Download Plane-Wave DFT Methods Applied to large Molecules and

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
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Large Molecules and NPs
(http://www.vchem3d.com)
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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)
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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
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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)
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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)
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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
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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
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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
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in silico modeling of organometallic NPs: a
challenge for computational chemists
strategy?
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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
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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
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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
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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
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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
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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
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Computational Strategies
1st Strategy. Scale Models = Small Inorganic Clusters
Ru4H3(C6H6)4CO
Ru6H2(CO)18
Ru6H(CO)18-
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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
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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”
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Computational strategies
2nd Strategy. PBC Calculations, i.e. surfaces
5 nm
(001)
CoNW (hcp)
infinite (0001) surface
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Computational strategies
3rd Strategy. Model Clusters at the Nanoscale. PBC as well...
Ru55 - MD
Ru55 - IC
H*, PH3**Ru55 - IC
Ru55 - HCP
Ru147 - IC
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Electronic structure of large metal systems
Quantum size effect: Electronic structure of metallic systems
Atom
Molecule
Cluster
NP
Solid (metal)
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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
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Tool of Choice: Periodic DFT Methods
within Plane Waves
DFT
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DFT calculations
DFT?
Zhao and Truhlar (2007), Acc. Chem. Res. 41, 157
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DFT
NPs = n-electrons systems
where:
WFT vs. DFT
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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
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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)
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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
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Tool of Choice: Periodic DFT Methods
within Plane Waves
Periodic Boundary Conditions
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Periodicity, Cells
2D lattice
1x1 unit cell
a2
spanning vectors
a1
2x2 supercell
a2
a1
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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...
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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)
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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.
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Periodicity
Linear conjugated polyenes (1D finite systems): π MOs
∞
∞
particle in a box
Limit conditions
→
→
, if
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Periodicity
Linear conjugated polyenes (1D finite systems): π MOs
∞
∞
HMO => Space discretization x → x = pa
, of the form:
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Periodicity
Linear conjugated polyenes (1D finite systems): π MOs
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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
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Periodicity
1D ∞ systems: Bloch functions
elementary cell
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Periodicity
1D ∞ systems: Band Structure
Γ
X
Band width
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Periodicity
1D ∞ systems
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Periodicity
1D ∞ systems: self assembly of PtL4 complexes
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Periodicity
1D ∞ systems: self assembly of PtL4 complexes
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Periodicity
2D ∞ systems
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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
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Surfaces
Periodicity. Slabs and Cells. Case of Ru(0001)
6 layers / cell
2x2 hcp cell
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Surfaces
Periodicity. Slabs and Cells. Case of Ru
step atoms
[3,3,2] cells
6 layers / cell
2x2 hcp cell
Ru(0001)
Ru(1015)
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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°
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Large metal clusters
Supercell approach
cubic cell
5x5x5
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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
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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
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How to setup a PBC calculation?
VASP tutorials and lectures
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Tool of Choice: Periodic DFT Methods
within Plane Waves
Plane Waves
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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:
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PAW
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Tool of Choice: Periodic DFT Methods
within Plane Waves
Accuracy
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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
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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
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DFT
CPU time & Speedup
64 Li-atoms + 1H defect
vasp 5.2
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Theory-experiments relationship
(i) geometry optimization
TS
R
P
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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
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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.:
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Standard algorithms
Steepest Descent & Conjugated Gradient
Steepest Descent
Conjugated Gradient
(where γk is a parameter)
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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:
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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
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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
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Theory-experiments relationship
(ii) shape
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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
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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
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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
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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
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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
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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
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Theoretical rationalization of the structural dependence
Combination of Wulff construction and DFT calculations
Bare surface
Ru(hcp)
bulk unitcell
Dressed Surface
Ru(0001)
unitcell
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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°
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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
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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
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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
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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
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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
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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
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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 ⟧
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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
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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)
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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
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Analysis of the electronic structure
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Density of states (DOS)
Benzene
DOS
=
number of states n(E) lying between E and E + dε
DOS ?
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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 ?
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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
►
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Density of states (DOS)
Stacked PtH42-
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Tool: where are the bonds ?
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Tool: where are the bonds ?
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Tool: where are the bonds ?
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DOS/COHP
LiLa9Mo16O35
Mo16O36
Possible reduction of LiLa9MO16O35 ?
J. Cuny (2011) Thèse de l'université de Rennes
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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
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Chemical activity
Multi-step chemical reactions
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Quantum chemistry and reactivity
Born-Oppenheimer approximation:
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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
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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
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Quantum chemistry and reactivity
Qualitative methods: frontier orbital theory
Transferability to the adsorbate / surface case ?
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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
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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
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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
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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.
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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
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Qualitative methods: frontier orbital theory
d-band center model for adsorption on ∞ surfaces
Application to Pt surfaces
The highest εd
→ the lowest ΔEads
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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
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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
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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
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