Direct Methods and Refinement in Electron and X-Ray Crystallography

Home , Electron crystallography, Phase problem

86 Z. Kristallogr. 225 (2010) 86–93 / DOI 10.1524/zkri.2010.1198 # by Oldenbourg Wissenschaftsverlag, Mu¨nchen

Direct methods and refinement in electron and X-ray crystallography – diketopiperazine revisited

Douglas L. Dorset*

Corporate Strategic Research, ExxonMobil Research & Engineering Co., Inc., 1545 Route 22 East, Annandale, NJ 08801 USA

Received June 26, 2009; accepted November 13, 2009

Electron crystallography / X-ray crystallography / with varying success. Eventually, the perturbation of the Direct methods / Refinement / Structure comparison intensities by multiple beam dynamical scattering was recognized to hinder the success of many ab initio structure Abstract. Texture diffraction data in pioneering Russian analyses (Cowley and Moodie, 1959). Subsequently, it be- electron crystallographic studies inspired the invention of came almost dogma to state that such determinations modern precession methods for collecting single crystal were, in fact, impossible (e.g., Eades, 1994). It was as- data. (Although oblique texture patterns contain over- sumed without proof that the kinematical (single scatter- lapped reflections, the similarity to precession methods is ing) limit should be closely approached by the experimen- a result of averaging intensities over an angular distribu- tal intensity data before they could be used for any ab tion of crystallites within the large area sampled by the initio analysis. incident electron beam.) Using texture diffraction ampli- A far different viewpoint was held in the East, princi- tudes originally measured by Vainshtein, the crystal struc- pally the Former Soviet Union, where, instead of electron ture of diketopiperazine has been re-determined using microscopes, electron diffraction cameras were employed automated direct methods (SIR97) and refined by block for collection of intensity data. With these cameras, data diagonal least squares. This determination was carried out were collected on film from millimeter diameters as obli- in parallel with an equivalent direct methods study of a que texture patterns, contrasting to the ca. micrometer (or larger X-ray data set from the same material published by less) diameters sampled by selected area diffraction in the Degeilh and Marsh, followed by least squares refinement, electron microscope. It was argued that the angular distri- reproducing the original results. After both refinements, bution of crystallites within the larger illuminated area the electron crystallographic bonding parameters for heavy would produce an integrated average intensity for each atoms are found to be similar to derived X-ray crystallo- reflection (Vainshtein, 1956), significantly reducing the graphic parameters. On the other hand, C––H and N––H multiple beam interactions characteristic of zonal single distances are more accurately determined by electron crys- crystal patterns. Any dynamical effects were assumed to tallography than by X-ray crystallography since the light be correctable by two-beam approximations (Vainshtein atom positions are more easily detected in ensuing maps. and Lobachev, 1956) which could be applied a priori Surprisingly the least squares refinement against the elec- without foreknowledge of the underlying crystal structure. tron diffraction data did not require a restraint on the mag- In an important review, Cowley (1967) partially agreed nitude of atomic incremental movement; moreover the with this assumption, stating that the angular averaging atomic temperature factors could be refined, producing re- would diminish non-systematic dynamical effects but not sults that were more reasonable than expected from the the systematic ones. Significantly, the recent introduction very low overall B value found from a Wilson plot. of precession diffraction methods for selected area diffraction data collection (Vincent and Midgley, 1994) simulates the same sort of angular averaging effects cited in the Introduction early Russian work. Again, non-systematic n-beam effects seem to be diminished by precession geometry but not the In the pioneering attempts to determine crystal structures systematic ones (Midgley, 2007). by electron diffraction, two sources of intensity data were One other historical problem facing the early electron generally consulted. In the West, where the electron micro- crystallographers was that it was not easy to determine a scope was commonly employed to produce selected area crystal structure from diffraction data from any radiation electron diffraction patterns from individual microcrystals, source prior to the 1950’s. Trial and error or Patterson John Cowley (1953a, b; 1956) and others explored the methods were the major methods employed by X-ray crys- kinematical approach for analysis of such intensity data tallographers for crystallographic phase determination until the development of direct methods by Hauptman and Karle (1953), Sayre (1952), and others. In addition, many * e-mail: [email protected] early electron crystallographic structures reported from the Crystal structure of diketopiperazine 87

