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NBS TECHNICAL NOTE 854

***eAu o* *

U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards

A System of Fortran IV

Computer Programs for

rystal Structure Computations

100

HI 5
NATIONAL BUREAU OF STANDARDS

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ttUIUUU HI iili.iyd.u: '"BAR 7 1975

A System of Fortran IV Computer Programs

for Crystal Structure Computations

>953

Larry W. Finger

Geophysical Laboratory

Carnegie Institution of Washington

Washington, D.C. 20008

and

^r^tMj^tCLoJ) n.- ^

E. Prince

Institute for Materials Research

National Bureau of Standards Washington, D.C. 20234

S** 0F cOA

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

Issued February 1975
National Bureau of Standards Technical Note 854

Nat. Bur. Stand. (U.S.), Tech. Note 854, 133 pages (Feb. 1975) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE WASHINGTON: 1975

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SD Catalog No. 03.46:854). Price $2.00
ABSTRACT

This report gives detailed descriptions and instructions for use of

a system of programs for crystallographic calculations, including leastsquares refinement with generalized systems of constraints, calculation

of bond distances and angles with errors, Fourier synthesis, plotting

of contours in Fourier maps, and preparation of structure factor tables

for publication.

Key Words:

Computer programs; contour plotting; constrained refinement; crystallographic calculations; Fourier section; Fourier synthesis; least

squares

This is also Paper Number 1651 from the Geophysical Laboratory,

Carnegie Institution of Washington, Washington, D. C.

in
TABLE OF CONTENTS

Page

RFINE4: A program for least squares refinement including 1 general constraints.

General Description 1

File and Record Structure 1

Description of Routines 2

Common Blocks and Variables 11

Dependent Parameter Input 14

Details of Data Input 17

BONDAN: A program for computing bond distances, angles, 27

and thermal ellipsoids , with errors

.

FOURIER • 29

LISTFC 32

CNTPLT 34

ARBSECT 37

References Cited 40

APPENDIX A - Overlay scheme for RFINE 41

APPENDIX B - BADTEA - alternate input for bond distances 42

and angles , and thermal ellipsoids

APPENDIX C - Listings of programs. 56

These programs, although tested on many problems and several

computing systems, may not give correct results in all circumstances.

Therefore, neither the authors nor their institutions will guarantee the

correctness of the program in all respects, but the authors will greatly

appreciate being informed of discrepancies in the code or desirable

'options not currently included.

IV
RFINE4: A program for least squares refinement

including general constraints

General Description

This program, called RFINE4, performs the calculations necessary

to refine crystal structures using the techniques of full-matrix

least-squares refinement. In addition, it can compute bond distances

and angles or thermal ellipsoids so that these functions may be

monitored throughout the various stages of a refinement. File

control capabilities have been designed s'o that parameters may

be saved at the end of a given computer run or, alternatively, virtually

any parameter may be changed at almost any time.

A scheme of derivative modification and parameter restoration

which eliminates the need for coding special subroutines to treat

constrained parameters has been included in this program. This method

will handle thermal and positional restrictions for atoms in positions

of special symmetry and, in conjunction with the occupancy refinement

capability, a multiposition, partially ordered structure may be con- strained to a particular chemical composition. Provision has been

made for incorporating parameters which are not members of the set

used in conventional least-squares refinement . Any set of linear

constraints may be specified, and also groups of atoms may be treated

as rigid bodies both with respect to position and orientation (shape

constraints) and with respect to thermal motion.

In its present form, the program allows a maximum of 60 atoms,

20 chemical species , 10 scale factors , 1320 parameter dependencies

and 200 refined parameters for a sfructure in any space group. With

the maximum number of parameters ' and no use of overlays the program

requires approximately 65000 words of core storage. However more restricted problems can be operated in smaller core sizes, and the program

can be further reduced in size by means of overlays. A suggested overlay

structure is given in Appendix A. Some operating systems allow for

dynamic allocation of core. Such provisions can be used to advantage.

File and Record Structure

In the following discussion a record is the unit of information

formally defined as a logical record, i.e. the amount of information

transferred in a single non-formatted read or write statement. This

description is independent of any physical device which may be associated

with the file and the program has been coded so as to operate fairly

efficiently with any type of auxiliary storage currently available.

Furthermore, a data set shall mean the collection of records needed

to describe a structure. Any reference to file or data labels should

not be confused with file control information for a particular computer

system.
Master File . The file referred to as ISC1 and assigned to logical

unit 10 is the main file of the program. This file consists of a data

set containing all the information required to resume least-squares

refinement . This data set has two label records , one or more structure

factor records, a constraint relations record, a variance-covariance

record, and a record containing variances of previously refined parameters

(useful in large problems where the refinement must be "blocked")

.

The label records contain a 12-character alphanumeric identification

tag, a cycle number and all the atom parameters, symmetry data and

scattering factor information needed for this structure. The reflection

data consist of the Miller indices, the observed structure factor on

a relative scale, the standard deviation of F bs

» tne value of sin2 8/A2 for the reflection, an integer specifying which scale factor applies

to this reflection, 3 —the extinction parameter, a flag to mark reflections

which had an intensity less than the minimum observable, the dif fractometer

angles x , $ and u + 6 in radians , and the value of each of the scattering

curves for this reflection. These quantities are packed into a buffer,

starting with word 2, until another reflection would require the buffer

to be larger than 1023 words. An end of reflection records marker,

which is non-zero only in the last block, is contained in word 1. The information for a single reflection is never split between two

records and all records, including the last which may be only partially

filled, are the same length. The end of the reflection list is marked

by a Miller index h having a value of 3000. If the number of scattering

curves is N, the number of reflections per record (n) may be found

from the integral part of 1022/ (N + 12). The length of the record

is then 1023 minus 1022 modulo n. The variance-covariance record

contains the variance-covariance matrix for the previous cycle of refinement

in packed, upper triangular format.

Intermediate File . This file, named ISC2 and assigned to logical

unit 29, contains the structure factor records for the current structure.

A distinct file allows the scattering factor information to be changed

without reading the structure factors from cards.

Auxiliary Files. A file named ISC3 and assigned to logical unit

11 is used as an alternative structure factor input. A file named

ISC4 and assigned to logical unit 12 is used to pass information to Fouier synthesis, structure factor tabulation and statistical analysis

programs. A file named ISC5 and assigned to logical unit 28 is used

for temporary storage of constraint relationships.

System Files . This program uses a file named IN, logical unit

5, as a card read file, a file named IOUT, logical unit 6, as the print

file, and a file named IPUN, logical unit 1, as the punch file. These

assignments may be changed for a particular system by a block data

subprogram.

Description of Routines

Main Program . The main program serves as a driver for the leastsquares section. It reads one card which contains a 12-character label.
If a new data set is being constructed, this label will be inserted

into that file. Control is then transferred to subroutine RFINE. If

the updated information is written back to file ISC1, the subroutine

exits normally. In that event, the main program prints a message stating

that the file has been updated.

Subroutine RFINE . This routine is the primary one for the structure

factor^and least-squares calculation and is responsible for controlling

the functions computed and the parameters used

.

When this routine is first entered, initialization is accomplished

and the main cycle loop is entered. The first step is to read the title

and control cards for the cycle. If this is the first cycle of the

current run, the new parameter indicators from the control card are interrogated and if all are non-zero, signifying that all parameters

are to be loaded from cards , the data set on file ISC1 is presumed not

to exist; however, if any of the indicators are zero, the first two records

of that data set are loaded into memory. Next, any new parameters required

are read from cards. As new parameters are read, the values are printed.

If the unit cell parameters are to be changed, that card is read

and the direct space metric tensor is prepared. This is a 3 x 3 symmetric

matrix whose elements are the scalar products of the unit cell vectors.

This matrix is then inverted using subroutine SYMINV described below.

The inverse matrix is then the reciprocal space metric tensor and the

determinant of the original matrix is the square of the unit cell volume.

Note: If the unit cell parameters are to be changed, the structure factors

must also be changed if the effects of the cell data are to be applied

to the values of sin 0/X .

New symmetry cards, if required, are now read. Each card lists

an equivalent position in very nearly the same form as Vol. I of the

International Tables with only one major difference; positions related

by a center of inversion or lattice centering must not be included since

they are internally generated. Using this scheme, all space groups

may be generated by using a maximum of 24 cards. Each symmetry data

card is transformed into matrices R and T satisfying the equation

X' = RX + T

with X

1 equal to the vector of transformed coordinates, X is the position

vector with components, xyz , R is a 3 x 3 rotational matrix and T is

a vector describing the translation associated with this symmetry operation,

At this point in the program, new scattering factor coefficients

are read, if desired. We must emphasize two points of caution concerning

the changing of the scattering factors. First, since the blocking of

structure factors depends upon the number of scattering factors used,

if that number is changed, new structure factors must be read. Violation

of this rule will lead to unpredictable, but probably catastrophic,

results. The second rule requires that the scattering factors be changed
only when file ISC2 is being written, that is, either on the first cycle

of a given run or when new structure factors are being read from cards.

If this restriction is violated, the new curves are loaded to core and

subsequently included in the data set for this structure, but they are never used to compute scattering factors for the reflections

.

The new atom parameters, if required, are now read from cards.

The scale factors are read first, followed by the extinction parameters and

the atom cards . The atom parameters needed are an identification label

of 6 characters, an equipoint fraction which is the multiplicity of

the position occupied by this atom divided by the multiplicity of an atom in the general position of this space group, the total occupancy

of the site (usually one) , the occupancy of species A (needed if more

than one species present in this site) , the identification of scattering

factor (s) to be applied to this atom, fractional coordinates for the

atom, and temperature factor information, as well as third and fourth

cumulant parameters, consistent with the temperature factor type card

described below. The forms of the cumulant expressions used in this

program are:

F =

Jf expti^xJ - ZZIh^h^) - Zl(hkh^£ -WhhH^)] J J kJlm k£ mn J atoms

• f.i- x~ is the k fractional coordinate of atom j , 3 is the array of

anisotropic thermal coefficients , and c_ and d_ are the arrays of third

and fourth cumulant coefficients . After the atom parameter cards have

been read, the special parameter cards are read. The first card gives

the total number of special parameters (up to 60) . Then one card for

each special parameter contains its name, three integers specifying

its type, and its initial value.

At this point in the program, the parameter refinement selection

information is read. A parameter of the model is varied in the leastsquares refinement if its associated indicator is one; parameters with

zero indicators are changed only if they are linked to refined parameters

by dependency cards. If new parameter selection information is read,

constraint data are also read. If no dependency relationships exist

for this structure, a blank card must be included in the data deck.

The method used to treat parameter constraints is from Finger (1969)

and essentially consists of the application of the chain rule to the

derivatives of the structure factor with respect to the parameters of

the model. The user must determine the dependent parameters of the

model using the rules of Levy (1956) for thermal coefficients or the

equations of Finger (1969) for chemical constraints and then prepare

the data cards . The program currently will not allow the constraining

of the total occupancy for a site with only one species present; however,

this may be accomplished by including a null scattering curve which

is made the second species present. The amount of the first species

may then be refined and constrained.
At this point, all structural data except the observed structure

factors have been read. If these are to be read, a data card containing a FORMAT specification is read. Two optional cards may be read, one to specify the wavelength if it is needed for the computation of extinction

corrections and the other to specify an "ignorance factor", p, to be used

in computing-weights according to the formula

w = l/[a 2+ (pF ) 2

]. obs

Subsequently, the subroutine INPUT is called to read and process a reflection card. The data are then added to a buffer which, if full, is output

on file ISC2 . If the old structure factors are desired and this is

the first cycle of a given computer run, the buffer is filled from file

ISC1, the scattering factors are recomputed if necessary and the buffer

is written on file ISC2. If this is not the first cycle of a given

run, the old structure factors are read from file ISC2 . After the data

for a reflection have been obtained, the calculated structure factor

and partial derivatives are computed in subroutine SFAC, the results

tested to determine if this reflection should be used in the normal

equations matrix, and the contributions of this reflection to the residuals

are calculated. Next, the structure factors are printed, if desired,

with the rejected data marked with a single asterisk in the right margin.

