Hirshfeld methods and Quantum Crystallography
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1 AvdL M. Giorgi & A. van der Lee Fédération des sciences chimiques de Marseille, Marseille Institut Européen des Membranes, Montpellier
2 AvdL Introduction Quantum Crystallography? See also: Genoni et al., Quantum Crystallography: Current developments and future perspectives, Chem. Eur. J., 2018, DOI: /chem
3 AvdL Introduction QC results and XRD as mutual constraints Clinton et al. (1969) Jayatilaka & Dittrich (2008)
4 vdl Introduction Doing more with your diffraction data - I Tarahhomi & Van der Lee, in press (2018)
5 Introduction Doing more with your diffraction data - II Classical and more advanced structure analyses Classical analysis atomic positions structure packing features interactions by analysis of atom-atom distances Electron density (ground state of the system - Hohenberg-Kohn theorem) total energy of the system molecular properties such as inter and intramolecular interactions/chemical bonds ρ(r) is a quantum-mechanical observable!
6 AvdL Contents Contents From theory to practice Electron density and chemical bonding Experimental and theoretical modelling of electron densities Hirshfeld analysis Hirshfeld surfaces Hirshfeld refinement Software Tonto & CrystalExplorer Tonto & OLEX2
7 Electron density and chemical bonding Contents From theory to practice Electron density and chemical bonding Experimental and theoretical modelling of electron densities Hirshfeld analysis Hirshfeld surfaces Hirshfeld refinement Software Tonto & CrystalExplorer Tonto & OLEX2
8 vdl Electron density and chemical bonding X-rays interact with electrons ECIPROCS BAG at Soleil - february Zheng, Legrand, Van der Lee et al. R
9 vdl Electron density and chemical bonding Residual density ECIPROCS BAG at Soleil - february Zheng, Legrand, Van der Lee et al. R
10 Electron density and chemical bonding History of electron density and chemical bonding - I It seems to me that the experimental study of scattered radiation, in particular from light atoms, should get more attention, since along this way it should be possible to determine the arrangement of the electrons in the atoms P. Debye - Zerstreuung von Röntgenstrahlen: Ann. Phys 351, (1915) An electron may form a part of the shell of two different atoms and cannot be said to belong to either one exclusively G. N. Lewis - The atoms and the molecule : J. Am. Chem. Soc. 38, (1916)
11 Electron density and chemical bonding History of electron density and chemical bonding - II Pauling (1939) - On the nature of the chemical bond application to the elucidation of complex substances Bader ( ) - Quantum theory of atoms in molecules chemical bonding based on the topology of electron density distribution Stewart (1976), Hansen & Coppens (1978) - Extension of refinement techniques to include effects of chemical bonding
12 AvdL Experimental and theoretical modelling of electron densities Contents From theory to practice Electron density and chemical bonding Experimental and theoretical modelling of electron densities Hirshfeld analysis Hirshfeld surfaces Hirshfeld refinement Software Tonto & CrystalExplorer Tonto & OLEX2
13 AvdL Experimental and theoretical modelling of electron densities Experimental determination of electron density
14 Experimental and theoretical modelling of electron densities The spherical and the aspherical model F (hkl) = j f j exp[2πi(hx j + y j + z j )] f j (q, E) = ρ j (r) exp[iq.r]d 3 r spherical electron distribution f j (q, E) = ρ j (r) exp[iq.r]d 3 r IAM: Independent Atom Model
15 AvdL Experimental and theoretical modelling of electron densities Calculation of spherical atomic scattering factors IAM uses quantum-mechanically calculated atomic scattering factors (ex-situ) as does Hirshfeld Atomic Refinement (in-situ).