Soviet Union had often been determined previously by X- s(Fo) ¼ 0.1jFoj was applied (C. J. Gilmore, personal com- ray crystallography. In fact, these investigators actually in- munication). tended to exploit the advantages of electron diffraction for, X-ray diffraction data from this material had been re- e.g., the detection of light atoms in the presence of heavier corded on films exposed on a Weissenberg camera as de- ones (Vainshtein, 1964a,b). Nevertheless, claims of valid scribed by Degeilh and Marsh (1959). From systematic structural results from electron diffraction analyses of any absences, the space group had been found to be P21/a, kind were often met with extreme skepticism. where a ¼ 5.228(2), b ¼ 11.554(2), c ¼ 3.980(5) A, One of the most complete intensity data sets ever col- b ¼ 97.98(2). Intensities of 1631 reflections had been es- lected from an organic molecule was that from diketopi- timated visually. For the compilation of data for direct perazine (2,5-piperazinedione) (Vainshtein, 1954, 1955). methods and refinement, either of two error estimates was Its structure had been determined earlier from zonal X-ray employed. One assumed that s(Fo) ¼ 0.05jFoj for all re- diffraction data (Corey, 1938) so that the electron crystal- flections, generated by the maXus software (see below) lographic analysis intended to give a more accurate view (Mackay et al., 2001). Another assumed that s(Fo) ¼ of hydrogen atom positions that had only been estimated 0.071jFoj for reflections above a certain jFoj¼2.9 thresh- previously from reasonable bonding geometry. Although old and s(Fo) ¼ 0.21 for the weaker reflections below the some dynamical scattering perturbations to the intensities threshold, as recommended by Degeilh and Marsh (1959). were noted (chiefly by the effect on the overall tempera- During the refinements, each weighting scheme was evalu- ture factor), direct methods were eventually used to solve ated separately. the structure from the electron diffraction amplitudes, first The original zonal X-ray diffraction data obtained by by symbolic addition (Dorset, 1991) and later by use of Corey (1938) were also considered for direct methods the tangent formula (Dorset and McCourt, 1994). Some trials. These comprise 80 unique hk0 and h0l reflections, improvement of the bonding geometry was realized in the again in space group P21/a, where a ¼ 5.20, b ¼ 11.53, second study when restrained least squares refinement was c ¼ 3.97 A, b ¼ 83. attempted via dampened movement of atomic positions. In this work, thermal parameters were not refined. In this communication, modern structure solution and Crystal structure analysis refinement software were employed for a comparative ana- Zachariasen (1952) stated that a successful determination lysis of X-ray and electron diffraction amplitudes from di- of any crystal structure requires finding accurate phase val- ketopiperazine. As will be shown, direct methods yield ues for a small number (ca. 15%) of reflections, among basically the same heavy atom structure, regardless of data the strongest normalized amplitudes. After that, the struc- origin. Further, a