In addition, the reflections having intensities less than the minimum

observable value are marked with a single asterisk next to the value

of F , and a double asterisk in the right margin.

If the reflection is to be included in the matrices, subroutine

MATRIX is entered; then the next observation is processed. As soon

as the end of the reflection list is encountered, the residuals for

this cycle are printed. Next, the normal equations matrices are converted

to correlation matrices to prevent overflow or underflow of the determinant

during the matrix inversion, which is accomplished by subroutine SYMINV,

adapted from Busing and Levy (1962). If the matrix is non-singular,

the inverse is converted to a variance-covariance matrix and the parameter

shifts are computed. Subroutine RESET is entered to compute the new

values of any dependent parameters. The ratios of the parameter shifts

to the standard deviations are tested to find the largest absolute value

and the magnitudes of the ratios are averaged. At this point in the

program, the correlation matrix, if requested, is printed. The twenty

off-diagonal elements of the correlation matrix with largest absolute

values are always printed. Then the optional bond functions are computed

by routine B0DAN1 and the thermal functions by routine ELVIB1.

Finally, a run continuation card is read. If column 1 is non-zero,

another cycle is to be performed. However, if this column is blank

or zero, the program prepares to terminate. If the new values for the

parameters are not to be saved, the program exits. On the other hand,

if these parameters are to be kept, a new label record is written on

ISC1, the structure factor records are copied from ISC2 to ISC1 and

the variance-covariance record is output . The current values of all

parameters may be punched on cards . This routine then returns control

to the main program.
Subroutine B0DAN1 . This subroutine uses the symmetry operators

supplied on cards combined with the action of centers of symmetry, if

present, and the effects of multiple lattice points and cell translations

to generate all atomic locations within a spherical shell centered on each atom in the asymmetric unit. The search for the contents of this

shell is done efficiently by enclosing the central atom with a unit

cell shaped parallelopiped tangent to the sphere. For each location

inside this box, a table of generated positions is checked to determine

if this location is unique and if it is, the bond distance is computed.

If this is within the desired range, the distance and coordinates are printed and this location is added to the table of generated positions.

If bond angles are also desired, each atom in the asymmetric unit

is used as the central atom with the atom positions generated in the

bond distance calculation considered. The peripheral atoms are identified

only by their alphanumeric tag and a number which points to an entry

in the bond distance output where the coordinates of the generated atom

are listed. Caution—if the maximum bond distance is large, very many

angles will be generated.

Subroutine ELVIB1 . For each anisotropic atom, the equivalent isotropic temperature factor of Hamilton (1959) , the root-mean-square (rms)

amplitudes of vibration (in A) of the principal axes and the orientations

of these axes with respect to the direct cell axes are computed, or if

the temperature factor matrix of an atom is nonpositive-definite, this

is reported. This routine will also compute the thermal corrections

to bonds using the formulae of Busing and Levy (1964) . To assist the

interpretation of the orientations of thermal ellipsoids relative to the bonding, the rms amplitude of the ellipsoid in the direction of

the bond is computed and printed. B0DAN1 is used to generate the necessary

atomic locations

.

Subroutine INPUT . This subroutine reads a reflection data card,

computes sinz 0/A z

, evaluates the scattering factors for this observation

and computes a-p f.or the observation. Maximum and minimum values of

Fobs anc* sin2 6/X 2

, are determined in this routine.

Subroutine MATRIX . This subroutine calculates the derivatives

of the residual from the derivatives of the calculated structure factor

and accumulates the sums required in the normal equations matrices.

Subroutine MODIFY.

and

to the calculated derivatives. A and jB are the real and imaginary portions

of the calculated structure factor, P is the independent parameter, p

This routine applies the equations

9A '

3 *n

= Zm 9A_!Zs

9p 9p m n

9B ' 3P„

= Em

_3B ^m

9p 9p

6
represents a dependent parameter, the summation is over all the dependencies, the unprimed derivative denotes the one calculated without regard

for dependencies and the primed derivative is needed to form the leastsquares matrices.

Subroutine RCALC . This subroutine accumulates the terms needed to calculate the discrepancy factors which are computed for all reflections

and separately for those data used in the refinement. A weighted residual,

R which is computed using the formula

2 Kw

Zw (|Fo| - |Fd )'2 w =

Z w I Fo I

for refinement on J_ or

R = Zw(|Fo| 2 - |F C | 2

) 2 w '

' ' 'L- Zw \?o\ h

for refinement on J_2

, is determined for each class of data. In addition,

the conventional discrepancy index

R= E{| F°l - |FcH

Z |Fo|

is computed for all reflections in each class. In addition, the ranges of

variation of F and sin2 8/X 2 are each divided into eight subranges and the sums needed for R are accumulated. (Note: the maximum and minimum values

are dynamically determined during any cycle in which new structure factors

are read and the distribution of residuals for such a cycle will be meaningless.) If the weighting scheme is correct, the expression (l/s) Zw(F -F ) 2 where the sum is taken over a subset of the full data set containing s reflections, should be independent of the composition of the subset. If we divide this expression by the constant (1/t) ZwF2

, where t is the

total number of reflections in the data set and the sum is taken over the

entire set, and take the square root of the quotient, the result should

also be independent of the composition of the subset. This number, which

can be considered the average contribution to weighted R for reflections

in the subset (Note that the expression is equal to weighted R if s=t.)

is printed in the output for each subset.

Subroutine RESET . This subroutine is entered once for special parameters and once for each atom to compute the updated values of all dependent parameters. Linear relationships and rigid-body parameters are com- puted directly. Other non-linear systems of constraints may be handled

by the user through subroutine SPVAL, supplied as a dummy.

Subroutine SFAC . This subroutine computes the structure factor and

partial derivatives with the symmetry operators applied to reciprocal space;

i.e. the Miller indices are transformed rather than the atomic coordinates.

If H

1 represents the transpose of the vector of Miller indices and X' .
represents the jth symmetry transformed coordinate for the n_th atom, then

the argument of the trigonometric functions needed to calculate the con- tribution of this atom to the structure factor may be written as HTX' nj.

Utilizing the expression for X' from page 3,

H

TX = H

T

(RX + T) = H

TRX + H

T

T.

The final form is the method used in this subroutine.

As the routine is entered for each reflection, the matrix products

H Rj = H! and HTTj are computed and saved. Then the structure factors

and partial derivatives are accumulated. If 3rd and 4th cumulant terms

are absent, the structure factor

with

and

F= E af ^ T A ? A n n n j nj nj

P . = exp [2tt1 (H

,TX + H

TT.)]

nj r n j

for anisotropic atoms or

T . = exp (-H'^ 3 H!)

nj r

j n 3

T . = exp (-B sin 2 9A2

) nj n

for isotropic atoms, the n summation is over the atoms in the asymmetric

unit, the j_ summation is over the symmetry operators, a is an occupancy

term, f is the value of the scattering curve for the n*h atom, i_ is /-I

and $ *s the matrix of anisotropic thermal coefficients. When anomalous

dispersion is included, the scattering factor becomes

f = f + f ' + if" n o n n

n with f_Q the wave length independent part , f^ the real part of the anomalous

dispersion and f^ the imaginary part. Including this expression in the

structure factor equation, applying Euler's rule and writing F_ in the form

F = A + iB, we find that

(f + f ') ? T .cos2tt(H! TX + H

T

T ) on

n n n' j nj j n j

- a f" ? T 4 sin2TT(H! TX + H

T

T.) n n j nj j n j J
and

= Z Ta (f

nL n o

r

+ a f" ? cos 2tt (H/ TX + H

T

T.) n n 3 n n j J

+ f ' ) E S in 2tt(h: TX + H

T

T.) n - j n j

The required partial derivatives are found by differentiating each of the

above equations with respect to the components of Xn and (3.= . After all

sums have been accumulated, the subroutine MODIFY is called to perform any

chainrule operations required. If this is to function properly, all

derivatives must be calculated although the refinement indicators would

suggest that certain derivatives are not needed.

Subroutine SFAC also includes an entry point named EXCALC which calculates the extinction factor of Zachariasen (1967) as extended by Coppens

and Hamilton (1970), and revised by Nelmes and Thornley (1973) . Using

this scheme, the structure factor is corrected for extinction by the

relationship

F*=F y1/2

c c

y = (1 + B(9)F

c

2 r*)

-1/2

-2 e

4 2

( + cos

4 26) dA

3(6) =

n2 c

4V2A sin 26 (1 + cos

2 29) dy

or x-rays

X2 dA - -t— for neutrons

V

2 A sin 20 dy

X_ is the wavelength, V is the cell volume, A is the transmission factor,

u_ is the linear absorption coefficient, and e_, m and c_ have their usual

meanings. The program allows three different forms for the extinction:

a) isotropic with r* refined, b) Type I anisotropic with r* =_/D'ZD) -1 ' 2 with zj[j refined and c) Type II anisotropic with r* = (N'W N) with_

W. . refined. If p is a unit vector parallel to the incident beam and s is a unit vector parallel to the diffracted beam, D is a unit vector parallel' to p x s and N = p x D.- The reference coordinate system for Z and

W is attached to the crystal and has its x-axis parallel to the incident beam and its z_ axis vertical when all diff Tactometer angles are 0.

Subroutine SYMINV . The method used by this matrix inversion routine

has been described by Busing and Levy (1962) and was coded by Mr. James

Mundstock of the Numerical Analysis Center of the University of Minnesota.

Subroutine REJECT . This routine may be used to exclude reflections

from the normal equations matrices for other than the standard reasons.

If a reflection is to be rejected, the variable NREJ in Common Block /F/

should be set to one.

9
Subroutine WEIGHT . This routine may be used to compute a non-standard

weighting scheme. Upon entry, the information contained in the standard

deviation field of the structure factor card will be in the variable WT

in Common Block /F/. This variable must be set to the desired weight

information when the routine is exited.

Subroutine RDEGN . This subroutine reads the dependent parameter information from cards, and rearranges the array of relationships in ascending order first of independent atom and parameter designators and then of

dependent atom and parameter designators. Special parameters are arbitraily assigned atom number 0.

Subroutine DERIVS . This, subroutine is entered once at the beginning

of each cycle . It computes the current values of derivatives of conventional parameters with respect to rigid body parameters. (Other non-linear

derivatives may be supplied by the user through subroutine SPDERI) . If

any independent parameters are themselves actually dependent on other parameters (such as when the values of the elements of the rigid-body motion

tensors are restricted by symmetry) the derivatives with respect. to the

true independent parameters are computed. Then the independent parameter

designators are replaced by their row indices in the normal equations

matrix, and the array is packed to eliminate any redundant relations.

Subroutine PAREXT . This subroutine reads the special parameter

cards, and prints the names and initial values of the parameters.

Subroutine TRNFRM. This subroutine is entered once at the beginning

of a run to compute the transformation matrices required to convert parameters from the crystal system to a standard orthonormal coordinate system

and back again.

Subroutine FILL IN . This subroutine stores special parameter values

in the proper positions in rigid-body parameter arrays

.

Function ERRCAL . This function subroutine determines the standard

deviation of a dependent parameter according to the approximate formula

* , n n 9P, 3 P. .1/2

K 1=1 j=l 3P4 3P. 1J

where n is the number of refined parameters and cr-j-j is the covariance of

the ith refined parameter with the j th refined parameter.

Subroutine EULER. This subroutine computes an orthogonal matrix

which is necessary to transform parameters from a general orthonormal

coordinate system to the standard orthonormal coordinate system defined

by x parallel to a and z parallel to c.*. The relative orientations are defined by Eulerian angles <j>,x> and u. Here oi is the angle through

which the special coordinate system must be rotated clockwise, as viewed

down the positive z axis of the standard system, about the z axis of

10
the standard system to bring the z axis of the special system into the

x-z plane of the standard system; x is a clockwise rotation about the y axis of the standard system (viewed down the positive y axis) to bring

the z axes of the two systems into coincidence; <J> is a clockwise rotation

about the common z axis to bring the x and y axes into coincidence.

Three additional entry points, DTDPHI, DTDCHI, and DTDOMG compute the

derivatives of the elements of the transformation with respect to (j>, x> and a) respectively.

