TRENDS IN STRUCTURE DETERMINATION BY POWDERDIFFRACTOMETRY

TRENDS IN STRUCTURE DETERMINATION BY POWDER DIFFRACTOMETRY

A. Le Bail

Université du Maine, Laboratoire des Fluorures, CNRS ESA 6010, Av e nue O.

Messiaen, 72085 Le Mans cedex 9, France - E-mail: [email protected]

Ab stract

Tech niques for solv ing struc tures from pow der dif frac tion data have con sid er ably evolved dur ing the last 10 years. Ten dencies are de duced from an ex haus tive re view of the SDPD-Database, a bib li og ra phy ex tend ing over more than 300 ex per i men tal cases.

Last year, it was said that the Rietveld method al lowed re fin ing struc tures more com plex than we could de ter mine solely from pow der data. Re cent ad vances show that we can now lo cate mol e cules more com plex than we could re fine with out con straints.

1. Introduction

Dur ing the past 10 years, the num ber of struc ture de ter mi na tions by pow der diffractometry (SDPD) in creased from 28 to more than 300. A com mented bib li og ra phy (the SDPD-Database), avail able on Internet [1], al lows draw ing some con clu sions on cur - rent ten den cies. Quan tity is not qual ity, so that you may not con sider that the lead ing meth ods de duced from the fol low ing sta tis tics are re ally rep re sen ta tive of the ways peo - ple can best solve their prob lems. Nev er the less, many meth ods and soft wares listed be - low have proven their ef fi ciency for suc ceed ing in the var i ous steps of the pro cess of solv ing struc tures from pow der dif frac tion data rou tinely. “Rou tine” means, in the au - thor opin ion, that the job can be done by any crys tal log ra pher in any lab in the world, not only by the method in ven tors. There was no rea son that ma jor trends in crys tal log ra phy would not have af fected the SDPD sub-discipline. One ir re vers ible ten dency is that nor - mal hu man brains are no lon ger able to re al ize the cal cu la tions needed for a non-trivial struc ture de ter mi na tion. More and more fast and pow er ful com put ers do the job. For tu - nately, hu mans hav ing not found limit in that do main con tinue to con ceive new soft - ware as well as hard ware. A trend in SDPD is thus clearly the emer gence of so phis ti cated meth ods that would not have been ap pli ca ble if the power of cheap com put ers had not in creased so much. How many times (at least, as found in the SDPD-Database) some meth ods and soft wares were ap plied to ex per i men tal cases will be given along the text, un der pa ren the ses. Note that sev eral meth ods for solv ing one par tic u lar step may have been used in the same study.

2. Statistics on productivity, peoples, reviews

If we con sider the UCl3 struc ture de ter mi na tion by Zachariasen [2] as the first work of this kind, it ap pears that 1998 is the SDPD ju bi lee. Ura nium and plu to nium-based com - pounds were the sub ject of three other pub li ca tions [3-5], till 1968. We waited 1977 for the first mod ern SDPD by Berg and Werner [6], mak ing use of com put ers for in dex ing (TREOR), ap ply ing the Patterson method to 120 in te grated in ten si ties (Guinier-Hägg film), lo cat ing 2 heavy (Mo) at oms, com plet ing and re fin ing the struc ture by suc ces sive Rietveld re fine ment and Fou rier cal cu la tions. Ac cord ing to the SDPD-Database [1], up to 1987, only 27 pub li ca tions coped with struc ture de ter mi na tion by pow der dif fracto - metry. The an nual pro duc tion in this early stage was rang ing ep i sod i cally from 0 to 4 pa - pers max i mum. Dur ing the year 1988, a peak oc curred with 12 pa pers, and then the an nual pro duc tion was more or less sta ble till 1992 show ing a new ex pan sion cor re - spond ing to 29 pub li ca tions. The

yearly pro duc tion os cil lated then, with 22, 40, 55, 34 and fi nally 46 pa pers in 1997. These pub li ca tion num bers cor re spond to slightly more struc tures de ter mined (Fig.

1). The ju bi lee co in cides with ex - ceed ing 300 struc ture de ter mi na - tions (at least 318 struc tures de scribed in 295 pa pers). These num bers are un der es ti mated be - cause some triv ial (in clud ing many struc ture de ter mi na tions based on isotypism) and pre-Rietveld struc tures were not in cluded in the SDPD- Database.

As many as 570 peo ples are au thors or co-authors of these 295 pa pers. Among them, 170 have co-authored 2 pa pers min i mum, 41 have co-authored at least 5 pa pers, and 17 at least 10 pa pers. These con tri bu tions at test for an in creas ing pro fes sion al ism and give an idea of the mass of re search ers mo ti vated by solv ing struc tures in the ab - sence of a suit able sin gle crys tal. The two lead ers (Clearfield and Poojary) have solved to - gether 33 pre vi ously un known struc tures. Ob vi ously, SDPD was and re mains hot topic since now 10 years. It is hard to say if the two abrupt vari a tions in pub li ca tion num bers (1988 and 1992) have some team-leader pre cise det o na tors. Pos si bly, fol low ing the Werner [7-11], Rossel [12], Noläng [13], Raveau [14-16], Smith [17] and Clearfield [18-19] early se ries of SDPDs, the 1988 peak could be due to a new se ries of pa pers pub - lished then in Na ture by Cheetham [20-21]. Pop u lar ity and tem per a ture of SDPD can be also mea sured by the num ber of re view pa pers al ready pub lished on the sub ject (33 ref - er ences are gath ered in the da ta base, of which only a se lec tion is pro posed here [22-44]).

