Introduction
UNEX is a programming environment for investigation of molecular structure. I develop new and existing experimental methods and combine them in order to increase the accuracy and precision of results. At the current stage the full support of gas electron diffraction (GED [1, 2, 3]) investigations is provided, from the calibration of instruments and data reduction to the refinement of molecular structure. Additionally, rotational constants can be used solely or in combination with GED data for determination of molecular geometry.
Cite UNEX as
Yury V. Vishnevskiy, 2021, UNEX 1.6, http://unexprog.org (access date) 
UNEX is not related to any version of KCED or other programs. This is an independent research project. Nevertheless, many implemented in UNEX methods and algorithms are based on investigations of other authors. 
The aim of this manual is not to teach how to investigate molecules but only to describe UNEX commands. Please remember that incorrect settings and inappropriate usage of different methods can lead to incorrect results! 
General ideas

Exploration of experimental possibilities for investigation of molecular structure.

Development of the gas electron diffraction (GED) method and its automation.

Development of spectroscopic methods for molecular structure investigation.

Providing the ability to carry out very accurate studies due to extended facilities and flexibility of the program interface.

Elaboration of joint methods for molecular structure investigations.
Capabilities

Investigation of molecular structure by means of GED method.

Refinement of geometrical parameters from rotational constants.

Combined refinements on the basis of GED data and rotational constants.

Static and 1Ddynamic models in GED.

Numerical and analytical parametric forms of potential function.

Support for relaxation of geometrical parameters, amplitudes and corrections.

Modelling of any mixtures of molecules with static and dynamic GED models.

Definition of molecular geometry in terms of Zmatrix.

Both internal geometrical parameters and Cartesian coordinates can be used as parameters.

Support for dummy atoms.

Powerful methods for functional minimization.

Robust minimization with iteratively reweighted experimental data.

Automatic calculation of uncertainties for dependent parameters.

Global minima search by grid scanning and MonteCarlo method (randomization).

Multidimensional scanning.

MonteCarlo calculation of total uncertainties for refined parameters.

Automatic determination of point group symmetry.

Methods for modeldependent multiplicative and additive GED backgrounds using cubic splines and polynomials.

Automatic calculation of scattering factors and atomic intensity.

Unlimited amount of GED data can be used in refinements.

Nonequal steps for intensity curves.

GED data reduction on the basis of TIFF images (8 or 16 bit grayscale) of diffraction patterns.

Calibration of electron wavelengths using gas standards: benzene C_{6}H_{6}, CO_{2}, CS_{2}, CCl_{4}.

Refinement of sector function from gas standard data and sector images.

Calibration of scanners.

Refinement of response functions for detectors.

Statistical thermodynamics.

Flexible and convenient input format.

Efficient usage of SMP (multiprocessor/multicore) systems.

Versions of program for Linux, Windows, macOS, OS/2, FreeBSD.
Conditions of program usage
UNEX is distributed for free. Conditions of its distribution are of "AS IS" type.
You use this program on your own risk.
To download and use UNEX you must accept the license agreement, see files license.html
and license.pdf
.
Malfunction
If you think you have found a bug or some incorrectness in UNEX the first thing to do is to check everything. Second, make sure you are using the latest version of UNEX1. If you still cannot find the source of the problem it is possible to write an email to the main developer of UNEX (see below). It is recommended to isolate the problem and to send a smallest possible input file generating incorrect result(s). Do not forget to provide UNEX version number and your operating system type/version.
Questions and comments
If you have questions or comments you can write an email to the main UNEX developer, Dr. Yury V. Vishnevskiy, using one the following addresses
yu.v.vishnevskiy@gmail.com yu.v.vishnevskiy@yandex.ru yu.v.vishnevskiy@mail.ru yu.v.vishnevskiy@web.de
Usage
General conventions
Reading this manual you can meet numbers expressed using scientific notation and symbol e
, for example 5.5e13
.
This corresponds to base10 exponentiation, so the number above is equivalent to 5.5×10^{13}.
Note, UNEX can read numbers in scientific notation.
In many examples you can see tripple dots …
. This does not reflect the input format but just indicates that further data may follow.
Otherwise the examples would be too long.
Installation
UNEX is a command line program, which is distributed together with some supplementary programs and other files.
For the installation there is no need to do any special actions.
Just copy files to any suitable directory. It is recommended to place all the files to one dedicated directory listed in the environment
variable PATH
so that the executables can be called from everywhere.
In this case it is also easier to update UNEX just by replacing old files in this particular directory.
Checking for new versions and updating can be also performed automatically by starting update.sh
(in Linux and macOS) or update.cmd
script (in Windows).
Basics
In order to use UNEX an input file should be prepared first (for details see below). Starting UNEX without any input file prints
general information about UNEX usage. In simplest case the only command line parameter is the name of input file.
After starting UNEX the input file remains unmodified and an output file is created with results.
If you do not define the output file name explicitly, UNEX automatically creates one named as the input file with added underline
symbol, a number and an extension .log
. The number indicates the version of the output and it increases with each run of UNEX
with the same input file. Thus, output files are never overwritten. Alternatively, you can define in command line the name for the
output file explicitly. All command line options can be printed by running UNEX with h
or help
options.
Syntax
Input for UNEX are usual text files. They contain control commands and data fields. Each type of command has its own syntax. Data fields are needed for introducing any input information. For arrangement of data fields socalled tags are used, i.e. a logically complete fragment of data is placed between two certain words which are called tags. In general they may contain any letters. A possible way is to use simple and clear constructions like
<my_info> Here goes my info/data... </my_info>
Here <my_info>
and </my_info>
are opening and closing tags, respectively.
In spite of considerable number of different commands and field types, all these elements follow similar pattern,
which is easy to understand and to use.
The sequence of commands is important.
It is not recommended to use very long strings in input files.
The total length of strings with commands is limited to 500 symbols.
Any string in input file can be commented out.
For this in the very first position of the line you should type symbol #
.
It is also possible to place a comment in the end of string, for example
# Run UNEX command in the next line COMMAND # This should start some procedure
Starting from version 1.61258 UNEX does not accept comment 
Control flow
GOTO
GOTO
command is used for unconditional jumps to commands coming after particular label.
Labels are defined using the command LABEL
.
The following example demonstrates the principle.
GOTO=MYLABEL COMMAND1 LABEL=MYLABEL COMMAND2
In the demonstrated code, when the GOTO
is executed, all subsequent commands (in the example only COMMAND1
) are skipped until the
required label (here is MYLABEL
) is found, which is defined in the command LABEL
. After this point the execution is continued,
so that COMMAND2
is started.
STOP
STOP
command terminates execution of UNEX.
Data input
Below are described commands used primarily for introducing and definition of data.
BASE
In most cases UNEX input files begin with introduction of basic information. For this purpose the BASE
command is used:
BASE=BASE,<BASE>,</BASE>
Here the first word BASE
is the name of the command. After the =
symbol goes the field format type (also named BASE
here).
The other two words are tags pointing to the start and end lines of the field containing basic information.
Thus, UNEX will try to find the field and read the corresponding information from the input file between the following tags
<BASE> Basic info goes here... </BASE>
BASE field can contain control keys with their values. Depending on the job type different keys can be used. Two of them are used most often: molecules
and imgfiles
.
The first is used when models of molecules are created and manipulated, for example in a structural analysis. The other is for definition of images, for example in GED data reduction.
Generally lines in a BASE field look like following: key name, space letter(s) and/or =
character, key value. Key values can be strings, integers or floating point numbers. Usually keys accept only one value with some exceptions (for example, molecules
accepts list of strings).
Sometimes it is useful to apply different settings for different stages of UNEX running. This can be achieved by calling BASE
command several times, for example
BASE=BASE,<BASE1>,</BASE1> #MaxIter is equal here to 20 BASE=BASE,<BASE2>,</BASE2> #MaxIter is equal here to 30 <BASE1> MaxIter=20 </BASE1> <BASE2> MaxIter=30 </BASE2>
Below is the list of keywords valid in BASE field:

Basic keywords
 molecules

Name(s) of molecule(s) participating in a model. In simple cases only one molecule is defined here. Sometimes several molecules must be defined. For GED this corresponds to a model of a mixture of molecules. Note, UNEX expects a special field for each molecule defined here. The opening and closing tags must correspond to the name of the molecule, for example:
<BASE> molecules=mol1 </BASE> # Special field for mol1 <mol1> mol1related info goes here... </mol1>
 imgfiles

Names of image files to be processed. UNEX can handle uncompressed Intel TIFF 8/16bit grayscale files. As in the case of
molecules
special fields for each image are expected.

Keywords related to refinement of parameters
 MaxIter

Maximal allowed number of iterations for leastsquares method in
MINIMIZE
. The default value is 20.  damp

Damping factor in leastsquares method for scaling of parameter additions. There are three options:

damp=[number]
— constant damping factor (for example,damp = 0.5
) 
damp=linear
— damping factor increased linearly up to 1.0 in the last iteration. 
damp=sigma
— damping factor increased sigmoidally to the value of 1.0 in the last iteration; this is default.

 LsqAddTol
 LsqGrdTol

threshold values for maximal relative addition and gradient used as convergence criteria in leastsquares procedure.
 LsqFunTol

threshold value for relative functional change as as convergence criterion in leastsquares procedure.
 LsqLamMaxInc

Maximal allowed number of consecutive increments of parameter
Lambda
in LevenbergMarquardt method for minimization of nonlinear leastsquares functionals. Default value is 5.  LsqLamDecFac

Decrement factor for parameter
Lambda
in LevenbergMarquardt method. Default value is 0.1.  LsqLamIncFac

Increment factor for parameter
Lambda
in LevenbergMarquardt method. Default value is 10.0.  MinOrthoParams

Turns on (
=1
) or off (=0
, default) refinement of orthogonal linear combinations of parameters.  GedVarAmplScale

Turns on (
=1
) or off (=0
, default) refinement of scale factors for ED vibrational amplitudes. Ratios of amplitudes within each group remain constant if scales are refined (GedVarAmplScale=1
), otherwise differences between amplitudes remain constant within one group.  MinMethod

method for minimization of functional value in
MINIMIZE
command. Three options available:
lsq
— leastsquares method (LSQ). 
goldsec
— golden section method. 
lsqgoldsec
— combination of leastsquares and golden section (activates automatically when LSQ fails) methods. This is default.

 MinRobMaxIter

Maximal allowable number of iterations in Robustminimization of the
ROBUSTM
command. The default value is 10.  MinPrintEllipsoid

Turns on (
=1
) or off (=0
, default) printing of functional (hyper)ellipsoid at the end ofMINIMIZE
procedure.  PrintSearchResults

Controls whether full table of results is printed (
=1
, default) or not (=0
) afterSEARCH
command.  ShowSearchInfo

Turns on (
=1
, default) or off (=0
) printing of progress status and speed ofSEARCH
procedure.  SearchTime

Total allowed time for
SEARCH=RAND
command in seconds. Default value is 3600.0, i.e. one hour.  SearchRngSeed

Seed (integer number) for random number generator used in
SEARCH=RAND
command. Default value is 0, meaning automatic generation of seed.  MaxDerErr
 MinDerErr

Maximal and minimal errors in numerical differentiation. By default, the maximal error is one order of magnitude less than the threshold value for derivatives (see stopping criteria for minimization). Minimal error is by default three orders of magnitude less than maximal error.
 PrintEsdFactor

Factor for printed standard deviations of parameters. By default it is 1.0.
 RegAlpha

Factor for the regularization term in leastsquares functional. Default value is 1.0.
 RotConstAlpha

Factor for the term with rotational constants in leastsquares functional. Default value is 1.0.
 RestrGeomAlpha

Factor for the term with restraining geometrical parameters in leastsquares functional. Default value is 1.0.
 DepSigmaCovar

Turns on (
=1
, default) or off (=0
) the usage of covariation matrix in calculations of standard deviations for dependent parameters.  MinAbsWeighting

Turns on (
=1
) or off (=0
, default) using absolute weights in leastsquares method for calculation of standard deviations of refined parameters. The weights are calculated from standard deviations of experimental data as .  CalcFuncProportion

Controls calculation of contributions from different parts of leastsquares functional into refined parameters using W12 [4] and W2 [5] methods. This can be done at the end of the
MINIMIZE
procedure. Possible values for this keyword are0
(calculate nothing, this is default),1
calculate values using W2 method,2
use W12 method,3
calculate values using both methods.  MinSigmaExcludeFunc

This keyword defines which types of LS functional should be excluded in calculation of experimental errors for refined parameters. The idea and the method are described in [4]. Calculated values can printed at the end of
MINIMIZE
procedure (requested by keywordCalcFuncProportion
), by calling PRINT=GEOMFUNCW2 and PRINT=GEOMFUNCW12. Particular types of LS functional are defined as bits at different positions but the keyword requires definition of respective integer values:1
,2
,4
and8
forGEDSMS
,ROTCONST
,REGPRM
andRESTRGEOM
, respectively. Combinations of types are also defined as integer, which must be the result of the respective bitwise inclusive OR operation. For example, the combination ofREGPRM
andRESTRGEOM
is defined as12
. This is the default value for this keyword.  MinPrintSensibility

Turns on (
=1
) or off (=0
, default) the first order sensibility analysis inMINIMIZE
procedure.

MonteCarlo simulations
 MCMaxIter

Maximal allowed number of iterations in MonteCarlo procedure
MCMIN
. The default value is 100000.  MCsMsSpread

Default standard deviation for sM(s) data used in MonteCarlo procedure
MCMIN
.  MCSimulateData

Turns on (
=1
) or off (=0
, default) simulation of experimental data on the basis of model by adding some random noise.  MCRandData

Turns on (
=1
) or off (=0
, default) randomization of data used for refinement of model.  MCRandParams

Turns on (
=1
) or off (=0
, default) randomization of parameters of model.  MCRandDataSeed
 MCRandParamsSeed

Seeds for random number generators used for data and parameters, respectively. By default they are initialized to random values based on current time and process ID. If you want deterministic results you have to define seeds with these keywords.
 MCPrintParams

Print (
=1
) or not (=0
, default) randomized values of parameters to output file.  MCCalcRotConsts

Calculate (
=1
) or not (=0
, default) rotational constants during simulation.  MCPrintInterResults

Number of steps to be done for printing intermediate results of simulation and testing for convergence. Default is 1000. Zero means no printing of intermediate results and no testing for convergence.
 MCUseExtData

A keyword to allow (
=1
, default) or not (=0
) using additional precalculated results of MonteCarlo simulations read in with theMCREAD
command.  RegAlphaMCgroup

Group number for
RegAlpha
parameter in MonteCarlo simulations.  RegAlphaMCmin
 RegAlphaMCmax

Maximal and minimal allowed values for
RegAlpha
parameter in randomization.  RotAlphaMCgroup

Group number for
RotConstAlpha
parameter in MonteCarlo simulations.  RotAlphaMCmin
 RotAlphaMCmax

Maximal and minimal allowed values for
RotConstAlpha
parameter in randomization.  MCSetStdDev

Turn on (
=1
, default) or off (=0
) assignment of determined in the simulation standard deviations to respective refined parameters.  MCApplyBias

Turn on (
=1
, default) or off (=0
) application of determined in the simulation biases to respective refined parameters.  MCAmplTmplExr
 MCCorrTmplExr

Percentages of extensions applied to ranges of amplitude and correlation values printed by
PRINT=AMPLMCTMPL
andPRINT=CORRMCTMPL
, respectively. Default value is 30.0 for both keywords.

