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Parst: A system of fortran routines for calculating molecular structure parameters from results of crystal structure analyses
0097~8485/8313.00+.00 Pmgamon Pm66 Ltd.
PARST: A SYSTEM OF FORTRAN ROUTINES FOR CALCULATING MOLECULAR STRUCTURE PARAMETERS FROM RESULTS OF CRYSTAL STRUCTURE ANALYSES M. NARDELL~ lstituto di Chimica Generale e Inorganica della Universita’ di Parma, Centro di Studio per la Strutturistica Diffrattometrica del C.N.R., via M.D’AzegEo 85, 43100 Parma, Italy (Receioed
28 December 1982)
Abstract-Given a set of atomic positional and thermal parameters in a crystal defined by the unit cell constants and the space group symmetry operations, the title program calculates: (1) the Niggli reduced cell, (2) V, M, D,, F(OO0) and p. (3) the orthogonal coordinates in A, (4) the principal axes of fhe thermal ellipsoids together with LIeqand I&, (5) all bond distances (uncorrected and corrected for thermal motion), (6) all bond angles, (7) all torsion angles, (8) coordinates of hydrogen atoms in typical groups, (9) weighted least-squares planes or straight lines through given subsets of atoms and the angles they form, (10) ring puckering coordinates and displacement asymmetry parameters, (I 1) spherical polar coordinates for stereographic projections of atomic environments, (12) interatomic contacts between atoms of the given set and between these and symmetry related atoms, (13) geometrical parameters of possible hydrogen bonds, (14) transformed coordinates of pairs of atomic subsets to check the possible presence of pseudo-symmetries or to find the atoms responsible for the departure from symmetry, etc. The estimated standard deviations of the derived quantities are calculated from the e.s.d.‘s of the atomic parameters and unit cell constants, neglecting the covariances between the individual parameters. INTRODUCHON
where
The relevant results of every crystal structure analysis are the positional and thermal (if the analysis is accurate enough) parameters together with the unit cell constants and the space group symmetry operations. Indeed from these data it is always possible to reconstruct every structural aspect interesting in connection with the chemical, physical, biological, etc. properties of the compound whose crystal structure has been determined. For this reason it is essential that these data, together with suitable programs for working them out, are easily available to the reader of a crystal structure paper to enable him to get out the information he needs even when the author of the paper has not considered it explicitly. With this in mind a set of computer routines, written to solve current laboratory problems, has been put together with the aim of making available a program which systematically carries out all the most commonly requested structural calculations.
A,=u(a)/a
&=sina(coscc-cos@cosy)a(a)
A2=a(b)/b
B,=sin/?(cosfi-cos~cosy)a(g)
A, = a(c)/c
& = sin y (cos y - cos a cos /3)&f)
and uz
ab cosy ( ac cos /?
SE=
ab cos y
b2 bc wsa
ac cos B bc cos a > c* (3)
This routine calculates also the reciprocal and its standard deviation:
angle CL+
cos a * = (cos j? cos y - cos a)/(sin fi sin y)
(4)
a*(ar:*) = {[sin GLU (a)J2 + [(sin B cos y + cos a * X cos/3 sinr)u(8)]2+[(cosj3
siny
-t cos 01* sin /I cos r)u(r)]*)/ TYPES OF CALCULATIONS, FORMULAE AND ALGORITHMS (1)
Unit ceil volume and metric
(sin dl* sin j3 sin v)*.
tensor
(2) Niggli reduced ceN If the lattice is not primitive, first it is transformed into a primitive one by subroutine PRMTRN, then the Niggli reduced cell and the Niggli matrix (Niggli, 1928; Azaroff ZL Buerger, 1958) are derived by subroutine NIGGLI which makes use of the Ciivj & Gruber (1976) algorithm.