These re views may pro vide dif fer ent col ors of SDPD, ac cord ing to the au thors main spe - cial ties. Solving metallo-organic phos phates [43], metal hy drides [35, 42], zeolites [27], co or di na tion com pounds [44] (...) crys tal struc tures with con ven tional X-ray sources [24, 33], syn chro tron [30, 31, 37] and neu tron data [28], by di rect method [41], Monte

Fig. 1

Structures determined by powder diffractometry

Carlo [40] or other meth ods do have some pe cu liar i ties. None of the pre vi ous re views gave the ex haus tive list of ref er ences avail able at their pub li ca tion date, al though it was much shorter than now. A com plete list of ref er ences on ex per i men tal works, meth ods and soft wares would be larger than 500 and will not be given here. The reader will find them in the SDPD-Database, in clud ing works in volv ing struc ture redeterminations for the pur pose of fea si bil ity dem on stra tion (20 pub li ca tions listed in the SDPD-D). The most fa mous of these al ready known (AK) com pounds is cimetidine [45], C10H16N6S, hav - ing long been given as the ex am ple of what could be done from syn chro tron data (1991), with 17 non-hydrogen at oms lo cated by di rect meth ods. This limit was at tained in cases of pre vi ously un known struc tures since, even from con ven tional X-ray data, though it was not much out per formed.

3. More statistics on periodicals, compounds, crystal systems, instruments

The 295 pa pers are dis trib uted in 51 pe ri od i cals of which 18 have pub lished more than 5 SDPD, and 9 have pub lished more than 10 SDPD. Two jour nals are largely head ing : the Jour nal of Solid State Chem is try and the Eu ro pean Jour nal of Solid State and In or ganic Chem is try, with 67 and 32 con tri bu tions, re spec tively. This tends to in di cate the main kind of prob lem solved by pow der dif frac tion, up to now. In deed, only 31 or ganic, 63 organometallic and 8 poly mers have been the sub ject of a SDPD. This may be con sid ered as ab nor mal be cause of the al most re versed or ganic/in or ganic 4/1 ra tio in the Cam bridge Struc ture Da ta base (CSD) and in the In or ganic Crys tal Struc ture Da ta base (ICSD, ex clud - ing com pounds with C-C bonds). Most of the 216 in or ganic phases are ox ides (159), the re main ing be ing mainly halides (43) and intermetallics (12). Ox ides based on tet ra he dra build ing units rep re sent 90 com pounds with 73 in the group of phos phates, sul fates and ar se nates, and 17 sil i cates. The next well-identified group are ni trates (12) stud ied mainly by Louër ‘s group. Why these com pounds are spe cial can di dates for a SDPD is an in ter est ing ques tion. Most of them were ob tained from syn the ses in volv ing hy dro ther - mal pro cess at me dium tem per a tures, un fa vor able to the grow ing of suf fi ciently large sin gle crys tal, or de hy dra tion lead ing to un avoid able frag men ta tion. It is not risky to pre - dict that the fu ture will very

prob a bly see the re ver sal of this cur rent 1/4 ra tio in or der to fit the 4/1 pro por tion of or ganic/in or - ganic com pounds ob served in the CSD/ICSD data banks. Ar gu - ments in this sense will be given.

All crys tal sys tems were the sub ject of SDPD, the monoclinic and orthorhombic sys tems be ing the more gen er ally stud ied with 42.5 and 29.7%, re spec tively

(Fig. 2). The crys tal sys tem re par - Fig. 2 - Crystal systems (%)

ti tion for these mod ern pow der stud ies does not seem to dif fer from sin gle crys tal stud - ies.

In stru ments se lected for these SDPD are tra di tion ally lo cated at ei ther neu tron re - ac tors (65) or at syn chro tron ra di a tion sources (64), or are in-laboratory con ven tional X-ray diffractometers with (94) or with out (143) in ci dent beam mono chro ma tor. Graph - ite re flected beam mono chro ma tor al lows the study of sam ples con tain ing 3d el e ment with a cop per X-ray tube. Mono chro matic con ven tional X-ray is ob vi ously re served to non-fluorescent sam ples (con tain ing ei ther light or very heavy el e ments, or both of them). How ever, if the use of Peltier-cooled solid-state de tec tors is gen er al ized, this dis - tinc tion could van ish. The Fig ure 3 shows the re par ti tion of in stru ment use by year. In fact these num bers do no re flect the pos si ble joint use of 2 or 3 of these in stru ments. Neu - trons were used scarcely alone (22), many neu tron cases cor re spond to stud ies of liq - uid-solid state phase tran si tions at low tem per a ture (Fitch and Cockcroft, [46-55]), or to deu te rium and lith ium-based com pounds. Ob vi ously, the most com plex struc tures will soon be solved from syn chro tron data. We should first agree on what is a com plex struc - ture and de fine com plex ity cri te ria. Any way, con ven tional X-rays have not re ally been out per formed yet by syn chro tron ra di a tion in quan tity (237 and 64 ap pli ca tions, re spec - tively) nor in qual ity (no real gap in com plex ity is ob served, al though it should be).

These points will be treated fur ther in this re view.

Fig. 3. Instruments

4. SDPD steps - summary

SDPD is a step by step op er a tion where com put ers are es sen tial at stages of (a) in dex - ation, (b) struc ture fac tor ex trac tion, (c) struc ture fac tors se lec tion and mas sag ing, (d) ap - pli ca tion of Patterson or di rect meth ods, (e) model com ple tion, and (f) Rietveld method re fine ment. Be side this clas si cal ap proach, when ini tial mod els are known from other tech niques than crys tal log ra phy (mol e cule from NMR), stage (d) may be re placed (d’) by us ing meth ods that try to find an op ti mum po si tion for a start ing model/frag ment in side the cell. Testing a mol e cule lo ca tion is re al ized by com par i son of the cor re spond ing cal - cu lated dif frac tion data ei ther to the ex tracted “|Fobs|” or to the whole (or large part of) ob served pow der pat tern (sup press ing the need of stages b and c). Only stages (a, b, c, f) are fully spe cific to pow der dif frac tion meth od ol ogy. Stages (d, d’, e) may be re al ized by us ing sin gle crys tal soft wares ap plied to ex tracted struc ture fac tors. Adapted sin gle crys - tal soft wares may work on the whole pow der pat tern in stead of ex tracted struc ture fac - tors. More over, some pow der-specific mod i fi ca tions were in tro duced for deal ing with over lap ping re flec tions.

