- Email: [email protected]
Simulation of molecular reorientation in crystals
Computer Physics Communications 39 (1986) 221—231 North-Holland, Amsterdam
221
SIMULATION OF MOLECULAR REORIENTATION IN CRYSTALS J.C.A. BOEYENS and D.C. LEVENDIS Department of Chemistry, University of the Witwatersrand, Johannesburg, South Africa Received 19 July 1985
PROGRAM SUMMARY Title of program: ORIENT Catalogue number: AADZ Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue) Computer: CDC Cyber 750; Installation: Centre for Computing Services, CSLR, Pretoria, South Africa Operating system: CDC NOS 2.2
total interaction of a rotating molecule with the whole lattice is often inconclusive and the physical process would be better simulated by reorientation in a non-static environment. Method of solution Reorientation of disc-like molecules in crystals are simulated by rotation about molecular plane normals with preservation of translational symmetry. To allow for the independent reorientation of symmetry-related molecules all simulations are done in space group P1. The lattice potential energy over a suitable fragment of interest is calculated at each rotational state and mapped in rotational space.
Programming language used: Fortran 77 High speed storage used: 62 100 words (CDC 750)
Restrictions on the complexity of the problem The program handles up to three independent molecular or
No. of magnetic tapes required: 9 tapes or disc files
rigid-group rotations about any specified vectors. In all successful applications to date however, the problems were defined in terms of various sets of two independent rotations about the plane normals of flat molecules. Potential energy is calculated
Otherperipherals used: line printer, terminal or card reader
at rotational stations only and minimization as a function of individual atomic coordinates is not provided for. Automatic
No. of bits in a word: 60 (CDC 750)
No. of lines in combined program and test deck: 1739
adjustment of cell parameters cannot be done.
Keywords: crystallography, molecular reorientation, lattice energy, concerted rotation, disorder, phase transition, structure simulation
Typical running time For 13 step rotations of two molecules of 52 atoms each and a cut-off at 5 A, 470 s of CDC 750 CPU time is required.
Nature of physical problem Phase transitions of the A-type are common in molecular crystals where disorder—order transformations are possible because of molecular reorientation. Simulation in terms of the
Unusual features of the program Energy minimization routines Cannot escape from local minima. This program can serve to locate subminima in potential energy surfaces, to be followed by minimization.
OO1O-4655/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
222
J. C.A. Boeyens, D.C. Levendis
/
Simulation of molecular reorientation in crystals
LONG WRITE-UP 1. Introduction Packing consideration used to be an extremely useful criterion for judging the acceptability of trial structures before the powerful new direct methods became established in crystallography [1]. A systematic procedure for generating alternative models compatible with the available structural information was described by Hirshfeld [2]. The effects of packing are also of importance in determining orientation in disordered molecular crystals. For the study of rotationally disordered molecular crystals it is therefore desirable to devise a systematic way of mapping lattice energy as a function of different molecular orientations. The program ORIENT was written for this purpose. It calculates the non-bonded potential energy environment of various rotational arrangements of molecules in a crystal. The rotational arrangements are systematically obtained by rotating up to three independent molecules or rigid groups about any specified vectors. The rotation vectors are defined in terms of fractional coordinates. There is also an option to calculate the rotation vector as the unit vector normal to the least-squares plane through chosen atoms. In this case the centroid of these positions is used as the origin of rotation. Input coordinates éould be taken from the observed crystal structure, preferably with idealised geometry of the molecules. Hydrogen atoms should be placed in calculated positions with C—H distances consistent with the atom—atom potentials used [3]. It is important to note that unless rotation vectors and grid size in rotation space are chosen with care detail in the potential energy surface can easily be overlooked. The program is most useful for crystals of orthorhombic or lower symmetry, with Z = 4 or 2. Typically the symmetry should be relaxed to P1 anyway, and pairs of units rotated independently. At each rotational state of the asymmetric unit, the potential energy is computed by summing all the non-bonded atom—atom interaction energies for contacts within the desired maximum radius. These energies are systematically”tab~~tedas a function of molecular rotation. Alternatively a
separate program can be used to contour the resulting potential energy surfaces. Three modes of calculation are offered: (i) isolated molecule(s), (ii) fixed lattice mode (keeping the environment fixed in the input configuration) and (iii) concerted rotation mode the molecules in the environment maintain the same orientation as those in the reference set. The output level can be specified: If required, details of all interatomic interactions at each rotational position are written to a separate output file. Also, coordinates at each rotational position are stored on a separate file. These are useful to recover as trial structures for predicted disorder models. ORIENT is organised into four logical units, shown in the flow diagram (fig. 1). The input is read into common block arrays using the free format read routine, FFREAD (adapted from similar routines in EENY [4], and SHELX [5]). The —
O~,J~NT CONTROL
PROGRAM
iaiuii susRouTINEsj DIRC0S. ROTM
M%’. VV.
I
__________________
/DAT1/. /cooR/.
/ITGRI/, /uToR2/. /poTN/. /AIIATR/.
/5TH!
______
Ef8~D] _____
_________
~Lfli~1 SET UP
_________________
7
8ff~.IE
GENERATES REQUIRED
7 ~
INITIAL
ROTATIONAL ARRANGE-
PARAMETERS
MENTS
PEE
NBDIST COMPUTES LEAST— SQUARES
GENERATES NEIGHBOURS AND COMPUTES [!~ON-BONDED DISTANCES
COMPUTES NON-BONDED ATOM-ATOM POTENTIAL
Fig. 1. Flow diagram of ORIENT.
4
1
TABULATES
J. CA. Boeyens, D.C. Levendis
/ Simulation of molecular reorientation
program uses 61 K of storage and 120 Kbytes of memory for an average run. As can be seen from fig. 1, an overlay or segmented tree structure could easily be implemented if space became a problem. The input, output and scratch tapes used are mitialised in subroutine FFREAD. They are: Tape Tape Tape Tape
1 2 3 6
Input, Output, Contour map data output (optional), Coordinates of each rotational position (“),
Tape 7 Tape 8J Tape 9
Scratch tapes to save coordinates used in subroutine ROTATE, Detailed output of interatomic distances (optional).
A summary of input instructions is included in the listing as subroutine INSTRU.
in crystals
223
Note: “input coordinates” refer to the unchanged input fractional (or orthogonal) coordinates, while “starting coordinates” refers to coordinates after rotation to the starting position. Matrix vector products are required at several points in the program. For example transformation of fractional coordinates X = (x, y, z) to orthogonal coordinates X0 = (xO, yO, zO) is accompushed by xo = M x where M is a 3 X 3 matrix Specifically Mu = a, M12 = b cos y, M13 M22 M23
= =
c cos b~
cos y/sin y, M33 =c(1 cos2fl_(M23)2)~’2, M2l = M31 = M32 = 0, =
c(cos a
—
cos
—
2. Description of the program ORIENT, consisting of only 6 lines, is simply the control program that calls FFREAD, SETUPI, ROTATE and PEEMAP in sequence. 2.1. Subroutine FFREAD This free format read routine has been adapted and modified from code in the programs SHELX [5] and EENY [4]. Each input record is identified by the first four characters, KEY. The remaining characters on the card are decoded into numerical form by the subroutine FREFOR and returned to FFREAD in the array FN(80). Each number in this array is then assigned to the correct variable, all in labelled common arrays, according to the code in KEY.
(1)
where a, b, c, a, $, y are the unit cell constants in àngströms and degrees. The product, M X, is computed by the subroutine MV simply by calling this routine with tfle required matrix (as a premultiplier) and the vector. Function VV calculates the dot product of two vectors. Subroutine DIRCOS calculatles the direction cosines, XD, of a vector, given the orthogonal coordinates of two points along it. The direction cosines and the required angle of rotation, p, are used in subroutine ROTM to compute the general rotation matrix, R [6]. Note that here, R is the transpose of the matrix in the International Tables [6] since it is used as a premultiplier. If XD = (1, m, n) are the direction cosines of the rotation vector, then Ru = cos p + 11(1 cos —
2.2. Subroutine SETUPI As the name implies, this subroutine sets up the initial parameters required for the job. It calculates the orthogonal coordinates, least-squares planes, coordinates of the rotation vectors, and rotation vectors, and rotates the molecules to the desired starting position using the utility subroutines.
R12=!m(1 —cos p)—n sinp, R13 = nl(1 cos p) + m sin p, —
R21
=
im (1
—
cos p)
+
n sin p.
R22=cosp+mm(1—cosp), R23
=
mn(1
—
cos p) —1 sin p,
(2)
224
J. CA. Boeyens, D.C. Levendis
R31
=
ln(1
R32
=
mn (1
—
cos p)
R33
=
cos p
+
nn(1
—
cos p)
—
—
/
Simulation of molecular reorientation in crystals
m sin p.
this plane then identifies the vector. Having ob-
/ sin ~
tamed these, the direction cosines and rotation matrix are calculated as before. The different coordinate sets are housed in four
+
COS
p).
In the calculations involving rigid bodies [7], the rotation matrix typically involves three angles, the Eulerian angles, which relate the internal molecular axes to the crystal axes. However, only one rotation is performed here at a time in order to systematically step through the rotational arrangements. If the rotation vector is itself rotated by one of the subsequent rotations, this must be specified by the user in the input. The vector coordinates are then rotated by the desired amount. It is for this reason that fractional coordinates of the rotation vector are required in the input. The first point of the rotation vector is taken as the origin of rotation. In the case of planar aromatic molecules, this point is usually the centroid of the molecule. If the origin and terminal points of the rotation vector are given by the orthogonal vectors VA and VB then the direction cosines are XD = 1/L(VB— VA), where L is the length of the vector. An atom with orthogonal coordinates xo can be rotated by ~ degrees to a new position XOS by the operation XOS = R( XO
—
VA) + VA.
