Charge Density from X-ray Diffraction. Methodology

Transcription

1 Charge Density from X-ray Diffraction. Methodoogy Ignasi Mata Master on Crystaography and Crystaization, 2012

2 Outine I. Charge density in crystas II. The mutipoar refinement III. Methodoogy IV. Exampe 2

3 I. Charge density in crystas II. The mutipoar refinement III. Methodoogy IV. Exampe 3

4 Charge density from X-ray diffraction In X-ray diffraction we see the eectrons. From the eectrons, we find where the nucei are. There is a ot of information on the eectron distribution about the interaction of the atoms in the crysta (chemica bonds, atomic and moecuar charges, intermoecuar interactions ). Charge density: - We are interested in ALL the charge distribution in the crysta, not just the nucei. - We are going to use X-ray diffraction for mapping the eectron distribution inside the crysta. 4

5 The intensity I of the refection H depends on the structure factor F F ( H) = f ( H) T ( H) exp ( 2πiH r ) a a ( H) F( H) 2 The structure factor depends on the crysta structure I a Atomic position in the crysta attice Atomic form factor Scattering from the atom Type of atom Therma vibration Atomic coordinates 5

6 In X-ray diffraction, the form factor depends on the eectron she f a ( H) ρa ( r) Cusp at the nucear position 2πi r = e H dr Eectron density: probabiity of finding an eectron at r Fast decay from the nucear position Spherica symmetry (for an isoated atom) 6

7 The independent atom approximation ρ a (r) in the crysta = ρ a (r) for the independent (isoated) atom The environment of the atom has no effect on its ρ a (r). f IA a ( ) IA H ρa ( r) 2πi r = e H dr Pt 5 1 0, sth 2 Ex. Phosphorus 0,01 0,0 0,5 1,0 1,5 2,0 With the IAM approximation: f a IA (H) has spherica symmetry. f a IA (H) depends ony on the chemica eement. 7

8 The interaction of an atom with its environment perturbs its ρ a (r). In a crysta: ρ a (r) does not present spherica symmetry. ρ a (r) is different for each atom in the asymmetric unit. In the IA approximation, we suppose this In the crysta, we have this Ex. P-atom in H 3 PO 4 Isodensity surface ρ(r) = constant 8

9 Bonding effects: Deviations of the IA approximation that introduce systematic errors in the crysta structure. Ex. Systematic underestimation of X-H bonding distances IA ρ a (r) IA r a (atomic position from X-ray diffration) Correct ρ a (r) Correct r a X H Bonding effects introduce sma errors in r a and U a in crysta structures from X- ray diffraction. A crysta structure from X-ray diffraction presents bonding effects. 9

10 Tota eectron density: Superposition of ρ a (r) Independent Atom Mode (IAM) of the eectron density: Superposition of ρ a IA (r) = ρ ρ( r) ( r ra ) a ρ IA = ρ IA ( r) ( r ra ) a ρ( r) = ρ IA ( r) + ρ( r) Deformation density: A the information about the effect of any kind of interatomic interactions (chemica bonding, intermoecuar interactions ) is here. 10

11 2D representation of the eectron density Isodensity contour: Constant eectron density Eectron density is aways positive Contours are truncated at a ρ max just to make the interpretation easier 11

12 ρ (r) IA = ρ (r) + ρ(r) ρ(r) It is the target of a charge density determination ρ>0 ρ<0 IA ρ (r) ρ(r) It is determined from the crysta structure. It is the main contribution to the X-ray structure factors. It cannot be determined from the crysta structure. Its contribution is ~3% of the X-ray structure factors. 12

13 Accurate structure factors Lower experimenta error Deformation density can be observed Experimenta ρ Experimenta deformation ρ 13

14 I. Charge density in crystas II. The mutipoar refinement III. Methodoogy IV. Exampe 14

15 F ( ) ( 2 T H = f ( H, p ) exp 2π H U H) exp ( 2πiH r ) a a a a a Generaized scattering factor: Specific for each atom in the asymmetric unit. Spherica symmetry can be reaxed. Atomic parameters: Specific for each atom in the asymmetric unit. Determined from a east squares fit against the experimenta structure factors. f a ( H, p ) a ρa ( r, pa ) 2πi r = e H dr ρ a IA ( r, p ) = ρ ( r) + ρ( r, p ) a a a 15

16 Atomic eectron densities superpose in the bonding regions. There are many partitions of the eectron density in atomic contributions. Assignment of portion of eectron density to an specific atom does not mind that eectrons beong to the atom. Instead of atoms : pseudoatoms ρ α ( r) ρ ( r) ρ α ( r) ( r) β ρ β Two possibe partitions of the tota eectron density in atomic eectron densities 16