ED intensity
 IModel

Model for the total ED intensity. At present this controls the type of background used for calculation of the total intensity. The available options are
mbgl
,a1bgl
anda2bgl
. The default option ismbgl
. For details see Models for ED intensity.  ImolAnhTermModel

Model for anharmonic terms of distances in calculation of ED molecular intensity. Available options are
Asym
(asymmetry parameters, default) andMorse
(Morse parameters). For details see chapter Models for ED intensity.  GFsmin
 GFsmax

Minimal and maximal svalues (in Å^{1}) for precalculated scattering factors. Default values are 0.0 and 60.0.
 GFstep

Step size on the sscale (in Å^{1}) for precalculated scattering factors. Default value is 0.1.
 BglApproxType

Type of approximating function for background lines. Available options are

Spline
— cubic spline, this is default, 
Polynom
— simple polynomial, 
ChebPolynom
— orthogonal Chebyshev polynomial.

 BglNinflThr

Global threshold number of inflection points for ED background lines. The default value of this number is 3.
 BglPolPow

Global polynom power for ED background lines. By default it is 3.
 BglPrintRaw

Turns on (
=1
) or off (=0
, default) printing of raw (before smoothing) background in theBGL
procedure.  RespFuncPolPower

Degree of polynomial function used in refinement of response function for ED detector in
RESPFUNC=CALCIDS
procedure. Default value is 10.  SFactorsDBVer

Version of database with atomic scattering factors. Old values are used when
SFactorsDBVer=1
. By defaultSFactorsDBVer=2
, this corresponds to factors published in [6].  BglSmoothReduced

Turns on (
=1
) or off (=0
, default) smoothing of the reduced (divided by the sector function and atomic scattering) multiplicative background inBGL
command.  BglRefScaleMaxIter

Maximal number of iterations (by default 0) in the refinement of the scale factor for sM(s) in the
BGL
procedure. For the case of additive backgrounds this keyword just turns on (any positive value) or off (=0
) the refinement of the t factor for the total intensity.  BglRefScaleTol

Relative change (0.001 by default) in scale factor as convergence criterion for procedure of refinement of sM(s) scale factors in the or tfactors of the total intensity in
BGL
procedure.  MinDs

Minimal allowed difference between svalues. Default value is 1e7 Å^{1}.
 BglPSDStrRmin
 BglPSDStrRmax

Minimal and maximal interatomic distances in the molecular model when power spectral density of background line is analysed. By default these parameters are negative, which indicates automatic determination of the corresponding values.
 BglPSDStrRminShift
 BglPSDStrRmaxShift

Shifting factors for the automatically determined minimal and maximal interatomic distances in the molecular model for the analysis of background line PSD. Default values are 0.2 and 1.0, respectively.
 BglPrintPSD

Turns on (
=1
) or off (=0
, default) printing of the power spectral density for background and experimental intensity in theBGL
procedure.

ED Sector
 SecModelType
 RegSecModelType

Type of model for sector function and regularization sector function, respectively. Possible values are
rpn
,sinpn
andconst
. For explanation see below section related to introduction of sector functions. Default isrpn
.  SecPrmA
 RegSecPrmA

Parameter A in the model for (regularization) sector function. Default value is 2π.
 SecPrmN
 RegSecPrmN

Parameter n in the model for (regularization) sector function. Default value is 3.0.
 SecPrmRmax
 RegSecPrmRmax

Parameter r_{max} (in mm) in the model for (regularization) sector function, see chapter ED sector function. The default value is 100.0.

ED Standards
 DefStdType

Default type of standard if it is not indicated explicitly in the input of ED intensities. Possible values are
CCl4
,C6H6
,CO2
andCS2
. Default isCCl4
.  StdLsqMaxIter

Maximal number of iterations in leastsquares refinement of parameters from ED gas standard data. The default value is 100.
 StdRegSecAlpha

Factor for regularization of refined sector function. By default it is 0.0, which indicates the absence of regularization.
 StdRegBglAlpha

Factor for regularization of refined background functions. Default value is 0.0.
 StdRegBglValue

Regularizing value for background. Default value is 0.0.
 StdVarLambda
 StdVarSector
 StdVarScale
 StdVarBgl

Keys turning on (
=1
) or off (=0
), refinement of electron wavelength, sector function, scale factors and additive background, respectively. By default, everything is on, except forStdVarLambda
.  StdPrintCorrs

Enables (
=1
) or disables (=0
, default) printing correlations between refined parameters inSTANDARD
procedure.  StdRegDBglAlpha

Prefactor for leastsquares term calculated as sum of squares of second derivatives of background lines. By default this factor is zero meaning that this term is not included.
 StdScanIter

Number of iterations in scanning of electron wavelength in
STANDARD
. By default it is zero, i.e. scanning is not performed.  StdScanLamMin
 StdScanLamMax

Minimal (default value is 0.039 Å) and maximal (default value is 0.120 Å) values of electron wavelength in scanning.
 StdRefLamMaxIter

Maximal number of iterations in refinement of electron wavelength. Default number is 50.
 StdRefLamTol

Convergence tolerance in relative change of refined lambda. Default value is 1.0e4.
 StdInitSecStep

Step size in mm for automatically initialized reduced sector function in LSQ refinement in
STANDARD
. Default value is 1.0 mm.  StdInitSecMin
 StdInitSecMax

Minimal and maximal allowed rvalues of the autogenerated reduced sector function for refinement in
STANDARD
. Default values of these keywords are negative, which means that the corresponding parameters should be determined automatically.  StdInitRefBgl

Initialize additive background using
BGL
procedure before LSQ refinement inSTANDARD
. Turned on (=1
) by default.  StdBglRefScaleMaxIter

Maximal number of iterations for refinement of scale or tfactors in background procedures used from
STANDARD
. Default number is 30.  StdCorrNegRefBgl

Correct refined in
STANDARD=LSQ
background if it gets negative. By default this is turned off (=0
).

ED Radial distribution functions
 RdfType

Method for calculation of radial distribution curves. There are three options:
old
,classic
(this is default) andmodern
. For details see description of theRDF
command.  RdfMultR

The key determines whether the Fourier curve is multiplied (
=1
, default) or not (=0
) by r. Multiplication by r produces a better approximation to P(r) function, but also increases difference curves.  RdfRto

Maximal value of r (in Å), for which radial distribution curves are calculated. By default
RdfRto
is determined automatically depending on the maximal interatomic distance in the model.  RdfRdr

Step size along rscale for calculation of radial distribution functions. Default value is 0.01 Å.
 RdfPruneRlen

Allowed distance between points along the radial distribution function. Default value is 0.02 Å. The value 0.0 turns off the prunning.
 RdfAdaptiveR

Turns on (
=1
) or off (=0
, default) the usage of an adaptive method for choosing points on the rscale for calculation of radial distribution functions.  RdfDamp

Coefficient in an exponential function used for multiplying sM(s) curves before Fourier transformation. By default it is calculated according to , where s_{max} is the maximal svalue of the transformed sM(s) function.
 RdfDivGf

This key enables (
=1
, default) or disables (=0
) the division of sM(s) curves by a gfunction (by default corresponding to a term with maximum contribution) before Fourier transformation.  RdfDivGfAtoms

Types of atoms (for example
=C,O
), for which the corresponding gfunction must be calculated and used for modification of sM(s) before Fourier transformation ifRdfDivGf=1
. By default this is initialized automatically so that the pair of atoms available in molecule(s) of the model have highest atomic numbers.  RdfTermDif

Influences results of the
PRINT=GRAPHTERMS
command. This parameter defines maximal allowed difference between distances of degenerate terms in calculation of their contributions. By default this key is negative, which turns off the searching for degenerate terms.  RdfTermDivAmpl

Influences results of the
PRINT=GRAPHTERMS
command. Turns on (=1
, default) or off (=0
) division of calculated term contributions on the respective amplitudes.  RdfIntegMethod

Method of numerical integration:
trapezoidal
(default, fast) orromberg
(slow but potentially a bit more accurate).  RdfCalcStdevs

Turns on (
=1
) or off (=0
, default) calculation of standard deviations for experimental radial distribution functions.  RdfMCEsdIter

Number of iterations in MonteCarlo procedure for calculation of standard deviations for experimental radial distribution functions. Default value is set 0, turning off this procedure.
 RdfPrintEsdIterval

Time interval in seconds for printing progress of calculations of RDF standard deviations. The default value is 60 seconds.
 RdfNconcat

Number of common points (by default 11) for experimental and model sM(s) when they are concatenated in RDF procedure. The larger this number the greater is the overlap of the sM(s) curves.

ED Data reduction
 IntScanIter

Maximal number of iterations in the leastsquares procedure of the
IMAGE=INTSCAN
command. The default value is 50.  WriteAsymBglImg
 WriteCurveImg
 WriteWeightsImg

Turn on (
=1
) or off (=0
) the creation of image files representing refined asymmetric additive background, intensity curve and weights of original data points. By default only images of intensity curves are created.  ImgPrintIntR

Refined in data reduction intensity curves are printed with corresponding rvalues (
=1
) or svalues (=0
, default).  ImgPrintBlc

Print optical density (
=1
) or relative electron scattering intensity (=0
, default) in the end of theIMAGE=INTSCAN
command.  ImgPrintIntCorrs

Determines whether correlations between refined intensity values should be printed (
=1
) or not (=0
, default).  ImgPrintAllCorrs

Determines whether correlations between all refined in data reduction parameters should be printed (
=1
) or not (=0
, default).  ImgPrintDataHistogram

Determines whether histogram of image should be printed (
=1
) or not (=0
, default) inIMAGE=INTSCAN
procedure.  IntScanRobNum

The parameter in the method of Tukeys bisquares weights used for rejection of image data points in the
IMAGE=INTSCAN
command. The default value is 4.685.

Other
 jobname

Any string, describing the input for UNEX. This is optional.
 SVDTol

Factor for calculation of threshold value for minimal singular number in SVD decomposition procedure. The threshold value determined as product of this factor and maximal singular number. Singular numbers less than the threshold value are discarded. Default value is 10^{3} times machine double precision, which usually corresponds to 2e13.
 SVDMaxIter

Maximal number of iterations in SVD procedure. Default value is 30.
 MoveWedArea

Depending on the setting of this parameter shapes and locations of areas in optical wedges are refined (
MoveWedArea=1
, default) or remain fixed (=0
) in theWEDGE=AUTO
andWEDGE=MANUAL
commands.  PotEUnits

Potential energy units on input, when data introduced in numerical form (
POTENTIAL=mol,PTL1,...
). Possible values are
au
— atomic units 
kcal
— kcal/mol 
kJ
— kJ/mol, default

 RotConstUnits

Input units for rotational constants. Possible values are

cm
— cm^{1} 
MHz
— MegaHertz, default

 Temperature

Temperature in Kelvins. This parameter affects calculations related to GED with dynamic models and calculations of thermodynamic functions with
CALCTHERMO
command. The default value is 298.15 K.  Pressure

Pressure in standard atmospheres (atm). It is used in calculations of thermodynamic functions. The default value is 0.986923267 atm.
 Wplot
 Hplot

Width and height (expressed as number of characters) of pseudographics produced by the
PLOT
command.  CpuNum

Number of threads used for parallel calculations in UNEX whenever possible. By default UNEX uses all available in system processors/cores.
 PrintMainInertXYZ

Cartesian coordinates of molecules are printed in system of principal axes of inertia (
PrintMainInertXYZ=1
, default) or in input/Zmatrix orientation (=0
).  MainInertOrient

Definition of the system of principal axes of inertia. Possible values are
xyz
,xzy
,yxz
,yzx
,zxy
,zyx
(the last one is default).  SymTol

Sensitivity factor for determination of symmetry elements in molecules. The default value is 1.0. The less this factor is, the more accurate must be molecular geometry.
 PrintSymUniqAtoms

Defines whether symmetrically unique atoms should be printed (
=1
) or not (=0
, default) byPRINT=SYMMETRY
command.  PrintEsdZMatrix

Defines whether standard deviations of Zmatrix parameters should be printed (
=1
) or not (=0
, default) byPRINT=SYMMETRY
command.  SmoothSplineEdge

Turns on (
=1
) or off (=0
, default) an alternative method for calculating edge points of smoothing splines.  F3cBlockCols

Number of columns for cubic force constants printed by
PRINT=F3CBLOCKS
command. The default value is 5.  GeomBondTol

Parameter to control detection of bonds between atoms. If distance between atoms is less than sum of their covalent radii plus
GeomBondTol*100%
then a bond is recognized. The default value is 0.15.  GvibSymmInput

Turns on (
=1
, default) or off (=0
) symmetrization of input ED vibrational parameters for molecules with determined symmetry.  CzmTmplStartGroup

Starting value for group numbers in
PRINT=CZMTMPL
. The default value is 1.  PrintEDIntHash

If greater than zero, print hashes of ED intensities and of other related functions in PRINT=INT, etc. This can be used for testing of data.
Molecules
For each molecule declared in BASE
a special field must be defined, which contains some general information about the molecule. The starting and ending tags of this field must be constructed as <name_of_molecule>
and </name_of_molecule>
, respectively. For example, for a molecule mymol
the corresponding field is
<mymol> mymolspecific info goes here... </mymol>
Possible keywords in field of molecule:
 formula

Empirical formula of the molecule, for example
formula=C6H12O6
. This is a mandatory keyword.  molx

Mole fraction of the respective molecule in a mixture, if several molecules are defined in
BASE
. The value must be in range 0.01.0. For the last molecule listed inBASE
this keyword is irrelevant because the corresponding mole fraction is calculated automatically.  varx

Group number for mole fraction. In order to refine the mole fraction a positive integer group number must be defined using this keyword. Note, this must be unique number, since mole fractions cannot be refined in groups with other parameters.
 sing

If the molecule is a pseudoconformer in a GED dynamic model, this keyword means its degeneracy. By default it is equal to 1.
 gedmodel

GED model for the molecule. Possible values are
static
(default) anddynamic
. In case of dynamic model, pseudoconformers must be defined usingpsconfs
keyword.  psconfs

List of pseudoconformers in the dynamic model of the molecule. Syntax is the same is for
molecules
inBASE
field.  pcnum

Total number of pseudoconformers in the dynamic model of the molecule. By default this equals to the number of pseudoconformers defined using
psconfs
keyword. However, additional pseudoconformers can be automatically generated and populated ifpcnum
has larger value.  allsing

Default degeneracy for all pseudoconformers.
 PotType

Type of potential function used in the dynamic model of the molecule. Possible options are

Spline
— cubic spline, no fittable parameters. This is default. 
Cos1
— parametric function . 
Cos2
— parametric function . 
Gauss
— parametric function, sum of gaussians . 
Polynom
— parametric function, polynomial .
For details on how to introduce potential functions see Potential functions.

