DLS-76A PROGRAM FOR THE SIMULATION OF CRYSTAL STRUCTURES BY GEOMETRIC REFINEMENT by Ch. Baerlocher, A. Hepp and W.M. Meier AUGUST 1977 INSTITUTE OF CRYSTALLOGRAPHY AND PETROGRAPHY, ETH SONNEGGSTRASSE 5 8092 ZUERICH SWITZERLAND

Revised version, March 1978 Address for copies of DLS-76 and inquiries regarding the program: Dr. Ch. Baerlocher Lab. of Crystallography ETH CH-8092 Zuerich, Switzerland

P R E F A C E DLS-76 is the successor of the original distance least squares program DLS by VILLIGER (1969). It is a much extended and revised version of DLS-74, a preliminary program set up by GUIGAS (1975) at the University of Karlsruhe, BRD. DLS-76 in- corporates all the features (such as linear constraints etc.) which have been developed and used in various modified versions since 1969. The present program is written in FORTRAN IV and has been extensively tested on the CDC 6400/6500 at ETH Zuerich. This program is distributed with the disclaimer that it is to be used at your own risk. Comments will be much appreciated. We want to thank many of our friends and colleagues, in parti- cular Prof. W. Baur, Drs. V. Gramlich, B. Guigas, S.L. Lawton, and E.L. Wu for helpful discussions and various contributions. We also want to express our appreciation to the Schweizerischer Nationalfonds and ETH Zuerich for financial aid.

CONTENTS -------- INTRODUCTION 1- 1 to 1- 3 PROGRAM DESCRIPTION Program Features and Applications 2- 1 to 2- 4 Program Flow and Operations 2- 5 to 2- 9 DATA INPUT Function and Format of Data Cards 3- 1 to 3-29 Parameter File 3-31 EXAMPLES 4- 1 to 4-16 GLOSSARY OF SYMBOLS Control Integers and Single Variables 5- 1 to 5- 3 Arrays 5- 3 to 5- 8 FORMULAE 6- 1 to 6- 8 REFERENCES 7- 1 to 7- 2 PROGRAM LISTING 8- 1 to 8-39

1-1 1. INTRODUCTION --------------- For a great many crystal structures interatomic distances and bond angles can be predicted within fairly narrow limits. For the present purpose, bond angles are conveniently expressed in terms of bonded and non-bonded distances. Especially for framework type structures the total number of predictable interatomic distances as a rule exceeds the number of ad- justable atom coordinates or positional parameters. In the 2-dimensional (hypothetical) example shown in Fig. 1 a total of at least 10 crystallographically non-equivalent distances(1 would presumably be predictable and could then be used to determine the 6 positional parameters of the structure since interatomic distances are a function only of the atom coordi- nates and the unit cell constants(2. It is evident that in such a case the positional parameters can be computed from prescribed interatomic distances Dj0 by a least-squares pro- cedure minimizing the residual function FORMULA in which Djm,n is the calculated distance of type j between atoms m and n, and Wj is the weight ascribed to the inter- atomic distance of type j. This method of geometric or DLS refinement, first described in detail by MEIER and VILLIGER (1969), produces optimized model structures with respect to (1 5 M-X and 5 X-X distances (2 The distance Dm,n between a pair of atoms m and n is FORMULA

1-2 Figure 1 prescribed interatomic distances (or ratios of interatomic distances) and unit cell constants for a given space group. Only very approximate starting coordinates are needed. Possible variables are atom coordinates and/or unit cell constants (or functions thereof). The weight w of each error equation is normally based on bonding considerations (cf. BAUR, 1977) or observed variations in bond length values. Applications of DLS include (i) Evaluation of hypothetical structures and preliminary refinement of trial structures. This is of particular use in structure determinations based on powder data (cf. BARRER and VILLIGER, 1969).

1-3 (ii) Study of geometrical constraints in framework type structures and determination of probable space group symmetry (cf. MEIER and VILLIGER, 1969). (iii) Analysis and refinement of pseudosymmetric crystal structures (GRAMLICH and MEIER, 1971; TILLMANNS, GEBERT and BAUR, 1973; DOLLASE and BAUR, 1976). (iv) Interpretation of symmetrized and of superimposed structures (MEIER, 1973). (v) Simulation of the response of complex crystal structures to changes in pressure and/or temperature, as well as estimation of likely changes in cell dimensions on isostructural substitution (DEMPSEY and STRENS, 1976; KHAN, 1976). Hard constraints are imposed upon the error equations by crystallographic symmetry elements. Other subsidiary conditions are encountered (e.g.) in analysis of pseudosymmetric structures. To ensure that a DLS model structure with reduced symmetry (hettotype or H-type structure) remains compatible with an experimentally determined structure of idealized high symmetry (aristotype or A- type structure) subsidiary conditions of the form FORMULA are applied where #xi are the displacements which generate the H- type structure from the A-type, d(i) the pseudosymmetry operations and n is the index of the subgroup relation (MEIER and VILLIGER, 1969). The weights of these soft constraints (which are treated like observational equations according to WASER, 1963) are based on the standard deviations of the coordinates of the A-type structure.

2-1 2. PROGRAM DESCRIPTION ---------------------- In the first part of this chapter a general description of the features of the program is given. In the second part the more specific operation and program flow is described together with information which may be helpful when installing the program on another computer. 2.1 Program features and applications The basic features of the present program as compared to previous versions are: - Simplified and flexible data input The input data are kept to a minimum and are checked as far as possible. In case an error is detected detailed error messages are printed. However, the program is stopped only after all input cards have been processed. Prescribed distances which happen to be symmetrically equivalent are eliminated by the program. For tetrahedral framework structures only the connectivity of the atoms must be specified and the program generates all error equations (T- O, O-O, and T-T distances) itself. (See description of TETCON card). - Random atom coordinates To further simplify the data input atom coordinates obtained from a random number generator can be used instead of punched coordinates. As shown by GUIGAS (1975) DLS computations converge in most instances even when random coordinates are used as starting parameters, in which case some 10 to 40 cycles are usually required. Random coordinates have two further advantages: (1) By testing around 5 to 10 different random parameter sets it is possible that the program may find two or more non-equivalent solutions which otherwise would remain undetected.

2-2 (2) Estimated starting parameters frequently tend to correspond to a higher symmetry than the one actually desired. This results in some very high correlations and the refinement can be inhibited or a solution will only be found in the higher symmetry. The same situation arises when a symmetry reduction is performed. In this case random coordinates should be used or the trick used in Example 2 (with NATOM cards) may be applied. - Refinement procedures Basically the program uses the least squares method. However, there is a choice of two different modes, the Newton-Raphson and the Gauss-Newton procedures. GUIGAS (1975) investigated the convergence behavior of these two procedures and his recommendations are as follows: In the Newton-Raphson procedure the first and the second derivatives of the distances are calculated. Normally this leads to a faster convergence. It is recommended for runs with invariant cell parameters, when using random starting coordinates, and when the other procedure does not converge. The Gauss-Newton procedure is the classical one, in which only the first derivatives are calculated. It is employed with advantage when cell parameters are also refined, when linear constraints are used, and when the Newton-Raphson method fails. - Refinement of cell parameters and of prescribed distances (constant ratio or R refinement) In addition to the atom coordinates the cell parameters can also be varied. This is useful for the determination of ideal cell parameters. In the constant ratio refinement the prescribed distances are also refined but their ratios are kept constant. In

2-3 this way a model with ideal polyhedra will result.Normally these refinements should only be tried using coordinates which have already been partially refined. - Linear restrictions The program also allows for linear restrictions on the coordinates which arise when the symmetry for the DLS refinement has been reduced. These restrictions can be included as soft and/or hard constraints and are simply punched on cards as equations. This application is illustrated in Example 2. - Adjustment of prescribed interatomic distances (APID) It has been shown (cf. BROWN, GIBBS and RIBBE, 1969) that in framework silicates e.g. T-O distances depend to some extent on the T-O-T angle. Such relationships can be included as polynominal functions. The program will adjust the prescribed interatomic distances after each set of refinement cycles according to this function. - Calculation of approximate eigenvalues of the matrix If the refinement has converged, the approximate eigenvalues of the matrix are printed for the last cycle. In the case of the Newton- Raphson procedure they are to be interpreted as follows: - All eigenvalues are positive: A minimum has been found. - All eigenvalues are negative: This corresponds to a maximum in the function and the parameters do not represent a proper solution of the least squares problem. - Positive and negative eigenvalues are present: This indicates a saddle point of the function and is again not an actual solution. - Some eigenvalues are extremely small (approaching zero): In this case no statement about the nature of the solution can be made.

2-4 - Difference vectors of reference structure and DLS model Frequently, one may wish to compare the DLS model with the coordinates obtained by X-ray analysis (reference structure). If the coordinates of the latter are supplied as starting values DLS-76 will calculate the difference vectors between the two structures and their magnitudes (in A). In space groups with no fixed origin one or more coordinates of an arbitrarily chosen atom have to be fixed in the DLS-refinement. In these cases the deviations between the DLS model and the reference structure are minimized by translating the DLS model along the respective axes. - Parameter file In order to divide a large job into several smaller jobs, the refined parameters of each cycle can be written on a parameter file. This file can also be used to select "prerefined" sets of coordinates which have resulted from a run with different sets of random starting coordinates. - Additional features There are a number of additional features in the program for special applications (e.g. variable and fixed damping factors, convergence test, tests whether distances lie within specified limits etc.). Information on these can be found in the description of the data input.

2-5 2.2 Program flow and operations This part is intended for users who would like to understand the detailed operation of the program and who may want to modify some parts of it. The general outlay of the program and the function of some of the more important subroutines will be described here. Numerous comment cards are included in the source deck which describe in detail the specific operations performed. General The standard version is dimensioned for 100 atoms (independent and dependent) 200 distances 150 variables The approximate memory requirements are therefore as follows program code, includinq system routines ca. 16K without I/O buffers arrays (labeled common) ca. 13K matrix array (blank common) 11.5K adding up to a total of about 40K without I/O buffers. The program uses 3 machine specific functions, namely DATE(DA) in the main program for the current date. The date is printed in subroutine KOPF SECOND(CP) in the main program which gives the CP time since start of job RANF(Y) in subroutine DATIN which returns a random number between 0 and 1 (see comment cards in DATIN) (CA, CP and Y are all dummy arguments) A simplified flow chart giving an overview of the program is shown in Fig. 2 on the next page. The calling sequence of the subroutines can be readily seen in this chart. The main program, described below

2-6 Simplified Flow Chart of DLS-76 Figure 2

2-7 in more detail, is marked with heavy lines. The subroutine calls are indicated by dotted lines. The subroutine names are given and their function is briefly explained. For simplicity only the more important subroutine calls are included. Main program As indicated in the flow chart this routine controls the program flow by calls to different subroutines according to the control flags set by the user. The parameter file is also written by this program but only minor calculations (such as R-values) are performed. Subroutine DATIN All input data is handled by this routine. The cards are read twice, first to determine their function and a second time to read the parameters. In the CDC version this is done by means of the DECODE statement. The input parameters and the dimension bounds are checked, error messages printed, index tables built up and some preliminary calculations performed. DATIN also contains the random coordinate generator and for this reason it is called each time a new set of coordinates is generated. The program is stopped by this routine when a FINISH card is encountered. Subroutine SYMOP The coded symmetry information on the ATOM cards (special positions) and the SYMEQ cards are decoded by this routine. The homogeneous, non-translational part is then stored in arrays B(K,K,N) and SIGN(K,K,NEQU) for the ATOM card number N and SYMEQ card number NEQU, respectively. Similarly BI(K,N) contains the invariant part of the restriction and SI(K,NEQU) the translational part of the symmetry transformation. SYMOP also calculates all dependent coordinates from the independent ones according to these symmetry transformations.

2-8 Subroutine SETUP SETUP is called only in case of tetrahedral structures when TETCON cards are supplied. Internally the connectivity of each tetrahedron is stored in ICON(NZA,ll). The routine generates all distances around the central atom NZA (i.e. T-O, O-O and T-T distances), evaluates their prescribed values using the information of the BONDIS cards and eliminates equivalent distances. It sets up the array MD(NZA,18) which contains the internal number of each distance around atom NZA in the order described in the Glossary of Symbols. This array is used in subroutine APID. Subroutine EQUI This routine is called to test if two distances are symmetrically equivalent. The test is done in the following way. First the calculated values of the two distances are compared. If the difference is larger than 10-8, the distances are considered as not equivalent. If they do agree within this limit, artificial shifts are applied to the atom parameters and the resulting distances are compared once more. If they still agree the distances are considered equivalent. Subroutine DATOUT The checked input data, i.e. the program control flags and the initial parameters (cell constants, atom coordinates, linear restrictions etc.) are printed by this routine. It has also a second entry (PAREX) which is called at the end of a run to print the final atom parameters in a special format. Subroutine DISDER The matrix and vector of the equations to be solved in each iteration cycle (the normal equations in the Gauss-Newton procedure) are set up by this routine. This involves mainly the calculation of the distances and their derivatives. The more important equations on which the calculations are based are derived in chapter 6.