Subroutine VOLMET calculates the unit cell.volume, V, and the metric tensor g; using the formulae: Y=dV
Jl-cos*a-cos*/?-coszy+2cosc(cos~cosy (1)
O(V)
(3) Molecular
=J_
absorption (i=1,2,3)
CAC vol. 7 No. L-A
(5)
weight, calculated coeficient
density,
F(OOO) and
These quantities are calculated by subroutine MOLABS which uses the IUPAC (1977) atomic weights
(2) 95
96
M. NARDELLI
and the atomic
total cross sections for the CuKa and MoKa radiations tabulated in International Tables for X-Ray CrysraNography (1974). (4) Atomic orthogonal coordinates in A Subroutine ORTHO carries out the using the orthogonalisation matrix:
o=
bcosy b sin y 0
z ( .O
calculation
c cos B - csinjlcosa* csin/?sina*
(7) Bond distances All the interatomic distances dU (i,j = 1, 2, _ . N; >
The effects of the errors in the unit cell parameters on the e.s.d.‘s of the atomic orthogonal coordinates are accounted for in the approximation expressed by the following formulae: = [k,a(a)]2
+ [k,cos
+ [k, cos PO (c#
vb(b)Jz
a*(Y)
applied In the e.s.d. calculations the errors in the unit cell parameters are taken into account, so the used are:
d,, : [(aAx)’
Ba(zll’
+ (bAy)’
+ 2(bcAyAz
ya(y)12 -t [k,c sin @(fl)]’
+ abAxAy
= [k, sin yfr(b)]* + [k, sin p cos cx*a(c)]’ + [b sin ya 0, )I2 + [c sin p cos E ‘a(z)]* + [k&J cos ra(y)]2
c’(dJ
= $. {(aAx
= [k, sin /I sin a*o(~)]~
+ (aAx
+ [c sin /l sin x *a(z)]’
+ [k,c sin b cos a *a(a *)I’
(8)
+ b Ay cos y + c AZ cos p)’
c In,-
x
([A~u(b)l* + b2[U2CvJ+ ~‘Cvj)l) (aAn
cos p + bA_y cos bl + cAz)’
and similar formulae for i = 1, 2, 3, . __IV and N = total the set.
k, and k,, with number of atoms of
(5) Coordinates of hydrogen atoms in calculated positions Subroutine CALCOH uses the algorithms already described (Nardelli, 1982) and, in addition, consider also the case of ethylenic hydrogens. The coordinates of the hydrogen atoms are then added to the set by subroutine COORDH and used in successive calculations. (6) Mean-square displacements along the principal axes qf the atomic thermal ellipsoids and isotropic equivalent thermal parameters The mean-square displacements along the principal axes of the atomic thermal ellipsoids are calculated for the atoms treated anisotropically, following Willis & Prior (1975). from the anisotropic thermal parameters, and their values are tabulated with those of the equivalent isotropic thermal coefficients as defined by Hamilton (1959). The e.s.d.‘s of all these magnitudes are calculated using
+ c2[u2(z,) + u2(zi)]) sin aa(
+ [OCAXAZ sin jlu(,fi)]’
2 1,
=(l/N)Cx,
+ a2(x,)])
+ CAZ CDS a)’
+
x ([Azu(c>]*
where
+ a2[a2(xi)
cos y + bAy
+ [bcAyAz
x
cos B
cos Y)]“~
i ([Axu(u)]’
(7)
+ [k,c cos jl sin a*o(j?)]’
k, = (l/N)
+ (cAz)~
cos a + acAxAz
+ [k-c cos /I cos a ‘u(/T)]’
+ [k,c sin /l sin a *u (a *)I2 a’(Z)
i cj) between the atoms of the set falling in the interval (r, + r, - 0.5) < dv < (r, + ri + 0.5) (where r, is the covalent radius of atom i in A) are considered, and in addition, if the anisotropic thermal parameters have been given, the corrections for the anisotropic motion proposed by Busing & Levy (1944) are
formulae
+ [au (.x )I*
+ [b cos yd_Y)12 + [c cos
+ [k,b sin
described by Swain et al. are carried out by subroutine THERPA. Provision is made to consider different types of thermal coefficients, i.e. Vi,, Bii, fl,,, b, etc. Isotropic atoms are recognized by the zero values of their thermal coefficients excepting the first
ones.
(6)
a’(X)
the Monte Carlo method (1980). These calculations
+ (abAxAy
sin yu(y)]‘j
(9)
where Ax = x, - x,, Ay = y, - y,, AZ = z, - zj are the differences between the fractional coordinates of the two atoms. All these calculations are performed by subroutine BOND1 which builds up also a bond matrix. In calculating the e.s.d.‘s the program does not take into account the correction which must be applied when the atoms involved in the contacts are related by a symmetry operation. (8)
Bond angles All possible bond angles are considered
and a matrix of the atomic triplets defining the angles (angle matrix) is built up by subroutine TERNE. For each triplet the angle, with its e.s.d., is calculated using the orthogonal coordinates and the formulae given in International Tables for X-Ray Crystallography (1959). (9) Torsion angles From the bond and angle matrices all the quadruplets defining the torsion angles are derived by subroutine QUART and, for each quadruplet, subroutine TORSZ calculates the angle with its e.s.d. The “right-hand rule” of Klyne & Prelog (1960) is followed and the e.s.d. is calculated in the isotropic
PARST: a system of FORTRAN approximation using the formula & Waser (1972).
