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B. DittrichB. Dittrich

Institute for Inorganic Chemistry, Institute for Inorganic Chemistry, Tammannstr. 4Tammannstr. 4

37077 G37077 Gööttingen, Germany ttingen, Germany

Introduction to Introduction to HirshfeldHirshfeld--atom refinementatom refinement

D. JayatilakaD. Jayatilaka

Chemistry, M313Chemistry, M313University of Western Australia, University of Western Australia,

6009 Crawley, Australia6009 Crawley, Australia

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Overview of aspherical atom Overview of aspherical atom approachesapproaches

11Pichon Pesme et al., Pichon Pesme et al., J. Phys. Chem. 99J. Phys. Chem. 99, , 19951995, 6242, 624222Volkov et al., Volkov et al., J. Phys. Chem. 108J. Phys. Chem. 108, , 20042004, 4283, 4283

33Dittrich et al., Dittrich et al., Angew. Chem. Int. Ed. 43Angew. Chem. Int. Ed. 43, , 20042004, 2718 , 2718

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LetLet‘‘s start from scratch s start from scratch ––the structure factor:the structure factor:

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The IA or promolecule modelThe IA or promolecule model

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Thermal motionThermal motion

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Using a quantum mechanical Using a quantum mechanical electron density formalismelectron density formalism

Quantum chemistry ED formalism has a physical basis Quantum chemistry ED formalism has a physical basis ––uses many center ED expressionuses many center ED expression

We want to use this QM ED description for XWe want to use this QM ED description for X--ray structure ray structure refinementrefinement

We need atomWe need atom--centering for a Xcentering for a X--ray scattering formalism ray scattering formalism because of the temperature factorbecause of the temperature factor

But: Each time we refine a structure a SCF calculation But: Each time we refine a structure a SCF calculation based on the geometry found in the crystal is required.based on the geometry found in the crystal is required.

Disorder can not currently be Disorder can not currently be handled.handled.

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What can we obtain?What can we obtain?

As we will model our experimental data with a quantum As we will model our experimental data with a quantum chemical entity we are not able to get an chemical entity we are not able to get an ““experimentalexperimental””ED like in a multipole refinement, unless we do ED like in a multipole refinement, unless we do ““wavefunction fittingwavefunction fitting””. However, our . However, our geometrygeometry and the and the ADPsADPs improve, since we describe aspherical ED.improve, since we describe aspherical ED.

In general we can obtain all In general we can obtain all propertiesproperties that can be that can be calculated from the wave function for the conformation calculated from the wave function for the conformation found in the crystal.found in the crystal.

Positions and ADPs in NeutronPositions and ADPs in Neutron--diffraction quality, diffraction quality, depending on the quality of the data.depending on the quality of the data.

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ED description in the QM worldED description in the QM world

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HirshfeldHirshfeld’’ss fuzzy boundary fuzzy boundary partitioning schemepartitioning scheme

Hirshfeld, Hirshfeld, TheorTheor. . ChimChim. Acta (. Acta (BerlBerl.) 44.) 44, , 19771977, 129, 129Authors who used Authors who used HirshfeldsHirshfelds method for density partitioning: method for density partitioning:

KoritsanszkyKoritsanszky & & VolkovVolkov, , Chem. Phys. Chem. Phys. LettLett. 385. 385, , 20042004, 431, 431

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HirshfeldHirshfeld--atoms refinementatoms refinement•• Using a QM electron density and by partitioning it withUsing a QM electron density and by partitioning it withHirshfeldsHirshfelds stockholder scheme we generatestockholder scheme we generateatomatom--centered noncentered non--spherical scattering factors.spherical scattering factors.

•• Point charges, dipoles and Point charges, dipoles and quadrupolesquadrupoles approximate approximate the crystal environment. They can be calculated fromthe crystal environment. They can be calculated fromHirshfeld partitioning.Hirshfeld partitioning.

•• Thermal motion can be treated as introduced and Thermal motion can be treated as introduced and leastleast--squares refinement of positional andsquares refinement of positional anddisplacement parameters becomes possible. displacement parameters becomes possible.

DonDon’’t confuse Hirshfeld partitioning with Hirshfeld surfaces.t confuse Hirshfeld partitioning with Hirshfeld surfaces.