The fractional coordinates, XR, of this position will be XR
=
M ‘XOS.
The complete operation to obtain a new set of fractional coordinates rotated by some angle 41 is thus (3) XR = M 1(R( MX VA) + VA). —
-
If the fractional coordinates of the rotation vector are not supplied in the input, they are calculated as points on the normal to a given plane by subroutine LSPLN. LSPLN calculates the least-squares plane through a given set of points according to the method of Schomaker, Waser, March and Bergman [8]. The centroid (average) of the positions specified for the least-squares plane calculation is taken as the origin of the rotation vector. The end point of the unit vector, normal to
arrays in common block COOR. These are X(200,3), XO(200,3), XS(200,3) and XOS(200,3) for the input fractional and orthogonalized coordinates and starting fractional and orthogonalized coordinates, respectively. The arrays X, XO and XS are unchanged during program runs, while XOS is rewritten for each rotation. An additional array, XR(200,3), is used in subroutine ROTATE for the fractional rotated coordinates of each position. 2.3. Subroutine ROTATE This routine, together with subroutines NBDIST and NBENG, constitutes the main body of the program. Up to three rigid bodies or molecules can be rotated as required by the ROTD input instructions. The rotations are performed in nested loops, where the third rotation given in the input forms the innermost ioop. Each rotational position is identified in two possible ways; (i) the degrees away from the initial position, (0, 0, 0), of each rotated unit, given as (~,412. 41~) and (ii) the integer NRID defined by NRID =
ioooo NRI + 100 NR2 + NR3
(4)
where NRI, NR2, NR3 refers to the rotational position of each unit. Thus, for example, if unit 1 was rotated from 15°to 15°in 50 steps and unit 2 from 10° to 100 in 2° steps, then NRID = 10101 refers to the starting position at (—15, 10, 0) and NRID = 40601 to the input position (0, 0, 0). Recall that for each rotation the direction cosines and rotation matrix need to be recalculated. After each rotation, the rotated fractional coordinate array, XR(200,3) is rewritten, so defining a new asymmetric unit. This new arrangement is passed to subroutine NBDIST to calculate all non-bonded distances and the potential energy of its environment. The process of operating with the rotation matrix on previously rotated coordinates at each step needs to be treated with caution. Table I summarises the problem. The first column —
—
—
J. CA. Boeyens, D.C. Levendis
/ Simulation
of molecular reorientation in crystals
225
position all Cartesian components of the coordinate difference is less than DMAX, the actual
Table I Correct rotational positions
Incorrect
10101 10201 10301 20101 20201 20301
10101 10201 10301 20301 20401 20501
lists the required rotational positions, as identified by NRID numbers, for two rotations of unit 1 and three of unit 2. If no precautions are taken the above process would yield the positions shown in the second column. What is needed, therefore, is simply to save the KOlOl coordinates and after all the unit 2 rotations are completed, to replace the XR(200,3) array with these. Thus for two rotations, one scratch tape (tape 7) is needed to save the coordinates at each rotation of unit 1. Two tapes are needed for three rotations (tape 8). In principle it should be possible to map more than three rotations in this matter. However, the time required becomes prohibitive. 2.4. Subroutine NBDJST The array XR(200,3) which defines a new asymmetric unit forms the input to NBDIST. Using the specified symmetry (preferably only translation) all interatomic distances less than a given maximum, DMAX, from each atom are computed. The algorithm used was described by Rollett [9]. Unwanted contacts such as bonded atoms and 1,3 intramolecular contacts are eliminated. In fact, all contacts within each unit are eliminated. Rollett’s algorithm can be summarised as follows: Given the fractional coordinates of one atom (taken from the array XR(200,3)) and the symmetry operators for the lattice, each symmetry position of the atom is generated in turn. It is then translated in the negative direction parallel to one coordinate axis by sufficient unit cells to make the separation between it and the reference atom greater than DMAX hngstroms. It is then returned, one unit cell at a time, until the difference exceeds DMAX in the other direction. If, at any intermediate
distance is calculated and compared with DMAX. By suitable modification to the algorithm it is possible to calculate the non-bonded distances (and hence the environment potential energy) for either an isolated asymmetric unit in fixed lattice mode or the concerted rotation model. One needs an unambiguous identification of atomic positions in neighbouring molecules. This can be done using the integer IDTAR, IDTAR = 10000 nx + 1000 ny + 100 nz + is, (5) where nx, ny, nz refer to the number of unit translations along the a, b, or c axes and “is” is the number of the symmetry operator. The number 55501 is taken as the reference position. Thus, 54501 and 65501 refer to unit translations in the negative y and positive x directions, respectively. The target atoms are completely identified by the number NRID and IDTAR. Calculations for an isolated unit take only interactions with IDTAR = 55501 into account. The concerted rotation mode is actually implicit in the algorithm. The translations and symmetry operations are applied to the new asymmetric unit at each step. Hence, the orientation of neighbours generated by this procedure must be symmetry related to that of the reference molecules. Clearly the choice of asymmetric unit is critical and for most cases this should be one complete unit cell. the fixed lattice mode is not that simple to simulate using this algorithm, but it is possible by indirect means. The method requires use of the original input fractional coordinates at (0, 0, 0) to generate the target atoms. The environment will then have the unchanged (fixed) conformation. A problem arises for the reference atoms though, since effectively two asymmetric units will be superimposed; the rotated set and the original set. Two loops are necessary to calculate the environment energy: one to compute the “interasymmetnc unit” energy and a second to calculate the interaction energy between atoms within the same asymmetric unit. For the first loop interactions with IDTAR = 55501 are ignored and thereafter IMODE is set to 3 (isolated unit) for the second loop. By changing the set used to generate the
226
J. C.A. Boeyens, D.C. Levendis
/ Simulation
fixed lattice environment, this procedure can easily be used to study the rotational behaviour of molecules in various environments. For every nonbonded distance found to obey the conditions of that particular mode, the non-bonded potential energy is computed by subroutine NBENG. 2.5. Subroutine NBENG
/~1/ 2 Z
— —
Having computed the atom—atom potential energy it is summed into the 3-dimensional array EPEE (NRI, NR2, NR3), which holds the total environment energy of each rotational position. In ORIENT the maximum number of increments set for rotations one and two is 25, and 5 for rotation 3. 2.6. Subroutine PEEMAP
The general form of the non-bonded interaction potential used here can be written as ENONB
of molecular reorientation in crystals
jf~A exp~
—
BR ~ j/r
—
the environment can quite easily be mapped as a potential function energies of molecular rotations.
Cr
+Kq1q2/r),
Finally, given the 3-dimensional array EPEE,
(6)
where A, B, C and d are constants, r is the interatomic distance, q1 and q2 are the charges on the target and reference atoms in units of protons and K is a dimensional constant converting energies to kJ mol The scale factor, Z, relates to the number of molecules in the asymmetric unit. For example it can be used to compare calculations done with one molecule, Z = 1, with a second set having Z = 2, which will have more neighbours. The scale of energies will then be approximately the same, but one should guard against making too many deductions from such comparisons since the number of intermolecular contacts for Z = 2 need not be twice that for Z = 1. The factor f is used to ensure that interactions within the asymmetric unit are treated correctly. Normally f = 1, but for all interactions with IDTAR = 55501, f= 0.5. Since interactions within the asymmetric unit will always be counted twice (e.g. Cl.. H2 and H2.. Cl), only half the energy is added for each interaction, Setting B = 0 or d = 0 transforms eq. (6) into Lennard-Jones [10] or Buckingham [11] potentials, respectively. It is possible to assign a charge to each individual atom in the input. This may be useful where these charges can be obtained from molecular orbital calculations. An option to include a harmonic torsional potential in the energy has been given in the current program. However, it is much more efficient to simply add these rotational energies directly to the array EPEE and this option is not really necessary.
For each increment of unit 3, the energies at each position are tabulated with 41~across and 412 down the page. The entire array is also written to tape 3 together with other information needed by the contour drawing programs.
~.
.
.
3. Description of the test run The test run is based on the analysis of concerted rotation of aromatic molecules in the anthracene—tetracyanobenzene charge-transfer complex, around plane normals through the molecular centroids, defined by fractional coordinates in CENI to CEN4 instructions. The CELL instruction inputs the monoclinic cell constants. NDAT specifies 2 types of rotation, 84 atoms per asymmetric unit, 2 symmetry operators and 6 atom—atom interaction potentials. MODE specifies a maximum 5 A non-bonded interaction distance, concerted rotation mode via the integer 1 and 2 formula units per asymmetric unit. ROTD declares the limits for each rotation, in the order: starting position, final position, increment (all in degrees), followed by specification of the rotation vector. The relevant numbers (— 81, etc.) refer to entries in the coordinate list. The negative sign stipulates the vector as the normal to a leastsquares plane, defined by LSPL instructions. Further numbers define coordinates to be rotated (atoms 1 through 24). PRNT selects suppression of printed energies at each step (2), printing of orthogonal coordinates, least-squares plane parameters, etc., for initial checking (1), transfer of calculated energy surfaces to a tape for plotting (1),
J. C.A. Boeyens, D.C. Levendis
/
Simulation of molecular reorientation in crystals
227
maximum bond distance of 2.4 A (0). CONN prevents interaction between atoms 81 and 82. The first part of the output produced by the test input serves primarily for ‘input control and lists details of interaction energies at each rotational step. The final two pages summarize the results of the calculation. Some typical output pages are reproduced at the end of the paper.
quences. It has been used to identify disorder in crystals where other methods have failed and also to simulate and predict order—disorder phase transformations. It has been used successfully for the analysis of phase relationships in crystals of ferrocene and nickelocene [13], tetracyanobenzene—anthracene [14], pyrene [15] and pyromellitic dianhydride—perylene [15].