given by Stanford
(10) Least-squares weighted planes and srraight lines Subroutines LSQPLA and LSQLIN perform the calculations using the method described by Nardelli et al. (1965) which essentially is that proposed by Schomaker et al. (1959) as modified by Blow (1960), using the diagonal approximation of the Hamilton’s (1961) weighting matrix. The atomic orthogonal coordinates are used and the calculated magnitudes are: the direction cosines, the distance from the origin, the distances of the atoms from the plane or the line, the e.s.d.‘s of these distances and the sum of the squared ratios [(distancc)/(e.s.d.)]* indicative of the significance of the deviations from planarity or linearity affecting the group of atoms. The e.s.d.‘s of the direction cosines are calculated using the previously quoted Monte Carlo method. Subroutine ANGDS calculates all the angles (with e.s.d.‘s) formed by the normals to the planes, by the lines and by the normals and lines.
(I I ) Ring puckering
coordinares The Cremer & Pople’s (1975) puckering coordinates are particularIy useful to give a quantitative description of the deviation from planarity of an atomic monocyclic ring. Subroutines CRTCRD and PUCK perform these calculations and, for every (up to 40-membered) ring, give the internal cartesian and puckering coordinates with their e.s.d.‘s, calculated using the already mentioned method of Swain et al. (1980). (12) Displacement asymmetry parameters These parameters are a modified version of those proposed by Dudx of al. (1976) to give a quantitative evaluation of how a ring deviates from ideal symmetry. Instead of using the torsion angles as proposed by Duax et al., the perpendicular displaccments of the atoms from the mean plane through them are considered (Nardelli, 1983). The subroutines ASYMMP, APAR and ROTRIN perform these calculations systematically and a table is given with the values of these parameters for all possible mirrors and 2-fold axes, so the symmetry of the ring can be easily deduced. (13) Cartesian and spherical polar coordinates for stereographic projections These coordinates are useful to present in a stereographic projection the spatial orientation of the bonds formed by a set of ligands with a central coordinating atom. Three options are considered: (i) The origin is at the first atom of the set and the z-axis lies along the bond that atom forms with the second one. (ii) The origin is at the first atom of the set and the z-axis is perpendicular to the mean plane through the first n atoms excepting the first one. (iii) The origin is at the centre of the set formed by the first n atoms and the z-axis is perpendicular to the mean plane through them. These calculations are performed by subroutine CRTPL.
97
routines
(14) Interatomic contacts Subroutine BOND1 is used also to calculate all the non-bonding contacts, in a required hi,, - d,,,. range, formed by the atoms of the set between themselves. If the equivalent positions are given, also the interatomic contacts ( -=zd,,,,,), formed by the atoms of the set with those generated by symmetry and translations, are calculated and the values (with their e.s.d.‘s) are tabulated together with the equivalent positions involved. The subroutines which perform these calculations are: INTERC, DISCON, DSTCHB, DECEQP and EQPDEC. (15) Possible hydrogen bonds If the coordinates of the hydrogen atoms are supplied or their calculation has heen requested, the program calculates the relevant parameters (with their e.s.d.‘s) describing the possible hydrogen bonds, i.e. the donor-hydrogen, donor _ . . acceptor, hvdrogen . __acceptor distances and the donor-hydrogen ___acceptor angle. The subroutines used for this purpose are: HB1 and HDRB. The contacts considered as possible hydrogen bonds require: (i) that the acceptor is one of the most electronegative atoms, i.e. N, 0, S, Se, F, Cl, Br, I; (ii) the distance hydrogen-acceptor is not greater than 2.8 A; (iii) the angle is larger than 120”. (16) Comparison of coordinates of atomic subsets In crystal structure analysis it is frequently necessary to compare subsets of atoms (molecules or part of molecules) present in the crystal and usually the comparison is carried out on bond distances and angles sometimes using statistical methods like normal probability plots (Abrahams & Keve, 1971). Subroutine