4.1 Indexation

Au to matic in dex ing is not ac tu ally a very in no va tive field. Three main well known pro - grams tra di tion ally oc cupy the mar ket : TREOR (111 ap pli ca tions to SDPD), ITO (90) and DICVOL (42). The next most ap plied tech nique is elec tron dif frac tion (11). Dif fi culties to in dex pow der pat terns of which you are not even sure than they rep re sent a pure phase has dis cour aged many peo ple to de ter mine struc tures from pow der data, in spite of the ex treme ef fi ciency of the above pro grams. Rec om men da tion is to use all of them.

DICVOL is ex haus tive in its search, but does not tol er ate unindexed re flec tions, which could be due to an im pu rity in weak pro por tion. In prin ci ple one can not go fur ther if this es sen tial step is not ful filled suc cess fully, so that we have here un doubt edly a well es tab - lished trend (but see struc ture de ter mi na tion from pack ing con sid er ations be low, at the d’ stage).

4.2 Structure factors extraction

At he roic times, struc ture fac tor ex trac tion by hand was not rare (cut ting peaks on the re - cord ing pa per by us ing scis sors, and weight ing them), or more so phis ti cated, by us ing a planimeter. In a quite early (1971) sys tem atic ap prox i mate method for the de ter mi na tion of struc ture fac tors from a pow der diffractogram, the pro file shapes were de scribed as tri an gles. The method was ap plied to the so lu tion of the struc ture of metavariscite (AK) [56]. Now a days, mainly the Pawley (43), Le Bail (134) or Rudolf and Clearfield (20) meth ods re al ize this stage. A trend is that the two first meth ods are used also in or der to check the cor rect ness of cells pro posed at stage (a), rather than us ing the clas si cal cell pa - ram e ter re fine ment from es ti mated re flec tion po si tions. The space group prop o si tion is now also gen er ally checked at this stage. Pos si bly due to a better sta bil ity and faster ex e - cu tion, the Le Bail method (LB) dom i nated over the last 6 years (Fig. 4).

Al though the method (P) was de scribed in 1981 [57], the first ap pli ca tion to an ex - per i men tal case occured in 1987 [58]. The LB method was mainly used by the au thor and co-work ers dur ing 4 years (ARIT Soft ware) [59-72], al though also soon avail able at the

ILL, in tro duced into the FULLPROF soft ware [73] and dis closed at the Pow der Dif frac - tion sat el lite meet ing of the IUCr XVth Con gress [74]. The abrupt in crease of the SDPD ap pli ca tion num ber in 1992 (13 →31) co in cides with the avail abil ity of this method in a se ries of Rietveld soft wares (com pare Fig ures 4 and 1), in clud ing GSAS, MPROF and some un named ones gath ered as “LBM”. One is thus tempted to at trib ute a part of the merit in the 1992 in crease in SDPD ap pli ca tions to the LB method avail abil ity, with 18/31 of those year ap pli ca tions. These ap pli ca tions in cluded some re mark able works like Ga2(HPO3)3·4H2O [75] with 29 in de pend ent at oms, the most com plex at this date, and from this point of view ; τ-AlF3 [76] with a com pletely new and un ex pected struc ture built ex clu sively from cor ner shar ing octahedra.

Both P and LB meth ods al low whole pow der pat tern fit ting with cell con straint.

The dif fer ence co mes from the con sid er ation of in ten si ties as in de pend ent refinable pa - ram e ters (P) or not (LB). Thus, ex tract ing 1000 re flec tions by the P method im poses 1000 pa ram e ters to be added to the usual (Rietveld) pro file pa ram e ters. In the LB method, the struc ture fac tors are ex tracted by it er at ing the Rietveld de com po si tion for mula giv ing the so-called “|Fobs|” which are used for es ti mat ing the Bragg RB as well as the RF re li abil - ity fac tors. Starting struc ture fac tors are con strained to be iden ti cal. This re la tion with the Rietveld method ex plains the easy in tro duc tion of the LB method in nu mer ous Rietveld codes. Those prin ci ples ap plied in the P and LB meth ods were thus adopted by many soft wares, the main of which are listed in Fig ure 5, giv ing the num ber of ap pli ca - tions for each of them, as found in the SDPD-Database.

Al ter na tive is to use soft wares ex tract ing struc ture fac tors with out cell con straint. The most pop u lar is MLE (Max i mum Like li hood Es ti ma tion) from Rudolf and Clearfield. On the Fig ure 6, PD is mean ing Pat tern De com po si tion, and cor re sponds to 15 pub li ca tions, which were not pro vid ing the name of the soft wares, but ex plic itly in di cated the method for ob tain ing in te grated in ten si ties. To be noted is the low score of POWLS (one ap pli ca - tion), a pro gram pro posed in the past for the so-called two-stage al ter na tive to the Rietveld method. Dis cus sions on the P and LB meth ods, in clud ing com par i sons of their re spec tive ef fi cien cies were pub lished [41, 77-81]. Im prove ments were pro posed in clud -

Fig. 4. Structure factors extraction : Pawley and Le Bail methods

ing ei ther a Bayesian ap proach that tack les the prob lem of highly cor re lated pos i tive and neg a tive in ten si ties in the P method [82] or an other sta bi li za tion al go rithm [83]. The cur - rent rate of use of the P and LB meth ods and of re lated al go rithms does not in di cate any ten dency to aban don them, on the con trary.

4.3 Structure factors selection, treatment of overlapping reflections

Many vari ants ex ist at this stage. Ei ther the whole data set (in 141 ex per i men tal cases) is used or a se lected set of un am big u ously in dexed re flec tions (80) is re served for the next (d) step. At this step, the prob lem of the in ten sity re par ti tion among re flec tions, which are more or less over lap ping, is con sid ered. No need to say that you can hardly ex pect to ob tain the cor rect in ten sity es ti ma tion of the ex actly over lap ping re flec tions. The de fault out put of the P and LB meth ods is the equipartition (but see later com ments about the GSAS pro gram im ple ment -

ing a vari ant of the LB method, and note the ten - dency of the P method to pro duce neg a tive in ten si ties when the slack con straints are not used ap pro pri ately).

