One-Electron Properties of Solids

Transcription

1 One-Electron Properties of Solids Alessandro Erba Università di Torino most slides are courtesy of R. Orlando and B. Civalleri

2 Energy vs Wave-function

3 Energy vs Wave-function

4 Density Matrix

5 Density Matrix

6 Density Matrix

7 Link between Density Matrix and 1-el Props DIRECT SPACE RECIPROCAL SPACE

8 Direct (real) and reciprocal spaces The solution of the Schrödinger equation contains more information than the energy, which can be extracted with appropriate analysis methods Analysis of the quantum-mechanical solution for solids can be done in: DIRECT SPACE 1-electron density ρ(r) (similar to molecular systems) RECIPROCAL SPACE energy bands density of states (DOS) structure factors electron momentum density (orbital energies in molecular systems associated with states with single-double occupation)

9 Wavefunction analysis and properties Band structure Total and projected density of states Electron charge and spin density maps Mulliken population analysis Structure factors Compton profiles Electron momentum density Electrostatic potential, electric field and electric field gradient Fermi contact, hyperfine and nuclear quadrupole coupling tensors Localized Wannier functions (LWF; Boys method) X-ray Structure factors

10 properties Getting started with properties performs the wavefunction analysis (e.g. one-electron properties) Parallel version available. At the end of the SCF process, data on crystalline system and wavefunction are stored in: fort.9 (unformatted) fort.98 (formatted) (keyword: RDFMWF) properties input ends with the keyword END

11 How to run properties Script: runprop17 filename1 filename2 where filename1.f9 and filename2.d3 Visualization tools CRYSPLOT: web-oriented application (

12 How to run properties Script: runprop17 filename1 filename2 where filename1.f9 and filename2.d3 Visualization tools CRYSPLOT: web-oriented application ( CrGra06 DLV: band structure, DOSs, 3D charge density, J-ICE: 3D charge density and mapped electrostatic potential

13 Direct Space

14 Electron Charge Density The Electron Charge Density (ECD) is a 3D function in direct space which exhibits the whole symmetry of the Space Group of the crystal. Bader s Atoms-in-Molecules theory shows that a wealth of information about the chemical features of the system can be obtained from its ECD through a topological analysis.

15 Analysis of the electronic density Visual 2D and 3D plots ECHG ECH3 Numerical Mulliken population analysis PPAN charge of orbital µ charge of atom A bond charge between atoms A and B

16 VISUAL ANALYSIS OF ρ(r) 2D PROJECTION ρ(r) in 2D Si SiO 2 AlPO 4 TOTAL Al P FRAMEWORK PROPERTIES OF ZEOLITES AND AlPOs 1) SELECT A REPRESENTATIVE PLANE 2) SUITABLE REFERENCE: ISOLATED FORMAL IONS (SPHERICAL SYMMETRY). Δρ Solid-ions ELECTR. POTENTIAL F: Corà

17 Δρ(r) deformation maps for crystalline urea Comparison between HF and DFT methods Δρ(r) = ρ(r) bulk - ρ(r) mol Typical pattern of the ECD deformation due to H-bond SVWN (LDA) tends to delocalize the charge density to a larger extent than other methods. This is mitigated when passing to GGA and hybrids (B3LYP and PBE0). Charge redistribution evident in the CO and CN bonds. Depletion around C, build-up on O and N. Charge transfer is confirmed by large changes in the atomic Born charges (in e): ΔZ(O)=-0.63 ΔZ(C)=+0.52 ΔZ(N)=-0.41

18 Hydrogen bonding in molecular crystals P. Ugliengo, D. Hugas ρ crystal Δρ: 0.01 e /a 0 3 ρ 0.12 e /a 0 3 Formic acid ρ crystal - ρ molecules Δρ: e /a 0 3 ρ 0.02 e /a 0 3

