J. Phys. Chem. 1995,99, 3765-3773
3765
Crystal Packing without Symmetry Constraints. 2. Possible Crystal Packings of Benzene Obtained by Energy Minimization from Multiple Starts K. D. Gibson and H.A. Scheraga" Baker Laboratory of Chemistry, Cornel1 University, Ithaca, New York 14853-1301 Received: July 18, 1994; In Final Form: November 16, 1994@
The crystal energy of benzene was minimized with no constraints other than the existence of a lattice, starting from 60 structures with Pbca symmetry. The initial lengths of the sides of the unit cell were varied from one start to another. Ten distinct energy-minimized ordered structures were obtained; four of these structures had orthorhombic space group symmetries, five had monoclinic symmetries, and one was tetragonal. The energies of nine of the structures fell within a range of 1.Ok c d m o l ; the energy of the 10th structure was 0.8 kcal/mol higher. One of the orthorhombic structures could be identified with the low-pressure orthorhombic crystal form of benzene, and one of the monoclinic structures could be identified with the high-pressure monoclinic form. With the empirical potential used for these energy minimizations, two of the monoclinic structures had lower energies than the orthorhombic structures. However, when energy minimization was carried out with two other potential energy functions taken from the literature, with or without an added term to reflect the effect of pressure, the relative energies of the orthorhombic and monoclinic structures favored the observed crystal structures at both low and high pressure. Comparison of the possible crystal structures of benzene found in this study with those found by Dzyabchenko (Dzyabchenko, A. V. J . Stmct. Chem. 1984, 25, 416) showed that four were the same but the other six were new.
1. Introduction The physicochemical properties of solid and liquid benzene have been the subject of a large number of computational studies that employ empirical or semiempirical potential energy There are many reasons for the popularity of benzene as a subject of study, including the simplicity of its chemical composition, the appealing symmetry of its geometry, and the detailed experimental knowledge of its physical chemistry. The latter is especially important because it provides a rigorous test of empirical potentials and computational methods for use with more complex molecules. In most studies, the benzene molecule is treated as rigid, planar, and perfectly Symmetric. Although some molecular mechanics simulations suggest that the benzene molecule is rather flexiblez8 and some computations have used flexible molecule^?^^^ the very accurate investigation of orthorhombic crystalline C a 6 by neutron diffraction suggests that the C atoms satisfy the criteria of planarity and symmetry closely, while the deviation of the D atoms is at most marginal.29 Thus, the evidence from the crystal supports the view that benzene can be considered as rigid and symmetric to a very good approximation; for this reason, computations involving benzene can use fairly simple techniques, whether they involve energy minimization, molecular dynamics, or Monte Carlo methods. On the other hand, empirical potentials for describing benzene-benzene interactions have become steadily more complex. Virtually all such empirical potentials use pairwise interactions between atoms or pseudoatoms, but the shapes of the atoms and the mathematical forms of the pairwise interactions vary considerably. Early studies used a Buckingham 6-exp potential or a Lennard-Jones 6-12 potential for nonbonded interactions, with the centers of attraction placed on the (spherical) at~rns.'*~,~,' However, it was soon realized that this potential was inadequate and that some account had to be taken of the inhomogeneous distribution of charge in the molecule. e Abstract published in Advance ACS Abstracts, February 15, 1995.
0022-3654/95/2099-3165$09.00/0
Addition of a single quadrupole at the centroid was partially successful,8-z1but a better solution was to place partial charges on each atom, thereby adding a Coulombic term to each interatomic i n t e r a c t i ~ n . ~ * ~One s ~ ~other - ~ ~ modification has proved popular and relatively successful; in this modification, the centers of charge and attraction for the H atoms are separated slightly, to reflect the ellipsoidal nature of these atom^.^^^^ This type of potential has had considerable success in computations of crystal structures. More complex, and specialized, potentials have been used by some workers, including a sum of five inverse powers of the interatomic d i s t a n ~ e , ' ~an J ~anisotropic ,~~ potential consisting of multipoles placed at six points,loJ9 a Gaussian overlap potentialFO and a generalized Buckingham potential with highly asymmetric atom^?^,^^ In this paper, we apply one of the 6-12-1 potentials to study the possible ways of packing benzene in a crystal, by energy minimization from different starting points. In most crystal packing studies, including most of the studies on benzene, the aim has been to reproduce known crystal structures rather than explore other possible structures. An early exception to this statement is the work of Hagler and L e i s e r ~ w i t z who , ~ ~ packed adipamide starting with several hypothetical space group symmetries to try to account for the anomalous hydrogenbonding pattern in the observed crystal. For benzene itself, an extensive study was carried out by Dzyabchenko, who initially explored the possible crystal packings by generating a benzene crystal with a specified space group symmetry and minimizing its energy with retention of that ~ y m m e t r y . ~ ~InJ ~subsequent J~ papers, he extended the work to include the effect of pressure;16 finally, he generated structures with no initial symmetry (space group P1) and minimized their energies, and in this way obtained some new structures as well as some that had been observed before.2z Our approach differs from that of Dzyabchenko, in that we maintain the Pbca symmetry of the orthorhombic benzene crystal in all starting points but vary the initial lengths of the sides of the unit cell. The structure is then subjected to energy minimization, using the algorithm described
1995 American Chemical Society
Gibson and Scheraga
3766 J. Phys. Chem., Vol. 99, No. 11, 1995 in the accompanying paper35(hereafter referred to as Paper I), in which the energy is minimized with no constraints other than the existence of a lattice.