Al ter na tively, the data may be fur ther mod i fied, af ter ex trac tion, by meth ods try - ing to ex trap o late knowl - edge from a se lected set of non-overlapping re flec tions

Fig. 5. Softwares for structure factors extraction with cell constraints (P = Pawley, LB = Le Bail)

Fig. 6. Softwares for structure factors extraction without cell constraints

in or der to dis crim i nate and change the in ten sity re par ti tion be tween those more or less over lap ping re flec tions. Many ar ti fices were used, in clud ing for in stance giv ing a ran - dom re par ti tion of in ten sity in stead of equipartition, or us ing the ex pected positivity of the Patterson map. Da vid [84] based his es ti mates on the en tropy max i mi za tion of an

|F|²Patterson func tion. Use of Patterson map is also made in the FIPS (Fast It er a tive Patterson Squaring) method [85]. In the DOREES pro gram [86], the ap proach is based on the trip let and quar tet re la tions from di rect meth ods and the the ory of the Patterson func tion. In the di rect meth ods SIRPOW pro gram [87-88], a spe cial treat ment is re served to the more or less over lap ping re flec tions. Tex ture was also used as an in ten - sity-separating ‘de vice’ [89] in an at tempt at par tial elim i na tion of the prob lem of over - lap ping re flec tions. A proba bil is tic method was in te grated with the LB al go rithm [90].

Finally, Patterson informations are also used in ref er ence [91]. The sim plest ap proach is the prep a ra tion of a se ries of data sets from which re flec tions hav ing a neigh bor ing one at less than δ(2θ) (with δ = 0.01°, 0.02°, 0.03°..., or frac tions of the FWHM vary ing from 0.1 to 0.5) are ex cluded. Di rect meth ods are then ap plied to these re duced data sets (with a limit cor re spond ing to 50% max i mum of the hkl ex cluded), or Patterson meth ods (up to 80-90% of the re flec tions can be ex cluded if 1 or 2 heavy at oms max i mum are to be lo - cated). All these meth ods may en hance the suc cess rate for struc ture so lu tion. Nev er the - less, a com par i son of the ef fi ciency of these ap proaches is lack ing. The SDPDRR (Struc ture De ter mi na tion by Pow der Diffractometry Round Robin) [92] could draw some con clu sions on this point. Any way, it seems that in al most half of the SDPDs (141 cases), the whole ex tracted “|Fobs|” data set, with equipartition, led to the suc cess with out any data mas sag ing.

4.4 Solving the structure by direct or Patterson methods

For this more than es sen tial SDPD step, au thors have sim ply adopted the gen eral trend in struc ture de ter mi na tion from sin gle crys tal data, by us ing Patterson (104 cases) and di rect meth ods (149 cases), in clud ing their most re cent im prove ments. For in stance, the new SHELXS-97 in cludes phase an neal ing di rect meth ods and new de vel op ments in Patterson in ter pre ta tion. SHELX pro -

grams are used in well over than 50%

of small mol e cule struc ture de ter mi - na tions from sin gle crys tal data as well as from pow der data (ver sions SHELX-76, SHELXS-86 or 97). The main soft wares used for Patterson and di rect meth ods are shown in the Fig ure 7.

It should be re al ized that in many cases, only one or two heavy at oms had to be lo cated for ob tain ing a par tial model al low ing to start re - fine ments, the lo ca tion of the re -

main ing at oms be ing then ob tained Fig. 7. Softwares for Patterson and direct methods

by Fou rier dif fer ence syn the ses. There are 99 of these sim plest cases in the SDPD-Database. In 17 of them, the heavy atom is zir co nium. Other fre quent oc cur rences are for lanthanides (7 cases), U and Bi (6 for each of them), Ba, Mo, Pd (4), Ag, Tl (3).

Solving these rather triv ial struc tures was even pos si ble by Patterson method ap plied to a very lim ited range of data : 30 or 40 in ten si ties may suf fice, ob tained from those re flec - tions un am big u ously in dexed of which the in te grated in ten si ties can be ex tracted by the old gen er a tion of pro file fit ting soft wares with out cell con straint (POWLS, PROFIT, MLE...). In the years 1948-87, Patterson method dom i nated the di rect meth ods with a ra - tio 16/6. Since 1988, the ten dency is re versed with a ra tio 79/139. The SDPD-Database con tains 18 struc tures guessed from some con ver gent informations (for in stance there could be 4 Pt at oms oc cu py ing ob vi ously only one pos si ble spe cial po si tion with all fixed co or di nates). Model build ing meth ods con cern the 51 re main ing struc tures in the Fig ure 8. With sin gle crys tal data, es ti mated stan dard de vi a tions of struc ture fac tors are used by most Patterson and di rect method soft wares only in or der to ex clude re flec tions with a limit gen er ally set to |Fobs| < 2 σ(|Fobs|). The LB method is re puted for not pro vid - ing cor rect esds. I rec om mend to avoid any “|Fobs|” elim i na tion based on esds, what ever the method used for struc ture fac tors ex trac tion, this would ex clude the very im por tant in for ma tion (for di rect meth ods) con tained in weak re flec tions.

Re cent re view ar ti cles on di rect meth ods ap plied to pow der data may be found [41, 93]. Use of Patterson and di rect meth ods rules for an al ter na tive to the equipartitionning of the over lap ping re flec tions was al ready dis cussed at the pre vi ous para graph. The dom i nant po si tion, in this field, of the pro gram SIRPOW is clear on Fig ure 7, how ever ap pli ca tion of con ven tional di rect and Patterson meth ods for sin gle crys tal data are still head ing (SHELX, and to a lesser ex tent, TEXSAN, MULTAN, MITHRILL...). Other spe - cial ad ap ta tion of di rect meth ods for solv ing struc tures from pow der dif frac tion data

