Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
QCLAMS. Winter School on Large Systems Plane-Wave DFT Methods Applied to large Molecules and Nanoclusters in their Electronic Groundstate: a Practical Guide Romuald Poteau [email protected] INSA-UPS-CNRS 135 avenue de Rangueil 31077 TOULOUSE CEDEX 4 - FRANCE Classical-CLAMS Quantum-CLAMS TCCM European Master 1/116 Large Molecules and NPs (http://www.vchem3d.com) TCCM European Master 2/116 Biomolecules - Molecular Mechanics (Force Fields) - Localized basis sets - Standard DFT - Linear Scaling - QM/MM DNA (486 atoms) Rapid ab initio electronic structure package Gaussian09 Serotonin Receptor (2795 atoms) Terachem (GPU) TCCM European Master 3/116 Chemistry of Nanoparticles (NPs) Field of intense research Structure Magnetic, Electric and Optical Properties Catalysts Synthesis and control of nanomaterials that offer advanced properties for novel applications Theory: need for bridging the gap between Molecules, Clusters ↔ NPs ↔ Surface Chemistry TCCM European Master 4/116 Experimental context: colloidal NPs Colloidal Ru NPs N-heterocyclic carbene L2 Lara et al. (2011) Angew. Chem., Int. ed. Eng. 50, 12080 CPK Model 1.8 nm hcp [Ru262] NP 8 L2 NHCs 1.5 H / Rusurf (~ exp. titration) TCCM European Master 5/116 Experimental context: colloidal NPs Colloidal Ru NPs N-heterocyclic carbene L2 Lara et al. (2011) Angew. Chem., Int. ed. Eng. 50, 12080 CPK Model 1.8 nm hcp [Ru262] NP 8 L2 NHCs 1.5 H / Rusurf (~ exp. titration) TCCM European Master 6/116 Experimental design of NPs. 1. Versatile nanocrystal growth processes. The Pt case Lacroix, L.-M.; Chaudret, B. ; Viau, G. and coll. (2012) Angew. Chem., Int. ed. Eng. 51, 4690 TCCM European Master 7/116 Experimental design of NPs. 2. Anisotropic Etching of Nanoparticles - new method to chemically control the shape of silver nanocrystals - tuning of the etchant strength and reaction conditions → new shapes which cannot be synthesized with conventional nanocrystal growth methods Etchant solution containing H2O2/NH4OH/CrO3 P. Yang and coll. (2010) J. Am. Chem. Soc. 132, 268 TCCM European Master 8/116 Nanocatalysis Toolbox property experiments theory Shape of the metal core WAXS, TEM ab initio thermodynamics Wulff construction (Gibbs energy) Adsorption sites of ligands Solution or solid-state NMR, IR AFM (surfaces only) NMR, IR Theoretical AFM Number of surface ligands Initial concentrations of the precursors, Titration Ab inito thermodynamics Adsorption strength of ligands Spectroscopy (IR, NMR) Energies (usually DFT) Catalytic activity Reaction mechanisms Identification of products, yields, TOF - Frontier orbital theory / Correlation diagrams / Conceptual DFT - Multi-step reaction studies (QST, NEB) Electronic density STM DOS, MOs TCCM European Master 9/116 in silico modeling of organometallic NPs: a challenge for computational chemists strategy? TCCM European Master 10/116 Electronic and Steric effects The case of tin-poisoned NPs Preformed RuNPs/Lsurf + (Bu)3SnH → Ru/Lsurf/Sn - modification of their surface chemistry - modification of their catalytic reactivity TEM image of Ru/PVP/Sn NPs (Bu)3SnH, bulky ligand Ru/dppb/Sn Sn Mössbauer ⇒ presence of Sn(IV)R3R’ species decomposition of (Bu)3SnH +RuNP → (Bu)Sn* + 3 (Bu-H) ? electronic effect of the surface 119 ?????? Philippot K., Poteau R., Chaudret B., to be published equiv. Sn H/Rusurf mean Ø 0 1.2 1.3 ± 0.2 nm 0.05 1 1.7 ± 0.3 nm 0.1 1.1 1.5 ± 0.3 nm 0.2 0.1 1.7 ± 0.5 nm 0.5 0.05 1.5 ± 0.3 nm Experimental titration of surface hydrides 0.1 equiv. molar per introduced Ru steric effect TCCM European Master 11/116 Electronic and Steric effects The case of tin-poisoned NPs. Steric effects “multi-scale” strategy ● coarse-grained method ● the metal core and the ligands are kept frozen ● Sn(tBu)3* 16 top-Sn(Bu)3 = 0.055 Sn / Rusurf ● ● a steric hindrance index is minimized in order to ● search for the optimal arrangement of ligands on the surface (super-atom - VSEPR) ● find the optimal number of ligands that could be grafted on the surface without being sterically discomforted the relative orientations and positions of the ligands are adjusted at every step by a Monte-Carlo process the geometry of each type of ligand was separately optimized (DFT-B3PW91 functional)s. Sn(tBu)* 