The Zürich School of Crystallography

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1 The Zürch School of Crystallograhy Unversty of Zürch June 9, 3 Otmzng the of a crystal structure Hans-Beat Bürg nsttute of Organc Chemstry, Unversty of Zürch, Swtzerland hans-beatbuerg@krstunbech Contents Model of the dffracton exerment Otmzaton condtons 3 Lnear regresson 4 on-lnear least-squares otmzaton 5 Restrants and constrants 6 Qualty of the 7 Weghtng schemes

2 8 Otmzng the of a crystal structure Model of the dffracton exerment Structure soluton by drect methods, nterretaton of the Patterson functon or other rocedures rovdes assgnments of atom tyes and estmates of coordnates for some or all atoms n the unt cell The relablty of these results s lmted and deends on the qualty of the ntal estmates of the hases of the structure factors n order to mrove the relablty of the structure determnaton, a mathematcal of the electron densty and more generally of the entre dffracton exerment s requred ts arameters must be adusted to reroduce otmally the exermental ervatons The electron densty n a unt cell s aroxmated as a sum of atomc denstes, as descrbed earler, but to be hyscally realstc t has to also take nto account atomc moton The motonless atomc denstes are usually assumed to be shercally symmetrc, but may also be chosen to account for devaton from shercal symmetry due to chemcal bondng n standard structure analyss shercally symmetrc densty functons calculated by Hartree-Fock methods for solated atoms are used The detals of atomc and molecular vbratons are usually very comlex t s therefore assumed that each atom moves ndeendently of all others n a quadratc otental whch may be sotroc or ansotroc The corresondng robablty densty functon takes the form of a 3D Gaussan dstrbuton Mathematcally the smearng of the statc atomc densty ρ, (r r ) by thermal moton s exressed n terms of a convoluton wth the Gaussan robablty densty: q ρ ( r) ρ, ( r r ) ex / r 3 / ( π ) U T [ r ( U ) / (see footnote 3 n ntroducton to crystals, electromagnetc radaton and nterference by Hans-Beat Bürg and Math refresher by Gervas Chaus) The elements of the symmetrc matrx U are the mean-square exectaton values of the atomc dslacements Δx, Δy, Δz from the mean atomc oston: U <Δx >a, U <Δy >b, U 33 <Δz >c, U <ΔxΔy>ab, U 3 <ΔxΔz>ac, U 3 <ΔyΔz>bc; U s the determnant of U Fourer transformaton leads to a structure-factor exresson, whch s now somewhat more comlcated F( q q f f,, ( S) ( S) o ex π ( r * o ex π ( h ex π( hx T a * + hka * b * U + ky + lz U r*) ex πr ) U + k b * U + hla * c * U r * 3 + l c * U 33 + klb * c * U The atomc scatterng factor f of my revous lectures (see Scatterng from crystals: dffracton by Hans-Beat Bürg) has been dvded nto three arts, f, beng the Fourer transform of the statc densty ρ, (r r ), o beng an occuaton factor and ex-π(r* T U r*) beng the Fourer transform of the 3D Gaussan dstrbuton functon descrbng atomc moton n the early stages of otmzaton an sotroc aroxmaton to atomc dslacement arameters s usually suffcent 3 )

3 F( q q f, f, ( S) o ( S) o ex π ( U ex π U so so r * T r*) ex πr (snθ / λ) r * ex π( hx + ky + lz ) For every atom there are thus three ostonal coordnates, x, y, z, to be otmzed, at least kl one, but usually sx dslacement arameters U and a oulaton arameter o to be determned, u to ten arameters n total An arorate for a dffracton exerment has to take nto account factors other than dffracton by shercally symmetrc electron densty dstrbutons These nclude a factor k scalng the structure factors calculated for one unt cell to the volume of the crystal secmen, twn arameters (BASF), a Flack arameter (f the absolute structure of the crystal secmen needs to be secfed), extncton arameter(s) (EXT, accountng aroxmately for the falure of the knematc scatterng theory used throughout) and several others The resultng for the scattered ntensty may be qute comlex: F k g( BASF, Flack, EXT, etc) Otmzaton condtons The condton used n crystal structure least-squares otmzaton s to mnmze the sum S of the squares of the dfferences between erved and sutable ntenstes The uncertanty of the measurements s accounted for by a weght factors S [ The mnmzaton condtons are [ F k g( BASF, Flack, EXT, etc) x y z 33 3, (,, q) 3 k These calculatons mly some straghtforward, but comlcated algebra The otmzaton rocess s therefore llustrated frst wth a much smler examle, lnear regresson 3 Lnear regresson Suose that consecutve measurements of a quantty A may be exected to ncrease an ntal value q by a constant amount A( A ( n + q (n,, ) n Fg Results of consecutve measurements of A( exec- 3 ted to ncrease lnearly wth n

