DLS-76
A 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