16 Experimental and theoretical modelling of electron densities Treatment of the aspherical atomic scattering factors in the multipole model N atoms N sym F (q) = i j 4 Φ il (q, κ l) l [ P i,core f i,core (q) + P i,val f i,val (q, κ)+ l m= l ] P ilm Ylm(q/ q ) i e iq.r ij.t i (q) Three contributions: core - spherical valence - aspherical valence electrons; f i,core and f i,val are derived from the classical spherical atomic scattering factors f (q, E) To be refined: population parameters P, expansion-contraction parameters κ and κ
17 Experimental and theoretical modelling of electron densities In between the IAM and the multipole model The different single-crystal refinement methods Independent atom model refinements (SHELXL, CRYSTALS, JANA) Intermediate approaches Maximum Entropy methods (BAYMEM) Database transfer (INVARIOM, MOPRO) Hirshfeld refinement (TONTO) Multipole refinements (XD, MOPRO, JANA)
18 Experimental and theoretical modelling of electron densities Quantum-mechanical determination of electron density Solve the Schrödinger equation Ĥψ = Eψ Two approaches Wave function based ab initio ρ is the square of the absolute value of the wave function Electron density based density functional treatment
19 Hirshfeld analysis Hirshfeld surfaces Contents From theory to practice Electron density and chemical bonding Experimental and theoretical modelling of electron densities Hirshfeld analysis Hirshfeld surfaces Hirshfeld refinement Software Tonto & CrystalExplorer Tonto & OLEX2
20 Hirshfeld analysis Hirshfeld surfaces The Hirshfeld surface Where are the interactions between molecules located in space? How is the surface of a molecule defined? Hirshfeld partitioning: w(r) = Nmolecule i=1 ρ i (r) Ncrystal i=1 ρ i (r) = ρ promolecule(r) ρ procrystal (r) where ρ i (r) is the spherically-averaged atomic electron density function. Hirshfeld surface w(r) = 1/2
21 Hirshfeld analysis Hirshfeld surfaces Intermezzo Hirshfeld did not invent the Hirshfeld surface, but he did invent the stockholder partitioning scheme for an atom in a molecule; Spackman generalized the concept to a molecule in a crystal and named it Hirshfeld surface.
22 AvdL Hirshfeld analysis Hirshfeld surfaces The Hirshfeld atom ρ i (r) w i (r) = Nmolecule i=1 ρ i (r) = ρ i (r) ρ promolecule where ρ i (r) is the spherically-averaged atomic electron density function. w i (r) defines the relative share of atom i to the total promolecule density at r. Hirshfeld atom w 2 (r) = 1/2
23 AvdL Hirshfeld analysis Hirshfeld surfaces The Hirshfeld surface visualized Not very interesting...
24 AvdL Hirshfeld analysis Hirshfeld surfaces The Hirshfeld surface visualized - transparency Nice but mapping a property on the surface is more interesting!
25 AvdL Hirshfeld analysis Hirshfeld surfaces Mapping properties on the Hirshfeld surface Mapped with the d norm property
26 Hirshfeld analysis Hirshfeld surfaces d i, d e and d norm Definitions d i : distance from the surface to the nearest nucleus internal to the surface d e : distance from the surface to the nearest nucleus external to the surface d norm : normalized contact distance d norm = d i ri vdw ri vdw + d e r vdw e r vdw e
27 Hirshfeld analysis Hirshfeld surfaces Hirshfeld surface versus Promolecule surface isosurface at 0.5 isosurface at a.u. Promolecule surface is rather a surface in the gaz state
28 Hirshfeld analysis Hirshfeld surfaces Fingerprint plots Each (d i,d e ) pair on the Hirshfeld surface is taken and plotted in a 2D plot, where the color indicates the frequency (number) of (d i,d e ) pairs found: white = no pairs; increasing pair frequency from blue to red. AvdL isosurface at 0.5
29 vdl Hirshfeld analysis Hirshfeld surfaces Decomposed fingerprint plots Contribution of O H interactions: 46.5 %.