careful refinement of the electron diffrac- ture can be completed usually in a straightforward manner tion data yields a reasonable model with perhaps a more by successive approximations, e.g. by Fourier refinement. accurate view of the hydrogen atom geometry than is pos- In ‘traditional’ probabilistic direct methods, often evaluated sible from X-ray data, supporting conclusions made in the for electron diffraction data sets (Dorset, 1995), one pre- original Russian work (Vainshtein, 1955). dicts the value w of the phase invariant sum, a linear combination of crystallographic phases for individual reflections within certain component index combinations Data and methods (Hauptman, 1972): w ¼ jh þ jk þ jhk, where h h1k1l1, k h2k2l2. (Shown is a three-phase invariant or tri- Intensity data from diketopiperazine plet. Four phase invariants or quartets are also useful as are seminvariant doublets and triplets that are defined by The collection of oblique texture electron diffraction data characteristics of a particular space group symmetry from diketopiperazine was described in the original paper (Hauptman, 1972).) After construction of a Wilson (1942) by Vainshtein (1955). They had been recorded on film. plot from the observed intensity data to obtain an overall The space group was determined to be P21/a, where temperature factor, the normalizedP structure factors are cal- a ¼ 5.20, b ¼ 11.45, c ¼ 3.97 A, b ¼ 81.9 . (In their con- 2 2 2 culated from jEhj ¼jFhj =eh fi .Theeh term weights version of Vainshtein’s original measurement in kX units i to A, Degeilh and Marsh (1959) assigned smaller standard reflections according to symmetry classes (e.g., see Gia- deviations than are normally found in electron diffraction covazzo (1998) p. 18). Prediction of w (which is either experiments. For the current study, it was assumed that all 0orp in the centrosymmetric case) depends on the normalized magnitudes of reflections at these Miller in- axial values have an 0.05 A error.) In all, there were 316 2 data reported, but three were estimated values for (0k0) dices, jEhj , and the associated concentration variable, reflections, unrecorded because of a ‘missing cone’ prob- 2ffiffiffiffi A ¼ p jEhEkEhþkj. In the preceding, fi are electron (or lem induced by the texture geometry (Vainshtein, 1964b). N For this analysis, these estimated (0k0) amplitudes were X-ray) scattering factors (Doyle and Turner, 1968) and N rejected to give a slightly smaller dataset with 313 reflec- is the number of atoms in the unit cell. Initial information tions. There were overlapping doublets for 25 measured for the phasing process can include a small number of reflections. The partitioning scheme described by Vainsh- phase values assigned to reflections with appropriate index tein (1955) was accepted; in any case the overlapped re- parity to define the unit cell origin (Rogers, 1980); simi- flections were mostly weak. As is often the case in auto- larly, a single reflection phase in some non-centrosym- mated electron crystallographic analyses, an estimated metric structures can be used to define the enantiomorph. 88 D. L. Dorset

