Prediction of crystal structures of organic molecules

Journal of

MOLECULAR STRUCTURE

ELSEVIER

Journal of Molecular

Structure 474 (1999) 13-23

Prediction of crystal structures of organic molecules Detlef W.M. Hofmanna3*, Thomas Lengauera3 b “Institute,fr,r Algorithms and Scientific Computing, German National Research Centre for Information Technology (GMD-SCAI), Birlinghoven, 53754 Sankt Augustin. Germany bDepartment of Computer Science, University of Bonn. RomerstruJe 164. 531 I7 Bonn, Germany Received 23 November 1997; received in revised form 15 April 1998; accepted

SchloJ

15 April 1998

Abstract We have developed a new algorithm for crystal structure prediction that takes a conformer of an organic molecule and produces a number of candidates of crystal structures for this molecule, together with a score that roughly approximates the energy of the respective crystal structure. The prediction method takes steric aspects (dense packing) as well as chemical interactions into account. The algorithm differs from existing methods in three aspects. First, we analyse a single molecule rather than a collection of identical molecules. The analysis yields candidates for symmetry operations that are suitable for making the crystal. Each of the candidates is generated by a chemical interaction between two versions of the molecule. Second, we model space discretely, currently as a mesh with size 1 A. Third, the scoring function representing energy is derived statistically from known crystal structures and tabulated. Our program FlexCryst computes a list of crystal structures ranked according to our scoring function. The new algorithm is currently implemented for the four space groups PI, Pi, P2,, and P2,2,2,. The three latter space groups are widespread in nature. The algorithm computes structural models of acceptable quality and shows excellent time performance. 0 1998 Elsevier Science B.V. All rights reserved. Keywords:

Crystal structure prediction;

Statistical potentials;

1. Introduction One of the most fundamental unsolved problems in chemistry is predicting how a molecule will pack in the solid state solely on the basis of its molecular structure [ 11. The problem is of high interest, because the crystal packing affects many physical properties of the solid, such as solubility, bioavailability and colour. But ab initio crystal structure prediction is still considered a long-term goal [2]. A formidable obstacle to such a prediction is the existence of a large number of local minima in the high dimensional potential energy surface of the crystal, which makes it * Corresponding

author; e-mail: [email protected].

Discrete molecular modelling

extremely difficult to locate the most stable structure [3]. Sometimes thousands of quite different local minima can fall within a narrow energy range (40 kJ/mole), as has been witnessed for monosaccharides [4]. This fact is reflected by the phenomenon of polymorphism [5] in nature, where many different crystal structures can often be found for one molecule. Even so, sometimes the problem can be rendered feasible. Sometimes the X-ray powder diffraction is available, but the structure cannot be solved from the diffraction data due to the phase problem. The phase problem implies that the cell parameters (the length of the translation vectors, the angles between them, and the space group) can be determined, but the

0022.2860/98/$ - see front matter 0 1998 Elsevier Science B.V. All rights reserved PII: SOO22-2860(98)00556-O

14

2. The FlexCryst

Fig

I.

Model of a hydrogen bond. The two interactions centres H

and 0 lie on the interaction surface of the other unit forming an interaction.

orientation of the molecule inside the cell cannot be calculated directly from the diffraction. In this case the diffraction of proposed crystal structures can be compared with the experimental data. Thus crystal prediction is satisfactory for this purpose, if the observed crystal structure is among the highest ranking solutions found by the algorithm. These structures can be used as good starting points for refinement procedures based on diffraction comparison and/or minimizing procedures with highly accurate intermolecular forces [6]. An overview on various approaches for the refinement procedures can be found in several textbooks [7, 81. Most approaches reported in the literature proceed in two steps. In a first step, a more or less random structure is generated using a Monte-Carlo procedure. In the second step, the initial structure is refined by energy minimization using a steepest descent [9] or a simulated annealing [IO] procedure. The existing algorithms require extensive amounts of runtime because the initially chosen structure tends to be far away from a local energy minimum. In addition, the configuration space is not sampled globally, so that the energy minimization ends up in a local minimum that cannot be further characterized and will not, in general, be the global minimum. We have developed a new program, FlexCryst, to solve the ab initio problem, but the program can be used also for interpreting powder diffraction. In this case the additional information from the diffraction can be exploited to improve the performance and the quality of the results. The program FlexCryst can be used via Internet calling the address http:// cartan.gmd.de/compchem/hofmann/home.html.