SYMMOL allows to compare directly the atomic coordinates and gives the averaged coordinates with the rotation matrices and the translation vectors which bring one subset of atoms into the other and vice uersa. The principles of the calculation are as follows. If x, = (x,;. x2,, x,,) is the position vector of atom i in one subset (defined by the fractional coordinates in the crystal space) and xi = (xii, x;, x&) is the same vector for the corresponding atom i in the second subset, then x; = Rx, + t (10) where R is a rotation matrix and t a translation vector. The elements of this matrix and vector are calculated by a least-squares technique. If RsTis the transposed of vector R, = (R,,, R,, R,,, t5) formed by the s-row of matrix R and by the s element of vector t, then: R,r=A-‘P,’ where A ’ is the reciprocal A whose elements are:
s =I,...4 of the symmetric
j,k=1,2,3
(11) matrix
(12)
98
M. NARDELLI
and P,’ is the s-column elements are:
vector
j,k=
of matrix
P whose
1,2,3
(13)
the sums being over i = 1, 2,. . N/2, with N = total number of atoms (of the two sets altogether). The output gives: (i) The symbols of the atoms of the two subsets. (ii) The fractional and orthogonal coordinates of the centroids of the two subsets with e.s.d.‘s. (iii) The rotation matrix and translation vector which bring subset 1 into subset 1’. (iv) The fractional coordinates of the atoms of subset I’ as obtained from those of subset 1, together with the differences with the original coordinates of subset 1’. (v) The averaged fractional coordinates of subset 1’ together with the differences with the original coordinates of the same subset 1’. (vi) The same as (iiij(v) but referred to subset 1’ which is now brought into subset 1. (vii) The internal coordinates, Xi”, in A units referred to two isooriented orthogonal systems of axes having their origins at the centroids of the two subsets, together with the e.s.d.‘s, the differences A, = X7 - Xp. for each couple of atoms and the ratios A#, = AJ[a’(x:) + .s*(~,))“*. (viii)
The sums Z (Ada;)’
the theoretical
for each coordinate
and
x2 Value. COMMENTS
The program has been implemented on a Gould Systems 32-77, (32-bit words computer), working under the MPX-32 multiuser operating system, but the routines had been written p;evious& fbr a Idbit minicomputer, so many statements needed to avoid stops due to underflow are present. The language is essentially FORTRAN IV with only a few multiple assignment statements.
Two versions, one for 100 maximum number of atoms, are number of statements is 4021, comments giving the instructions program. Copies of the listing, ples of input and output, are author on request.
the other for 200 available. The total of these 159 are for the use of the together with examavailable from the
Acknowledgement-The author is indebted to Mr. G. Pasquinelli for his valuable help in implementing the program on the computer. REFERENCES Abrahams, S. C. & Keve. E. T. (1971), Acfa Crysf. A27, 157. Azaroff, L. V. & Buerger, M. J. (195S), The Powder Method in X-Ray Crystallography. New York, McGraw-Hill. Blow. D. M. (1960). Acca Crvsz. 13. 168. Bus&, W. R: & L&, H. A: (1964j, Acra Crysr. 17, 142. Cremer, D. & Pople, J. A. (1975), J. Am. Chem. Sot. 97, 1354. Duax, W. L., Weeks, C. M. & Robrer, D. C. (1976) in Topics in Stereochemistry, Eliel, E. L. % Allinger, N., Eds., New York, Wiley, Vol. 9. Hamilton, W. C. (1959), Acto Cry.vt. 12, 609. Hamilton. W. C. (1961). Aera Crvst. 14. 18.5. Intematidnal Tables f& X-Ray’ Crys&ography (1959), Vol. II, pp. 60 and 331. International Tables for X-Ray Crystallography (1974), Vol. IV. b. 55. Klyne, W. & Prelog, V. (1960), Experientia 16, 521. Kiivj,, I. & Gruber, B. (1976), Acra Crysr A32, 297. Nardelli, M. (1982), Compur. Chem. 6, 139. Nardelli, M. (1983), Actn Crust. C in press. Nardelli, M., Musatti, A., Domiano, P. & Andreetti, G. (1965), Ric. Sci. 15(11-A), 807. Niggli, P. (1928), Hundbuch der Experimenfulphysik, Vol. 7, Part 1. Leipzig, Akademische Verlagsgeselschaft. Schomaker, V., Waser, J., Marsh, R. E. & Bergman, G. (1959), Acrn tryst. 12, 600. Standford, R. H. Jr & Waser, J. (1972), Ac~a Crysr. AZ& 213.
Swain, C. G., Swain, M. S. & Strong, P. F. (1980), J. Chem. hf. Compur. Sci. 20, 51. Willis, B. T. M. % Prior, A. W. (1975), Thermal Vibrations in Crystallography. Cambridge University Press.