4 Gven a seres of measurements A ( wth relablty, what are the best estmates of and q? The quantty to be mnmzed and the otmzaton condtons are S n n [ A [ A q ( A ( ( n q The dervatve calculatons lead to two equatons n the unknowns, q The equatons are called lnear because and q occur to the frst ower S qot q ot n n n n + q + q n n n n n na A ( ( Fg Parabolc deendence of S on the arameters and q for a lnear least-squares roblem The coordnates of the mnmum of the arabola are ot and q ot These equatons are more convenently exressed n matrx notaton: n n n n n n n ( ) q w n n n na A ( ( abrevated as Δ The otmzed arameters are Δ A grahcal reresentaton of ths rocedure s gven n Fg The functon S beng quadratc n and q, s reresented by a 3D arabolc basn The coordnates of ts mnmum reresent the otmal values of the arameters ote that the calculaton of ot and q ot requre the known quanttes A (, n and only (x, y, z ) (U kl ) 4 on-lnear least-squares otmzaton The structure factor F( and thus ( s clearly a non-lnear functon n the arameters x, y, z, U kl The quantty S s therefore no longer arabolc and most mortantly may show several mnma mlyng that the mnmzaton condtons Fg 3 Schematc contour-lne reresentaton of S as a functon of the arameters x, y, z, U kl for a non-lnear least-squares roblem The absolute mnmum s ndcated by a cross The crcle ndcates a set of aroxmate structural arameters lkely to refne to the absolute mnmum of S; the square ndcates a set of aroxmate structural arameters lkely to refne to the nearby subsdary mnmum of S 4

5 may lead to more than one soluton Ths s shown schematcally n Fg 3, where S s reresented by contour lnes The roblem s to fnd the absolute mnmum of S (cross n Fg 3) From the ntal soluton of the hase roblem aroxmate values of the arameters x, y, z, U kl are avalable f these arameters are close to the absolute mnmum (crcle n Fg 3) a modfed form of lnear regresson s hghly lkely to fnd ths mnmum f the ntal arameters are close to one of the subsdary mnma (eg the square n Fg 3), the otmzaton rocess may well end u n the subsdary mnmum and roduce a defcent or even erroneous descrton of the crystal structure From the ont of vew of actual calculatons, the non-lnearty of the structure-factor equaton mles that the smle matrx aroach gven for lnear regresson cannot be used as t stands The soluton to the roblem s to lnearze t The ntenstes ( are exanded nto a Taylor seres u to frst order terms n the changes of the structural arameters Δ relatve to the ont characterzed by the set of P aroxmate arameter values x,, y,, z,, U kl,, k, etc abbrevated below as the vector (crcle n Fg 3) P ( ; ( ; ) + Δ nstead of determnng the arameters, only the shfts of the arameters Δ need to be found From the shfts, mroved arameters + Δ are obtaned The shae of S around, and ts absolute mnmum s usually not arabolc, not even aroxmately Lnearzaton and determnaton of the arameter shfts must therefore be reeated untl convergence s reached, e untl the arameter shfts are smaller than a secfed lmt The functon to be mnmzed s S( Δ( ; ) ( ; P and the mnmzaton condtons are Δ (,, P) The resultng P lnear equatons n Δ are P ( ; Δ( ; n matrx notaton, these equatons are, Δ Δ ) ) P ( ; ( ; ( ; ( ; Δ Δ Δ (,, P) as revously The element of n row and column s 5