4. Discussion
References
From the description of the program it is seen to be supprisingly easy to construct a concerted rotation model. Admittedly it is limited to cases of low symmetry and preferably to cases with at most four rotating units in the unit cell. Nevertheless, it is a powerful tool for investigating and characteristing rotational disorder in crystals. The qualitative descriptions of disorder resulting from this model are quite different from results of other models such as the fixed lattice or Monte Carlo methods. Molecular positions at the predicted subminima can be used as trial structures in the crystallographic refinement of a disordered structure. The lattice energy of each position could also be minimised with respect to molecular orientation using a program such as WMIN [12] to yield more accurate relative energies at the subminima. Mapping the concerted rotation of molecules in a unit cell of lower symmetry has some important conse-
[1] Al. Kitaigorodsky, Molecular Crystals and Molecules (Academic Press, New York, 1973). [2] FL. Hirshfeld, Acta Cryst. A24 (1968) 301. [3] D.F. Williams, Spectr. Acta 37A (1981) 835. [4] S. Motherwell, EENY, Unpublished computer program, Cambridge (1974). [5] G.M. Sheidrick, in: Computing in Crystallography,eds. H. Schenk et al. (Deift Univ. Press, Delft, 1978). [6] International Tables for Crystallography, vol. 2 (Kynoch Press, Birmingham, 1972) p. 63. [7] B.T.M. Willis and A.W. Pryor, Thermal Vibrations in Crystallography (Cambridge Univ. Press, London, 1975). [8] [9] [10] [11]
V. Schomaker et al., Acta Cryst. B24 (1968) 63. J.S. Rollett, Computing Methods in Crystallography (Pergamon Press, Oxford, 1965). J.E. Lennard-Jones, Physica 4 (1937) 941. R.A. Buckingham, Proc. Roy. Soc. (Lond) A168 (1938) 264. [12] W.R. Busing, ORNL-5747, WMIN (1981). [13] D.C. Levendis and J.C.A. Boeyens, J. Cryst. Sped. Res. 15 (1985) 1.Boeyens and D.C. Levendis, J. Chem. Phys. 80 [14] J.C.A. (1984) 2681. [15] J.C.A. Boeyens and D.C. Levendis, J. Chem. Phys. 83 (1985) 2368.
228
J. C.A. Boeyens, D.C. Levendis
/ Simulation of molecular reorientation
in crystals
TEST RUN OUTPUT SETUP INITIAL
PARAMETERS FOR
A*N*T*H* ~
MATRI CES DL 9.5850 .0808 .8808 12.7480 .0808 .0088
OLI —.3171 .8080 7.4182
.1852 .0808 .0008
ORTHOGONAL
COORDINATES BEFORE ROTATION
Cl C2 C3 C4 C12 C13 C14 CM1 CM2 CM3 CM4 CM12 CMI3 CMI4 Hi H3 H4 Hi3 H14 HMI HM3 HM4 +113 1+114 C*i C*2 C*3 C*4 C*12 C*13 C*14 C:1 C:2 C:3 C:4 C:12 C:13 C:14 H*i H*3 H*4 H*13 H*14 H~I H:3 H:4 H:i3 H:14
.08808 1.15292 2.32299 3.44286 —1.15276 —2.32936 —3.44648 .08127 1.15293 2.32918 3.44648 —1.15292 —2.32380 —3.44286 —.08177 2.32556 4.27914 —2.33698 —4.28691 .08526 2.33645 4.28699 —2.32557 —4.27924 4.75250 5.98542 7.87549 8.19536 3.59974 2.42314 1.38602 4.75377 5.90543 7.08168 8.19898 3.59958 2.42950 1.30964 4.75073 7.87886 9.83164 2.41552 .46559 4.75776 7.88895 9.83949 2.42693 .47326
1 I 1 1 1 1 1 1 1 I 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2
1.48751 .71733 1.48891 .71657 .71529 1.40139 .78458 —1.40751 —.71529 —1.48139 —.70458 —.71733 —1.40891 —.71657 2.42722 2.42849 1.22483 2.42897 1.28928 —2.42722 —2.42097 —1.20928 —2.42849 —1.22483 7.78151 7.09133 7.78291 7.09057 7.08929 7.77539 7.07858 4.96649 5.65871 4.97261 5.66942 5.65667 4.96589 5.65743 8.88122 8.80249 7.59883 8.79497 7.58328 3.94678 3.95303 5.16472 3.94551 5.14917
.08888 —.40349 -.84i28 —1.22306 .48853 .84343 1.22113 .08371 —.48823 —.84373 —1.22113 .40349 .84136 1.22306 —.88296 -.87826 —1.51043 .87278 1.503~i —.8118’a —.87374 —1.50272 .87048 1.51843 .08880 —.48349 —.84128 —1.22306 .40853 .84343 1.22113 .08371 —.40823 —.84373 —1.22113 .40349 .84136 1.22386 —.00296 —.87826 —1.51843 .87278 1 .50301 —.01186 —.87374 —1.50272 .87848 1.51843
U
.8800 .0784 .0080
*i *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *2 *2 *2 *2 ‘2 *2 *2 ‘2 *2 *2 *2 *2 *2 *2 *2 *2 *2 *2 *2 *2 *2 *2 *2 *2
.0045 .0000 .1349
ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE
.1052 .8088 .0845
FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM
—i5.0 —15.0 —15.8 —15.8 —15.0 —15.0 —15.0 —15.8 —15.8 —15.0 —15.8 —15.8 -15.8 —15.0 —15.0 —15.0 —15.8 —15.0 —15.8 —15.8 —15.8 —15.8 —15.8 —15.8 —15.8 —15.8 —15.8 —15.8 —15.8 —15.8 —15.8 —15.0 —15.8 —15.8 —15.8 —15.0 —15.0 —15.0 —15.8 —15.8 —15.8 —15.0 —15.0 —15.0 —15.8 15.8 -15.8 —15.0
HKL SPACINGS
.8000 .0784 .0080
TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO
i5.0 15.0 15.8 15.0 15.0 15.8 15.0 15.8 15.8 15.8 15.0 15.8 15.8 15.8 15.8 15.8 15.8 15.0 15.0 15.8 15.8 15.8 15.0 15.0 15.8 15.8 15.8 15.0 15.8 15.8 15.0 15.0 15.8 15.8 15.8 15.0 15.8 15.8 15.0 15.0 15.0 15.8 15.8 15.8 15.8 15.8 15.8 15.0
.8888 .8800 .1349
DES. DES. DES. DES. DES. DES. DES. CES. DES. DEli. DES. DEG. DES. DES. DES. DES. DES. DEO. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES.
ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ASOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT
9.4963 12.7488 7.4182
VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR ‘.‘ECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR
—Si —81 —81 —81 -81 —81 —81 —81 —81 —81 —81 —81 —81 —81 —81 —Bi -81 —81 —81 —81 —81 —81 —81 —81 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83
TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO
—82 —82 —82 —82 —82 —82 —82 —82 —82 —82 —82 —82 —82 —82 —82 —82 —82 —82 -82 —82 —82 —82 —82 —82 -84 —84 —84 —84 —84 —84 —84 —84 —84 —84 —84 —84 -84 —84 —84 —84 —84 —84 —84 —84 —84 —84 —84 —84
J. C.A. Boeyens, D.C. Levendis CS MS C6 C7 NB C16 CL? N18 CMS HM5 CM6 CM? NM8 CM16 CM17 MIIB C*5. H*5 C*6 C*7 N*8 C*16 C*17 N*1B C:5 H:S C:6 C:7 N:8 C:i6 C:17 N:18 CEN1 CEN2 CEN3 CEN4 LEAST ATOM Cl C2 C3 C4 C12 C13 C14 CMI CM2 CM3 CM4 CM12 CMI3 CM14
1 2 1 1 3 I 1 3 1 2 1 1 3 1 1 3 1 2 1 1 3 1 1 3 1 2 1 1 3 1 1 3 0 8 0 0
—.15853 —.15953 .98078 2.16816 3.09348 —1.29876 —2.48684 —3.41851 —.15853 —.15853 .98078 2.16816 3.89348 —1.29876 —2.48604 —3.41851 4.59397 4.59397 5.73328 6.92066 7.84598 3,45374 2.26646 1.34199 4.59397 4.59397 5.73328 6.92866 7.84598 3.45374 2.26646 1.34199 .88088 .88088 .88808 5.93029
1.40228 2.42212 .69987 1.42813 2.000,16 .69987 1.42013 2.08016 —1.48228 —2.42212 —.69987 —1.42813 —2.08016 —.69987 —1.42013 —2.00016 7.77628 8.79612 7.07387 7.79413 8.37416 7.07387 7.79413 8.37416 4.97172 3.95188 5.67413 4.95387 4.37384 5.67413 4.95387 4.37384 .80000 .88808 .08008 6.37480
3.78511 3.70511 3.33534 2.95297 2.65286 4.87414 4.47651 4.75662 3.78511 3.70511 3.33534 2.95297 2.65286 4.87414 4.47651 4.75662 3.78511 3.70511 3.33534 2.95297 2,65286 4.87414 4.47651 4.75662 3.78511 3.70511 3.33534 2.95297 2.65286 4.07414 4.47651 4.75662 .00080 .80080 .88080 3.38718
SQUARES PLANE FOR ROTATION WEIGHT DEVIATION 1.0808 .0000 1 .8800 .8872 1.0080 —.8119 1.8888 .0847 1.8888 —.8827 1.0808 .8117 1.0008 —.0088 1 .8888 .0034 1.0800 .8825 1.0080 —.8126 1.0808 .8875 1.8088 —.0078 1.8800 .8114 1.8088 —.8053
SUM WT*DEV
/ 0 8 8 0 8 0 0 8 0 8 0 0 8 0 8 8 0 8 8 8 8 0 0 0 0 0 8 8 0 0 0 8 8 0 0 0
Simulation of molecular reorientation in crystals KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEP’r KEPT KEPT KEFT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT
FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED
1
.88008
SUM IJT*DEV**2 =
.00887 PARAMETERS OF PLANE L M N 0
LAMBDA
=
= = =
.33680 .08828 .94186 .08828
.00887
ROTATION 1 FROM CEN1 TO CEN2 WITH NEW COORDINATES .0800
.0000
.8808
AND
.0394
.8000
.1271
229
230
J.C.A. Boeyens, D.C. Levendis
/ Simulation
of molecular reorientation in crystals
~**~**rr**wr*****~n**w*~
BREAKDOWN OF THE POTENTIAL ENERGY ENVIRONMENT AT EACH ROTATIONAL POSITI ON FOR A*N~T*H*P*A~C*E*N~E *T*E~T*R*A*C~Y*A*N*O~B*E*N*Z*
INTERACTION TYPE
1... 2... 2... 3... 3... 3...