 PotCoefNum

Total number of parameters in potential function including free term if applicable. Makes no sense in case of splines.
 DynRelaxPln

Number of coefficients in relaxational polynomials used for generation of additional pseudoconformers. Default value is 5, which corresponds to polynomials of power 4.
 DynImolModel

Model for molecular intensity when dynamic GED model is used, i.e.
gedmodel=dynamic
. Possible options areinteg
(default) andsum
. For details see section Models for ED intensity.  SpinMult

Spin multiplicity of the molecule. The default value is 1.
 isotopmols

List of isotopologues related to the molecule. This keyword is similar to
molecules
inBASE
. Each isotopologue must have its own field just like normal molecules. Theisotopmols
is generally used for refinement of molecular structure from rotational constants of parent molecule and its isotopologues.  ExpRotA
 ExpRotB
 ExpRotC

Experimental rotational constants of the molecule.
 SigRotA
 SigRotB
 SigRotC

Standard deviations of the corresponding experimental rotational constants.
 DeRotA
 DeRotB
 DeRotC

Corrections to rotational constants. Usually B_{e}B_{0}.
 MCStdRotA
 MCStdRotB
 MCStdRotC

Standard deviations of the corresponding experimental rotational constants for the MonteCarlo method.
 MCvarx

Group number for mole fraction in MonteCarlo simulations.
 MCxMin
 MCxMax

Minimal and maximal allowed values for mole fraction in MonteCarlo simulations.
 MCwriteXYZ

Turn on (
=1
) or off (=0
, default) writing of molecular Cartesian coordinates on each step of MonteCarlo simulations to a special file. The file is created in current directory with a name consisting of the name of molecule, data seed andxyz
extension.
Images
Images are defined in UNEX similar to molecules — i.e. image file names are listed using imgfiles
keyword in BASE
.
Accordingly, for each image file can be defined a field with starting and ending tags constructed from the name of this file. For example
<BASE> imgfiles=img1.tif </BASE> # Special field for img1 <img1.tif> img1related info goes here... </img1.tif>
Valid keywords in image fields are:
 XResolution
 YResolution

Resolution of image along X and Y directions, corresponding to width and height of the image. These keywords are not obligatory since TIFF files contain this information and UNEX can read it. However, the nominal resolution values may not represent real resolution, for example due to imperfections in scanning device. In such cases true resolution can be defined explicitly with these keywords.
 Xc
 Yc

Coordinates of the center of diffraction pattern in pixels. In GED data reduction procedure these values can be further refined.
 Xs
 Ys

Coordinates of the center of rotating sector device in pixels. Similar to
Xc
andYc
these values play role in GED data reduction and can be refined. By default they are equal toXc
andYc
, respectively.  fog

Optical density corresponding to zero level of measured electron diffraction intensity. By default, 0.02.
 NozToPlate

Distance (in mm) from nozzle to detector in GED experiment.
 SecToPlate

Distance (in mm) from sector to detector in GED experiment.
 UseSector

Turns on ('=1') or off (
=0
, default) usage of sector function in data reduction of this image.  IntRfr
 IntRto

Smallest and largest distances (in mm) to the center of diffraction pattern. The diffraction pattern within this range will be used in data reduction procedure.
 IntStep

Step size for intensity curves refined from diffraction pattern in data reduction. The units for this keyword depend on the
IntStepType
keyword. IfIntStepType=sconst
step size is in reverse Angstroms, ifIntStepType=rconst
the step size is in mm.  IntStepType

Type of step increments for intensity curve in data reduction:

sconst
— constant step size on sscale, this is default. 
rconst
— constant step size on rscale.

 IntLambda

Electron wavelength (in Angstroms) for the diffraction pattern used in data reduction.
 MinT
 MaxT

Minimal and maximal level values. Only pixels with levels within this range are processed. By default these keywords correspond to the full range of possible levels.
 IntVarCentre
 IntVarSecCentre
 IntVarAsymBgl

This group of keywords is for control of data reduction. They turn on (
=1
) or off (=0
) the refinement of the center of diffraction pattern, center of rotating sector device and asymmetric background, respectively. By default all types of parameters refined. Note, ifIntVarCentre=0
then the center of sector is also fixed!  BitsPerPixel

Number of bits per pixel. Normally this is determined automatically. If not, this can be defined here. Allowed values are 8 or 16.
 ImgNbglX
 ImgNbglY

Number of anchor points for additive asymmetric background along the X and Y axes.
 MaxBgl

Maximal allowed value of additive background in percent of average signal value on the diffraction pattern. The default value is 10.0.
Molecular geometry
Zmatrices
In UNEX geometrical structure of molecules can be defined by means of Zmatrices. For this purpose ZMATRIX
command is used
ZMATRIX=mol,FREEZM,<ZMAT>,</ZMAT>
Here mol
is the name of molecule, FREEZM
is the format and the rest are the tags of the respective field in input file.
The FREEZM
format is rather flexible so that Zmatrices from many other programs can be transferred without problems. Positions of atoms can be defined by independent internal geometrical parameters (bond lengths, angles and torsional angles), Cartesian coordinates or combination of both. Usually a Zmatrix consists of two sections, body of Zmatrix and a list of its parameters. Elements in each line of Zmatrix can be separated by spaces, commas and/or tabulation characters. Same variable(s) can be used multiple times within one Zmatrix. In the most general case, definition of an atom in body of Zmatrix is as follows:
number of atom, symbol of atom, atomic mass, 1st reference atom, 1st parameter, 2nd reference atom, 2nd parameter, 3rd reference atom, 3rd parameter, type of definition
All items must be in one line. Number of atom, atomic mass and type of definition are optional. First three atoms require less reference atoms (see examples). Parameters can be explicitly defined as floating point numbers (in this case they cannot be refined) or as names of variables. The list of variables goes after the main body of Zmatrix. In the very end of the line the type of definition can be defined as an integer. Possible types are

0
, default type, three internal parameters are used for the definition of atom position: bond length, angle and torsional (dihedral) angle. 
1
or1
, expected parameters are bond length, and two angles. There are two equvivalent mirrorsymmetric positions of atom corresponding to this set of internal parameters, therefore this type can be positive (+1) and negative (1). The sign of the type corresponds to the sign of the torsional angle constructed on the defined atom and 1st, 2nd and 3rd reference atoms. 
2
or2
, similar to the type above, internal parameters are bond length, 2nd bond length and an angle. 
3
or3
, similar to the types above, internal parameters are three bond length to three reference atoms, respectively. 
4
, expected internal parameters are bond length, 2nd bond length and a torsional angle.
In case of using Cartesian coordinates, positions of atoms are defined as
Number of atom, symbol of atom, atomic mass, first parameter, second parameter, third parameter
Here the parameters are Cartesian coordinates of the atom.
As in the case of bond lengths, angles and torsional angles, explicit values of Cartesian coordinates or names of variables can be defined here.
If variables are used, a minus sign 
can be prepended to a variable,
indicating that in calculations of the atom position negated value of the corresponding parameter must be used.
Note, it is impossible to use both Cartesian coordinates and internal geometrical parameters for definition of the same atom.
However, within the same Zmatrix different atoms can be defined using both internal parameters and Cartesian coordinates.
Atoms can be also defined in centroids. For this the keyword centroid
should be indicated after the definition of the atom and a list of reference atoms should be given.
See below for particular examples.
The second part of Zmatrix is the list of variables, their values and, optionally, group numbers. Values of bond lengths cannot be negative or equal to zero. Values for angles must be between 0 to 180 degrees. Extreme values (0 or 180) are possible only in some special cases. Torsional angles can have any values.
Group number of a parameter is an integer value indicating the group, in which the parameter can be refined. Differences between values of parameters within one group are fixed during refinement. It is impossible to combine in same group

bond lengths and (torsional) angles,

Cartesian coordinates and bond lengths,

Cartesian coordinates and (torsional) angles.
In GED dynamic models, nonrigid coordinates (for example, torsional angles) must be labeled by negative group numbers. This is special case, not an indication of refinable parameter. In dynamic models there is no need to specify groups in Zmatrices for each pseudoconformer; it is enough to specify group numbers only for parameters of the first pseudoconformer.
Examples of Zmatrices:

Simplest example
Figure 1. Molecular structure of H_{2}O_{2}.H O 1 0.960 O 2 1.480 1 120.0 H 3 0.960 2 120.0 1 120.0
This is a simplest example. First three atoms are defined in a special way, the fourth one is defined in a general way. To specify the first atom no parameters are required, position of the second atom is determined by the H—O bond length, position of the third atom is determined by the O—O bond length and the H—O—O angle. The fourth atom is determined by a triple of parameters: a bond length, an angle and a torsional angle. The values of all parameters are given explicitly in the Zmatrix body. However, in many cases it is more convenient to define variables in a second part of Zmatrix and use them in the first part:
H O 1 Roh O 2 Roo 1 Aooh H 3 Roh 2 Aooh 1 Fhh Roh=0.960 1 Roo=1.480 1 Aooh=120.0 2 Fhh=120.0 3
In this example it is also demonstrated how group numbers are assigned to parameters. Here two bond lengths
Roh
andRoo
are in group1
, the angleAooh
is in group2
and the torsion angleFhh
is in group3
. The parameters with assigned group numbers can be processed and refined, for example inMINIMIZE
procedure. 
Alternative way for definition of third atom
Figure 2. Molecular structure of H_{2}O.Usually (see first example) position of third atom is determined by a distance to second atom and by an angle to the first atom. Here is an example of an alternative way for definition of the third atom, where a distance to the first atom and an angle of 3—1—2 atoms are used:
H O 1 Roh H 1 Rhh 2 Ahho Roh=0.960 Rhh=1.663 Ahho=30.0

Definition of third atom by means of two distances
Figure 3. Molecular structure of cyclopropane.C C 1 Rcc C 1 Rcc 2 Rcc 3 Rcc=1.508
Only distances can be used to define position of atoms. For a third atom only two distances are required. Here to define position of the third carbon distances to the first and to the second carbons are used and a special integer key
3
is given in the very end of the corresponding line. 
Definition of atoms with distance and two angles
Figure 4. Molecular structure of carbon tetrachloride.Cl C 1 R1 Cl 2 R1 1 A1 Cl 2 R1 1 A1 3 A1 1 Cl 2 R1 1 A1 3 A1 1 R1 1.7724 A1 109.47122063
Here is an example of carbon tetrachloride defined with T_{d} symmetry. The last two chlorine atoms are defined using bond lengths and angles, represented as
R1
andA1
variables. This type of definition is indicated with1
or1
keywords in the very end of the corresponding lines. The sign of this keyword always corresponds to the sign of the respective tosion angle X—A—B—C, where X is the defined atom and A, B and C are the first, second and third anchor atoms, respectively. In this example positions of the last two atoms are determined by exactly the same parameters but with two different by sign keys1
and1
. 
Definition of atoms using two distances and one angle
Figure 5. Fragment of a molecule with 5membered ring.C C 1 Rcc O 2 Rco 1 Acco C 3 Rco 2 Acoc 1 F1 N 1 Rcn 4 Rcn 3 Anco 2 Rcc 1.51 Rco 1.53 Rcn 1.40 Acco 106.0 Acoc 108.0 Anco 90.0 F1 0.0
In this example the 5th atom is defined using two distnaces C1—N5 and C4—N5 (both equal to
Rcn
) and an angle O3—C4—N5 (parameterAnco
). This type of definition is indicated by the integer key2
in the very end of the line. Sign of this parameter corresponds to the sign of the torsional angle N5—C1—C4—O3. In this case the configuration with parameter2
would be symmetrically equivalent to the presented structure. 
Definition of atoms using two distances and a torsional angle
Figure 6. Fragment of another molecule with 5membered ring.C C 1 Rcc C 2 Rcc2 1 Accc C 3 Rcc 2 Accc 1 F1 N 1 Rcn 4 Rcn 3 F2 4 Rcc 1.5152 Rcc2 1.53 Rcn 1.40 Accc 106.0 F1 0.0 F2 90.0
This is an example of geometry definition of a fivemembered ring in envelope conformation with symmetry C_{s}. Here the 5th atom is defined using two distnaces C1—N5 and C4—N5 (both equal to
Rcn
) and a dihedral angle C3—C4—C1—N5 (parameterF2
). This is indicated by the key4
in the very end of the respective line. There is no4
type since the geometrical configuration is already defined by the sign of the dihedral angle. 
Definition of atoms using three distances
Figure 7. Tetrahedral structure of P_{4} molecule.P P 1 Rpp P 1 Rpp 2 Rpp 3 P 1 Rpp 2 Rpp 3 Rpp 3 Rpp 1.60
This is an example of a Zmatrix for the tetrahedral P_{4} molecule with only one independent geometrical parameter within T_{d} point group. The third atom is defined as in example 3. The fourth atom is defined using three distances and a key
3
. The sign of the key defines geometrical configuration and corresponds to the sign of the dihedral angle P3—P2—P1—P4. 
Using Cartesian coordinates
Figure 8. Structure of cubane carbon skeleton.C xx yy zz C xx yy zz C xx yy zz C xx yy zz C xx yy zz C xx yy zz C xx yy zz C xx yy zz xx = 0.8 yy = 0.8 zz = 0.8
In UNEX it is possible to define molecular geometry using Cartesian coordinates within Zmatrix. In this example a cubane carbon skeleton is defined using only Cartesian coordinates. Here formally three independent parameters are used:
xx
,yy
andzz
. However, for the octahedral symmetry they are all equal and can be reduced to just one parameter. Note also the usage of minus signs before variables in some places. 
Mixing Cartesian coordinates and internal parameters
Figure 9. Molecular structure of cubane.C xx yy zz C xx yy zz C xx yy zz C xx yy zz C xx yy zz C xx yy zz C xx yy zz C xx yy zz H 2 Rch 3 Rch 4 Rch 3 H 1 Rch 6 Rch 7 Rch 3 H 3 Rch 4 Rch 8 Rch 3 H 1 Rch 5 Rch 7 Rch 3 H 2 Rch 4 Rch 8 Rch 3 H 5 Rch 6 Rch 7 Rch 3 H 2 Rch 3 Rch 8 Rch 3 H 1 Rch 5 Rch 6 Rch 3 xx = 0.8 yy = 0.8 zz = 0.8 Rch = 2.4
In UNEX it is also possible to define molecular geometry using Cartesian and internal coordinates together. The cubane skeleton from the previous example is supplemented here with hydrogen atoms using the ±3 type of definition (three distances). This is only for illustration purposes. In real practice for this molecule in the case of octahedral symmetry it would be more simple to use Cartesian coordinates for definition of hydrogens just like for carbons.