Subroutine MULMM. This subroutine forms the product of two 3x3

matrices. An alternate entry, MULMV, multiplies a three component

column vector by a 3 x 3 matrix.

Subroutine TFM . This subroutine has two entry points. TFM2 transforms a second cumulant tensor in an orthonormal system to a set of anisotropic temperature factors in the crystal system. TFM3 does the same for

a third cumulant tensor.

Subroutine TRNSPS . This subroutine generates the transpose of a 3x3 matrix.

Function IPAK . This function packs 4 integer variables into a single computer word. The version given is appropriate only to the UNIVAC

1108. Other versions must be used for other machines. If the integer

word length of the machine is shorter than 32 bits other adjustments

must be made.

Functions SKTLS , D2KDT , D2KDL , and D2KDS . These functions compute

the second cumulants for an atom and their derivatives with respect to the

rigid body motion parameters T, L, and S/ .

Functions TKTLS , D3KDL , and D3KDS . compute the third cumulants

and their derivatives with respect to L and S.

Subroutine CORECT . This subroutine applies libration corrections to the atomic positions in a rigid body.

Function DETERM and subroutine SOLVE are used by CORECT. Subroutine

VXV forms the vector product of two vectors . Functions AIJ , BIJK, CIJKL , DIJKLM , EIJ , FIJ, AXA , AXAXB , PIJKLM , EL4 . are a set of functions used

in the rigid body relations.

Common Blocks and Variables

1. Common Block /LSMAT/

B(200) Normal equations vector

IEXT Extinction type indicator

IPARA (1000) Atom Parameter refinement indicators

MRANK Order of the normal equations matrices

KCYL Cycle number for the data set LABEL (3) 12 character label for the data set

11
JSCRF (10)

NRSF

SIGMA

MDEGN

A(22740)

Common Block /A/

:

BETA(6,60)

G(3,3)

IS0T(60)

NATOM

TAG (2, 60)

TITLE (18)

XYZ(3,60)

PICR(10,60)

DR(15,60)

Refinement switches for scale factors

Number of scale factors refined

Standard deviation of a parameter

Twice the number of constraint relationships

Constraints array and normal equations matrix

or inverse. The dimensions are a maximum.

The actual number for a given problem is

given by the expression MDEGN+ (MRANK* (MRANK

+l))/2. If, as on the UNIVAC 1108, the

operating system allows dynamic core allocation, this number can be determined at run time

.

Temperature factor coefficients

Real space metric tensor

Temperature factor type indicators

Number of atoms in the asymmetric unit

Alphanumeric atom tags or labels

Alphanumeric title information

Atomic positions

Value of tt Third cumulant coefficients

Fourth cumulant coefficients

3. Common -Block /C/:

CELLP(3,4)

ICENT

JCELL

NSYM

RMAT(3,3,24)

TRANS (3, 24)

4. Common Block /D/i

Coordinates of lattice points

Centric indicator

Number of lattice points

Number of symmetry operators

Rotational symmetry operators

Translational symmetry operators

ACALC

AIND(60)

ANOM(2,20)

BCALC

BIND (60)

DADB(6,60)

DADO (60)

DADX(3,60)

DBDB(6,60)

DBDO(60)

DBDX(3,60)

Real part of calculated structure factor

Individual atom contributions to ACALC

Anomalous dispersion corrections

Imaginary part of calculated structure factor

Individual atom contributions to BCALC

Derivatives of ACALC with respect to temperature factors

Derivatives of ACALC with respect to occupancy

Derivatives of ACALC with respect to position

Derivatives of BCALC with respect to temperature factor

Derivatives of BCALC with respect to occupancy

Derivatives of BCALC with respect to position

12
H(3)

IH(3)

ISCAT(2,60)

ISCL

MFSEL

OCCA(60)

OCCUP(60)

SCALE (10)

SCAT (20)

SITE(60)

DADC(10,60)

DBDC(10,60)

DADD(15,60)

DBDD(15,60)

AMINUS

BMINUS

DRDE(6)

NDEGN

5. Common Block /E/

:

DMAX

DMIN

RECELL(6)

EXANG(3)

6. Common Block /F/

i

DELTA

FCALC

FOBS

FOMAX

FOMIN

IREJ

MREF

MREJ

NREJ

NUMBER(36)

RDEN(36)

RHO

RHOMN

RHOMX

RNUM(36)

Miller indices in real representation

Miller indices in integer representation

Table used to select atomic scattering factor

Number of scale factor for this reflection

Function selector

Occupancy of species A

Total occupancy of site

Scale factors

Values of scattering factor for current

reflection

Equipoint 'fraction for atom site

Derivatives of ACALC with respect to 3rd

cumulant

Derivatives of BCALC with respect to 3rd

cumulant

Derivatives of ACALC with respect to 4th

cumulant

Derivatives of BCALC with respect to 4th

cumulant

ACALC for opposite absolute orientation

BCALC for opposite absolute orientation

Derivatives of r* (extinction factor) with

respect to coefficients of tensor

Number of constraint relationships

Maximum bond distance

Minimum bond distance

Reciprocal cell lengths and cosines

4>, X and0+ a) (in radians) for current

reflection

Weighted residual for current reflection

Calculated structure factor

Observed structure factor

Maximum observed structure factor

Minimum observed structure factor

Rejection mode indicator

Refinement mode indicator

Minimum observable indicator

Used to indicate whether or not a reflection

is rejected

Counters for subdivisions of R factors

Denominators for R factors

Current value of sin2 0/X 2 Minimum value of RHO

Maximum value of RHO

Numerators for R factors

13
WT

EXBETA

EXX

EXY

FISGN

IFSGN

EXTIN(6)

KEXTRF(60)

DELMAX

7. Common Block /G/

:

BUFF (1023)

FCURVE(9,20)

FRMT(18)

IWT NSCAT

BETFAC

WVLNTH

WTFAC

8. Common Block /TAPE/:

Weight for current reflection

Factor (3(9) for extinction correction

Factor B(9)F2 r* used in extinction

-1/5

(1 + EXX) ' Absolute orientation parameter

Orientation determination indicator

Extinction parameter matrix

Refinement switches for extinction parameters

Maximum permitted value of AF/ct

Array used primarily for blocking structure

factor records

Scattering curve coefficients

Variable format specification for reading

structure factors

Weight calculation ,type indicator

Number of scattering factors

Used in extinction correction

Wavelength of radiation

"Ignorance factor" in computation of weights

ININN

IOUT

ISC1

ISC2

ISC3

ISC4

ISC5

IPUN

Logical unit

Logical unit

input unit

Logical unit

Logical unit

file (10)

Logical unit

file (29)

Logical unit

Logical unit

Logical unit

list (28)

Logical unit

number of the card read unit (5)

number of the structure factor

(5 or 11)

number of the print file (6)

number of the least-squares

number of the intermediate

number of the auxiliary file (11)

number of the Fourier file (12)

number for saving constraints

number for punch file (1)

Dependent Parameter Input

In certain structures, the parameters of the model may not be independently variable. If they are merely constrained to have a fixed value,

there are no difficulties as the refinement selection variable for this

parameter is set to zero. However, if a parameter of the model is

required to be equal to a linear combination of variable quantities, the

situation is greatly complicated. In the least-squares refinement of

such a structure, the contribution of the dependent parameter to the

derivative of the structure factor with respect to the independent

variable must be computed if the shifts are to be properly evaluated.

14
Similarly, the correct value of the dependent parameter must be restored

at the end of the cycle if the structure factors are to be properly

computed in the next cycle . This program utilizes a new algorithm to perform the derivative modification and parameter restoration. The method

of preparing the input data cards will be shown with the aid of examples

.

Example 1: Atom 6 is an anisotropic atom on a 3-fold axis parallel

to c_. " Employing the rules of Levy (1956) , the following relationships

are required for this atom:

13 M 23

=

22

2 3 12

The first equations relate parameters to a constant and thus may be

ignored. However, this is not the case for the second, and since 3n

must be the independent variable, the following equations are applied as constraints to the least-squares solutions:

22 11

312

= 1/2 3n.

The resulting Parameter Constraint Cards (See page 24) for this situation

would be:

Dependent Independent Constant Comments (not on card)

6 6 6 5 1.0

0.5

6 22 = 3-1 1 for atom 6

312

= o.5 e u

End of list

Example 2 : Grunerite has four independent sites (atoms 1 to 4) for

Fe and Mg, but the total amount of each is known from the chemical analysis. Therefore, we want to refine the values of the Mg-occupancies for

the first three atoms and constrain the fourth to agree with the bulk

chemistry. In the notation of Finger (1969) , the constraining equation is

E b a = C m m mn n

with bju the multiplicity of the mth site, a

jnn the fractional occupancy

for the" nth species in the mth site, and C ""the total number of atoms of

species n per unit cell. In grunerite, sites 1 through 4 have multiplicities 4, 4, 2, and 4 respectively and there are two formula units per unit

cell. If we constrain the formula to be Feg 2^0.8^8^22 ^*^2 anc^ su^~

stitute into the above equation, we find that

15
Solving for a4Mg

»

4alMg + 4A

2Mg + 2a3Mg + 4a4Mg = 1 * 6

a4Mg

= °' 4 " alMg" a 2Mg " 1/2 a

3Mg'

The Parameter Dependency Cards needed to constrain this chemistry would be:

Comments (not on card)

Initialization

Site 1 dependence

Site 2 Site 3 End of list

Dependent Indep<sndent Constant

4 1 0.40

4 1 1 1 -1.0

4 1 2 1 -1.0

4 1 3 1 -0.5

Alternatively, the latest version allows up to 5 independent parameters

to affect one dependent one on the same card, so the first four cards

above may be replaced by the card as follows

:

4 10 0.40 1 1 -1.0 2 1 -1.0 3 1 -0.5

Note that when occupancies are constrained, the dependent value must

first be initialized with a card containing zeros in the independent parameter fields.

Example 3 : Like many chemical groups , phosphate groups often have

thermal vibration that can be represented by the motions of a rigid body,

which can be expressed as the elements of the T, L,, and S_ tensors introduced by Schomaker and Trueblood (1969) . Expressions for the second and

third cumulants of the probability distribution functions for individual

atoms (Johnson, 1969) as functions of the elements of £ and L are given

by Prince and Finger (1973) , and the corresponding expressions including

_S can be readily derived (Prince, unpublished but given as a FORTRAN

expression in this program) . If atom 3 is an oxygen atom of a phosphate

group and special parameters 1 through 24 are 3 coordinates of the

origin to which the motions are referred, 6 elements of ^T, 6 elements of

L, and 9 elements of S, the constraint cards will be set up as follows:

First, because of the so called "trace of S singularity", the three

diagonal elements of £ are not independent, and S33 can be set equal to

"S ll " S 22 by means o? the card 18 16 -1. 17 -1.

The anisotropic temperature factors may depend on all of the rigid

body elements, so there must be six cards of the form

3 5 4 0. 0-24

3 6 4 0. 0-24

3 7 4 0. 0-24

3 8 4 0. 0-24

3 9 4 0. 0-24

3 10 4 0. 0-24

16
Here the minus sign before the 24 indicates that the dependent parameter

depends on special parameters 4 through 24 inclusive. The coefficients

are given as zeros because, being non-linear, they change from cycle to cycle and are therefore computed in each cycle according to the current

values of the refined parameters. Third cumulants are independent of T,

but may be dependent on any element of L or S. Therefore a typical

constraint card for a third cumulant would have the form

3 12 10 -0. 0-24 0. The program uses the information on the constraint cards to allocate

space in any array and then, in non-linear cases, fills in the space at the

beginning of each cycle. Therefore if a given relationship is not listed

it will never be used whether or not its value is actually zero. Because

of symmetry some relationships, particularly those involving T, may be

identically zero. It will save space and almost certainly time to omit

these from the list.

Details of Data Input

1. File Name Card: Columns 1-12 contain an alphanumeric label to be

written into a new data set. This label will be

printed when the file is read

.

2. Title Card: Any desired Hollerith information in columns 1-72. This

information appears at the top of most pages of the output.