Fig. 8 Methods for structure solution

were pro posed, namely a multisolution method of phase de ter mi na tion by com bined max i mi za tion of en tropy and like li hood (pro gram MICE) [94-99], di vid ing data in two sets (over lap ping and non-overlapping). It was scarcely used for the de ter mi na tion of un - known struc tures, up to now [100 - 101]. Spe cial ad ap ta tions of the Patterson method were ap plied, us ing max i mum en tropy Patterson maps [102] and sym me try min i mum func tion [103 - 104], and a tan gent for mula was de rived from Patterson-function ar gu - ments [105]. As pects of struc ture de ter mi na tion from pow der data us ing anom a lous scat ter ing were con sid ered [106-109], re quir ing ac cess to syn chro tron ra di a tion and re - cord ing at mul ti ple wave lengths. Ap pli ca tion to di rect meth ods of the early find ing of pre ferred ori en ta tion was ex am ined [110]. Di rect de ter mi na tion of poly mer crys tal struc - tures from fi bre and pow der X-ray data was con sid ered [111]. A weight ing scheme low - er ing the prob a bil ity of phase re la tions for in ten si ties of over lap ping re flec tions was in tro duced in the op ti mal sym bolic ad di tion pro gram SIMPEL88 [112]. But no trend is ob vi ous, for this step, that the clas si cal ap proach (i.e. us ing di rectly soft wares de vel oped for sin gle crys tal data) will be soon re moved from its dom i nant po si tion by some clearly more ef fi cient ap proach.

4.5 Model building (MB), locating fragments of known or guessed geometry When stan dard meth ods fail (Patterson and di rect meth ods), the al ter na tive is to build a model and to lo cate it in the cell. This could be ei ther a sim ple or a for mi da ble task, de - pend ing on the ex is tence or not of prior informations. Lo cating frag ments of known ge - om e try was long ago used in sin gle crys tal stud ies of or ganic com pounds. Prior in for ma tion may con sist in the whole con nec tiv ity scheme of a mol e cule from NMR data. On the other hand, in or ganic phases al low also some guess when the ba sic struc - tural units are known or ob vi ous (tet ra he dra, octahedra...) and more over if the con nec - tiv ity of these units is ev i dent (for in stance ex clu sive cor ner shar ing of [SiO4] tet ra he dra in com pounds hav ing ba si cally SiO2 for mula, like in zeolites). It is in this (d’) stage that the last ten years of SDPD were par tic u larly in no va tive. No less than 20 new meth ods or trans po si tion of ex ist ing (sin gle crys tal or mo lec u lar mod el ling) meth ods re cently emerged for solv ing struc tures from pow der data. The list, as found in ti tles of re cent pa - pers, is re viewed be low, but the reader should keep in mind that the pres ent au thor has never used any of them ! Some meth ods have evolved and im proved over the years. This (d’) sub-topic is quite hot be cause phar ma ceu ti cal or other eco nom i cally im por tant com - pounds are po ten tially in volved. Is model build ing/lo cat ing spe cific to pow der dif frac - tion? Ab so lutely not, one could ap ply these meth ods also to sin gle crys tal data set, of course. A to tal of 51 cases were found in this cat e gory in the SDPD-Database, since 1988.

A ten dency to ex pan sion is ob vi ous, with 12 cases in 1997. The ba sic dif fi culty in these ap proaches is in fact to de fine the start ing model ; most of those model-location meth ods have a role to play only when this dif fi culty has been solved. We are be tween prior chem i cal knowl edge and full model pre dic tion.

1- Early stud ies. In pi o neer ing works, mod els were guessed (more or less) from con - ver gent ev i dences. Once mod els are built, they can be op ti mized by var i ous means like the dis tance least squares pro gram () [113] or some en ergy minimization tools. It is not al ways easy to un der stand how the mod els were built be fore be ing op ti mized in some

works clas si fied in this MB cat e gory [46, 114]. Packing con sid er ations on the heavy at - oms were used to solve the struc ture of two lan tha num pal la dium ox ides [115]. The 4 SDPD cited above were pub lished be fore 1988 and are the first ones in this cat e gory.

Many of the sub se quent ap proaches of this kind can not be eas ily sum ma rized, here is an ex am ple for a metal-substituted alu mi num phos phate cat a lyst [116] which was “solved on the ba sis of ev i dence gar nered from high-resolution elec tron mi cros copy, elec tron dif frac tion, and other meth ods in clud ing en ergy minimization”..."A con sid er ation of pos si ble frame works of ap pro pri ate di men sions and with the ob served size and spa tial dis tri bu tion of unidimensional, large-pore chan nels sug gests a trial struc ture with ide al - ized sym me try Cmcm". In this case, DLS and METAPOCS pro grams [117] were used for the model op ti mi za tion. Nu mer ous other SDPD suc ceeded by this kind of model build - ing [118-126], for which the op ti mi za tion of the start ing mod els was de rived ei ther from the above pro grams or from MNDO cal cu la tions [129], or CERIUS [130], or THEO, or INSIGHT-II [131].

2- Model lo ca tion with out en ergy minimization. When an ini tial model is se lected, which should be suf fi ciently large for lead ing to cal cu lated struc ture fac tors or Patterson map or phases eval u a tion near of the ob served ones, then the only prob lem is to lo cate this ini tial model into the cell. The PATSEE pro gram [132] is an old SHELX com pan ion.

It was ap plied to pow der dif frac tion data [133-134]. The pro gram re quires ex tracted struc ture fac tors and at tempts to com bine the mer its of both Patterson and di rect meth - ods in or der to po si tion a frag ment of known ge om e try in the unit cell. Ran dom ro ta tions may con cern one frag ment and the pro gram al lows one tor sional de gree of free dom. The ran dom trans la tion search may lo cate up to two in de pend ent frag ments of any size. Early works in this model-location cat e gory con sisted in the brute force : search ing for a mo - lec u lar po si tion and ori en ta tion fol low ing a sys tem atic grid search, as in the cases of solid CF3I [53], CFCl3 [55], or RS-camphor [135]. Some times, re tain ing a prop o si tion be - fore try ing a Rietveld re fine ment was based on in ter atomic dis tances cri te ria : i.e. pack - ing con sid er ations. The ROTSEARCH pro gram [136-137] al lows ro ta tion and trans la tion Patterson searches, checked against ex tracted in ten si ties. It was ap plied to the so lu tion of ze o lite and or ganic com pounds [138-140]. A re cent ver sion (ROTS96 [141]) was able to lo cate 3 in de pend ent mol e cules in C16H22N6 [142]. P-RISCON is a real-space scav en ger pro gram [143] ca pa ble of set ting an ini tial rigid model by re fin ing the frac tional co or di - nates of its cen ter of mass and its an gu lar ori en ta tion, us ing ex tracted in ten si ties for check ing [144-146]. A search for the model po si tion may also be found in the struc ture de ter mi na tion of C60Br24(Br2)2 [147]. More so phis ti cated ap proaches in op ti mi za tion of known frag ment lo ca tion in clude sim u lated an neal ing or the re lated Monte Carlo (MC) al go rithm. The MC ef fi ciency was dem on strated by the de ter mi na tion of known [148-151] as well as un known [149, 152-158] struc tures us ing pro grams as sess ing the suit abil ity of the model lo ca tion on the ba sis of the agree ment with the ex per i men tal dif - frac tion data (no need to ex tract struc ture fac tors). Vari ants have in cor po rated re strained re lax ation of the mo lec u lar ge om e try or a high de gree of mo lec u lar flex i bil ity or were said to be gen er al ized (OCTOPUS96 and OCTOPUS97 pro grams). Do not for get that the model has to be known when deal ing with these meth ods. Even more so phis ti cated, maybe, are meth ods ap ply ing sim u lated an neal ing, pos si bly through a ge netic al go rithm