11. Methods
E = [e, e2 e3IT E = [C, C2 &IT
(5)
are related by
A. Energy Functions. The energy functions used in this work were all sums of pairwise interatomic interactions, in which each interaction was taken to be the sum of a Coulombic term and a term for nonbonded interactions. The Coulombic terms took the form shown in Paper I, eq 9, and their sum over the lattice was computed by Ewald summation as described in Paper I, section II.C.2. The nonbonded terms were either Lennard-Jones 6-12 potentials, as in Paper I, eq 10, or Buckingham potentials of the form
E = ~ e - -~c / ~r $ i ~
C~ i j a i j
For primitive lattices, all Mij must be integers; for centered lattices the Mij may be fractions whose denominator depends on the type of centering. If we set
a = [a,, 82, 4 I T
(3') \
,
eq 1 can be rewritten in matrix form as
A=MA
where
Q = U ~ M ~
(7)
(for the definition of U see Paptr I, eq 6). Atomic coordinates 13&"), referred to the frame E, are related to the atomic
coordinates x&")referred to the frame E by
(1)
where rij is the distance between atoms i and j . All energies were expressed in kilocalories/mole (1 cal = 4.18635 J). Nonbonded interactions of either type were cut off with a cubic spline from Pa to Qa, to ensure continuity of the function and its f i s t derivative (see Paper I, section II.C.1; in that paper, A and B take the place of P and Q, respectively). Here 0 is the value of rij at which E, = 0 and E', < 0; for potentials of the form in eq 1 this value was calculated using Newton's method. In all the work described here, P and Q, which describe the cutoff values (see Paper I, section II.C.l), were set to 6.0 and 6.5, respectively. In some computations, a term was added to the energy to simulate the effect of pressure.36 This term took the form P(V - Vo),where Vo is the volume of a suitably chosen unit cell at zero pressure. B. Determination of Symmetry Relations between Molecules. This determination was carried out as described in Paper I, section ILF and Appendix B, with certain modifications that are necessitated by the planarity and point group symmetry of the benzene molecule; these modifications are described in Appendix A of this paper. C. Relation between Observed and Conventional Unit Cell. The final lattice parameters obtained after energy minimization will define three independent basis vectors, but these need not be the basis vectors that correspond to a conventional unit cell. In the following we distinguish all quantities related to the conventional unit cell with a caret. Let al, a2, a3 be the observed basis vectors, and 8I1, 812, 8I3 the conventional ones; then
ai =
E=QE
(4)
It is always possible to choose the conventional unit c5ll so that det M =- 0. The two sets of kasis vectors A and A are related to Euclicean frames E and E through lower triangular matrices L and L whose components are given in eq 2 of Paper I. The Euclidean frames E and E, with unit vectors
If the matrix M is known, the lattice parameters of the conventional cell can be found from the vectors &, L2, A3 obtained by inverting eq 4. Deduction of the symmetry relations between molecules in the conventional unit cell depends on the determinant of M. We distinguish three cases. I , Det M = 1 . The volumes of the observed and conventional unit cells are equal; the number of molecules in the conventional unit cell is equal to the number in the observed unit cell. If the coordinates of molecukes m and n in the unit cell, referred to the two frames E and E, are related by
then the type and order of the transformations {S,s} and {SC} are the same; this is proved in Appendix B. The space group of the conventional unit cell can be deduced from the symmetry transformations of the observed unit cell. 2. Det M = N, Where N Is an Integer Greater than I . The conventional unit cell contains ZIN molecules; the remaining molecules divide into cosets, each of which contains translational images of the molecules in the conventional unit cell. I f f and f denote the vectors of fractional coordinates of a point with respect to the observed unit cell and conventional unit cell, respectively, then
= MTf
(10)
and the components of ? for any atom in a coset must differ from the corresponding components for its image atom in the conventional unit cell by integers. Symmetry relations for the conventional unit cell may not be obvious until it has been identified. 3. Der M = UP, Where P Is an Integer Greater than I . The conventional unit cell is centered and contains PZ molecules. The observed lattice vectors must contain enough vectors ending at centering points to generate the entire centered lattice. The molecules in the observed unit cell must generate at least one primitive unit cell, and the symmetry relations between them should reflect this fact. In the work presented here, it was possible to determine the matrix M by inspection in all cases. Examples of all three types of relationship between the observed and conventional unit cell are presented in section 111. D. Computational Details. Gay's routine SUMSL (secanttype unconstrained minimization ~olver),~'with rescaling as described,38was used throughout to minimize the energy. With
J. Phys. Chem., Vol. 99, No. 11, 1995 3761
Crystal Packing without Symmetry Constraints SUMSL, the search path in the space of independent variables depends to some extent on the size of the initial step. In all computations with Lennard-Jones potentials this was set equal to 1/(M l), where M is the number of independent variables. However, in computations with Buckingham potentials, it was found necessary to divide the initial step by a factor varying from 10 to 105, depending on the particular computation, to prevent the search from ending at a false minimum with ro 0 for one or more pairs of atoms.
+
-