3. Control Card:

Column Variable Description

1 MFSEL ' Function selector

- Structure factors only

1 - Least-squares refinement

2 MSF Structure factor output selector

- No structure, factor output

1 - Printed structure factors

2 - Printed and written (for a Fourier program)

on file ISC4

3 - Written on file ISC4

3 MCORR Correlation matrix*

- Not printed

1 - Printed

4 MBODAN Bond distance, angle selector

- None computed

1 - Distances only

2 - Distances and angles computed

5 MELVIB Ellipsoids of vibration

- None computed

*The values of the 20 largest off-diagonal correlation coefficients will

be printed for every least-squares cycle.

17
3. Control Card (Cont.)

Column Variable Description

8 NEW(l)

9 NEW(2)

10 NEW(3)

11 NEW (4)

12 NEW (5)

13

14

15

1 - Ellipsoids computed for anisotropic atoms

2 - Ellipsoids and' thermal corrections to bonds

computed

6 MREF Refinement type indicator

- Refine on F„

1 - Refine on F

7 IFILE Rewrite data set indicator

- Data set rewritten if this is the last cycle

of the current run

1 - Data set not rewritten

2 - Data set rewritten. On input the standard

deviations of parameters refined in previous

cycles will be read in.

3 - Data set not rewritten. , Standard deviations

read in.

If non-zero , read new cell parameters*

If non-zero, read new bond distance limits

If non-zero, read new symmetry operators

If non-zero, read new scattering factorst

If non-zero, read new scale factors and atom parameters. If equal to one, all atoms will be read

If equal to two, only selected atoms will be read.

If equal to 3 only scale and extinction will be

read. If equal to 4 only special parameters will

be read.

NEW(6) If non-zero, read temperature factor type card

NEW(7) If non-zero, read parameter selection card(s) and

parameter dependency card(s)

NEW(8) If non-zero, read new structure factors. Values 1,

2 , and 3 correspond to various options for the

order of the list, as explained below.

For all zero values in columns 8-15 , the old values are read from file

ISC1 or the values in memory are used . The above integers are required for

all cycles.

16-20 DMIN

21-25 DMAX

Minimum bond distance in Angstroms. Needed only if

NEW(2) 4 0. Maximum bond distance in Angstroms. Needed only

if NEW(2) 4 0.

*Note that if the new cell parameters will affect sinS/X, and therefore

the atomic scattering factors for various reflections, structure factors

must be read in.

tWarning: New scattering factor curves must be read only in the first

cycle of a given computer run or when new structure factors are being

read in, or they will never be used in computation.

18
3. Control Card (Cont.):

Column Variable Description

26 ICENT Centric indicator, needed only if NEW(3) 4

- Centric

1 - Acentric

27 ICELL Cell type indicator, needed only if NEW(3) 4

- Primitive cell

1 - A-centered cell

2 - B-centered cell

3 - C-centered cell

4 - I-centered cell

5 - F-centered cell

6 - R-centered cell

28 MANOD Anomalous dispersion indicator, needed only if

NEW(4) 4

- No corrections are to be used

1 - Include anomalous dispersion corrections

29-30 NSCAT Number of scattering curves. This is needed only

if NEW (4) ^0. (0 < NSCAT < 21)

31-32 NATOM Number of atoms in asymmetric unit . This is needed

only if NEW (5) ^ 0. (0 < NATOM < 61)

33-34 NSF Number of scale factors. This is needed only if

NEW (5) 4 0. (0 < NSF < 11)

35 NEWEXT Extinction change indicator (needed only if NEW(5)

4 0)- New extinction parameters not read

1 - Extinction parameters read. from cards

36 IWT Weight type, needed only if NEW(8) 4 0. - Standard deviation of F on cards

1 - Standard deviation of F

2 on cards

2 - Weight on cards

3 - Sigma for F is to be computed in WEIGHT

4 - Compute weight in subroutine WEIGHT

5 - Standard deviation of F on cards will be

altered by "ignorance factor" by

a = (a 2 +(PF) 2 )!/ 2 37 INN Structure factor source indicator, needed if

NEW(8) 4 0. - Structure factors on cards

1 - New structure factors from file ISC3

38 IREJ Rejection parameter, needed only if NEW(8) 4 0. 0,5 - No reflections rejected

1.6 - Reject if I less than the minimum obs.

2.7 - Reject if I less than the minimum obs. or | Fo-Fc | hy greater than DELMAX

3.8 - Reject if |Fo-Fc|/a greater than DELMAX

In both of these cases weights will be

a^red by w' = wCl^x^x^) , where

x = | Fo-Fc | / (aF*DELMAX)

4.9 - Use special routine REJECT

19
3. Control Card (Cont.)

Column Variable

39-43

61

NOTE:

DELMAX

44-53 JSCRF

54-59 KEXTRF

60 JXPAR

IFSGN

Description

Values of IREJ from 5 to 9 allow the rejection mode

to be changed even though NEW(8) = 0.

Maximum allowable difference between observed and

calculated structure factors on an absolute

scale. This is needed only if NEW (8) 4 and

IREJ is 2 or 3, or if IREJ is 7 or 8. Scale factor refinement switches (needed only if

NEW(7) 4 0)

Extinction factor refinement switches (needed only

if NEW(7) 4 0)

Extra parameter indicator

- No extra parameters

1 - Extra parameters will be read

(needed only if NEW(5) 4 0)

Orientation determination parameter

- Do not determine the orientation

1 - Change to orientation with lower weighted

r (needed only if NEW(7) 4 0)

4. Cell Card : Ne<

Columns Variable

1-8 A(l)

9-16 A(2)

17-24 A(3)

25-32 A(4)

33-40 A(5)

41-48 A(6)

Needed only if NEW(l) 4 0.

Description

a_ axis length in angstroms

b_ axis length

c_ axis length

Alpha in degrees

Beta in degrees

Gamma in degrees

5. Symmetry Cards: Needed only if NEW(3) 4 0,

Columns Variable

1-3 JTNS (1)

5-6 IRMAT(l.l)

8-9 IRMAT(2,1)

11-13 JTNS (2)

15-16 IRMAT(1,2)

18-19 IRMAT(2,2)

21-23 JTNS (3)

25-26 IRMAT(1,3)

28-29 IRMAT(2,3)

34 IEF

Description

Translational part of x_' First positional part of x' Second positional part of x.' Translational part of v_' First positional part of y_' Second positional part of y_' Translational part of z/

First positional part of z/

Second positional part of z_' End flag, non-zero on last card only

- Another symmetry card follows

1 - This is the last card

20
The translational operators are of the following form:

bbb 1/2 1/3 2/3 1/4 3/4 1/6 or 5/6. The positional operators are of

the form bW = W or -W with W equal to X, Y, or Z and b signifies a blank. The symmetry position X Y Z must be included.

6. Scattering Factor Cards: Needed only if NEW(4) 4 0.

These coefficients must

satisfy the equation

*

2 f (x) = c + E a exp(-b x ) i=l

Columns Description

1-8

c

9-16

17-24

25-32

33-40

41-48

49-56

57-64

65-72

73-80 Identifier

with x = sine

A

Coefficients for scattering curves of this form have been tabulated

by Cromer and Waber (1965) for relativistic Dirac-Slater wave functions,

by Cromer and Mann (1968) for Hartree-Fock wave functions and by Doyle

and Turner (1968) for relativistic Hartree-Fock wave functions. If

anomalous dispersion coefficients are being used (MANOD = 1) , the above

card is followed by one with the real part of the anomalous dispersion

in col. 1-8 and the imaginary part in col. 9-16. Coefficients for the

hydrogen atom are given by Forsyth and Wells (1959)

.

NOTE: The scattering curves should be changed only on the first

cycle of a given computer run or when NEW(8) = 1. New structure factors

must be read (NEW(8) = l)if the number of scattering curves is to be

changed

.

7. Scale Factor Cards: Needed only if NEW(5) 4 0.

The NSF scale factors are punched with 8 columns per value and 9 values per card and should be the values needed to place the calculated

structure factors on the same scale as the observed

.

8. Extinction Cards: Needed only if NEW (5) 4 and NEWEXT 4 0. Format

(6E12.6)

Columns Value

1-12 r* for isotropic extinction or r* for anisotropic

13-24 r*

25-36 r*it

37-48 r*f;J 49-60 r*r^

61-72 r*g

21
9. Atom Parameter Cards: Needed only if NEW (5) 4 0.

If NEW (5) = 1, the entire atom list is read with' two cards per atom.

If NEW(5) = 2, the Atom Number is read with Format (12) followed by

the two cards for that atom. Card reading is terminated by a blank

Atom Number card.

Card

C olumns

1-6

7-12

13-18

19-24

Card 2 1-2

3-4

9-15

16-22

23-29

30-36

37-43

44-50

51-57

58-64

65-71

73

Variable

TAG

SITE occur OCCA

ISCAT(l)

I SCAT (2)

XYZ(l)

XYZ(2)

XYZ(3)

BETA(l)

BETA (2)

BETA (3)

BETA (4)

BETA (5)

BETA (6)

IEF

1-8

9-16

17-24

25-32

33-40

41-48

49-56

57-64

65-72

73-78

80

:iu

:222 :333

;m

;i22

;ii3

;i33

:223

:233

Description

Alphanumeric identifier

Equipoint fraction

Total occupancy of site

Occupancy of species A in this site

Needed only if multiple occupancy

Number of scattering curve for species A

Number of scattering curve for species B (blank if single element in site)

.

x of atom in fractional coordinates

Z

z_

B_ for an isotropic atom or (3, , for an anisotropic atom

6 22

e 33

3 12

Jl3 P 23 Indicator of third cumulant

Card 3 - Needed if IEF 4 0,

Fourth cumulant ind.

Card 4-5 - Needed if IEF 4 0,

Punch dim, d

2222 , d

3333 , dm2 , dm3 , dm2 , dn33 , d1222> d1223>

d

2

, d..-., d

222 , d2233» d2333 in order with 8 columns per value.

10. Special parameter cards: Needed only if NEW(5) 4 and JXPAR 4 0.

Card 1: Columns 3-4 contain the number of special parameters, NPAR.

22
If NPAR>0 the succeeding cards contain the names, initial values, and

identifiers according to the following form.

Columns

Columns

1-8

9-10.

Columns 11-12

Column 13-14

Name of parameter, up to 8 alphanumeric characters.

Type of parameter, 0- linear function with coefficient

given on constraint card.

1-rigid body system

2-defined by user in subroutines SPVAL and SPDERI

Integer indicating which rigid-body system, if more than

one , this parameter refers to

.

Parameter designator. If the type is 1 the code is as follows

1-X coordinate of origin (in crystal axes system)

2-Y coordinate of origin

3-Z coordinate of origin

4-to angle for transformation, to standard, orthonormal

coordinate system

5~x angle for transformation

6-<J> angle for transformation

7-X coordinate of an atom in the special system

8-Y coordinate of an atom

9-Z coordinate of an atom

10-T,

11-T

12 -T 13-T

14-T

15-T

16-L

17-L

18-L

19-L

20-L

21-L

22-S

23-S

24-S

25-S

26-S

27-S

28-S

29-S

30-S

11

22

33

12

13

23

11

22

33' 12

13

23

11

22

33

12

13

23

21

31

32

Columns 15-24 initial value of this parameter.

11. Extinction and Temperature Factor Type Card: or NEW(6) i 0. Needed only if NEW(5)

Column 1 contains the extinction factor code:

- Isotropic, do not change

1 - Anisotropic Type I (mosaic spread dominates)

23
2 - Anisotropic Type II (domain size dominates)

3 - Now Isotropic, change to Type I

4 - Now Isotropic, change to Type II

5 - Now Type I, change to Isotropic

6 - Now Type II, change to Isotropic

7 - Now Type I, change to Type II

8 - Now Type II, change to Type I

The temperature factor type code is punched with one column per atom beginning in Col. 2 with the following code:

,0 - Now isotropic, do not change

1 - Now anisotropic, do not change

2 - Now isotropic, change to anisotropic

3 - Now anisotropic, change to isotropic

4 - Now 3rd cumulant, do not change

5 - Now anisotropic, change to 3rd cumulant

6 - Now 3rd cumulant , change to anisotropic

7 - Now 4th cumulant , do not change

8 - Now 3rd cumulant, change to 4th

9 - Now 4th cumulant, change to 3rd

12. Parameter Selection Cards. Needed only if NEW(7) 4 0. If JXPAR ± the first card contains refinement selection for the special parameters in the first NPAR columns. A zero in a column means that the

corresponding parameter will not be refined and a one means that it

will." After this cards with the refinement indicators for the

conventional parameters punched in columns 1-80. If the number of

atom parameters is such that more than 80 values are required, the

parameter selection information should be continued in column 1 of

the next card. The order of parameters is:

a) occupancy - If single occupancy, the total occupancy is refined.