6 The element of Δ n row s Δ( ; ( ; ) ( ; ( ; The arameter shfts and the new arameters become Δ - Δ + Δ The smlcty of the fnal result s decevng as may be seen from the followng consderaton For a relatvely small structure wth atoms a wth ansotroc dslacement arameters contans at least 9 + arameters The number of elements of and Δ to be calculated s / + 55 Assumng a tycal rato of arameters to ervaton of : there wll be ervatons mlyng that values of (; and values of (; / need to be comuted and roerly summed u 55 tmes Kee these numbers n mnd when you do your own leastsquares refnement, robably wth many more arameters and ervatons and n as lttle tme as a few seconds to a few mnutes Exerence ths blessng of modern-day comutng ower knowngly! 5 Restrants and constrants t sometmes haens that the structural chosen s hyscally unreasonable or that the nformaton contaned n the dffracton ntenstes s nsuffcent to determne all arameters of a structural n such cases the matrx of normal equatons may become sngular or near-sngular, e the determnant s zero or very small (on the order of the numercal accuracy of the rocessor n your comuter) As a consequence the nverse matrx - s undetermned and the arameter shfts Δ cannot be obtaned Oscllatng values of one or more elements Δ and unreasonably large standard uncertantes of the arameters (see below) are sgns of such roblems The roblem must be remeded ether by choosng a more reasonable structural or by codng ndeendent external nformaton on the crystal structure nto restrants or constrants Restrants are condtons q res mosed on features q ( of the structural that must be fulflled wthn a certan tolerance Such condtons are called seudo-ervatonal equatons and are added to the mnmzaton functon S S' S + w [ q ( ) q They affect and - f roerly chosen - may remove the sngularty 5 Restrants res n asymmetrc unts contanng several chemcally dentcal unts (sometmes related by seudo-translaton or other seudo-symmetry oeratons), one may wsh to restran the geometry of these unts to be closely smlar For dstances d n unts and the restrants take the form 6

7 S' S + w [ d ( ) d,, ( f an nteratomc dstance or other geometrcal arameter d s oorly determned by the dffracton data, but known from other exermental or theoretcal sources, ts value d res may also be ntroduced nto the of refnement n terms of a restrant Examles nclude C-H, Cl-O or B-F dstances, the latter n dsordered ClO 4 or BF 4 ons, devatons from lanarty n fragments wth more than three atoms, etc: S' S + w [ d ( ) d res For a nearly sngular matrx of normal equatons the arameter shfts between cycles k and k+ from k to k+ may become unreasonably large, thus cataultng the from an area near the absolute mnmum to one near a subsdary mnmum (Fg 3) Ths effect can sometmes be avoded by shft-lmtng restrants S' m m, k + m S + w [ m, k Ths restrant translates nto addng w m to the dagonal element of corresondng to m n some sace grous the orgn n one or more drectons can not be fxed on symmetry elements Examles nclude P, P, P, P3, Pna, etc Most software n common use fx the orgn wth a restrant descrbed by HD Flack & D Schwarzenbach (988) (Acta Cryst A44, ) 5 Constrants A constrant s an exact relatonsh between features of a structural, q res q ( A constrant may be smulated n terms of a restrant by ncreasng ts weght w to a very large value, e decreasng the tolerance for devatons of q ( from q res Charge neutralty n onc comounds s an examle that s convenently treated n ths way Exact constrants may be exressed n two ways On one hand a constrant arameter may be assgned a fxed value whch s not refned, e t s not consdered n the dervatve calculatons of the mnmzaton condtons Examles nclude secal values of atomc coordnates and atomc dslacement arameters requred for certan secal ostons n some sace grous Alternatvely and more generally constrants are exressed wth the hel of LaGrange multlers (see any advanced text book of mathematcs) 6 Qualty of the There are several ways to gauge the qualty of a structural Some qualty ndcators, such as R-factors and Goodness-of-ft (GOF) are global quanttes They gve only lmted nformaton on the qualty of a structure determnaton Others, such as the standard uncertantes of the refned arameters, whch wll be ntroduced below, refer to secfc detals of the structural 6 Global qualty ndcators: R-factors and Goodness-of-ft Many dfferent R-factors have been nvented Here we menton only those n general use: R, wr and GOF 7