III. Results and Discussion A. GenerationofEnergy-M' * * d Structures. Aplanar symmetric benzene molecule, whose C-C and C-H distances were 1.398 8, and 1.090 A, was generated and placed with its centroid at the origin of the experimentally observed unit cell of the orthorhombic crystal of benzene at 270 K,39in such a way that the normal to the plane of the ring coincided with the normal to the least-squares plane of the ring in the crystal, and the C1 atom lay in the same plane as the normal and the C1 atom in the crystal. Starting structures for energy minimization were obtained by setting the initial values of a, b, and c to values ranging from 1.O to 2.0 times the values a,, bo, and c, observed in the crystal, and generating a unit cell with Z = 4 and Pbca symmetry, as described in Paper I, section 1I.E. The energy function used for these energy minimizations was the one described in Paper I, section II.G and Table 2. Where necessary, the final lattice parameters were subjected to the analysis described in section II.C of this paper to determine the conventional unit cell and its parameters. Out of 60 different starts, 8 led to collapse of the lattice before energy minimization was complete. The remaining 52 computations converged to 10 different energy-minimized ordered structures (Table 1). Four of these structures (01 to 04 in Table 1) had orthorhombic space groups, five (ml to 1x15) had monoclinic space groups, and one (tl) had a tetragonal space group. The total energies of nine of these structures fell within a range of 1.0 kcdmol; the 10th structure, tl, which appeared to have the highest degree of symmetry, had a significantly higher energy. In most cases, the basis vectors of the final lattice either coincided with the basis vectors of the corresponding conventional lattice, or were permutations or simple linear combinations of them. However, for three monoclinic structures, ml, m2, and m3, the relationship between the final basis vectors and the conventional basis vectors was occasionally complex. As shown below, the number of molecules in the conventionalunit cell of these structures differs from the number in the final computed unit cell. Despite the apparent variability of the final structure, the final lattice parameters after energy minimization were reasonably consistent with the choice of starting values. Two examples are presented in Table 2. In the first example, the initial values of the lattice parameters a and c were held constant at 1.254, and 1.IC,, respectively, and the initial value of b was increased in steps from bo to 2b,. The final ordered structures fell into categories 01, ml, and m3 in no obvious order; however, the final values of the observed lattice parameters showed a steady rise in b, paralleling the rise in the starting value of this parameter, and accompanied by a decline in the final value of c. In the second example, the initial value of a was held constant at a,, and the initial values of b and c were increased in steps together from 1.25b0, 1 . 2 5 to ~ ~1.5b0, 1 5 , . The final ordered structures fell into categories m2, ml, and 01, but the final values of b rose in parallel with the starting values. B. Deduction of Symmetry. The space group symmetry of the 10 ordered structures was deduced from the symmetry
relations of the molecules in the conventional unit cell after the latter had been determined as described in section 1I.C. As discussed in Appendix A, there are many ways of superposing two planar symmetric benzene molecules, and more than one superposition may satisfy the requirements for a crystallographic transformation. The possible crystallographic relations between the molecules in the conventional unit cell were examined, and a set, consisting of one relation per pair of molecules, was chosen in such a way as to make up a group. The transformations of this group were transcribed into the symbolic forms used in the International Tablesfor Crystallography (ref 40, p 794), and a search was made for a space group with these transformations, either as the entire group or as an appropriate subgroup. To facilitate the search, the origin of the conventional cell was moved from the centroid of all the molecules to an appropriate point as needed. An example of a final set of transformations, for one of the energy-minimized ordered structures in each of the 10 categories, is presented in Table 3. Two pairs of ordered structures had the same space group symmetry. One pair was 01 and 04, which both had space goup Pbca with Z = 4; the other pair was ml and m2, which both had space group P2Jc with Z = 2. In all four of these structures the asymmetric unit consisted of half a molecule. The energies of these four ordered structures were all different (Table 1). The two remaining orthorhombic ordered structures had space groups P2lcn and Pna21, each with Z = 4 and an asymmetric unit consisting of one molecule. The transformations listed in Table 3 for the ordered structure m3 are the first four transformations of space group C2/c; the remaining four transformations are related to these by the centering translation t('/2,*/2,0). In the ordered structure m, the asymmetric unit consisted of two molecules (nos. 1 and 3); each of the other two molecules was related to one of these by the same axial glide. C. Parameters of Conventional Unit Cells. Lattice parameters and fractional coordinates of the centroid of molecule 1, for each of the 10 ordered structures are presented in Table 4. There are marked differences between the lattice parameters of the structures 01 and 04, which have the same space group; the same is true for the two structures ml and m2. Together with the differences in energies, this underscores the fact that each of these ordered structures is distinct. To verify the identification of space groups and lattice parameters for the ordered structures, we minimized their energies again starting from the values for the parameters in Table 4. Cartesian coordinates of the molecule(s) in the asymmetric unit were generated from their positions and attidues in the conventional unit cell. Coordinates for the remaining Z - 1 (or Z - 2) molecules, where 2 is given in Table 3, column 7, were generated by applying the transformations of the space group in Table 3, column 6, as described in Paper I, section 1I.E. Energy minimization was started using the rigid body variables deduced from their set of coordinates. If the lattice parameters and space group for each energy-minimizedordered structure are correct, these starting structures should be very close to local energy minima, although numerical roundoff errors would prevent them from being exactly at local energy minima. Energy minimization with an efficient secant-type algorithm, such as the one used in this study, should result in convergence within N, or at most lSN, iterations, where N = 62 6 is the number of independent variables; furthermore, the final lattice parameters, energies and space groups should agree with those in Tables 1, 3, and 4. These expectations were met in all cases. The number of iterations to convergence ranged from 0.3N to 1.2N, with most values lying in the range 0.5-0.8N. The final