28 μ3-Sn(Bu) = 0.10 Sn / Rusurf Theoretical titration of Sn species Philippot K., Poteau R., Chaudret B., to be published TCCM European Master 12/116 Electronic and Steric effects The case of tin-poisoned NPs. Steric effects “multi-scale” strategy ● ● minimization of a steric hindrance index = Global Optimization → Cost Function? Mathematical problem ≈ distributing many points on a sphere Saff, E. & Kuijlaars, A. (1997) Distributing Many Points on a Sphere, Math. Intelligencer 19, 5-11 Sn(tBu)3* 16 top-Sn(Bu)3 = 0.055 Sn / Rusurf ● Sn(tBu)* Solution to this problem: MC method → simulated annealing (=biased random walk) (a) N. Metropolis and S. Ulam (1949) The Monte-Carlo method , J. Am. Stat. Assoc. 44, 335 ; (b) Kirkpatrick, S.; Gelatt Jr., C. D. & Vecchi, M. P. (1985) Optimisation by simulated annealing, Science 220, 671-680 ; (c) W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery (1992) Numerical Recipes, Cambridge University Press, Chapter 10 28 μ3-Sn(Bu) = 0.10 Sn / Rusurf Theoretical titration of Sn species TCCM European Master 13/116 Electronic and Steric effects The case of tin-poisoned NPs. Electronic effects Thermodynamics & Kinetics? Adsorption (1) HSnR3 + 2H* → SnR3*, 3H* Decomposition (2) SnR3*, 3H* → SnR2*, 2H* + RH (3) SnR2*, 2H* → SnR*, H* + RH (4) SnR*, H* → Sn* + RH Philippot K., Poteau R., Chaudret B. and coll, to be published TCCM European Master 14/116 Metal NP = faceted crystal with surface ligands TEM image of a single Ru NP Ruthenium NP = faceted hcp crystal step atoms Diameter ≈2.9 Å 990 atoms (Surface: 402, Core: 588) Nørskov, J. K. and coll. (2005) Ammonia Synthesis from First-principles Calculations, Science, 307, 555-558 TCCM European Master 15/116 Metal NP = faceted crystal with surface ligands Examples 2.9 nm Nørskov, J. K. and coll. (2005) Ammonia Synthesis from Firstprinciples Calculations, Science, 307, 555-558 RuNP (hcp) 5 nm 7 nm 5nm CoNW (hcp) Chaudret, B. and coll. (2009) Iron Nanoparticle Growth in Organic Superstructures, J. Am. Chem. Soc. 131, 549-557 FeNP (bcc) Soulantica, K. (2013) Private Communication TCCM European Master 16/116 Computational Strategies 1st Strategy. Scale Models = Small Inorganic Clusters Ru4H3(C6H6)4CO Ru6H2(CO)18 Ru6H(CO)18- TCCM European Master 17/116 Computational Strategies 1st Strategy. Scale Models = Small Inorganic Clusters. DFT performs well (local basis sets; hybrid functionals - PBE0, B3PW91) H NMR 1 Ru4H3(C6H6)4CO Spectroscopy. NMR (13C, 1H, 2H) Electronic structure Ru6H2(CO)18 Coordination Mode of Ligands on the Surface & Binding Energies H-Ru Spectroscopy. IR del Rosal I., Poteau R. and coll (2009) Ligands effects on the NMR, vibrational and structural properties of tetra- and hexanuclear ruthenium hydrido clusters: a theoretical investigation, Dalton Trans., 2142-2156 TCCM European Master 18/116 Computational Strategies Larger Metal Clusters. Local Basis Sets: Possible Strong Convergence Issues HOMO(α) total DOS Small HOMO-LUMO GAP → convergence issues of the SCF → costly calculation with an inefficient QC software HOMO(β) Ru13 G03 Localized Basis Set RECPs 2000 SCF cycles 1 day / core (8 cores) Outcome: giant atom model DOS insubstantial... ⇒ Which alternative? “d AOs” “s AOs” TCCM European Master 19/116 Computational strategies 2nd Strategy. PBC Calculations, i.e. surfaces 5 nm (001) CoNW (hcp) infinite (0001) surface TCCM European Master 20/116 Computational strategies 3rd Strategy. Model Clusters at the Nanoscale. PBC as well... Ru55 - MD Ru55 - IC H*, PH3**Ru55 - IC Ru55 - HCP Ru147 - IC TCCM European Master 21/116 Electronic structure of large metal systems Quantum size effect: Electronic structure of metallic systems Atom Molecule Cluster NP Solid (metal) TCCM European Master 22/116 Quantum-Size Effects Ruthenium. PBE-PBC calculations (PAW) Energy relative to the lowest structure / ΔEads(H) Special sites / -3.3 / -10.1 / -27.3 / -26.4 / -20.1 / -13.6 / -17.0 I.C. Gerber & R. Poteau (2013), in Nanocatalysis (Serp and Philippot Eds), Wiley energies in kcal/mol TCCM European Master 23/116 Tool of Choice: Periodic DFT Methods within Plane Waves DFT TCCM European Master 24/116 DFT calculations DFT? Zhao and Truhlar (2007), Acc. Chem. Res. 41, 157 TCCM European Master 25/116 DFT NPs = n-electrons systems where: WFT vs. DFT TCCM European Master 26/116 DFT Hohenberg and Kohn Theorem “The external potential is (to within a constant) a unique functional of ; since in turn fixes the Hamiltonian we see that the full manyparticle ground state is a unique functional of “ For a given GS density, it is possible to calculate the corresponding GS wavefunction (Ψ0 is a unique functional of ρ0) The GS energy is calculated as: The functional has its minimum relative to small variations of the density at ρ0(r): P. Hohenberg, W. Kohn (1964), Phys. Rev. B 136, 864 TCCM European Master 27/116 DFT Total Energy Hartree energy (ee repulsion) kinetic energy exchange-correlation energy external potential (eN interaction) The exact form of T[ρ] and Exc[ρ] is unknown ► ► LDA: T[ρ] and Exc[ρ] approximated by the corresponding energies of a homogeneous electron gas of the same local density Kohn-Sham Theory: Parametrize the particle density in terms of a set of one-electron orbitals, φi, representing a non-interacting reference system: Determine the optimal one-electron orbitals using the variational condition: NB. P. Hohenberg, W. Kohn (1964), Phys. Rev. B 136, 864 , but the Koopmans theorem is not valid (on the contrary to HF) TCCM European Master 28/116 DFT Generalized Gradient Approximation. GGA The inhomogeneity of the electron gas in compounds is taken into account f is often fit (!!) to a large dataset of exactly known binding energies of atoms and molecules Hybrid Functionals Mixing of the exact-exchange (i.e. Hartree-Fock) and local-density energies For example, in the case of B3LYP P. Hohenberg, W. Kohn (1964), Phys. Rev. B 136, 864 TCCM European Master 29/116 Tool of Choice: Periodic DFT Methods within Plane Waves Periodic Boundary Conditions TCCM European Master 30/116 Periodicity, Cells 2D lattice 1x1 unit cell a2 spanning vectors a1 2x2 supercell a2 a1 TCCM European Master 31/116 Periodicity, Cells Translational invariance implies the existence of a corresponding quantum number, usually called the Bloch wave vector k. All electronic states can be indexed by this quantum number. cubic piece of bulk Cobalt (fcc ) Sounds weird to introduce periodicity in this case: wait... TCCM European Master 32/116 Periodicity Bloch theorem ● ● ● In a one-electron theory (e.g. DFT-KS), one can introduce a second index, corresponding to the oneelectron band n The Bloch theorem states that the one-electron wavefunctions obey the equation: k is usually constrained to lie within the first Brillouin zone in the reciprocal space (R is a translational vector that leaves the hamiltonian invariant) Brillouin Zone (denominator = volume of the real space, ΩR) TCCM European Master 33/116 Periodicity - The evaluation of many key quantities, e.g. charge density, density-of-states, and total energy) requires integration over the first BZ. The charge density ρ(r), for instance, is given by: where fnk are the occupation numbers - The integration over k is approximated by a weighted sum of a discrete set of points: Calculating Ψ(r1, ...,rN ) has been reduced to calculating ψnk(r) at a discrete set of points {k} in the first BZ, for a number of bands that is of the order of the number of electrons per unit cell. TCCM European Master 34/116 Periodicity Linear conjugated polyenes (1D finite systems): π MOs ∞ ∞ particle in a box Limit conditions → → , if TCCM European Master 35/116 Periodicity Linear conjugated polyenes (1D finite systems): π MOs ∞ ∞ HMO => Space discretization x → x = pa , of the form: TCCM European Master 36/116 Periodicity Linear conjugated polyenes (1D finite systems): π MOs TCCM European Master 37/116 Periodicity Linear conjugated polyenes (1D finite systems): π MOs 2 4 6 8 10 12 14 16 18 20 EF valence N W = band width → 4|β| 0 Energy / β units énergie / unité β conduction C20 C70 towards the crystal TCCM European Master 38/116 Periodicity 1D ∞ systems: Bloch functions elementary cell TCCM European Master 39/116 Periodicity 1D ∞ systems: Band Structure Γ X Band width TCCM European Master 40/116 Periodicity 1D ∞ systems TCCM European Master 41/116 Periodicity 1D ∞ systems: self assembly of PtL4 complexes TCCM European Master 42/116 Periodicity 1D ∞ systems: self assembly of PtL4 complexes TCCM European Master 43/116 Periodicity 2D ∞ systems TCCM European Master 44/116 Surfaces Periodicity. Slabs and Cells. Case of Ru(0001) Ru: Hexagonal close-packed (hcp) hexagonal lattice Ru(001) unit cell Ru(100) Ru(010) hcp structure (Bravais Lattice) Unit cell = lattice with a two-atom basis a=b≠c α = β = 90°; γ = 120° bulk Ru TCCM European Master 45/116 Surfaces Periodicity. Slabs and Cells. Case of Ru(0001) 6 layers / cell 2x2 hcp cell TCCM European Master 46/116 Surfaces Periodicity. Slabs and Cells. Case of Ru step atoms [3,3,2] cells 6 layers / cell 2x2 hcp cell Ru(0001) Ru(1015) TCCM European Master 47/116 Surfaces hcp-001 a = 2.490Å b = 2.490Å c = 27.197Å α = 90° β = 90° γ = 120° Periodicity. Slabs and Cells. Case of Co hcp-010 a = 2.492Å b = 4.037Å c = 29.953Å α = 90° β = 120° γ = 90° ε-001 a = 6.059Å b = 6.059Å c = 28.118Å α = 90° β = 90° γ = 90° TCCM European Master 48/116 Large metal clusters Supercell approach cubic cell 5x5x5 TCCM European Master 49/116 How to setup a PBC calculation? Sampling the BZ ► The charge density ρ(r), should be given by: ► The integration over k is approximated by a weighted sum of a discrete set of k-points: (w is related to symmetry) ► ► Special k-points = k-point meshes Monkhorst-Pack: equally spaced mesh of k-points in the BZ bi = reciprocal lattice vectors kp = total number of k-points in direction p construction rule VASP tutorials and lectures Monkhorst, H. J. & Pack, J. D. (1976) Special Points for Brillouin-Zone Integrations, Phys. Rev. B 13, 5188-5192 TCCM European Master 50/116 How to setup a PBC calculation? Case of Metals → Smearing Methods ► ► in metallic systems BZ integrals over functions are discontinuous at the Fermi-level replace step function fnk (=0, 1, 2) by a smoother function i.e. fnk become fractional occupancies Two main possibilitiés Fermi-Dirac function Broadening of energy levels with Gaussian functions = Gaussian smearing ● ● ● ● ● ● σ = smearing factor, no physical meaning (Fermi-Dirac → temperature) σ → 0 ⇔ step function it turns out that the total energy is no longer variational (or minimal) in this case. It is necessary to replace the total energy by some generalized free energy !! forces are calculated as derivatives of the variational quantity F(σ) E(σ→0) ≈ ½ [F(σ) + E(σ)] For large σ, extrapolation to σ→0 is less accurate than small σ values TCCM European Master 51/116 How to setup a PBC calculation? VASP tutorials and lectures TCCM European Master 52/116 Tool of Choice: Periodic DFT Methods within Plane Waves Plane Waves TCCM European Master 53/116 Why plane waves? Historical reasons: Many elements exhibit a band-structure that can be interpreted in a free electron picture (metallic s and p elements). Pseudopotential theory was initially developed to cope with these elements. Practical reason: “The total energy expressions and the Hamiltonian H are easy to implement.” Martijn Marsman (VASP team) Computational reason: The action H|ψnk> can be efficiently evaluated using FFT’s. where: - Exchange-correlation: easily obtained in real space: - FFT to reciprocal space: - add all contributions: - FFT to real space: TCCM European Master 54/116 PAW TCCM European Master 55/116 Tool of Choice: Periodic DFT Methods within Plane Waves Accuracy TCCM European Master 56/116 DFT … describes trends well Experiments : Toyoshima, G. Somorjai (1979), Catal.Rev.Sci.Eng. 19, 105 Theory : B. Hammer & J. Nørskov (2000), Adv. Catal 45, 71 TCCM European Master 57/116 DFT … but do not expect systematic accuracy Adsorption of CO on surfaces. Experiments: atop adsorption is preferred / hollow adsorption shortcomings of semi-local and hybrid functionals learned from surface science studies “B3LYP and BLYP functionals seem to be the overall best choice for describing adsorption on metal surfaces, but they simultaneously fail to account well for the properties of the metal, vastly overestimating the equilibrium volume and underestimating the atomization energies” Stroppa, A. & Kresse, G., New J. Phys., 2008, 10, 063020 TCCM European Master 58/116 DFT CPU time & Speedup 64 Li-atoms + 1H defect vasp 5.2 TCCM European Master 59/116 Theory-experiments relationship (i) geometry optimization TS R P TCCM European Master 60/116 Introduction ► ► ► ► Geometry optimization is a key component of most computational