Dummy atoms
Figure 10. Molecular structure of NH_{3} with a dummy atom.X N 1 1.0 H 2 Rnh 1 Ahnx H 2 Rnh 1 Ahnx 3 Dx H 2 Rnh 1 Ahnx 3 Dx Rnh=1.1 Ahnx=110.0 Dx=120.0
Dummy atoms can be utilized for definition of molecular structure. The
X
symbol must be used for them. Also note the possibility to apply negative sign to the dihedral angleDx
. 
Centroids
C H 1 RCH C 1 RCC 2 120.0 H 3 RCH 1 120.0 2 0.0 C 3 RCC 1 120.0 4 180.0 H 5 RCH 3 120.0 4 0.0 C 5 RCC 3 120.0 6 180.0 H 7 RCH 5 120.0 6 0.0 C 7 RCC 5 120.0 8 180.0 H 9 RCH 7 120.0 8 0.0 C 9 RCC 7 120.0 10 180.0 H 11 RCH 9 120.0 10 0.0 X centroid 1 3 5 7 9 11 RCH=1.08105 1 RCC=1.39157 2
Here the last dummy atom is defined in the centroid of the ring of atoms 1,3,5,7,9 and 11.

Explicit numeration of atoms
If default numbering is not acceptable atom numbers can be given explicitly in Zmatrix:
5 X 1 N 5 1.0 2 H 1 Rnh 5 Ahnx 3 H 1 Rnh 5 Ahnx 2 Dx 4 H 1 Rnh 5 Ahnx 2 Dx Rnh=1.1 Ahnx=110.0 Dx=120.0
Here the first defined dummy atom is in fact the 5th in the list of atoms.

Definition of atom masses
By default UNEX uses masses of the most stable isotopes of atoms. However, masses of individual atoms can be defined right in Zmatrix, like in the example for D_{2}O below
H 2.0141 O 1 Roh H 2.0141 2 Roh 1 Ahoh Roh=1.0 Ahoh=109.0

Definition of standard deviations
In UNEX there is a possibility to define values of Zmatrix parameters together with their respective standard deviations. This can be useful if you want to calculate propagation of the defined specific errors to some other dependent geometrical parameters. In the example below the distance
Rnh
has the value1.1
and standard deviation0.001
, the angleAhnx
is defined to be110.0
degrees with standard deviation0.2
, while the parameterDx
is defined with a standard deviation equal to0.0
. For the latter it is also possible just to omit the value0.0
. Note, for calculation of errors for other dependent geometrical parameters it is necessary to assign group numbers to parameters in Zmatrix, otherwise they will not participate in calculation even if their standard deviations are not zero.X N 1 1.0 H 2 Rnh 1 Ahnx H 2 Rnh 1 Ahnx 3 Dx H 2 Rnh 1 Ahnx 3 Dx Rnh=1.1 0.001 1 Ahnx=110.0 0.2 2 Dx=120.0 0.0
After definition of Zmatrix it is possible to modify its parameters.
For this purpose SET
commands can be used. The example below demonstrates their usage:
SET=BOND,mol, 2,3, 1.3 SET=ANGLE,mol, 1,2,3, 90.0 SET=TORSION,mol, 1,2,3,4, 180.0
Here the first command changes the value of a parameter in a Zmatrix, which corresponds to the distance between atoms 2 and 3. The value of this parameters will be set to 1.3 Å. Analogously, the other two commands set parameters corresponding to the angle 1—2—3 and torsion angle 1—2—3—4 equal to 90 and 180 degrees, respectively.
Cartesian coordinates
In some cases to perform required computations it is sufficient to define
molecular structure only in form of Cartesian coordinates.
For this purpose MOLXYZ
command can be used:
MOLXYZ=mol,format,otag,ctag
Here mol
is the name of molecule, format
must be either XYZUNEX
or XYZGAUSSIAN
, otag
and ctag
are opening and closing tags of the corresponding data field to be read.
XYZUNEX
is a flexible format. In the most complete form each line defines atom number, atom symbol, mass (in amu) and Cartesian coordinates:
<xyz> 1 O 16.0 0.000000 0.000000 0.115719 2 H 1.0 0.000000 0.748790 0.462876 3 H 1.0 0.000000 0.748790 0.462876 </xyz>
Numeration must not be sequentially ordered. The following is also possible:
<xyz> 3 H 1.0 0.000000 0.748790 0.462876 1 O 16.0 0.000000 0.000000 0.115719 2 H 1.0 0.000000 0.748790 0.462876 </xyz>
In the simplest form, numeration and masses can be omitted. In this case sequentially ordered numeration and default masses are assumed.
If masses are not given in the data field explicitly and the structure has been already defined earlier, then the original masses are not redefined. 
Default units for Cartesian coordinates are Angstroms. With Units
keyword also Bohrs can be used:
<xyz> Units=Bohr O 0.00000000 0.00000000 0.12236619 H 0.00000000 1.41500832 0.97102012 H 0.00000000 1.41500832 0.97102012 </xyz>
The other possible format is XYZGAUSSIAN
. In can be used for data printed by Gaussian [7] program, for example
<xyz2> 1 1 0 0.000000 0.000000 1.539305 2 6 0 0.000000 0.000000 0.458150 3 17 0 0.000000 1.678636 0.084082 4 17 0 1.453741 0.839318 0.084082 5 17 0 1.453741 0.839318 0.084082 </xyz2>
Note, Gaussian can print Cartesian coordinates with or without atomic types (zeros in the example above). Both cases are recognized by UNEX automatically, so the following data can be read using exactly the same command:
<xyz2> 1 1 0.000000 0.000000 1.539305 2 6 0.000000 0.000000 0.458150 3 17 0.000000 1.678636 0.084082 4 17 1.453741 0.839318 0.084082 5 17 1.453741 0.839318 0.084082 </xyz2>
Upon reading of data atoms can be automatically renumbered using RENUM
command,
which must be given inside the data field:
<xyz1> RENUM=12,23,31 C 12.0 1.03693735 0.02315941 0.76526551 C 12.0 0.02512688 1.12502827 0.77820594 C 12.0 0.02512688 1.12502827 0.77820594 </xyz1>
Here the renumbering works as C1→C2, C2→C3 and C3→C1.

Potential functions
Starting from UNEX 1.6990 the input numeration of parameters for potential functions starts from zero! 
In order to construct a dynamic model for molecular part of electron diffraction intensity
a potential function must be introduced.
This can be done using POTENTIAL
command:
POTENTIAL=mol,format,otag,ctag
here mol
is the name of molecule, format
can be PTL1
or FUNC
, otag
and ctag
are
opening and closing tags of data field as usually.
The PTL1
format is for the case when potential function is introduced in numerical form, for example
POTENTIAL=mol,PTL1,<pot>,</pot> <pot> 0.0 52.759317 10.0 52.265898 20.0 50.701285 30.0 47.791786 40.0 43.252047 50.0 37.535062 60.0 31.751394 70.0 23.594173 80.0 13.194505 90.0 5.294997 100.0 1.090070 110.0 0.000000 </pot>
Here in the first column are the values of the geometric parameter corresponding to the dynamic coordinate
In the second column are the respective energy values; their units can be defined by PotEUnits
keyword in BASE.
After reading the data UNEX performs two major actions.
First, all the values are shifted so the the minimal value equals to zero.
Second, the data are approximated with a function.
Type of the function depends on the PotType
setting in the field of the respective molecule.
If it is a parametric function then the PotCoefNum
keyword should be set to a proper value.
Note, this keyword defines total number of parameters in the potential function including free term, if applicable.
For example, the combination PotType=Cos1
and PotCoefNum=3
defines the following potential function
(note the numeration of parameters V_{i}):
Similarly, for PotType=Polynom
and PotCoefNum=5
the potential function will be
The model Gauss
for potential function has no free term and the combination PotType=Gauss
and PotCoefNum=6
gives
Data field in PTL1
format may contain two special keywords: POTCOEFV
and POTCOEFG
.
The first one is for setting initial values for parameters of model potential function.
They are used in approximation procedure.
Generally, for cosine series they are not required but for a sum of Gaussians it is very advisable
to set them to some reasonable values, which will be refined further by UNEX.
The other keyword, POTCOEFG
, is for setting group numbers for those parameters,
which should be refined in MINIMIZE
and related commands.
The example below demonstrates both keywords
<pot> POTCOEFG=031,132,233,334,435,536 POTCOEFV=053.0,10.3,21.0,32.0,42.8,51.0 0.0 2104.2041440921 10.0 2104.2043320255 20.0 2104.2049279551 30.0 2104.2060361246 40.0 2104.2077652201 50.0 2104.2099427046 60.0 2104.2121455875 70.0 2104.2152525088 80.0 2104.2192135333 90.0 2104.2222222971 100.0 2104.2238238691 110.0 2104.2242390548 120.0 2104.2240176886 130.0 2104.2236761259 140.0 2104.2234703951 150.0 2104.2234587732 160.0 2104.2235955815 170.0 2104.2237665581 180.0 2104.2238429817 </pot>
Several important points can be mentioned for this example.