17 ρ ( r) = ρ ( r) ( r) core + ρ va = + ρ core ( r) ρ va ( r) Very arge peaks at the atomic positions. Not perturbed by the atomic environment It does not depend on the atomic parameters Diffuse in the space. Perturbed the atomic environment. It depends on the atomic parameters. 17

18 a f a (H) is decomposed in core and vaence contributions ( r p ) = ρ ( r) ρ (, p ) f ( H p ) = f ( H ) + f (, p ) ρ +, a a, core a, va r a a, a a, core a, va H a ,1 0,01 0,0 0,5 1,0 1,5 2,0 Ex. P-atom Ony the vaence f a (H) is perturbed by the atomic environment Core contribution: Bonding effects are negigibe Vaence contribution: Bonding effects can be significant 18

19 As the atom is arger, the contribution of the vaence eectron density to the atomic scattering factor takes pace at ower anges. is a smaer fraction of the tota scattering. This puts a imit to the size of the atoms whose eectron density can be determined. C S Ti Zr Tota Core Vaence 19

20 The suitabiity factor: estimate of the suitabiity of a given crysta for X- ray charge density anaysis. S V unit ce = 2 Ncore As S is smaer, the determination of the experimenta eectron density is more chaenging. S fas as the size of the heaviest atom in the crysta increases. Schiøtt, Int J Quant Chem, 96 (2004) 23 20

21 The Hansen-Coppens mode of the eectron density f a ( H) = f core ( H ) + P v f va H + κ max = 0 m= 4πi P m j H Y κ' m ( β, γ ) Kappa term With spherica symmetry Mutipoar term Without spherica symmetry ρ max 3 3 ( r) = ρ ( r) + Pκ ρ ( κr ) + κ ' P R ( κ ' r) Y ( θ, φ) p ρ a = core Pv κ,κ ' Pm v va = 0 m= max 3 3 ( r) = Pκ ρ ( κr ) + κ' P R ( κ' r) Y ( θ, φ) N ρ ( r) v va Number of eectrons Expansion/contraction Non-spherica deformation = 0 m= m m m m Eectrons in the vaence she of the neutra atom v va 21

22 max = 0 m= max = 0 m= H 4π i Pm j Y m, κ ' κ' 3 P m R Mutipoar term ( κ' r) Y ( θ, φ) m ( β γ ) Expansion in spherica harmonics. Spherica harmonics are simiar to atomic orbitas. Each deformation consists in transferring eectrons from the negative to the positive regions of the spherica harmonic. Coppens, X-Ray Charge Densities and Chemica Bonding,

23 In the mutipoar refinement, differences between experiment and IA mode are taken as the contribution of ρ to the X-ray scattering. ( ( H) kf ( H) ) 2 obs ca H F obs ( H) kf ( H) ca H F ( H) kf ( ) or ( 2 2 F ) 2 obs ( H) kf ( H) IAM The mutipoar refinement is the minimization of H F ( ) H with and f ( H, p i ) = f F ca core ( H) = f ( H, p ) T ( U, u) exp ( 2πiH r ) ( H ) + P i v f i va i H + κ i max = 0 m= 4πi P m j i H Y κ' m ( β, γ ) 23

24 ρ( r) Two kind of parameters ρ(r) Structura parameters r, U α α Atomic parameters P, κ, κ ', v P m Bonding effects: Contribution of ρ is aready in the structura parameters. Structura parameters have more weight in the east squares refinement. Contribution of ρ must be removed from the mode before starting the mutipoar refinement. Before the mutipoar parameter, bonding effects must be removed. 24

25 Residua maps ρ res = 1 V H 1 k F obs ( H) F ca ( H) e e iϕ ca ( H) 2πiH r Structure with bonding effects. ρ is in the structura parameters. Bonding effects corrected. ρ appears in the residua maps. After mutipoar refinement. ρ is in the atomic parameters 25

26 Method 1 for removing bonding effects C High order refinement: Refections above a threshod ange ony High order refections ony present core contribution. sinθ λ im ~ Å -1 Tota Core Vaence At the end of the refinement: IA mode without bonding effects. The choice of the threshod vaue is a compromise between Minimization of vaence she contribution. Data set arge enough. 26

27 Hydrogen atoms have no core H Hydrogen atoms do not contribute to the high order refections. Reiabe structura parameters for H-atoms cannot be obtained from X-ray diffraction data. ρ(r) around H-atoms is inaccurate. Tota Core Vaence Choices: a) Stay with the approximate ρ(r) for the H- atoms. b) Use additiona information for a correct estimation of structura parameters of H-atoms 27

28 Method 2 for removing bonding effects Neutron diffraction: Eastic scattering of neutrons by the atomic nucei F ( H) = b T ( U, u) exp ( 2πiH r ) i Neutron scattering ength. Depends on the nuceus i i i X-ray diffraction F s F.T. Eectron distribution Nucear distribution (structure) Independent atom approximation Neutron diffraction F s F.T. Nucear distribution (structure) Bonding effects are reated to deviations from this approximation No bonding effects in the neutron diffraction structure 28