chemistry studies The notion of molecular structure and potential energy surfaces are outcomes of the Born– Oppenheimer approximation (separation of the motion of electrons from the motion of the nuclei) For a given structure, R, and electronic state, Ψ(r;R), a molecule has a specific energy E(R) A potential energy surface describes how the energy of the molecule in a particular state varies as a function of the structure of the molecule (R) potential energy surface = hilly landscape, with valleys, peaks, and mountain passes TCCM European Master 61/116 Standard algorithms Steepest Descent & Conjugated Gradient F(X) = function to be optimized. Representation of F(X) as an infinite sum of terms (Taylor series) G = gradient H = hessian - define a starting point (X0) X0 - follow the slope p0 - at any point Xk: - follow pk, i.e.: TCCM European Master 62/116 Standard algorithms Steepest Descent & Conjugated Gradient Steepest Descent Conjugated Gradient (where γk is a parameter) TCCM European Master 63/116 Standard algorithms Newton-Raphson It is possible to simply show that: Problem: explicit calculation of the Hessian H at any step Solution: Quasi-Newton methods, which start with an inexpensive approximation to the Hessian, H(X0), and then: Schlegel, H. B. (2011) Geometry optimization, WIRES. Comput. Mol. Sci. 1, 790-809 Broyden-Fletcher–Goldfarb–Shanno update: TCCM European Master 64/116 A very efficient algorithm... … unfortunately usually not available in standard periodic-DFT packages ► ► ► A redundant internal coordinate system for optimizing molecular geometries is constructed from all bonds, all valence angles between bonded atoms, and all dihedral angles between bonded atoms Relevant for molecules. Probably less relevant for metal bonds? Kudin, K. N.; Scuseria, G. E. & Schlegel, H. B. (2001)A redundant internal coordinate algorithm for optimization of periodic systems, J. Chem. Phys. 114, 2919-2923 ● Bučko, T.; Hafner, J. & Ángyán, J. G.(2005) Geometry optimization of periodic systems using internal coordinates, J. Chem. Phys. 122, 124508 ● ► Periodic systems → no technical or methodological bottlenecks ► Definition of the Wilson B matrix ↔ correspondence between cartesian (x) and internal (q) coordinates → H q , Gq the Newton step is given by: can be coupled to eigenvector following algorithms ► ► Peng, C.; Ayala, P. Y.; Schlegel, H. B. & Frisch, M. J. (1996) Using redundant internal coordinates to optimize equilibrium geometries and transition states, J. Comp. Chem. 17, 49-56 TCCM European Master 65/116 A very efficient algorithm... … unfortunately usually not available in standard periodic-DFT packages ► For TS, the success of the optimization depends on the topology of the surface, on the starting point, and on the initial hessian Newton and quasi-Newton algorithms are the most efficient single-ended methods for optimizing transition structures if the starting geometry is within the quadratic region of the transition structure with suitable techniques for controlling the optimization steps such as eigenvector following, these methods will also converge to a transition structure even if they start outside the quadratic region Schlegel, H. B. (2011), Geometry optimization, WIRES. Comput. Mol. Sci. 1, 790-809 TCCM European Master 66/116 Theory-experiments relationship (ii) shape TCCM European Master 67/116 Organometallic NPs : diversity of geometries The gold case 1.4 mL of added water Kim et al (2010), CrystEngComm 12, 116 2 mL of added water TCCM European Master 68/116 Do not try to optimize the geometry of NPs with realistic size... Single point / DOS calculation / VASP 98 SCF cycles 432 nodes 200 hours (user time) Ru288 TCCM European Master 69/116 Theoretical rationalization of the structural dependence Relation between surface energy and structure of organometallic NPs In liquids, shapes are governed by minimisation of the surface free energy (surface tension) In general droplets adopt a spherical geometry and display no directional dependence The shape a crystal adopts during (equilibrium) growth is determined by the directional dependence of the surface tension TCCM European Master 70/116 Theoretical rationalization of the structural dependence Relation between surface energy and structure of organometallic NPs 1878: Gibbs For each facet : surface energy per unit area x surface area 1901: Wulff's theorem → for an equilibrium crystal there is a point in the interior such that its perpendicular distance hi from the ith face is proportional to the surface energy γi Wulff construction determination of the equilibrium shape of a crystal with fixed volume V so that its Gibbs free energy is minimized by assuming a configuration of low surface energy Wulff (1901), Z. Kristallogr. Mineral. 