The energy values are given in atomic units so
PotEUnits=au
should be defined in BASE. 
This potential is best described with a sum of two Gaussians, so
PotType=Gauss
should be defined in molecular data field. 
Two Gaussians require in total six parameters, so
PotCoefNum=6
should be defined. 
POTCOEFG
defines group numbers for potential function parameters in formatParameterNumberGroupNumber
. In this example the parameters get group numbers 31—36. 
POTCOEFV
defines initial values for potential function parameters in formatParameterNumberValue
. For negative values the format includes the second minus signParameterNumberValue
. In this example the initial values for parameters are 53.0 for V_{1}, 0.3 for Δ_{1}, 1.0 for w_{1}, 2.0 for V_{2}, 2.8 for Δ_{2} and 1.0 for w_{2}. See above for the example of analytical expression of the sum of two Gaussians. 
Numeration of parameters in
POTCOEFG
andPOTCOEFV
starts from zero. 
Not necessarily all parameters should be declared in
POTCOEFG
andPOTCOEFV
.
With a 
The other available format FUNC
is for the case of introducing particular values of parameters for the potential function.
The following example demonstrates the usage of this format.
POTENTIAL=mol,FUNC,<pot>,</pot> <pot> 0 0.0 1 1.5 101 2 0.2 102 3 0.001 4 0.04 103 </pot>
In the data field at least two columns must be defined. The first column is for parameter indices,
the second column contains values of respective parameters.
The optional third column can contain group numbers for respective parameters.
In this example the introduction of values for first five parameters is done.
As in the examples above the keys PotType
and PotCoefNum
here also should be defined before reading the data.
Additionally, group numbers 101, 102 and 103 are assigned to parameters 1, 2 and 4, respectively.
Note, the numeration of parameters starts from zero. The scheme for numeration is as described above.
For Gauss
models special scheme is used: the parameters 0, 1 and 2 correspond to V, Δ, and w of the first Gaussian in the sum.
The next tripple, 3, 4 and 5, correspond to parameters V_{2}, Δ_{2} and w_{2} (parameters of the second Gaussian) and so on.
It is possible to execute several POTENTIAL
commands with FUNC
format introducing different values
of parameters and/or group numbers if required.
Also it is not necessary to set all parameters in the data field. It is possible to define values only for selected parameters.
In this case the other parameters will not be affected.
There is no general scheme for units of parameters for all types of parametric potential functions. Thus UNEX reads values of parameters without any internal conversion. It is user’s responsibility to define values so that potential function gives energy in kJ/mol for dynamic coordinate in radians. 
Vibrational data
Force constants
In UNEX there is a possibility to introduce force constants for molecules.
This can be required for some types of calculations or
can be used for transformations of the force constansts and/or printing in different formats.
Harmonic force constants in Cartesian coordinates are introduced with F2C
command:
F2C=mol,format,otag,ctag
Reading of data can be done in three formats, ARCHGAUSSIAN
, FREEFC
and CFOURFCM
.
The first one is the format of archive area in the very end of Gaussian [7] output files.
Here is an example for water molecule:
F2C=mol,ARCHGAUSSIAN,<data>,</data> <data> 1\1\GINCCHEOPS10401\Freq\RPBE1PBE\631G(d,p)\H2O1\YVISHNEV\04Nov201 5\0\\#P PBE1PBE/631G(d,p) Freq\\H2O\\0,1\O,0.,0.,0.118417\H,0.,0.7569 23,0.473669\H,0.,0.756923,0.473669\\Version=EM64LG09RevD.01\State= 1A1\HF=76.3369645\RMSD=8.979e09\RMSF=4.171e06\ZeroPoint=0.0217171\ Thermal=0.024552\Dipole=0.,0.,0.8151838\DipoleDeriv=0.7285751,0.,0., 0.,0.4461319,0.,0.,0.,0.3542551,0.3642876,0.,0.,0.,0.223066,0.078652 ,0.,0.1104485,0.1771276,0.3642876,0.,0.,0.,0.223066,0.078652,0.,0.11 04485,0.1771276\Polar=3.0234315,0.,7.2833047,0.,0.,5.4419574\PG=C02V [ C2(O1),SGV(H2)]\NImag=0\\0.00011233,0.,0.69921020,0.,0.,0.47159438,0. 00005616,0.,0.,0.00006270,0.,0.34960510,0.20818553,0.,0.38113336,0., 0.27346025,0.23579719,0.,0.24082289,0.22485174,0.00005616,0.,0.,0.00 000654,0.,0.,0.00006270,0.,0.34960510,0.20818553,0.,0.03152826,0. 03263736,0.,0.38113336,0.,0.27346025,0.23579719,0.,0.03263736,0.0109 4545,0.,0.24082289,0.22485174\\0.,0.,0.00000976,0.,0.00000261,0.00000 488,0.,0.00000261,0.00000488\\\@ </data>
The option FREEFC
is used when data are not formatted in any particular manner.
UNEX reads floating point numbers line by line from left to right and correspondingly fills in lowerleft triangle of force constants matrix.
The same data for water molecule can be introduced as
F2C=mol,FREEFC,<data2>,</data2> <data2> 0.00011 0.00000 0.69921 0.00000 0.00000 0.47159 0.00006 0.00000 0.00000 0.00006 0.00000 0.34961 0.20819 0.00000 0.38113 0.00000 0.27346 0.23580 0.00000 0.24082 0.22485 0.00006 0.00000 0.00000 0.00001 0.00000 0.00000 0.00006 0.00000 0.34961 0.20819 0.00000 0.03153 0.03264 0.00000 0.38113 0.00000 0.27346 0.23580 0.00000 0.03264 0.01095 0.00000 0.24082 0.22485 </data2>
(reduced number of digits is given for compactness) or, for example
F2C=mol,FREEFC,<data3>,</data3> <data3> 0.00011233 0.00000000 0.69921020 0.00000000 0.00000000 0.47159438 0.00005616 0.00000000 0.00000000 0.00006270 0.00000000 0.34960510 0.20818553 0.00000000 0.38113336 0.00000000 0.27346025 0.23579719 0.00000000 0.24082289 0.22485174 0.00005616 0.00000000 0.00000000 0.00000654 0.00000000 0.00000000 0.00006270 0.00000000 0.34960510 0.20818553 0.00000000 0.03152826 0.03263736 0.00000000 0.38113336 0.00000000 0.27346025 0.23579719 0.00000000 0.03263736 0.01094545 0.00000000 0.24082289 0.22485174 </data3>
Format CFOURFCM
is used for reading force constants as they are written by Cfour program [8] in FCM
files.
For example:
F2C=mol,CFOURFCM,<data1>,</data1> <data1> 2 12 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 2.0658524394 0.0000000000 0.0000000000 2.0658524394 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 2.0658524394 0.0000000000 0.0000000000 2.0658524394 </data1>
Note, in this format Cfour prints full matrix of force constants and UNEX reads all data.
Cubic force constants in system of Cartesian coordinates are introduced with F3C
command:
F3C=mol,format,otag,ctag
However, in this case the only available format is the ARCHGAUSSIAN
.
Gaussian calculates and prints cubic force constants in system of Cartesian coordinates if Freq=cubic
is given in its input file.
Instead of copying data to UNEX input file it is also possible to read this data directly from other files. For this, instead of giving opening and closing tags, name of the file with the data must be provided as F3C=mol,ARCHGAUSSIAN,cubic.log 
Vibrational modes
For introducing data on vibrational modes there is a general command VMOD
, which can be used in several variants.
If only vibrational frequencies are required, then the following command can be used:
VMOD=mol,RFREQ,otag,ctag
For example:
VMOD=h2o,RFREQ,<freq>,</freq> <freq> 1737.01 3988.50 4145.43 </freq>
ED terms
A term in GED is a pair of atoms with associated parameters like distance, vibrational amplitude, correction and asymmetry (anharmonic, phaseshift) constant.
Vibrational amplitudes, corrections and asymmetry constants required for calculation of ED molecular intensities can be introduced with AMPLITUDES
command.
The syntax of the command is as usual:
AMPLITUDES=mol,format,otag,ctag
where format
can be either FREEU
, SHRINKU
or ELDIFF
.
An example of data field in FREEU
format:
<ampl> #a1 a2 comment l corr a g # O1 H2 0.9591 0.0675 0.0126 2.0000 1 O1 H3 0.9591 0.0675 0.0126 2.0000 1 H2 H3 1.5063 0.1125 0.0129 1.0000 2 </ampl>
Here each line includes a pair of atoms, arbitrary comment (here the distance between atoms is used as a comment),
amplitude l, vibrational correction corr, asymmetry constant a and group number g.
Amplitudes and corrections must be given in Å.
By default the type and units for asymmetry constants depend on the current setting of the ImolAnhTermModel
keyword (for details see see chapter Models for ED intensity).
For ImolAnhTermModel=Asym
the default expected type is here asymmetry parameter κ and the expected units are Å^{3}.
In case of ImolAnhTermModel=Morse
the expected type is Morse parameter and the units are Å^{1}.
However, it is possible to define explicitly the type and units for the input aparameters with the local keyword aUnits
as in the example:
<ampl> aUnits=kA3 O1 H2 0.9591 0.0675 0.0135 8.0e6 O1 H3 0.9591 0.0675 0.0135 8.0e6 H2 H3 1.5063 0.1125 0.0137 1.8e6 </ampl>
The setting kA3
for aUnits
indicates here that the input avalues are in fact asymmetry constants κ in Å^{3}.
If the global keyword ImolAnhTermModel
is set to Morse
, then these values are converted internally to Morse constants
according to the formula
where l is the respective amplitude taken from the same current input. Another option is kpm3
if κ are given in pm^{3}.
Yet another option is aUnits=c3pm3
, indicating that the input values are c_{3} parameters from cumulant approximation.
They are converted internally to asymmetry parameters as κ = 10^{6}c_{3}/6.
Again, if ImolAnhTermModel=Morse
, than Morse constants are calculated automatically as described above.
Also it is possible to indicate explicitly the input of Morse constants in Å^{1} by defining aUnits=MoAm1
.
If at this point the current setting is ImolAnhTermModel=Asym
, then the input Morse constants are automatically recalculated
into asymmetry parameters κ according to the formula above.
Group numbers g may be defined for amplitudes. However, if you do not need to assign group numbers (no need to refine amplitudes), there is no need to set them to zeros explicitly. Instead, they may be simply omitted
<ampl> #At1 At2 comment l corr a g # O1 H2 0.9591 0.0675 0.0126 2.0000 O1 H3 0.9591 0.0675 0.0126 2.0000 H2 H3 1.5063 0.1125 0.0129 1.0000 </ampl>
If aparameters are zero, they also can be omitted. Note, group numbers may still be defined
<ampl> #At1 At2 comment l corr a g # O1 H2 0.9591 0.0675 0.0126 1 O1 H3 0.9591 0.0675 0.0126 1 H2 H3 1.5063 0.1125 0.0129 </ampl>
If a correction is zero and the respective aparameter is also zero, then both numbers may be omitted. But if the correction is zero and the aparameter is not zero, then both numbers must be defined explicitly.
<ampl> #At1 At2 comment l corr a g # O1 H2 0.9591 0.0675 O1 H3 0.9591 0.0675 1 H2 H3 1.5063 0.1125 0.0000 1.0000 </ampl>
Another format option is SHRINKU
. It is implemented for reading data produced by the Shrink [9] program.
Specifically UNEX expects data as it is printed by Shrink in tables for the second approximation:
<ampl> #Amplitudes and corrections at 0010 K, second (harmonic) #approximation, local centrifugal distortions included; #<dr()> are deviations from equilibrium distances # # Atoms Distance Amplitude <dr(loc)> <dr(har)> K # 1 O1 H2 0.9591 0.0675 0.0011 0.0000 0.00365 1 2 O1 H3 0.9591 0.0675 0.0011 0.0000 0.00365 2 3 H2 H3 1.5063 0.1125 0.0001 0.0038 0.01201 3 </ampl>
From the given data UNEX reads amplitudes (from the fifth column) and corrections
(Shrink prints them in the column K
, in the second approximation they correspond to r_{h1}r_{a}).
The values in columns <dr(loc)>
and <dr(har)>
are ignored.
Note, in Shrink output the integers in the last column are simply indices of the corresponding atom pairs.
In contrast, UNEX interprets them as group numbers. In reality it is very unlikely that one would leave them as they are.
Normally you need to delete this column and assign group numbers manually or in another way, as needed.
If you want to use anharmonic corrections from Shrink, the column K
must be substituted with the column r_er_a
from the very last table produced by Shrink, so the format remains for UNEX unchanged.
In the example below corrections to equilibrium structure are used and the integers are removed as it is normally done.
<ampl> 1 O1 H2 0.9591 0.0675 0.0011 0.0000 0.0126 2 O1 H3 0.9591 0.0675 0.0011 0.0000 0.0126 3 H2 H3 1.5063 0.1125 0.0001 0.0038 0.0129 </ampl>
The third available format ELDIFF
is for introducing data produced by the ElDiff [10] program.
Here is a shortened example of such data field:
<ampl> #At.pair Num Re Rg Ra Dr Dr(har) Dr(kin) Dr(dyn) Ampl. c3/6 Cl1C2 ( 1) 1.7609 1.7687 1.7672 0.0078 0.0027 0.0009 0.0042 0.0517 2.59 555 Cl1Cl3 ( 1) 2.8755 2.8857 2.8841 0.0102 0.0018 0.0017 0.0101 0.0668 3.76 </ampl>
Vibrational corrections are calculated from the input data as differences Re
Ra
.
Amplitudes and asymmetry parameters c3/6
are read as they are given.
Note, the input c3/6
values in pm^{3} are converted internally to asymmetry parameters κ in Å^{3}.
However, they can be internally further converted to Morse parameters if the ImolAnhTermModel
keyword is set to Morse
.
As usually, optional integers in the very end of lines are the group numbers for
refinements of the respective amplitudes or their scale factors.
If due to some reason the numeration of atoms in the input data does not coincide with already defined numeration
(for example, in Zmatrix), the RENUM
command can be used:
<ampl> RENUM=12,23,31 1 O1 H2 0.9591 0.0675 0.0011 0.0000 0.0126 2 O1 H3 0.9591 0.0675 0.0011 0.0000 0.0126 3 H2 H3 1.5063 0.1125 0.0001 0.0038 0.0129 </ampl>
It works in the same way as in the case of reading Cartesian coordinates.
Namely, in the example above the following numeration will be done: O1→O2, H2→H3 and H3→H1.
RENUM
works in fields of all format types.
Note, multiple RENUM
commands can be defined in the same data field and UNEX will process all of them.
This can be useful when large amount of atoms must be renumbered and the corresponding list would be too long for a single RENUM
command.
As already mentioned integer numbers in the very end of lines are interpreted by UNEX as group numbers for amplitudes so that they can be refined.
In case of dynamic GED models group numbers can be defined only for the first pseudoconformer in the psconfs
list (see above).
Group numbers for amplitudes of other pseudoconformers are assigned automatically and coincide with those for the first pseudoconformer.
In general AMPLITUDES
allows introducing vibrational amplitudes, corrections,
asymmetry constants and group numbers for amplitudes. It is not guaranteed that
with this command interatomic distances can be introduced.
Another limitation is the imposibility to define group numbers for r_{a} distances,
when there is a need to refine them as independent parameters.
Thus the AMPLITUDES
command is used mostly for geometrically consistent models,
when molecular structures are defined in terms of Zmatrices or Cartesian coordinates.
There is, however, an alternative command GEDTERMS
for defining all required parameters
for calculation of ED molecular intensity curves
GEDTERMS=mol,GTRM,otag,ctag
It accepts only one type of format GTRM
, which assumes that each line contains a pair of atoms,
r_{a} distance, amplitude, correction (normally zero), asymmetry constant and group numbers for the distances and amplitudes
as shown in the example below
<gtrm> #At1 At2 r_a l corr a Gr Gl # C2 Cl1 1.7625 0.05485 0.00 0.00001 3 5 C2 Cl3 1.7625 0.05485 0.00 0.00001 3 5 C2 Cl4 1.7625 0.05485 0.00 0.00001 3 5 C2 Cl5 1.7625 0.05485 0.00 0.00001 3 5 Cl1 Cl3 2.8871 0.06844 0.00 0.00002 4 6 Cl1 Cl4 2.8871 0.06844 0.00 0.00002 4 6 Cl1 Cl5 2.8871 0.06844 0.00 0.00002 4 6 Cl3 Cl4 2.8871 0.06844 0.00 0.00002 4 6 Cl3 Cl5 2.8871 0.06844 0.00 0.00002 4 6 Cl4 Cl5 2.8871 0.06844 0.00 0.00002 4 6 </gtrm>
Note, the definition of aparameters follows here the same rules as for the AMPLITUDES
command,
likewise it is possible to use the local RENUM
command and the aUnits
keyword. See above for details.
For investigation purposes with UNEX it is possible to refine r_{a} distances independently,
resulting in a socalled geometrically inconsistent structure.
The corresponding group numbers for r_{a} distances can be defined only with the GEDTERMS
command.
Both 
UNEX can automatically check values of parameters for symmetrically equivalent terms.
This is done if symmetry has been determined for the molecule before introducing parameters for terms, for example by calling 
ED scattering factors
For calculation of theoretical molecular contribution to the total electron diffraction intensity scattering factors are required. In chapter Models for ED intensity these factors are denoted as ffunctions, which characterize scattering ability of atomic pairs. Also in some cases it is convenient to operate with so called gfunctions defined as ratio
where I_{at} is the atomic contribution to the total electron scattering intensity.
In UNEX calculation of all these functions can be done using GF
command:
GF=mol,AUTO,0.0500
Here the last number is the electron wavelength (in Å) for which the quantities must be calculated.
The wavelength is automatically recalculated into accelerating voltage U.
UNEX uses builtin tables of scattering aplitudes and phases defined for
accelerating voltages 10, 40, 60, 90 kV and for svalues up to 60 Å^{1}.
In general, twodimensional interpolating cubic splines are used for calculation
of required parameters from tabulated values.
The version of tables can be chosen using SFactorsDBVer
keyword in BASE.
By default the most recent published in [6] values are used.
Note, GF
command can be called multiple times for the same molecule.
The corresponding functions will be recalculated for the defined electron wavelengths.
ED intensities
UNEX has a special system for referencing of all types of intensity curves.
Each curve is defined as a pair of integer numbers, for example 11
represents in the input syntax a particular curve.
This system is designed to simplify categorization of curves and is closely related to the format (see below) of their definition in UNEX input files.
The most common and natural for GED basis for categorization of curves is the distance between nozzle and detector.
There can be, however, other considerations how to form groups of curves, by methods of data reduction and processing, by dates of experiments, this is up to user.
Total intensity
To introduce total intensity curves INT
command is used:
INT=READ,otag,ctag
The simplest example of the data field is
<INT> INT1 7.4000000000 5.8921150540 7.6000000000 5.6976865331 7.8000000000 5.5221522042 8.0000000000 5.3696527781 8.2000000000 5.2437855620 8.4000000000 5.1021824331 8.6000000000 4.9709462648 8.8000000000 4.8373286097 9.0000000000 4.7019566330 9.2000000000 4.5680270813 9.4000000000 4.4361494355 9.6000000000 4.3235065367 9.8000000000 4.2099473887 10.0000000000 4.1088546902 </INT>
In the first column are svalues, in the second column are values of total electron scattering intensity.
Here only one intensity curve is defined. In UNEX syntax it can be referenced as 11
curve.
The input format requires that each data set starts with an INT
keyword followed by an integer number.
In the example above it is INT1
, which means first goup. This is first definition of an INT1
curve, therefore it has the 11
reference code.
You can input several data sets in the first group as in the following example
<INT> INT1 7.4000000000 5.8921150540 7.6000000000 5.6976865331 7.8000000000 5.5221522042 INT1 9.6000000000 4.3235065367 9.8000000000 4.2099473887 10.0000000000 4.1088546902 </INT>
Here the first curve from above is 11
, the second one gets the code 12
.
In the same way a second and so on groups of intensities can be introduced:
<INT> INT1 7.4000000000 5.8921150540 7.6000000000 5.6976865331 7.8000000000 5.5221522042 INT1 9.6000000000 4.3235065367 9.8000000000 4.2099473887 10.0000000000 4.1088546902 INT2 28.1000000000 1.0475326418 28.2000000000 1.0393546185 28.3000000000 1.0276683596 INT2 28.0000000000 1.0475326418 28.2000000000 1.0393546185 28.3000000000 1.0276683596 30.0550000000 1.0159231825 </INT>
Here curves 11
, 12
, 21
and 22
are defined.
Note, UNEX assumes no constant step on the sscale. The only requirements are that

s and intensity values must be positive,

svalues must be sorted in ascending order,

there must be no equal svalues, differences smaller than
MinDs
are not allowed.
The UNEX input format also allows introducing standard deviations for intensity values:
<INT> INT1 6.2000000000 7.4146108106 0.0039183246 6.4000000000 7.1488355268 0.0021688941 6.6000000000 6.8938017418 0.0021065446 6.8000000000 6.6412539602 0.0020678461 7.0000000000 6.3619711300 0.0020204005 7.2000000000 6.1273418113 0.0019858103 7.4000000000 5.8921150540 0.0019449769 7.6000000000 5.6976865331 0.0019131756 7.8000000000 5.5221522042 0.0018765991 8.0000000000 5.3696527781 0.0018468983 </INT>
Here as usually in the first and second columns are s and intensity values and in the third column are the respective standard deviations of the intensity values. They can be used by UNEX in different procedures, for example in leastsquares refinement method. Standard deviations are optional and can be defined only for particular points:
<INT> INT1 6.2000000000 7.4146108106 6.4000000000 7.1488355268 2.0 6.6000000000 6.8938017418 6.8000000000 6.6412539602 2.0 </INT>
Here we define standard deviations 2.0
for two points, for the rest default value is assumed. The default value can be defined by using GStdev
keyword.
Also note that in this example we define standard deviations for total intensity. However, using respective setting for the keyword TypeStdev
(see below) it is possible to define standard deviations for molecular intensity sM(s), which may be later calculated from the input total intensity.
UNEX allows setting special parameters related to particular intensity curves. They are defined after corresponding INT
keywords like in the example
<INT> INT1 Parameter1=value Parameter2=value 6.2000000000 7.4146108106 6.4000000000 7.1488355268 6.6000000000 6.8938017418 6.8000000000 6.6412539602 </INT>
Note, there must be no space characters in pairs Parameter=value.
Set of parameters, which should be defined for each intensity curve depends on the particular UNEX application. Different procedures in UNEX may require different parameters to be defined. 
Possible parameters are
 Scale