Jayatilaka & Dittrich, Acta Jayatilaka & Dittrich, Acta A64A64, , 20082008, 383, 383--393393

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Least squares refinement with HALeast squares refinement with HA

•• Two different Two different minimisationminimisation algorithms have been algorithms have been implemented in TONTO; a conjugate gradient implemented in TONTO; a conjugate gradient minimisationminimisationand a and a minimisationminimisation viavia normal equations; in the latter normal equations; in the latter 22 is is minimized, rather than minimized, rather than

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Least squares refinement:Least squares refinement:derivativesderivatives

•• Derivatives used in the normal equations areDerivatives used in the normal equations are(example of (example of rr derivative):derivative):

•• Extinction and scale factor are determined separatelyExtinction and scale factor are determined separately

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Why has it not been done before?Why has it not been done before?

In my opinion without having had a library that In my opinion without having had a library that ““knewknew”” about about symmetry and crystallography as well as a symmetry and crystallography as well as a fully functional fully functional QM programQM program at hand (TONTO) it was just too much workat hand (TONTO) it was just too much work……

Other examples for nonOther examples for non--trivial algorithms necessary for the trivial algorithms necessary for the procedure:procedure:

•• Numerical integration of the integralsNumerical integration of the integrals**

(also used for exchange(also used for exchange--correlation integrals in DFT)correlation integrals in DFT)•• Fourier transform of Gaussian functionsFourier transform of Gaussian functions##

(also needed for SCF)(also needed for SCF)

**BeckeBecke, A.D., , A.D., J. Phys. Chem. 88J. Phys. Chem. 88, , 19881988, 2547, 2547##Jayatilaka, D., Jayatilaka, D., Phys. Rev. Phys. Rev. LettLett. 80. 80, , 19981998, 798, 798

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Practical stepsPractical steps

0. Measure I, get Fo and s0. Measure I, get Fo and s1.1. Obtain a starting molecular structure (r, ADPs) Obtain a starting molecular structure (r, ADPs) 2.2. Generate symmetry related atoms from unique Generate symmetry related atoms from unique

atoms (choose stabilizer group)atoms (choose stabilizer group)3.3. Select the unique molecule for the molecular crystal, Select the unique molecule for the molecular crystal,

choose QM method and basis.choose QM method and basis.4.4. Perform SCF calculation and Hirshfeld partitioningPerform SCF calculation and Hirshfeld partitioning5.5. Calculate asphericalCalculate aspherical--atom scattering factors; use atom scattering factors; use

symmetrysymmetry6.6. Calculate complex model structure factors and Calculate complex model structure factors and

derivativesderivatives7.7. Calculate parameter shifts and symmetrise themCalculate parameter shifts and symmetrise them8.8. Update aspherical atom scattering factorsUpdate aspherical atom scattering factors

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Practical stepsPractical steps

To obtain the best result it is advisable to To obtain the best result it is advisable to repeat the steps 4 to 8 several times.repeat the steps 4 to 8 several times.Although changes in geometry/ ADPs areAlthough changes in geometry/ ADPs areminiscule, especially bond distances tominiscule, especially bond distances tohydrogen atoms can still change.hydrogen atoms can still change.Usually five cycles are enough for selfUsually five cycles are enough for self--consistency.consistency.

When shifts are small the structure is refined.When shifts are small the structure is refined.

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Real life examplesReal life examples

A A ““normalnormal”” structure: Amoxicillin structure: Amoxicillin 3H3H22O*O*ReRe--measured at HASYLAB measured at HASYLAB beamlinebeamline D3 at T = 9 KD3 at T = 9 KHAR with STOHAR with STO--3G, 63G, 6--31G*, DZP, cc31G*, DZP, cc--pVDZpVDZ and ccand cc--pVTZpVTZ

HAR of Benzene and Urea HAR of Benzene and Urea –– HH--ADPsADPs

HAR of HAR of LincomycinLincomycin hydrochloride with point charges, hydrochloride with point charges, dipoles and dipoles and quadrupolesquadrupoles –– better results than better results than multipole refinement.multipole refinement.

*Boles *Boles et al.et al., , Acta Cryst. B34Acta Cryst. B34, , 19781978, 461, 461

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Application to Amoxycillin Application to Amoxycillin 3H3H22OO•• AmoxycillinAmoxycillin is a widely used is a widely used ββ--lactamlactam antibiotic. Like all of antibiotic. Like all of these molecules it is a suicide inhibitor that stops bacterial these molecules it is a suicide inhibitor that stops bacterial cellcell--wall synthesis.wall synthesis.

•• AmoxycillinAmoxycillin is a weak is a weak scattererscatterer, crystals only grow in long , crystals only grow in long needles.needles.

•• A multipole refinement is not possible since it does notA multipole refinement is not possible since it does notscatter well even at the synchrotron.scatter well even at the synchrotron.