30 AvdL Hirshfeld analysis Hirshfeld atomic refinements Contents From theory to practice Electron density and chemical bonding Experimental and theoretical modelling of electron densities Hirshfeld analysis Hirshfeld surfaces Hirshfeld refinement Software Tonto & CrystalExplorer Tonto & OLEX2
31 Hirshfeld analysis Hirshfeld atomic refinements Hirshfeld refinements Using tailor-made aspherical atomic electron densities, which are extracted from a crystal-field embedded quantum-chemical electron density using Hirshfeld s scheme AvdL Diborane (B 2 H 6 ); left - IAM B-atom; right - Hirshfeld B-atom; blue and red - deformation Hirshfeld density for bridging H Result of HAR: 50% probability level
32 AvdL Hirshfeld analysis Hirshfeld atomic refinements Hirshfeld Atomic Refinement - I Step 1 - creating initial density - ED cycle 1. calculate QM initial density from IAM positions 2. divide QM density in Hirshfeld atoms 3. calculate charges and moments of these Hirshfeld atoms 4. calculate electric crystal field 5. calculate new QM density 6. go to 2 until convergence in molecular energy
33 Hirshfeld analysis Hirshfeld atomic refinements Hirshfeld Atomic Refinement - II Step 2 - structure refinement - HAR cycle 1. Fourier transform the Hirshfeld atoms in order to obtain non-spherical atomic scattering factors 2. perform a structure refinement 3. go to 2 until convergence Then alternate ED cycle and HAR cycle until complete convergence Note: the spherical atomic scattering factors used in IAM refinements have been obtained by theoretical calculations... Thus Hirshfeld refinement is a natural extension of IAM refinement.
34 AvdL Hirshfeld analysis Hirshfeld atomic refinements Quantum-mechanical electron density calculations Details Hartree-Fock (rhf) and Kohn-Sham (rks/blyp) density functional theory wavefunction is calculated self-consistently (SCF) possibility of inclusion of cluster of charges and dipoles during the SCF calculation in order to account for the crystal environment def2-svp or cc-pvdz Gaussian basis sets perform well, no need for more elaborate sets
35 Modélisation moléculaire? Modèles Mathématiques Structure(s)-3D d un système moléculaire énergie, réactivité, propriétés électroniques, Comprendre Prévoir Copyright : Pr. Didier Siri, ICR, Aix-Marseille Université
36 Méthodes Quantiques Méthodes Classiques Photochimie Méthodes Multiréférences Spectroscopie Méthodes ab initio et DFT Réactivité QM/MM Méthodes Semi-empiriques Propriétés Structurales Mécanique Moléculaire Polymères Macromolécules Materiaux Dynamique Moléculaire Biochimie Biologie Mesoscale 2 Copyright : Pr. Didier Siri, ICR, Aix-Marseille Université
37 Quantum methods ĤΨ = EΨ Photochimie Méthodes Multiréférences Spectroscopie Méthodes ab initio et DFT Réactivité Méthodes Semi-empiriques Déterminant simple Electrons de valence, solvant, AM1 PM3 Déterminant simple Tous les électrons, solvant, 6-31G(d) (3df,3pd) B3LYP M08 Déterminants multiples Tous les électrons, (solvant), CI CASSCF CC Ψ e ( n) = 1 n! ϕ 1ϕ 2 ϕ 3...ϕ n Ψ e ( n) = 1 n! ϕ 1ϕ 2 ϕ 3...ϕ n Ψ e = C 0 Ψ 0 + C r r a Ψ a + C ab rs Ψ rs ab +... ra a<b r<s 3 Copyright : Pr. Didier Siri, ICR, Aix-Marseille Université
38 L équation de Schrödinger ĤΨ = EΨ Associe la fonction d onde ψ à l énergie E du système par l opérateur Hamiltonien H Fonction d onde = orbitales atomiques (moléculaires), mouvement des électrons dans le champs d un ou plusieurs protons constituant les noyaux. Ĥ = ˆT N + ˆT e + ˆV en + ˆV ee + ˆV NN + V ext T termes cinétiques, V termes coulombiens solution exacte seulement pour l atome d hydrogène! Approximations
39 Ĥ = ˆT N + ˆT e + ˆV en + ˆV ee + ˆV NN L équation de Schrödinger ĤΨ = EΨ + V ext Approximation de Born-Oppenheimer : traite uniquement le cas des électrons en considérant les noyaux fixes dans l espace (justifié par la différence de masse 1880 plus faible) H = T e + V en + V ee + V NN et considérant V NN constant (noyaux fixes), H = H e + V NN et corollaire : la fonction d onde totale peut-être approchée par le produit des solutions monoélectroniques : ψ totale = ψ(1). ψ(2) ψ(n) À partir de là plusieurs méthodes pour calculer l énergie du système