Scale factor for molecular intensity curve. The default value is 1.0.
 VarSc

Group number in
MINIMIZE
and alike procedures. If this parameter equals to a positive integer number then the respective scale factor will be refined.  NtoP

NozzletoPlate — distance from diffraction point to the detector in mm. Input svalues must correspond to this value.
 NewNtoP

New nozzletoplate value in mm. Input svalues will be recalculated using defined
NtoP
andNewNtoP
.  StoP

SectortoPlate — distance from rotating sector defice to the detector in mm.
 Lam

Lambda — electron wavelength in Å. This parameter tells UNEX for which electron wavelength correspond input svalues.
 NewLam

New value of electron wavelength in Å. This should be defined together with
Lam
if you want to recalculate input svalues for the new value of electron wavelength.  Tfac

tfactor — a value proportional to the exposure time in the measurement of the corresponding diffraction pattern. This parameter can be used in models for the total intensity. For description of models see introduction to the theory of background lines.
 Std

Set this parameter to
CCl4
,C6H6
,CS2
orCO2
if the intensity corresponds to one of these compounds and you want to use it for determination of electron wavelength withSTANDARD
command.  BglNinflThr

Threshold for the number of inflection points on background line. By default the value of this keyword is negative, indicating that the global setting must be used (see keyword
BglNinflThr
).  RBPSDThr

Threshold for the relative power spectral density when smoothing background.
 BglPolPow

Polynom power for the background line. In negative (this is default), then the global setting is used, see the global keyword
BglPolPow
.  MCLam

Group number for electron wavelength in MonteCarlo procedure
MCMIN
. Positive value indicates that electron wavelength will be randomized inMCMIN
.  MCLamS

Standard deviation for electron wavelength in Å. It is used for randomization of electron wavelength in
MCMIN
.  MCBglQG

Group number for functional
Q
used in procedure for approximation of background line with cubic splines. Positive value indicates thatQ
will be randomized inMCMIN
and used for recalculation of the background line.  MCBglQMin

Minimal value of functional
Q
randomized inMCMIN
.  MCBglQMax

Maximal value of functional
Q
randomized inMCMIN
.  ReadStdev

Integer keyword allowing (
ReadStdev=1
, default) or disabling (ReadStdev=0
) reading of standard deviations for intensity values.  GStdev

Default value for standard deviations of intensity if they are not given explicitly in input. By default
GStdev=1.0
.  TypeStdev

Type of standard deviations on input. By default they are given for total intensity (
TypeStdev=int
) but you can also provide standard deviations directly for sM(s) curves (TypeStdev=sms
).  LvlInfl

Number of inflection points of spline used for levelling of total intensity and background.
 LvlPow

Power of polynomial used for levelling of total intensity and background.
 StdModBglPow

Power of Chebyshev polynomial, which is used in
STANDARD=LSQ
procedure as a model for additive background in leastsquares refinement. The default value is 3.  IModel

Type of model for total intensity, one of:
none
(undefined model),mbgl
,a1bgl
anda2bgl
. For details see Models for ED intensity and ED background lines.
Molecular intensity
Reduced molecular intensity sM(s) can also be directly introduced into UNEX using SMS
command
SMS=READ,otag,ctag,int
where int
is the intensity identificator, otag
and ctag
are opening and closing tags for the corresponding data field, respectively.
Format of data field is simple — just two columns with s and sM(s) values. Example of the command and data field:
SMS=READ,<sms>,</sms>,11 <sms> 6.0000 1.2163543885 6.2000 1.0629013696 6.4000 0.6487354615 6.6000 0.1055888595 6.8000 0.4035742744 7.0000 0.7359056260 </sms>
If the data set 11
has been already defined and the original number of data points coincides with the number of data in the SMS
field
then the newly introduced data just overwrites old values. Otherwise this data set will be (re)initialized and will contain only sM(s) data.
Experimental background
Experimental background can be directly introduced using the BGL
command
BGL=READ,otag,ctag,int [num] [keywords]
where otag
and ctag
are the tags of the data field, int
is the data set identificator,
the optional integer number num
is used when smoothing of the input data is required.
Optional keywords can be also specified for additional control of the smoothing procedure.
They are explained below in ED background lines section.
The format of the data field normally requires two columns, with values for s and respective background:
<bgl> 10.0 0.208101 10.1 0.208028 10.2 0.207974 10.3 0.207913 10.4 0.207872 10.5 0.207835 </bgl>
Note, the svalues must be exactly the same as in the data set. There is also a possibility to introduce background values without respectivev svalues. In this case the data are just in one column and no checks are performed:
<bgl> 0.208101 0.208028 0.207974 0.207913 0.207872 0.207835 </bgl>
In this case it is users responsibility to ensure that the input values correspond to the svalues in the original data set.
After reading and optional processing the background data are used for calculation of experimental molecular intensity function.
For details about smoothing procedure and derivation of the molecular intensity see sections below regarding models of ED intensity and background.
Below are some examples.
Read background and calculate sM(s) for 11
:
BGL=READ,<mybgl>,</mybgl>,11
Read background, smooth it (by default cubic splines are used and here 3
means
maximal number of inflection points) and calculate sM(s):
BGL=READ,<mybgl>,</mybgl>,11,3
Note, the type of background (multiplicative, additive) can be indicated using the IModel
keyword for the BGL
command.
Details on keywords and examples are given below.
The described 
ED sector function
Rotating sector is a special device in electron diffraction experiment for levelling intensity of scattered electrons so that detector can measure it in wide range of angles. Thus, sector modifies primary data, which is mathematically equivalent to multiplication of primary intensity by some function. This function depends on the shape of the sector and is called sector function.
In UNEX sector function is decomposed into two parts, analytical model part and a possible numerical correction.
The model part is controlled by the SecModelType
keyword. For SecModelType=rpn
the model sector is
for SecModelType=sinpn
the model is
and if SecModelType=const
the model is
The parameters A, n and r_{max} are controlled by the SecPrmA
, SecPrmN
and SecPrmRmax
keywords.
By default, the model sector function is
The other part of the sector function is the socalled reduced sector function. By default it is initialized to constant 1. However, it can be defined in the numerical form and the total sector function is then calculated as
where S is the total sector function, F is the model sector function and f is the reduced sector function.
In numerical form sector function can be introduced with the SECTOR
command
SECTOR=format,otag,ctag
Here format
can be READTOTAL
or READREDUCED
.
The READTOTAL
format is used to input total sector function
SECTOR=READTOTAL,<sec>,</sec> <sec> 1.0 0.00000463 10.0 0.00462963 20.0 0.03703704 30.0 0.12500000 40.0 0.29629630 50.0 0.57870370 60.0 1.00000000 </sec>
The data field in the example above contains distances in mm from the senter of sector and respective values of the total sector function. The reduced sector function is calculated from these values based on the current model.
The other option is the READREDUCED
format. It is used for introducing of the reduced sector function directly.
Below is the respective example
SECTOR=READREDUCED,<rsec>,</rsec> <rsec> 8.250 1.8684053318 8.500 1.7616089813 8.750 1.6663497956 9.000 1.5813559000 9.250 1.5064034587 9.500 1.4407076490 9.750 1.3836521842 10.000 1.3344986762 </rsec>
In UNEX there is also a special sector function which can be used as regularization data in STANDARD=LSQ
procedure for refinement of sector function from intensity data.
This function can be introduced by the REGSEC
command, which has exactly the same syntax as SECTOR
:
REGSEC=READTOTAL,<sec>,</sec>
or
REGSEC=READREDUCED,<sec>,</sec>
The model for the regularization sector function is controlled by keywords with the RegSec
prefix: RegSecModelType
and others.
By default they are initialized to the same values as for the normal sector function.
Together with values of sector function it is also possible to introduce respective standard deviations. For this the data field must contain a third column with standard deviations:
<rsec> 8.250 1.8684053318 0.1 8.500 1.7616089813 0.2 8.750 1.6663497956 0.2 9.000 1.5813559000 0.2 9.250 1.5064034587 0.2 9.500 1.4407076490 0.2 9.750 1.3836521842 0.2 10.000 1.3344986762 0.1 </rsec>
Note, if total sector function is introduced the standard deviations must also correspond to the total sector function.
After introducing sector function it is possible to smooth it using natural Bsplines. This is done by command
SECTOR=SMOOTH
Analogous command exists for smoothing regularizing sector function:
REGSEC=SMOOTH
Data processing
Detector calibration
Optical density
Diffraction images can be measured on photo materials (plates or films), which must be afterwards digitized. Devices for this purpose, microphotometers and scanners, must be calibrated for optical density of blackness. This kind of calibration is represented as a relationship between true density values D and respective values provided by a particular device. This is especially important for commercially available optical scanners, because they usually underestimate large D values. Calibration of optical density is usually done by scanning a special standard target also known as optical wedge (see Figure below), which provides areas corresponding to particular accurately known D_{std} values. Comparing of calculated from scanned image Dvalues with respective D_{std}values shows how accurate is the scanner. This data can be afterwards used as calibration for calculation of accurate optical densities of pixels.
For processing of images of calibration targets WEDGE
command can be used. There are two modes of operation, automatic and manual.
For the automatic mode syntax of the command is as follows
WEDGE=AUTO,img,list of standard Dvalues
where img
is the name of image file to be processed. Note, this file must be defined in the imgfiles
keyword in BASE.
For example, command
WEDGE=AUTO,wedge.tif,0.05,0.20,0.33,0.46,0.61,0.77,0.92,1.06,1.19,1.33,1.48
will process wedge.tif
image and try to recognize automatically areas with indicated optical densities.
However, this mode is generally not recommended, since it can be unstable for complicated and noisy images.
A better option is to use the manual mode
WEDGE=MANUAL,img,otag,ctag
where otag
and ctag
are opening and closing tags of a field with supplementary information, for example
WEDGE=MANUAL,wedge.tif,<wed>,</wed> <wed> 0.05 26 128 0.20 86 124 0.33 150 124 </wed>
Here wedge.tif
is processed and the data between <wed>
and </wed>
tags contain information about standard areas in format D_{std} X Y
.
D_{std} are standard values of optical density, X and Y are coordinates of centers of respecive areas.
Origin of coordinate system for images in UNEX is always in the upperleft corner. 
Even better approach is to define not only centers of standard areas but also coordinates of their four corners:
WEDGE=MANUAL,wedge.tif,<wed>,</wed> <wed> 0.05 26 128 44 4 4 2 2 264 46 270 0.20 86 124 110 6 64 2 64 254 108 258 0.33 150 124 168 6 124 6 122 246 168 256 </wed>
Here after D_{std} for each area pairs of coordinates X Y are given for

center

upperright corner

upperleft corner

lowerleft corner

lowerright corner
In this case coordinates of centers should not necessarily be accurate.
Note, both automatic and manual modes can be influenced by MoveWedArea
keyword from BASE.
In fully manual mode if coordinates of corners were accurately defined it is recommended to fix them by setting MoveWedArea=no
.
After processing the image of calibration target a new image is created and saved under name of original image with added _proc
suffix and .tif
extension.
This image shows areas processed by UNEX and noisy pixels excluded from processing. Results, including calibration data, are printed in output file.
The obtained calibration can be used later in data reduction.
Introducing of calibration curve is done with help of the same WEDGE
command using its special syntax:
WEDGE=READ,otag,ctag
The corresponding data field is simple as in the example:
WEDGE=READ,<wed>,</wed> <wed> # Dstd Dscanner 0.050 0.053 0.200 0.152 0.330 0.215 0.460 0.274 0.610 0.336 0.770 0.415 0.920 0.481 1.060 0.540 </wed>
Here in the first and second columns are given standard and respective scanner values of optical densities.
Response function
In general, response function determines relationship between detector signal and level of stimulus, which is detected. In ED detector is irradiated by scattered electrons. For photomaterials response function is the relationship between measured optical density and intensity of electrons. In analogy response functions exist for other types of detectors.
In UNEX there is a possibility to refine response functions from series of experimental total intensity functions using the method of Kochikov [11]. For this, for following command should be used:
RESPFUNC=CALCIDS,int1,int2,int3,...
where int1
, int2
etc, are identificators of experimental total intensity functions, which should be processed.
The intensity functions should meet special requirements:

They must be measured in equal conditions.

They must be measured with different exposure times.

The number of different exposures must be as large as possible.
According to experience, the best strategy is to measure background patterns
with explicitly marked center (using primary beam; this is to increase reliability of data reduction)
starting with some minimal exposure time and doubling it (or increasing in some other way) for each subsequent measurement.
This should ensure stability of the primary electron beam and of the residual gas in the diffraction chamber for the series of the measurements.
The number of measurements should be as large as possible and the measured patterns should contain signal values spanning as large as possible range.
Sector device is irrelevant for the procedure. Most importantly, sector function does not change during the measurements.
Note, for each of the intensities participating in this procedure the exposure time must be defined with the Tfac
keyword.
The units can be any suitable for the particular case, for example seconds. Most importantly, they must be proportional to real time.
The other important issue is that the refined response function is defined in a form of a polynomial.
The degree of the polynomial should be defined using keyword RespFuncPolPower
.
Note, the best value for this parameter depends on the particular detector and data set.
It can be determined only in trialanderror procedure.
Keep in mind that too large degrees can lead to unstable and not reliable solutions.
Too low degrees can poorly describe the response function.
The refined response function is printed in numerical form in the end of the procedure. This data can be read later for using in other calculations as
RESPFUNC=READ,otag,ctag
where the numerical data must be located between corresponding opening and closing tags, for example:
RESPFUNC=READ,<resp>,</resp> <resp> 1.0 1.1 2.0 2.1 3.0 3.1 4.0 4.1 ... </resp>
The introduced response function can be used for correction of experimental total intensity functions with the INT=RESPCORR
command.
Note, the currently active response function can be printed as PRINT=RESPFUNC
.
Sector images
For determination of sector functions it is possible to use images of sector devices. A sector should be scanned with highest possible spatial resolution. The image must be in 8 or 16bit grayscale uncompresssed TIFF format. Before processing the image must be prepared so that pixels of the sector surface have values corresponding to exactly black color and the rest of pixels must be exactly while. The intermediate grayscale values are not allowed.
For the processing of the image several parameters should be defined:

Coordinates (in pixels) of the center of the sector. They can be estimated from the image manually.