1 1 2 1 2 3
TOTALS
EREP 342.084
INTERACTION TYPE
1 1 2 1 2 3
TOTALS EREP 295.819
NO. OF INTERACTIONS CLOSER THAN 5.08 ANOSTROMS
Sl1-1S OF THE NON—BONDED INTERACTION ENERGY/ KJ/IIOL
NINTER 556 674 174 258 134 14
NINTRA 188 143 25 51 29 1
ECOUL .800
440<
ETORI .880
INTERACTION TYPE
1 1 2 1 2 3
TOTALS
2258)
ETOR2 .008
—15.0
.0
INTERASYM. —29.605 10.316 7.143 —5.431 1.148 —.977
INTRAASYM. —12.536 —1.470 —.176 —1.276 —.263 —.078
TOTALS —42.141 8.846 6.966 —6.707 .885 —1.847
—11.37
—18.80
—30.18
ETOR3 .000
EPEE —30.176 **
**
BREAKDOWN OF INTERACTIONS AT <
—15.0
NO. OF INTERACTIONS CLOSER THAN 5.00 ANGSTROMS
SUMS OF THE NON—BONDED INTERACTION ENERGY/ KJ/MOL
HINTER 562 672 182 258 140 14
NINTRA 190 141 25 53 28 1
1828
EATT —388.888
1... 2... 2... 3... 3... 3...
—15.0
1810
EATT —402.436
1... 2... 2... 3... 3... 3...
BREAKDOWN OF INTERACTIONS AT (
ECOUL .080
440< ETOR1 .880
2268) ETOR2 .808
-10.0
.0
INTERASYM. —34.8BB 4.279 7.312 —5.688 1.287 —.977
INTRAASYM. —12.604 —1.481 —.169 —1.303 —.260 —.070
TOTALS —46.692 2.798 7.143 —6.991 1.027 —1.047
—28.00
—18.93
—46.93
ETOR3 .000
EPEE —46.931
**
BREAKDOWN OF INTERACTIONS AT (
—15.0
NO. OF INTERACTIONS CLOSER THAN 5.00 ANGSTROMS
51.8-IS OF THE NON—BONDED INTERACTION ENERGY/ KJMIOL
NINTER 554 670 176 258 142 14 1814
NINTRA 192 137 25 53 29 I 440<
2254)
-5.0
**
.8
INTERASYM. —35.787 —.312 7.132 —5.816 1.637 —.977
INTRAASYM. —12.701 —1.489 —.159 —1.313 —.249 —.070
TOTALS —48.488 —1.801 6.974 —7.129 1.389 —1.847
—38.61
—i9.03
-57.64
SLH~?ARY OF THE POTENTIAL ENERGY E~.8.JIRC~.lIENTS(EPEE /K3/MOL) AT THE 49 ROTAT I ONAL POSITIONS FOR A’N~T~H~R~A’C’ E*N~ E ‘T~E~T’R’A~ C~Y~A*N~ O
PEE MAP WITH PHI3 = PHIl ACROSS PHI2 DOWN
.88
J.C.A. Boeyens, D.C. Levendis
1 —15.8 2 —18.8 3 —5.0 4 .0 5 5.0 6 10.0 7 15.8
/
Simulation of molecular reorientation ‘in crystals
1 —15.0
2 —18.0
3 —5.8
4 .8
5 5.0
6 18.8
7 15.8
—38.2 —46.9 —57.6 —66.2 —72.3 —76.4 —75.9
—47.8 —S8.8 —66.8 —73.0 —77.1 —79.5 —77.2
—57.8 —66.9 —72.7 —76.3 —77.8 —77.6 —74.0
—66.3 —73.0 —76.2 —77.8 —76.7 —74.3 —69.2
—72.3 —77.8 —77.7 —76.6 —74.1 —69.7 —62.6
—76.3 —79.3 —77.4 —74.1 —49.6 —63.5 —54.5
—75.6 —76.8 —73.5 —68.8 —62.3 —54.3 —41.4
ORDERED VALUES OF PEE IN MAP(S)
28681
EMAX
—30.2 AT NRID
18181
60201
EMAX
—41.4 AT NRID
70701
38581
EMAX
—46.9 AT NRID
=
10281
48401
EMAX
—47.8
AT NRID
=
20101
=
50301
EMAX
—54.3 AT NRID
=
70601
=
38601
El-lAX
—54.5 AT NRID
=
60301
El-lAX
—57.6 AT NRID
20781
EMAX
—57.8 AT NRID
=
38101
=
28581
El-lAX
—58.8 AT NRID
=
28281
=
58281
EMAX
—62.3 AT NRID
=
78581
—76.8 AT NRID
=
70281
EMAX
—62.6 AT NRID
—76.7 AT NRID
=
48581
EMAX
-83.5 AT NRID
EMIN
—76.6 AT NRID
=
58481
EMAX
—66.2 AT NRID
EMIN
—76.4 AT NRID
=
10681
El-lAX
EMIN
—76.3 AT NRID
=
30401
EMIN
-76.3 AT NRID
=
Er-1IN
—76.2 AT NRID
EMIN
—79.5 AT NRID
EMIN
—79.3 AT NRID
EMIN
—77.8 AT NRID
EMIN
—77.8 AT NRID
=
EMIN
—77.7 AT NRID
EMIN
—77.6 AT NRID
EMIN
—77.4 AT NRID
EMIN
—77.2 AT NRID
EMIN
—77.1 AT NRID
EMIN
—77.8 AT NRID
EMIN EMIN
=
68701 18381
58781 =
68601
—66.3 AT NRID
=
40181
El-lAX
—68.8 AT MRID
=
20301
60101
El-lAX
-66.9 AT NRID
=
38201
48301
EMAX
—68.8 AT NRID
=
70401
18481
EMIN
-75.9 AT NRID
=
10781
El-lAX
-69.2 AT NRID
=
48701
EMIN
—75.6 AT NRID
=
78101
El-lAX
—69.6 AT NRID
=
68501
EMIN
—74.3 AT NRID
=
40681
El-SAX
—69.7 AT NRID
=
50681
20 SEVERE CONTACTS THAT OCCUR IN THE
FROM
TO
C:3 H13 H*13 H:3 H3 C:13 H*3 H:13 H~13 H3 C:13 H:13 11-113 C*3 H~13 H;3 H~l3 +113 H*3 H~13
H13 C:3 01-13 C13 C:I3 H3 01113 C3 C3 C:13 H3 C3 C*3 1*113 H3 H13 H3 H*3 11113 H3
NRID 18101 10101 18181 18101 78701 70791 70701 70701 78601 70681 78401 60701 70681 78681 60601 30101 58701 70501 70S01 70501
IDTAR 65501 45501 56501 65501 55581 55501 66501 55581 55501 55501 55501 55501 44581 66501 55501 65501 55501 44501 66501 55501
01ST 2.257 2.257 2.257 2.258 2.259 2.259 2.259 2.259 2.383 2.333 2.333 2.333 2.333 2.333 1.875 1.884 1.887 1.887 1.887 1.887
ROTATIONS OF
ENONB 3.0S4 3.054 3.854 3.046 3.033 3.033 3.033 3.033 2.532 2.248 2.248 2.239 2.239 2.239 1.845 1.783 1.759 1.759 1.759 1.759 END OF JOB
A*N~T~H*R~A*C*E~N*E *T*E~T*R*A*C~Y~A*N*O*B*E*N*2~
231
221
SIMULATION OF MOLECULAR REORIENTATION IN CRYSTALS J.C.A. BOEYENS and D.C. LEVENDIS Department of Chemistry, University of the Witwatersrand, Johannesburg, South Africa Received 19 July 1985
PROGRAM SUMMARY Title of program: ORIENT Catalogue number: AADZ Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue) Computer: CDC Cyber 750; Installation: Centre for Computing Services, CSLR, Pretoria, South Africa Operating system: CDC NOS 2.2
total interaction of a rotating molecule with the whole lattice is often inconclusive and the physical process would be better simulated by reorientation in a non-static environment. Method of solution Reorientation of disc-like molecules in crystals are simulated by rotation about molecular plane normals with preservation of translational symmetry. To allow for the independent reorientation of symmetry-related molecules all simulations are done in space group P1. The lattice potential energy over a suitable fragment of interest is calculated at each rotational state and mapped in rotational space.