2-9 Subroutine MATOUT This routine may be called to print the matrix and vector set up by subroutine DISDER. It is also used to calculate and print the correlation matrix, the estimated standard deviations (in the case of the Gauss-Newton procedure) and to print the approximate eigenvalues. Subroutine INVERT and INCH These two routines are used to calculate the inverse matrix. They are based on the procedure for inverting large symmetric matrices described by BUSING and LEVY (1962). As a further option a diagonal matrix approximation is also available. Subroutine APID Subroutine APID adjusts in an iterative manner the prescribed interatomic distances in tetrahedral structures according to a function supplied on the BONDIS card. This function expresses the dependence of the T-O distances on the observed T-O-T angle. Prescribed O-O distances are calculated using an ideal tetrahedral angle and the adjusted prescribed T-O distances. For calculating the T-T distances the T-O-T angle supplied on the BONDIS card is used. The routine also checks whether the observed distances are whithin prescribed ranges and it can adjust the weight of "out of bound distances" in order to force them back. The calculations are mainly controlled by the array ICON(NZ,ll) which contains the connectivity around each tetrahedral atom and the array MD(NZA,18) which contains the number of all distances of each tetrahedron (see also Glossary of Symbols). At the end a table is printed of all old and new prescribed D 's and of the distances and angles in the model which are of likely interest.

3-1 3. DATA INPUT ------------- The data input for DLS-76 is similar to that of the well-known X-RAY-SYSTEM. Each data card has a name consisting of up to six characters which determines its function. Currently the following cards are accepted: 1) TITLE Page 3- 3 2) DLS-76 3- 5 3) FACTOR 3- 7 4) RANGES 3- 9 5) CELL 3-11 6) ATOM 3-13 7) SYMEQ 3-15 8) NOREF 3-17 9) BONDIS 3-19 10) TETCON 3-21 11) DISTAN 3-23 12) LINRES 3-25 13) FILES 3-27 14) END 3-29 15) FINISH 3-29 With the exception of the FILES card, which can appear anywhere in the deck, the data cards should be in the above order. Generally not all these data cards will be used depending on the specific problem.

3-2

3-3 3.1 Function and Format of Data Cards TITLE card (optional) FORMAT (A6,18A4) Cols. Specified punching or function of the field ----- ------------------------------------------- 1 - 5 TITLE 7 - 78 Alphanumeric text which will be printed as heading on each page.

3-4

3-5 DLS-76 card FORMAT (A6,1X,I3,14I2,2X,5I2) Cols. Specified punching or function of the field ----- ------------------------------------------- 1 - 6 DLS-76 8 - 10 0/1/-N: use Gauss-Newton/use Newton-Raphson procedure/do first N cycles Gauss-Newton and finish with Newton-Raphson. 11 - 12 0/1: full matrix/diagonal matrix approximation. 13 - 14 0/N: do not/generate N sets of coordinates. Coordinates which are invariant are not gene- rated and the value supplied on the ATOM card is used. 15 - 16 0/N: do not/do make N cycles of distance refinement. 17 - 18 0/N: do not/do make N cycles of prescribed distance refinement (constant ratio refinement). 19 - 20 0/N: do not/do make N "APID" cycles (adjusted prescribed interatomic distances; see BONDIS and TETCON cards). 21 - 22 0/1: do not/do refine cell parameters 23 - 24 NC} } select parameters of cycle NC of APID cycle } NA of set number NS (if random coordinates } have been used), when reading from file 25 - 26 NA} } NFILEA (see FILES card). 27 - 28 NS} 29 - 30 0/1: do/do not make a convergence test (see FACTOR card). 31 - 32 0/N: do not/use variable damping factor for parameter shifts. (If in a particular cycle the agreement factor increases, all calculated parameter shifts will be halved N times or at least until the new agreement factor is smaller than the old one).

3-6 DLS-76 card (Cont.) Cols. Specified punching or function of the field ----- ------------------------------------------- 33 - 34 0/1: do not/translate refined DLS model along x to minimise average deviations from initial coordinates. (This can only be used when the space group requires the x coordinate of an atom to be arbitrarily fixed.) 35 - 36 0/1: do not/translate refined DLS model along y. 37 - 38 0/1: do not/translate refined DLS model along z. 39 - 40 not used. Output control: --------------- 41 - 42 0/1: print parameters after first and final cycle/after each cycle. 43 - 44 0/1: print distances after first and final cycle/after each cycle. 45 - 46 0/1/2: do not/print matrix and vector of final cycle/of each cycle. 47 - 48 0/1/2/-1/-2: do not/print elements of corre- lation matrix of magnitude greater than the value specified on the FACTOR card after final cycle/after each cycle/print complete corre- lation matrix after final cycle/after each cycle. Note: when using Newton-Raphson procedure, the correlation matrix will only be calculated after the final cycle (using Newton-Gauss procedure). Each time the correlation matrix is calculated the estimated standard deviation of the atom coordinates will also be printed. 49 - 50 0/1/2: do not/write parameters of final cycle/ write parameters of every cycle in card format on file NFILEA (see FILES card).

3-7 FACTOR card (optional) FORMAT (A6,4X,3F5.2,I5,4E10.4) Cols. Specified punching or function of the field ----- ------------------------------------------- 1 - 6 FACTOR 11 - 15 Damping factor to be applied to coordinate changes (default 1.0). 16 - 20 Damping factor for cell parameter changes (default 1.0). 21 - 25 Damping factor for changes of prescribed distances (default 1.0). 26 - 30 Starting number for random number generator (integer, default 0). 31 - 40 Factor for convergence test (default 0.0001). Refinement stops when all parameter changes are smaller than this value (see col. 30 on DLS-76 card). 41 - 50 Multiply all weights of linear restrictions (see LINRES card) with this factor (default 1.0). 51 - 60 Multiply all weights of distances which are outside a given range with this factor (see RANGES card). The distances are only checked and this factor applied during an APID cycle (default 1.0). 61 - 70 Minimum absolute value for correlation matrix printout (default 0.5). Only the elements with an absolute value greater than this one will be printed if requested on column 48 of DLS-76 card. Note: If the card is not supplied or a field is left blank, the default values are used.

3-8

3-9 RANGES card (optional) FORMAT (A6,4X,5F5.2) With this card the range for each type of distance in a tetrahedral framework can be defined. In an APID run the program will mark with an asterisk those distances which fall outside this range and will multiply the appropriate weight by the factor supplied on the FACTOR card (col. 51-60). When this factor is not equal to 1.0, the adjustment of the corresponding prescribed distance will be suppressed. Cols. Specified punching or function of the field ----- ------------------------------------------- 1 - 6 RANGES 11 - 15 Relative deviation of the T-O distance (default: 0.03) 16 - 20 Lower limit of O-T-O angle (default: 104.5) 21 - 25 Upper limit of O-T-O angle (default: 114.5) 26 - 30 Lower limit of T-O-T angle (default: 115.5) 31 - 35 Upper limit of T-O-T angle (default: 175.5) Note: If the card is not supplied or the field is left blank, the default values are used.

3-10

3-11 CELL card FORMAT (A6,1X,A4,9X,6F10.4) Cols. Specified punching or function of the field ----- ------------------------------------------- 1 - 4 CELL 8 - 11 Axial system specification TRIC triclinic MON1 monoclinic, first setting MON2 monoclinic, second setting ORT orthorhombic TET tetragonal HEX hexagonal and trigonal RHO rhombohedral CUB cubic 21 - 30 lattice constant a in Angstroem units 31 - 40 b 41 - 50 c 51 - 60 angle #a in degrees or cos #a 61 - 70 #b or cos #b 71 - 80 #g or cos #g Note: Only the parameters which are independent for a given system must be supplied, e.g. for the hexagonal system only a and c must be punched. Angles in degrees and cosines may be mixed. Values <1 are taken to be cosines. When the cell refinement is used, the symmetry restrictions on the cell parameters are set by the program.

3-12

3-13 ATOM card FORMAT (A6,1X,A6,3F8.5,2X,A3,1X,30A1) Cols. Specified punching or function of the field ----- ------------------------------------------- 1 - 4 ATOM 8 - 13 Atom label 14 - 21 x coordinate 22 - 29 y coordinate 30 - 37 z coordinate 40 - 42 Atom type (used in connection with BONDIS card) 44 - 73 If the atom is in a special position this field contains in free format the relation between the coordinates and/or the values of the fixed coordinate in a form similar to that commonly used (see below). Note: For each symmetrically independent atom an ATOM card must be supplied. The atom type must be stated when the distances are specified by BONDIS and TETCON cards. Atoms which are symmetrically equivalent to these-atoms must be listed on SYMEQ cards. Special positions: Special positions are written in the form commonly used, e.g. X,2X,Z whereby the following special rules must be observed: 1) Coordinates considered as independent may not have a sign, a coefficient or an additional constant. 2) An independent coordinate must appear at its correct place, e.g if Y is considered to be the independent coordinate, it must appear in the second position, after the first comma. 3) Coordinates fixed by symmetry are punched in the form 0,1/2,-3/4 etc. (and not as floating point numbers).

3-14 ATOM card (Cont.) The following symbols may be used to describe a special position: 0 1 2 3 4 5 6 7 8 9 + - / , X Y Z Blanks may be included anywhere, the comma is used as separator. Examples: X,2X-1,1/2 correct -X,1-2X,1/2 not allowed (violates rule 1) Y,2Y-1,1/2 not allowed (violates rule 2) X, -X,Z correct 2X, X,Z not allowed (must be written as 2Y,Y,Z) X,1/2X,Z not allowed (no fractional coefficients allowed) 1/4, 1/4, 1/4 correct -Y, Y, 0 correct The coordinate fields (cols. 14 - 37) of dependent or fixed coordinates can be left blank.

3-15 SYMEQ card FORMAT (A6,1X,A6,1X,A6,1X,40A1) This card is used to specify symmetrically equivalent atoms to the atoms supplied on ATOM cards. Cols. Specified punching or function of the field ----- ------------------------------------------- 1 - 5 SYMEQ 8 - 13 Atom label of independent atom (must appear on an ATOM card). 15 - 20 Atom label of the symmetrically equivalent atom to the atom in cols. 8-13. This label should not appear on an atom card. 22 - 61 Symmetry operation which transforms the independent atom (cols. 8-13) into the de- pendent atom (cols. 15-20). The symmetry operation can be punched in free format in the same manner as the special position on the ATOM card. The transformations of the "general positions" in the International Tables are used throughout (also for atoms in special positions). For atoms in neighbouring cells additional translations must be added. The following symbols may be used to state the symmetry operation: 1 2 3 4 5 6 7 8 9 + - / , X Y Z Blanks may be included anywhere, the comma is used as separator. If a translation is a fraction of a cell edge, it must be given as a quotient n/m, where n and m are integers. The order of the terms is free, but additions of translations are not performed, i.e. 1 + 1/4 must be punched as 5/4 (see examples 2 and 3).

3-16 SYMEQ card (Cont.) Examples: (all acceptable) -Z, X, -Y 1/4+Z, 3/4+Y, 3/4-X Z+5/4, 3/4+Y, -X+3/4 X-Y, -Y, 1/2+Z -X+1, -X+Y-1, Z-1/2

3-17 NOREF card Using this card parameters (atom coordinates, cell parameters) which are not already invariant or dependent due to symmetry restrictions (special positions) can be set constant. There are two formats for this card, one for atom coordinates and one for cell parameters. a) Atom coordinates FORMAT (A6,1X,A6,3(A1,2X)) Cols. Specified punching or function of the field ----- ------------------------------------------- 1 - 5 NOREF 8 - 13 Atom label 14 X, Y and/or Z, depending which coordinate 17 should be kept invariant. X, Y and Z may be 20 in any order. If the cols. 14, 17 and 20 are left blank, all three coordinates are set invariant. The card NOREF ATOMS, where ATOMS is punched in columns 8-12 forces all coordinates invariant. In this case only cell parameters and prescribed distances can be refined.

3-18 NOREF card (Cont.) b) Cell parameters FORMAT (A6,1X,A6,6A4) Cols. Specified punching or function of the field ----- ------------------------------------------- 1 - 5 NOREF 8 - 11 CELL 14 - 17 These fields may contain any of the following 18 - 21 words in any order: 22 - 25 Abbb, Bbbb, Cbbb, ALFA, BETA, GAMA (b=blank) 26 - 29 30 - 33 The parameters of the words appearing are set 34 - 37 invariant. Note: A NOREF CELL card is only necessary in case of cell refinement when parameters not already invariant or dependent due to crystal system requirements are to be fixed.