8 R wr GOF F F ( ; F [ ( ;, / S( [ [ [ ( ; / / S( n n ar n n ar The smlest of them, R, measures the fracton of the sum of the structure-factor amltudes that s not exlaned by the wth arameters t s often multled by and gven as a ercentage value The quantty wr s based on a smlar dea; t comares the sum of the squared dfferences between erved and ntenstes wth the sum of squared ntenstes, both weghted wth the relablty factors The two quanttes wr and GOF are closely related to the functon S mnmzed durng refnement and reresent relable measures of qualty The values of R, wr and GOF to be exected n a structure analyss deend on several factors: the qualty of the crystal and thus of the dffracton data, on one hand, and the adequateness of the, on the other hand For molecular crystals (excludng rotens) R ~ 5 s not unreasonable; as a rule of thumb wr ~ -3 R The exectaton value of GOF s, but deends strongly on the relablty factors as dscussed n chater 7 on weghtng schemes 6 Relablty of structural, standard uncertantes and correlaton The results of a least-squares refnement are ncomlete wthout a quanttatve statement of ts uncertanty Such a statement reflects a lack of knowledge due to random and systematc defects n the erved data and to defcences n the Least-squares refnement rovdes not only otmzed atomc coordnates and dslacement arameters but also ther standard uncertantes (su) and correlatons between su s n terms of an uncertanty matrx U U GOF - The dagonal elements of U are u, the off-dagonal elements are u u cor The u s are the su s of the arameters (revously called estmated standard devatons) The convergence of the least-squares otmzaton may be exressed n terms of the ratos between arameter changes and ther sus, Δ /u Otmzaton s generally contnued untl all Δ /u < The quanttes cor are the mutual uncertanty coeffcents (revously called correlaton coeffcents) Ther meanng wll be exlaned below Snce atomc coordnates are dmensonless fractons of the cell constants, so are ther su s Thus the accuracy of atomc ostons s the roduct of the coordnate su s wth the cell constants For an average structure analyss t s a few thousands of an Å Due to ther large number of electrons, heavy atoms contrbute more to the total scatterng than the lghter atoms wth fewer electrons Therefore the accuracy n oston tends to be hgher for the heavy atoms The ostons of hydrogen atoms wth ther sngle electron are only oorly defned n X-ray structure analyss Ther ostonal uncertantes are usually a few hundredths /, 8

9 to tenths of an Å Much more recse locatons of hydrogen atoms are obtaned from neutron dffracton exerments On decreasng the temerature, the su s of the arameters U kl generally decrease because the dffracton ntenstes at hgh values of snθ/λ become stronger and thus easer to measure relably ote that the uncertanty matrx U deends on the, ncludng constrants and restrants f a arameter s constraned, ts uncertanty s necessarly zero f a arameter or a derved quantty s restraned, ts uncertanty u( ) or u(d ) s aroxmately gven by the weght of the restrant, namely as ~w -/ t s therefore mortant to reort all restrants and constrants used n the structure exlctly The meanng of the mutual uncertanty coeffcents cor s best llustrated grahcally Fg 4 shows roectons of S( onto the lane, for two dfferent stuatons On the left an examle of ostve mutual uncertanty s shown The artal uncertanty matrx u uu cor U(, ) σ σ uu cor u s reresented as an ellsod n the left half of Fg 4 the uncertanty n and s assumed to be about equal, whereas t s larger n the drecton + and smaller n the drecton Ths s due to a ostve mutual uncertanty coeffcent cor On the rght of Fg 4 the case of negatve mutual uncertanty s llustrated; the uncertanty s larger n the drecton and smaller n the drecton + Partcularly strong mutual uncertanty exsts between arameters related by a symmetry oeraton Consder and -, whch are assumed to be related by a centre of symmetry or a twofold axs Any change n mles a change of the same magntude, but of ooste sgn n The mutual uncertanty coeffcent between the two quanttes s! ote that large su and mutual uncertanty coeffcents are often assocated wth a near-sngular, lldetermned matrx of normal equatons The general exresson for multdmensonal ellsods analogous to those n Fg 4 s σ Fg 4 Parameter correlaton Left: ostve correlaton coeffcent, the dfference between and s better defned than ther sum and than the ndvdual arameters Rght: negatve correlaton coeffcent, the sum of and s better defned than ther dfference and than the ndvdual arameters σ S T Δ U Δ where Δ ot are devatons from otmzed arameters Qute generally, mutual uncertanty has consequences on quanttes derved from and Suose a bond dstance deends on the dfference between and For ostve correlaton such a dstance wll be more accurately determned than ndcated by u( ) and u( ) alone The converse holds for negatve mutual uncertanty n general the nfluence of mutual uncertanty on the su of a quantty d deendng on the set of arameters s gven by T u [ d( ( d / V( ( d / The elements of the vector d/ are d/, e dervatves of the derved quantty d wth resect to the arameters that d deends on 9