algorithm

Currently, the program can handle the four space grous Pl, Pi, P2,, and P2,2,2,. Some adaption to each space group has to take place, since different space groups are determined by different sets of symmetry operations. In the worst case of space group Pi, four independent symmetry elements (three translations and one inversion centre) define the crystal structure. Our program requires the longest computation time for this space group with 12 free variables. For the other space groups the procedure can omit a few steps and the program shows better performance. The algorithm presupposes that the molecule is rigid and that the conformation is known. This is justified for pigments, which are often fixed by the enlarged rr systems, and for steroids, which are fixed by the high connectivity of the ring systems. At the moment the program handles only crystal structures with one molecule per asymmetric unit. Fortunately most crystals observed in nature meet this condition. An extension of the algorithm to several molecules per asymmetric unit increases the search space by six degrees of freedom per molecule. The corresponding variables determine the translation of the molecule and its orientation in the asymmetric unit. In the following we give a description of the FlexCryst algorithm. A figurative description of the procedure has been published [ 111. Step 1. The input is a 3D conformation of an organic molecule in mo12 format. Step 2. The molecule is automatically supplemented with hydrogen atoms. For this step we use SYBYL 1121. Step 3. The molecule is searched for active centres by using the program Flex [ 13-161. We calculate potential interaction surfaces around these centres. If an interaction between two groups is formed, the interaction centre of the first group has to lie on the interaction surface of the second group and vice versa (Fig. 1). The geometry of the interaction surfaces is tailored to the interaction to be modelled. A complete description and an on-line version of Flex is available via Internet [http:// cartan.gmd.de/flex-bin/Flex]. Step 4. The interaction centres and the interaction

D. W.M. Hofmann, T. Lengauer / Journal of’Molecular Structure 474 (1999) 13-23 Table I Flowchart

for the construction

15

of the crystals in the various space groups.

Pl

fi

p2,

p21212,

Search for translation Energy constraint Add translation Add translation Add translation

Search for inversion Energy constraint Search for translation Energy constraint Add translation Add translation Add translation

Search for screw axis Energy constraint Search for translation Angle constraint Add translation Energy constraint Add translation

Search for screw axis Angle constraint Add screw axis

surfaces are discretized, i.e. the molecule is placed into a 3D grid and each interaction point and interaction centre is shifted to the nearest grid point. In order to trade off calculation time and accuracy our mesh size is 1 A. Larger mesh sizes significantly reduce the number of interaction points, which reduces the runtime in the subsequent steps, but decreases the accuracy of the prediction. In a preliminary investigation we analysed the accuracy needed for starting points for Rietveld refinement. For this purpose we used the DBWS module of the Cer-ius’ package. In our experience the structure has to be accurate to 0.3 A. A detailed description of our measure of accuracy will be found in step 9. To use our program for crystal structure determination, therefore, it becomes necessary to improve the structures. At present we improve the generated structures by simulated annealing and refine them by the Rietveld procedure. Work is in progress to omit the simulated annealing stage. Originally the concept of Rietveld refinement assumes that the angles and unit cell vectors are already quite exactly known by indexing the diffraction data. This is not the case for the computationally generated crystal structures. One approach to bypass the deficiency is described in Ref. [17]. The peaks of the diffraction pattern are broadened. This makes the Rietveld procedure for larger differences between experimental unit cell and starting cell still successful. Step 5. Possible crystal structures are generated. This step differs for each space group. The computation of this step can be decomposed into modules that perform tasks such as searching for a certain symmetry operation, adding an additional symmetry element, applying the energy constraint,

and applying the angle constraint 1. ’ The detailed procedure for each space group is depicted in Table 1. (a) Search for symmetry operation: proper symmetry operations for crystal structures are determined, analysing interaction centres and interaction points found in step 1. Likely symmetry operations are those which create strong molecular interactions between molecules. The interaction centres and interaction points are just chosen for this purpose. Always if an interaction point falls onto an interaction centre of a nightbouring molecule an interaction is formed. Therefore we are searching for a symmetry element which maps one or more points p onto interaction centres c. Currently this step is implemented for the translation, the inversion centre, and the twofold screw-axis. A detailed description can be found in the Appendix 1. (b) Add symmetry operation: this module adds an additional symmetry element to a structure that is already partially defined by a number of symmetry elements. In this way the crystals are constructed step by step. The number of symmetry elements necessary to define a crystal uniquely depends on the space group and varies from two (P2,2,2{) up to four (pi). (c) Energy constraint: to apply the energy constraint, first the symmetry elements are applied to the molecule, mapping it onto images.