(GA), be cause a set of in ter nal co or di nates de fines the ge om e try of the mol e cule (the con - for ma tion), al low ing tor sional an gles to vary (in ad di tion to the usual ex ter nal de gree of free dom de fin ing po si tion and ori en ta tion). The use of GA was de vel oped in de pend ently by two teams and ap plied ei ther to al ready known [159-160] (pro gram “GAP”) or un - known com pounds [161-162]. Testing the agree ment be tween the pos tu lated struc ture and the ex per i men tal dif frac tion data was ei ther on ex tracted struc ture fac tors (us ing the Pawley method) or on the full pow der pat tern (pro gram “GAPSS”).

3- Model lo ca tion from crys tal pack ing con sid er ations. Com pu ta tional meth ods, which pre dict pos si ble crys tal struc tures on the ba sis of the mo lec u lar struc tures, were ap plied to pow der data [163]. Even more, the A mod i fi ca tion of guanylhydrazone struc - ture could be de ter mined with out knowl edge of the lat tice pa ram e ters and crys tal sys - tem. For a pack ing-based struc ture de ter mi na tion, in dex ing is not es sen tial, how ever it is very use ful in re duc ing the amount of com pu ta tions by re strict ing the pack ing to a lim - ited num ber of space groups. Nev er the less, the pow der pat tern re mains es sen tial for a con fir ma tion of the model, of course. An in con ve nience of this method is that the whole mo lec u lar struc ture should be known oth er wise pack ing con sid er ation would not ap ply.

In this cat e gory, but us ing the prior knowl edge of cell di men sion and spacegroup, may be clas si fied var i ous meth ods for com puter pre dic tion of mol e cule lo ca tion (still with out the need of in ten si ties), by pack ing en ergy cal cu la tion [164] (on AK sam ples) ; com pu ta - tional chem is try tech niques [165-167] (3 AK sam ples, one with pre vi ously large R=20%

pow der re sult) ; minimization of the crys tal-lattice po ten tial en ergy cal cu lated with semi-empirical atom-atom po ten tials [168] us ing the PMC pro gram [169]. Finally, the sys tem atic rank ing of all po ten tial pack ing ar range ments on the ba sis of lat tice en er gies was shown to be ef fi cient on AK sam ples [170].

4- Ab in itio pre dic tion. Most ex perts be lieve that crys tal struc ture pre dic tion from a given com bi na tion of el e ments is still a far away goal. Easier, pre dic tion by sim u lated an - neal ing start ing from unit

cell di men sions and con tent has pro gressed for sim ple sys tems [171-172]. The more com plex struc tures are pre - dicted when the prior knowl - edge of sym me try is added [173].

To be clas si fied as hy - brid ap proaches or new con - cepts are the com bi na tion of chem i cal in for ma tion and pow der dif frac tion data in an au to mated struc ture de ter mi - na tion pro ce dure for zeolites (Fou rier re cy cling with a spe - cial ized to pol ogy search)

[174-175] and the use of a pe - Fig. 9. Model building/location, softwares and methods

ri odic nodal sur face cal cu lated from a few strong, low- index re flec tions to fa cil i tate struc ture so lu tion [176]. A sum mary of these 51 ap pli ca tions of model build ing-location meth ods to un known struc ture is shown in Fig ure 9.

The dom i nant trend is given by Monte Carlo and Patterson-search meth ods. These pro grams for mol e cule lo ca tion are sel dom in the pub lic do main. How ever, they are quite re cent and could be meth ods for the fu ture. Most of them are spe cif i cally de signed for or ganic com pounds and they can not be ap plied if at least a large part of the struc ture is not al ready known. Trends are in add ing flex i bil ity to the start ing mod els. The use of ge netic al go rithm ap pears cur rently to be the most so phis ti cated ap proach, giv ing free - dom to non-rigid mol e cule parts. Any way, the im pact of these meth ods on the rou tine anal y sis is small. Some meth ods were ap plied to the so lu tion of AK struc tures [177] and still not to any un known one. Each of the above method is claimed to pos sess ad van tages that are con sid ered as in con ve niences by oth ers. For in stance : avoid ing struc ture fac tor ex trac tion is times to times con sid ered as be ing an ad van tage be cause the use of the full pat tern over comes over lap ping prob lems ; work ing on ex tracted struc ture fac tors is con - sid ered as an ad van tage for speed ; those pre dict ing the po si tion of mol e cules with out the need of in ten si ties at all find this an ad van tage (al though few peo ple will trust such re sults alone) ; those work ing on mol e cule pack ing un der line that they even do not need the cell pa ram e ters (any way, they al ways make a Rietveld fi nal re fine ment). Finally, some cur rent so phis ti cated model lo ca tion ap proaches are no more than the old good

“trial and er ror” pro cess, mod ern ized for mak ing high num bers of tri als in small times.