+
Gibson and Scheraga
3768 J. Phys. Chem., Vol. 99, No. 11, 1995 TABLE 1: Lattices Obtained by Energy Minimization of Benzene
energy (kcal/mol)" starting" lattice
finalblattice
ao&o,co
01
uo,bo;l.lco l.lao;bo,co 1.5ao;borco 1.25ao;bo,co 1.25a,,b,;c, 1.25a,,c0;b, 1.25a,,b,,c, a,,b,;l .5c0 1.25a,;b,; 1.IC, 1.25a,;b,; 1.5c0 1.25a,;b0; 1.75, 1.5&;1.25b0,c0 1.5a,;b,;1.75cO 1.5ao,b,;l.lc, 1.25a0;l.75b,;c, 1.25a,; 1.5b,;c0 a,;l .5b,,c, a,; 1.25b0;1.4c0 a.,;1.25b,;1.75c0 a,;1.4b,;1.75co ao&o;2co 1Sa,; 1.25b0;c, 2a,;1.256,;c0 a0;l.4b,,co a,;l Sb,; 1.IC, ao;2bo,co 1.25a,,b,;l.lc0 1.2~a,;1.7~b0;l.~c, 1.5ao,co;bo 1Saok~o~o 1.75a,;1.5b0;l.lc0
01
observed' basis vectors 6L1&,& al,&9&
01
AI,&.&
01
81,&.& 11,&,63
bo;bo,co 2u,;1.75b,;c0
ml ml
01 01
01
8l,a2,?3 blr82&3 ~ l & ~ 3 a3&1,&
01
&,PI&
01 01
ha1,h
01
P3raldZ
01
al,a2,83
01 01
blrPZ,& al,&-a1,13
02
&1$2&3
02
gl$l+&~&
03
P1&&
01 01
03 03
PI&&
04
Bl,aZ,& &$lr&2
04
81,P2,&
04
81&& -82,291+83$3
ml ml ml
ml ml ml ml ml
kotbo;Co
ml
2u,,bo;1.IC,
ml m2 m2 mz
UO,~,; 1.25C0 ~;l.lb0;1.25c0 n,;l .lb0;1.5c, no;1.25b,,c, a0;1.25b,;1.5c, ao;2bo;co 1.25a0;1.5b0;1.IC, 1.25a,;2bO;l.IC, 1.5a,;1.75bO;l.IC, 1.5a0;2b0;l.IC, 1.75a,,b0;l .IC, 1.75a,;2bo;c, 2u,;1.5b,;c0 ai;l .25b0;2c0 a0;1.5b,;1.25c0 1.5a,;l .25b0;l.IC,
&981,&
m2
mz m3 m3 m3 m3
-&?,al+a3ra3
-&,2&-2Pl-&,& Al+&&-fil,al+& &,211+&+&& ~2-gll,-(Bl+~3),-(91+~2)
-2&,-(&+83)$1 -2hI,-(&l+&+&),-& 2Pl+&&B3 Pl-b2+~3,2&2,-~I bl -&+&,&I +%,,-a1 81+&,-2(Pl+~Z),dl -a2,2Al+a3ra3 fit,-aI-&&
&,-2Pl-P3,& -&2,261+63&3 &2,-281-&,a3 &,&-'/Z(hI+&),-& 13,g3-1/2(P1+312),-a2 &%'/Z(~I-&)r-62
m3
L3,83-'/Z(P1+3P2),-4 83,13-'/Z(P1+3bZ),-b2
m3
&,&+'/Z(%l+&)&
m3 m3
m4
L3,-glS-'/Z(bl+gZ),-bZ P3,'lZ(bl -Pz),bz -82$3r-(Pl+&)
m5
&,-&,-PI
tl
Plr&,&
NB
ES
-11.820 -11.820 -11.820 - 11.820 -11.820 -11.820 -11.820 - 11320 - 11.820 -11.820 - 11.820 -11.820 -11.820 -11.820 -11.820 -11.820 - 12.063 - 12.063 -11.665 -11.665 -11.655 -11.567 -11.567 -1 1.567 -12.511 -12.511 -12.511 -12.51 1 -12.512 -12.511 -12.51 1 -12.511 -12.5 11 - 12.511 -12.511 - 12.511 -11.989 -11.989 - 11.989 -11.989 - 11.989 -12.338 -12.338 -12.338 -12.338 -12.339 -12.338 -12.338 -12.339 - 11.646 -11.918 - 10.783
- 1.229 -1.229 -1.229 - 1.229 -1.229 - 1.229 - 1.229 - 1.229 -1.229 -1.229 -1.229 -1.229 - 1.229 -1.229 - 1.229 - 1.229 -0.716 -0.716 - 1.076 - 1.076 -1.076 -0.819 -0.819 -0.819 -0.770 -0.770 -0.770 -0.770 -0.769 -0.770 -0.770 -0.770 -0.770 -0.770 -0.770 -0.770 -1.123 -1.123 -1.123 -1.123 -1.123 -0.664 -0.664 -0.664 -0.664 -0.664 -0.664 -0.664 -0.664 -1.002 -0.431 -0.769
total -13.049 -13.049 -13.049 -13.049 -13.049 -13.049 -13.049 -13.049 -13.049 -13.049 -13.049 -13.049 -13.049 -13.049 -13.049 -13.049 -12.779 -12.779 -12.741 -12.741 -12.741 -12.386 -12.386 -12.386 -13.281 -13.281 -13.281 -13.281 -13.281 -13.281 -13.281 -13.281 -13.281 -13.281 -13.281 -13.281 -13.112 -13.112 -13.112 -13.112 -13.112 -13.002 -13.002 -13.002 -13.002 -13.003 -13.002 -13.002 -13.003 -12.648 -12.349 -11.552
a a, = 7.46 A, bo = 9.67 A, co = 7.03 A.3g 01-04, structures with orthorhombic space groups; m1-1115, structures with monoclinic space groups; tl, structure with a tetragonal space group. Basis vectors (alra2,a3)of the final observed unit cell, expressed as linear combinations of the basis vectors (&,&,&) of the conventional unit cell. NB,nonbonded energy; ES, Coulombic energy.
values of a, b, and c were within f0.0005 A of the starting values; the final values of a,/I, and y were within ztO.005" oTf the starting values; the final values of each component of the energy were within f0.0005 kcal/mol of the averages of the values in Table 1. We conclude that, at least with the potential energy function described in Paper I, section II.G, each of these ordered structures is stable or metastable and could correspond to a possible crystal packing of benzene. The possible relationships between the 10 ordered structures described here and the structures found by Dzyabchenko' 1,17,22 are discussed in section III.F.1. D. Identification and Ordering of Structures. From the values of the lattice parameters, the space group, and the fact that energy minimization staging from the experimental value of the lattice parameters of orthorhombic benzene at 270 K led
TABLE 2: Observed Lattice Parameters of Some Energy-Minimized Structures final lattice parameters' finalb startinglattice lattice a b c a ,9 1.25a,;b,;l.lc0 01 6.945 7.492 9.511 90.00 90.00 1.25a,;1,25b,;l.lc0 ml 8.016 8.016 8.181 106.54 73.46 1.25ao;1.5b,;l.lc, m3 7.633 11.377 5.801 75.23 90.00 1.2~n0;1.75bo;l.lc, ml 7.827 12.982 5.708 63.92 90.00 1.25ao;2bo;l.lco m3 7.633 16.222 5.801 57.57 90.00 a,;1.25b,;1.25c0 m2 6.944 9.633 7.374 83.30 90.00 n,;l .4b,; 1.4c0 ml 5.707 11.660 7.829 67.28 90.00 a,;1.5b,;1.5cO 02 5.750 12.982 7.329 89.98 90.00
y
90.00 89.21 87.30 69.71 59.77 90.00 90.00 63.71
a a, = 7.46 A; bo = 9.67 A; co = 7.03 A. 01, 02, orthorhombic space groups; ml, m2, m3, monoclinic space groups. a, b, c in A; a, ,9, y in deg.
J. Phys. Chem., Vol. 99, No. 11, 1995 3769
Crystal Packing without Symmetry Constraints
TABLE 3: Space Group Symmetries of the Energy-Minimized Structures of Benzene characteristics of conventional unit cell transformationsb starting lattice fiial molecule 2 molecule 3 molecule 4 parameters lattice" ao&o,co 1.25aJ .5b0.co a,,1.25b0,1.4c, 2a,, 1.25b,,c, a,, 1.4b,, 1.4c, a,, 1.25b,, 1.4c, bo91%o&o a,,1.25b,,2c0 a,, 1.5b0,1.25c, 1.5a,,l .25b,,l. IC,
01 02
b n(1/2,1/2,0)
03 04
ml
t( 1,0,0)[4Id
m2
t(- 1,0,0)[4id
m3 m4
m5 tl
'/4,Y,Z
C
X,'/4,Z
c
X,I/4,Z
b
x,y,l/4 OY,Z
1/4,Y,Z
c c t(l,O,l)[lId tl- 1,o,- 1)[ 1Id 1
X,'/4,Z
b 2 a[lld 2 2
OJ,% x,o,z O,Y,% o,o,z
1 1
4+(O,O,z);
X,'/4,Z
o,o,o o,o,o O,O,O
a 2('/2,0,0) 2(0,0,'/2) a 2(0,'/2,0) 2(0,'/2,0) c
space group
g
x , ~ / ~ , z Pbca x,O,O P2lcn O,O,Z Pbc21 x,y,l/4 Pbca O,y,'/4 EI/C O,y,I/4 Ei/C x,o,z a/c x,O,z Pa x,o,z Pyc
4-(0,O,z);
O,O,O
@Id
P4
Z
asymmetricuniF
4 4 4 4 2 2
112 1 1 112
112 112
8
4 4 4
1 2 1 1
" 01-04, structures with orthogonal lattices; ml-rns, structures with monoclinic lattices; t l , stmcture with a tetragonal lattice. Transformation corresponding to molecule 1 was taken to be the identity transformation in all cases. Number (fraction) of molecules in the asymmetric unit. Related by the indicated symmetry transformation to the molecule in brackets.
