Geometry optimization of solids

Transcription

1 The Minnesota Workshop on ab Initio Modeling in Solid State Chemistry with CRYSTAL Minneapolis, MN(U.S.A.) 9-14 July Geometry optimization of solids Bartolomeo Civalleri Dip. di Chimica IFM, Via P.Giuria 5 Università di Torino bartolomeo.civalleri@unito.it Some slides are courtesy of C.M. Zicovich-Wilson, A. Erba, A. Rimola Geometry Optimization for: 2 - Finding equilibrium nuclear configurations; - Finding and characterizing Transition States; - Including the effect of Pressure on structural properties of crystals; - Accounting for nuclear-relaxation effects on tensorial response properties (elastic, piezoelectric, photoelastic, etc.); - 1

2 Chemical structures 3 Molecule/Cluster (0D) Polimer (1D) Carbon nanotube (9,9) τ Surface (2D) Crystal (3D) τ 1 τ 2 (001) τ 1 τ 2 τ 3 Slab model of the (001) MgO surface Urea bulk How can one define a chemical structure? Geometries 4 The same chemical compound (same atoms, same bonds) may have many nuclear configurations. In general these nuclear configurations have different energies, and experimentally they appear as a statistical distribution of nuclear arrangements. Which of such configurations (geometries) can be adopted as a suitable representative of the compound in order to compute its properties and compare them with observations? 2

3 Cation location in proton and alkali-exchanged chabazite Experimental data O1 O4 O3 O2 SIII SIII SII SI Three general cation positions from experiments: 1. SI center of hexagonal prism 2. SII six-membered ring of the prism 3. SIII and SIII eight-membered ring Neutron powder diffraction, MAS-NMR, MQMAS-NMR measurements [1-3] on H, Li, Na and K chabazites show: H - two sittings: O1 (SIII ) and O3 (SII) Li - two sittings: SII and SIII Na - two sittings: SII and SIII K - located at SIII No cations were found at the SI site. [1] Smith, L.J.; Davidson, A.; Cheetham, A.K. Catal. Lett. 1997, 49, 143 [2] Smith, L.J.; Eckert, H.; Cheetham, A.K. J. Am. Chem. Soc. 2000, 122, [3] Smith, L.J.; Eckert, H.; Cheetham, A.K. Chem. Mater. 2001, 13, 385. Instability of site SI in Na-exchanged chabazite Sodium is not stable at site SI and moves towards site SII, A.M. Ferrari, M. Llunell, R. Orlando, M. Mérawa, P. Ugliengo, Chem. Mater.,

4 7 Geometry Optimization Definition (I) Potential Energy Surface (PES) x i : Cartesian coordinates of the N nuclei E=f(x 1,x 2,x 3, x 3N ) PES describes the energy of the system as a function of its geometry ( hypersurface) PES arises naturally from the Born-Oppenheimer approximation PES makes it possible to discuss chemical structures 8 Internal degrees of freedom E=f(x 1,x 2,x 3, x 3N ) Invariant under translations and rotations internal degrees of freedom Molecules: M = 3N 3 3(2) Cartesian translational rotational 3D crystals: M = 3N 3 + ( 9 3 ) lattice 4

5 9 Coordinate systems There are infinite choices to define the M internal coordinates The shape of the PES depends on the adopted coordinate system The best one should be the one that makes easiest the study of the potential energy function (to be discussed later) Let's assume that v = (v 1, v 2,...,v M ) is the nuclear configuration vector in a given coordinate choice PES: relevant information 10 Minimum for product A Transition state A Second order saddle point Minimum for reactant Second order saddle point Transition state B m Valleyridge Inflection point E=f(v 1,v 2,v 3, v M ) Minimum for product B How to describe the features of the PES? Relevant information (invariant under transformations of the coordinate system) is provided by critical points 5

6 PES: relevant information 11 Minimum for product A Transition state A Second order saddle point Minimum for reactant Second order saddle point Transition state B m Valleyridge Inflection point E=f(v 1,v 2,v 3, v M ) Minimum for product B H ij 2 E = vi vj v Relevant information (invariant under transformations of the coordinate system) is provided by critical points 0 12 Geometry Optimization Definition (II) Chemical Structures Equilibrium geometries: are local minima on the PES (e.g. conformers, isomers, polymorphs, reactants, intermediates and products of a chemical reaction) Transition states: correspond to saddle points (firstorder) on the PES (e.g. transition structures of a chemical reaction) 6

7 PES and related quantities (derivatives) 13 E x, a ) Energy differences ( i E E ; Forces (geometry optimisation) x a i 2 E Vibrations (Phonons) x i x j 2 E Elastic constants a i a j E Pressure V 2 E 2 Bulk modulus V x: atomic coordinates a i : lattice constants V: cell volume 14 Which is the meaning of Geometry Optimization? It is the process of finding minima and saddle points on the PES How to do that? 7