4.5 Structure completion

Struc ture com ple tion is usu ally per formed by Fou rier syn the ses ap plied to “|Fobs|”

pro vided by the Rietveld de com po si tion for mula. Few us ers (147) are re ally ex plicit on soft wares used to re al ize this step. Not many Rietveld soft wares are able to give a Fou rier syn the sis as out put. GSAS and XRS-82 are part of them; this ex plains their pres ence in the Fig ure 10. The great ma jor ity of the other soft wares are di rectly is sued from sin gle crys tal pack ages. Again SHELX dom i nates (SHELX-76, -93 or -97, or -TL). When a big mol e cule is lo cated from 100 or 300

re flec tions, and the struc ture is rigid-body re fined, or is re fined with many re straints up to RB < 5%, then it is nor mal that Fou rier dif fer - ence syn the sis does not re veal any re sid ual. Some times, Fou rier syn - the ses re main also si lent with large RB value, so that one may have to guess the po si tion and na ture of a few last at oms. Using soft ware like DLS-76 or soft wares al low ing holes lo ca tion or mak ing use of max i mum en tropy [178] pos si bly help the guess.

Fig. 10. Softwares for Fourier syntheses

4.6 Final Rietveld refinement

GSAS (with 84 ap pli ca tions) and FULLPROF (57) pro grams dom i nate the Rietveld [179-180] re fine ment last stage, as shown in Fig ure 11. Five other soft wares totalize more than 6 ap pli ca tions, and 20 soft wares (whose names are not re ported here) were ap plied 1 to 5 times. In 12 cases, there was no Rietveld re fine ment at all, the fi nal least squares be ing re al ized on the in te grated in ten si ties by us ing sin gle crys tal soft wares. At the struc ture fac tor ex trac tion stage, the GSAS/FULLPROF ra tio was re versed (22/46 in stead of 84/57 here), this point mer its some ex pla na tion. The LB method is sup posed to it er ate the Rietveld de com po si tion for mula start ing from a set of all iden ti cal |F| val ues, and this is achieved in many Rietveld soft wares in clud ing FULLPROF. An other ap proach is used by GSAS, that con sists in gen er at ing the ini tial |F|s from in ten si ties cor re spond ing to a dummy atom, the po si tion of

which de pends on the user de ci - sion. Using an atom in gen eral po si - tion is highly rec om mended, be cause the LB method may en - coun ter dif fi cul ties to change null struc ture fac tors (they will stay at zero af ter each it er a tion un less an ad di tional pro cess changes their value). Any way, the ex actly over - lap ping re flec tions will keep an in - ten sity ra tio equal to that gen er ated by the dummy atom. The equi - partition of strictly over lap ping re - flec tions is there fore not re spected.

This may be the rea son of this GSAS/FULPROFF ra tio re ver sal, and pos si bly in sta bil i ties when pat - terns in clud ing the K al pha dou blet are pro cessed with GSAS.

You should have the last im prove ments in cor po rated in your Rietveld pro gram for deal ing with strong asym me try, or pre ferred ori en ta tion or anisotropic line broad en ing.

De vel opers com pete for up dat ing their pro grams. The ten dency to un der take large struc - tures leads to the need of ef fi cient pro grams in clud ing soft con straints and rigid-body re - fine ment. Trends are to deal with more and more com plex struc tures us ing low-resolution data. The re sults will be of poor qual ity, with large es ti mated stan dard de vi a tions on atomic co or di nates, du bi ous in ter atomic dis tances (the fact that they are con strained will not be a suf fi cient rea son for trust ing them).

Fig. 11. Softwares for final Rietveld refinement

5. Complexity

There is a per pet ual race for the pub li ca tion of the most com plex struc ture ever de - ter mined from pow der dif frac tion data. Each time the max i mum num ber of in de pend ent at oms is over come, those words oc cur in the pa per in tro duc tion “X is the most com plex struc ture ever solved from SDPD”. Things are not so sim ple that the com plex ity would be the to tal num ber of in de pend ent at oms. More over, the at tain able com plex ity level is not just de fined by the struc tures al ready de ter mined. The cur rent max i mum com plex - ity should be de fined by the most com plex struc tures de ter mined by us ing the low est res o lu tion. For in stance, find ing 10 in de pend ent at oms by the di rect meth ods with a pat - tern show ing min i mum FWHM as large as 0.25° (2θ), ex tend ing at least up to 100°, for a wave length ~1.5 Å, is pos si ble (sam ple 1 of the SDPD Round Robin [92]). Then, you can ex tend pro por tion ally this prop o si tion to syn chro tron data with 0.05° FWHM and ob tain that solv ing a 50 at oms struc ture should be fea si ble as well, or a 250 at oms struc ture if the min i mum FWHM was lower than 0.01° (as ob tained re cently at ESRF). Nev er the less, does com plex ity is cur rently in creas ing ? Not re ally, the world of SDPD is con cerned by mod er ately com plex struc tures. The max i mum of at oms si mul ta neously lo cated by di - rect meth ods is not larger than 18. The to tal num ber of in de pend ent at oms is near of 60.

On the other hand, rarely more than one in de pend ent mol e cule was lo cated by model-building non-conventional meth ods (d’). For or ganic com pounds, the trend is to use geo met ri cal re straints in the fi nal re fine ment. A kind of limit has been at tained re - cently for C16H22N6 with 70 atomic co or di nates rigid-body-refined from 104 re flec tions for 3 in de pend ent mol e cules lo cated by Patterson search [142]. Con fi dence in crys tal struc ture ac cu racy de pends on the re flec tion/pa ram e ter ra tio which is ad mit tedly ≥10 when sin gle crys tal data are in volved and should be ≥ 20 when pow der data are con - cerned (due to over lap ping). One could doubt about some de tails of the struc ture when this ra tio is not much larger than 1. Mod ifying tor sion an gles would be of lit tle in flu ence on the fi nal fit. Lo cating hy dro gen at oms could not be se ri ously un der taken.

The SDPD-Database sorts com pounds ac cord ing to 4 com plex ity cri te ria : C1 = Num ber of in de pend ent at oms in the asym met ric unit.

Nc = Num ber of re fined atomic co or di nates at the fi nal stage.