Determination of unknown crystallographic phase values involves an optimization step, including enforcement of from starting phase values proceeds by solving a system map positivity, that is discussed by Bricogne and Gilmore of simultaneous equations (i.e., the phase invariant (1990). The FT of the map should also predict the close fit obs ME sums). Algebraic direct methods, such as the Sayre of jUhj to jUh j and the entropy maximization evalu- (1952) equation, behave in a similar fashion, since the ates this via a reduced c2 statistic, whose optimal value invariant relationships are generated by the convolution: should be unity provided that viable errors can be estab- ðhÞ P lished for the unitary structure factors. It is therefore bene- Fh ¼ FkFhk. (The q(h) term is a function of V k ficial to insert n algebraic phase terms into the basis set atomic scattering factors and V is the unit cell volume.) for other strong reflections in the data set. If there are n(c) The success of this phasing approach in electron crys- centrosymmetric phases (whose values are restricted to 0, tallography, especially in the guise of symbolic addition p) and n(a) non-centrosymmetric phases (whose values are (Karle and Karle, 1966) or with the tangent formula estimated by quadrant permutations: p=4; 3p=4) to be P inserted into the basis set, one then generates 2n(c),4n(a) whjEkjjEhkj sin ðjk þ jhkÞ possible phase combinations and obtain multiple basis Pkr tan jh ¼ sets, each one of which needs to be subjected to entropy whjEkjjEhkj cos ðjk þ jhkÞ maximization. kr TheP entropy optimization subjects the map entropy, (Karle and Hauptman, 1956) or the Sayre (1952) equation, S ¼ pi ln pi (where pi are normalized map pixel val- i is based on the observation that many of the strong struc- ues), to a constrained maximization in which constraints ture factor magnitudes often remain strong even after mul- are the unitary structure factors of the relevant basis set. It tiple scattering perturbations (Dorset, 2003). (The weight- is necessary to select the optimum phase solution from the ing term is often assumed to be wh ¼ 1.0 but can be multiple phase sets. Map density flatness, a quantity re- modified as discussed by Hauptman (1972).) This is lated to maximum entropy, has often been cited as a suita- equivalent to saying that the experimental Patterson func- ble FOM to recognize viable phase combinations (Luzzati tion still contains most of the information present in the et al., 1987; Sato, 1992). However, a likelihood measure is actual crystal autocorrelation function (Dorset, 2003). preferred for this identification (Gilmore et al., 1990; Strict adherence to the kinematical scattering limit is not Sayre, 1993). One can define the likelihood for the centro- required, therefore, and a few erroneous phase estimates symmetric case: can be tolerated, although, in some cases, multiple scatter- () ing can cause the appearance of spuriously strong reflec- obs obs 2 ME 2 2 jUkj 1 ðjUK j Þ þðjUK jÞ tions to lead the phase determination astray. This condi- LK ¼ 2 exp 2 pð2eK S þ K Þ 2 2eK S þ K tion had been applied in another way in a demonstration ! that direct methods could be derived from the Patterson obs ME jUK j jU j function (Hauptman and Karle, 1962).) Other causes for cosh K : 2e S þ 2 unsuccessful determinations include incomplete data sets K K leading to poor vector connectivity via Miller indices, a (In this expression, S measures the statistical dispersion of problem with some electron diffraction intensity sets the distribution of observed moduli (see Bricogne and Gil- (Dorset, 1995). more (1990).) This is compared to the null estimate L0 , Another way of approaching the phase problem that is K L not restricted by vector connectivity of reflections in the ME K evaluated for jUK j¼0 with the quantity: LK ¼ log 0 , data set (i.e. to construct invariant sums) is to simply carry P LK out a series of phase permutations and combinations for via the global log likelihood gain LLG ¼ LK . Some- K an identified number n of strong reflections. This simple times an FOM combining the LLG and the entropy value device was used effectively by Robertson and White S is evaluated as NS þ LLG, but the scale factor N can (1945) in the analysis of the coronene structure well be- be difficult to estimate. fore direct methods were invented. In the approach utiliz- As is apparent, a large number of trial phases (n(a) þ ing maximum entropy and likelihood (Bricogne and Gil- n(c)) can, by permutation and combination, lead to a very more, 1990; Gilmore et al., 1990) to evaluate the phasing large number of phase sets, even if the structure is centro- trials, the computer program MICE (Maximum Entropy in symmetric. For this reason various error-correcting codes a Crystallographic Environment) (Gilmore et al., 1990) are employed to reduce the number of trials (Gilmore was employed. It is first assumed that there are a small et al., 1999), accepting that a very small number of phases number of reflections in a basis set {H} whose phases are might be in error. For example, a Hadamard-Hamming known a priori. (These can include origin-defining phases code (Reed, 1954) can be evaluated, in which only 16 as before.) The task then is to find the phases of reflec- phase trials are evaluated instead of 28 256 permutations tions in unknown set {K} from the known set {H}. Defin- for 8 reflections. (This use of coding theory had been dis- ing the unitary structureffiffiffiffi factor in space group P1, covered independently by Woolfson (1954).) The best obs obs p jUhj ¼jEhj = N, the task is to extrapolate the known phase set has two or fewer errors. Similarly, a Nordstro¨m- information into the unknown set. If the number of start- Robinson (1967) code can be explored, permitting 16 ing reflections is small, the phasing power of a maximum phase terms to be evaluated. Instead of exploring 216 entropy map qME(x) is rather weak when its Fourier trans- 65 536 permutations, only 256 trails need be considered form (FT) is calculated. Entropy maximization of the map with a maximum of 4 errors in the best solution. Finally, a Crystal structure of diketopiperazine 89