8 Geometry Optimization Tools (I) 15 Characterization of the PES finding the critical (or stationary) points First derivatives Gradient (vector) In classical mechanics: F x =- V/ x force acting on the atoms Second derivatives Hessian matrix Condition: E v i v0 = 0 v 0 is a stationary point ij 2 E = vi vj v v i is a nuclear configuration vector in a given coordinate choice H 0 Eigenvalue equation Geometry Optimization Tools (II) 16 Topological analysis of the PES Minima local or global Equilibrium structures forces=0 Hessian matrix positive definite all the eigenvalues must be positive Critical points of order 0 Saddle points forces=0 one or more eigenvalues of the Hessian matrix are negative Critical points of index 1 (one negative eigenvalue; all the others positive) first order saddle points Transition State structures N-order saddle points n negative eigenvalues (N=M maximum) 8

9 Geometry Optimization Methods (I) 17 Finding an equilibrium geometry involves an unconstrained optimization on the PES Algorithms for energy minimization Energy-based methods (e.g. line optimization,...) Simple, widely applicable, very slow convergence Gradient-based methods (e.g. conjugate gradient, quasi-newton,...) Good convergence, expensive Second derivative methods (e.g. Newton, Newton- Raphson,...) Very fast, very expensive Geometry Optimization: Methods (II) 18 Strategy (in Newton and quasi-newton (QN) methods): PES is approximated as a quadratic function (Taylor expansion truncated to second-order local quadratic region) E(v) = E 0 + g 0 t Δv + (1/2) Δv t H 0 Δv g(v) = g 0 +H 0 Δv where Δv = v - v 0 ; g is the gradient and H is the Hessian g(v) = g 0 +H 0 Δv = 0 (at the minimum) Δv = -H -1 g (Newton s step) 9

10 QN methods: some remarks 19 For a quadratic PES, the minimum can be found in one step if the gradient vector and the Hessian matrix are known Few cycles are required to reach the minimum even on non-quadratic PESs The Hessian calculation is computationally expensive, so QN methods are based on the use of gradients (analytically computed) E ( ) gi = 0 = E v 0 < < < 2 E E Hij = vi v vi vj 0 v 0 QN methods: variable metric 20 During the optimization process, the changes in the gradient between steps provide useful information on the surface curvature (second derivatives) An approximation to the Hessian matrix, H, or its inverse, H -1 is built at the beginning and updated during the optimization process (variable metric) by using information from displacements and gradients. Several Hessian updating schemes exist: Fletcher-Powell Murtagh-Sargent (SR1) Powell-Symmetric-Broyden (PSB) Broyden-Fletcher-Goldfarb-Shanno (BFGS) Note: some of them impose a Hessian matrix positive definite 10

11 Flow chart for Quasi-Newton algorithms Choose coordinate system Input starting geometry Obtain initial estimate of Hessian Calculate energy and gradient 21 Update the Hessian Use Hessian and gradient to take a step If necessary, restrict step size Check for convergence on gradient and displacement no Check for maximum cycles no Update geometry yes yes Done Stop 22 Efficiency of a geometry optimization The efficiency (i.e. number of steps) depends on several factors: 1) Initial geometry (experiment, GUI/Molecular Modelling, Molecular Mechanics, lower level QM) 2) Coordinate system (Cartesian, internal (redundant, natural,...)) 3) Hessian initial guess (Identity, Molecular Mechanics, lower level QM, numerical) 4) Hessian updating (Berny, MS, DFP, BFGS,...) 5) Step size control (simple rescaling, trust radius, RFO,...) 11

12 23 Estimating the Hessian matrix A good estimate of the initial Hessian can drastically reduce the optimization process L Identity Matrix [HESSIDEN] No structure information, hence very inefficient K Numerical Hessian [HESSNUM] Close to exact, hence fast convergence. However, high calculation cost (good for difficult cases, e.g. transition state searches) J Model Hessian [HESSMOD1,HESSMOD2] Good and cheap approximation. Based on valence force fields (Lindh, Schlegel). Significant improvement with respect to the Identity matrix (>50% speed-up). Step size control 24 When the step size is too large a simple scaling can be applied (too crude) In non-quadratic functions the Newton step may not be the best choice and should be controlled: Level-shift trust region: A trust radius of an hyper-sphere in which the function is expected to behave quadratically. A parameter, µ, is computed so that the displacement: v i+1 v i = (H i -Iµ) -1 g(v i ) is kept within the trust region. Ensured to be the best direction within the hyper-sphere of radius, τ = v i+1 v i Line search: A scale factor α i is computed as to reach a minimum along the Newton step direction, according to: v i+1 v i = - α i H i -1 g(v i ) 12