Programming language used: Fortran 77 High speed storage used: 62 100 words (CDC 750)
Restrictions on the complexity of the problem The program handles up to three independent molecular or
No. of magnetic tapes required: 9 tapes or disc files
rigid-group rotations about any specified vectors. In all successful applications to date however, the problems were defined in terms of various sets of two independent rotations about the plane normals of flat molecules. Potential energy is calculated
Otherperipherals used: line printer, terminal or card reader
at rotational stations only and minimization as a function of individual atomic coordinates is not provided for. Automatic
No. of bits in a word: 60 (CDC 750)
No. of lines in combined program and test deck: 1739
adjustment of cell parameters cannot be done.
Keywords: crystallography, molecular reorientation, lattice energy, concerted rotation, disorder, phase transition, structure simulation
Typical running time For 13 step rotations of two molecules of 52 atoms each and a cut-off at 5 A, 470 s of CDC 750 CPU time is required.
Nature of physical problem Phase transitions of the A-type are common in molecular crystals where disorder—order transformations are possible because of molecular reorientation. Simulation in terms of the
Unusual features of the program Energy minimization routines Cannot escape from local minima. This program can serve to locate subminima in potential energy surfaces, to be followed by minimization.
OO1O-4655/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
222
J. C.A. Boeyens, D.C. Levendis
/
Simulation of molecular reorientation in crystals
LONG WRITE-UP 1. Introduction Packing consideration used to be an extremely useful criterion for judging the acceptability of trial structures before the powerful new direct methods became established in crystallography [1]. A systematic procedure for generating alternative models compatible with the available structural information was described by Hirshfeld [2]. The effects of packing are also of importance in determining orientation in disordered molecular crystals. For the study of rotationally disordered molecular crystals it is therefore desirable to devise a systematic way of mapping lattice energy as a function of different molecular orientations. The program ORIENT was written for this purpose. It calculates the non-bonded potential energy environment of various rotational arrangements of molecules in a crystal. The rotational arrangements are systematically obtained by rotating up to three independent molecules or rigid groups about any specified vectors. The rotation vectors are defined in terms of fractional coordinates. There is also an option to calculate the rotation vector as the unit vector normal to the least-squares plane through chosen atoms. In this case the centroid of these positions is used as the origin of rotation. Input coordinates éould be taken from the observed crystal structure, preferably with idealised geometry of the molecules. Hydrogen atoms should be placed in calculated positions with C—H distances consistent with the atom—atom potentials used [3]. It is important to note that unless rotation vectors and grid size in rotation space are chosen with care detail in the potential energy surface can easily be overlooked. The program is most useful for crystals of orthorhombic or lower symmetry, with Z = 4 or 2. Typically the symmetry should be relaxed to P1 anyway, and pairs of units rotated independently. At each rotational state of the asymmetric unit, the potential energy is computed by summing all the non-bonded atom—atom interaction energies for contacts within the desired maximum radius. These energies are systematically”tab~~tedas a function of molecular rotation. Alternatively a
separate program can be used to contour the resulting potential energy surfaces. Three modes of calculation are offered: (i) isolated molecule(s), (ii) fixed lattice mode (keeping the environment fixed in the input configuration) and (iii) concerted rotation mode the molecules in the environment maintain the same orientation as those in the reference set. The output level can be specified: If required, details of all interatomic interactions at each rotational position are written to a separate output file. Also, coordinates at each rotational position are stored on a separate file. These are useful to recover as trial structures for predicted disorder models. ORIENT is organised into four logical units, shown in the flow diagram (fig. 1). The input is read into common block arrays using the free format read routine, FFREAD (adapted from similar routines in EENY [4], and SHELX [5]). The —
O~,J~NT CONTROL
PROGRAM
iaiuii susRouTINEsj DIRC0S. ROTM
M%’. VV.
I
__________________
/DAT1/. /cooR/.
/ITGRI/, /uToR2/. /poTN/. /AIIATR/.
/5TH!
______
Ef8~D] _____
_________
~Lfli~1 SET UP
_________________
7
8ff~.IE
GENERATES REQUIRED
7 ~
INITIAL
ROTATIONAL ARRANGE-
PARAMETERS
MENTS
PEE
NBDIST COMPUTES LEAST— SQUARES
GENERATES NEIGHBOURS AND COMPUTES [!~ON-BONDED DISTANCES
COMPUTES NON-BONDED ATOM-ATOM POTENTIAL
Fig. 1. Flow diagram of ORIENT.
4
1
TABULATES
J. CA. Boeyens, D.C. Levendis
/ Simulation of molecular reorientation
program uses 61 K of storage and 120 Kbytes of memory for an average run. As can be seen from fig. 1, an overlay or segmented tree structure could easily be implemented if space became a problem. The input, output and scratch tapes used are mitialised in subroutine FFREAD. They are: Tape Tape Tape Tape
1 2 3 6
Input, Output, Contour map data output (optional), Coordinates of each rotational position (“),
Tape 7 Tape 8J Tape 9
Scratch tapes to save coordinates used in subroutine ROTATE, Detailed output of interatomic distances (optional).
A summary of input instructions is included in the listing as subroutine INSTRU.
in crystals
223
Note: “input coordinates” refer to the unchanged input fractional (or orthogonal) coordinates, while “starting coordinates” refers to coordinates after rotation to the starting position. Matrix vector products are required at several points in the program. For example transformation of fractional coordinates X = (x, y, z) to orthogonal coordinates X0 = (xO, yO, zO) is accompushed by xo = M x where M is a 3 X 3 matrix Specifically Mu = a, M12 = b cos y, M13 M22 M23
= =
c cos b~
cos y/sin y, M33 =c(1 cos2fl_(M23)2)~’2, M2l = M31 = M32 = 0, =
c(cos a
—
cos
—
2. Description of the program ORIENT, consisting of only 6 lines, is simply the control program that calls FFREAD, SETUPI, ROTATE and PEEMAP in sequence. 2.1. Subroutine FFREAD This free format read routine has been adapted and modified from code in the programs SHELX [5] and EENY [4]. Each input record is identified by the first four characters, KEY. The remaining characters on the card are decoded into numerical form by the subroutine FREFOR and returned to FFREAD in the array FN(80). Each number in this array is then assigned to the correct variable, all in labelled common arrays, according to the code in KEY.
(1)
where a, b, c, a, $, y are the unit cell constants in àngströms and degrees. The product, M X, is computed by the subroutine MV simply by calling this routine with tfle required matrix (as a premultiplier) and the vector. Function VV calculates the dot product of two vectors. Subroutine DIRCOS calculatles the direction cosines, XD, of a vector, given the orthogonal coordinates of two points along it. The direction cosines and the required angle of rotation, p, are used in subroutine ROTM to compute the general rotation matrix, R [6]. Note that here, R is the transpose of the matrix in the International Tables [6] since it is used as a premultiplier. If XD = (1, m, n) are the direction cosines of the rotation vector, then Ru = cos p + 11(1 cos —
2.2. Subroutine SETUPI As the name implies, this subroutine sets up the initial parameters required for the job. It calculates the orthogonal coordinates, least-squares planes, coordinates of the rotation vectors, and rotation vectors, and rotates the molecules to the desired starting position using the utility subroutines.
R12=!m(1 —cos p)—n sinp, R13 = nl(1 cos p) + m sin p, —
R21
=
im (1
—
cos p)
+
n sin p.
R22=cosp+mm(1—cosp), R23
=
mn(1
—
cos p) —1 sin p,
(2)
224
J. CA. Boeyens, D.C. Levendis
R31
=
ln(1
R32
=
mn (1
—
cos p)
R33
=
cos p
+
nn(1
—
cos p)
—
—
/
Simulation of molecular reorientation in crystals
m sin p.
this plane then identifies the vector. Having ob-
/ sin ~
tamed these, the direction cosines and rotation matrix are calculated as before. The different coordinate sets are housed in four
+
COS
p).
In the calculations involving rigid bodies [7], the rotation matrix typically involves three angles, the Eulerian angles, which relate the internal molecular axes to the crystal axes. However, only one rotation is performed here at a time in order to systematically step through the rotational arrangements. If the rotation vector is itself rotated by one of the subsequent rotations, this must be specified by the user in the input. The vector coordinates are then rotated by the desired amount. It is for this reason that fractional coordinates of the rotation vector are required in the input. The first point of the rotation vector is taken as the origin of rotation. In the case of planar aromatic molecules, this point is usually the centroid of the molecule. If the origin and terminal points of the rotation vector are given by the orthogonal vectors VA and VB then the direction cosines are XD = 1/L(VB— VA), where L is the length of the vector. An atom with orthogonal coordinates xo can be rotated by ~ degrees to a new position XOS by the operation XOS = R( XO
—
VA) + VA.