3-19 BONDIS card FORMAT (A6,1X,3(A3,1X),1X,4E10.4,3F5.0) This card can be used in conjunction with the TETCON card to specify the prescribed interatomic distances of tetrahedral atoms and their weights. Additionally it is used to specify the dependence of the T-O distance as a function of the T-O-T angle. If an APID run (adjusted prescribed interatomic distances) is requested (Col. 20 on DLS-76 card) this function is used to calculate new prescribed distances according to the T-O-T angles in the model. The card contains the bond type, the distance function for this bond type and the weights for the various types of distances. The function has the form DO = A + B(TOT - #w) + C(TOT - #w)**2 where DO : prescribed interatomic distance TOT : actual angle at bridging atom #w : standard T-O-T angle (e.g. 145 Degree) A,B,C : constants Cols. Specified punching or function of the field ----- ------------------------------------------- 1 - 5 BONDIS 8 - 10 Atom type of tetrahedral atom (central atom) 12 - 14 Atom type of bridging atom 16 - 18 Atom type of outer T-atom 21 - 30 Parameter A of distance function 31 - 40 B 41 - 50 C 51 - 60 Angle #w in degrees (default 145 Degree) 61 - 65 Weight for T-O bond of this bond type 66 - 70 Weight for O-O bond 71 - 75 Weight for T-T bond

3-20 BONDIS card (Cont.) Note: This card must be supplied if the error equations (DISTAN-cards) are generated by the program from the connectivity specifications and of course always when an APID run is to be performed.

3-21 TETCON card FORMAT (A6,7X,9(A6,1X)) This card serves to specify the connectivity of tetrahedral atoms, i.e. the way these atoms are connected in the tetrahedral framework. On one card the central atom and the atoms of its first and second coordination are stated. Together with the information from the BONDIS card, the program generates all independent T-O, O-O and T-T distances and assigns the proper prescribed distances and weights. This saves punching all these DISTAN cards. However, distances generated in this way can always be overwritten by supplying a DISTAN card, and, of course, other distances can be added with DISTAN cards. In an APID run the TETCON card supplies all the necessary information for the calculation of the adjusted prescribed interatomic distances of tetrahedral framework structures. Cols. Specified punching or function of the field ----- ------------------------------------------- 1 - 6 TETCON 14 - 19 Label of the T-atom 21 - 26 Label of first bridging atom (O-atom) 28 - 33 Label of second, 35 - 40 third, 42 - 47 and fourth bridging atom 49 - 54 Label of outer T-atom bonded to first, 56 - 61 second, 63 - 68 third, 70 - 75 and fourth bridging atom. Note: All labels of atoms considered must appear either on an ATOM or a SYMEQ card.

3-22 TETCON card (Cont.) Special positions: Only those T-O bonds which are symmetrically independent must be specified. However, make sure that all O-atoms are punched which are necessary for the specification of all independent O-O distances. For each independent T-O bond the outer T-atom must also be given (for the determination of bond type). Bonds or distances which are symmetrically equivalent are eliminated by the program.

3-23 DISTAN card FORMAT (A6,1X,2(A6,1X),2F10.5,A1,F10.5) This is the standard card to specify an error equation. Each card contains the label of two atoms, their prescribed distances and the weight assigned to these distances. In addition a reference distance can also be supplied in case a constant (distance) ratio refinement is to be performed. Cols. Specified punching or function of the field ----- ------------------------------------------- 1 - 6 DISTAN 8 - 13 Atom label of first atom 15 - 20 Atom label of second atom 22 - 31 Prescribed interatomic distance between the two atoms. If the field is left blank, the value of the previous card is used. 32 - 41 Weight assigned to this distance. If the field is left blank, the weight will be taken as 1.0. However, if the distance field is also blank, the weight of the previous card will be used. 42 Blank/any alphanumeric character: This pre- scribed distance is invariant/this prescribed distance will be refined in a constant ratio refinement. All prescribed distances having the same character in this field depend on the same reference distance. Their ratio to the reference distance will be kept constant during the re- finement (see example). 43 - 52 Reference distance. If this field is left blank (and column 42 is not blank) the prescribed distance for this card will be taken as re- ference distance, i.e. the distance ratio will be 1.0.

3-24 DISTAN card (Cont.) Example (with constant ratio refinement) DISTAN A1 A2 2.0 1.0 R 2.0 DISTAN A1 A3 3.0 1.0 R 2.0 DISTAN A1 A4 DISTAN A1 A5 2.5 1.0 R 2.0 All four prescribed distances will be varied, but not independently because all cards have the same character R in column 42 (the third card has the same values as the second card). According to these specifications the ratio of distance 2 to distance 1 and distance 3 to distance 1 is always 3:2 and the ratio of distance 4 to distance 1 is 2.5:2. As a consequence of this, distance 4 and 3 will have a ratio of 2.5:3. Note: Constant ratio refinement and APID refinement can not be combined. In the latter case a prescribed distance can be held constant (not adjusted by the APID function) by supplying a DISTAN card for this distance and punching a C in column 42.

3-25 LINRES card FORMAT (A6,1X,F8.1,1X,5(F4.0,2A1,A6,1X,A1,A6),1X,A1,1X) This card allows to impose linear restrictions on the shifts of the atom coordinates in the form of hard and/or soft constraints. These restrictions take the form 2.0 * #X (ATOM1) - 1.0 * #Z (ATOM3) = 0 as an example. Cols. Specified punchinq or function of the field ----- ------------------------------------------- 1 - 6 LINRES 8 - 15 Weight or sigma of this restriction for soft constraints. If this field is left blank, the re- striction will be taken as a hard constraint and the variable of the last term of this restriction is eliminated. Values preceded by a minus sign are interpreted as weights, otherwise they are taken as sigmas and the weight for the restriction is calculated as follows: Weight = 1/(sigma * nr. of terms in restriction) 17 - 20 First coefficient of the restriction. If the coefficient is +1.0, the field can be left blank. 21 Multiplication sign * (optional) 22 X, Y or Z 23 - 28 Atom label Further terms of the restriction are punched in the same manner in the columns 29-32, 41-44, 53-56, 65-68 coefficients 33, 45, 57, 69 multiplication sign (optional) 34, 46, 58, 70 X, Y or Z 35-40, 47-52, 59-64, 71-76 atom label

3-26 LINRES card (Cont.) In this way up to 5 terms can be punched per card. The restriction can be continued on further cards using the same format (the weight can be omitted). On the last card of the restriction the two characters "=0" must be punched in columns 78-79. A restriction may contain up to 20 terms. Blank fields may be left on cards, i.e. one can use as many cards as are desired. Note: A hard constraint on a single coordinate (e.g. #X (ATOM1) = 0) must be put in using a NOREF card.

3-27 FILES card FORMAT (A6,I3,3X,I3) The FILES card is used to change at any time the logical number of the input file and the parameter output file (not the printing file). In this way, the input or part of it can be read from a mass storage file and not from the card reader, and the parameter output can be diverted either to the card. punch or a disk file. Although the FILES card can be placed anywhere in the data deck it will mainly be used in connection with the transfer of parameters from one job to another. In this case, it has to be inserted after the ATOM cards to read the new parameters from the file (which are on NATOM cards) and also after the TETCON card (if used) in place of or before the DISTAN cards. Please note that when a refinement is divided into several jobs, the card deck is not changed except for the insertion of the FILES cards. Only the DISTAN cards are removed if in the previous job the prescribed distances were altered, either by a constant ratio refinement or an APID cycle. (For further details see section on the format of the parameter file.) Cols. Specified punching or function of the field ----- ------------------------------------------- 1 - 5 FILES 7 - 9 logical number NTIN of the input file 13 - 15 logical number NFILEA of the parameter file (output file) or card punch

3-28 FILES card (Cont.) The default values set by the program (in subroutine DATIN) are NTIN = 5 Input file (normally card reader) NFILEA = 8 Parameter output file NTOUT = 6 Output file (printer) If a field is left blank, the file number is not changed. In the present version, the file numbers 7, 8, 9 and 10 can be used. If number 8 is chosen as input file the file will be rewound each time the FILES 8 cardis encountered. The file number 7 is assigned to the card punch.

3-29 END card FORMAT (A6) The END card signals the end of a data deck and starts the computation. After this card another complete job may follow. Cols. Specified punching ----- ------------------ 1 - 3 END FINISH card FORMAT (A5) After reading this card, the program is immediately stopped. It must occur at the very end of a data deck. Cols. Specified punching ----- ------------------ 1 - 6 FINISH

3-30

3-31 3.2 Parameter File When requested in column 50 of the DLS-76 card the refined parameters (atom coordinates and also cell constants and distances if their values have been changed in the run) are written on a separate file which can be used as input of a subsequent run. The file consists of coded card records and the card format is similar or identical to the normal input cards. Currently the following card types will be written: Name Content (Col. 1-6) TITLE title as supplied on input card CYCLE FORMAT (A6,I4,2I5) NC, NA, NS (see DLS-76 card, cols. 23-28) All parameter sets are identified by such a card. When reading the file the program can thus select the requested set. CELL Format like normal input card(1 NATOM new atom parameters, format as on ATOM card (coordinates only) DISTAN Format like normal input card(1 FILES N Here N is the logical number of the card reader. This card appears at the end of a set of parameter cards and switches the input back to the normal input device. (1 These cards are only written when the values of the respective parameters have been changed.

3-32

3-33 BOUND card (CDC-version only) FORMAT (A6,4X,4E10.4) With this card limits or bounds can be specified for the weighted deviations of linear restrictions and/or distances. If the residual of a particular restriction or distance exceeds these limits the corresponding weight will be multiplied by the absolute value of the residual divided by the "modified bound". The "modified bound" has to be smaller than the specified limit and is calculated according to the following equation: "modified bound" = 3.0 * bound/(3.0+t) where t is supplied on the card. Cols. Specified punching or function of the field ----- ------------------------------------------- 1 - 5 BOUND 11 - 20 Bound for linear restrictions defined in multiples of sigmas (see LINRES card). (default: 3.0) 21 - 30 Term t for the calculation of the "modified bound" in linear restrictions, a suitable value is 1.0. If left blank no bound checks for linear restrictions are made. 31 - 40 Bounds for weighted deviations in distances. (default: 0.05) 41 - 50 Term t for the calculation of the "modified bound" for the distances. Blank: no bound check. Note: The correction factors will also be written on the parameter file on WEIMOD cards and can thus be applied in a subsequent run.

3-34

3-35 GENER card (optional) (CDC-version only) FORMAT (A6,4X,3I5) When this card is encountered the symmetry information for the specified space group is read from the file and all required atoms outside the asymmetric unit (i.e. atoms on the SYMEQ-cards) are generated and their connectivity determined (i.e. the information contained on the TETCON cards). Thus SYMEQ cards and TETCON cards are not required in this case. The program assumes the four shortest T-O and T-T bonds to be the correct connections and therefore the atom coordinates should already be sufficient accurate. Cols. Specified punchinq or function of the field ----- ------------------------------------------- 1 - 5 GENER 11 - 15 Space group number as in International Tables of X-Ray Crystallography, Vol. I. If two orientations are listed (e.g. space group 125, P4/nbm) the first listed is positive (+) and the second is negative (-). The symmetry cards are read from the master data file of program POWD (Smith Plot Program). 16 - 20 File number of the symmetry data file (default 8) 21 - 21 0/1: Do not/print all information used to set up the connectivity tables. This may be used as a debugging aid in case the automatic setting up has failed (e.g. because of inaccurate starting coordinates). Note: The GENER card should appear after the ATOM cards in place of the SYMEQ cards. The TETCON cards must also be omitted.However, DISTAN cards may still be added.

4-1 4. EXAMPLES ----------- Two test examples are provided to illustrate the operation and output of DLS-76. The first example is a straight-forward DLS-refinement and can be used to check the basic operations of the program. The second example is somewhat more elaborate and makes use of a number of special features of DLS-76. Some background information and a description of the two examples is given below. Example 1: Low-quartz-type structure of AlPO4 AlPO4 has a quartz-type structure which was refined in space group P3121 by D. SCHWARZENBACH (1966). The following data are required to set up the basic DLS job: Unit cell parameters: hexagonal system a = 4.9429 A, c = 10.9476 A Atomic positions: There are 4 atoms in the asymmetric unit, i.e. Al in 3a x, 0, 1/3 P in 3b x, x, 1/2 O1 in 6c x, y, z 02 in 6c x, y, z For each of these atoms an ATOM card has to be punched, containing the atom label, approximate coordinates and, in case of Al and P, a specification of those parameters which are fixed by symmetry. Further atoms supplied on SYMEQ cards have to be included in order to be able to specify all independent distances. Besides a new atom label each of these cards contains the label of the symmetrically

4-2 related atom on the respective ATOM card and the transfor- mation (including translational components where applicable) which generates the coordinates of this additional atom. Interatomic distances: In the present example the inter- atomic distances are supplied by DISTAN cards(1. For each independent distance (T-O and O-O distances, T = Al, P) one DISTAN card is punched. Each such card contains the atom labels of two atoms, the prescribed value of their distance and the weight associated with this distance. The values for the T-O distances used here are those given by LOUISNATHAN and GIBBS (1972) and the O-O distances are cal- culated assuming an ideal tetrahedral angle. The input and output for this example using the Newton- Raphson procedure is reproduced below. The refinement con- verges after 6 cycles. The parameters, the shifts and the resulting interatomic distances are printed for the first and the last cycle only, as specified on the DLS-76 input card. For the intermediate cycles only the R-values are printed which serve as an indication of the progress of the refinement. (1 Alternatively, TETCON and BONDIS cards could be used in this case.