C2 = Num ber of in de pend ent at oms in the ini tial model (Patterson or di rect meth ods).

C3 = Num ber of in de pend ent frag ments lo cated by model build ing tech niques.

Looking at Fig ure 12, show ing the vari a tion of the max i mum and mean C1 val ues dur ing the last 10 years, one is tempted to con clude that the com plex ity of struc tures, which can be de ter mined from SDPD, is slightly in creas ing. The max i mum C1 val ues are still be low the num ber of at oms that one may ex pect to re fine by the Rietveld method.

This num ber de pends on the in stru men tal res o lu tion for a well-crystallized com pound.

From syn chro tron data with FWHM ~ 0.03°(2θ), ex tend ing over 120°, 4000 hkl could be

“half-resolved”, al low ing the re fine ment of 400 atomic co or di nates, cor re spond ing to more than 130 at oms in gen eral po si tion.

The num ber of re fined atomic co or di nates (Nc) is a com plex ity cri te rion giv ing sim i lar re sults as C1 (Fig ure 13). The mean Nc value in 1997 (45.6) is al most twice the value ob served in the 1948-87 pe riod (23.5). These C1 and Nc ten den cies need to be con - firmed in 10 years. Any way, this evo lu tion is re ally ex pected be cause of in creas ing in - stru men tal res o lu tion at syn chro tron sources (FWHM as low as 0.01° 2θ was at tained at ESRF) and also in lab o ra to ries (FWHM as low as 0.04° 2θ is pro duced by us ing vari able slits).

From the C2 cri te rion, the mean (noted M1) num ber of lo cated at oms was cal cu - lated and re ported on Fig ure 14 along years. A par tial mean (noted M2) was also re - ported, ex clud ing the 99 struc tures for which 1 or 2 at oms were found to rep re sent a suf fi cient start ing model. In fact, only the max i mum shows a ten dency to in crease.

Fig. 12. Complexity criterion C1 : Number of independent atoms

Fig. 13. Nc complexity criterion : Number of refined coordinates

The C3 cri te rion was con sid ered to be ap pli ca ble to meth ods lo cat ing mol e cules or frag ments. Trial and er ror pro cess used in try ing to lo cate one or sev eral heavy at oms were dis carded. Only 26 un known struc tures were de ter mined since 1993 cor re spond - ing to the C3 cri te rion, so that sta tis tics are hardly pos si ble. Build ing pro teins from mol e - cules will need the abil ity to lo cate si mul ta neously much more frag ments than is pos si ble at pres ent.

6. Accuracy

When no sin gle crys tal is avail able, it is hardly pos si ble to have an idea of the ac cu - racy of a struc ture de ter mi na tion solely from pow der dif frac tion data. How ever, as a gen - eral rule, we know that ac cu racy will al ways be lower than if the same struc ture had been de ter mined from sin gle crys tal data. The many SDPD from AK (al ready known) com pounds have pro duced a set of in ter est ing ref er ences for com par i son. Very prob a bly a trend is that AK com pounds

will con tinue to be used as test sam ples for new meth ods. Care to keep com mon sense will be nec es sary. Cred i bil ity of struc - tures should not in crease when the data/pa ram e ters ra tio de creases. IUCr tends to make dif fi cul ties for pub lish ing sin - gle crys tal data when this ra tio is less than 10, and R-values larger than 5%. Over lapping prob lems ob vi ously should lead to a re vi sion of these lim - its for SDPD.

Fig. 14. C2 complexity criterion : Number of atoms located by Patterson or direct methods

Fig. 15. C3 complexity criterion : Number of independent molecules or fragments simultaneously located

7. Conclusion

SDPD cre ated a need for the de vel op ment of spe cific meth ods or for adapt ing sin gle crys tal tech niques. Many meth ods are even com mon to both pow der and sin gle crys tal data with out any need for ad ap ta tion. There is not so much dif fer ence be tween crys tal - log ra phers ei ther try ing to cope with bad sin gle crys tal data or fac ing pow der dif frac tion data. A cleav age is ap par ent be tween the un knowns for which it will be pos si ble to guess a mol e cule or a suf fi ciently large frag ment (or sev eral ones) and those for which no suf fi - cient prior in for ma tion will be avail able. In the lat ter case, the ab in itio meth ods will con tinue to be ap plied (Patterson and di rect meth ods). In the for mer case, not nec es sar - ily re stricted to pow der prob lems, var i ous sim u lated an neal ing ap proaches have yet proven their abil ity. The def i ni tion of a “pre vi ously un known” com pound is dif fer ent for in or ganic and or ganic com pounds. In most cases, the mo lec u lar struc ture is avail able for the lat ter, and only the po si tion and ori en ta tion in the cell is un known.

Some time ago, it was stated that we were un able to de ter mine struc tures as large as those we could re fine by the Rietveld method. The new par a dox is that we can lo cate now mol e cules in much big ger cells than we could re fine with out con straints. Due to re - sis tance to change, hab its listed in the pres ent re view have chances to sur vive and give us the ten den cies for the years to come. A few soft wares dom i nate each step of the SDPD whole pro cess, they will prob a bly ex tend their dom i na tion un less more ef fi cient ones ap pear. Not all soft wares are in the pub lic do main so that some meth ods are the ex clu siv - ity of de vel op ers or teams re peat ing struc ture de ter mi na tions by their own way. SDPD will not ex pand faster be fore a larg est dis tri bu tion of these new soft wares. Al ter na tive should be re mem bered as well : struc ture de ter mi na tion from very small sin gle crys tal [181] and struc ture de ter mi na tion from elec tron dif frac tion [182]. The for mer pos si bil ity co mes in lab o ra to ries with im ag ing plate data re cord ing, or at syn chro tron ra di a tion sources for crys tal size as low as 20 µm (sin gle crys tal data for sam ple 2 of the SDPD Round Robin were ob tained from a 40x30x20 µm se lected from the “pow der”). A trend is even that many of the first SDPD can now be com pared to sin gle crys tal stud ies (the way to ob tain a sin gle crys tal was found later). The last trend will be that over lap ping peaks in pow der diffractometry will con tinue to over lap, for ever.

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