The fractional coordinates, XR, of this position will be XR
=
M ‘XOS.
The complete operation to obtain a new set of fractional coordinates rotated by some angle 41 is thus (3) XR = M 1(R( MX VA) + VA). —
-
If the fractional coordinates of the rotation vector are not supplied in the input, they are calculated as points on the normal to a given plane by subroutine LSPLN. LSPLN calculates the least-squares plane through a given set of points according to the method of Schomaker, Waser, March and Bergman [8]. The centroid (average) of the positions specified for the least-squares plane calculation is taken as the origin of the rotation vector. The end point of the unit vector, normal to
arrays in common block COOR. These are X(200,3), XO(200,3), XS(200,3) and XOS(200,3) for the input fractional and orthogonalized coordinates and starting fractional and orthogonalized coordinates, respectively. The arrays X, XO and XS are unchanged during program runs, while XOS is rewritten for each rotation. An additional array, XR(200,3), is used in subroutine ROTATE for the fractional rotated coordinates of each position. 2.3. Subroutine ROTATE This routine, together with subroutines NBDIST and NBENG, constitutes the main body of the program. Up to three rigid bodies or molecules can be rotated as required by the ROTD input instructions. The rotations are performed in nested loops, where the third rotation given in the input forms the innermost ioop. Each rotational position is identified in two possible ways; (i) the degrees away from the initial position, (0, 0, 0), of each rotated unit, given as (~,412. 41~) and (ii) the integer NRID defined by NRID =
ioooo NRI + 100 NR2 + NR3
(4)
where NRI, NR2, NR3 refers to the rotational position of each unit. Thus, for example, if unit 1 was rotated from 15°to 15°in 50 steps and unit 2 from 10° to 100 in 2° steps, then NRID = 10101 refers to the starting position at (—15, 10, 0) and NRID = 40601 to the input position (0, 0, 0). Recall that for each rotation the direction cosines and rotation matrix need to be recalculated. After each rotation, the rotated fractional coordinate array, XR(200,3) is rewritten, so defining a new asymmetric unit. This new arrangement is passed to subroutine NBDIST to calculate all non-bonded distances and the potential energy of its environment. The process of operating with the rotation matrix on previously rotated coordinates at each step needs to be treated with caution. Table I summarises the problem. The first column —
—
—
J. CA. Boeyens, D.C. Levendis
/ Simulation
of molecular reorientation in crystals
225
position all Cartesian components of the coordinate difference is less than DMAX, the actual
Table I Correct rotational positions
Incorrect
10101 10201 10301 20101 20201 20301
10101 10201 10301 20301 20401 20501
lists the required rotational positions, as identified by NRID numbers, for two rotations of unit 1 and three of unit 2. If no precautions are taken the above process would yield the positions shown in the second column. What is needed, therefore, is simply to save the KOlOl coordinates and after all the unit 2 rotations are completed, to replace the XR(200,3) array with these. Thus for two rotations, one scratch tape (tape 7) is needed to save the coordinates at each rotation of unit 1. Two tapes are needed for three rotations (tape 8). In principle it should be possible to map more than three rotations in this matter. However, the time required becomes prohibitive. 2.4. Subroutine NBDJST The array XR(200,3) which defines a new asymmetric unit forms the input to NBDIST. Using the specified symmetry (preferably only translation) all interatomic distances less than a given maximum, DMAX, from each atom are computed. The algorithm used was described by Rollett [9]. Unwanted contacts such as bonded atoms and 1,3 intramolecular contacts are eliminated. In fact, all contacts within each unit are eliminated. Rollett’s algorithm can be summarised as follows: Given the fractional coordinates of one atom (taken from the array XR(200,3)) and the symmetry operators for the lattice, each symmetry position of the atom is generated in turn. It is then translated in the negative direction parallel to one coordinate axis by sufficient unit cells to make the separation between it and the reference atom greater than DMAX hngstroms. It is then returned, one unit cell at a time, until the difference exceeds DMAX in the other direction. If, at any intermediate
distance is calculated and compared with DMAX. By suitable modification to the algorithm it is possible to calculate the non-bonded distances (and hence the environment potential energy) for either an isolated asymmetric unit in fixed lattice mode or the concerted rotation model. One needs an unambiguous identification of atomic positions in neighbouring molecules. This can be done using the integer IDTAR, IDTAR = 10000 nx + 1000 ny + 100 nz + is, (5) where nx, ny, nz refer to the number of unit translations along the a, b, or c axes and “is” is the number of the symmetry operator. The number 55501 is taken as the reference position. Thus, 54501 and 65501 refer to unit translations in the negative y and positive x directions, respectively. The target atoms are completely identified by the number NRID and IDTAR. Calculations for an isolated unit take only interactions with IDTAR = 55501 into account. The concerted rotation mode is actually implicit in the algorithm. The translations and symmetry operations are applied to the new asymmetric unit at each step. Hence, the orientation of neighbours generated by this procedure must be symmetry related to that of the reference molecules. Clearly the choice of asymmetric unit is critical and for most cases this should be one complete unit cell. the fixed lattice mode is not that simple to simulate using this algorithm, but it is possible by indirect means. The method requires use of the original input fractional coordinates at (0, 0, 0) to generate the target atoms. The environment will then have the unchanged (fixed) conformation. A problem arises for the reference atoms though, since effectively two asymmetric units will be superimposed; the rotated set and the original set. Two loops are necessary to calculate the environment energy: one to compute the “interasymmetnc unit” energy and a second to calculate the interaction energy between atoms within the same asymmetric unit. For the first loop interactions with IDTAR = 55501 are ignored and thereafter IMODE is set to 3 (isolated unit) for the second loop. By changing the set used to generate the
226
J. C.A. Boeyens, D.C. Levendis
/ Simulation
fixed lattice environment, this procedure can easily be used to study the rotational behaviour of molecules in various environments. For every nonbonded distance found to obey the conditions of that particular mode, the non-bonded potential energy is computed by subroutine NBENG. 2.5. Subroutine NBENG
/~1/ 2 Z
— —
Having computed the atom—atom potential energy it is summed into the 3-dimensional array EPEE (NRI, NR2, NR3), which holds the total environment energy of each rotational position. In ORIENT the maximum number of increments set for rotations one and two is 25, and 5 for rotation 3. 2.6. Subroutine PEEMAP
The general form of the non-bonded interaction potential used here can be written as ENONB
of molecular reorientation in crystals
jf~A exp~
—
BR ~ j/r
—
the environment can quite easily be mapped as a potential function energies of molecular rotations.
Cr
+Kq1q2/r),
Finally, given the 3-dimensional array EPEE,
(6)
where A, B, C and d are constants, r is the interatomic distance, q1 and q2 are the charges on the target and reference atoms in units of protons and K is a dimensional constant converting energies to kJ mol The scale factor, Z, relates to the number of molecules in the asymmetric unit. For example it can be used to compare calculations done with one molecule, Z = 1, with a second set having Z = 2, which will have more neighbours. The scale of energies will then be approximately the same, but one should guard against making too many deductions from such comparisons since the number of intermolecular contacts for Z = 2 need not be twice that for Z = 1. The factor f is used to ensure that interactions within the asymmetric unit are treated correctly. Normally f = 1, but for all interactions with IDTAR = 55501, f= 0.5. Since interactions within the asymmetric unit will always be counted twice (e.g. Cl.. H2 and H2.. Cl), only half the energy is added for each interaction, Setting B = 0 or d = 0 transforms eq. (6) into Lennard-Jones [10] or Buckingham [11] potentials, respectively. It is possible to assign a charge to each individual atom in the input. This may be useful where these charges can be obtained from molecular orbital calculations. An option to include a harmonic torsional potential in the energy has been given in the current program. However, it is much more efficient to simply add these rotational energies directly to the array EPEE and this option is not really necessary.
For each increment of unit 3, the energies at each position are tabulated with 41~across and 412 down the page. The entire array is also written to tape 3 together with other information needed by the contour drawing programs.
~.
.
.
3. Description of the test run The test run is based on the analysis of concerted rotation of aromatic molecules in the anthracene—tetracyanobenzene charge-transfer complex, around plane normals through the molecular centroids, defined by fractional coordinates in CENI to CEN4 instructions. The CELL instruction inputs the monoclinic cell constants. NDAT specifies 2 types of rotation, 84 atoms per asymmetric unit, 2 symmetry operators and 6 atom—atom interaction potentials. MODE specifies a maximum 5 A non-bonded interaction distance, concerted rotation mode via the integer 1 and 2 formula units per asymmetric unit. ROTD declares the limits for each rotation, in the order: starting position, final position, increment (all in degrees), followed by specification of the rotation vector. The relevant numbers (— 81, etc.) refer to entries in the coordinate list. The negative sign stipulates the vector as the normal to a leastsquares plane, defined by LSPL instructions. Further numbers define coordinates to be rotated (atoms 1 through 24). PRNT selects suppression of printed energies at each step (2), printing of orthogonal coordinates, least-squares plane parameters, etc., for initial checking (1), transfer of calculated energy surfaces to a tape for plotting (1),
J. C.A. Boeyens, D.C. Levendis
/
Simulation of molecular reorientation in crystals
227
maximum bond distance of 2.4 A (0). CONN prevents interaction between atoms 81 and 82. The first part of the output produced by the test input serves primarily for ‘input control and lists details of interaction energies at each rotational step. The final two pages summarize the results of the calculation. Some typical output pages are reproduced at the end of the paper.
quences. It has been used to identify disorder in crystals where other methods have failed and also to simulate and predict order—disorder phase transformations. It has been used successfully for the analysis of phase relationships in crystals of ferrocene and nickelocene [13], tetracyanobenzene—anthracene [14], pyrene [15] and pyromellitic dianhydride—perylene [15].