PARST: A SYSTEM OF FORTRAN ROUTINES FOR CALCULATING MOLECULAR STRUCTURE PARAMETERS FROM RESULTS OF CRYSTAL STRUCTURE ANALYSES M. NARDELL~ lstituto di Chimica Generale e Inorganica della Universita’ di Parma, Centro di Studio per la Strutturistica Diffrattometrica del C.N.R., via M.D’AzegEo 85, 43100 Parma, Italy (Receioed
28 December 1982)
Abstract-Given a set of atomic positional and thermal parameters in a crystal defined by the unit cell constants and the space group symmetry operations, the title program calculates: (1) the Niggli reduced cell, (2) V, M, D,, F(OO0) and p. (3) the orthogonal coordinates in A, (4) the principal axes of fhe thermal ellipsoids together with LIeqand I&, (5) all bond distances (uncorrected and corrected for thermal motion), (6) all bond angles, (7) all torsion angles, (8) coordinates of hydrogen atoms in typical groups, (9) weighted least-squares planes or straight lines through given subsets of atoms and the angles they form, (10) ring puckering coordinates and displacement asymmetry parameters, (I 1) spherical polar coordinates for stereographic projections of atomic environments, (12) interatomic contacts between atoms of the given set and between these and symmetry related atoms, (13) geometrical parameters of possible hydrogen bonds, (14) transformed coordinates of pairs of atomic subsets to check the possible presence of pseudo-symmetries or to find the atoms responsible for the departure from symmetry, etc. The estimated standard deviations of the derived quantities are calculated from the e.s.d.‘s of the atomic parameters and unit cell constants, neglecting the covariances between the individual parameters. INTRODUCHON
where
The relevant results of every crystal structure analysis are the positional and thermal (if the analysis is accurate enough) parameters together with the unit cell constants and the space group symmetry operations. Indeed from these data it is always possible to reconstruct every structural aspect interesting in connection with the chemical, physical, biological, etc. properties of the compound whose crystal structure has been determined. For this reason it is essential that these data, together with suitable programs for working them out, are easily available to the reader of a crystal structure paper to enable him to get out the information he needs even when the author of the paper has not considered it explicitly. With this in mind a set of computer routines, written to solve current laboratory problems, has been put together with the aim of making available a program which systematically carries out all the most commonly requested structural calculations.
A,=u(a)/a
&=sina(coscc-cos@cosy)a(a)
A2=a(b)/b
B,=sin/?(cosfi-cos~cosy)a(g)
A, = a(c)/c
& = sin y (cos y - cos a cos /3)&f)
and uz
ab cosy ( ac cos /?
SE=
ab cos y
b2 bc wsa
ac cos B bc cos a > c* (3)
This routine calculates also the reciprocal and its standard deviation:
angle CL+
cos a * = (cos j? cos y - cos a)/(sin fi sin y)
(4)
a*(ar:*) = {[sin GLU (a)J2 + [(sin B cos y + cos a * X cos/3 sinr)u(8)]2+[(cosj3
siny
-t cos 01* sin /I cos r)u(r)]*)/ TYPES OF CALCULATIONS, FORMULAE AND ALGORITHMS (1)
Unit ceil volume and metric
(sin dl* sin j3 sin v)*.
tensor
(2) Niggli reduced ceN If the lattice is not primitive, first it is transformed into a primitive one by subroutine PRMTRN, then the Niggli reduced cell and the Niggli matrix (Niggli, 1928; Azaroff ZL Buerger, 1958) are derived by subroutine NIGGLI which makes use of the Ciivj & Gruber (1976) algorithm.