’In an earlier description we applied in addition a triangle constraint to avoid collapsing of the crystal. Such structures are screed out faster by the density constraint and we do not use this constraint any more.

D. W.M. Hofmunn, T. Lrnpw

/ Jourrml of Molecular Structure 474 (1999) 13-23

Then the interaction energies between the images and the reference molecule are evaluated and summed. The structures are sorted according to their energy and only the highest ranking structures are retained. For scoring structures we use the widespread atom-pair energy approach. The interaction energy E,i,,,(r,J) between two molecules I and J is assumed to be the sum of atom-pair energies &Xn(r,G).

In contrast to most other force fields, the atompair potentials used were derived by analysing known crystal structures. From these, we obtain probabilities for the contacts between the different atoms. The energy function is derived from these probabilities by the inverse Boltzmann equation. Eatom(r,i,j) = -0.68

kcal/mole + NLkTlog

P(4.0 Ah’ if y ~ 4 A

Step 7. The crystal structures are sorted according to our scoring function. Step 8. The energy constraint is applied to the crystal structures. The 2000 highest ranking structures are retained. Step 9. The crystal structures are compared with the experimental structure. Comparison of crystal structures is difficult, because an infinite number of representations for the unit cell are possible for each crystal structure. Various approaches to checking the similarity of two cells have been published. Some are based on comparison of the simulated diffraction pattern [ 171. In others the two unit cells are first normalized [23] and the square deviation between the atoms in the two unit cells is calculated [24, 251. For our purposes, we propose a third method that exploits the fact that we are always dealing with the same molecule, which is rigid and fixed in space. The translation vectors are compared first, and the origins of the cell are compared next. The similarity of the translation vectors is checked in the same way for all space groups. Assuming one base B defined by three vectors b,, bz, and b3 B = (bl>b>bd

P(r)4.0 A2

I

0 kcahole

if r > 4 .&

The derivation has been described in detail in a previous publication [ 181. This deductive approach was introduced first for protein structure prediction 119-211 and, later, has been justified theoretically [22]. The atom-pair energies for the stances Y are tabulated to avoid timeconsuming recalculation. (d) Angle constraint: if the space group is not triclinic the angles between the axis are not arbitrary. They have to be 90% or 60”. Step 6. The atom density constraint is applied to the crystal structure. This restraint requires the density of atoms per A’ to be in the range from 0.7 to 1.3. It is well known that the mass density of organic molecule crystals varies only little. We found that using the number of atoms per unit volume rather than the mass leads to an even higher correlation between density and volume.

the three vectors b I,, b h. and b /3 of the second cell are expressed as linear combinations of ti of the vectors of the first base B. b’; = Bti

The distance Yi between the vectors bl, and the nearest grid point Pi of the grid [P(B)} defined by B is given by r, = IIB(lt; + 0.51 - UII To compare the origins we have to distinguish between the space groups. The grid points {P(B)] defined by the translation vectors are always a subset of the grid (S} spanned by the possible . origms, {P(B)] C ISI e.g. in space group Pl the crystal structure

is not

D. WM.

Hofmann,

T. kngauer/

0

Journd

----i);2 7------

??

IIs(lrorigin

+

o.5J

-

13-23

17

a

0

index for the space group Pl in a 2D projection

influenced at all by the choice of the origin, in space group Pi and P2,2,2, the basis vectors of the grid {S} are just half of the translation vectors b, and in space group P2, two vectors are half of the translations and the third dimension is continuous. The distance of the origin to the next possible grid point of the superset is calculated in the same way as before.

rorjgin =

474 (1999)

??

rl?

;’

Fig. 2. Similarity

Structure

0

0

b;

of Molecular

toriginll

As similarity index we choose the maximum four distances.