29 F ( H) = b T ( U, u) exp ( 2πiH r ) i i i i Scattering power does not depend on the atom size. No specia treatment is required to H- atoms. Coppens, X-Ray Charge Densities and Chemica Bonding, 1997 Anisotropic therma parameters for H-atoms 29

30 I. Charge density in crystas II. The mutipoar refinement III. Methodoogy IV. Exampe 30

31 1.- The experimenta structure factors ρ(r) It is ~3% of the X-ray structure factors. We need very high quaity structure factors High quaity crystas Accurate experiment High data redundancy High order data Longer time per frame Use of synchrotron radiation Compex data reduction Integration of measured intensities Accurate correction of absorption Accurate equivaent merging DREADD package for data reduction oriented to charge density studies 31

32 Low temperature In some cases, heium cooing (~ 20 K) is needed. F ( ) ( 2 T H = f ( H, p ) exp 2π H U H) exp ( 2πiH r ) a a a U a decrease with T, F(H) increases with T a a ρ res for an IAM at 120 K ρ res for an IAM at RT T increases, ρ becomes more diffuse 32

33 Use of short waveength / high energy radiation The size of the Ewad sphere increases. Refections shift to ower anges More refections to be measured λ = 2H sin θ Zr More refections with significant contribution of ρ va Tota Core Vaence Asanov, Crystaographic Instrumentation, 1998 Energies up to 100 KeV (0.015 Å) have been used in charge density studies 33

34 2.- The independent atom mode If using neutrons, X-ray and neutron therma parameters shoud match. Both X-ray and neutron diffraction data must be coected at the same temperature (same ce parameters) Experimenta errors such as absorption and extinction are absorbed into the U a. Anisotropic differences Good agreement Temperature differences Coppens, Acta Cryst A84 (1984)

35 The treatment of the hydrogens Approximate therma vibration. Approximate ρ(r) around H-atoms. Anisotropic therma parameters for H- atoms aow a detaied mode of ρ(r) around these atoms. Accurate ρ(r) around H-atoms. Munshi Acta Cryst. A64 (2008)

36 2.- The mutipoar refinement X-X method U iso for H-atoms F obs (H) Mutipoar refinement High order refinement r a and U a non-h atoms H-atoms: Bonding distance fixed to the average neutron bonding distance U iso estimated from the bonded atom. Step 1: Low ange refections P v, κ, κ, P m Step 2: A refections r a, U a, P v, κ, κ, P m H-atoms: Bonding distance as in the high order refinement. U iso refined. ρ( r) 36

37 2.- The mutipoar refinement X-N method F ( ) ( ) X, obs H F N, obs H Mutipoar refinement Step 1: Low ange refections P v, κ, κ, P m Step 2 (optiona): A refections r a, U a, P v, κ, κ, P m H-atoms fixed Crysta structure determination r a and U a a atoms ρ( r) 37

38 2.- The mutipoar refinement X-(X+N) method F X, obs ( H) F N, obs ( H) High order refinement r a and U a non-h atoms Mutipoar refinement Step 1: Low ange refections P v, κ, κ, P m Step 2: A refections r a, U a, P v, κ, κ, P m H-atoms fixed Crysta structure determination r a and U a H-atoms ρ( r) 38

39 Therma eipsoid scaing Neutron 1.- Get transformation by comparing therma eipsoids of non-hydrogen atoms. Minimization of ( U qu U) 2 X N High order X-ray 2.- Appy transformation to hydrogen-atom therma eipsoids U X = qu N + U 39

40 ρ max 3 3 ( r) = ρ ( r) + Pκ ρ ( κr ) + κ ' P R ( κ ' r) Y ( θ, φ) core v va A non-spherica atom has spatia orientation. A oca coordinate axis must be defined for each atom. In most cases, axis are oriented aong chemica bonds. = 0 m= m m 40

41 ρ max 3 3 ( r) = ρ ( r) + Pκ ρ ( κr ) + κ ' P R ( κ ' r) Y ( θ, φ) core v va = 0 m= Ex. r a, U a P v, κ, κ P m Parameters O-atom = 27 P-atom = 36 m m Restrictions are often introduced in order to reduce the number of parameters in the mutipoar refinement: Same k and k for atoms with simiar chemica environment. Same popuation parameters for atom with the same chemica environment. Fix tota charge of ions and moecues. Loca symmetry conditions. Besides this: Neutraity: not a constraint but a condition that must fufi the tota eectron density. 41

42 ρ max 3 3 ( r) = ρ ( r) + Pκ ρ ( κr ) + κ ' P R ( κ ' r) Y ( θ, φ) core Some P m can be set to zero by oca symmetry conditions. Ex. P in the H 3 PO 4 v va = 0 m= m m P P P fod symmetry aong axis z P m 0 m = 3i i = K 1,0,1,K P22+ P30 8 from 24 P m parameters nonzero P31 P 32 P