4. Discussion
References
From the description of the program it is seen to be supprisingly easy to construct a concerted rotation model. Admittedly it is limited to cases of low symmetry and preferably to cases with at most four rotating units in the unit cell. Nevertheless, it is a powerful tool for investigating and characteristing rotational disorder in crystals. The qualitative descriptions of disorder resulting from this model are quite different from results of other models such as the fixed lattice or Monte Carlo methods. Molecular positions at the predicted subminima can be used as trial structures in the crystallographic refinement of a disordered structure. The lattice energy of each position could also be minimised with respect to molecular orientation using a program such as WMIN [12] to yield more accurate relative energies at the subminima. Mapping the concerted rotation of molecules in a unit cell of lower symmetry has some important conse-
[1] Al. Kitaigorodsky, Molecular Crystals and Molecules (Academic Press, New York, 1973). [2] FL. Hirshfeld, Acta Cryst. A24 (1968) 301. [3] D.F. Williams, Spectr. Acta 37A (1981) 835. [4] S. Motherwell, EENY, Unpublished computer program, Cambridge (1974). [5] G.M. Sheidrick, in: Computing in Crystallography,eds. H. Schenk et al. (Deift Univ. Press, Delft, 1978). [6] International Tables for Crystallography, vol. 2 (Kynoch Press, Birmingham, 1972) p. 63. [7] B.T.M. Willis and A.W. Pryor, Thermal Vibrations in Crystallography (Cambridge Univ. Press, London, 1975). [8] [9] [10] [11]
V. Schomaker et al., Acta Cryst. B24 (1968) 63. J.S. Rollett, Computing Methods in Crystallography (Pergamon Press, Oxford, 1965). J.E. Lennard-Jones, Physica 4 (1937) 941. R.A. Buckingham, Proc. Roy. Soc. (Lond) A168 (1938) 264. [12] W.R. Busing, ORNL-5747, WMIN (1981). [13] D.C. Levendis and J.C.A. Boeyens, J. Cryst. Sped. Res. 15 (1985) 1.Boeyens and D.C. Levendis, J. Chem. Phys. 80 [14] J.C.A. (1984) 2681. [15] J.C.A. Boeyens and D.C. Levendis, J. Chem. Phys. 83 (1985) 2368.
228
J. C.A. Boeyens, D.C. Levendis
/ Simulation of molecular reorientation
in crystals
TEST RUN OUTPUT SETUP INITIAL
PARAMETERS FOR
A*N*T*H* ~
MATRI CES DL 9.5850 .0808 .8808 12.7480 .0808 .0088
OLI —.3171 .8080 7.4182
.1852 .0808 .0008
ORTHOGONAL
COORDINATES BEFORE ROTATION
Cl C2 C3 C4 C12 C13 C14 CM1 CM2 CM3 CM4 CM12 CMI3 CMI4 Hi H3 H4 Hi3 H14 HMI HM3 HM4 +113 1+114 C*i C*2 C*3 C*4 C*12 C*13 C*14 C:1 C:2 C:3 C:4 C:12 C:13 C:14 H*i H*3 H*4 H*13 H*14 H~I H:3 H:4 H:i3 H:14
.08808 1.15292 2.32299 3.44286 —1.15276 —2.32936 —3.44648 .08127 1.15293 2.32918 3.44648 —1.15292 —2.32380 —3.44286 —.08177 2.32556 4.27914 —2.33698 —4.28691 .08526 2.33645 4.28699 —2.32557 —4.27924 4.75250 5.98542 7.87549 8.19536 3.59974 2.42314 1.38602 4.75377 5.90543 7.08168 8.19898 3.59958 2.42950 1.30964 4.75073 7.87886 9.83164 2.41552 .46559 4.75776 7.88895 9.83949 2.42693 .47326
1 I 1 1 1 1 1 1 1 I 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2
1.48751 .71733 1.48891 .71657 .71529 1.40139 .78458 —1.40751 —.71529 —1.48139 —.70458 —.71733 —1.40891 —.71657 2.42722 2.42849 1.22483 2.42897 1.28928 —2.42722 —2.42097 —1.20928 —2.42849 —1.22483 7.78151 7.09133 7.78291 7.09057 7.08929 7.77539 7.07858 4.96649 5.65871 4.97261 5.66942 5.65667 4.96589 5.65743 8.88122 8.80249 7.59883 8.79497 7.58328 3.94678 3.95303 5.16472 3.94551 5.14917
.08888 —.40349 -.84i28 —1.22306 .48853 .84343 1.22113 .08371 —.48823 —.84373 —1.22113 .40349 .84136 1.22306 —.88296 -.87826 —1.51043 .87278 1.503~i —.8118’a —.87374 —1.50272 .87048 1.51843 .08880 —.48349 —.84128 —1.22306 .40853 .84343 1.22113 .08371 —.40823 —.84373 —1.22113 .40349 .84136 1.22386 —.00296 —.87826 —1.51843 .87278 1 .50301 —.01186 —.87374 —1.50272 .87848 1.51843
U
.8800 .0784 .0080
*i *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *1 *2 *2 *2 *2 ‘2 *2 *2 ‘2 *2 *2 *2 *2 *2 *2 *2 *2 *2 *2 *2 *2 *2 *2 *2 *2
.0045 .0000 .1349
ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE ROTATE
.1052 .8088 .0845
FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM FROM
—i5.0 —15.0 —15.8 —15.8 —15.0 —15.0 —15.0 —15.8 —15.8 —15.0 —15.8 —15.8 -15.8 —15.0 —15.0 —15.0 —15.8 —15.0 —15.8 —15.8 —15.8 —15.8 —15.8 —15.8 —15.8 —15.8 —15.8 —15.8 —15.8 —15.8 —15.8 —15.0 —15.8 —15.8 —15.8 —15.0 —15.0 —15.0 —15.8 —15.8 —15.8 —15.0 —15.0 —15.0 —15.8 15.8 -15.8 —15.0
HKL SPACINGS
.8000 .0784 .0080
TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO
i5.0 15.0 15.8 15.0 15.0 15.8 15.0 15.8 15.8 15.8 15.0 15.8 15.8 15.8 15.8 15.8 15.8 15.0 15.0 15.8 15.8 15.8 15.0 15.0 15.8 15.8 15.8 15.0 15.8 15.8 15.0 15.0 15.8 15.8 15.8 15.0 15.8 15.8 15.0 15.0 15.0 15.8 15.8 15.8 15.8 15.8 15.8 15.0
.8888 .8800 .1349
DES. DES. DES. DES. DES. DES. DES. CES. DES. DEli. DES. DEG. DES. DES. DES. DES. DES. DEO. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES. DES.
ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ASOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT ABOUT
9.4963 12.7488 7.4182
VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR ‘.‘ECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR VECTOR
—Si —81 —81 —81 -81 —81 —81 —81 —81 —81 —81 —81 —81 —81 —81 —Bi -81 —81 —81 —81 —81 —81 —81 —81 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83 —83
TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO TO
—82 —82 —82 —82 —82 —82 —82 —82 —82 —82 —82 —82 —82 —82 —82 —82 —82 —82 -82 —82 —82 —82 —82 —82 -84 —84 —84 —84 —84 —84 —84 —84 —84 —84 —84 —84 -84 —84 —84 —84 —84 —84 —84 —84 —84 —84 —84 —84
J. C.A. Boeyens, D.C. Levendis CS MS C6 C7 NB C16 CL? N18 CMS HM5 CM6 CM? NM8 CM16 CM17 MIIB C*5. H*5 C*6 C*7 N*8 C*16 C*17 N*1B C:5 H:S C:6 C:7 N:8 C:i6 C:17 N:18 CEN1 CEN2 CEN3 CEN4 LEAST ATOM Cl C2 C3 C4 C12 C13 C14 CMI CM2 CM3 CM4 CM12 CMI3 CM14
1 2 1 1 3 I 1 3 1 2 1 1 3 1 1 3 1 2 1 1 3 1 1 3 1 2 1 1 3 1 1 3 0 8 0 0
—.15853 —.15953 .98078 2.16816 3.09348 —1.29876 —2.48684 —3.41851 —.15853 —.15853 .98078 2.16816 3.89348 —1.29876 —2.48604 —3.41851 4.59397 4.59397 5.73328 6.92066 7.84598 3,45374 2.26646 1.34199 4.59397 4.59397 5.73328 6.92866 7.84598 3.45374 2.26646 1.34199 .88088 .88088 .88808 5.93029
1.40228 2.42212 .69987 1.42813 2.000,16 .69987 1.42013 2.08016 —1.48228 —2.42212 —.69987 —1.42813 —2.08016 —.69987 —1.42013 —2.00016 7.77628 8.79612 7.07387 7.79413 8.37416 7.07387 7.79413 8.37416 4.97172 3.95188 5.67413 4.95387 4.37384 5.67413 4.95387 4.37384 .80000 .88808 .08008 6.37480
3.78511 3.70511 3.33534 2.95297 2.65286 4.87414 4.47651 4.75662 3.78511 3.70511 3.33534 2.95297 2.65286 4.87414 4.47651 4.75662 3.78511 3.70511 3.33534 2.95297 2,65286 4.87414 4.47651 4.75662 3.78511 3.70511 3.33534 2.95297 2.65286 4.07414 4.47651 4.75662 .00080 .80080 .88080 3.38718
SQUARES PLANE FOR ROTATION WEIGHT DEVIATION 1.0808 .0000 1 .8800 .8872 1.0080 —.8119 1.8888 .0847 1.8888 —.8827 1.0808 .8117 1.0008 —.0088 1 .8888 .0034 1.0800 .8825 1.0080 —.8126 1.0808 .8875 1.8088 —.0078 1.8800 .8114 1.8088 —.8053
SUM WT*DEV
/ 0 8 8 0 8 0 0 8 0 8 0 0 8 0 8 8 0 8 8 8 8 0 0 0 0 0 8 8 0 0 0 8 8 0 0 0
Simulation of molecular reorientation in crystals KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEP’r KEPT KEPT KEFT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT KEPT
FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED
1
.88008
SUM IJT*DEV**2 =
.00887 PARAMETERS OF PLANE L M N 0
LAMBDA
=
= = =
.33680 .08828 .94186 .08828
.00887
ROTATION 1 FROM CEN1 TO CEN2 WITH NEW COORDINATES .0800
.0000
.8808
AND
.0394
.8000
.1271
229
230
J.C.A. Boeyens, D.C. Levendis
/ Simulation
of molecular reorientation in crystals
~**~**rr**wr*****~n**w*~
BREAKDOWN OF THE POTENTIAL ENERGY ENVIRONMENT AT EACH ROTATIONAL POSITI ON FOR A*N~T*H*P*A~C*E*N~E *T*E~T*R*A*C~Y*A*N*O~B*E*N*Z*
INTERACTION TYPE
1... 2... 2... 3... 3... 3...