Golay (1949) code will generate 4096 phase combinations for 24 reflections instead of 224 16 777 216 trials with a maximum number of 4 incorrect assignments. From the first set of reflections assigned phases by direct methods, a trial electrostatic potential map was generated to look for atomic positions. Optimally, a Fourier refinement could follow to improve the structural model. Initially, trial atom positions would be used for a partial structure factor calculation, to generate a new phase set and thus a new map, until allP atoms are found andP the crystal- calc obs calc lographic residual, R ¼ jjFh jkj Fh jj= jFh j,is suitably minimized.

Fig. 1. Phase determination of X-ray zonal data from Corey (1938). Crystallographic software In line with the previous section, two types of direct phas- Apparently, no ab initio phasing technique had been ing approaches were attempted. In one case, via the used to determine the structure directly from the extensive maXus software package (Mackay et al., 2001), phase in- X-ray data published by Degeilh and Marsh (1959). They variants and seminvariants (Hauptman, 1972; Giacovazzo, relied on Corey’s heavy atom coordinates as a starting 1998) were analyzed, mainly using the computer program point for refinement. With the program sequence in SIR97 (Altomare et al., 1999). Likely structural solutions maXus, the solution and refinement was virtually auto- were recognized by a combined figure of merit. As stated matic. The space group was correctly identified after the above, a radically different approach employed maximum false indication of a non-centrosymmetric structure from entropy and likelihood methods (Bricogne and Gilmore, the intensity statistics was rejected. Structure solution pro- 1990) via the computer program MICE (Gilmore et al., ceeded automatically via SIR97 finding the positions of 1990). After definition of the crystallographic origin by four heavy atoms. The chemical assignments, however, assigning phase values to a small group of reflections with were incorrect for three heavy atoms, requiring that these symmetry-specific Miller index parity (Rogers, 1980; Gia- should be readjusted. Because of the favorable ratio (Ladd covazzo, 1998), a multiple solution regime was carried out and Palmer, 1993) of observed intensity data to 37 para- via permutation of phase signs for reflections within a meters (14.6 for the most intense reflections), full matrix small subset (e.g. 16), assuming that they could be reason- least squares refinements was carried out. After least ably predicted by an error correcting code (e.g., Nord- squares refinement with isotropic temperature factors, fol- stro¨m-Robinson – see above). From the parent set, exten- lowed by refinement with anisotropic temperature factors, sion to a larger reflection set was made after entropy theoretical hydrogen atoms were added to the tetrahedral maximization of associated Fourier maps. The best phase carbon and the nitrogen atoms. After a final refinement solutions were identified by a maximum likelihood figure cycle, R ¼ 0.0785 for the 482 most intense reflections and of merit (see above). R ¼ 0.0830 for the 542 reflections used for the refinement. Structure refinement of atomic positions and thermal This most successful refinement utilized the reflection parameters was carried out via block diagonal (using the program LSQ within maXus) or full-matrix (calling up the program SHELXL (Sheldrick, 2001) from maXus) least squares. Unlike the previous attempts on the electron diffraction data (Dorset and McCourt, 1994), no restraints were placed on atomic movement, nor were the temperature factors fixed.

Results

Analysis of X-ray data Corey (1938) had originally determined the structure of diketopiperazine from two zonal X-ray patterns by model building. Phasing the 80 combined h0l and hk0 data by MICE assigns values to 58 reflections. There were several solutions where the structure could be seen embedded in the density profiles of the ensuing maps (Fig. 1) even though there were extraneous peaks. A representative solution would have, for example, 17 phase errors for 58 re- Fig. 2. Molecular packing in structures determined from 3-D electron flections. diffraction and X-ray diffraction data by maXus. 90 D. L. Dorset

Table 1. Results of X-ray diffraction study of diketopiperazine.

Atom x=ay=bz=cUiso Biso O 0.3319(4) 0.1328(2) 0.0955(5) 0.0361(7) 2.8 C1 0.1822(4) 0.0697(2) 0.7168(7) 0.0238(7) 1.9 C2 0.0513(5) 0.1230(2) 0.5158(6) 0.0262(7) 2.1 N 0.2204(4) 0.0430(2) 0.3096(5) 0.0279(7) 2.2 H1 0.372 0.078 0.187 0.033 2.6 H2 0.002 0.179 0.362 0.031 2.4 H3 0.154 0.160 0.667 0.031 2.4