Subroutine VOLMET calculates the unit cell.volume, V, and the metric tensor g; using the formulae: Y=dV
Jl-cos*a-cos*/?-coszy+2cosc(cos~cosy (1)
O(V)
(3) Molecular
=J_
absorption (i=1,2,3)
CAC vol. 7 No. L-A
(5)
weight, calculated coeficient
density,
F(OOO) and
These quantities are calculated by subroutine MOLABS which uses the IUPAC (1977) atomic weights
(2) 95
96
M. NARDELLI
and the atomic
total cross sections for the CuKa and MoKa radiations tabulated in International Tables for X-Ray CrysraNography (1974). (4) Atomic orthogonal coordinates in A Subroutine ORTHO carries out the using the orthogonalisation matrix:
o=
bcosy b sin y 0
z ( .O
calculation
c cos B - csinjlcosa* csin/?sina*
(7) Bond distances All the interatomic distances dU (i,j = 1, 2, _ . N; >
The effects of the errors in the unit cell parameters on the e.s.d.‘s of the atomic orthogonal coordinates are accounted for in the approximation expressed by the following formulae: = [k,a(a)]2
+ [k,cos
+ [k, cos PO (c#
vb(b)Jz
a*(Y)
applied In the e.s.d. calculations the errors in the unit cell parameters are taken into account, so the used are:
d,, : [(aAx)’
Ba(zll’
+ (bAy)’
+ 2(bcAyAz
ya(y)12 -t [k,c sin @(fl)]’
+ abAxAy
= [k, sin yfr(b)]* + [k, sin p cos cx*a(c)]’ + [b sin ya 0, )I2 + [c sin p cos E ‘a(z)]* + [k&J cos ra(y)]2
c’(dJ
= $. {(aAx
= [k, sin /I sin a*o(~)]~
+ (aAx
+ [c sin /l sin x *a(z)]’
+ [k,c sin b cos a *a(a *)I’
(8)
+ b Ay cos y + c AZ cos p)’
c In,-
x
([A~u(b)l* + b2[U2CvJ+ ~‘Cvj)l) (aAn
cos p + bA_y cos bl + cAz)’
and similar formulae for i = 1, 2, 3, . __IV and N = total the set.
k, and k,, with number of atoms of
(5) Coordinates of hydrogen atoms in calculated positions Subroutine CALCOH uses the algorithms already described (Nardelli, 1982) and, in addition, consider also the case of ethylenic hydrogens. The coordinates of the hydrogen atoms are then added to the set by subroutine COORDH and used in successive calculations. (6) Mean-square displacements along the principal axes qf the atomic thermal ellipsoids and isotropic equivalent thermal parameters The mean-square displacements along the principal axes of the atomic thermal ellipsoids are calculated for the atoms treated anisotropically, following Willis & Prior (1975). from the anisotropic thermal parameters, and their values are tabulated with those of the equivalent isotropic thermal coefficients as defined by Hamilton (1959). The e.s.d.‘s of all these magnitudes are calculated using
+ c2[u2(z,) + u2(zi)]) sin aa(
+ [OCAXAZ sin jlu(,fi)]’
2 1,
=(l/N)Cx,
+ a2(x,)])
+ CAZ CDS a)’
+
x ([Azu(c>]*
where
+ a2[a2(xi)
cos y + bAy
+ [bcAyAz
x
cos B
cos Y)]“~
i ([Axu(u)]’
(7)
+ [k,c cos jl sin a*o(j?)]’
k, = (l/N)
+ (cAz)~
cos a + acAxAz
+ [k-c cos /I cos a ‘u(/T)]’
+ [k,c sin /l sin a *u (a *)I2 a’(Z)
i cj) between the atoms of the set falling in the interval (r, + r, - 0.5) < dv < (r, + ri + 0.5) (where r, is the covalent radius of atom i in A) are considered, and in addition, if the anisotropic thermal parameters have been given, the corrections for the anisotropic motion proposed by Busing & Levy (1944) are
formulae
+ [au (.x )I*
+ [b cos yd_Y)12 + [c cos
+ [k,b sin
described by Swain et al. are carried out by subroutine THERPA. Provision is made to consider different types of thermal coefficients, i.e. Vi,, Bii, fl,,, b, etc. Isotropic atoms are recognized by the zero values of their thermal coefficients excepting the first
ones.
(6)
a’(X)
the Monte Carlo method (1980). These calculations
+ (abAxAy
sin yu(y)]‘j
(9)
where Ax = x, - x,, Ay = y, - y,, AZ = z, - zj are the differences between the fractional coordinates of the two atoms. All these calculations are performed by subroutine BOND1 which builds up also a bond matrix. In calculating the e.s.d.‘s the program does not take into account the correction which must be applied when the atoms involved in the contacts are related by a symmetry operation. (8)
Bond angles All possible bond angles are considered
and a matrix of the atomic triplets defining the angles (angle matrix) is built up by subroutine TERNE. For each triplet the angle, with its e.s.d., is calculated using the orthogonal coordinates and the formulae given in International Tables for X-Ray Crystallography (1959). (9) Torsion angles From the bond and angle matrices all the quadruplets defining the torsion angles are derived by subroutine QUART and, for each quadruplet, subroutine TORSZ calculates the angle with its e.s.d. The “right-hand rule” of Klyne & Prelog (1960) is followed and the e.s.d. is calculated in the isotropic
PARST: a system of FORTRAN approximation using the formula & Waser (1972).