1 1 2 1 2 3
TOTALS
EREP 342.084
INTERACTION TYPE
1 1 2 1 2 3
TOTALS EREP 295.819
NO. OF INTERACTIONS CLOSER THAN 5.08 ANOSTROMS
Sl1-1S OF THE NON—BONDED INTERACTION ENERGY/ KJ/IIOL
NINTER 556 674 174 258 134 14
NINTRA 188 143 25 51 29 1
ECOUL .800
440<
ETORI .880
INTERACTION TYPE
1 1 2 1 2 3
TOTALS
2258)
ETOR2 .008
—15.0
.0
INTERASYM. —29.605 10.316 7.143 —5.431 1.148 —.977
INTRAASYM. —12.536 —1.470 —.176 —1.276 —.263 —.078
TOTALS —42.141 8.846 6.966 —6.707 .885 —1.847
—11.37
—18.80
—30.18
ETOR3 .000
EPEE —30.176 **
**
BREAKDOWN OF INTERACTIONS AT <
—15.0
NO. OF INTERACTIONS CLOSER THAN 5.00 ANGSTROMS
SUMS OF THE NON—BONDED INTERACTION ENERGY/ KJ/MOL
HINTER 562 672 182 258 140 14
NINTRA 190 141 25 53 28 1
1828
EATT —388.888
1... 2... 2... 3... 3... 3...
—15.0
1810
EATT —402.436
1... 2... 2... 3... 3... 3...
BREAKDOWN OF INTERACTIONS AT (
ECOUL .080
440< ETOR1 .880
2268) ETOR2 .808
-10.0
.0
INTERASYM. —34.8BB 4.279 7.312 —5.688 1.287 —.977
INTRAASYM. —12.604 —1.481 —.169 —1.303 —.260 —.070
TOTALS —46.692 2.798 7.143 —6.991 1.027 —1.047
—28.00
—18.93
—46.93
ETOR3 .000
EPEE —46.931
**
BREAKDOWN OF INTERACTIONS AT (
—15.0
NO. OF INTERACTIONS CLOSER THAN 5.00 ANGSTROMS
51.8-IS OF THE NON—BONDED INTERACTION ENERGY/ KJMIOL
NINTER 554 670 176 258 142 14 1814
NINTRA 192 137 25 53 29 I 440<
2254)
-5.0
**
.8
INTERASYM. —35.787 —.312 7.132 —5.816 1.637 —.977
INTRAASYM. —12.701 —1.489 —.159 —1.313 —.249 —.070
TOTALS —48.488 —1.801 6.974 —7.129 1.389 —1.847
—38.61
—i9.03
-57.64
SLH~?ARY OF THE POTENTIAL ENERGY E~.8.JIRC~.lIENTS(EPEE /K3/MOL) AT THE 49 ROTAT I ONAL POSITIONS FOR A’N~T~H~R~A’C’ E*N~ E ‘T~E~T’R’A~ C~Y~A*N~ O
PEE MAP WITH PHI3 = PHIl ACROSS PHI2 DOWN
.88
J.C.A. Boeyens, D.C. Levendis
1 —15.8 2 —18.8 3 —5.0 4 .0 5 5.0 6 10.0 7 15.8
/
Simulation of molecular reorientation ‘in crystals
1 —15.0
2 —18.0
3 —5.8
4 .8
5 5.0
6 18.8
7 15.8
—38.2 —46.9 —57.6 —66.2 —72.3 —76.4 —75.9
—47.8 —S8.8 —66.8 —73.0 —77.1 —79.5 —77.2
—57.8 —66.9 —72.7 —76.3 —77.8 —77.6 —74.0
—66.3 —73.0 —76.2 —77.8 —76.7 —74.3 —69.2
—72.3 —77.8 —77.7 —76.6 —74.1 —69.7 —62.6
—76.3 —79.3 —77.4 —74.1 —49.6 —63.5 —54.5
—75.6 —76.8 —73.5 —68.8 —62.3 —54.3 —41.4
ORDERED VALUES OF PEE IN MAP(S)
28681
EMAX
—30.2 AT NRID
18181
60201
EMAX
—41.4 AT NRID
70701
38581
EMAX
—46.9 AT NRID
=
10281
48401
EMAX
—47.8
AT NRID
=
20101
=
50301
EMAX
—54.3 AT NRID
=
70601
=
38601
El-lAX
—54.5 AT NRID
=
60301
El-lAX
—57.6 AT NRID
20781
EMAX
—57.8 AT NRID
=
38101
=
28581
El-lAX
—58.8 AT NRID
=
28281
=
58281
EMAX
—62.3 AT NRID
=
78581
—76.8 AT NRID
=
70281
EMAX
—62.6 AT NRID
—76.7 AT NRID
=
48581
EMAX
-83.5 AT NRID
EMIN
—76.6 AT NRID
=
58481
EMAX
—66.2 AT NRID
EMIN
—76.4 AT NRID
=
10681
El-lAX
EMIN
—76.3 AT NRID
=
30401
EMIN
-76.3 AT NRID
=
Er-1IN
—76.2 AT NRID
EMIN
—79.5 AT NRID
EMIN
—79.3 AT NRID
EMIN
—77.8 AT NRID
EMIN
—77.8 AT NRID
=
EMIN
—77.7 AT NRID
EMIN
—77.6 AT NRID
EMIN
—77.4 AT NRID
EMIN
—77.2 AT NRID
EMIN
—77.1 AT NRID
EMIN
—77.8 AT NRID
EMIN EMIN
=
68701 18381
58781 =
68601
—66.3 AT NRID
=
40181
El-lAX
—68.8 AT MRID
=
20301
60101
El-lAX
-66.9 AT NRID
=
38201
48301
EMAX
—68.8 AT NRID
=
70401
18481
EMIN
-75.9 AT NRID
=
10781
El-lAX
-69.2 AT NRID
=
48701
EMIN
—75.6 AT NRID
=
78101
El-lAX
—69.6 AT NRID
=
68501
EMIN
—74.3 AT NRID
=
40681
El-SAX
—69.7 AT NRID
=
50681
20 SEVERE CONTACTS THAT OCCUR IN THE
FROM
TO
C:3 H13 H*13 H:3 H3 C:13 H*3 H:13 H~13 H3 C:13 H:13 11-113 C*3 H~13 H;3 H~l3 +113 H*3 H~13
H13 C:3 01-13 C13 C:I3 H3 01113 C3 C3 C:13 H3 C3 C*3 1*113 H3 H13 H3 H*3 11113 H3
NRID 18101 10101 18181 18101 78701 70791 70701 70701 78601 70681 78401 60701 70681 78681 60601 30101 58701 70501 70S01 70501
IDTAR 65501 45501 56501 65501 55581 55501 66501 55581 55501 55501 55501 55501 44581 66501 55501 65501 55501 44501 66501 55501
01ST 2.257 2.257 2.257 2.258 2.259 2.259 2.259 2.259 2.383 2.333 2.333 2.333 2.333 2.333 1.875 1.884 1.887 1.887 1.887 1.887
ROTATIONS OF
ENONB 3.0S4 3.054 3.854 3.046 3.033 3.033 3.033 3.033 2.532 2.248 2.248 2.239 2.239 2.239 1.845 1.783 1.759 1.759 1.759 1.759 END OF JOB
A*N~T~H*R~A*C*E~N*E *T*E~T*R*A*C~Y~A*N*O*B*E*N*2~
231