2 illustrates the similarity index for the case of space group PI. One cell is spanned up by the vectors b, and bZ, the other by the vectors b/, and b 12.The formula given above rapidly calculates the distance rl and r, between the vectors and the nearest grid point (balls) of the grid B. The maximum of these distances is our similarity index. Step 10. The internal description of the crystal structures by symmetry operations is converted to the common description of crystals by unit cell vectors, angles and fractional coordinates. A detailed description is given in Appendix 2

of the 3. Results

s = max] r),

t-2, t-3, rorigin

I

For our calculations we consider structures to be 0 similar if s is less than 1.8 A. This is the distance of two mesh points about the edges of the cube. Fig. Table 2 The results for the different space groups Space group

#free variables

#tests

#hits

Percentage

Time

PI

9 9 IO 12

131 98 100 95

128 94 73 66

98% 96% 73% 69%

1.13 34.5 51.8 12.7

p2,2,21 P2I Pi

For validation we extracted about 100 experimental structures from the Cambridge Structure Database (CSD) [26] for each implemented space group. We selected the alphabetically first organic crystals containing only the elements H, C, N, 0, F, P, S, and Cl. The crystals were required to contain only one molecule per asymmetric unit. The molecular data were input to the program FlexCryst. The output of the program, 2000 crystal structures for each of the 100 molecules, was compared with the experimental structure stored in the CSD. We were interested in two different aspects of the quality of our results. We first investigated whether

D. W.M. H@irmnn, T. Lengauer / Jo~tmal of Molecular

Structure 474 (1999) 13-23

150

loo5‘ kii iii t” 50 -

OL 0

Fig. 3. Rank of experimental

100

200

rank of similar structure

structures

among the proposed

the experimental structure was among the proposed crystal structures at all. The results are presented in Table 2. The first column shows the space group. The space group Pl is the simplest, and the one most extensively used for further developments of the program. The other three space groups are often found in nature and are most important for practical applications. Fortunately 75% of all observed crystals are described by just five space groups [27]. This suggests one should restrict future extensions of the program to the space groups P2,/c (35%), Pi (19%), P2,212, (9%), C2/c (7%), and P2, (6%). The second column contains the number of free variables that uniquely determine the unit cell. This number ranges from 9 (Pl) to 12 (Pi), if we consider only one rigid molecule in the asymmetric unit. In the column ‘#tests’ we report the number of structures extracted from the CSD. In the case of PI we used all available structures, while for the other groups we limited our sets to 100 structures. Some of these structures were flagged erroneous and we dropped them manually before the validation. We observed about 50 structures in space group Pl which in reality belonged to space group Pi. A few structures contained metals, but the structure were not marked as organometalics. We report such problems as soon as possible to CSD and the structures are corrected within a few days.

300

> IO

structures for space group PI.

The column ‘#hits’ gives the number of experimental structures found among the proposed crystals. The next column gives the same result in percentages. The last column shows the average runtime per molecule for the particular space group. This time varies from 2 min up to 1 h on a SUNTMULTRATM’ workstation. The runtime rises significantly with the number of free variables. The crystals of space group P2, produce a low rate of hits compared with the other groups. This results from the construction of the crystals. The actual implementation requires the screw-axis to be a leading symmetry element of the crystal but, for some crystals, the most important packing pattern incorporates a translation and the procedure fails. To find such crystals our construction procedure has to be extended. The sequence of first determining the screw-axis and then the translation has to be reversed. Secondly, we looked for the rank of the experimental structure among our ranked list of structures. As expected, the best results were achieved for space group Pl (Fig. 3). For this group all experimental structures are found among the first 100 structures, most of them (111 of 129) among the first 10 structures. With increasing number of free variables to be determined this pattern becomes more and more diffuse. For the space group P2,2,2, most of the

D. WM. HoJ%mm, T. Lengctuer/ Journal of Molecular

Structure 474 (1999) 13-23

19

60

40 6 5 B .E 20

0

rId,

I

I.

