E E HF E corr Many-body expansion of the correlation energy per unit cell with corr E the indices i,j,k denote groups of orthogonal localized orbitals (e.g. atoms) in finite embedded clusters (i: unit cell; j,k: whole system) i i j ij ij i i j ij i j k etc.... ijk Does the dynamical behaviour of the electrons govern the unusual structure of zinc in position? Zinc in momentum : 4s 2 -valence shell: Reduction of the anisotropy (change of the Fermi surface) 3d 10 -shell: Enhancement of the anisotropy (dynamical correlation) a proper embedding scheme allows for: Correlation treatment: CCSD(T) -improved localizability -prevention of surface charging -a disappearing band gap H. Stoll, Phys. Rev. B 46 (1992) 6700 B. Paulus, Phys. Rep. 428 (2006) 1 E. Voloshina, B. Paulus, SPR Chemical Modelling: Application and Theory 6, M (2009) 13

14 Zinc, Cadmium Cohesive Energies Zinc Energy Landscape beyond DFT Zn Cd Correlation energy computed with the Method of Increments Phys. Rev. Lett. 100, (2008) expt MOI s 2 d 10 -correlation MOI s 2 -only-correlation DFT (LDA) DFT (PBE) DFT (B3PW) DFT (B3LYP) MOI: Method of Increments, values given for the expt-like minimum 4s 2 3d 10 -correlation only 4s 2 -correlation The bonding in zinc is exclusively due to dynamical correlation of the electron 3-body terms lead to different bonding interactions within und between the hexagonal planes. The closed 3d 10 -shell contributes significantly to the bonding. The cohesive energy is obtained quantitatively by the Method of Increments. (MOI: ev; expt.: ev) Phys. Rev. B 75, (2007) Phys. Rev. Lett. 100, (2008) Phys. Chem. Chem. Phys. 12, 681 (2010) high pressure data: K. Takemura Phys. Rev. B56 (1997) 5170 A modification of zinc with an ideal c/a ratio may exist. 14