10 7 Weghtng schemes The qualty of exermental ervatons vares wdely deendng on the qualty of the crystal, the qualty of the measurng equment, exermental desgn and on the tme and effort nvested n an exerment Ths s the reason for ntroducng a relablty or weghtng factor n structure-factor least-squares calculatons The quanttes fnally used n such calculatons are usually obtaned n several stes The frst ste consders ndvdual reflectons The quantty ( s not drectly measured t s the dfference between the total measured ntensty tot and a sutably scaled background ntensty s B corrected for Lorentz- and olarzaton effects L (see Data rocessng by Bernhard Sngler and Mchael Wörle): o tot s L B The outcomes of many measurements of the same quantty under the same condtons are never exactly the same, but always affected by some random error The dstrbuton of measurements and thus ther relablty may ether be determned exermentally by multle measurements or aroxmated by a theoretcal The dstrbuton of ntensty measurements or number of hotons regstered (k) s usually aroxmated by a Posson dstrbuton k P( k) ex( ) k! The exectaton or mean value of k s <k>, and the standard devaton of the mean s σ() () / Wth ths assumton and takng nto account the roagaton of uncertantes as descrbed n the recedng secton the su of ( becomes u[ tot + s L B / When multle measurements ( are avalable the mean ntensty and ts standard uncertanty become / u [, / u [ u [ [ / u [ / u [ / f there s a choce between the two, the larger of the two quanttes s used to defne the weght Mn of / u [ or / u [

11 Wth modern area detectors some reflectons and ther symmetry equvalents may have been measured several tmes n dfferent runs, whereas others may have been erved only once or twce n such cases the multle measurements are used to generate a u'[ ( whch should be alcable to all ervatons rresectve of ther frequency of ervaton The frequently-erved reflectons are dvded nto grous accordng to ther ntensty ( Standard uncertantes u[ and u[ are averaged wthn each grou The grou uncertanty u [ ( s exressed n terms of these averages and of a correcton term roortonal to the grou average ( accordng to u' [ k{ u [ + [ g The constants k and g are determned by least-squares otmzaton f the u [ ( are bona fde estmates, the constant g should equal and k should equal Accetable values of g are n the range of -, of k n the range 7-3 Devatons from these aroxmate values ndcate roblems wth the ntensty measurements, scalng and absorton correcton of the data or wth some other systematc defcency of the data The weghts n the structure-factor least squares are usually chosen as ( ) ( ) + / u' + + ( ) M a 3 b To determne the constants a and b the data are dvded nto ca equally oulated ranges of resoluton and of the rato F c / F c (max) The constants are chosen such that n every range the GOF s as close to as ossble deally a and b should be zero n ractce values of a ~ 5 and b ndcate nadequaces of the of the electron densty For very good data sets the devaton from deal behavour may be due to the use of shercal atomc scatterng factors whch neglect effects of chemcal bondng For dsordered structures the of dsorder may be ncomlete or nsuffcently flexble Fnally one should also consder the ossblty that a and b comensate for nadequaces of u from the error and thus for unrecognzed systematc errors n the exerment and the data rocessng f results of the hghest accuracy and recson attanable from a gven data set are asred to, the questons of weghtng should be gven careful consderaton t s nearly always ossble to choose values of a and b whch roduce a GOF of However, f ths condton s fulflled at the rce of unreasonable values of the constants a, b, k and g the standard uncertantes of the atomc coordnates, ansotroc dslacement arameters and of derved quanttes such as bond lengths and angles should be consdered wth some suscon Comarson of dstances and angles obtaned from many ndeendent studes of the same chemcal comounds n the same crystal modfcaton ndcate ndeed that ther reorted standard uncertantes are underestmated by a factor of -3 To gve the reader an arecaton of the relablty of standard uncertantes from a least-squares refnement, t s recommended to reort the values of the constants a, b, k and g together wth R-factors and GOF-values Unfortunately ths s not a general ractce } Acknowledgment: thank HD Flack for mrovng the above text and correctng many msrnts (July 9)