Atom U11 U22 U33 U23 U13 U12

O 0.046(1) 0.029(1) 0.028(1) 0.0010(7) ––0.0158(8) 0.0026(7) C1 0.026(1) 0.024(1) 0.020(1) ––0.0006(8) ––0.0020(8) ––0.0012(8) C2 0.030(1) 0.022(1) 0.023(1) ––0.0037(8) ––0.0088(9) 0.0016(8) N 0.035(1) 0.026(1) 0.018(1) ––0.0023(7) ––0.0096(8) 0.0004(8)

Bond distances D & Ma Bond angles D & Ma

N––C1 1.323(3) 1.325 C1––N––C2 125.7(2) 126.0 N––C2 1.452(3) 1.449 N––C2––C1 115.3(2) 115.1 C1––C2 1.496(3) 1.499 O––C1––N 122.4(2) 122.6 O––C1 1.241(3) 1.239 O––C1––C2 118.6(2) 118.5 N––H1 0.96 0.86 N––C1––C2 118.9(2) 118.9 C2––H2 0.96 0.95 C1––N––H1 119.9 123.0 C2––H3 0.96 0.93 C2––N––H2 114.4 111.0 N––C2––H3 106.8 113.0 C1––C2––H3 109.5 104.0 N––C2––H2 106.3 108.0 C1––C2––H2 109.3 107.0 H3––C2––H2 109.5 109.0 a: values from Degeilh and Marsh, 1959 weights derived by Degeilh and Marsh (1959); if the solution was found at the 18th ranked likelihood value. maXus generated weights were used, the agreement be- Match of the map density profiles to the known structure tween calculated and observed intensities was somewhat are shown in Fig. 3. worse. The molecular packing model is shown in Fig. 2. Analysis of the electron diffraction data (Vainshtein, The atomic coordinates of the heavy atoms (Table 1) 1955) with maXus did not proceed automatically and re- were within 0.004 A of the positions found originally quired sequential application of successive calculation (Degeilh and Marsh, 1959). Hydrogen positions were steps. The space group determination, nevertheless, indi- within 0.1 A. (Note that the equivalent isotropic thermal parameters were in the range B ¼ 1.9 to 2.9 A2, somewhat low for typical organics (Ladd and Palmer, 1993).) Although hydrogen positions had been located in difference maps in the original refinement (Degeilh and Marsh, 1959), no direct evidence for them was found. Geometric parameters of our model also agreed very closely to those found earlier (Table 1). A more complete analysis of anisotropic thermal vibration had been carried out for this structure by Lonsdale (1951), but this aspect is beyond the scope of the current study.

Analysis of electron diffraction data If MICE was used to determine the structure of diketopiperazine from all of the published electron diffraction data (Vainshtein, 1955), a correct structure was found at the phase combination giving the second highest likelihood estimate. If the unmeasured (0k0) data were removed, the Fig. 3. Phase determination of 3-D electron diffraction data by MICE. Crystal structure of diketopiperazine 91

Table 2. Results of electron crystallographic study of diketopiperazine.

Atom x=ay=bz=cUiso Biso O 0.338(3) 0.131(1) 0.393(3) 0.021(2) 1.7 C1 0.185(2) 0.071(1) 0.207(3) 0.007(2) 0.6 C2 0.049(2) 0.121(1) 0.010(3) 0.012(3) 1.0 N 0.223(2) 0.046(1) 0.194(3) 0.019(3) 1.5 H1 0.348(10) 0.083(6) 0.366(13) 0.016(12) 1.3 H2 0.003(10) 0.183(6) 0.150(15) 0.026(15) 2.0 H3 0.158(11) 0.163(6) 0.205(15) 0.020(13) 1.6