given by Stanford
(10) Least-squares weighted planes and srraight lines Subroutines LSQPLA and LSQLIN perform the calculations using the method described by Nardelli et al. (1965) which essentially is that proposed by Schomaker et al. (1959) as modified by Blow (1960), using the diagonal approximation of the Hamilton’s (1961) weighting matrix. The atomic orthogonal coordinates are used and the calculated magnitudes are: the direction cosines, the distance from the origin, the distances of the atoms from the plane or the line, the e.s.d.‘s of these distances and the sum of the squared ratios [(distancc)/(e.s.d.)]* indicative of the significance of the deviations from planarity or linearity affecting the group of atoms. The e.s.d.‘s of the direction cosines are calculated using the previously quoted Monte Carlo method. Subroutine ANGDS calculates all the angles (with e.s.d.‘s) formed by the normals to the planes, by the lines and by the normals and lines.
(I I ) Ring puckering
coordinares The Cremer & Pople’s (1975) puckering coordinates are particularIy useful to give a quantitative description of the deviation from planarity of an atomic monocyclic ring. Subroutines CRTCRD and PUCK perform these calculations and, for every (up to 40-membered) ring, give the internal cartesian and puckering coordinates with their e.s.d.‘s, calculated using the already mentioned method of Swain et al. (1980). (12) Displacement asymmetry parameters These parameters are a modified version of those proposed by Dudx of al. (1976) to give a quantitative evaluation of how a ring deviates from ideal symmetry. Instead of using the torsion angles as proposed by Duax et al., the perpendicular displaccments of the atoms from the mean plane through them are considered (Nardelli, 1983). The subroutines ASYMMP, APAR and ROTRIN perform these calculations systematically and a table is given with the values of these parameters for all possible mirrors and 2-fold axes, so the symmetry of the ring can be easily deduced. (13) Cartesian and spherical polar coordinates for stereographic projections These coordinates are useful to present in a stereographic projection the spatial orientation of the bonds formed by a set of ligands with a central coordinating atom. Three options are considered: (i) The origin is at the first atom of the set and the z-axis lies along the bond that atom forms with the second one. (ii) The origin is at the first atom of the set and the z-axis is perpendicular to the mean plane through the first n atoms excepting the first one. (iii) The origin is at the centre of the set formed by the first n atoms and the z-axis is perpendicular to the mean plane through them. These calculations are performed by subroutine CRTPL.
97
routines
(14) Interatomic contacts Subroutine BOND1 is used also to calculate all the non-bonding contacts, in a required hi,, - d,,,. range, formed by the atoms of the set between themselves. If the equivalent positions are given, also the interatomic contacts ( -=zd,,,,,), formed by the atoms of the set with those generated by symmetry and translations, are calculated and the values (with their e.s.d.‘s) are tabulated together with the equivalent positions involved. The subroutines which perform these calculations are: INTERC, DISCON, DSTCHB, DECEQP and EQPDEC. (15) Possible hydrogen bonds If the coordinates of the hydrogen atoms are supplied or their calculation has heen requested, the program calculates the relevant parameters (with their e.s.d.‘s) describing the possible hydrogen bonds, i.e. the donor-hydrogen, donor _ . . acceptor, hvdrogen . __acceptor distances and the donor-hydrogen ___acceptor angle. The subroutines used for this purpose are: HB1 and HDRB. The contacts considered as possible hydrogen bonds require: (i) that the acceptor is one of the most electronegative atoms, i.e. N, 0, S, Se, F, Cl, Br, I; (ii) the distance hydrogen-acceptor is not greater than 2.8 A; (iii) the angle is larger than 120”. (16) Comparison of coordinates of atomic subsets In crystal structure analysis it is frequently necessary to compare subsets of atoms (molecules or part of molecules) present in the crystal and usually the comparison is carried out on bond distances and angles sometimes using statistical methods like normal probability plots (Abrahams & Keve, 1971). Subroutine