TABLE 4: Parameters of 10 Energy-Minimized Structures of Benzene lattice parametersb lattice" a b C a B 01(16) 7.491 9.511 6.946 90.00 90.00 (0.002) (0.001) 5.750 11.639 7.329 89.99 90.00 (0.01) O@) 5.412 10.535 8.683 90.00 90.00 (0.001) 5.580 90.00 90.00 8.139 11.051 04(3) (0.01) 5.707 7.828 90.00 107.18 md12) 5.629 (0.001) (0.001) 7.374 90.00 123.06 5.726 6.944 m(5) md8) 22.002 5.801 7.633 90.00 92.79 (0.001) m4(1) 10.701 5.372 10.009 90.00 120.38 m d 1) tl(1)
11.458 9.703
6.163 9.703
7.120 5.497
90.00 90.00
98.17 90.00
coordinates of asymmetric unir'
Y
VC
90.00
123.7
(0.0, 0.0,O.O)
90.00
122.6
(0.0,0.068, -0.250)
90.00
123.8
(-0.269,0.115, -0.250)
90.00
125.5
(0.0, 0.0,O.O)
90.00
120.1
(O.O,O.O,0.0)
90.00 90.00
122.9 121.6
(-0.127, -0.252,0.123)
90.00
124.1
(0.0, -0.250,0.250)
90.00 90.00
124.4 129.4
(0.0, 0.0,O.O)
(0.0,0.250, -0.250) (0.253,0.263,0.062) (-0.250, -0.250, -0.250)
Number in parentheses indicates number of structures used in taking the average and SD. a, b, c in A; a,B, y in deg. Number in parentheses is SD,where this differs from zero. Volume per molecule in A3. Fractional coordinates of centroid of molecule(s) in asymmetric unit, referred to origin chosen for conventional cell. (I
to this ordered structure, we are led to identify 01 with the experimentally observed low-pressure crystal structure of benzene. To determine whether any of the remaining ordered structures corresponds to the high-pressure monoclinic form of benzene, we placed a symmetrical molecule into a unit cell with the observed space group symmetry and lattice parameters of monoclinic b e n ~ e n e ? ~in, ~the ~ manner described in section III.A.l, and minimized the energy. The final lattice parameters, space group, and energy agreed with those of the ordered structure ml. As confirmation, we minimized the energies of all 10 ordered structures with the addition of a pressure term, with the reference volume set to the volume for the 01 structure at zero pressure (Table 4, column 8). We also minimized the energy with this pressure term, starting from the experimentally observed monoclinic crystal structure. The energy, lattice parameters, and space group of the energy-minimized structure obtained from the experimentally observed monoclinic crystal structure coincided exactly with those obtained from the ml structure and disagreed with the results obtained from the other nine ordered structures (Table 5 ) . Thus, the ordered structure ml corresponds to the high-pressure monoclinic crystal structure of benzene. It may be noted that, with this potential at 25 kbar, energy minimizations starting from the 0 2 and 03 structures led to the same final ordered structure, whose symmetry corresponded to space group P2lcn. Further cases in which changing the potential leads to instability of a particular ordered
structure appear in subsequent sections; Dzyabchenko12.22has noted the same phenomenon. If it is assumed that differences in vibrational free energy between the ordered structures may be neglected, differences between the observed energies of the structures represent differences between their lattice energies and should be a measure of the relative stabilities of these molecular packings at 0 K. The results in Table 5 would then suggest that at high pressure benzene should crystallize in the monoclinic P21/c form of the ml structure, in agreement with observation. However, the results in Table 1, column 6, do not support the orthorhombic Pbca form of the 01 structure as the preferred low-pressure crystal, even though the 01 structure obtained by minimization of the potential energy of Paper I reproduces the observed lattice parameters of orthorhombic benzene at 218 K within 1%. A possible explanation for this discrepancy was that we had chosen the wrong geometry for the benzene molecule, since our choice was based on the crystal structure of C a 6 rather than C6H6. Therefore, a symmetrical benzene molecule was generated whose C-C and C-H distances were equal to the average C-C and C-H distances, determined by neutron diffraction, in the benzene crystal at 218 K;43these are slightly smaller than the distances in section III.A. Energy minimization with this molecule from all starting structures except nu led to final structures that retained their starting symmetry; the final structure obtained from was a double-cell version of the final structure obtained from ml. Apart from this, there was no
Gibson and Scheraga
3770 J. Phys. Chem., Vol. 99, No. 11, 1995 TABLE 5: Energy Minimizationat 25 kbar final latticeb starting structure' a b C B 7.292 9.237 6.551 90.00 01 02 5.730 11.287 6.718 90.00 03 5.730 11.287 6.716 90.00 04 7.968 10.690 5.388 90.00 ml 5.424 5.521 7.433 105.90 m2 5.498 6.631 7.104 122.35 m3 21.373 5.639 7.188 93.30 m4 10.467 5.205 9.577 122.14 m5 11.437 5.914 6.789 108.92 tl 9.403 9.403 5.147 90.00 MC 5.424 5.521 7.432 105.90
space group Pbca P2lcn P2lcn Pbca P2llC P21Ic C21C Pa P2Ic P4
NB -9.663 -10.035 - 10.037 -9.366 -10.518 -9.845 -10.341 -9.571 - 10.228 -8.397
energy (kcal/mol)' ES PAV -1.474 -4.816 -0.759 -5.426 -0.759 -5.424 -0.966 -4.619 -0.890 -5.995 -1.345 -5.141 -0.741 -5.603 -1.154 -4.762 -0.530 -5.433 -1.040 -3.578
P2IlC
-10.518
-0.890
-5.995
total -15.953 -16.220 -16.220 -14.950 -17.403 -16.331 -16.684 -15.487 -16.191 -13.015 -17.403
a 01-04. ml-ms, tl: lattice structures discovered by energy minimization. MC: experimentally observed crystal structure of monoclinic benzene. a., b., c in A: B in dee. Values for a and y were 90.00' in all cases. Final space group determined as described in section III.A.2. NB,nonbonded energy; ES, Coulombic energy; PAV, pressure term.36
b
I .