If multiple occupancy, the occupancy of species A is refined.

b) x coordinate

c) y_ coordinate

d) z_ coordinate

e) thermal parameters, one for isotropic atoms, six for anisotropic

atoms. The thermal coefficients occur in the same order as on

the Atom Parameter Card 2

.

f

)

3rd and 4th cumulant parameters (if needed)

.

13. Parameter Constraint Cards (s) : Needed only if NEW(7) i 0.

a) Columns Description

1-3 Integer specifying the dependent atom number.

5-6 Dependent parameter number. This number is 1 for occupancy, 2-4 for x through z respectively, 5 for B^ or

B n, and 6, 7, 8, 9, 10 for 3 2 2> 833 » &i 2

» ^13 and

3„_ respectively. Values of 11 to 20 treat 3rd

cumulant parameters and 21-35 are for the 4th cumulants,

24
13. Parameter Constraint Cards(s) (Cont.):

a) Columns Description

7-9 Atom number of first independent parameter. Special

parameters are arbitrarily indicated as atom number 0. 10-12 Parameter number of first independent parameter. Atom

parameter designates an initializing constant in a linear constraint function.

13-20 Coefficient of first relationship.

21-23 Atom number of second independent parameter

24-26 Second independent 'parameter number

27-34 Second coefficient

35-37 Third independent atom number

38-40 Third independent parameter number.

41-48 Third coefficient

49-51 Fourth independent atom number

52-54 Fourth independent parameter number

55-62 Fourth coefficient

63-65 Fifth independent atom number

66-68 Fifth independent parameter number

69-76 Fifth coefficient.

If the independent atom number is zero, designating a special parameter,

and the constraints are non linear, the second and fourth independent

parameter numbers may be negative to denote the end of an inclusive list

beginning with the first (third) independent parameter number.

b) One (1) blank card or a card with zero in column 3.

14. Structure Factor Cards: Needed only if NEW(8) 4 0.

a) Format Card: This card contains the format of the input structure

factor data. The number and order of variables on a card (or tape

record) depends on the value of NEW(8) , as explained in detail below.

All records must contain at least four integer fields, for the indices

and an observed/unobserved indicator, and two floating point fields

for F bs and ap or other weight information. They may contain

other information, such as a scale factor group identifier, the factor

6(6) used in extinction corrections, and the setting angles, needed

for anisotropic extinction corrections.

b) Only if IWT=5 include a card with the "ignorance factor" punched as

a real number in the first 8 columns.

c) Only if NEW(8)=3 include the wavelength in A in the first 8 columns.

For neutron data only an approximation to 6(9) will be computed ac- cording to the formuja $(0) = [ (2X 2 /(V2 sin2 0) ] x 10 8

, where V is the

unit cell volume in A

3

. The refined quantity then will be the product

Tr*, with T assumed to be approximately constant.

25
d) Data Cards in a form consistent with Format Card:

list is in this order.

If NEW(8)=1 the

Variable Description

IH(1) h for reflection

IH(2) k

IH(3) 1 FOBS Observed structure factor

SIGMA Weight information consistent with IWT on Control Card

EXBETA Factor g(6) for extinction

I Number of scale factor for this reflection. If zero, the

scale factor is the same as the preceding reflection.

Scale factor 1 is assumed for the first reflection.

MREJ Minimum Observable Parameter

1 - Intensity above minimum observable.

2 - Intensity below minimum observable.

IEF End-of-list marker, zero on all data cards

TTH 20 These angles are needed only if anisotropic

OMEG co extinction is to be calculated and must

CHI X conform to the diffractometer conventions

PHI $ given by Busing and Levy (1967, p. 458).

If NEW(8) = 2 the input list is XH(1), IH(2), IH(3), MREJ ,1 , FOBS , EXBETA

,

SIGMA, with the same meanings as above. The end of the data must be

marked by an end-of-file.

If NEW(8) = 3 the list is IH(1), IH(2), IH(3), MREJ, FOBS, SIGMA.

EXBETA is computed, and again the end of data must be marked by an end-of-file.

e) End-of-list card - A card with a non-zero value in the field of

variable IEF or an end-of-file card appropriate to the system.

15. Run Continuation and Parameter Punch Card: Needed for all cycles.

Column Variable

ICONT

3-4

JPUN

ISC1

Description

- No more cycles . The data set on file ISCI

will be updated if IFILE from the Control

Card is zero

- Read a new title card and start another cycle

- Do not punch any atom parameters

- Punch atom cards in form required by 'X-RAY67'

- Punch atom cards in form required by this

program

3 - Punch atom cards in both forms

If this field is non-zero its value will be substituted as the unit number for the binary

master file, making it possible to try new

courses of refinement without destroying

previous stages

.

26
BONDAN: A program for computing bond distances, angles,

and thermal ellipsoids, with errors

This program uses the master binary file created by RFINE to compute bond distances and angles, and also thermal ellipsoids, with

associated standard deviations. If the rigid body model is used,

the atomic positions of the rigid body will be corrected for libration before calculation of distances and angles. The program will

also accept input from cards by means of the program BADTEA. The

complete description of BADTEA is included in Appendix B. The

details of data input are as follows:

1. Input format indicator card

Col. 1-2 1-input from cards via BADTEA.

2-input from binary file created by RFINE.

If the input format indicator is 1, see appendix B for further input.

If the input format indicator is 2 , the card input continues as follows

.

2. Title Card: Any desired hollerith information in columns 1-72.

3. Search Card: Format (12 ,3A4)

If column 2 is zero or blank, a search of the data tape will be made

to find a data set with a label the same as columns 3-14. The error calculation will be performed on that data set. If column 2 is non-zero,

the first data set on the tape will be used in the error calculation.

4. Parameter Card: Format(2Il,3F8.0)

Column Variable Description

1 MBODAN Bond distance, angle selector

- None computed

1 - Distances only

2 - Distances and angles computed

2 MELVIB Thermal ellipsoid selector

- No ellipsoids computed

1 - Thermal ellipsoids computed for anisotropic

atoms

3-10 DLIMIT(l) Minimum bond distance output

11-18 DLIMIT(2) Maximum bond distance used in angles output

19-26 DLIMIT(3) Maximum bond distance output

5. Cell variance-covariance matrix: Format (6E12 .6)

This matrix is the matrix output by the lattice constant program of

C. W. Burnham.

27
Column Description

Card 1 1-12 Variance of _a (in angstroms**2)

13-24 Covariance of a_ with b

25-36 Covariance of a_ with c 37-48 Covariance of a_ with alpha (in angstrom-radians)

49-60 Covariance of a_ with beta

61-72 Covariance of a_ with gamma

Card 2 1-12 Variance of b

13-24 Covariance of b_ with c_ 25-36 Covariance of b_ with alpha

37-48 Covariance of b_ with beta

49-60 Covariance of b_ with gamma

61-72 Variance of c_ Card 3 1-12 Covariance of £ with alpha

13-24 Covariance of c with beta

25-36 Covariance of c with gamma

37-48 Variance of alpha (in radians**2)

49-60 Covariance of alpha with beta

61-72 Covariance of alpha with gamma

Card 4 1-12 Variance of beta

13-24 Covariance of beta with gamma

25-36 Variance of gamma

If the errors in the cell constants have a small effect on the

distances, the last 4 cards may be replaced by blanks.

28
FOURIER

This program uses the binary structure factor file created by RFINE

(ISC4-unit 12) as input for a Fourier synthesis. The map is written on unit 11 for -input to the plotting program CNTPLT or the arbitrary section

program ARBSECT. The details of data input are as follows:

1. Title Card: Any desired •alphanumeric information in columns

1-72. This information heads all map sections.

2. Parameter Card: Format (611, 2X, 914, 3F8 0.4)

Columns

2.

9-12

Variable

ITYPE

ISECT

INCENT

KREJ

IPLT

LSPC

IX

13-16 IXI 17-20 IXF 21-24 IY

25-28 IYI 29-32 IYF 33-36 IZ

37-40 IZI 41-44 IZF 45-52 EXXAG

53-60 F000

Description

Map type indicator

- Fourier map

1 - Difference Fourier

Section parameter

- x across , v_ down and z_ sections

1 - x " £

Z

z

X

£

X

£

x_

x_ £

Centric indicator

- Centric

1 - Acentric

- Ignore rejection flag and include

all reflections.

1 - Do not include any reflections

with the rejection flag set.

If non-zero, map sections will be

written to unit S2 for plotting or calculation of an arbitrary section.

If zero, the lines of the map will be

double spaced. A non-zero value indicates single spacing.

Number of divisions along horizontal

axis (>0)

Initial point

Final point

Number of division along vertical axis

(>0)

Initial line

Final line

Number of divisions along section axis

(>0)

Initial section

Final section

Map exaggeration desired

Number of electrons in unit cell for

Fourier

.

29
2. Parameter Card (cont.):

Zero for difference Fourier.

61-68 TMULT Multiplicity of a general reflection

3. Multiplicity Card: Format (12F6.6

Columns Description

1-6 Multiplicity of hOO or OkO

7-12 Multiplicity of 001

13-18 Multiplicity of hkO

19-24 Multiplicity of hOl

25-30 Multiplicity of Okl

31-36 Multiplicity of hhO

37-42 Multiplicity of hOh

43-48 Multiplicity of Okk

49-54 Multiplicity of hhh

55-60 Multiplicity of hhl

61-66 Multiplicity of hkk or hkh

67-72 Multiplicity of hkl

C. Size of Program:

This program uses the array FBUF in common block ABLE as pooled

storage. The size of this array, S, is governed by the map to be calculated and may be estimated using:

S = 3N , + 2(h' + 2) (2k* +1) ref max max + 2(h* + l)(P

v + 1) max x

where Nre f in the number of reflections included, h' max is the maximum

value of the index associated with the horizontal axis, k' xis the

maximum value of the index associated with the axis plotted down the page,

and P is the number of points calculated in the horizontal direction.

If the Fourier sections are to be saved for later plotting, S should

be increased by P P . If the computer operating system supports dynamic

core allocation, the program may be converted to automatically request

the proper memory size.

D. Output:

Each map point is converted to a four-digit integer before printing with zero values replaced by blanks. Each point will occupy an area

4 columns by 1 or 2 rows depending upon the spacing selected. For most

printers, the area is 0.4 x 1/6 inches for single spacing or 0.4 x 1/3

inches for double spacing of the lines. If the map requires more than

30 points, the section will be printed in strips with 30 points per strip.

30
E. Additional Problems:

If additional maps are to be run using the same structure factor

list, a complete set of cards beginning with the Title Card should be

included. An end-of-file in the card stream terminates the program.

31
LISTFC

This program is an adaption of the XRAY67 program LIST FC (Stewart

et al., 1967). It has been altered to stand alone and read the structure

factor file (ISC4 - unit 12) created by RFINE. The output of the program

has been changed so that the "phase in millicycles" has been replaced by

sigma (Fobs). Input is as follows:

Card 1.Col. 1-6 compound identification; Col. 7-78 any title.

Card 2.Col. 14 (l)/(2)/(3) for h index varies (most) /(next most)/

(least) rapidly

15 Same for k index

16 Same for I index

(Note that the sum and product of Cols. 14, 15, and 16

must equal 6.)

17-20 Number of lines per list page.

21 (blank) /(A) for (DO) /(DO NOT) internally change the

value of the number of lines in order to make the

bottom of the last page as even as possible.