Bond lengths Va Bond angles Va

C1––C2 1.47(2) 1.44 C2––C1––O 121.8(12) 122.7 C1––N 1.36(2) 1.37 C2––C1––N 118.5(10) 117.9 C1––O 1.22(2) 1.22 O––C1––N 119.6(11) 119.4 C2––N 1.41(2) 1.40 C1––C2––N 119.1(11) 119.8 N––H1 0.97(6) 0.98 C1––N––C2 122.3(10) 122.2 C2––H2 1.13(6) 1.11 C1––C2––H2 105.0(30) 110.1 C2––H3 1.00(6) 1.09 C1––C2––H3 111.0(30) 105.0 N––C2––H2 108.0(30) 107.0 N––C2––H3 105.0(30) 103.1 H2––C2––H3 109.0(30) 111.5 C1––N––H1 119.0(40) 119.2 C2––N––H1 117.0(40) 116.7 a: values calculated from Vainshteins (1955) coordinates cated that the structure should be centrosymmetric and pared in Table 2 to those from the previous electron dif- correctly identified P21/a. Structure analysis via SIR97 fraction determination. Average heavy atom positions are was successful, finding the positions of four heavy atoms within 0.02 A of values derived from Vainshtein’s (1955) and two hydrogens. Again, the chemical identity of three coordinates and hydrogen position within 0.08 A. heavy atoms required reassignment. The initial packing model identified in this determination was equivalent to the one found from the X-ray data, except that the origin Discussion was moved to another (permissible) location (Hauptman, 1972; Hahn, 1995), i.e., from the center of symmetry at Automated crystal structure analysis from electron diffrac- 1 (0,0,0) to the one at (0,0, =2). Since the ratio of useful data tion data is known to be possible (Dorset, 1995), given to 29 refineable parameters was somewhat low (9.69), that pains are taken to collect the best possible data. Even block diagonal isotropic least squares refinement with jFj with the modest electron accelerating voltages used in values was carried out via the program LSQ. Unlike in early Russian work (e.g., 40 to 60 kV), it is instructive to previous work (Dorset and McCourt, 1994), no restrictions find how useful intensity data can result from careful ex- were placed on the temperature factor, nor was a damping periments on hand-made equipment, even when recorded factor placed on atomic movement. After the first applica- on photographic film (but also with knowledge of non-lin- tion of LSQ (8 cycles), two other atomic positions were ear dose response (Vainshtein, 1964b)). For light atom found in the peak list. One of these corresponded closely structures, there is sufficient applicability of the two-beam to a hydrogen position on the CH2 moiety and was as- dynamical or even the kinematical model, facilitated by signed as such; the other was excluded. A second isotro- the averaging over various orientations of the crystallites pic application of LSQ (8 cycles) resulted in R ¼ 0.224 for illuminated over a large area (e.g., 1 mm diameter). More the 281 largest reflections or 0.283 for the entire set of recent work on another organic material containing heavy 313 reflections. No further improvement was noted in atoms indicates that the same improvement can be realized further refinement attempts. In addition, it was not possi- (Dorset, 2007) from precession experiments (Vincent and ble to carry out anisotropic refinement because one atom Midgley, 1994) on single crystals. was assigned physically meaningless values. In previous use of the tangent formula to solve this The coordinates of the refined structure are listed in structure (Dorset and McCourt, 1994) there was some dif- Table 2. Even though the initial Wilson (1942) plot indi- ficulty in identifying the correct phase set when NQEST cated an overall temperature factor of B ¼ 0.0 A2, the iso- (Detitta et al., 1975) was used as a figure of merit. This tropic temperature factors (Table 2), although somewhat FOM relies on accurate estimates of negative four phase low, are not too different from the ones found in the X-ray invariants involving weak normalized structure factor mag- analysis. The experimental bonding parameters are com- nitudes. With the combined figure of merit (CFOM)in 92 D. L. Dorset