4-3 Input for Example 1: TITLE *** EXAMPLE 1 : ALPO4 *** SPACE GROUP P3(1)21 DLS-76 1 10 CELL HEX 4.9429 10.9476 ATOM AL .4 X,0,1/3 ATOM P .6 X,X,1/2 ATOM O1 .5 .4 .3 ATOM O2 .8 .7 .6 SYMEQ O1 O1* X-Y,-Y,2/3-Z SYMEQ O1 O1** Y,X,1-Z SYMEQ O2 O2* Y-X+1,1-X,Z-1/3 SYMEQ O2 O2** Y,X-1,1-Z SYMEQ O2 O2*** Y,X,1-Z DISTAN AL O1 1.748 2. DISTAN AL O2* 1.748 2. DISTAN O1 O1* 2.8545 1. DISTAN O1 O2* 2.8545 1. DISTAN O1 O2** 2.8545 1. DISTAN O2* O2** 2.8545 1. DISTAN P O1 1.538 2. DISTAN P O2 1.538 2. DISTAN O1 O1** 2.5115 1. DISTAN O1 O2 2.5115 1. DISTAN O1 O2*** 2.5115 1. DISTAN O2 O2*** 2.5115 1. END FINISH

4-4 4-5 4-6 ------------------------------------------------------------------------ DLS-76 *** EXAMPLE 1 : ALPO4 *** SPACE GROUP P3(1)21 DATE: 24-MAY-95 PAGE 1 PROGRAM SPECIFICATIONS ---------------------- REFINEMENT MATRIX RANDOM CONST DIST CONST RATIO PRESCRIBED DO CELL TYPE INVERSION COORDINATES REFINEMENT REFINEMENT ADJUSTMENT REFINEMENT NEW RAPHS FULL NO 10 CYCLES NO NO NO PARAMETERS FROM CONVERGENCE USE VARIABLE TRANSL FINAL (CYC./APID/SET) TEST DAMP FACTOR COORDINATES CARDS YES NO NO OUTPUT CONTROL -------------- LIST LIST LIST MATRIX LIST CORREL WRITE PARAMETERS PARAMETERS DISTANCES AND VECTOR COEFFICIENTS ON NFILEA ( 8) FIRST/LAST FIRST/LAST NO NO NO CONTROL FACTORS --------------- DAMPING FACTORS STARTING NR CONVERGENCE LINRES WEIGHTS WTS FACTOR FOR CORR MATRIX COORDINATES CELL DISTANCES RANDOM GENER TEST FACTOR MULTIPLIED BY OUT OF RANGE D'S TEST FACTOR 1.000 1.000 1.000 0 0.0001000 1.00000 1.00 0.50 PRELIMINARY STATISTICS ---------------------- NUMBER OF NUMBER OF NUMBER OF NUMBER OF NUMBER OF INDEPEND.ATOMS DEPEND.ATOMS DISTANCES VARIABLES RESTRICTIONS 4 5 12 8 0 HARD/ 0 SOFT DLS-76 *** EXAMPLE 1 : ALPO4 *** SPACE GROUP P3(1)21 DATE: 24-MAY-95 PAGE 2 INITIAL PARAMETERS ------------------ R=REFINE D=DEPENDENT IN A SPECIAL POSITION I=INVARIANT L=DEPENDENT DUE TO A LINEAR RESTRICTION CELL PARAMETERS A B C ALPHA BETA GAMMA VARIABLES 4.9429 I 4.9429 D 10.9476 I 90.00 I 90.00 I 120.00 I 0 - 0 ATOM PARAMETERS ATOM X Y Z TYPE N SPECIAL POSITIONS AL 0.40000 R 0.00000 I 0.33333 I 1 X,0,1/3 P 0.60000 R 0.60000 D 0.50000 I 2 X,X,1/2 O1 0.50000 R 0.40000 R 0.30000 R 3 O2 0.80000 R 0.70000 R 0.60000 R 6 (N=NO OF THE FIRST VAR. IN THIS LINE) SYMMETRICALLY DEPENDENT ATOMS SYMMETRY OPERATIONS O1* 0.10000 -0.40000 0.36667 X-Y,-Y,2/3-Z O1** 0.40000 0.50000 0.70000 Y,X,1-Z O2* 0.90000 0.20000 0.26667 Y-X+1,1-X,Z-1/3 O2** 0.70000 -0.20000 0.40000 Y,X-1,1-Z O2*** 0.70000 0.80000 0.40000 Y,X,1-Z INTERATOMIC DISTANCES BEFORE CYCLE 1 ATOM 1 ATOM 2 CALCULATED D PRESCRIBED DO DO-D WEIGHT W*(DO-D) AL O1 1.8192 1.7480 -0.0712 2.0000 -0.1423 AL O2* 2.2748 1.7480 -0.5268 2.0000 -1.0536 O1 O1* 3.5015 2.8545 -0.6470 1.0000 -0.6470 O1 O2* 2.6409 2.8545 0.2136 1.0000 0.2136 O1 O2** 3.7287 2.8545 -0.8742 1.0000 -0.8742 O2* O2** 2.2500 2.8545 0.6045 1.0000 0.6045 P O1 2.3509 1.5380 -0.8129 2.0000 -1.6259 P O2 1.3898 1.5380 0.1482 2.0000 0.2965 O1 O1** 4.4619 2.5115 -1.9504 1.0000 -1.9504 DLS-76 *** EXAMPLE 1 : ALPO4 *** SPACE GROUP P3(1)21 DATE: 24-MAY-95 PAGE 3 O1 O2 3.6035 2.5115 -1.0920 1.0000 -1.0920 O1 O2*** 2.0323 2.5115 0.4792 1.0000 0.4792 O2 O2*** 2.3510 2.5115 0.1605 1.0000 0.1605 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.32529685 BEFORE CYCLE 1 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 1.63613963 PARAMETERS AFTER CYCLE 1 PARAMETER OLD CHANGE NEW PARAMETER OLD CHANGE NEW ATOM AL ATOM P X 0.40000 -0.11670 0.28330 X 0.60000 -0.12685 0.47315 Y 0.00000 0.00000 0.00000 Y 0.60000 -0.12685 0.47315 Z 0.33333 0.00000 0.33333 Z 0.50000 0.00000 0.50000 ATOM O1 ATOM O2 X 0.50000 -0.08358 0.41642 X 0.80000 0.14474 0.94475 Y 0.40000 -0.01377 0.38623 Y 0.70000 -0.17291 0.52709 Z 0.30000 0.06864 0.36864 Z 0.60000 -0.06723 0.53277 SYMMETRICALLY DEPENDENT ATOMS ATOM X Y Z O1* 0.03020 -0.38623 0.29803 O1** 0.38623 0.41642 0.63136 O2* 0.58235 0.05525 0.19943 O2** 0.52709 -0.05525 0.46723 O2*** 0.52709 0.94475 0.46723 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.20469035 BEFORE CYCLE 2 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 1.02952731 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.06937429 BEFORE CYCLE 3 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 0.34893057 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.01674912 BEFORE CYCLE 4 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 0.08424272 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.00697363 BEFORE CYCLE 5 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 0.03507515 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.00664027 BEFORE CYCLE 6 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 0.03339842 CONVERGENCE TEST POSITIVE AFTER CYCLE 6 DLS-76 *** EXAMPLE 1 : ALPO4 *** SPACE GROUP P3(1)21 DATE: 24-MAY-95 PAGE 4 APPROXIMATE EIGENVALUES OF MATRIX (LAST CYCLE): 0.1951E+04 0.1798E+04 0.2355E+03 0.2006E+03 0.8508E+02 0.7552E+02 0.6704E+02 0.4515E+02 PARAMETERS AFTER CYCLE 6 PARAMETER OLD CHANGE NEW PARAMETER OLD CHANGE NEW ATOM AL ATOM P X 0.46171 0.00004 0.46175 X 0.53424 -0.00003 0.53421 Y 0.00000 0.00000 0.00000 Y 0.53424 -0.00003 0.53421 Z 0.33333 0.00000 0.33333 Z 0.50000 0.00000 0.50000 ATOM O1 ATOM O2 X 0.40947 0.00001 0.40948 X 0.84793 -0.00005 0.84788 Y 0.29672 -0.00006 0.29666 Y 0.58520 0.00002 0.58521 Z 0.39445 0.00002 0.39448 Z 0.54941 0.00002 0.54943 SYMMETRICALLY DEPENDENT ATOMS ATOM X Y Z O1* 0.11282 -0.29666 0.27219 O1** 0.29666 0.40948 0.60552 O2* 0.73734 0.15212 0.21610 O2** 0.58521 -0.15212 0.45057 O2*** 0.58521 0.84788 0.45057 INTERATOMIC DISTANCES BEFORE CYCLE 7 ATOM 1 ATOM 2 CALCULATED D PRESCRIBED DO DO-D WEIGHT W*(DO-D) AL O1 1.7447 1.7480 0.0033 2.0000 0.0066 AL O2* 1.7447 1.7480 0.0033 2.0000 0.0067 O1 O1* 2.8711 2.8545 -0.0166 1.0000 -0.0166 O1 O2* 2.8474 2.8545 0.0071 1.0000 0.0071 O1 O2** 2.8248 2.8545 0.0297 1.0000 0.0297 O2* O2** 2.8784 2.8545 -0.0239 1.0000 -0.0239 P O1 1.5393 1.5380 -0.0013 2.0000 -0.0026 P O2 1.5393 1.5380 -0.0013 2.0000 -0.0026 O1 O1** 2.5042 2.5115 0.0073 1.0000 0.0073 O1 O2 2.5528 2.5115 -0.0413 1.0000 -0.0413 O1 O2*** 2.4876 2.5115 0.0239 1.0000 0.0239 O2 O2*** 2.4957 2.5115 0.0158 1.0000 0.0158 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.00663989 BEFORE CYCLE 7 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 0.03339653 DLS-76 *** EXAMPLE 1 : ALPO4 *** SPACE GROUP P3(1)21 DATE: 24-MAY-95 PAGE 5 TOTAL PARAMETER SHIFTS AFTER LAST CYCLE PARAMETER INITIAL CHANGE FINAL SHIFT PARAMETER INITIAL CHANGE FINAL SHIFT ATOM AL ATOM P X 0.40000 0.06175 0.46175 0.30522 X 0.60000 -0.06579 0.53421 -0.32520 Y 0.00000 0.00000 0.00000 0.00000 Y 0.60000 -0.06579 0.53421 -0.32520 Z 0.33333 0.00000 0.33333 0.00000 Z 0.50000 0.00000 0.50000 0.00000 MAGNITUDE 0.30522 MAGNITUDE 0.32520 ATOM O1 ATOM O2 X 0.50000 -0.09052 0.40948 -0.44743 X 0.80000 0.04788 0.84788 0.23664 Y 0.40000 -0.10334 0.29666 -0.51078 Y 0.70000 -0.11479 0.58521 -0.56739 Z 0.30000 0.09448 0.39448 1.03429 Z 0.60000 -0.05057 0.54943 -0.55358 MAGNITUDE 1.14119 MAGNITUDE 0.90479 ------------------------------------------------------------------------ *** FINISH ***

4-7 Example 2: Desymmetrization of the crystal structure of analcime Analcime is a framework silicate and is normally described as cubic with space group Ia3d and a = 13.73 R . It has a remarkably constant unit cell composition of Na16Al16Si32096 . 16 H2O, which, if fully ordered, is in- compatible with cubic symmetry. MEIER (1973) therefore proposed on the basis of DLS calculations that the symmetry should at least be reduced to I41/acd which would allow for Si, Al ordering assuming a likely distribution scheme. A recent neutron-diffraction study (FERRARIS et al., 1972) based on space group Ia3d led to the following atomic co- ordinates of the framework atoms (estimated standard de- viation in parentheses): T(Si,Al) in 48g .16208(15) .08792(15) .125 (x,1/4-x,1/8) O in 96h .10428(14) .13440(16) .21932(12) (x,y,z) On reducing the symmetry to I41/acd these two positions would split up into the following 5 independent positions(1 (origin in -1): T1(Al) in 16f .16208 .08792 .125 (x,1/4-x,1/8) T2(Si) in 32g .08792 .12500 .33792 (x,y,z) O1 in 32g .10428 .13440 .21932 O2 in 32g .14572 .03068 .38440 O3 in 32g .13440 .21932 .39572 These parameters are punched on ATOM cards and could be used as starting parameters. However, they still possess cubic (1 Compared to the setting in the Int. Tables (Vol. I, page 142) the origin has been shifted to 0,0,1/2.