22-24 Number of list columns per page. Note that the product of the number of lines and the number of

columns cannot exceed 2000 . That is no more

than 3000 items for the number of reflections and

headings (including spaces) per page.

26 Number of blank print columns before the LISTFC

column.

28 Number of print columns for most rapidly changing

index.

30 Number of print columns for F ^ s 32 Number of print columns for Fca^ c 34 Number of print columns for ap

36 Special flag for unobserved reflections (blank=*)

38 Special flag for severely extinct reflections

(blank=E)

40 Special flag for special reflection (JC0DE=4)

The next eight fields are (blank) /(l) for (do not) /do)

42 Print symbol for unobserved or severely extinct

reflections.

This symbol switch adds one print column

44 For centric structures, attach sign of A to F . 46 Double space the lines

48 Restore each LISTFC page to the top of the printer

page

50 Print title at the top of each printer page

52 Punch a set of "FCARD" cards

56 Print a minus sign on F b s for unobserved reflections

58 Write a separate copy of FC list on unit 7 62 Number of times to try to get headings at the top of

all columns

32
64 (l)/(2)/(3) for headings separated by (no blank

lines) /(l blank line above) /(l above and 1 below) (blank=3)

65-72 factor Fs are to be multiplied by (blank=10)

73-80 factor ct s are to be multiplied by (blank=10)

Note that all F and a values are printed as integers,

so that scale factor must be sufficient to preserve a sufficient number of significant figures

.

33
CNTPLT

CNTPLT is a contour plotting program for Fourier maps and other

similar functions. The routines contained in this program have been

derived from multiple sources with the original version of subroutine

ICONT written by S. H. Zisk and N. M. Brenner of M.I.T. Lincoln Lab.

Subroutine INTPLT uses the rational polynominal spline interpolation

of D. V. Ahujo, I.B.M. Systems Journal , p. 208-217, (1968). The other

routines have been modified from code prepared by G. Ford, Purdue University.

The present version of these routines have been written to be as' machine independent as possible. It is believed that only the call to the plotter function PLOTS (Sequence No. A5300) and the two functions

MARK and UNMARK called by ICONT are machine specific. This is the main

contribution of the present author (LWF)

.

Method

The main contouring subroutine, ICONT, takes the function values

at the grid points of the map and computes the coordinates of the intersection of a specified contour line with the lines of the grid. Linear

interpolation is used in this process. The second contouring subroutine,

CPLOT, coordinates the drawing of the contours and produces any desired

contour labels. The third contouring routine, INTPLT, produces a smooth

curve through the intersections located by ICONT. Rational cubic polynomials are used to fit a spline-like curve through neighboring points

.

Two such curves meet smoothly because the first and second derivatives

are single valued along the composite curve. The plotter line is drawn

as a series of straight line segments with the number of such segments

specified by the user.

The main program includes various options which allow labeling of

plot sections and/or grid points, drawing of the bounding parallelogram,

and other similar functions.

Input

The values of the function to be plotted are read from a binary

input file (unit 10) with each section occupying two records. The

first contains the variables MA, MB, JZ, MZ where MA and MB are the

number of grid points along x and y respectively. The second record

contains the data for the map section at JZ/MZ. These data are assumed

to be in real form and are packed in an array which is MAxMB. The

output file produced by program FOURIER is in this form.

Input

1. Title Card: Format (20A4)

2. Control Card: Format (F5.0, 15 ,3F5. 0,615, F5. 0,215)

34
Columns Variable Description

1-5

6-10

11-15

16-20

21-25

TLEHGT

NLABL

A

ALP

Plotting height of title in inches. If

<0, no title will be plotted

Number of vertex labels to be plotted

Width of plot (in- inches) parallel to x-axis of Fourier. If A>0, an 11'

plotter is assumed. If a 30" machine

is available, A should be less than zero. Length of plot (in inches) parallel to

v_-axis of Fourier which is parallel to plotter x-axis.

Angle between Fourier axes (in degrees)

26-30 NA

31-35 NB

36-40 NP

41-45 NCONT

46-50 NLINE

51-55 NCROSS

56-60 SCROSS

61-65 NSECT

66-70 NZ

Number of grid points parallel to Fourier

x_-axis

Number of grid points parallel to Fourier

v_-axis

Number of interpolation points per spline.

The default value is 4

.

Number of sets of contour levels (NCONT

£10)

Number of line vertices specified

(NLINE<20)

Number of crosses to be placed on plot

(NCROSS<20)

Size of cross (in. inches)

Number of sets of sections specified

Grid spacing of section levels.

3. Contour Card(s) : Format (3F5.0,I5)

There should be NCONT of these cards,

Columns

1-5

6-10

11-15

16-20

Variable

SBEG

CSTEP

SLIM

NSIG

Description

Height of initial contour

Step between contours

Height of final contour

Number of digits after decimal point in

contour label. If NSIG<0, contours will

not be labelled.

4. Line Vertex Card (s) : Format (16F5.0)

The positions of the line vertices in map grid coordinates should

be punched. A line will be drawn from point i_ to point i+1 if x^+

,HD.

No line will be drawn if xi+-i<0. Omit these cards if NLINE<0.

Note: The first grid point is (1,1), not (0,0). i.e. add (1,1) to scaled coordinates.

5. Cross Card(s): Format (16F5.0)

35
The map grid coordinates of the crosses should be punched sequentially.

Omit these cards if NCROSS<0.

6. Section Selection Cards: Format (315)

Columns Variable Description

1-5

6-10

11-15

INITZ

ISTEPZ

IFINZ

Sections from INITZ

to IFINZ in steps

of ISTEPZ will selected and contoured.

There should be NSECT of these cards.

7. Vertex Label Card (s) : Format (5F5.0,A6)

Columns

1-5

6-10

11-15

16-20

21-25

26-31

Variable

WX

WY

DX

DY

HGT

LABL

Description

x-grid coordinate

y-grid coordinate

Plotter x-axis offset

Plotter y-axis offset

Label height in inches

Alphanumeric label

There should be NLABL of these cards

.

Tape output for Calcomp plotter is on Unit 12,

36
ARBSECT

This program calculates an arbitrary section of a Fourier map from

a complete set of sections calculated in regular intervals parallel to the crystallographic axes. The input sections may be computed with the

regular Fourier program and are assumed to be in file No. 11. The plane

of the output section is specified by the coordinates of the plane to be calculated.

Program ARBSECT has several options available. If only a fraction

of the unit cell is used as input, it 'will calculate the electron density

in the accessible portion only with the remainder replaced by asterisks

in the printout. However, if the map for a complete unit-translation has

been computed, the program will use the translation symmetry to calculate

a complete section. No other symmetry operations are used. Other options

include (a) describing the points used to define the plane in terms of

map or fractional coordinates as desired, (b) saving the resulting

section in file 12 for later plotting, (c) computing the entire section

or selecting a portion of it, and (d) calculating and printing the geometry

including axial lengths and interaxial angle, of the output section.

Method

The input coordinates of the three points defining the plane are initially converted to map coordinates if necessary. The coefficients

of the plane in the equation a,x + a, y + a,z + a, = may be found from

the determinant equation

x y z 1 x

l yl

z lX x

2 y2

Z

2 1 x

3 y3

Z

3 1

where x^, y^, and z^ are the coordinates of the i'th point. This plane

will have a line of intersection with each section of the original map.

For the values of y_ and z_ corresponding to each line and section of the

output map, the x-coordinate of this point in the plane is determined.

A cubic function of the form ax3 + bx2 + ex + d is fitted to the four

adjacent grid points on the line, and the value of this function at the

intersection point is used for the output at that y and z. This process

is repeated until all output points have been determined.

Input

The input map sections are read from file 11 and the resulting map

is output to file 12. These assignments may be changed by redefining

variables ISC and ISC1 respectively.

37

=
1. Title Card - Format (18A4)

Columns 1-72 of this card may contain any desired information.

2. Control Card - Format (813)

columns

1-3

Variable

10

6

9

12

IZO

ISPC

IMRK

13-15 IYI 16-18 IYF

19-21 IZI 22-24 IZF

Description

If IC#0, the output section will be

written on Unit 12.

Section number of output

If ISPC = the output map will be

single spaced; double spaced if ISPC=1.

If IMRK=0, the y- and z- limits of the

input sections will be used as the x-, and y-limits, respectively, of the output

section. If IMRK ^0, "these values are read from this card.

Initial value of x in transformed section

Final value of x in transformed section

Initial value of y in transformed section

Final value of y in transformed section

3. Cell Card - Format (6F8.0)

Columns Description

o 1-8 Length (in A) of cell axis across page in original section

9-16 Length of cell axis down page in original section

17-24 Length of cell axis out of page in original section

25-32 Angle (in degrees) between axis 1 and 2 33-40 Angle between axes 1 and 3 41-48 Angle between axes 2 and 3

The above length and axes are required only if the program is to compute and print the axial length and interaxial angle of the transformed

section. If not, this card may be blank.

4. Plane Card - Format (9F8.0, 12)

Columns Description

1-8 x- coordinate of first point

9-16 yii it ii 17-24

25-32

z- " x- "

ii it ii

,, second „ 33-40 yii ii ii 41-48

49-56

z- " x- "

ii ii it „ third „ 57-64 y- "

ii ii ti

38
4. Plane Card - (Cont.)

Columns Description

65-72 z- coordinate of third point

74 Coordinate type indicator. If zero, the coordinates are

given in grid spacings. If non-zero, the points are given in fractional coordinates.

Additional arbitrary slices may be computed by supplying additional

data decks beginning with the Title Card.

Form of Input Sections

The input Fourier sections are read from Unit 11 with unformatted

reads. There must be two records per section as follows:

Record 1:

Word Variable

1 JXA

2 JYA

3 KZC

4 IZ

5 IX

6 IXI

7 IXF

8 IY

9 JYI

10 JYF

11 JZI

12 JZF

Record 2

Description

Number of grid points calculated parallel to x Number of grid points calculated parallel to y

Section number

Number of sections per unit cell

Number of intervals per unit cell parallel to x

Initial grid points parallel to x Final grid point parallel to x Number of intervals per unit cell parallel to y

Initial grid point parallel to y

Final grid point parallel to y

Initial grid point parallel to z Final grid point parallel to z

This record consists of the JXA*JYA values of the map in this

section. These real values must be packed in an array with the x co- ordinate as the first subscript and y as the second subscript.

39
REFERENCES CITED

Busing, W. R. , and H. A. Levy (1962) A procedure for inverting large

matrices. Comm. ACM , 5_, 45-46.

and (1964) The effect of thermal motion on the estimation

of bond lengths from diffraction measurements. Acta Crystallogr .

,

17, 142-146.

and (1967) Angle calculations for 3- and 4-circle x-ray

and neutron diffractometers. Acta Crystallogr ., 22 , 457-464.

Coppens, P., and W. C. Hamilton (1970) Anisotropic extinction corrections in the Zachariasen approximation. Acta Crystallogr ., A26 ,

-

71-83.

Cromer, D. T., and J. T. Waber (1965) Scattering factors computed from

relativistic Dirac-Slater wave functions. (Acta Crystallogr . , 18 , 104-109.

and J. B. Mann (1968) X-ray scattering factors computed from

numerical Hartree-Fock wave functions. Acta Crystallogr ., A24 , 321-324.

Doyle, P. A., and P. S. Turner (1968) Relativistic Hartree-Fock X-ray

and electron scattering factors. Acta Crystallogr ., A24 , 390-397.

Finger, L. W. (1969) Determination of cation distribution by leastsquares refinement of single-crystal X-ray data. Carnegie Inst . Wash. Year Book 67, 216-217.

Forsyth, J. B., and M. Wells (1959). On an analytic approximation to the atomic scattering factor. Acta. Crystallogr . , 12 , 412.

Hamilton, W. C. (1959) On the isotropic temperature factor equivalent

to a given anisotropic temperature factor. Acta crystallogr ., 12 , 609-610.

Levy, H. A. (1956) Symmetry relations among coefficients of the

anisotropic temperature factor. Acta Crystallogr . , 9_, 679.