Y
TABLE 6: Energy Minimization with AMBER energy (kcallmol)" starting space group (Z) structure of final structure NB ES total volb(A3) 01 Pbca(4) -10.706 -3.110 -13.816 112.5 112.1 -11.032 -1.854 -12.886 02 P211b (4) 112.1 -11.032 -1.854 -12.886 03 P21/b(4) 04 Pbca (4) -10.556 -2.176 -12.732 114.2 109.5 ml P211c (2) -11.498 -1.950 -13.448 111.5 mz P21/~(2) -10.954 -2.771 -13.725 111.2 -11.320 -1.736 -13.056 m3 C2/~(8) 113.5 m4 Pa (4) -10.490 -2.721 -13.211 111.7 -11.220 -1.792 -13.021 m5 P2/c(4) 117.5 tl p4 (4) -9.901 -1.865 -11.766 NB,nonbonded; ES, Coulombic. Partial charges of f0.15 were placed at each atomic center.32 Volume per molecule.
TABLE 7: Energy Minimization with AMBER at 25 kbar space group energy (kcdmol)" starting (Z) of final NB ES PAV total volb(A3) structure structure -8.574 -3.586 -4.131 -16.291 103.5 99.6 -9.220 -1.868 -4.648 -15.736 99.6 -9.220 -1.868 -4.648 -15.736 -8,477 -2.443 -3.969 -14.889 101.5 98.1 -9.645 -2.154 -5.912 -16.991 -8.827 -3.237 -4.474 -16.538 100.1 99.5 -9.504 -1.830 -4.801 -16.135 -8.406 -3.080 -3.988 -15.474 101.4 99.7 -9.282 -2.020 -4.601 -15.903 -7.550 -2.320 -3.012 -12.882 104.1 Volume NB,nonbonded; ES, Coulombic; PAV, pressure per molecule.
change in the order of the final energies. We conclude that varying the molecular geometry within reasonable limits is unlikely to affect the relative stabilities of these structures. E. Tests of Other Potentials. Examination of Table 1 reveals considerable variation in the fraction of the total energy that is due to Coulombic interactions, from 8-9% for the 01 and m2 structures, down to less than 6% for the ml and m3 structures. In the potential described in Paper I, Table 2, which was used for the calculations in Table 1 of this paper, the partial charge placed on each atom by the PEOE method4 is f0.093e. Other potentials designed for general use, such as AMBER32 or the potential of Williams and his collaborator^^^^^ place larger charges on these atoms. It therefore seemed important to examine the relative stabilities of the 10 ordered structure with these two potentials. 1. AMBER. The results of energy minimization with the AMBER potential are shown in Tables 6 and 7. For these computations, the C-C and C-H distances of the benzene molecule were changed to 1.40 and 1.08 A, in line with the geometry assumed for phenylalanine and tyrosine,32 and the value of VO,for use in the pressure term of the potential, was set equal to the volume of the unit cell of the (AMBER) energyminimized 01 structure at zero pressure. At zero pressure, the final structure with the lowest energy was the Pbca structure 01; at 25 kbar, the final structure with the lowest energy was the P2dc structure ml. Those are the two energy-minimized structures that bear the closest resemblance to the observed crystal structures of benzene at low and high pressure, respectively. The final values for the lattice parameters in the structures in Tables 6 and 7 were fairly close to those in Table 4 and 5 (a, b, and c within 5%; /3 within 4O), except for the structures whose final space group changed during energy minimization (02 and 03 in Table 6; 03 in Table 7). Energy
TABLE 8: Energy Minimization with Williams' Potential energy (kcallmol)" space group (Z) NB ES total volb (A3) of final structure 121.6 Pbca (4) -9.659 -2.908 -12.567 120.8 -9.964 -1.842 -11.806 P2lcn (4) 120.8 -9.964 -1.843 -11.807 P 2 m (4) 123.8 Pbca (4) -9.403 -2.110 -11.513 118.9 -10.299 -1.886 -12.185 P21lc (2) 120.7 -9.920 -2.528 -12.448 P.21Ic (2) C2Ic (8) -10.184 -1.659 -11.843 120.5 122.7 -9.457 -2.509 -11.966 Pa (4) -10.117 -1.604 -11.721 121.1 P2Ic (4) 127.3 -8.956 -1.669 -10.625 p4 (4) NB,nonbonded; ES, Coulombic. Volume per molecule. a
(1
,
,
(I
minimization with the AMBER potential resulted in a satisfactory ordering of the energy of the ten structures. The extent of agreement of the final computed lattice parameters with the observed ones is discussed in section m.F.2. 2. Williams' Potenrial. The potentials examined so far were designed to be as simple as possible, since they are intended for use in computationally intensive studies of macromolecular systems. A somewhat more complicated potential, which was designed for the investigation of crystal structures and has had wide application in studies of crystal packing of aromatic molecules, is the 6-exp-1 potential of Williams and his coll e a g u e ~ . This ~ ~ ~potential ~ was applied to the 10 ordered structures of benzene, using the geometry and parameters employed by Yashonath et al. (ref 19, Table 1, column 2), with the results shown in Tables 8 and 9. As with the other potentials, the reference volume VOfor use in the pressure term was set equal to the volume of the unit cell in the energyminimized structure obtained from the 01 ordered structure at zero pressure.