4-8 symmetry and are unlikely to refine in the tetragonal space group. Instead, somewhat desymmetrized or random coordinates are needed for starting the refinement. The values on the atom cards are then used as reference coordinates only in the calculation of the linear restrictions and in the cal- culation of the total parameter shifts. To enable actual checking in this test example "pseudo-random" coordinates supplied by NATOM cards are used here. Such cards are usually only read from the parameter file and overwrite the starting coordinates but not the reference coordinates. As pointed out in Section 1 the symmetrized DLS coordinates should still agree (say within 3#) with the experimentally determined coordinates of the high-symmetry reference structure. Therefore, restrictions have to be placed on the total coordinate shifts. These restrictions are applied as weighted constraints. In this example the 3-fold axes are removed and the constraints are accordingly: #x(T1) - #z(T2) - #x(T2) = 0 #x(O1) - #x(O2) - #z(O3) = 0 #y(O1) + #z(O2) + #x(O3) = 0 #z(O1) - #y(O2) + #y(O3) = 0 These equations are punched on LINRES cards. The weights used here are the reciprocal values of the estimated standard deviations of the respective coordinates as ob- tained by FERRARIS et al. (1972). These weights are all multiplied by a factor of .012 (punched on the FACTOR card) to scale them to the weights of the distance error equations. The distance error equations for this example are generated from two TETCON cards (one for each tetrahedron). For the prescribed interatomic distances a function is supplied (on a BONDIS card) for each bond type giving the relation- ship between T-O bond distance and T-O-T angle which is taken into account in an APID cycle after a maximum of 15 cycles of DLS refinement using standard distance values.

4-9 The input and complete output is reproduced below. The linear restrictions, bond distance functions and (as a check) the bonding scheme for each tetrahedron as stated on the TETCON card are also printed. Despite the near-random starting coordinates used in this example, the Newton-Gauss refinement converges in 12 cycles, i.e. the shifts in the coordinates become all less than 0.0001 after 12 cycles. The parameter shifts, the interatomic distances, the dif- ference vectors relative to the initial (reference) coordi- nates and the residuals of the linear restrictions are again only printed for the first and final cycle. For the inter- mediate cycles only the R-values are printed and in addition the sum RHO over all squared residuals resulting from the distances and the linear restrictions. Once the convergence test becomes positive (after 12 cycles), the prescribed distances are adjusted according to the function supplied and the T-O-T angles in the model. A list of all the actual distances in the refined model, the original distances as well as the new prescribed distances are printed, and another refinement with the new set of pres- cribed distances is started. Convergence is reached after 3 cycles and all final parameters and values are then printed. The differences in the coordinates of the refined DLS model and the reference structure are given in the last table of the output.

4-10 Input for Example 2: TITLE *** EXAMPLE 2 : ANALCIME *** I4(1)/ACD (WITH LINEAR RESTRICTIONS) DLS-76 15 1 FACTOR .012 CELL TET 13.73 13.73 ATOM T1 .16208 AL X,1/4-X,1/8 ATOM T2 .08792 .125 .33792 SI ATOM O1 .10428 .1344 .21932 O ATOM O2 .14572 .03068 .3844 O ATOM O3 .1344 .21932 .39572 O NATOM T1 .11 .14 .125 NATOM T2 .21 .22 .23 NATOM O1 .31 .32 .33 NATOM O2 .41 .42 .43 NATOM O3 .51 .52 .53 SYMEQ T1 T1* X,-Y,1/2-Z SYMEQ T2 T2* 1/4-Y,1/4+X,3/4-Z SYMEQ T2 T2** Y-1/4,1/4-X,3/4-Z SYMEQ T2 T2*** 1/4+Y,1/4-X,Z-1/4 SYMEQ O1 O1* 1/4-Y,1/4-X,1/4-Z SYMEQ O2 O2* 1/4+Y,1/4-X,Z-1/4 SYMEQ O2 O2** X,-Y,1/2-Z SYMEQ O3 O3* Y-1/4,1/4-X,3/4-Z BONDIS SI O SI 1.620 -.0004 0. 145. 2. 1. .1 BONDIS SI O AL 1.593 -.0004 2. 1. .1 BONDIS AL O SI 1.740 -.0004 2. 1. .1 TETCON T1 O1 O2* O1* O2** T2 T2*** TETCON T2 O1 O2 O3 O3* T1 T1* T2* T2** LINRES 2223. 1. *XT1 -1. *ZT2 -1. *XT2 =0 LINRES 2380. 1. *XO1 -1. *XO2 -1. *ZO3 =0 LINRES 2083. 1. *YO1 1. *ZO2 1. *XO3 =0 LINRES 2776. 1. *ZO1 -1. *YO2 1. *YO3 =0 END FINISH