Nelmes, R. J., and Thornley, F. R. (1973) Anisotropic extinction in

a synthetic boracite. To be published.

Zachariasen, W. H. (1967) A general theory of x-ray diffraction in

crystals. Acta Crystallogr . , 23 , 558-564.

40
Appendix A

Overlay scheme for RFINE

When program RFINE is run under operating systems which charge

according to the product of core allocation and CPU time, substantial

savings can be achieved by using an overlay structure which causes only

that portion of the program which is actually needed to be resident in

core. Furthermore , there can be large savings if the core allocated

for the dependency list and the normal equations matrix can be adjusted

as appropriate to a particular problem. On UNIVAC machines operating

under EXEC 8 this latter function is accomplished dynamically at run

time by means of the subroutine CORSZ, which makes use of executive

requests ER MCOREg and ER LCOREg . On other machines which don't have

a similar function capability, this can be achieved by a run time

recompilation of the short main program, with the dimension of the

array A in common block LSMAT appropriately adjusted.

The program has been run successfully with a structure consisting of

a main stem and three overlay segments as follows:

Main stem: main program, subroutines RFINE and CORSZ.

Overlay 1: Subroutines RESET, DERIVS , RDEGN , and PAREXT plus

auxilliary functions called by them, including DIJKLM.DETERM,

EL4,PIJKLM,BIJK,CIJKL,AIJ,VXV,S0LVE,EIJ,FIJ,IPAK,PDEGN,D3KDS,

D2KDS,P3KDL,P2KDL,D2KDT,TKTLS,TFM,SKTLS,CORECT,MULMM,TRNSPS,

EULER, FILLIN,SPVAL,ERRCAL.

Overlay 2: Subroutines INPUT,SFAC,RCALC,MATRIX,MODIFY, and SYMINV.

Overlay 3: Subroutines TRNFRM , B0DAN1 , and ELVIB1.

41
Appendix B BADTEA-Alternate input for bond distances a-nd angles, and thermal ellipsoids

A. IDENTIFICATION

TITLE: Crystallographic error analysis

NAME: U M BADTEA

PROGRAMMER: Larry W. Finger

DATE: December 27, 1965 (revised November 1968)

B. PURPOSE

U M BADTEA (University of Minnesota program for computing Bond

angles and Distances, and Thermal Ellipsoids with error Analysis)

will calculate bond distances, bond angles, thermal ellipsoids

and the errors associated with each computed value. The program

was converted to Fortran IV with the assistance of C. T. Prewitt.

C. USAGE

1. Input deck order:

Title card

Cell card

Function card

Symmetry card(s)

*Standard Deviations of cell parameters

Correlation matrix of cell parameters

Atom cards

*Temperature Factdr Standard Deviations

*Temperature Factor Correlation matrix

*Atomic Position Standard Deviations

*Atomic Position Correlation Matrix

*These data may not be needed. The next section describes in

detail the conditions requiring these input data. All input

from Atom Cards to the end of the deck may come from a modified ORFFF tape.

2, Description of input deck:

a. Title Card FORMAT (18A4)

Columns Description

1-72 Identifier for problem. This should

be any title which identifies the

compound

.

42
b. Cell Card FORMAT (6F10.5)

Columns

1-10

11 - 20

21 - 30

31 - 40

41 - 50

51 - 60

Description

o

a_ axis length, in A

b_ axis length,

c_ axis length* a in degrees

Function Card

Columns

1-4

5-8

9-12

13 - 16

17 - 20

24

31 - 40

41 - 50

51 - 60

d. Symmetry Cards

Columns

1

- 3

5 - 6 8 - 9 11 - 13

15 - 16

18 - 19

21 - 23

25 - 26

28 - 29

FORMAT (514, 3X.A1.6X, 3F10.5)

Description

Number of atoms in asymmetric unit (<60)

Centric Indicator

- Structure is centric

1 - Structure is non-centric

Bond Distance, Angle Parameter

- No angles or distances calculated

1 - Distances only

2 - Angles and Distances

Ellipsoid Indicator

- None calculated

1 - Ellipsoids calculated for anisotropic atoms

Error Indicator

- Errors not calculated

1 - Errors calculated

.

2 - Errors calculated with structure

information read from modified

ORFFE tape.

Lattice Type Code - This column should

be punched with a P, A, B, C, F, I, or R depending on the lattice type.

Minimum bond distance to be printed

Maximum distance used in angle calculation,

Maximum bond distance to be printed.

FORMAT (3 (A3, 2 (IX, A2), IX), 14)

Description

Translational part of x_' First positional part of jc' Second positional part of x_' Translational part of y_' First positional part of y_' Second positional part of y_' Translational part of z_' First positional part of z_

? Second positional part of a* 43
Columns

34

Description

End of Symmetry Deck Indicator

Blank - more symmetry cards follow

1 - this is the last card

Note: The translational parts of the symmetry operators are of the

form: bbb 1/2 1/3 2/3 1/4 3/4 1/6 or 5/6 where b signifies a blank

The positional parts are of the form: bw + w - w where w may be

X, Y or Z. Symmetric positions which are related by a center of

inversion or a multiple lattice point must hot be included since

these positions are internally generated. However, position x_ y_ z_ must be included and must be first .

e. Standard Deviations of Cell Parameters FORMAT (6F10.8)

This card and the cards in section f . below are needed

only if errors are to be calculated.

Columns

1-10

11 - 20

21 - 30

31 - 40

41 - 50

51 - 60

Description

Standard Deviation of a_ Standard Deviation of b_ Standard Deviation of c_ Standard Deviation of a in degrees

Standard Deviation of 3 Standard Deviation of y

f. Cell Parameter Correlation Matrix FORMAT (6F8.5)

Columns Corredat ion C

Card 1: 1-8 a with a_ (1.0)

Card 2: 1-8 a with b

9-16 b with b (1,0)

Card 3: 1-8 a with c_ 9-16 b with c_ 17 - 24 c_ with c^ (1.0)

Card 4: 1-8 a with a 9-16 b with a

17 - 24 c_ with a

25 - 32 a with a (1.0)

Card 5: 1-8 a with 6 9-16 b with 3 17 - 24 c with 3 25 - 32 a with 3 33 - 40 3 with 3 (1.0)

44
Cell Parameter Correlation Matrix (Cont)

.

Columns Correlatioii Coefficieixt of

Card 6: 1-8 a with y 9-16 b_ with Y

17 - 24 c_ with Y

25 - 32 a with Y

33 - 40 3 with Y

41 - 48 Y with Y (1.0)

Atom Cards FORMAT (2A3, 7X, 3F7.5, I2/6F8.5)

There should be a set of 2 atom cards for each atom

in the asymmetric unit.

Position Card:

Columns Description

1-6 Identification for atom

14-20 x_ in fractional coordinates

21-27 y_ in fractional coordinates

28-34 z_ in fractional coordinates

36 Temperature factor type

- isotropic

1 - anisotropic

Temperature factor card

1-8 6 or B 9-16 B„

17 - 24 3„

25 - 32 B??

33 - 40 B7,

41 - 48 0g

The anisotropic coefficients should be those for which the

temperature factor expression is

exp [-(h2 $ u

+ k

2

3 22 + 1

2 3 33 + 2hk3

12 + 2hl313 + 2kl3

23 )].

h. Temperature Factor Standard Deviations FORMAT (6F10.8)

These cards are needed only for anisotropic atoms when

ellipsoids and errors are to be calculated.

Columns Description

1-10 Standard Deviation of B,-, 11 - 20 Standard Deviation of B 22

21 - 30 Standard Deviation of B„

45
Temperature Factor Standard Deviations (Cont)

Columns Description

31 - 40 Standard Deviation of g_

2 41 - 50 Standard Deviation of g-„ 51 - 60 Standard Deviation of Rl"

i. Temperature Factor Correlation Matrix FORMAT (6 F8.5)

These cards are needed only for anisotropic atoms when

ellipsoids and errors are to be calculated.

Columns Correlation Coefficient of

Card 1: 1-8

Card 2

:

1-8

9-16

Card 3: 1-8

9-16

17 - 24

Card 4: 1-8

9-16

17 - 24

25 - 32

Card 5: 1-8

9-16

17 - 24

25 - 32

33 - 40

Card 6: 1-8

9-16

17 - 24

25 - 32

33 - 40

41 - 48

Each anisotropic atom has a Standard Deviations card and a Correlation Matrix.

j. Standard Deviations of Atomic Positions FORMAT (3F10.8)

These cards are needed only if bond distances or angles

are to be calculated with errors

.

46

;.. with hi (1.0)

!- 1 with

L- with R 22

3 22

(1.0)

i- . with

? 33

\\ with

; 33

With

B 33

3 33

(1.0)

I-, with

ol2

; with 2"

oo with P^ n 12

with ft 12

312

(1.0)

,

, with 3i 11 ol3

22 with

ol3

oo with ^3 .a with 37o

*3 with 13

3 13

(1.0)

with Boo o23

2 with ??3

with g"

it with 3„

zt with

"Po 23

With R 23

3 23

(1.0)
Standard Deviations of Atomic Positions (Cont)

Columns Description

1-10 Standard deviation of x

11 - 20 Standard deviation of y_ 21 - 30 Standard deviation of z_

There should be one card of this type for each atom in the

input list.

k. Atomic Positions Correlation Matrix FORMAT (10F8.5)

These cards are needed only if bond distances or angles are

to be calculated with errors. The lower triangle of this

matrix is read by rows with > each row starting on a new

card. When a row contains more than ten elements, it should

be continued on the next 'card.

Correlation Coefficient of

x_ of atom 1 with x_ of atom 1 (1.0)

x of atom 1 with y_ of atom 1

y_ of atom 1 with y_ of atom 1 (1.0)

x_ of atom 1 with z^ of atom 1

y_ of atom 1 with x_ of atom 2

z_ of atom 1 with z_ of atom 1 (1.0)

x of atom 1 with x of atom 2

y_ of atom 1 with x. of atom 2 x of atom 1 with x of atom 2

x_ of atom 2 with x of atom 2 (1.0)

This scheme should be continued until all elements of the

matrix have been punched.

Columns

Card 1:

1 - 8 Card 2:

1

-- 8

9 - 16

Card 3:

1 - 8

9

-- 16

17 - 24

Card 4:

1

- - 8

9

- - 16

17 - - 24

25 - - 32

D. METHOD

Bond distances

If X is a column matrix of fractional atomic coordinates for an atom in the asymmetric unit and Y is the column matrix of the

coordinates of a symmetry related atom, then Y and X satisfy

the following equation:

Y = RX + T (1)

where R is a rotation-like operation in that it involves axis

interchanges and T is a translation operation which includes

47
glide and screw components as well as origin shifts. Defining

Z as a column matrix whose coefficients are the differences in

fractional coordinates between an atom at X and an atom at Y,

then

Z = X - Y (2)

If G is defined as the real space metric tensor which has elements

gij = ^ . a. (3)

where the a^ s are the real space axis vectors, then G is the

following matrix:

a ab cos y

G =( ab cos y ° be cos a ) (4)

,ac cos 3 be cos a

T

If Z denotes the transpose of Z which involves an interchange of

the rows and columns of Z , it may be shown that the square of the

bond distance, s^, between the atoms is determined by the following matrix equation:

2 T

s = Z GZ (5)

In the program, all atoms within a spherical shell centered on each atom in the asymmetric unit are located and the distances

are calculated. The size of the spherical shell is determined

by the minimum and maximum distances to be considered which are

input parameters. The search for the contents of the shell is

made efficient by enclosing the central atom with a unit cell

shaped parallelopiped tangent to the sphere with all atomic

locations outside this parallelopiped excluded. The position

of those atoms which lie within the spherical shell are stored

in a table. This table also includes the atom number and the

number of the R matrix used to generate this position which

is needed to properly compute the partial derivatives of the

bond distance with respect to the positional coordinates. These

two numbers are stored in packed form in the same storage

location. Each time a new atom is generated, this table is

searched to determine whether this atom has already been included.

This search assures that atoms in special positions are properly

treated although they may be generated in several different ways

.