40 Méthodes ab initio Principe d exclusion de Pauli : 2 électrons ne peuvent pas se trouver simultanément dans le même état quantique -> introduction d une fonction pour les propriétés de spin : χ = spinorbitale La fonction d onde ψ s écrit sous la forme d un déterminant de Slater Ψ =1/ n! det χ1 χ2 χn Le nouvel opérateur prenant en compte ces spinorbitales = opérateur de Fock et équations d Hartree-Fock Méthodes ab initio (ou HF, Hartree-Fock) : toutes les intégrales sont calculées Pour les systèmes à couche fermée, simplification RHF (Restricted Hartree-Fock) : contributions identiques des 2 électrons de même couche, de spins opposés Radicaux : UHF (Unrestricted Hartree-Fock) Problème méthode HF : corrélation électronique pas prise en compte méthodes post-hf : CI, CASSCF
41 Méthodes DFT Autre méthode moins couteuse pour prendre en compte effets de corrélations électroniques : la DFT Thomas et Fermi, Hohenberg et Kohn : dans son état fondamental l énergie d un système est complètement déterminée par une fonctionnelle de sa densité électronique ρ(r) (Une) variable principale : gain sur le temps de calcul Problème : forme mathématique de la fonctionnelle inconnue! Développement de méthodes par Kohn et Sham Méthodes semi-empiriques Hartree-Fock simplifié : e - internes négligés, e - de valence représentés par une base minimale, paramètres empiriques pour certaines intégrales Méthodes AM1, PM3, SAM1 non utilisées par HARt
42 Fonction d onde, orbitales atomiques et moléculaires Fonctions de base, ou jeu de bases (basis set) Orbitales moléculaires = combinaison linéaire des orbitales atomiques Orbitales atomiques Orbitales de type Slater (Slater-type Orbital : STO) - précises à r 0 ou très loin - intégrales bi-électroniques à 3-4 centres analytiquement impossible Orbitales Gaussiennes (Gaussian-type Orbital : GTO) - intégrations plus facile - maniement plus simple (produit de gaussienne = gaussienne) - moins précises à r 0 - majoritairement utilisées distance électron-noyau (r) / bohr Combinaison de plusieurs primitives = représentation des orbitales atomiques : fonctions de base ϕ(r) rayon de Bohr STO 1GTO (STO-1G) 2GTO (STO-2G) 3GTO (STO-3G)
43 Exemples ST0-3G : 3 primitives gaussiennes sont utilisées pour décrire une orbitale de type Slater 3-21G, 4-31G, 6-311G : bases à valence splittée - ex G : 4 primitives gaussiennes pour les e - de cœur, 3 orbitales de valence (3, 1, 1 : triple zéta) représentées respectivement par 3, 1 et 1 gaussiennes cc-pvdz, cc-pvtz : cc-p = correlation-consistent polarized, V = valence orbitals only, D T Q = valences splittées double, triple, quadruple zéta def2svp, def2tzvp, def2tzpp : nouvelle définition (def2) des bases d Ahlrichs et al., valence splittée (simple, double n zéta), polarisation (P) mais pas diffusion Parfois ajouts de fonctions nécessaires (si non intégrées à la fonction de base) - fonction de polarisation (modification de la densité électronique autour du noyau) notée par *, ex. 6-31G* - fonction diffuse (modification de la densité électronique à longue distance du noyau, ex. atomes chargés ou radicalaires), notée +, ex G Attention : plus les fonctions de base sont complexes plus le calcul est long : en Hartree-Fock le temps de calcul est proportionnel à N 4, N étant le nombre de primitives présentes dans la base
44 Mais en pratique métaux de transition mal (ou non) traités par HARt, idem élements lourds (Br = erreur)
45 Serveur Lame HP Proliant BL460c : 388 cœurs de calcul, procs. Intel Xeon Centre Régional de Compétences en Modélisation Moléculaire, Aix-Marseille université Ecryst377 (C5 H4 O3) RHF Base 1 processeur 4 processeurs 8 processeurs 16 processeurs diffusion (n proc) (cluster et nbr. proc) STO-3G 2,6 cc-pvdz 8,6 cc-pvtz cc-pvqz >6h 36,4 def2-svp 5 def2-tzvp 32 10,6 def2- TZVPP 42,5 RKS cc-pvtz 29 def2-svp 5,4 def2-tzvp 20,3 40,5 (cluster 8, 8 proc) Temps processeur (min.)