Crystal Packing without Symmetry Constraints
TABLE 9: Energy Minimization with Williams' Potential at 25 kbar space group energy (kcal/moly starting (Z)offimal structure structure NB ES PAY total volb(A3) -6.918 -3.321 -5.984 -16.223 104.9 -7.371 -2.063 -6.377 -15.811 103.9 -7.633 -1.605 -6.437 -15.675 103.7 -6.620 -2.387 -5.734 -14.740 105.7 -7.714 -2.189 -6.802 -16.705 102.7 -7.063 -3.105 -6.127 -16.297 104.6 -7.628 -1.851 -6.461 -15.944 103.6 -7.714 -2.189 -6.802 -16.705 102.7 -7.489 -1.864 -6.250 -15.603 106.4 -5.836 -2.197 -4.683 -12.716 108.6 a NB,nonbonded; ES, Coulombic; PAV, pressure term.36 Volume per molecule. Observed lattice was (-2i31-83,-&~,&). As with the AMBER potential, the ordered structure 01 had the lowest energy at zero pressure with Williams' potential, and the ordered structure ml had the lowest energy at 25 kbar. Thus, this potential also favors the correct crystal symmetry at these two pressures. However, there were significant differences, amounting to about lo%, between the volumes of the unit cells of the energy-minimized structures obtained with AMBER and with Williams' potential. It may also be noted that, with Williams' potential, as with AMBER, some of the ordered structures adopted new space group symmetries after energy minimization. F. Comparisons of the Present Work. 1. Comparison with Results of Dzyabchenko. Since most of Dzyabchenko's computations were performed with Williams' potential, it is pertinent to compare the results in Tables 8 and 10 with his results. Exact agreement should not be expected, because the geometry of the benzene molecule that he used was slightly different from ours. Furthermore, Dzyabchenko used accelerated c ~ n v e r g e n c eto~ ~compute accurately the 11P attractive terms in the nonbonded potential (eq 1) and truncated the exponential repulsive terms, whereas we combined the two terms and reduced them smoothly to zero together. For this reason, the energies that he obtained should be somewhat lower than the energies in Tables 8 and 9. The final lattice constants for the 01 and ml ordered structures that we obtained by minimization of Williams' potential energy function agreed with the values found by Dzyabchenkoll for his Pbca (I) and P211c (E) structures, respectively, within 0.15 A and OS", and the energies agreed within 0.3 kcdmol; we are therefore led to identify our 01 and ml structures with those two structures. The 01 and ml structures are the two structures that correspond most closely to the low-pressure and highpressure crystal forms of benzene, respectively (see section DI.D). Two other structures found by Dzyabchenko" agree with structures found by us within the same tolerances; these are his P21/c (I) structure, which we identify with our m2 structure, and his Pbca (11) structure, which we identify with our 0 4 structure. However, there does not seem to be any obvious correspondence between any of the other structures reported by Dzyabchenko and the other six ordered structures reported here. DzyabchenkoZ2found a Pca21 structure using Williams' potential, but this is not the same as the 03 (Pbc21) structure found by us since the latter acquired a different symmetry in our hands with this potential. None of the C21c structures found by Dzyabchenkoz2 can correspond to our m3 (C21c) structure, since all his structures had 2 = 4, whereas ours had 2 = 8. Dzyabchenko does not appear to have found any Pc, P2/c2 or_P4 structures; on the other hand, we have not found any R3, P1,or a 2 2 1 structures, as reported by him.l1Sz2
J. Phys. Chem., Vol. 99, No. 11, 1995 3771 Taken together, our results and his show that there are very many possible ways of packing benzene molecules into a regular three-dimensional structure, involving a variety of space groups and having energies that lie in a rather narrow range. The problem of predicting, by computational methods, the crystal packings that are observed e ~ p e r i m e n t a l l y is ~ ~a .complex ~~~~~ multiple-minima problem, such as is found in other branches of theoretical physical ~ h e m i s t r y . ~ - ~ * 2. Comparison with Experiment. One criterion that is used frequently for assessing empirical potentials is the degree of agreement between lattice constants predicted by energy minimization and experimentally observed lattice constants. This test is rather simple to apply but, as has been noted by other a ~ t h o r sit, ~does not lack pitfalls. One important caveat is that since the potential function to be tested is strictly a potential energy, it should reproduce the crystal structure at 0 K only; however, experimental crystal structures cannot often be extrapolated to that temperature. The experimentally observed structure is an average one; because of the anharmonicity of lattice and intramolecular vibrations, the size of the average unit cell will expand with increasing temperature, and experimental values of the lattice parameters a, b, and c, measured at any realistic temperature, will be greater than the values that ought to be used for comparison with theoretical predictions. On the other hand, even with available sh0rtcuts,4~the computational effort that would be required to calculate a lattice structure, by f i s t obtaining a zero-point structure and then applying anharmonic vibrations and averaging, could be excessive. In principle, molecular dynamics or Monte Carlo methods can solve this problem directly, but the computational effort needed for great accuracy is again large.10919Accordingly, it has become common to match empirical potentials to crystal structures at temperatures that approximate the temperatures at which the potentials are to be used for more general calculations, and to regard the potentials obtained in this way as including an implicit allowance for the effects of thermal perturbation^.^ For a potential designed to reproduce conformations of biological macromolecules, the appropriate crystal structures to match should be structures at 250-300 K; potentials to be used to study crystal packing at lower temperatures should be matched to low-temperature crystal structures. With this preamble, we compare in Table 10 the lattice parameters of the 01 and ml ordered structures obtained after minimizing each of the three potentials used here, with the experimental lattice parameters for c6&,and c&. The parameters a, b, and c predicted for the 01 structure at zero pressure, using the potential in Paper I, section II.B, match the lattice parameters of C6H6 at 218 K rather closely, and are not greatly different from the parameters of C&6 at 138 or 270 K (Table 10). With this potential, the agreement between the lattice parameters predicted for the ml structure at 25 kbar, and the lattice parameters of the experimental monoclinic structure at high pressure is less satisfactory (Table 10); in particular, the value of is in error by over 4" and the volume is too high by nearly 4%. Taken together with the fact that, at low pressure, this potential predicts the wrong energetic order for the 10 ordered structures (section DI.D), we conclude that the potential might satisfactorily predict the structures of proteins (one purpose for which it was intended), but could not be relied on to differentiate the relative stabilities of different structures. The situation with AMBER is somewhat different. The predicted lattice constants for the 01 structure with this potential come closest to the observed lattice constants of C& at 0 K, and are quite far from the lattice constants of c&6 at 139 or 270 K (Table 9); in particular, the volume of the predicted unit
Gibson and Scheraga
3772 J. Phys. Chem., Vol. 99, No. 11, 1995 TABLE 10: Experimental and Computed Lattice Parameters of Benzene Orthorhombic-Experimentalb
comwund C&
T(K)
a
b
C
vol
270 218 138 123 80 52.6 15 0
7.46 7.44 7.39 7.398 7.377 7.367 7.360 7.357
9.67 9.55 9.42 9.435 9.402 9.386 9.375 9.373
7.03 6.92 6.81 6.778 6.735 6.715 6.703 6.701
126.8 122.9 118.5 118.3 116.8 116.1 115.6 115.5)
~
C& c6D6 c6D6 c6D6 c6D6 (c6D6 compound C&
Orthorhombic--Computed potential' a b 9.511 Test 7.491 AMBER
ws
C& c6H6
9.169 9.232
7.430 7.491
C
vol
6.946 6.607 7.034
123.7 112.5 121.6
Monoclinic-Experimentald T (K)
a
294
5.417
b 5.376
C
B
vol
7.532
110.00
103.1
Monoclinic-Computed' uotentialC Test AMBER
ws
a
5.424 5.258 5.330
b 5.521 5.350 5.346
C
B
vol
7.433 7.210 7.569
105.90 104.74 107.79
107.0 98.1 102.7
a, b, c, in A; /3 in deg; volume in A3/molecule. Data for C& from refs 39 and 43; data for c@6 from ref 29 (data at 0 OK, extrapolated). Test, potential from Paper I, Table 2; AMBER, potential from ref 31; WS, Williams and Stan potential taken from ref 19. *Pressure approx 25 kbar; all data refer to c6&.e Pressure term at 25 kbar added.36
cell is about 10% lower than that of the C a 6 crystal unit cell. Comparison of the predicted lattice constants for the ml structure at 25 kbar with the observed lattice constants at high pressure also shows discrepancies; for example, the error in the predicted value of B is more than 5" and the volume is again too low. The results indicate that the AMBER potential (based on the part of it examined here) is a zero-temperature potential; leaving aside questions of excessive molecular fle~ibility?~*~' structural predictions for macromolecules by energy minimization with AMBER are likely to be more compact than observed structures. On the other hand, our results suggest that relative stabilities predicted with this potential might be satisfactory. The best overall agreement between predicted and observed lattice constants, together with relative stability, was obtained with Williams' potential. Dzyabchenko12 also considered Williams' potential to be the most satisfactory in these respects, out of several that he tested. It should be noted that Williams' potential was not optimized for benzene alone, but for a set of 18 hydrocarbon structures. The major discrepancy that we observed with Williams' potential involved the predicted value for the lattice constant b for the 01 structure, which in our hands was lower than the lowest value observed for any of the orthorhombic c&i or C.@6 crystals. The agreement between the predicted lattice constants for the ml structure at 25 kbar and the lattice constants observed in the high-pressure crystal was better than with either of the other two potentials. This potential is well suited for predictions of crystal structure at low and moderate temperatures. It should be noted that, in computations by Williams and his collaborators, as in those of Dzyabchenko, the attractive part of the nonbonded potential is computed accurately by accelerated con~ergence:~while the repulsive part is truncated; therefore, their results may not agree exactly with ours.