4-11 4-12 4-13 4-14 4-15 4-16 ------------------------------------------------------------------------ DLS-76 *** EXAMPLE 2 : ANALCIME *** I4(1)/ACD (WITH LINEAR RESTRICTIONS) DATE: 24-MAY-95 PAGE 1 PROGRAM SPECIFICATIONS ---------------------- REFINEMENT MATRIX RANDOM CONST DIST CONST RATIO PRESCRIBED DO CELL TYPE INVERSION COORDINATES REFINEMENT REFINEMENT ADJUSTMENT REFINEMENT NEW GAUSS FULL NO 15 CYCLES NO 1 CYCLES NO PARAMETERS FROM CONVERGENCE USE VARIABLE TRANSL FINAL (CYC./APID/SET) TEST DAMP FACTOR COORDINATES CARDS YES NO NO OUTPUT CONTROL -------------- LIST LIST LIST MATRIX LIST CORREL WRITE PARAMETERS PARAMETERS DISTANCES AND VECTOR COEFFICIENTS ON NFILEA ( 8) FIRST/LAST FIRST/LAST NO NO NO CONTROL FACTORS --------------- DAMPING FACTORS STARTING NR CONVERGENCE LINRES WEIGHTS WTS FACTOR FOR CORR MATRIX COORDINATES CELL DISTANCES RANDOM GENER TEST FACTOR MULTIPLIED BY OUT OF RANGE D'S TEST FACTOR 1.000 1.000 1.000 0 0.0001000 0.01200 1.00 0.50 PRELIMINARY STATISTICS ---------------------- NUMBER OF NUMBER OF NUMBER OF NUMBER OF NUMBER OF INDEPEND.ATOMS DEPEND.ATOMS DISTANCES VARIABLES RESTRICTIONS 5 8 19 13 0 HARD/ 4 SOFT DLS-76 *** EXAMPLE 2 : ANALCIME *** I4(1)/ACD (WITH LINEAR RESTRICTIONS) DATE: 24-MAY-95 PAGE 2 INITIAL PARAMETERS ------------------ R=REFINE D=DEPENDENT IN A SPECIAL POSITION I=INVARIANT L=DEPENDENT DUE TO A LINEAR RESTRICTION CELL PARAMETERS A B C ALPHA BETA GAMMA VARIABLES 13.7300 I 13.7300 D 13.7300 I 90.00 I 90.00 I 90.00 I 0 - 0 ATOM PARAMETERS ATOM X Y Z TYPE N SPECIAL POSITIONS T1 0.11000 R 0.14000 D 0.12500 I AL 1 X,1/4-X,1/8 T2 0.21000 R 0.22000 R 0.23000 R SI 2 O1 0.31000 R 0.32000 R 0.33000 R O 5 O2 0.41000 R 0.42000 R 0.43000 R O 8 O3 0.51000 R 0.52000 R 0.53000 R O 11 (N=NO OF THE FIRST VAR. IN THIS LINE) SYMMETRICALLY DEPENDENT ATOMS SYMMETRY OPERATIONS T1* 0.11000 -0.14000 0.37500 X,-Y,1/2-Z T2* 0.03000 0.46000 0.52000 1/4-Y,1/4+X,3/4-Z T2** -0.03000 0.04000 0.52000 Y-1/4,1/4-X,3/4-Z T2*** 0.47000 0.04000 -0.02000 1/4+Y,1/4-X,Z-1/4 O1* -0.07000 -0.06000 -0.08000 1/4-Y,1/4-X,1/4-Z O2* 0.67000 -0.16000 0.18000 1/4+Y,1/4-X,Z-1/4 O2** 0.41000 -0.42000 0.07000 X,-Y,1/2-Z O3* 0.27000 -0.26000 0.22000 Y-1/4,1/4-X,3/4-Z LINEAR RESTRICTIONS NR WEIGHT 1 0.222E+04 0 = 1.0*DX (T1 ) - 1.0*DZ (T2 ) - 1.0*DX (T2 ) 2 0.238E+04 0 = 1.0*DX (O1 ) - 1.0*DX (O2 ) - 1.0*DZ (O3 ) 3 0.208E+04 0 = 1.0*DY (O1 ) + 1.0*DZ (O2 ) + 1.0*DX (O3 ) 4 0.278E+04 0 = 1.0*DZ (O1 ) - 1.0*DY (O2 ) + 1.0*DY (O3 ) DLS-76 *** EXAMPLE 2 : ANALCIME *** I4(1)/ACD (WITH LINEAR RESTRICTIONS) DATE: 24-MAY-95 PAGE 3 BOND DISTANCE FUNCTIONS BOND - TYPE FUNCTION W(T-O) W(O-O) W(T-T) SI - O ... SI DO = 1.62000 + -0.4000E-03 * (TOT - 145.0) + 0.0000E+00 * (TOT - 145.0)**2 2.0000 1.0000 0.1000 SI - O ... AL DO = 1.59300 + -0.4000E-03 * (TOT - 145.0) + 0.0000E+00 * (TOT - 145.0)**2 2.0000 1.0000 0.1000 AL - O ... SI DO = 1.74000 + -0.4000E-03 * (TOT - 145.0) + 0.0000E+00 * (TOT - 145.0)**2 2.0000 1.0000 0.1000 DO( SI - SI ) IS CALCULATED USING AN ANGLE OF 145.0 DEGREES DO( SI - AL ) IS CALCULATED USING AN ANGLE OF 145.0 DEGREES DO( AL - SI ) IS CALCULATED USING AN ANGLE OF 145.0 DEGREES CONNECTIVITY OF TETRAHEADRAL ATOMS T2*** T1* I I O2* O2 I I T2 - O1 - T1 - O1* - T1 - O1 - T2 - O3 - T2* I I O2** O3* I I T2** INTERATOMIC DISTANCES BEFORE CYCLE 1 ATOM 1 ATOM 2 CALCULATED D PRESCRIBED DO DO-D WEIGHT W*(DO-D) T1 O1 4.6444 1.7400 -2.9044 2.0000 -5.8088 T1 O2* 8.7552 1.7400 -7.0152 2.0000 -14.0305 O1 O2* 8.4915 2.8414 -5.6501 1.0000 -5.6501 O1 O1* 9.2807 2.8414 -6.4393 1.0000 -6.4393 O1 O2** 10.8563 2.8414 -8.0148 1.0000 -8.0148 O2* O2** 5.2695 2.8414 -2.4281 1.0000 -2.4281 T2 O1 2.3781 1.5930 -0.7851 2.0000 -1.5702 T2 O2 4.7562 1.5930 -3.1632 2.0000 -6.3264 T2 O3 7.1343 1.6200 -5.5143 2.0000 -11.0286 T2 O3* 6.6431 1.6200 -5.0231 2.0000 -10.0462 O1 O2 2.3781 2.6014 0.2233 1.0000 0.2233 O1 O3 4.7562 2.6234 -2.1328 1.0000 -2.1328 O1 O3* 8.1239 2.6234 -5.5005 1.0000 -5.5005 O2 O3 2.3781 2.6234 0.2453 1.0000 0.2453 DLS-76 *** EXAMPLE 2 : ANALCIME *** I4(1)/ACD (WITH LINEAR RESTRICTIONS) DATE: 24-MAY-95 PAGE 4 O2 O3* 9.9587 2.6234 -7.3353 1.0000 -7.3353 O3 O3* 11.9861 2.6454 -9.3406 1.0000 -9.3406 T1 T2 2.2738 3.1790 0.9053 0.1000 0.0905 T1 T2*** 5.5027 3.1790 -2.3237 0.1000 -0.2324 T2 T2* 5.7289 3.0900 -2.6388 0.1000 -0.2639 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 2.39539957 BEFORE CYCLE 1 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 11.65664482 RHO=SUM(W*(DO-D))**2 + SUM(W*COND)**2 = 815.26428223 PARAMETERS AFTER CYCLE 1 PARAMETER OLD CHANGE NEW PARAMETER OLD CHANGE NEW ATOM T1 ATOM T2 X 0.11000 -0.07915 0.03085 X 0.21000 -0.18001 0.02999 Y 0.14000 0.07915 0.21915 Y 0.22000 -0.13128 0.08872 Z 0.12500 0.00000 0.12500 Z 0.23000 0.09707 0.32707 ATOM O1 ATOM O2 X 0.31000 -0.35343 -0.04343 X 0.41000 -0.30886 0.10114 Y 0.32000 -0.11035 0.20965 Y 0.42000 -0.56107 -0.14107 Z 0.33000 0.05129 0.38129 Z 0.43000 0.27581 0.70581 ATOM O3 X 0.51000 -0.18869 0.32131 Y 0.52000 -0.60853 -0.08853 Z 0.53000 -0.04337 0.48663 SYMMETRICALLY DEPENDENT ATOMS ATOM X Y Z T1* 0.03085 -0.21915 0.37500 T2* 0.16128 0.27999 0.42293 T2** -0.16128 0.22001 0.42293 T2*** 0.33872 0.22001 0.07707 O1* 0.04035 0.29343 -0.13129 O2* 0.10893 0.14886 0.45581 O2** 0.10114 0.14107 -0.20581 O3* -0.33853 -0.07131 0.26337 DLS-76 *** EXAMPLE 2 : ANALCIME *** I4(1)/ACD (WITH LINEAR RESTRICTIONS) DATE: 24-MAY-95 PAGE 5 DIFFERENCE VECTORS TO INITIAL COORDINATES ATOM DX DY DZ T1 -0.13123 0.13123 0.00000 T2 -0.05793 -0.03628 -0.01085 O1 -0.14771 0.07525 0.16197 O2 -0.04458 -0.17175 0.32141 O3 0.18691 -0.30785 0.09091 LINEAR RESTRICTIONS AFTER CYCLE 1 NO OF CONDITION C WEIGHT*C WEIGHT*C/NO OF TERMS 1 -0.06246 -138.84245 -46.28082 2 -0.19404 -461.81577 -153.93858 3 0.58358 1215.59656 405.19885 4 0.02586 71.79914 23.93305 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 1.81678200 BEFORE CYCLE 2 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 8.84093952 RHO=SUM(W*(DO-D))**2 + SUM(W*COND)**2 = 715.98822021 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.84998506 BEFORE CYCLE 3 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 4.13625145 RHO=SUM(W*(DO-D))**2 + SUM(W*COND)**2 = 105.00346375 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.92296815 BEFORE CYCLE 4 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 4.49140596 RHO=SUM(W*(DO-D))**2 + SUM(W*COND)**2 = 121.04497528 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.44245443 BEFORE CYCLE 5 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 2.15309978 RHO=SUM(W*(DO-D))**2 + SUM(W*COND)**2 = 28.06721878 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.13950098 BEFORE CYCLE 6 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 0.67884851 RHO=SUM(W*(DO-D))**2 + SUM(W*COND)**2 = 2.80475926 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.04515689 BEFORE CYCLE 7 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 0.21974532 RHO=SUM(W*(DO-D))**2 + SUM(W*COND)**2 = 0.28997770 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.01493201 BEFORE CYCLE 8 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 0.07266308 RHO=SUM(W*(DO-D))**2 + SUM(W*COND)**2 = 0.03533342 DLS-76 *** EXAMPLE 2 : ANALCIME *** I4(1)/ACD (WITH LINEAR RESTRICTIONS) DATE: 24-MAY-95 PAGE 6 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.01297724 BEFORE CYCLE 9 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 0.06315065 RHO=SUM(W*(DO-D))**2 + SUM(W*COND)**2 = 0.02759915 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.01291212 BEFORE CYCLE 10 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 0.06283379 RHO=SUM(W*(DO-D))**2 + SUM(W*COND)**2 = 0.02718676 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.01292431 BEFORE CYCLE 11 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 0.06289312 RHO=SUM(W*(DO-D))**2 + SUM(W*COND)**2 = 0.02716171 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.01292994 BEFORE CYCLE 12 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 0.06292051 RHO=SUM(W*(DO-D))**2 + SUM(W*COND)**2 = 0.02715991 CONVERGENCE TEST POSITIVE AFTER CYCLE 12 APPROXIMATE EIGENVALUES OF MATRIX (LAST CYCLE): 0.6960E+04 0.6195E+04 0.4177E+04 0.3183E+04 0.2711E+04 0.2441E+04 0.2144E+04 0.1359E+04 0.1004E+04 0.9461E+03 0.4662E+03 0.2203E+03 0.1314E+03 PARAMETERS AFTER CYCLE 12 PARAMETER OLD CHANGE NEW PARAMETER OLD CHANGE NEW ATOM T1 ATOM T2 X 0.15809 0.00001 0.15809 X 0.08416 0.00000 0.08416 Y 0.09191 -0.00001 0.09191 Y 0.12989 -0.00001 0.12988 Z 0.12500 0.00000 0.12500 Z 0.33776 0.00001 0.33777 ATOM O1 ATOM O2 X 0.10676 -0.00002 0.10674 X 0.13832 0.00000 0.13832 Y 0.15000 -0.00004 0.14997 Y 0.03356 0.00000 0.03356 Z 0.22574 0.00000 0.22574 Z 0.37331 0.00003 0.37334 ATOM O3 X 0.12875 0.00001 0.12876 Y 0.21675 0.00000 0.21675 Z 0.40422 -0.00002 0.40420 DLS-76 *** EXAMPLE 2 : ANALCIME *** I4(1)/ACD (WITH LINEAR RESTRICTIONS) DATE: 24-MAY-95 PAGE 7 SYMMETRICALLY DEPENDENT ATOMS ATOM X Y Z T1* 0.15809 -0.09191 0.37500 T2* 0.12012 0.33416 0.41223 T2** -0.12012 0.16584 0.41223 T2*** 0.37988 0.16584 0.08777 O1* 0.10003 0.14326 0.02426 O2* 0.28356 0.11168 0.12334 O2** 0.13832 -0.03356 0.12666 O3* -0.03325 0.12124 0.34580 DIFFERENCE VECTORS TO INITIAL COORDINATES ATOM DX DY DZ T1 -0.00399 0.00399 0.00000 T2 -0.00376 0.00488 -0.00015 O1 0.00246 0.01557 0.00642 O2 -0.00740 0.00288 -0.01106 O3 -0.00564 -0.00257 0.00848 LINEAR RESTRICTIONS AFTER CYCLE 12 NO OF CONDITION C WEIGHT*C WEIGHT*C/NO OF TERMS 1 -0.00008 -0.17073 -0.05691 2 0.00138 3.27439 1.09146 3 -0.00114 -2.36720 -0.78907 4 0.00097 2.69342 0.89781 INTERATOMIC DISTANCES BEFORE CYCLE 13 ATOM 1 ATOM 2 CALCULATED D PRESCRIBED DO DO-D WEIGHT W*(DO-D) T1 O1 1.7452 1.7400 -0.0052 2.0000 -0.0105 T1 O2* 1.7440 1.7400 -0.0040 2.0000 -0.0080 O1 O2* 2.8543 2.8414 -0.0129 1.0000 -0.0129 O1 O1* 2.7694 2.8414 0.0721 1.0000 0.0721 O1 O2** 2.8962 2.8414 -0.0548 1.0000 -0.0548 O2* O2** 2.8204 2.8414 0.0210 1.0000 0.0210 T2 O1 1.5932 1.5930 -0.0002 2.0000 -0.0003 T2 O2 1.5939 1.5930 -0.0009 2.0000 -0.0019 T2 O3 1.6215 1.6200 -0.0015 2.0000 -0.0029 T2 O3* 1.6202 1.6200 -0.0002 2.0000 -0.0003 O1 O2 2.6172 2.6014 -0.0158 1.0000 -0.0158 O1 O3 2.6336 2.6234 -0.0101 1.0000 -0.0101 O1 O3* 2.5626 2.6234 0.0608 1.0000 0.0608 O2 O3 2.5540 2.6234 0.0694 1.0000 0.0694 DLS-76 *** EXAMPLE 2 : ANALCIME *** I4(1)/ACD (WITH LINEAR RESTRICTIONS) DATE: 24-MAY-95 PAGE 8 O2 O3* 2.6724 2.6234 -0.0490 1.0000 -0.0490 O3 O3* 2.7037 2.6454 -0.0582 1.0000 -0.0582 T1 T2 3.1363 3.1790 0.0427 0.1000 0.0043 T1 T2*** 3.2503 3.1790 -0.0713 0.1000 -0.0071 T2 T2* 3.0258 3.0900 0.0642 0.1000 0.0064 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.01293159 BEFORE CYCLE 13 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 0.06292854 RHO=SUM(W*(DO-D))**2 + SUM(W*COND)**2 = 0.02715969 DLS-76 *** EXAMPLE 2 : ANALCIME *** I4(1)/ACD (WITH LINEAR RESTRICTIONS) DATE: 24-MAY-95 PAGE 9 INTERATOMIC DISTANCES AND BOND ANGLES AFTER APID CYCLE 1 ATOM 1 ATOM 2 BOND TYPE OBSERVED D OLD DO CHANGE NEW DO DO-DO(START) BOND ANGLE T1 O1 AL O SI 1.7452 1.7400 0.0020 1.7420 0.0020 T1 - O1 - T2 139.9 T1 O2* AL O SI 1.7440 1.7400 -0.0035 1.7365 -0.0035 T1 - O2* - T2*** 153.7 O1 O2* 2.8543 2.8414 -0.0012 2.8403 -0.0012 O1 - T1 - O2* 109.8 O1 O1* 2.7694 2.8414 0.0033 2.8448 0.0033 O1 - T1 - O1* 105.0 O1 O2** 2.8962 2.8414 -0.0012 2.8403 -0.0012 O1 - T1 - O2** 112.2 O2* O1* 2.8962 2.8414 -0.0012 2.8403 -0.0012 O2* - T1 - O1* 112.2 O2* O2** 2.8204 2.8414 -0.0057 2.8358 -0.0057 O2* - T1 - O2** 107.9 O1* O2** 2.8543 2.8414 -0.0012 2.8403 -0.0012 O1* - T1 - O2** 109.8 T1 T2 3.1363 3.1790 0.0039 3.1830 0.0039 T1 T2*** 3.2503 3.1790 -0.0066 3.1724 -0.0066 T2 O1 SI O AL 1.5932 1.5930 0.0020 1.5950 0.0020 T2 - O1 - T1 139.9 T2 O2 SI O AL 1.5939 1.5930 -0.0035 1.5895 -0.0035 T2 - O2 - T1* 153.7 T2 O3 SI O SI 1.6215 1.6200 0.0028 1.6228 0.0028 T2 - O3 - T2* 137.9 T2 O3* SI O SI 1.6202 1.6200 0.0028 1.6228 0.0028 T2 - O3* - T2** 137.9 O1 O2 2.6172 2.6014 -0.0012 2.6002 -0.0012 O1 - T2 - O2 110.4 O1 O3 2.6336 2.6234 0.0040 2.6274 0.0040 O1 - T2 - O3 110.0 O1 O3* 2.5626 2.6234 0.0040 2.6274 0.0040 O1 - T2 - O3* 105.8 O2 O3 2.5540 2.6234 -0.0005 2.6230 -0.0005 O2 - T2 - O3 105.2 O2 O3* 2.6724 2.6234 -0.0005 2.6229 -0.0005 O2 - T2 - O3* 112.5 O3 O3* 2.7037 2.6454 0.0046 2.6501 0.0046 O3 - T2 - O3* 113.0 T2 T1 3.1363 3.1790 0.0039 3.1830 0.0039 T2 T1* 3.2503 3.1790 -0.0066 3.1724 -0.0066 T2 T2* 3.0258 3.0900 0.0054 3.0954 0.0054 T2 T2** 3.0258 3.0900 0.0054 3.0954 0.0054 INTERATOMIC DISTANCES BEFORE CYCLE 1 OF APID CYCLE 1 ATOM 1 ATOM 2 CALCULATED D PRESCRIBED DO DO-D WEIGHT W*(DO-D) T1 O1 1.7452 1.7420 -0.0032 2.0000 -0.0064 T1 O2* 1.7440 1.7365 -0.0075 2.0000 -0.0150 O1 O2* 2.8543 2.8403 -0.0141 1.0000 -0.0141 O1 O1* 2.7694 2.8448 0.0754 1.0000 0.0754 O1 O2** 2.8962 2.8403 -0.0560 1.0000 -0.0560 O2* O2** 2.8204 2.8358 0.0154 1.0000 0.0154 T2 O1 1.5932 1.5950 0.0019 2.0000 0.0038 T2 O2 1.5939 1.5895 -0.0044 2.0000 -0.0088 DLS-76 *** EXAMPLE 2 : ANALCIME *** I4(1)/ACD (WITH LINEAR RESTRICTIONS) DATE: 24-MAY-95 PAGE 10 T2 O3 1.6215 1.6228 0.0014 2.0000 0.0027 T2 O3* 1.6202 1.6228 0.0026 2.0000 0.0053 O1 O2 2.6172 2.6002 -0.0170 1.0000 -0.0170 O1 O3 2.6336 2.6274 -0.0061 1.0000 -0.0061 O1 O3* 2.5626 2.6274 0.0648 1.0000 0.0648 O2 O3 2.5540 2.6230 0.0689 1.0000 0.0689 O2 O3* 2.6724 2.6229 -0.0495 1.0000 -0.0495 O3 O3* 2.7037 2.6501 -0.0536 1.0000 -0.0536 T1 T2 3.1363 3.1830 0.0466 0.1000 0.0047 T1 T2*** 3.2503 3.1724 -0.0779 0.1000 -0.0078 T2 T2* 3.0258 3.0954 0.0696 0.1000 0.0070 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.01309653 BEFORE CYCLE 1 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 0.06373119 RHO=SUM(W*(DO-D))**2 + SUM(W*COND)**2 = 0.02776967 PARAMETERS AFTER CYCLE 1 PARAMETER OLD CHANGE NEW PARAMETER OLD CHANGE NEW ATOM T1 ATOM T2 X 0.15809 0.00034 0.15843 X 0.08416 0.00025 0.08441 Y 0.09191 -0.00034 0.09157 Y 0.12988 -0.00016 0.12972 Z 0.12500 0.00000 0.12500 Z 0.33777 0.00009 0.33786 ATOM O1 ATOM O2 X 0.10674 0.00048 0.10721 X 0.13832 0.00028 0.13861 Y 0.14997 0.00004 0.15001 Y 0.03356 0.00006 0.03362 Z 0.22574 0.00003 0.22576 Z 0.37334 -0.00022 0.37312 ATOM O3 X 0.12876 0.00020 0.12896 Y 0.21675 0.00004 0.21679 Z 0.40420 0.00021 0.40440 SYMMETRICALLY DEPENDENT ATOMS ATOM X Y Z T1* 0.15843 -0.09157 0.37500 T2* 0.12028 0.33441 0.41214 T2** -0.12028 0.16559 0.41214 T2*** 0.37972 0.16559 0.08786 O1* 0.09999 0.14279 0.02424 O2* 0.28362 0.11139 0.12312 O2** 0.13861 -0.03362 0.12688 O3* -0.03321 0.12104 0.34560 DLS-76 *** EXAMPLE 2 : ANALCIME *** I4(1)/ACD (WITH LINEAR RESTRICTIONS) DATE: 24-MAY-95 PAGE 11 DIFFERENCE VECTORS TO INITIAL COORDINATES ATOM DX DY DZ T1 -0.00365 0.00365 0.00000 T2 -0.00351 0.00472 -0.00006 O1 0.00293 0.01561 0.00644 O2 -0.00711 0.00294 -0.01128 O3 -0.00544 -0.00253 0.00868 LINEAR RESTRICTIONS AFTER CYCLE 1 NO OF CONDITION C WEIGHT*C WEIGHT*C/NO OF TERMS 1 -0.00008 -0.17064 -0.05688 2 0.00136 3.24038 1.08013 3 -0.00112 -2.32772 -0.77591 4 0.00098 2.70768 0.90256 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.01303584 BEFORE CYCLE 2 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 0.06343582 RHO=SUM(W*(DO-D))**2 + SUM(W*COND)**2 = 0.02749679 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.01302805 BEFORE CYCLE 3 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 0.06339794 RHO=SUM(W*(DO-D))**2 + SUM(W*COND)**2 = 0.02749515 CONVERGENCE TEST POSITIVE AFTER CYCLE 3 APPROXIMATE EIGENVALUES OF MATRIX (LAST CYCLE): 0.6954E+04 0.6194E+04 0.4178E+04 0.3184E+04 0.2706E+04 0.2442E+04 0.2142E+04 0.1362E+04 0.1004E+04 0.9471E+03 0.4687E+03 0.2216E+03 0.1360E+03 PARAMETERS AFTER CYCLE 3 PARAMETER OLD CHANGE NEW PARAMETER OLD CHANGE NEW ATOM T1 ATOM T2 X 0.15839 0.00000 0.15839 X 0.08440 0.00000 0.08440 Y 0.09161 0.00000 0.09161 Y 0.12974 0.00001 0.12976 Z 0.12500 0.00000 0.12500 Z 0.33783 -0.00001 0.33783 ATOM O1 ATOM O2 X 0.10726 0.00002 0.10728 X 0.13857 0.00000 0.13857 Y 0.15015 0.00003 0.15018 Y 0.03358 0.00000 0.03358 Z 0.22576 0.00000 0.22576 Z 0.37300 -0.00003 0.37297 DLS-76 *** EXAMPLE 2 : ANALCIME *** I4(1)/ACD (WITH LINEAR RESTRICTIONS) DATE: 24-MAY-95 PAGE 12 ATOM O3 X 0.12892 0.00000 0.12892 Y 0.21676 0.00000 0.21676 Z 0.40449 0.00002 0.40450 SYMMETRICALLY DEPENDENT ATOMS ATOM X Y Z T1* 0.15839 -0.09161 0.37500 T2* 0.12024 0.33440 0.41217 T2** -0.12024 0.16560 0.41217 T2*** 0.37976 0.16560 0.08783 O1* 0.09982 0.14272 0.02424 O2* 0.28358 0.11143 0.12297 O2** 0.13857 -0.03358 0.12703 O3* -0.03324 0.12108 0.34550 DIFFERENCE VECTORS TO INITIAL COORDINATES ATOM DX DY DZ T1 -0.00369 0.00369 0.00000 T2 -0.00352 0.00476 -0.00009 O1 0.00300 0.01578 0.00644 O2 -0.00715 0.00290 -0.01143 O3 -0.00548 -0.00256 0.00878 LINEAR RESTRICTIONS AFTER CYCLE 3 NO OF CONDITION C WEIGHT*C WEIGHT*C/NO OF TERMS 1 -0.00008 -0.17228 -0.05743 2 0.00137 3.25666 1.08555 3 -0.00113 -2.35053 -0.78351 4 0.00098 2.71016 0.90339 INTERATOMIC DISTANCES BEFORE CYCLE 4 OF APID CYCLE 1 ATOM 1 ATOM 2 CALCULATED D PRESCRIBED DO DO-D WEIGHT W*(DO-D) T1 O1 1.7473 1.7420 -0.0052 2.0000 -0.0105 T1 O2* 1.7405 1.7365 -0.0040 2.0000 -0.0080 O1 O2* 2.8520 2.8403 -0.0118 1.0000 -0.0118 O1 O1* 2.7707 2.8448 0.0740 1.0000 0.0740 O1 O2** 2.8963 2.8403 -0.0560 1.0000 -0.0560 O2* O2** 2.8162 2.8358 0.0195 1.0000 0.0195 T2 O1 1.5952 1.5950 -0.0002 2.0000 -0.0004 T2 O2 1.5905 1.5895 -0.0010 2.0000 -0.0019 T2 O3 1.6244 1.6228 -0.0015 2.0000 -0.0030 T2 O3* 1.6230 1.6228 -0.0002 2.0000 -0.0004 DLS-76 *** EXAMPLE 2 : ANALCIME *** I4(1)/ACD (WITH LINEAR RESTRICTIONS) DATE: 24-MAY-95 PAGE 13 O1 O2 2.6140 2.6002 -0.0137 1.0000 -0.0137 O1 O3 2.6356 2.6274 -0.0081 1.0000 -0.0081 O1 O3* 2.5661 2.6274 0.0613 1.0000 0.0613 O2 O3 2.5554 2.6230 0.0676 1.0000 0.0676 O2 O3* 2.6741 2.6229 -0.0511 1.0000 -0.0511 O3 O3* 2.7091 2.6501 -0.0590 1.0000 -0.0590 T1 T2 3.1377 3.1830 0.0453 0.1000 0.0045 T1 T2*** 3.2451 3.1724 -0.0726 0.1000 -0.0073 T2 T2* 3.0297 3.0954 0.0658 0.1000 0.0066 R=SQRT( SUM(W*(DO-D))**2 / SUM(W*DO)**2 )= 0.01302656 BEFORE CYCLE 4 SIGMA=SQRT( SUM(W*(DO-D))**2 / (M-NV) ) = 0.06339065 RHO=SUM(W*(DO-D))**2 + SUM(W*COND)**2 = 0.02749504 DLS-76 *** EXAMPLE 2 : ANALCIME *** I4(1)/ACD (WITH LINEAR RESTRICTIONS) DATE: 24-MAY-95 PAGE 14 TOTAL PARAMETER SHIFTS AFTER LAST CYCLE PARAMETER INITIAL CHANGE FINAL SHIFT PARAMETER INITIAL CHANGE FINAL SHIFT ATOM T1 ATOM T2 X 0.16208 -0.00369 0.15839 -0.05071 X 0.08792 -0.00352 0.08440 -0.04835 Y 0.08792 0.00369 0.09161 0.05071 Y 0.12500 0.00476 0.12976 0.06530 Z 0.12500 0.00000 0.12500 0.00000 Z 0.33792 -0.00009 0.33783 -0.00129 MAGNITUDE 0.07171 MAGNITUDE 0.08126 ATOM O1 ATOM O2 X 0.10428 0.00300 0.10728 0.04123 X 0.14572 -0.00715 0.13857 -0.09814 Y 0.13440 0.01578 0.15018 0.21670 Y 0.03068 0.00290 0.03358 0.03985 Z 0.21932 0.00644 0.22576 0.08845 Z 0.38440 -0.01143 0.37297 -0.15691 MAGNITUDE 0.23766 MAGNITUDE 0.18931 ATOM O3 X 0.13440 -0.00548 0.12892 -0.07529 Y 0.21932 -0.00256 0.21676 -0.03520 Z 0.39572 0.00878 0.40450 0.12058 MAGNITUDE 0.14645 MAGNITUDE ------------------------------------------------------------------------ *** FINISH ***