34, 449 ; von Laue (1943), Z. Kristallogr. 105, 124 TCCM European Master 71/116 Theoretical rationalization of the structural dependence Relation between surface energy and structure of organometallic NPs Wulff construction 1. Draw a group of vectors from a common origin whose length is proportional to the surface tension of the crystal face (the direction is perpendicular the face) 2. Draw at the end of each vector a plane perpendicular to the vector direction → The shape enclosed by the planes gives the equilibrium shape of the crystal TCCM European Master 72/116 Theoretical rationalization of the structural dependence Combination of Wulff construction and DFT calculations DFT → γAu(hkl) DFT-based Wulff construction (energetic contribution from edges and corners have been ignored) Barnard et al. (2005), J. Phys. Chem. B 109, 24465 TCCM European Master 73/116 Theoretical rationalization of the structural dependence Combination of Wulff construction and DFT calculations Bare surface Ru(hcp) bulk unitcell Dressed Surface Ru(0001) unitcell TCCM European Master 74/116 Theoretical rationalization of the structural dependence Combination of Wulff construction and DFT calculations Dressed Surface Reuter, K.; Stampfl, C. & Scheffler, M (2005) Yip, S. (ed.) Handbook of Materials Modeling Ab initio atomistic thermodynamics and statistical mechanics of surface properties and functions, Springer, 1, 149-194 ● I. del Rosal, L. Truflandier, R. Poteau, I. C. Gerber (2011) JPCC 115, 2169 ● Calculation of the Gibbs free energy of adsorption can be approximated by ΔE or ΔG (weak dependence of μsolids on pressure, and cancellation of temperature dependence) The surrounding medium acts as a reservoir of ligands: Thermodynamic database tables → S°, H° → Reservoir = gas-phase → aL = pL / p° TCCM European Master 75/116 Theoretical rationalization of the structural dependence Wulff constructions. The case of FeNPs DFT issue? Thermodynamics or kinetics?... Fischer G., Poteau R., Gerber I. C. (2014) to be published TCCM European Master 76/116 Theory-experiments relationship (ii) local structure: IR/Raman/NMR Spectroscopy Feedback with experimentalists Informations about local surface states Solution NMR Solid-State NMR Coordination of hydrides TCCM European Master 77/116 Structural properties of Ru NPs ? Ru(COD)(COT) + PVP H2(3bar) THF; r.t. RDF profile [Ru]0/PVP WAXS Piece of hcp Method valid for bimetallic particles of controlled composition d m = 1.2 nm TCCM European Master 78/116 2 H Solid State NMR Analysis → separation of the contributions Pake Doublet 1 Ru/HDA 2 3 3 1 = C Q 1− Q 2 3 2 = C Q 1 Q 2 3 3= C Q 2 Quadrupole coupling constant CQ Asymmetry parameter Q TCCM European Master 79/116 2 H Solid State NMR A typical experimental spectrum Ru NP/HDA 200K Solid state 45.7 MHz 2H NMR spectra of static samples of Ru/HDA particles after H-D exchange performed in the solid state Electric quadrupole moment of quadrupolar nuclei (I=1) interacts with the electric field gradient (EFG) This interaction is related to the quadrupole coupling constant CQ TCCM European Master 80/116 H Solid State NMR 2 Analysis → separation of the contributions (spectrum = superposition of Pake doublets) Ru/HDA ~ ↔ shape of the electronic density CQ=66 kHz and ηQ=0.3 CQ=160 kHz and ηQ=0.02 TCCM European Master 81/116 Calculation of 2H Solid State NMR parameters Electric quadrupole moment of quadrupole nuclei interacts with the electric field gradient (EFG) This interaction is related to the quadrupole coupling constant CQ DFT → calculation of the EFG tensor V Diagonalization of V → V11, V22, V33 with |V11| < |V22| <|V33| V11+V22+ V33=0 V33 = principal component Calculation of the quadrupole coupling constant: (Q = nuclear quadrupole moment) C Q kHz =672.V 33 a.u e.Q.V 33 C Q= h ⟦ V 22 ⟧ −⟦ V 11 ⟧ ηQ = ⟦ V 33 ⟧ TCCM European Master 82/116 Interactions in Solid State NMR ; nuclei with I=1 Representation of the Electric Field Gradient (EFG) D-C D-Ru CpRu(D)3PPh3 Ru NPs: Where are the hydrogen atoms ? L. Trufflandier, I. del Rosal, B. Chaudret, R. Poteau, I. Gerber, ChemPhysChem 10 (2009) 2939 Assignment of an experimental spectrum assisted by QC Ru/HDA According to DFT coexistence of 3-H H on top -H Periodic-DFT, PBE, CASTEP commercial code Theoretical/experimental interplay NMR = interesting tool for probing organometallic complexes and more generally to understand surface chemistry In particular 2H NMR is a useful tool, in conjunction with DFT calculations ! No need of an explicit description of NPs We have provided reference theoretical 2H NMR data Contrarily to general accepted ideas, hydrogen atoms are not exclusively coordinated at μ3 sites, but a coexistence of μ3, μ and η hydrogen atoms takes place on the surface Spectroscopy H NMR 1 1 H NMR Main Problems ►Calculation of NMR properties of TM compounds : Reliability of theoretical (DFT!) calculations ? ►Motion of the hydrides / dynamic properties NMR spectra may not correspond to a single geometry (the global minimum on the PES) Calculation of chemical shielding tensors (pb of finite-size basis sets GIAO) σ xx σ xy σ xz σ=(σ yx σ yy σ yz ) σ zx σ zy σ zz σ 11 0 0 σ=( 0 σ 22 0 ) 0 0 σ 33 Calculation of chemical shifts δ = σ ref - σ complex 1 σ iso = (σ 11+σ 22 +σ 33) 3 σ aniso=σ 33−σ iso σ 22−σ 11 η= σ aniso an agreement of 10-20% with experiments is often considered as good DFT 1H NMR: validation of the method σtheo vs. δexp Test of reliability for clusters Good agreement σtheo = 31.2 – 0.89 δexp δextrap = (31.2 – σtheo)/0.89 ► calculation of σ satisfactorily accurate Hydridic shift ( < 0) I. del Rosal, L. Maron, R. Poteau and F. Jolibois Dalton Trans. (2008) 3959 I. del Rosal, L. Maron, F. Jolibois and R. Poteau Dalton Trans. (2009) 2142 DFT 1H NMR: calibration of the method Interstitial hydrides can also be probed by 1H NMR Octahedral site subsurfacic H strongly unshielded : δ exp +17.0 ppm δ theo +18.1 ppm Ru6(CO)18(μ6-H) I. del Rosal, L. Maron, F. Jolibois and R. Poteau Dalton Trans. (2009) 2142 VASP NMR chemical shifts Chemical shift GIAO (G03) GIPAW (VASP 5.3) H -2.9 -5.0 H2 -10.5 -8.3 CH3 CH 17.5 14.2 17.4 139.6 14.7 147.8 Linear response method limited to diamagnetic and insulator systems C.J. Pickard, F. Mauri, Phys. Rev. B 63, 245101 (2001), J.R. Yates, C.J. Pickard, F. Mauri, Phys. Rev. BTCCM 76, 024401 European(2007) Master 90/116 Chemical shifts of adsorbed species 1 H NMR calculations on surfaces or NPs ? fcc C.J. Pickard, F. Mauri, Phys. Rev. B 63, 245101 (2001), J.R. Yates, C.J. Pickard, F. Mauri, Phys. Rev. B 76, 024401 (2007) TCCM European Master 91/116 NMR. Summary NMR = interesting tool for probing organometallic complexes and more generally to understand surface chemistry In particular 2H NMR is a useful tool, in conjunction with DFT calculations ! No need of an explicit description of NPs Half Spin nuclei NMR: GIPAW extended to metals! We have provided reference theoretical 2H and 1H NMR data >> Conclusion J. Am. Chem. Soc., 2010, 132, 11759-11767 TCCM European Master 92/116 Analysis of the electronic structure TCCM European Master 93/116 Density of states (DOS) Benzene DOS = number of states n(E) lying between E and E + dε DOS ? TCCM European Master 94/116 Density of states (DOS) εj / unité β Conjugated polyenes -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 10 20 30 40 50 j 60 70 80 90 100 DOS ? TCCM European Master 95/116 Density of states (DOS) Periodic Systems ► The DOS depends on the k-point grid Related to the number of electrons, N: ► Can be used as a population analysis tool when Projected on AOs (χp) or atoms (I) = PDOS ► Other case: projection of the DOS on the d AOs of a metal atom, α ► TCCM European Master 96/116 Density of states (DOS) Stacked PtH42- TCCM European Master 97/116 Tool: where are the bonds ? TCCM European Master 98/116 Tool: where are the bonds ? TCCM European Master 99/116 Tool: where are the bonds ? TCCM European Master 100/116 DOS/COHP LiLa9Mo16O35 Mo16O36 Possible reduction of LiLa9MO16O35 ? J. Cuny (2011) Thèse de l'université