2 . Bond angles

:

All possible bond angles about each atom in the asymmetric unit

are computed using the table of positions prepared during the bond

distance calculation. Each pair of atoms from this table is

checked to determine whether their distances from the central atom

are within the limits, and if so, the bond angle is computed. If

48
X denotes the column matrix of the central atom with Y-, and Y2

denoting the column matrices of the peripheral atoms, the bond

angle for the angle Y^ - X - Y2 can be shown to satisfy the

following relationship

(Y -X)

T G(Y -X) cos = ^ 5 (6)

b lX b 2X

Where G is the real space metric tensor (see equation (4))

S. is the bond distance from the atom of X to the

atom at Y and

S„ is the bond distance from the atom at X to the

atom at Y„

.

3. Ellipsoids of Vibration:

The conversion of anisotropic thermal coefficients is a problem

of finding the characteristic values and characteristic vectors

of a particular matrix. Note that characteristic value, proper

value, principal value and eigenvalue are equivalent terms. The

matrix equation that must be solved is the following:

(B - AG

_1

)Q = (7)

where B is the matrix of anisotropic temperature coefficients

X is a characteristic value of the equation

G is the reciprocal space metric tensor with elements

g.. = b. • b. where the b's are the reciprocal space axis

vectors and

Q is a column matrix of the coefficients of the characteristic

vector. This is a vector in reciprocal space.

If equation (7) is premultiplied by G, it is transformed into the

standard form of a characteristic value problem.

(GB - XI) Q = (8)

where I is the identity matrix and

G is the real space metric tensor defined in equations (3)

and (4).

Equation (8) is a set of homogeneous equations of order 3 which

have a solution if the determinant of the coefficients is zero. Thus the characteristic values may be found from the equation

49
det (GB - Al) = (9)

Solution of this equation gives three values of A which

satisfy equation (8) . In order to have a meaningful physical

situation, it is necessary that the A's which are the roots of

equation (9) be real and positive. If they are not, the set

of temperature coefficients is said to be non-positive definite.

This is sometimes referred to as complex amplitudes of vibration

or negative temperature factors.

Each of the roots of equation (9) can then be inserted into equation (8) and the resulting equation solved for the characteristic

vector by the method of cofactors. After the coefficients of

Q are determined, they may be used to form the reciprocal space

vector q_ and the angle, 0^, between this vector q_ and the real

space axis vector a^ may be found by forming the dot product

of the two vectors

:

% • £= lajlsj cos

± (10)

This equation may be simplified since

%•£=%• fajki +

^2-2 + q 3-3 ) (11)

and

thus

, 1 for i = j , 10 v a. . b, = w. . = _ ,. ./. (12)

-d. ^ ij for i + j

a. = q, (13)

Equation (12) is the definition of the reciprocal space axis

vectors. Equation (10) can now be rewritten

cos Q ± = q

i (14)

IsJIal

The root-mean-square amplitude of vibration along the i'th characteristic vector is called y and is found from the i'th root, A of

equation (9) by the following relationship:

i±

= CX±

4. Error Analysis;

y, = (A 4 /27T 2

) 1/2

(15)

If y is a computed value which is a function of experimentally

determined variables x,12, x„,....x such that: n

y= y (xj^, x

2

Xj^) (16)

50
2 then the variance of y, denoted by a is found from the equation

n n

a = E I 3y 9y e s ,,_,. y • t

• i a T~~ cov(x x.) (17) 1=1 J1=1 dx.dx.

i

11

j

J

where cov(x ,x.) is the ij'th element in the variance-covariance

matrix of the x's.

In crystallographic calculations, the variables for which the

variance-covariance matrix elements are known include the unit

cell parameters, the atomic positions and the temperature factor

coefficients. If the dependence of the bond distance on temperature factors is neglected, as is done in this program, the error

in the bond distance will depend only on the cell parameters and

the atomic positions. Similarly the ellipsoids will depend only

on the unit cell parameters and the temperature factor coefficients.

Thus, for crystallographic problems, equation (17) may be rewritten

in the form:

a

2 = £ I r-* %— cov(a.,a.)

y j i i 3 aJ 9a. 1=1 j=l i j

(18)

where the a_' s are the unit cell parameters and the z_'s are the

atomic parameters. The variance-covariance matrix is the inverse

of the normal equations least squares matrix. The error formula

in equation (18) is a "full matrix" error formula. The so-called

"diagonal" form includes only the terms in equation (18) which

have i=j . This approximation usually gives a smaller error. All

partial derivatives required for computing the errors are analytically evaluated in this program. This procedure is better than

numerical differentiation because it is more accurate, easier and

faster to do and allows singularities in the derivatives to be

detected easier. The necessary derivatives are evaluated using

the chain rule of derivatives.

E. OUTPUT

1. General Output:

The first output consists of the information contained on the

Title Card. Then the real space axis lengths, the cosines of the

real space angles , the reciprocal space axis lengths and the cosines of the real space angles, the reciprocal space axis lengths

and the cosines of the reciprocal angles are printed. The real

space cell volume is also printed. Then the lattice type and

51
centric indicator is printed followed by the symmetry operations.

Next, if errors are to be calculated, the standard deviations

of the cell parameters and the cell parameter correlation

matrix are printed. Then the atomic positions and temperature

factors are printed. If ellipsoids and errors are to be calculated, the standard deviations of the thermal parameters and

temperature factors are printed. If ellipsoids and errors are

to be calculated, the standard deviations of the atomic positions

and the correlations matrix for the positions are then printed.

This completes the general output.

2. Bond Distance and Angle Output:

For each atom in the asymmetric unit , all atoms within the bond

distance limits are generated. These atoms are added to a table

of generated positions. For each bond distance within the allowable range, the atom identifier, the fractional coordinates,

the bond distance, the error, if calculated, and the number of

this atom in the table are printed. In the bond angle section

of the program, all atoms except those in the asymmetric unit

will be referenced by this atom number rather than by coordinates.

This number makes it easy to determine quickly what angle has

been calculated.

The bond angle section output consists of the atom identifier

and coordinates of the central atom which is one of those in

the asymmetric unit. The peripheral atoms are each referenced

by an atom identifier and an atom table number. The bond angle

and its error, if calculated, complete the output for each angle.

3. Ellipsoid of Vibration Output:

For each anisotropic atom, the equivalent isotropic B and its error are calculated and printed. Then for each of the characteristic values, the r.m.s. amplitude of vibration and the

angles that the associated characteristic vector makes with the

real cell axes as well as the errors in these quantities are printed. No output occurs for isotropic atoms.

A. Page Estimates:

The main program will produce approximately one page of output

plus 1/2 page per atom if errors are to be calculated. The

bond distance and angle subroutine will require about one page

for each 60 distances or angles calculated. However, the user should keep the bond distance limits as small as possible since

if n distances are within the limits, the number of angles com- puted will be approximately 1/2 n . The ellipsoids of vibration subroutine will require one page for each three atoms.

52
PROGRAM ERRORS AND ERROR STOPS

1. There are two kinds of errors which return control to the

monitor through a STOP statement. If a symmetry card cannot

be interpreted because the information is not of the correct

form or because it is punched in the wrong columns, the programs executes a STOP 11 statement and returns to the monitor.

The card in error will be the last symmetry qard listed on the

output. If the user attempts to read more than 24 symmetry

cards, the program will return to monitor with a STOP 22 instruction. A different kind of error can occur in the bond distance

calculation. If the distance limits are such that more than

1000 atom positions are generated, the program prints a diagnostic message and returns to the main program. This error

does not terminate the program and ellipsoids of vibration may

be calculated correctly after this condition occurs. However,

bond angles will not be calculated.

COMMON VARIABLES AND DESCRIPTION

XYZ(3,60) - Fractional coordinates of atoms in asymmetric unit

BETA(6,60) - Temperature factor coefficients

A (3) - Real cell axis lengths

ANG(3) - Cosines of the real cell angles

AS (3) - Reciprocal axis lengths

CS(3) - Cosines of the reciprocal cell angles

ATOM(2,60) - Atom identifiers

SIGA(6,6) - Variance-covariance matrix for real cell parameters

SIGB(6,6,60) - Variance-covariance matrix for thermal coefficients

RS(3,3,48) - Rotational matrix part of symmetry operator

TS(3,48) - Translational part of symmetry operator

TP(4,3) - Location of lattice points in fractional coordinates

ITF(60) - Temperature factor type indicator

CORR(16290) - Variance-covariance matrix for atomic positions.

This is stored in packed form.

XYZP(3,1000) - Fractional coordinates of atoms generated in the

bond distance subprogram.

IATP(IOOO) - Atom identifier of atoms generated in bond distance calculation.

NA - Number of atoms in asymmetric unit

NSS - Number of symmetry operations

VOL - Volume of real space unit cell

DLIMIT(3) - Array of bond distance limits

PI - The value of the constant it G(3,3) - Real space metric tensor

NLP - Number of lattice points in the unit cell

IERR - Error calculation indicator

TITLE (18) - Alphanumeric title information

53
H. PROGRAM NAMES AND DESCRIPTION

1. U M BADTEA:

This is the main program which handles the input and calls

the bond distance and ellipsoid of vibration programs.

2. BODAN(IBOD):

This subprogram calculates bond distances and errors and if

the argument IBOD is equal to 2 , calculates the bond angles

and errors

.

3. ELVIB:

This subprogram calculates the ellipsoid of vibration parameters and associated errors.

4. VARIAN(A,B,C,D,E,II):

This subprogram calculates the variance of a computed value

where the partial derivatives of the value with respect to the cell parameters are stored in array B, the partial derivatives with respect to the structure parameters are stored

in array C, the cell convariance matrix is stored in array

D, the structure parameters covariance matrix is in array E,

the argument II contains the atom number when the variance

of ellipsoid parameters is calculated otherwise it contains

1 and the computed variance is output in argument A.

5. CELLD(B,A,AN,R,CS,VOL)

:

This subprogram computes the partial derivatives of the re- ciprocal cell parameters with respect to the real cell

parameters. These derivatives are used in computing the

ellipsoid errors. Array B contains the partial derivatives

after execution. Arrays A, AN, R and CS are the input values

of the real cell axis lengths, the real cell cosines, the

reciprocal axis lengths and cosines of the reciprocal cell

angles respectively. The argument VOL is the input value of

the real cell volume.

6. DERDET(A,B,C) :

This subprogram calculates the partial derivatives of matrix

B with respect to the thermal coefficients and stores them

in array A. Array C contains an intermediate set of partial

derivatives.

54
7. C0RR1(SIGZ,NN,II,JJ,C0RR,KK,LL,MM)

This subprogram prepares the atomic positional correlation

matrix for atoms II and JJ from the packed matrix CORR of rank

MM and stores it at the KK,LL position of the array SIGZ which

is of rank NN.

SYSTEM REQUIREMENTS

U M BADTEA requires two tape units, a standard input tape and

a standard output tape. All input is BCD from Logical Unit

Number 5 while output is BCD to Logical Unit Number 6. The

program requires approximately 32K storage locations. If the

modified ORFFE tape is to be used, it is read from logical

unit 4.

55
Appendix C

LISTINGS OF PROGRAMS

Fortran listings of all programs and subprograms are included

herein. Example runstreams, with the output they produce, are available

on request from either author.

56
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128
NBS-114A (REV. 7-73)

1. PUBLICATION OR REPORT NO. WBS TN-85I+

U.S.' DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE A System of Fortran IV Computer Programs for Crystal

Structure Computations

5. Publication Date February 1975

6. Performing Organization Code

7. AUTHORISE Larry W. Finger and E. Prince 8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS NATIONAL BUREAU OF STANDARDS DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No.

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Same

13. Type of Report & Period Covered

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15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.)

This report gives detailed descriptions and instructions for use of a system

of programs for crystallographic calculations, including least-squares refinement

with generalized systems of constraints, calculation of bond distances and angles

with errors, Fourier synthesis, plotting of contours in Fourier maps, and preparation of structure factor tables for publication.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons) Computer programs; contour plotting; constrained refinement; crystallographic

calculations; Fourier section; Fourier synthesis; least squares.

18. AVAILABILITY [j£ Unlimited

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