5-1 5. GLOSSARY OF SYMBOLS ---------------------- The dimensions of the arrays are given in the description of their index variables. The index limits are defined in the subroutine DATIN. Symbols marked with an asterisk are read as input data. 5.1 Control integers and single variables The control integers on the DLS-76 card are explained in Section 3. * CORR see FACTOR card cols. 61 - 70 * CVGTST see FACTOR card cols. 31 - 40 IA Number of BONDIS cards (maximum value is IDIMAP = 15 ) * ICAL DLS-76 card col. 30 * ICOR DLS-76 card col. 48 * IDIS DLS-76 card col. 44 * IDOB DLS-76 card col. 18 * IGIT DLS-76 card col. 22 * IMAT DLS-76 card col. 46 * INEW DLS-76 card col. 10 * IORT DLS-76 card col. 42 * IPCH DLS-76 card col. 50 * IRNG DLS-76 card col. 14 * IRNGl Starting number of random generator (FACTOR card cols. 26 - 30) ISYS Internal crystal system indicator (see DATIN) * IVF DLS-76 card cols. 32

5-2 * LSYS Crystal system indicator as on CELL card M Number of distances (maximum value is IDIMM = 200) N Number of atoms (N1 + N2) N1 Number of atoms in asymmetric unit N2 Number of atoms outside asymmetric unit * NA DLS-76 card col. 26 * NAPID DLS-76 card col. 20 * NC DLS-76 card col. 24 * NCYCLE DLS-76 card col. 16 * NDIAG DLS-76 card col. 12 NDO Number of different prescribed ratios of interatomic distances (maximum number of ratios is IDIMRT = 40) NEQU Number of SYMEQ cards (maximum value is IDNEQU = 60) * NFILEA Logical number of parameter file NRE Number of linear restrictions (maximum value is IDIMLI = 35) NRH Number of hard constraints * NS DLS-76 card col. 28 * NTIN Logical number of card reader NTOUT Logical number of line printer NV Total number of variables (NVO + NVG + NVD) (maximum value is IDIMNV = 150) NV1 NVO + NVG NVD Number of distance variables

5-3 NVG Number of lattice constant variables NVO Number of coordinate variables NZA Number of TETCON cards (maximum value is IDIMNZ = 40) RDATE Date * RDD RANGES card cols. 11 - 15 * ROTOL RANGES card cols. 16 - 20 * ROTOU RANGES card cols. 21 - 25 * RTOTL RANGES card cols. 26 - 30 * RTOTU RANGES card cols. 31 - 35 * WFAC See FACTOR card cols. 51 - 60 * WRF See FACTOR card cols. 41 - 50 5.2 Arrays * A(6) Lattice constants * AOLD(6) Lattice constants (of previous cycle) B(K,K,N) (K = 1,3) homogeneous part of the restrictions resulting from special positions (for atoms in the asymmetric unit only) BEDING(NRE) Residual of the linear restriction NRE BI(K,N) (K = 1,3) invariant part of restrictions given by a special position (see B(K,K,N)) BT(IA) Bond type. Its value is calculated from the atom type numbers of the 3 atoms defining the bond type of a T-O bond. (see subroutine DATIN)

5-4 * CAPID(IA,4) Parameters of distance function (see BONDIS card) D(M) Calculated interatomic distances DELTAD(M) DOB(M) - D(M) DELV(NV) Vector of parameter shifts of current refinement cycle DGEL(NV) Approximate eigenvalues of matrix * DOB(M) Prescribed interatomic distances * DOBIN(M) Initial values of prescribed interatomic distances DOBOLD(M) Prescribed interatomic distances of previous APID cycle * FUDGE(3) Damping factors as stated on FACTOR card G(3,3) Metric tensor GABL1(3,3,6) First derivatives of metric tensor GABL2A(3,6,6)} } GABL2B(3,6,6)} Second derivatives of metric tensor } GABL2C(3,6,6)} IBT(NZA,4) Bond types of the four T-O distances of central atom NZA ICON(NZA,11) Tetrahedral connectivity of central atom NZA ICON(NZA,1 to 9) : internal representa- tion of TETCON card ICON(NZA,10) : number of non-equivalent T-O distances ICON(NZA,11) : number of non-equivalent O-O distances * ID(N) Label of atom N