100

II,

I

200

I_

300

)O

rank of similar structure Fig. 4. Rank of experimental

structures

among the proposed

structures (55) are still found among the 10 highest ranking candidates, but a few of them (11) occupy a rank 400 or greater (Fig. 4). Some of the outliers are caused by an incorrect automatic addition of hydrogens. Since it is not always possible to determine experimentally the position of the hydrogens, they

structures for space group P2,2,2,

are sometimes omitted in the CSD. These structures were supplemented automatically with help of the program SYBYL [12]. At this step the crystal structure is not included and, therefore, sometimes the orientation of the hydrogens is essentially random. During the construction process of the crystals these

“”

40

111

I

I

100

200

300

5 IO

rank of similar structure Fig. 5. Rank of experimental

structures

among the proposed structures

for space group P2,

20

D.W.M. Hnfmann, T. Lmprr

of Molecular

/ Journal

Structure 474 (1999) 13-23

1

I

v

I,

100

200

rank of similar structure Fig. 6. Rank of experimental

structures

among the proposed

supplemented H-atoms can cause bad contacts. The distribution for the space group P2, (Fig. 5) is similar to that for P212,2,. The lowest rate of ‘hits’ was obtained for space group Pi (Fig. 6). Many of the

.

.

??

? ?

.

??

.

??

??

??

.

.

??

??

Fig. 7. Molecule surrounded interaction centres (balls).

.

? ?

’

??

.

.

??

??

??

.

??

??

??

.

.

’

points (points)

for space group Pi

crystals (33) are still found among the 10 highest ranking candidates. This reflects the large number of degrees of freedom.

??

??

by interaction

structures

>400

4. Conclusions

. ??

300

and

We have presented a discrete algorithm that detects the experimentally observed crystal structure of organic molecules among the computed candidates. Almost always, the experimental structure is found for the simple case of PI with one molecule in the unit cell, and for the space groups Pi with two and P2,2,2, with four molecules in the unit cell. For the space group P2r with two molecules per unit cell, the structure is detected in a large percentage of cases and the percentage might increase further as the program develops. The program is several orders of magnitude faster than existing algorithms. Three ingredients are essential for this efficiency: analysing the intermolecular interaction as a proprocessing step, using a discrete configuration space, and reducing the energy calculation to looking up a simple table.

D. WM. Hofinann. T. Lengauer / Journd of Moleculm

Acknowledgements The authors would like to thank Dr P. Erk (BASF) and W.D.S. Motherwell (CSD) for fruitful discussions. Furthermore, we are grateful to C. Lemmen, B. Kramer, C. Oligschleger, M. Rarey, and S. Weting for helpful comments on the paper. This research was performed as part of GMD’s contribution to HLRZ (High Performance Computing Centre).

Structure 474

following

(1999)

equations

13-23

21

has to solved.

(J/::j~I)(~)+(Ej=(zJ Resolving gives:

the equation

for the free parameters

vv

w=c+p

Appendix A

A.3. Twofold screw-axis

If the molecule is analysed and the interaction surfaces and interaction centres are discretized, the program ends up with the molecule surrounded by potential interaction points and interaction centres (Fig. 7). Interactions between two molecules are created, if centres fall on points. Likely symmetry operations of a crystal structure might be those which create strong interactions between the molecules. Therefore we search for symmetry operations M, which map centres to points.

The twofold screw-axis MrotzI has five degrees of freedom. To find the proper axis for the crystal structure two pairs of centres and points have to be considered simultaneously (i E { 1,2 1). In general a rotation about the axis 1 is described by the equation

t 21.; x,1, - 1 21,1,

M C-P

At the moment the program searches for translations, inversions, glide planes, and a twofold screwaxes. Each symmetry element M can be described by a rotation W and a translation w. Mapping one point onto a centre amounts to fulfilling the following equation:

21; 21,1, - 1 24.1,

24.1, 2&l, - 1 )( 217

Pi? Pi-K Pt )( +

k”.V w, w; 1

Ci.r

=(1 Cl?

c,:

The symmetry operation has to be a unitary transformation, which requires det(W) = 1. For the solution of the equation we get: ,=

PI

-P2

+cl

IPI -pz+c,

-c2 -c2l

The translation

expressed gives:

M:= c, - wp, A. 1. Translation In the case of a pure translation M,,,,, the rotational part reduces to the unit matrix, and the solution of the condition gives for the remaining parameters w: w=c-p

The condition that the transformation be rewritten as: IP, -

P21 =

ICI -

is unitary can

c21

A.4. Glide plane The equations for the glide plane Mglide can easily be derived by observing that