19 From the dimers to the formic acid crystal ΔE : B3LYP binding energy (kj/mol) per H-bond Δω : B3LYP OH Symm. harmonic frequency shift (cm -1 ) linear ΔE = 25 polymer ΔE = 27 Δω = -813 cyclic gas-phase Δω = -373 ΔE = 21 Δω = -744 Crystal (010) (011) ΔE = 32 Δω = -870

20 Dimers, chain and crystal vs molecules Cyclic dimer Δρ: e /a 0 3 ρ 0.02 e /a 0 3 Linear dimer increasing density decreasing density null density Chain Crystal

21 Large unit cell system: Crambin Small structural protein extracted from Abyssinian cabbage Crystal structure characterized to very high precision by XRD studies (0.52 Å) P2 1 with two chains per unit cell and 46 residues per chain (1284 atoms) Hartree-Fock calculation STO-3G (3948 GTFs) 6-31G (7194 GTFs) 6-31G(d,p) (12354 GTFs) I.J. Bush (CLRC-DL)

22 Crambin electron charge-density Charge density isosurface coloured according to potential: possible chemically active groups Isosurface of the charge density at 0.1Å resolution: 0.1 e-/unit cell

23 With CRYSTAL14 B3LYP calculation Crambin electron charge-density Exploiting MPP version for SCF and P version for properties 500x500x500 grid of points

24 Crambin electron charge-density About 23 days if run on 1 CPU 10 hours if run on 64 CPUs

25 ρ(r) in 3D: Electrostatic potential in CPO-27-M CPO-27-M: M 2 (dhtp)(h 2 O) 2 8H 2 O dhtp=2,5-dihydroxyterephtalic acid + - Unsaturated metal site CPO-27-Mg

26 ECHG 0 65 COORDINA MARGINS END Electron charge density 2D plot: ECHG E.g.: MgO fcc cubic cell RHF/EBS Notes: Atomic positions can also be used to define A, B and C Window can be made rectangular ECD from a different density matrix: PDIDE, PBAN ECD difference from atoms: PATO Mulliken population analysis order of the derivatives charge density gradients number of point along the B-A segment Cartesian coord.s of points A,B,C defining the 2D window Cartesian coordinates of point A (Å) Cartesian coordinates of point B (Å) Cartesian coordinates of point C (Å) margins are added to the window (order: AB,CD,AD,BC) width of the margins (order: AB,CD,AD,BC) End of properties input

27 ECH3 Electron charge density 3D plot: ECH3 Notes: Grid of point in the primitive unit cell Data stored in fortran unit: fort.31 (DLV) Also saved as cube file: DENS_CUBE.DAT (J-ICE)

28 POT3 Electrostatic Potential 3D plot: POT3 Notes: Data stored in fortran unit: fort.31 (DLV) Also saved as cube file: POT_CUBE.DAT (J-ICE) It can be mapped on a ECD isosurface computed with ECH3

29 J-ICE: A Cloud Visualization Tool for CRYSTAL

30 Analysis of the electronic density Visual 2D and 3D plots ECHG ECH3 Numerical Mulliken population analysis PPAN charge of orbital µ charge of atom A bond charge between atoms A and B

31 PPAN PPAN END Mulliken population analysis: PPAN Mulliken population analysis End of properties input Printed information: AO population Shell population Overlap population

32 ECD: Topological Analysis CP = Critical Point TOPOND is a program that performs the topological analysis of the ECD as computed by CRYSTAL. Now merged into CRYSTAL14

33 ECD: Topological Analysis

34 Experimental Charge Density

35 Static Structure Factors Fourier Transform of the electron charge density

36 Dynamic Structure Factors

37 XFAC Debye-Waller Atomic Thermal Factors

38 Reciprocal Space

39 Reciprocal space properties Band structure Density of states spectrum of one-electron energy levels number of states available at each energy BAND DOSS E LUMO HOMO band gap E conduction band E F valence band molecules solids