5-5 IFAK(K,N) Can have values D,I,L,R indicating whether the coordinate K of atom number N is de- pendent, invariant, dependent by a linear restriction! or to be refined, respectively IFELD(6) Number of cell parameters which are to be refined (up to 6) IGIN(6) Can have values D,I,R indicating whether the corresponding cell parameter is de- pendent, invariant, or to be refined, respectively * IMOV(3) See DLS-76 card cols. 34 - 38 * ISYMB(M) Symbol on DISTAN card which identifies the reference distance in a constant ratio refinement * ITYPE(N) Number of atom type of atom number N IV(NV) Can have values of 1,2,3 indicating whether the variable NV is an X, Y or Z coordinate, respectively IV1(NV) Contains the number of the atom to which the variable NV belongs IZ1(NRE) Number of terms in the linear restriction NRE KA(K,NEQU) (K = 1,2) Atom number of both atoms on SYMEQ card NEQU KARI(NRE,IZ1) Variable number corresponding to term IZ1 of the linear restriction NRE KATOR(M) Flag for the calculation of derivatives of distance M (determined by subroutine DATIN) 0 if both atoms are in the asymetric unit and not on a special position

5-6 -1 if atoms are not related and at least one of them is not in the asymmetric unit, or else at least one of them is in a secial position MYS>0 if both atoms are symmetrically related (see SYMSIG(K,K,40)) KVAR(K,I1) (K = 1,3). Contains the variable number of coordinate K of atom N. If KVAR is zero the coordinate is not to be refined. KTYPE(NT) Atom type symbol LDR(NDO) Variable number of prescribed distance NDO in a constant ratio refinement LIND(NRE) Term number of the variable which is eliminated in the hard constraint NRE LINH(NRE) If the linear restriction NRE is a hard constraint, LINH(NRE) is 1, otherwise it is 0. LJ(M) DOB(LJ(M)) is the prescribed distance for distance D(M) MD(NZA, 18) Index to the D and DOB arrays of all distances around central tetrahedral atom NZA (inclusive outer O-T and T-T distances) MD(NZA, 1 to 4): T-01, T-02, T-03, T-04 MD(NZA, 5 to 10): 01-02, 01-03, 01-04 02-03, 02-04, 03-04 MD(NZA,11 to 14): 01-T1, 02-T2, 03-T3, 04-T4 MD(NZA,15 to 18): T-T1, T-T2, T-T3, T-T4

5-7 MSYMAB(N) Number of atom in asymmetric unit, sym- metrically equivalent to atom N MTR(N) MTR(N) = N indicates that the atom N is in a special position or outside the asymmetric unit (otherwise MTR(N) = O) MW(M) MW(M) = 2 if distance M is outside spe- cified ranges, otherwise MW(M) = 1 NN(M,K) (K = 1,2). Number of the first or second atom on DISTAN card number M NN11(M,K) (K = 1,2). Number of the atom in the asymmetrie unit to whieh atom K on DISTAN eard M is symmetrically related OTO(NZA,6) O-T-O angles at eentral atom NZA (in same order as 0-0 distanees in MD(NZA,18) array) RAT(NDO) Prescribed distance ratio (prescribed distance/reference distance on DISTAN card) * RESKO(NRE,IZ1) Coefficient of term IZ1 of the linear restriction NRE,(IZ1 IDIMIz = 20) S(NV*(NV+3)/2) Matrix array. Only upper triangle is stored as one-dimensional array SI(K,NEQU) (K = 1,3). Translational part of symmetry transformation of SYMEQ card number NEQU SIGN(K,K,NEQU) (K = 1,3). Non-translational (homogeneous) part of symmetry transformation of SYMEQ card number NEQU * SYMOPS(10,N) Symmetry transformation (coded) as punched on ATOM and SYMEQ cards SYMSIG(K,K,40) (K = 1,3). This variable is only defined when both atoms on a DISTAN card are symmetrically equivalent, in which case

5-8 SYMSIG(K,K,MYS) = B1 - B2 where B1 and B2 are the non-translational parts of the symmetry transformations (including restrictions due to special positions) leading to ATOM1 and ATOM2, respectively T(NV) Normal vector (gradient) TOT(NZA,4) T-O-T angles observed at central atom NZA (in same order as T-T distances in MD (NZA,l8) array) * W(M) Weight of DOB (M) * WEIGHT(IA,K) (K = 1,3). Weights assigned to T-O, O-O and T-T distances, respectively (see BONDIS card) WR (NRE) Weight of linear restriction NRE * X(K,N) X, Y, Z coordinate (K = 1,3) of atom N * XINI (K,N) X, Y, Z coordinate (input values) XOLD(K,N) X, Y, Z coordinate of previous cycle

6-1 6. FORMULAE ----------- In the following the more important formulae which form the mathematical basis of DLS-76 are briefly surveyed. The symbols used here and their names in the program are listed in Table 1 at the end of this section. Given a set of m weighted distance error equations and q weighted linear restrictions or soft constraints (treated as additional equations; cf. WASER, 1963) the following function #p(#v) must be minimized. FORMULA where FORMULA #v shall represent a vector consisting of the variable atomic coordinates zr' cell parameters as and (in a constant ratio refinement) of variable prescribed distances Dt. If in addition to the q soft constraints some coordinates are also subject to hard constraints (i.e. restrictions which must be exactly fulfilled), then an interatomic distance Dj may depend on additional coordinates other than those of the two atoms directly involved. The equation for the shift ##v towards a minimum of #p may be given as FORMULA This can be approximated by FORMULA

6-2 or, if we set FORMULA by FORMULA The iteration FORMULA then takes the form FORMULA in the Newton-Raphson procedure and FORMULA in the Gauss-Newton procedure. In both procedures the vector #T contains in general the partial derivatives of #p with respect to the coordinates zr', the cell parameters ar and the variable prescribed distances Dr0: FORMULA where FORMULA

6-3 An element Srs of the matrix S is given by FORMULA in the Newton-Raphson procedure and as FORMULA in the Gauss-Newton procedure where #pr and #ps are variables of the above three types. The following is a summary of all possible Srs. Type of Newton-Raphson Gauss-Newton Variables FORMULA

6-4 The distance D(P,Q) between the two atoms P and Q having coordinates #x and #x' can be expressed by FORMULA Generally P and Q are outside the asymmetric unit and #x and #x' can be given as FORMULA If P or Q occupy a special position, then B, #b or B', #b' contain the information of the symmetry transformation leading to P or Q as well as the relations describing the special positions. (Symmetry information on SYMEQ-card and on corresponding ATOM-card). If a component yi of #y is not set invariant then yi = zk for a given k, or FORMULA for a hard linear constraint l. Hence, #x = #x - #x' may be understood as dependent on a vector #z = (z1,...,zn)**T of n variable coordinates. If we define FORMULA and FORMULA we may write FORMULA where A is a (3 x n)-matrix.

6-5 Hence, we obtain the following derivatives of D with respect to coordinates and cell parameters. FORMULA In the calculation of A the program distinguishes three cases (i) P and Q are both in the asymmetric unit and in general position (indicated by KATOR(J) = 0)0 Then A has the form (1 0 0-1 0 0) A = (0 1 0 0-1 0) (0 0 1 0 0-1) (ii) P and Q are symmetrically equivalent (KATOR(J)>0) Then #y = #y' and R = R' and A can be written as A = (B - B')R (B - B' corresponds to SYMSIG(I,K,KATOR(J))) (iii) P and/or Q are not in the asymmetric unit and/or in special positions (KATOR(J)<0): A = BR - B'R'

6-6 TABLE 1 ------- Symbol Meaning Designation in text in program A matrix containing the deriva- ALIN(I,K) tives of #x with respect to all variable coordinates on which #x depends ar variable cell parameter A(I) number r B matrix containing the homoge- B(I,K,N) (includes neous part of the symmetry SIGN(I,K,NEQU)) information on a SYMEQ card and the corresponding ATOM card #b vector representing the in- SIGN(I,K,NEQU)*BI(K,N) homogeneous part of the sym- + SI(I,NEQU) metry information cl linear constraint number l BEDING(L) Dj calculated interatomic di- D(J) stance Dj0 prescribed distance DOB(J) Dr0 variable prescribed distance DOB(LJ(J)) djr ratio of Dj0 to Dr0 RAT(LJ(J)) #f,#p auxiliary quantities in the representation of #p as a sum of two scalar products gik metric tensor G(I,K) hlr coefficient of term r in RESKO(L,I) constraint l

6-7 Symbol Meaning Designation in text in program m number of distance equations M n number of variable coordinates KK on which a particular distance depends #v number of terms in constraint l IZ1(L) q number of weighted (soft) NRE-NRH constraints R auxiliary quantity: matrix containing the derivates of the yk with respect to all variable coordinates on which the yk depend #p function to be minimized ROV S normal matrix/Jacobian matrix S(KLM) of #p' #T vector of constants in system T(I) represented by S ul2 squared weight of soft con- WR(L) straint #v vector composed of variables zr, as, Dt0 #v shift of #v to be calculated in DELV(I) an iteration cycle wj2 squared weight of distance j W(J) #x coordinates of an atom X(K,N)

6-8 Symbol Meaning Designation in text in program #x difference of the coordinates DELTAX(K) of two atoms belonging to a particular distance #y auxiliary quantity: vector composed of the free para- meters(1 which describe the position of a particular atom #z vector containing all va- riable coordinates on which a particular distance depends zr variable coordinate number r zr0 initial value of zr (1 In the sense that a dependence due to a linear restriction is allowed.

7-1 7. REFERENCES ------------- BARRER R.M. and VILLIGER H. (1969): The Crystal Structure of Synthetic Zeolite L. Z.Kristallogr. 128, 352-370. BAUR W.H. (1977): Computer Simulation of Crystal Structures. Phys. Chem. Minerals. 2, 3-20. BROWN G.E., GIBBS G.V. and RIBBE P.H. (1969): The Nature and Variation in Length of the Si-O and Al-O Bonds in Framework Silicates. Am. Mineral. 54, 1044-1061. BUSING W.R. and LEVY H.A. (1962): A Procedure for Inverting Large Symmetric Matrices. Comm. ACM 5, 445-446. DEMPSEY M.J. and STRENS R.G.J.(1976): Modelling Crystal Structures. In 'Physics and Chemistry of Minerals and Rocks' (R.G.J. Strens, Ed.; Wiley), pg. 443-458. DOLLASE W.A. and BAUR W.H. (1976): The Superstructure of Meteoritic Low Tridymite Solved by Computer Simuiation. Am. Mineral. 61, 971-978. FERRARIS G., JONES D.W., and YERKESS J. (1972): A Neutrondiffraction Study of the Crystal Structure of Analcime, NaAlSi2O6 . H2O. Z.Kristallogr. 135, 240-252. GRAMLICH V. and MEIER W.M. (1971): The Crystal Structure of Hydrated NaA: A Detailed Refinement of a Pseudosymmetric Zeolite Structure. Z.Kristallogr. 133, 134-149. GUIGAS B. (1975): Verfeinerung von Kristallstrukturen mit dem Distance Least Squares-Verfahren: Behandlung von Konvergenzfragen und kristallographische Anwendungen. Doctoral Dissertation, University of Karlsruhe, BRD. KHAN A.A. (1976): Computer Simulation of Thermal Expansion of Non- Cubic crystals: Forsterite, Anhydrite and Scheelite. Acta Cryst. A32, 11-16.

7-2 LOUISNATHAN S.J. and GIBBS G.V. (1972): Bond Length Variation in TO4 Tetrahedral Oxyanions of the Third Row Elements: T = Al,Si,P,S and Cl. Mat. Res. Bull. 7, 1281-1292. MEIER W.M. (1973): Symmetry Aspects of Zeolite Frameworks. Adv. Chem. Ser. 121, 39-51. MEIER W.M. and VILLIGER H. (1969): Die Methode der Abstandsverfeinerung zur Bestimmung der Atomkoordinaten idealisierter Gerueststrukturen. Z.Kristallogr. 129, 411- 423. SCHWARZENBACH D. (1966): Verfeinerung der Struktur der Tiefquarz- Modifikation von AlPO4. Z.Kristallogr. 123, 161-185. TILLMANNS E., GEBERT W. and BAUR W.H. (1973): Computer Simulaticn of Crystal Structures Applied to the Solution of the Superstructure of Cubic Silicondiphosphate. J. Sol. State Chem. 7, 69-84. VILLIGER H. (1969): DLS-Manual. Institut fuer Kristallographie und Petrographie, ETH Zuerich. WASER J. (1963): Least-Squares Refinement with Subsidary Conditions. Acta Cryst. 16, 1091-1094.

Ch. Baerlocher, Laboratory of Crystallography, ETH Zürich, Switzerland. ch.baerlocher@kristall.erdw.ethz.ch