•• It is a suited candidate for HirshfeldIt is a suited candidate for Hirshfeld--atom refinement.atom refinement.

•• Only ultraOnly ultra--low temperature (9K) and intensive synchrotron low temperature (9K) and intensive synchrotron radiation gave decent single crystal Xradiation gave decent single crystal X--ray data to 0.6 ray data to 0.6 ÅÅ

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Amoxycillin Amoxycillin 3H3H22O: crystal dataO: crystal data

Space group Space group P2P21122112211, Z = 4, , Z = 4, a a = 15.58, = 15.58, b b = 18.71, = 18.71, c c = 6.64 = 6.64 ÅÅ, , V= 1936 V= 1936 ÅÅ, T= 9 K, resolution in sin , T= 9 K, resolution in sin / / = 0.81 = 0.81 ÅÅ--11, , dd = 0.6 = 0.6 ÅÅ. . 8492 unique, 8259 with F > 8492 unique, 8259 with F > 33 FF,, coverage 99.9 %;coverage 99.9 %; R(R(FF) ) InvariomsInvarioms 4.0 % (multipole model), 328 parameters. 4.0 % (multipole model), 328 parameters.

ADPs after refinement shown as ORTEP representation.ADPs after refinement shown as ORTEP representation.

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Amoxycillin Amoxycillin 3H3H22O: refinementO: refinement

44.7-1788.541.723.79 %cc-pVDZ

45.2-1789.091.713.78 %DZP

44.7-1788.431.723.81 %6-31G*

39.6-1764.442.034.38 %STO-3G

Mol. dipole

EnergyGoFR(F)Basis

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Amoxycillin Amoxycillin 3H3H22O: ESPO: ESP

ESP (shown), deformation densities, dipole moments and ESP (shown), deformation densities, dipole moments and various other properties can be obtained directly after refinemevarious other properties can be obtained directly after refinement.nt.

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Amoxycillin Amoxycillin 3H3H22O: def. den.O: def. den.

Deformation densities vary depending on the method/basisDeformation densities vary depending on the method/basis--setsetused. The ones given here are of similar quality.used. The ones given here are of similar quality.

HF/6HF/6--31G*31G*HF/ccHF/cc--pVDZpVDZ HF/DZPHF/DZP

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Hydrogen ADPs directly from Hydrogen ADPs directly from refinement for Urea and Benzenerefinement for Urea and Benzene

For benzene, a DFT calculation improves the ADPs fro HA refinement as shown by difference to Multi-temperature ADPs from Buergi et al.

MultiMulti--temperature Benzene ADPs from Buergi et al., temperature Benzene ADPs from Buergi et al., Chem. Eur. J., 8Chem. Eur. J., 8, , 20022002, 3512 , 3512 Urea ADPs from neutron data from Swaminathan et al., Urea ADPs from neutron data from Swaminathan et al., Acta Cryst.Acta Cryst. B40B40, , 19841984, 300, 300Benzene ADPs from neutron diff. from Jeffrey et al., Benzene ADPs from neutron diff. from Jeffrey et al., Proc. Roy. Soc., 414Proc. Roy. Soc., 414, , 19871987, 47, 47

For urea, including point charges improves agreement with neutron diffraction ADPs from Swaminathan et al.

ADP differencs visualised with PEANUTHummel et al., J. Mol. Graphics, 1990, 214

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Lincomycin: R(F) and dataLincomycin: R(F) and data

25.125.1——2.522.522.22 %2.22 %InvariomInvariom

————1.0*1.0*2.54 %2.54 %shelxlshelxl

36.636.6——1.941.941.89 %1.89 %MultipoleMultipole

26.1326.13--2193.592193.592.022.022.03 %2.03 %cccc--pVTZpVTZ

26.6026.60--2193.182193.182.042.042.04 %2.04 %cccc--pVDZpVDZ

30.5230.52--2166.872166.872.782.782.68 %2.68 %STOSTO--3G3G

Mol. Mol. dipoledipole

EnergyEnergyGoFGoFR(F)R(F)BasisBasis

Space group Space group P2P211221122 measured reflections 365496 of which 32637 measured reflections 365496 of which 32637 unique, resolution in sin unique, resolution in sin / / == 1.19 1.19 ÅÅ..

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Lincomycin: HLincomycin: H--ADPsADPs

DFT method/ basisDFT method/ basis--setset : BLYP/cc: BLYP/cc--pVDZ gives accurate XpVDZ gives accurate X——H bond distances H bond distances and Hand H--ADPs, missing in free multipole refinements. ADPs, missing in free multipole refinements.