40 The language of band structure To make sense of the marvelous electronic properties of the solid state, chemists must learn the language of solid state physics, of band structure R. Hoffmann Traslation invariance in crystalline systems leads to a band structure: Electronic bands Vibrational bands (phonons) Spin bands (spin waves)... Main features of a band structure are: Band width Band gap Fermi level (BWIDTH, ANBD)

41 Band structure representation: α-quartz conduction band Fermi energy gaps 2p O Energy (hartree) 2s Si 1s O 2p Si } 3p Si 2s O valence bands 1s Si top valence band k path

42 BAND MgO END Band structure: BAND E.g.: MgO fcc cubic cell RHF/EBS Band structure Title 4: number of path in the BZ 8: shrinking factor to define the extreme of the path 60: number of k-points along the path 1: first band to be saved 18: last band to be saved 1: plotting option (1=fort.25) 0: printing options (no) Γ X X W Extremes of the paths in the BZ W L as multiple of the shrinking factor L Γ End of properties input Information needed: Path in the BZ Number of bands to be computed. Suggestion: all

43 Bilbao Crystallographic Server For all 230 space groups List of special k-points in the BZ Plot of the BZ E.g.: for shrinking factor 8 ½, 0, ½ ½, ¼, ¾ 4 2 6

44 A simple example: an infinite linear chain a

45 A simple example: an infinite linear chain Minimal basis set one Bloch function per CO 1 elettron/cell monodimensional matrices By using translation invariance π/a α-2β 4β π/a k α+2β -1.0 E

46 Band gap in solids: DFT methods (II) Calculated band structure for cubic KNbO 3 along the Γ-X direction of reciprocal space, as a function of the Hamiltonian. The values indicate the mixing parameter α in global hybrid HF-BLYP scheme: E X = αe X HF + (1-α)E X BLYP. F. Corà, M. Alfredsson, G. Mallia, D.S. Middlemiss, W.C. Mackrodt, R. Dovesi, R. Orlando, Structure and Bonding 113 (2004) 171

47 Band gap in solids: DFT methods (I) Comparison of experimental band gaps for the SC/40 set with values computed with four different generations of DFT. The local spin-density approximation (LSDA), a generalized-gradient approximation (PBE), a meta- GGA (TPSS), and the hybrid functional HSE03. Accuracy : Hybrid > mgga > GGA > LDA SC/40: 40 semiconductors J. Heid, J.E. Peralta, G.E. Scuseria, R.L. Martin, J. Chem. Phys. 123 (2005)

48 Density of states The density of states can be projected onto the atomic orbitals (PDOS): V BZ : volume of the Brillouin zone δ: Dirac s delta function The density of states projected onto atom A: (based on Mulliken partition of charges) Total density of states: In the integration we lose information: DOS is less informative than band structures, but simpler for big systems and PDOS allows to pick AO contribution to bands.

49 WO3: bulk, surfaces, defects DOS, PDOS (How many levels available at each energy, N(E)) Surface states kink states defect states ~200 atoms: impossible to Highlight in band-structure.

50 Density of States: DOSS E.g.: MgO fcc cubic cell RHF/EBS NEWK DOSS END Band structure Shrinking factors: IS1, ISP Fermi energy; no printing options 2: nr. of projections (total DOS always computed) 200: number of points along energy axis 7: first band to be saved 14: last band to be saved 1: plotting option (1=fort.25) 12: degree of the polynomial used for DOSs expansion 0: printing options (no) -1 1: projection onto all the AOs (-1) of Mg (1st atom) -1 2: projection onto all the AOs (-1) of O (2nd atom) End of properties input Information needed: NEWK must be always run before DOSS Projection can be also done on selected atomic orbitals

51 Compton Effect

52 Compton Profiles A directional Compton profile of crystalline urea.

53 Compton Profiles

54 DFT Failure in Momentum Space

55 Compton Profiles with CRYSTAL BIDIERD