de Rennes TCCM European Master 101/116 What about PW calculations? No AOs! ► ► spdf-projected wavefunction character of each band is needed projection of the orbitals onto spherical harmonics that are non-zero within spheres of a given radius (for example Wigner-Seitz radius) around each ion ⇔ minimal basis set Ru55-CO I. C. Gerber, R. Poteau, tools4VASP, to be available – one day - as an opensource project TCCM European Master 102/116 Chemical activity Multi-step chemical reactions TCCM European Master 103/116 Quantum chemistry and reactivity Born-Oppenheimer approximation: TCCM European Master 104/116 On the role of defects N2 adsorption: the DFT point of view Experimental and theoretical evidence for the presence of step atoms B5 es s it 2.5 nm Nørskov and coll. (2005), Science 307, 555 TCCM European Master 105/116 Quantum chemistry and reactivity Nudged Elastic Band(NEB) TS search algorithms: TS, QST, NEB Liotard & Penot. (1981), in Numerical Methods in the Study of Critical Phenomena, Springer, 213 Liotard (1992), Int. J. Quant. Chem. 44, 723 Sheppard et al (2008), J. Chem. Phys. 128, 134106 Henkelmann et al (2000), J. Chem. Phys. 113, 9901 Peng & Schlegel (1993), Israel J. Chem. 33 449 TCCM European Master 106/116 Quantum chemistry and reactivity Qualitative methods: frontier orbital theory Transferability to the adsorbate / surface case ? TCCM European Master 107/116 Qualitative methods: frontier orbital theory Transferability to the adsorbate / surface case R. Hoffmann (1988), Solids and surfaces. A chemist's view of bonding in extended structures, Wiley B. Hammer & J. Nørskov (2000), Adv. Catal 45, 71 I.C. Gerber & R. Poteau (2012), in Nanocatalysis (Serp and Philippot Eds), Wiley TCCM European Master 108/116 Qualitative methods: frontier orbital theory d-band center model for adsorption on ∞ surfaces Confirmation: The highest εd → the lowest ΔEads Experiments : Toyoshima, G. Somorjai (1979), Catal.Rev.Sci.Eng. 19, 105 Theory : B. Hammer & J. Nørskov (2000), Adv. Catal 45, 71 TCCM European Master 109/116 Relationship with reactivity? Brönsted-Evans-Polanyi (BEP) relationship relates the change in activation energy of a reaction to the change in its reaction energy widely applied in the analysis of surface elementary reaction steps overall rate of a catalytic reaction (kinetics!) ↔ strength of the adsorbate chemical bonds (thermodynamics) van Santen, R. A.; Neurock, M. & Shetty, S. G. (2010) Chem. Rev. 110, 2005-2048 TCCM European Master 110/116 Why are adsorption energies so important? Science 332 (2011) 224-228 The search for more efficient heterogeneous catalysts remains critical to the chemical industry. The Sabatier principle of maximizing catalytic activity by optimizing the adsorption energy of the substrate molecule could offer pivotal guidance to otherwise random screenings. Here we show the chemical shift value of an adsorbate (formic acid) on metal colloid catalysts measured by 13C nuclear magnetic resonance (NMR) spectroscopy in aqueous suspension constitutes a simple experimental descriptor for adsorption strength. TCCM European Master 111/116 Qualitative methods: frontier orbital theory d-band center model for adsorption on ∞ surfaces Application to strained overlayers The highest εd → the lowest ΔEads Ruban et al. (1997), J.Mol.Catal. A 115, 421 TCCM European Master 112/116 Qualitative methods: frontier orbital theory d-band center model for adsorption on ∞ surfaces Application to Pt surfaces The highest εd → the lowest ΔEads TCCM European Master 113/116 Qualitative methods: frontier orbital theory d-band center model for adsorption on ∞ surfaces better adsorption on steps Rate-determining step + it seems that this general property can be extrapolated to (at least large) NPs TCCM European Master 114/116 Quick-and-Dirty Conclusion ► QC calculations on NPs = challenging domain ► Ruled by quantum-size effects ► Several strategies Facets → Surfaces Large Clusters Even small clusters for some spectroscopic data ► Beyond numbers, don't forget to build conceptual models TCCM European Master 115/116 Some interesting tools or websites Vesta Graeme Henkelman research group's web site → nice utilities & methods VASP site & the on-line Hands-On sessions jMol TCCM European Master 116/116