46 AvdL Hirshfeld analysis Hirshfeld atomic refinements Example - ecryst377 IAM Refinement R1 = ; ρmin = 0.611; ρmax = Hirshfeld Atomic Refinement R1 = ; ρmin = 0.282; ρmax = Isocontour levels from -0.6 to 0.5 eå 3 in steps of 0.05 eå 3 (red negative; blue positive) HAR QM calculations: Hartree-Fock; spinorbital: restricted; basis set: def2-svp Experimental data resolution (according to Dauter - Acta Cryst. D55, , 1999): 0.77 Å
47 AvdL Hirshfeld analysis Hirshfeld atomic refinements Benefits of Hirshfeld Atomic Refinement free refinement of hydrogen positions possible hydrogens at neutron positions free refinement of hydrogen isotropic or anisotropic ADP s possible but (ecryst377): classical refinement: Npar = 78; N ref (I > 2σ(I)) = 836 Hirshfeld refinement: Npar = 109; N ref (I > 2σ(I)) = 836 ADP s of all atoms become neutron like modeling of density inside chemical bonds
48 Hirshfeld analysis Hirshfeld atomic refinements Validation of Hirshfeld Atomic Refinement results do not use CheckCif! are R-factors lower? Criteria is there less residual electron density? are hydrogens at neutron positions? hydrogen ADP s: Hirshfeld rigid body test the relative vibrational motion of a pair of bonded atoms has an effectively vanishing component in the direction of the bond z 2 AB (A) zba 2 (B) < 5σ use platon: calc TMA HINCL
49 AvdL Hirshfeld analysis Hirshfeld atomic refinements PLATON listing file
50 Hirshfeld analysis Hirshfeld atomic refinements Limitations of Hirshfeld Atomic Refinement Limitations Computational cost - 3 hours and 16 processors for 20 atoms - scales with N up to N 2.5 Molecular wavefunctions are used, so coordination polymers or inorganic structures not feasible Supramolecular features not well reproduced (polarization...) Hirshfeld partitioning not unique Disorder can not be treated, nor aperiodicity Only isolated molecules, no extended polymeric-like structures
51 Hirshfeld analysis Hirshfeld atomic refinements Hirshfeld Atomic Refinement vs Multipole modelling What is better? Multipole modelling uses the experimental density Hirshfeld Atomic Refinement uses a calculated density Anisotropic refinement of hydrogens (nearly) not possible with multipole modelling ADP s hydrogens from HAR and neutron refinements comparable Will Hirshfeld Atomic Refinement replace IAM refinement and even multipole refinement? Fugel,..., Puschmann et al., IUCrJ 5, (2018)
52 AvdL Software Contents From theory to practice Electron density and chemical bonding Experimental and theoretical modelling of electron densities Hirshfeld analysis Hirshfeld surfaces Hirshfeld refinement Software Tonto & CrystalExplorer Tonto & OLEX2
53 Software Tonto, CrystalExplorer & OLEX2 Tonto characteristics For quantum chemistry and quantum crystallography Free under the LGPL license Coded in Fortran, interfaced using the Foo scripting language Hartree-Fock and DFT calculations Backend for CrystalExplorer (GUI) OLEX2 - interfaces to the Hirshfeld Atomic Refinement part of TONTO
54 AvdL Software Time for the demos!
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