SIR97 (Altomare et al., 1999) identification of the correct the original Vainshtein (1955) model. The refined electron structure is easier even though the negative quartet esti- crystallographic heavy atom positions differ from the re- mates form a component of the CFOM. fined X-ray position by 0.05 A whereas the hydrogen po- Least squares refinement of organic structures is also sitions differ by 0.15 A. possible with electron diffraction data beyond the expecta- In support of the claim originally made by Vainshtein tions of earlier, more cautious, attempts. Early studies had (1964a), the biggest success of the electron diffraction applied damping factors to restrain atom movements study is its facility to find hydrogen atom positions. Exam- (Dorset and McCourt, 1994; Dorset and Gilmore, 2000) ination of electrostatic potential maps during the direct and have neglected to refine thermal parameters. With structure solution from electron diffraction amplitudes im- block diagonal refinements, these restraints may not be mediately located two of the three positions so that SIR97 necessary and thermal parameters can be included. included them in the initial atom list. The third hydrogen How good is the direct electron crystallographic analy- was found readily during least squares refinement. Map sis of diketopiperazine? We have, first of all, shown that density revealing their presence is shown in Fig. 4. Aver- the original X-ray structural parameters can be closely ap- age C––H bond lengths in the CH2 group at 1.06 A is proximated in an independent determination. From the X- close to an ideal value of 1.09 A (Ladd and Palmer, ray determination, the C1––C2 bond length value of 1993); the measured N––H bond length of 0.97 A is also 1.50 A matches an independent X-ray study by Kartha near the ideal value (1.01 A) (Ladd and Palmer, 1993). et al. (1981) on the formic acid complex with this mole- These results are also consistent with previous electron cule and is somewhat shorter than the value found for two crystallographic determinations of C––H (Vainshtein and forms of glycine (1.52–1.53 A) (Itaka, 1958), or the stand- Pinsker, 1950; Lobachev, 1954; Vainshtein et al., 1958) ard C(sp4)––C(sp3) bond length (Ladd and Palmer, 1993). and N––H (Kuwabara, 1959; Lobachev and Vainshtein, Refined electron crystallographic measurements yield a va- 1961; Dvoryankin and Vainshtein, 1960, 1962) values. lue of 1.47 A, longer by 0.03 A than the value originally It is much more difficult to find hydrogen positions in quoted by Vainshtein (1955). The carbonyl C¼O distance electron density maps after an X-ray determination. From has been found to be 1.24 A from the X-ray analysis of difference maps, Degeilh and Marsh (1959) reported va- diketopiperazine but is 0.01 A longer in the formic acid lues of 0.96 A for the C––H distances of the CH2 moiety complex. The electron diffraction bond distance is shorter: and a very short 0.86 A for the N––H distance. In our 1.22 A, but still consistent with a typical carbonyl double direct structure determination, these positions were not bond (Ladd and Palmer, 1993). The CH2 carbon bond link identified by the SIR97 program from phase generated to nitrogen measures 1.45 A in the two X-ray analyses. maps, so that we used theoretical positions to end up with Values for glycine (Itaka, 1958) are slightly longer. Again, a nearly equivalent fit to the diffraction data. In the X-ray the electron diffraction result is somewhat shorter study of the formic acid complex, Kartha et al. (1981) cite (1.41 A). For the bond between the carbonyl carbon and values 0.92 and 0.99 A for the C––H lengths and 0.96 A nitrogen, the X-ray studies of diketopiperazine and its for- for the N––H length, again calculated from positions lo- mic acid complex agree (1.32 A). The refined electron dif- cated in difference maps. fraction bond length, although shorter than the value re- Although anisotropic refinement of temperature factors ported by Vainshtein (1955) is 1.36 A; closer to the was not possible with the electron diffraction data, isotro- standard C(sp3)––N(sp4) bond length of 1.40 A than the X- pic values are better than anticipated from the overall B ray studies. Some minor discrepancies are also seen for value derived from the Wilson (1942) plot. The average the major bond angles. The two X-ray structures agree value for the heavy atoms is 1.2 A2 vs. 2.2 A2 from the closely whereas the electron crystallographic results devi- X-ray determination. For the hydrogen positions, the elec- 2 ates from the X-ray determination, but not so much as did tron diffraction refinement yields BH ¼ 1.6 A . The average value quoted for the X-ray analysis of the formic acid complex (Kartha et al., 1981) is 4.7 A2. Again, dynamical scattering may have some effect on these values from the electron diffraction analyses but at least BH > BC, N, O.

Acknowledgments. I am grateful for helpful comments from Prof. Car- melo Giacovazzo on the use of SIR97 and to Prof. C. J. Gilmore for making the program MICE available for our use.

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