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Progress when compared to the Progress when compared to the multipole modelmultipole model

•• When we use multipoles for invariom modeling we have so When we use multipoles for invariom modeling we have so far only used isolatedfar only used isolated--molecular densities molecular densities

•• With TONTO we can include With TONTO we can include point charges, dipolespoint charges, dipoles and and quadrupolesquadrupoles of surrounding molecules from Hirshfeld of surrounding molecules from Hirshfeld partitioning in our SCF and the subsequent HA refinement partitioning in our SCF and the subsequent HA refinement

•• This way we can, given good lowThis way we can, given good low--order Xorder X--ray data, also ray data, also obtain obtain HH--ADPsADPs directly by refinementdirectly by refinement

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Computational speedComputational speed•• TONTO can be compiled with the TONTO can be compiled with the gfortrangfortrancompiler together with MPICH as a compiler together with MPICH as a parallelisedparallelised version.version.Both tools are free for academic users. Both tools are free for academic users.

•• A SCF on A SCF on AmoxycillinAmoxycillin with the DZP basis with the DZP basis takes a few takes a few minutes in parallel on sixteen 2.93 GHz minutes in parallel on sixteen 2.93 GHz intelintel XeonXeonprocessors, i.e. the total time for 6 cycles with refinement processors, i.e. the total time for 6 cycles with refinement is in the order of an hour. is in the order of an hour.

•• For For LincomycinLincomycin with BLYP/ccwith BLYP/cc--pVTZ the refinement took pVTZ the refinement took 5 days 11 hours.5 days 11 hours.

•• I.e. when point charges etc. are included and the basis I.e. when point charges etc. are included and the basis set is better, the time required can increase significantlyset is better, the time required can increase significantly

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Conclusion Conclusion HirshfeldHirshfeld--atom refinement can be used for typicalatom refinement can be used for typical

small molecule problems and does not require small molecule problems and does not require highhigh--resolution dataresolution data

Improvements in theory and their importance canImprovements in theory and their importance canbe seen in be seen in FOMsFOMs from from experimental data experimental data (correlation, relativistic effects)(correlation, relativistic effects)

Method can be considered to provide the best Method can be considered to provide the best ccurrently available Xurrently available X--ray geometries and ADPsray geometries and ADPs

Better RBetter R--Factors than from multipole model w/oFactors than from multipole model w/oparameter refinementparameter refinement

Feasibility of calculation limits the size of the problem Feasibility of calculation limits the size of the problem

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What else could be done? What else could be done?

•• We aim to create a database of asphericalWe aim to create a database of asphericalscattering factors with TONTO that couldscattering factors with TONTO that couldbe used in SHELXL. This would require a FT of the be used in SHELXL. This would require a FT of the real space QM density the invariom database real space QM density the invariom database is based upon.is based upon.

•• These require a definition of a local atomic coordinateThese require a definition of a local atomic coordinatesystem.system.

•• Tabulated aspherical atoms could then be used Tabulated aspherical atoms could then be used routinely in LSQ refinements, providing selected QMroutinely in LSQ refinements, providing selected QMproperties and improving the model currently used forproperties and improving the model currently used forinvariom refinement.invariom refinement.

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How can TONTO be obtained? How can TONTO be obtained?

•• TONTO can be obtained free of charge under GNU TONTO can be obtained free of charge under GNU copyleftcopyleftfrom from http://http://sourceforge.netsourceforge.net (Project (Project tonto_chemtonto_chem))

•• Just download and compile it.Just download and compile it.

•• The TONTO citation is: Jayatilaka, D & The TONTO citation is: Jayatilaka, D & GrimwoodGrimwood, D.J., , D.J., Computational Science Computational Science –– ICCS,ICCS, 20032003, 2660, 142, 2660, 142--151151

•• Citation for HirshfeldCitation for Hirshfeld--atom refinement:atom refinement:Jayatilaka & Dittrich, Acta Jayatilaka & Dittrich, Acta A64A64, , 20082008, 383, 383--393 393

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AcknowledgementsAcknowledgements

‘‘Australian Synchrotron Australian Synchrotron Research ProgramResearch Program’’

E. Chen for testing of HAR on E. Chen for testing of HAR on HomoserineHomoserine

W. W. MorgenrothMorgenroth , F.P.A. , F.P.A. FabbianiFabbiani,,D. Stalke for D. Stalke for diffractometerdiffractometer accessaccess

G. Sheldrick for interest and discussions.G. Sheldrick for interest and discussions.