SYMMOL allows to compare directly the atomic coordinates and gives the averaged coordinates with the rotation matrices and the translation vectors which bring one subset of atoms into the other and vice uersa. The principles of the calculation are as follows. If x, = (x,;. x2,, x,,) is the position vector of atom i in one subset (defined by the fractional coordinates in the crystal space) and xi = (xii, x;, x&) is the same vector for the corresponding atom i in the second subset, then x; = Rx, + t (10) where R is a rotation matrix and t a translation vector. The elements of this matrix and vector are calculated by a least-squares technique. If RsTis the transposed of vector R, = (R,,, R,, R,,, t5) formed by the s-row of matrix R and by the s element of vector t, then: R,r=A-‘P,’ where A ’ is the reciprocal A whose elements are:
s =I,...4 of the symmetric
j,k=1,2,3
(11) matrix
(12)
98
M. NARDELLI
and P,’ is the s-column elements are:
vector
j,k=
of matrix
P whose
1,2,3
(13)
the sums being over i = 1, 2,. . N/2, with N = total number of atoms (of the two sets altogether). The output gives: (i) The symbols of the atoms of the two subsets. (ii) The fractional and orthogonal coordinates of the centroids of the two subsets with e.s.d.‘s. (iii) The rotation matrix and translation vector which bring subset 1 into subset 1’. (iv) The fractional coordinates of the atoms of subset I’ as obtained from those of subset 1, together with the differences with the original coordinates of subset 1’. (v) The averaged fractional coordinates of subset 1’ together with the differences with the original coordinates of the same subset 1’. (vi) The same as (iiij(v) but referred to subset 1’ which is now brought into subset 1. (vii) The internal coordinates, Xi”, in A units referred to two isooriented orthogonal systems of axes having their origins at the centroids of the two subsets, together with the e.s.d.‘s, the differences A, = X7 - Xp. for each couple of atoms and the ratios A#, = AJ[a’(x:) + .s*(~,))“*. (viii)
The sums Z (Ada;)’
the theoretical
for each coordinate
and
x2 Value. COMMENTS
The program has been implemented on a Gould Systems 32-77, (32-bit words computer), working under the MPX-32 multiuser operating system, but the routines had been written p;evious& fbr a Idbit minicomputer, so many statements needed to avoid stops due to underflow are present. The language is essentially FORTRAN IV with only a few multiple assignment statements.
Two versions, one for 100 maximum number of atoms, are number of statements is 4021, comments giving the instructions program. Copies of the listing, ples of input and output, are author on request.
the other for 200 available. The total of these 159 are for the use of the together with examavailable from the
Acknowledgement-The author is indebted to Mr. G. Pasquinelli for his valuable help in implementing the program on the computer. REFERENCES Abrahams, S. C. & Keve. E. T. (1971), Acfa Crysf. A27, 157. Azaroff, L. V. & Buerger, M. J. (195S), The Powder Method in X-Ray Crystallography. New York, McGraw-Hill. Blow. D. M. (1960). Acca Crvsz. 13. 168. Bus&, W. R: & L&, H. A: (1964j, Acra Crysr. 17, 142. Cremer, D. & Pople, J. A. (1975), J. Am. Chem. Sot. 97, 1354. Duax, W. L., Weeks, C. M. & Robrer, D. C. (1976) in Topics in Stereochemistry, Eliel, E. L. % Allinger, N., Eds., New York, Wiley, Vol. 9. Hamilton, W. C. (1959), Acto Cry.vt. 12, 609. Hamilton. W. C. (1961). Aera Crvst. 14. 18.5. Intematidnal Tables f& X-Ray’ Crys&ography (1959), Vol. II, pp. 60 and 331. International Tables for X-Ray Crystallography (1974), Vol. IV. b. 55. Klyne, W. & Prelog, V. (1960), Experientia 16, 521. Kiivj,, I. & Gruber, B. (1976), Acra Crysr A32, 297. Nardelli, M. (1982), Compur. Chem. 6, 139. Nardelli, M. (1983), Actn Crust. C in press. Nardelli, M., Musatti, A., Domiano, P. & Andreetti, G. (1965), Ric. Sci. 15(11-A), 807. Niggli, P. (1928), Hundbuch der Experimenfulphysik, Vol. 7, Part 1. Leipzig, Akademische Verlagsgeselschaft. Schomaker, V., Waser, J., Marsh, R. E. & Bergman, G. (1959), Acrn tryst. 12, 600. Standford, R. H. Jr & Waser, J. (1972), Ac~a Crysr. AZ& 213.
Swain, C. G., Swain, M. S. & Strong, P. F. (1980), J. Chem. hf. Compur. Sci. 20, 51. Willis, B. T. M. % Prior, A. W. (1975), Thermal Vibrations in Crystallography. Cambridge University Press.