One final question concerns whether any of the other ordered structures that we found corresponds to any observed structure of benzene. Dzyabchenko and Basilevskii16suggested that their P2Jc (I) structure, which corresponds to our m2 structure, might be an intermediate between the known low-pressure and highpressure crystal forms of benzene. However, the lattice parameters of the computed structure do not agree with those of any known benzene crystal; in particular, the lattice parameters assigned by Thiery and U g e r to the benzene I1 crystal at intermediate pressures52 do not resemble those of any of our ordered structures. It remains to be seen whether future experimental work will identify a benzene lattice that corresponds to another of the structures found by us or by Dzyabchenko. Acknowledgment. This work was supported by grants from the National Science Foundation (DMB90-15815 and INT900411), from the National Institute of General Medical Sciences (GM-14312) of the National Institutes of Health, U.S. Public Health Service, and from Hoffmann-La Roche, Inc. Nearly all the computations were carried out using .the Cornel1 National Supercomputer Facility, a resource of the Center for Theory and Simulation in Science and Engineering at Cornel1 University, which is funded in part by the National Science Foundation, New York State, the IBM Corporation, and members of its Corporate Research Institute. The coordinates of the 10 ordered structures have been deposited with QCPE (No. 664). Appendix A. Superposition of Planar Molecules with Point Symmetry Superposition of molecule m on to molecule n involves determining the singular value decomposition (SVD)of a certain matrix C formed from the atomic coordinates of the two molecules (Paper I, Appendix B). The formula for C is given in Paper I, Appendix B, eq B1. Let the SVD of C be
where dl 1 d2 1 d3 2 0. If pi and qi (1 I i I3) are the column vectors of P and Q,respectively, then
cq,= dipi and the matrix of the superposition is
If d3 > 0, the vector p3 is uniquely determined through eq A2 once the vector q3 has been determined. However, if d3 = 0 the sign of the vector p3 is no longer determined when q3 is known, and there exists an alternative superposition whose matrix is
This situation will occur whenever the two molecules to be superposed are planar. In general, R and R' will be very different matrices; if R satisfies the conditions for a crystallographic transformation (Paper I, section II.F), there is no reason why R' should satisfy them, and vice versa. In order to determine whether a transformation between two planar molecules is crystallographic it is necessary to examine R and R' separately, using the tests in Paper I, section II.F and Appendix B.
Crystal Packing without Symmetry Constraints
J. Phys. Chem., Vol. 99,No. 11, 1995 3773
For molecules with internal symmetry, a further complication arises from the fact that there are several ways of obtaining a one-to-one correspondence between the atoms in the two molecules. In general, some of these correspondences will give rise to crystallographic transformations and others will not, and each manner of superposition must be examined separately. For example, two planar symmetrical benzene molecules can be superposed in 12 different ways. There will be two superposition matrices to examine for each of these correspondences, giving a total of 24 possible superpositions to be tested. With such a large number of superpositions, there is a significant chance that more than one of them will satisfy the conditions for a crystallogrphic transformation.
Appendix B. Equivalence of Symmetry Relations in Observed and Conventional Unit Cells. In this appendix, we show that if det M = 1, the symmetry relations between two molecules are of the same type whether their coordinates are referred to the frame of the observed unit cell or to that of the conventional unit cell. Here, M is the matrix relating the observed and conventional lattice vectors (see eqs 2 and 4 in the text). We note f i s t that all symmetry elements of the space group must be represented in the final structure. It follows that all components of M must be integers; this is obvious if the conventional unit cell is primitive. If the conventional unit cell is centered it must have a copy of the asymmetric unit at each centering point; if, also, one of the observed lattice vectors ends at a centering point, generation of the entire lattice using this vector would put two copies of the asymmetric unit at that centering point. Thus, the observed lattice vectors cannot end at centering points, which implies that, for centered cells, all components of M are integers also. Since M is composed of integers and det M = 1, all components of M-' are integers. Refemng to eqs 9 in section II.C, application of eq 8 to these equations leads to
S =Q ~ S Q with Q being an orthogonal matrix. If {S,s} is a crystallographic symmetry transformation, the matrix N defined by
N = USLT
(B2)
is a matrix of integers (Paper I, section II.F). From eq 7 (section n.C) and eq B 1, the corresponding matrix
Since the components o,f all the matrices on the right hand side of eq B3 are integers, N is composed of integers. In addition, the trace and determinant of S are equal to the trace and determinant of S; hence, the two matrices define the same crystallographic point transformation. Finally,
frS = MTUs Hence, Ui$will have integral or fractional components according as Us does. The type an! order of the crystallographic transformations {S,s} and {S,i$} are the same.
References and Notes (1) Williams, D. E. J. Chem. Phys. 1966, 45, 3770. (2) Evans, D. J.; Watts, R. 0. Mol. Phys. 1976, 31, 83. (3) Williams, D. E.; Star, T. L. Comput. Chem. 1977, 1 , 173. (4) Rerat, B.; Rerat, C. J . Chim. Phys. Chim. Biol. 1981, 78, 109.
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