43 f ρ max 3 3 ( r) = ρ ( r) + Pκ ρ ( κr ) + κ ' P R ( κ ' r) Y ( θ, φ) core v va Eectron configuration of the vaence she = 0 m= m m Shape of the deformation terms max H H ( H, pi ) = fcore ( H ) + Pv fva + 4πi Pm j Y m + κ = 0 m= κ' Anharmonic therma vibration F ca Anomaous scattering ( H) = f ( H, p ) T ( U, u, c) exp ( 2πiH r ) Ange dependence of the scae factor i i i i Extinction mode ( β, γ ) + f ' if '' i H ( ( ( ) ) ) F ( H) ky F H, g F ( H) 2 obs ca ca A ot to things to check, correct, optimize 43

44 Phases are very important Competey different modes can present amost identica F ca In the case of acentric crystas, restrictions must be imposed for avoiding unphysica modes Ex. ADP P Q = +0.1(2) Q = Q = Residuas in the P-O-P pane. Q = Moecuar charge of NH 4. Pérès Acta Cryst A55 (1999) 1038 Q = +0.7 Q =

45 3.- Vaidating the mode Good statistics (R factors, goodness of fit ) Residua maps Experimenta deformation maps 1 ρexp = ) V ρ res ( iϕ H i H H H ) ca ( ) ϕiam ( ) Fca ( ) e FIAM ( e H e 2πiH r ρ res ρexp 45

46 3.- Vaidating the mode U iso (H-atom) > U iso (bonded atom) Rigid Bond Test < Å 1 2 Bonding distance between non-hydrogen covaenty bonded atoms remains constant through therma vibration. Vibration ampitudes aong the bond shoud be equa for the bonded atoms. 46

47 3.- Vaidating the mode Comparison with theoretica cacuations In most cases, agreement is quaitative because of Measurement errors in the experimenta eectron density. Method-inherent errors in the cacuations. Best resuts observed for Periodic ab initio cacuations Purey organic moecuar crystas Ex. P-nitroaniine Experimenta Theoretica Vokov Acta Cryst A56 (2000)

48 The eectron density is the starting point for further anaysis Deformation density Topoogica anaysis Lapacian Eectrostatic potentia 48

49 I. Charge density in crystas II. The mutipoar refinement III. Methodoogy IV. Exampe 49

50 2-methy-4-nitroaniine Howard et a. J Chem. Phys. (1992) C 7 H 8 N 2 O 2 Ia (Monocinic) V Å 3 T 125 K Ref. measured 7348 Ref. independent 3743 Ref. observed 2045 (sin(θ)/λ) max 1.08 Å -1 R int (F 2 ) Loca axis C,N,O = 1, 2, 3 H = 1 H-atoms - d(x-h) fixed - U iso fixed -κ=

51 I II III IV V VI VII Crysta structure High order Kappa without structura parameters Kappa with structura parameters Mutipoe without oca symmetry restrictions Mutipoe with oca symmetry restrictions VALRAY mutipoar mode 51

52 Fina resuts Residuas Deformation Dipoe moment ( r) r r µ = ρ d T Kappa Mutipoar Theory µ 48 D 25 D 9 D Rigid bond test max = Å 2 Quaitative agreement with theoretica cacuations Large increase attributed to crysta fied effects 52

53 2-methy-4-nitroaniine revisited Whitten et a. J. Phys. Chem. (2006) X-rays Neutrons V (8) Å (4) Å 3 T 100 K 100 K Ref. measured Ref. independent Ref. observed (sin(θ)/λ) max 1.27 Å Å -1 R int (F 2 )

54 Fina resuts Residuas Deformation Dipoe moment Experimenta 11.3 D Theory ( r) r r µ = ρ d T Periodic 11.7 D Crysta geom. 9.0 D Isoated 7.1 D R=0.016 for 5055 obs. ref. and 383 pars. Exceent agreement with theoretica cacuations Increase in dipoe moment due to crysta fied effects. 54

55 Fina remarks - The methodoogy for the mutipoar refinement is we estabished for crystas of sma organic moecues in centric space groups. - Mutipoar refinement can be performed in crystas with transition metas. However there is some compexity invoved in the treatment of these atoms. - If hydrogen atoms are reevant, anisotropic vibration parameters for these atoms are needed. - In the case of acentric space group, there is a risk of mode indeterminacy. Restrictions shoud be imposed with care. - Neutron diffraction is very hepfu for the treatment of the therma vibration (H-atoms, anharmonicity). - Theoretica cacuations are usefu for vaidating the experimenta eectron density. However, theoretica eectron densities are not necessariy better than the experimenta ones. 55