13 Choice of coordinate system 25 Good Performance Quadratic behaviour Coordinate transformation: v i = v i (s 1, s 2,..., s M ) Transformation of F(v) results in a new function F': F'(s 1, s 2,..., s M ) = F(v 1 (s 1, s 2,..., s M ), F(v 2 (s 1, s 2,..., s M ),..., F(v M (s 1, s 2,..., s M ) Ideal transformation gives function F' that behaves quadratically close to the critical point Coordinate systems suitable for crystals: Fractionary coordinates + elastic distorsions (symmetry adapted) Internal valence coordinates (redundant) Coordinate systems for crystals (I) 26 Fractionary coordinates + elastic cell distortions Atom coordinates defined in the basis set of the lattice vectors. Cell parameters obtained from elastic distortions (expansion matrix applied on the lattice vectors) J Easy to keep special positions and cell parameters by imposing symmetry K Delocalised by definition. Not useful for chemical reactivity. L In general, non-quadratic L Low dependancy between coordinates and lattice parameters 13

14 Coordinate systems for crystals (II) 27 Redundant internal coordinates All valence coordinates (bond distances, angles and torsions) are considered to form a set of redundant coordinates (N RC > M) Connectivity is usually defined from vicinity criteria (R AB < R vdw,a +R vdw,b ) Cell parameters are embedded into the redundant coordinates Hessian, gradient and geometry displacements are built in terms of the redundant coordinates The redundancies are eliminated to obtain the actual geometry (in Cartesian coordinate space) using numerical approximations (back-transformation) J Quadratic behaviour J Easy choice of geometrical parameters J Easy to constrain 'chemical' degrees of freedom (bond lengths, angles or dihedrals) Good for reactivity studies L The size of redundant space may be very large (improved in CRYSTAL17) L The back-transformation from redundant to non-redundant space is difficult (improved in CRYSTAL17) Comparison between Frac.+Cell and Redundant System Dim Theor. Method Frac+Cell Redundant C-R Nstep Nstep ΔE (µha) α-quartz 3D B3LYP β-quartz 3D LDA All silica Faujasite 3D PBE KNbO 3 tetragonal 3D BLYP BaTiO 3 orthoromb. 3D LDA Bohemite 3D PW Calcite 3D B3PW NaNO 2 3D BP TiO 2 3D PW ZnGeP 2 3D LDA/ECP ZrO 2 3D PBE Formamide 3D B3LYP Ice 3D PW Oxalic acid 3D B3LYP Oxalic acid 3D HF Al 2 O 3 (0001) 12-layer 2D B3LYP EDI(100)/NH 3 2D B3LYP H 2 O infinite chain 1D PBE Li doped PA 1D B3LYP Tot

15 Geometry optimizer in CRYSTAL 30 Quasi-Newton algorithm BFGS Hessian updating Initial model Hessian (Schlegel) Trust Radius (TR) Newton s step only Linear search (if not TR) Symmetry adapted fractional coordinates (symmetrized directions) Symmetry adapted elastic deformations (symmetrized deformations) Analytical forces for Hartree-Fock and DFT 0D, 1D, 2D, 3D systems All-electron and Effective Core Potential (ECP) Atomic positions: K. Doll, V.R. Saunders, N.M. Harrison, IJQC 82 (2001) 1 Cell parameters: K. Doll, R. Dovesi, R. Orlando, TCA 112 (2004) 394 Geometry optimization type Unconstrained optimization of cell and atomic positions Transition state search (atoms, cell) Constraint optimization: constant volume, fragment, Geometry optimization input block 31 E.g.: α-quartz - hexagonal cell RHF/STO-3G a-qua (in. geom. expt.) CRYSTAL OPTGEOM Optional keywords END ENDG Title Dimensionality of the system Crystallographic information (3D only) Space Group (Fm3m 154) Shift of the origin Lattice parameters (hexagonal) Number of non equivalent atoms Atomic number and fractional coordinates Geometry optim. input block Optimization of atomic positions + cell End of geom. opt. input block End of geometry input section Optional keywords are related to: Modification of the algorithm (Initial Hessian, Hessian updating, ) Convergence criteria and tolerances (TOLDEE, TOLDEX, TOLDEG) Constrained geometry optimization See CRYSTAL Tutorials (Geometry Optimization) 15

16 Geometry optimization input block 32 E.g.: α-quartz - hexagonal cell RHF/STO-3G Redundant internal coordinates a-qua (in. geom. expt.) CRYSTAL OPTGEOM INTREDUN Optional keywords END ENDG Full geometry optimization of atomic positions and lattice param.s 33 Geometry Optimization is important for Equilibrium bond lengths and angles Equilibrium cell parameters Discriminating between competing structures Surface relaxation and reconstruction Pressure-driven phase transitions Starting point for many advanced studies 16

17 12/07/17 38 Constrained geometry optimization Optimization in Cartesian/fractionary coordinates (default) 1. Fixing Cartesian coordinates (FRAGMENT) 2. Constant volume geometry optimization (CVOLOPT) 3. Fixing a lattice parameter (CRYDEF FIXDEF) 4. Constant pressure geometry optimization (EXTPRESS) Optimization in redundant internal coordinates (INTREDUN) 1. Fixing a set of internal coordinates (FREEZINT) 2. Adding a new internal coordinate (DEFLNGS, ) 3. Fixing a given internal coordinate (LNGSFROZEN, ) 48 A special acknowledgement Claudio M. Zicovich-Wilson ( ) 17