A.2. Inversion M rot21@ Equal to translations inversions Mi,, have three free parameters. To determine proper inversions the

=

Minv

Mglide

=

Wrot21 winv

+

Wrot21 Winv

+

“‘ro121

D. W.M. Hq%mn.

22 Table 3 Internal description

T. Lengcruer / Journul of Molecular Structure 474 (1999) 13-23

Space group

pi:

of the crystal structures

b, = w2, b2 = w3, bj = w4 space group

MI

Mz

Mi

PI

f 2, I

t

pw,2, P2l

t 2, 21

Pi

i

t

t

MJ

Space group P21: b, = W,w, + wl, b2 = w2, bj = wj

I t

Space group P212121: Like the twofold screw-axis, the glide plane has five degrees of freedom. A glideplane is expressed by:

b, = W,w, + wI, b2 = W,w, + wz, b, = W,w, + %

i-. 1-241, - -241, 21: c,.r

In a second step we calculate the origin of the crystal. Space group PI : the origin is arbitrary and set to 0.

“i?.

o=o

1--21Jz -21,1, 21;

1-- -244 21.,1, 21,’ii

PiJ Pix Pi: i

+ ( w, WX W, 1

= 0 c,;

The solution of this equation including the condition det(W) = - 1 gives the same result as for the twofold screw-axis: 1=

Space group Pi: the origin has to be an inversion centre. Mathematical expressed: 0 “2 0 + 0 = w,,,/2

PI - P2 + Cl - c2 Ip, -P2+c,

-c2(

bv=CI

-wPl

Besides the presented symmetry operations, only a few other symmetry operations are possible in crystals. We have not implemented these symmetry operations yet. They are observed much less frequently in nature.

Appendix

Space group P2,: the origin lies on a screw-axis. MZI 0 - 0 + W&W,, + W& -

0 = w,,/2

Space group P2,2,2,: the origin is the midpoint three non-intersecting pairs of parallel 2, axes.

of

B

Crystal structures are described internally of the algorithm by symmetry operations M. Table 3 contains the ordered symmetry elements, which were used to construct the crystal step by step. Translations are abbreviated with t, inversion centres with i, and twofold screw-axes with 2,. For further processing conversion to the common presentation by the unit cell and its contents is necessary. First, we calculate from the symmetry operations M the translation vectors b. Space group Pl: b, = w,, b2 = w2, b3 = w3

+‘I 2

W’lr - 22-v

WI? -

(bt

(bi

x b2I.r

bb

w2,

x b2A

b2y lib, x b2IIIIh2II

WI: (b,

w2z

x b,),

h

bl llb,ll

The length of the unit cell vectors a, b, and c, often referred just by unit cell vectors, is the length of the translation vectors. The angles can easily be evaluated from the vector product between the translation vectors. One has to note that the system has to be right-handed. Finally, the molecule coordinates X,

D. W.M. H&ann,

are transformed

fc =

-1

-cosy

u

asin y

o0

with r =

to fractional

T. Len,quurr / Journal

coordinates

cosacos y -

co@

arsiny

1

cospcos y - coscz

bsin y

brsin y

0

siny cr

fC.

1 (xc - 0)

1 - cos2~ - cos2/3 - cos* y + 2coScwcos~Cosy

References [I] S. Zimmermann, Science 276 (1997) 543. [2] A. Gavezotti, Accounts Chem. Res. 27 (1994) 309. [3] R. Wawak, K. Gibson, A. Liwo, H. Scheraga, Proc. Nat1 Acad. Sci. USA 93 (1996) 1743. [4] B. van Eijk, W. Mooij, K. Kroon, Acta Cryst. B51 (1995) 99. [5] T. Threlfall. Analyst 120 (I 995) 2435. [6] D. Willock, S. Price. M. Leslie, C. Catlow. J. Comp. Chem. I6 (1995) 628. [7] J. Glusker, K. Trueblood, Refinement of the Trial Structure, 2nd ed., University Press, Oxford, 1985. [8] W. Kleber, Einfiihrung in die Kristallographie Verlag Technik, 16th ed., Berlin, 1983.

r!f Molecular

Structure

474 (1999) 13-23

23

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