Range of distances (in mm) from the center of sector, which should be processed.

Step size (in mm) for the processed distances.
Additionally, particular resolution values can be defined for the image, if they differ from the ideal values. Note, internally UNEX determines numerically the total sector function, which has values in the range [0,1]. This function is converted to the reduced sector function on the basis of the current sector model. Therefore it is advised to define explicitly a sector model, which fits your sector most closely. Remember, the determined reduced sector function is valid only for the model, which was defined in the processing of the sector image.
Below is an example of UNEX input for processing of sector image.
BASE=BASE,<BASE>,</BASE> IMAGE=SECTOR,Image0005.tif PRINT=SECTOR STOP <BASE> imgfiles=Image0005.tif SecModelType=sinpn SecPrmN=2.0 SecPrmRmax=75.0 SecPrmA=1.412283 </BASE> <Image0005.tif> Xc=2950 Yc=3707 XResolution=1200.0 YResolution=1200.0 IntRfr=2.5 IntRto=75.0 IntStep=0.05 </Image0005.tif>
Data reduction
UNEX implements procedures for data reduction, i.e. for transformation of 2D images of measured diffraction patterns
to profile curves of experimental electron scattering intensity functions.
Images must be in uncompressed 8 or 16bit grayscale TIFF format with littleendian byte order.
For introduction of images in UNEX the imgfiles
keyword in BASE
must be used.
The major procedure for data reduction [12] is started by command
IMAGE=INTSCAN,imgfile
where imgfile
is the name of image TIFF file to be processed.
Before processing it is highly recommended to clean images. The cleaning procedure is essentially the setting of absolute white grayscale level to pixels, which do not correspond to the diffraction pattern itself (shadows, etc) or represent areas, which should not be processed due to any other reason (defects, etc). The Figure below demonstrates on the left side an initial image of a diffraction pattern with shadows due to construction elements in the diffraction chamber and a resulted image after cleaning. Note, graphical software used for these purposes should not alter values of valid pixels or change the bit depth. Check this before using the software in real investigations!
The other important issue in data reduction is the usage of correct calibration for the sanning device.
If the scanner produces uncorrected images the calibration should be taken into account on the stage of data reduction.
This includes the spatial calibration, i.e. setting the true resolution values in the input file using
keywords XResolution
and YResolution
. Note, the deviations of true from nominal resolution values can be different
in different scanning modes. This must be carefully investigated for the particular scanning device and scanning modes.
The other part of calibration is the correction of output signal.
For optical scanners the calibration of optical density is essential.
In this case optical wedges are processed for obtaining calibration curve specific for the particular device
(and possibly for the particular mode of operation), which is introduced in the input file for the data reduction, see Optical density section.
In effect, this kind of calibration must be done for any other type of scanning devices, for example for imaging plate scanners.
However, in this case instead of optical density the signal is calculated from pixel values using special formulae and
for calibration purposes appropriate standards are required.
IMAGE=INTSCAN
procedure implements an iterative method [12] for refinement of parameters of diffraction patterns.
The maximal number of iterations can by adjusted with the keyword IntScanIter
. For the processed image the following parameters should be defined:

Nozzletodetector distance,
NozToPlate
. 
Step size for profile intensity grid,
IntStep
. 
Electron wavelength,
IntLambda
. 
Minimal and maximal distances from the center of diffraction pattern,
IntRfr
andIntRto
. 
Initial values for the coordinates of the center of the diffraction pattern,
Xc
andYc
. 
If required, initial values for coordinates of the sector center can be defined, see
Xs
andYs
. Otherwise they are initialized as equal toXc
andYc
. 
Optionally the signal (optical density) of unexposed areas can be defined using
fog
keyword. 
Parameters for the model of background,
ImgNbglX
,ImgNbglY
andMaxBgl
. 
With keywords
MinT
andMaxT
it is possible to define a range of valid pixel values.
Note, some of these parameters can have reasonable default values while other must be defined explicitly. In the refinement a model of the pattern is created, which depends on the following parameters:

Coordinates of the diffraction pattern center.

Coordinates of the center of rotating sector device.

Values of total intensity on different distances from the center of diffraction pattern. They constitute profile of the diffraction pattern.

Values of background, defined on a grid (see keywords
ImgNbglX
andImgNbglY
) of the image.
Whether parameters of one or other type will be refined depends on settings of keywords IntVarCentre
, IntVarSecCentre
and IntVarAsymBgl
.
Intensity values are always refined. The program also detects invalid pixels and sorts them out automatically.
The result of this procedure can be checked by inspecting image with weights of pixels.
In the end of diffraction image processing IMAGE=INTSCAN
can create several images in TIFF format with refined profile,
background and weights for original image points. The images are created, if the respective parameters were refined and the
keywords WriteAsymBglImg
, WriteWeightsImg
and WriteCurveImg
are set to appropriate values.
The names for images are constructed automatically as <basename>_profile.tif
, <basename>_bg.tif
and <basename>_w.tif
, where
<basename>
is the basename of the original input file of the processed image.
Note, these image files are created in the current directory.
Other operations with images
Histograms
UNEX can print histograms of images in numerical form, which can be useful for detailed analysis of data. For this the following command should be used
IMAGE=HISTOGRAM,imgfile
If at least one of keywords MinT
or MaxT
were defined for the processed image, then a new image is created with the name
<basename>_lvl.tif
, where <basename>
is the basename of the original image file.
In the created image the levels are adjusted by zeroing pixel values outside the range from MinT
to MaxT
whereas the values of other pixels are rescaled to the full range depending on the bit depth of the original image.
The figure below demonstrates the effect.
Comparing images
Comparison of images can be done with the command
IMAGE=COMPARE,img1,img2,...
where img1
and img2
are names of image files, which should be defined in BASE with keyword imgfiles
as usually.
At least two images must be defined in the command. All images must be of the same size, bit depth, and photometric interpretation.
The procedure does a pixelwise averaging of pixel values and calculation of respective standard deviations.
In the processing two new images are created and saved in files <basename>_average.tif
and <basename>_stdev.tif
,
which correspond to the averaged image and the image representing standard deviations of pixels.
The <basename>
is the name of the first processed file without extension.
Modification of ED intensity
The INT
command can be used not only for introducing total experimental electron scattering intensity functions but also for their modification.
In this case the general syntax is
INT=jobtype,int1,[int2,int3,...]
where jobtype
is the type of modification, int1
and possibly int2
and so on are intensity identificators.
The types of the modification are

S4MLT
— multiplication by s^{4} function. 
INORM
— calculates integral value of the total intensity and normalizes the intensity on this value. 
RSECDIV
— divides the total intensity by the reduced sector function. 
SPLDIV
— total intensity approximated by cubic spline and divided by this spline. By default number of inflection points for the spline is zero. This number can also be defined in the command. 
SMOOTH
— smoothing total intensity using natural Bsplines. 
SCALE
— more than one curve must be defined, the first curve remains unchanged, the other curves scaled so that they fit the first curve best. 
TSCALE
— intensity values are divided by tfactors from respective data sets. 
RESPCORR
— correction using response function. The response function must be already refined or introduced with theRESPFUNC
command. 
COPYMODEL
— copy data from the model I(s) intensity. This makes experimental I(s) equal to the calculated model I(s). 
ADDGNOISE
— add Gaussian noise to existing experimental I(s). Standard deviations for the Gaussian distributions can be defined in input (see ED intensities) or calculated in averaging procedures.
Note, if standard deviations were defined for the total intensity, then their values are also modified accordingly.
For modifying experimental molecular intensity functions a similar command SMS
with the same syntax exists
SMS=jobtype,int1,[int2,int3,...]
The available variants for jobtype
are

SMOOTH
— smoothing of molecular sM(s) intensity using natural Bsplines. 
COPYMODEL
— calculate and copy data from model sM(s) intensity. This makes experimental sM(s) equal to calculated model sM(s). 
ADDGNOISE
— add Gaussian noise to existing experimental sM(s). Standard deviations for the Gaussian distributions can be defined in input (see ED intensities), calculated by various procedures (for example in averaging, combining and background procedures, see this manual) or estimated inMINIMIZE
.
Models for ED intensity
In independent atom approximation total electron diffraction intensity can defined as [13]:
where I_{mol} is its molecular part (i.e. function depending on molecular dynamics and geometry) and I_{at} is the atomic part — a function depending only on properties of atoms but not on their relative positions. In reality the measured total intensity also contains some additional extraneous additive background B:
Note, the total intensity and all its components are here per unit time, i.e. they are flux values. In real experiments signal is accumulated over finite time, so in general case the model for the total intensity must include a tfactor proportional to exposure time:
In this case I_{tot} can be directly compared with experimental data. If rotating sector is used, it modifies the measured total intensity. Mathematically this can be defined using sector function S as
where M is the reduced molecular scattering intensity and β is the reduced additive background:
This model of I_{tot} corresponds to the setting IModel=a1bgl
.
For details on how sector function is defined and calculated see chapter ED sector function.
The other possibility is
which corresponds to the setting IModel=a2bgl
.
In case of using multiplicative background (see chapter Multiplicative background) the model is set to IModel=mbgl
and the total intensity is calculated as
where k is the scale factor for the molecular intensity M and Φ is the multiplicative background.
Calculation of molecular intensity I_{mol} depends on the setting of the keyword ImolAnhTermModel
.
If ImolAnhTermModel=Morse
then the formula is
where ij are the indices for the pair of atoms i and j, f is the scattering factor (see chapter ED scattering factors), l is the mean amplitude of interatomic vibrations, r_{a} is the thermalaverage interatomic distance, a is the parameter of the Morse anharmonic potential. The summation is performed for all pairs of atoms.
For ImolAnhTermModel=Asym
the model is
The symbols here have the same meaning as above, except that the asymmetry constants κ are used. This is the default approximation. Note, the c_{3} constants from cumulant3 approximation (as printed by, for example, the ElDiff program) are related to κ parameters as κ = c_{3}/6.
Dynamic models
The formulae above for I_{mol} are for semirigid molecules.
For nonrigid molecules dynamic models can be constructed using pseudoconformers
and defining potential energy function (see chapter Potential functions).
The molecular intensity is calculated according to DynImolModel
setting for the respective molecule.
If DynImolModel=sum
then I_{mol} is defined as a weighted sum of molecular intensities of pseudoconformers:
here P_{i} and I_{mol,i} are weighting factor and molecular intensity function for the pseudoconformer i.
The summation is performed over all N pseudoconformers.
The pseudoconformers are considered as semirigid and their respective I_{mol} are calculated as described above.
UNEX explicitly implements onedimensional dynamic models,
i.e. it is possible to choose one geometrical parameter as a coordinate φ for largeamplitude vibrations.
This is done assigning negative group number (usually 1
) to the respective parameter in Zmatrix of the first pseudoconformer
(the assignment for others is done automatically).
All pseudoconformers have different fixed values of this dynamic coordinate and it is excluded from all refinements.
Its values are used for calculation of potential energies of respective pseudoconformers.
The weighting factors are calculated from potential energies using a variant of Boltzmann distribution:
where k_{i} is the degeneracy factor of the pseudoconformer (see keyword sing
),
Q is the normalization denominator (sum of all P_{i} calculated with Q=1),
V(φ_{i}) is the pontential energy of the pseudoconformer with dynamic coordinate φ=φ_{i},
R and T are universal gas constant and temperature, respectively.
The other possibility is DynImolModel=integ
. In this case the molecular intensity of a nonrigid molecule
is defined as the integral [14]:
In UNEX the integration is done numerically (using trapezoidal rule) from φ_{min} to φ_{max}
defined by respective pseudoconformers. Here P is the same Boltzmann distribution as above.
This last integ
variant is computationally slightly more expensive but also more consistent in comparison to sum
above.
It is recommended to use integ
.
Mixtures
When the probe is a mixture of different species then the molecular and atomic scattering intensities are defined as weighted sums
where x_{i} is the mole fraction of species i, I_{mol,i} and I_{at,i} are molecular and atomic scattering functions of species i.
Values for mole fractions can be defined using keyword molx
in information fields of respective molecules.
Note, they must be in range 0.0 — 1.0 and are constrained as
The complete reduced molecular intensity M(s) is defined in UNEX as
which is generally not equal to the less accurate approximation
Below is an extreme example of 1:1 mixture of CH_{4} and CI_{4} showing differences between two methods for calculation of complete M(s) function.
The figures demonstrate that the later equation significantly overweights the contribution of CH_{4} in the total sM(s). This can be even better seen on the corresponding radial distribution functions below.
The figure on the left side correctly depicts the dominant contribution of CI_{4}, whereas the signals from CH_{4} can hardly be seen. On the right is the RDF from molecular intensity calculated using the last inaccurate equation. Clearly the contribution of CH_{4} is significantly overestimated.
Note, both methods are equal for models of mixtures of species with equal empirical formulae. For example, this is the case for models of conformational mixtures.
ED background lines
Structural analysis in gas electron diffraction is performed on the basis of the molecular part of the total ED intensity. The molecular intensity is obtained by applying background elimination procedure. In UNEX are implemented models for two major types of backgrounds, multiplicative and additive. The later can be additionally defined in two variants.
Multiplicative background
Lets assume that the total intensity can be described by the model (see above)
This can be further modified to
From here an exact expression for the sM function can be easily obtained as
where function Φ is the so called multiplicative background, defined as
The advantage of the last expression for the sM is that it has no sector function S in explicit form. If the sector function is smooth (i.e. the sector device is of good quality) and experimental conditions result in smooth background β then the multiplicative background Φ must also be smooth. However, separation of the two smooth functions, β and Φ is a very illposed problem and in real practice an approximate formula is used for calculation of experimental sM function:
This expression becomes exact if extraneous background β is zero, which is never achieved in real experiments. However, in structural analysis, when model sM functions are fitted to the experimental data, scale factors for the sM curves are usually refined to compensate for this problem. If experimental sM curves are obtained by accounting for multiplicative background as shown above then the refined scale factors k can be defined as
Clearly, in real cases they are less than 1 due to positive β. The smaller is the background, the closer is k to 1. If refined scale factors are larger than 1, this is strong indication of deficiency of the theoretical model. It should also be mentioned that the presence of strong background does not necessarily lead to significant inaccuracies in the obtained experimental sM curve. Much worse is a possible deviation of the forms of the functions B and I_{at}, making β nonconstant in the range of observable diffraction angles and, as such, not allowing for compensation with a single scale factor k.
Technically, multiplicative background is estimated on the basis of the calculated model molecular intensity sM_{mod}. First, the model background is obtained from the equation
In real practice the calculated Φ_{mod} is not smooth and can contain oscillations due to inexact sM_{mod} and experimental noise from the total intensity I_{tot}. On the next step Φ_{mod} is smoothed using cubic splines [15] or fitted with a polynom by minimizing the functional
The obtained in this way smooth line is called experimental background Φ_{exp} and is used for calculation of the experimental molecular intensity
which can be afterwards used in structural analysis, for example in leastsquares method. Thus, the background Φ_{exp} is in fact modeldependent and, as a consequence, the so called experimental molecular intensity sM_{exp} is to some degree also modeldependent. To overcome this problem it is recommended to refine the model on the basis of the obtained sM_{exp} and to use the updated sM_{mod} for calculation of a new experimental background and a more accurate experimental sM_{exp}. This procedure should be iteratively repeated until selfconsistency.
In UNEX there is a command BGL
for obtaining smoothed experimental
multiplicative background lines and corresponding experimental molecular intensities.
The general syntax of the command is as follows:
BGL=int1,int2,int3,...,intn [number] [keywords]
Here int1, int2 and so on are the identificators of the data sets, whose intensities need to be processed.
With the optional integer number
can be defined the required amount of inflection points
for splines (see details in [16], this is default) or, alternatively, the
order of polynomial function used for smoothing of the background.
The less is the number, the smoother will be the calculated background line.
BGL
command can also accept one or more keywords with respective values:

IModel
— keyword for choosing the model of the background. Can accept optionsmbgl
,a1bgl
anda2bgl
. 
CalcSigma
— boolen keyword (can accepttrue
orfalse
) for turning on or off the calculation of standard deviations of experimental sM(s). 
ApproxType
— type of background approximation. Can be one ofSpline
,Polynom
andChebPolynom
. 
CubSplQ
— cubic spline Qfunctional value, floating point number. 
Anchor
— for defining anchor points. 
RBPSDThr
— definition of threshold RBPSD for smoothing cubic spline.
Note, these keywords are optional and the particular settings for the BGL
procedure are determined at run time according to the priority scheme (from highest to lowest):

Keywords of the
BGL
command. 
Settings of the data set, see ED intensities chapter.

Global settings.
For example, the most important setting, background model, can be defined in the BGL
command using the IModel
keyword.
If it is not indicated, the setting from the data set is checked. If it is also not defined,
then the setting of the global keyword IModel
is used.
Note, not for all local keywords exist global or specific for data sets analogs.
CalcSigma
can be used if there are reasonable standard deviations
(introduced on input or calculated at run time) for the experimental total intensity,
which should be propagated into standard deviations of the calculated in BGL
procedure experimental sM(s)
BGL=11,3 CalcSigma=true
ApproxType
can be used for explicit choosing of the approximation type for the background.
However, in most cases the default Spline
setting from the global keyword BglApproxType
is the
most appropriate choice, unless something special is happening.
CubSplQ
can be defined for spline approximation, if no other criteria for the background line
smoothness are required. With this keyword a particular functional value Q can be given,
for which a spline must be calculated.
No integer number should be given in the BGL
command when CubSplQ
keyword is used,
for example
BGL=11 CubSplQ=1.0e5
Another criterion for the background smoothness is its relative power spectral density (RBPSD) in the range of structural frequencies.
This can be activated by using the RBPSDThr
keyword, for example
BGL=11 RBPSDThr=20.0
In this case a cubic spline will be used for approximation of the background and its smoothness will be adjusted iteratively
so that the final RBPSD will be not larger than the requested value (20.0 in the example above).
UNEX also provides a possibility to control approximation of background lines with help of anchor points,
which can be defined in BGL
commands using the Anchor
keyword as triples of numbers. For example, the command
BGL=11,2 Anchor=1.0/0.5327/1.0;18.4/0.4851/1.0
starts the procedure for the calculation of background for the intensity curve 11
.
Here with the Anchor
keyword two triples of numbers are indicated. These are the anchor points.
The format of the Anchor
keyword does not allow spaces or commas. The numbers may be separated by slashes \
and semicolons ;
.
In each triple the first number is the argument svalue, the second is the background anchor value
and the third is its weighting factor.
Thechnically, the anchor points simply substitute corresponding points of Φ_{mod},
so the anchor svalues must be also in the intensity data set.
The procedure should approximate background so that its calculated values are close to anchor points.
The larger are the weights of the archor points, the closer the approximated background line will be to them.
Note, this is true for cubic spline and simple polynomial approximations, but not for Chebyshev polynomial approximation.
Also the final result may be influenced by possible constraint(s) on the amount of inflection points,
polynomial power, etc.
Finding optimal anchor points is in general a manual trialanderror procedure.
The number of anchor points is not limited.
If several intensity curves are defined in one command they will be processed sequentially and independently. For example, the command
BGL=11,12,2
is equivalent to two sequential commands
BGL=11,2 BGL=12,2
As has been pointed out above the BGL
command calculates modeldependent background
because of the usage of the model sM(s) function.
The recalculated background, and as a consequence, updated experimental sM(s)
can be used to refine model parameters, including scale factor(s) k for molecular intensity(ies).
In UNEX it is possible to refine the best scale factors k internally in the BGL
command.
This is an iterative procedure, which can be turned on by setting the keyword
BglRefScaleMaxIter
to a some positive value, for example 30.
The convergence criterium of this procedure can be controlled by the optional keyword BglRefScaleTol
.
The accuracy of refined molecular parameters directly depends on the accuracy of the used
experimental molecular intensity function [17].
In turn, the molecular intensity is the result of the background elemination procedure
and it is very important that the backgound is reasonably smooth.
Otherwise the frequencies in the experimental molecular intensity can be
significantly biased, which can naturally lead to biased refined parameters.
The quality of the background line depends on how the total intensity is levelled.
This property strongly depends on the form of the sector device used in the experiment.
Best of all if values of the total intensity are in a narrow range of values
for all observable diffraction angles.
In this case the default procedure for smoothing of background lines works well
and the amount of inflection points serves as a good indicator of the background quality.
However, this criterion is not significant if the intensity curve changes too quickly.
A possible solution of this problem is to smooth reduced
(divided by sector function and atomic scattering) background by setting BglSmoothReduced=1
.
This requires introduction of a sector function.
Note, for this particular procedure the sector function must not be necessarily very accurate.
It is enough to define calculated values for your type of sector.
The most important is to ensure that this function is smooth and together with atomic scattering
lead to a well levelled reduced total intensity. The figures below demonstrate a case for the total intensity of CCl_{4}.
The first two demonstrate results of the default procedure when only 3 inflection points are allowed.
It is hardly possible to assess the quality of the total background line (on the left side).
However, in the reduced form (on the right side) the relatively low quality of the background is getting clear.
In contrast, smoothing of the reduced background results in the line of much
superior quality as the figure below shows.
It should, however, be noted that higher quality background lines naturally lead to higher Rfactors.
Additive background
Two types of additive background can be calculated in UNEX.
For the first type the model a1bgl
of total intensity is used:
In can be calculated using the BGL
command with the IModel
keyword as,
for example
BGL=11 2 IModel=a1bgl
Note, the calculated background is
For the second type the model a2bgl
is used, which is defined as
The corresponding UNEX command is, for example
BGL=11 2 IModel=a2bgl
In this procedure the background B is calculated.
Defining background type
Each invocation of any background command sets the respective type of model for the respective total intensity. For example, the commands
BGL=11,1 IModel=mbgl BGL=12,2 IModel=a1bgl BGL=13,3 IModel=a2bgl
internally set mbgl
, a1bgl
and a2bgl
as models for the total intensities of the data sets 11
, 12
and 13
, respectively.
The same happens when background lines are introduced in numerical form using the command BGL=READ
.
Thus the initial setting of the keyword IModel
for the respective data set can be overriden.
A special form of the BGL
command can be used for redefining the type of the model for total intensities:
BGL=SET,int
where int
is the data set identificator.
For example, the commands
BGL=SET,11 IModel=mbgl BGL=SET,21 IModel=a1bgl BGL=SET,31 IModel=a2bgl
set the models mbgl
, a1bgl
and a2bgl
for the total intensities in the sets 11
, 21
and 31
, respectively.
Averaging ED data
In UNEX there is a possibility for averaging ED intensity curves with an AVERAGE
command.
The general syntax of the command is as follows:
AVERAGE=itype,int1,int2,...,intn [> oint]
Here itype
is the type of intensity data to be averaged;
int1
, int2
up to intn
are identificators of intensity curves as input data for averaging;
> oint
is the optional argument with explicit identificator for output averaged data.
Output data set oint
must not necessarily by initialized before calling AVERAGE
.
However, if it was already initialized, the data will be overwritten.
If you do not define oint
explicitly then a new set of data will be initialized and its identificator will be printed to output.
itype
can be one of the following:

INT
— averaging total intensity curves 
INTS
— averaging total intensity with calculation of standard deviations 
SMS
— averaging experimental sM(s) intensity curves 
SMSS
— same asSMS
but with calculation of standard deviations
Experimental sM(s) must be initialized, i.e. introduced in UNEX or determined in background procedure from total intensity.
If background procedure is used for curves with unrefined scale factors, it is advisable to set the BglRefScaleMaxIter
keyword
in BASE to some positive value so that scale factors are adjusted to some reasonable values.
Note, averaged intensity values are calculated for s values of the first curve int1
defined in the command.
If points of the other curves are defined not in the same s values then interpolation with cubic splines is used.
In addition to averaging of data and optional calculation of standard deviations AVERAGE
command also calculates
experimental Rfactors [21].
For each of the curve, participating in averaging procedure, individual experimental Rfactors are calculated as
where is the ith point of the intensity curve, for which the Rfactor is calculated,
is the corresponding point of the averaged intensity with weight w_{i},
N is the total number of intensity points.
Intensity I is the total intensity if INT
or INTS
is defined, or sM(s) in case of SMS
or SMSS
.
Individual experimental Rfactors show how much each of intensity curves deviates from the average curve.
This information allows to sort out curves of low quality. An average value of individual R_{exp} is also printed.
Regarding weighting, UNEX calculates two types of experimental Rfactors:

with account of weights w, which are calculated from respective standard deviations of the averaged curve as .

without weighting, assuming all w_{i} = 1.
Note, in INT
and SMS
modes standard deviations are not calculated so both types of experimental Rfactors are equal.
Weighting works in SMSS
and INTS
modes, when standard deviations for the average curve are calculated.
Next UNEX calculates total experimental Rfactor as
Here summation is performed for all points of all M intensity curves. I can also be total or molecular sM(s) intensity, depending on averaged data type.
The total experimental Rfactor allows to represent numerically the overall reproducibility of experimental data and their general quality.
Weighted and nonweighted (setting all w_{i} = 1) total and average experimantal Rfactors are calculated.
Note, the weighted experimental Rfactors are calculated even if the standard deviations were not computed in this particular procedure.
They could have been defined or calculated earlier for the averaged data set.
Run PRINT=INTS
or PRINT=SMSS
for the averaged data set to see the values of standard deviations used for the calculation
of the weighted experimental Rfactors.
The advantage of experimental Rfactors based on total intensities is that no molecular model is needed for their calculation. However, the absolute values of such R_{exp} are generally meaningless. They can mostly be useful for comparison of data sets produced only by the same experimental setup. In contrast, experimental Rfactors on the basis of sM(s) curves are directly comparable with structural Rfactors:
where and are the experimental and model sM(s), respectively.
R_{str} indicates level of disagreement of the model with experimental data, while R_{exp} indicates reproducibility of experimental data. There can be several situations:

R_{str} >> R_{exp}: data are reproducible but model cannot describe them; the model should be improved.

R_{str} << R_{exp}: model describes data too well, probably not reproducible data features are fitted; better data are needed.

R_{str} ≈ R_{exp}: optimal solution if both values are small.
Note, if both R_{str} and R_{exp} are large, then something went completely wrong, first of all in experiment and/or in data reduction.
Also note that weighted R_{str} (named wRd
in UNEX output) should be compared with weighted R_{exp}, likewise nonweighted R_{str} should be compared with nonweighted R_{exp}.
Below are several examples of averaging commands.

Simple averaging of intensity curves
11
,12
and13
. A new average curve is created and assigned to an automatically chosen identificator.AVERAGE=INT,11,12,13

Same as above, but also standard deviations are calculated for the averaged data.
AVERAGE=INTS,11,12,13

Similar to the first example. Here the output average curve is accessible using the
14
identificator.AVERAGE=INT,11,12,13 > 14

Averaging experimental sM(s) curves
21
,22
,23
and24
. The output data is written to31
.AVERAGE=SMS,21,22,23,24 > 31

Averaging experimental sM(s) curves with calculation of standard deviations. The output data are written to a new automatically generated set with a new identificator. It is written in log file.
AVERAGE=SMSS,21,22,23,24
Combining ED data
Another possibility of converting multiple ED curves into a single curve provides COMBINE
command.
It has exactly the same syntax as the AVERAGE
COMBINE=itype,int1,int2,...,intn [> oint]
but in general case does a different job creating a curve with svalues present in all input curves
(in contrast, AVERAGE
takes points s only from the first input curve).
Thus, curves with different sranges can be combined together. For the overlapping areas averaged values are calculated.
If standard deviations were initialized for the respective input data sets, weighted averaging is used,
where weights are calculated as reverse squares of the respective standard deviations.
The Figure below demonstrates how COMBINE
works for two experimental sM(s) curves obtained from different nozzletodetector distances.
The respective command was
COMBINE=SMS,11,21 > 31
Note, in COMBINE
experimental Rfactors are calculated in a similar manner is in AVERAGE
(see above).
There are, however, some peculiarities.
The weighted Rfactors are also calculated if standard deviations were not requested for calculation.
However, in contrast to AVERAGE
, standard deviations are still calculated internally, used for computation of wRexp,
but are not saved for the averaged data set. In this case calling PRINT=INTS
or PRINT=SMSS
is useless
for getting respective standard deviations. Also it should be noted that standard deviations are set to 1e99
if only one data set has contribution to the combined data in the respective point.
ED radial distribution functions
Radial distribution functions in UNEX are calculated and printed by the RDF
command:
RDF=int1,int2,...,intn
The argument(s) of the command is a list of one or more ED intensity curves with initialized sM(s) functions. The calculation is essentially a sineFourier transformation of the respective experimental and model sM(s) curves:
If several curves are provided in the list of arguments, they are concatenated before Fourier transformation, for example
RDF=11,21
Note, the curves must have common range of svalues.
The concatenation procedure includes relative scaling of curves, reinterpolation to common argument values (if required) and weigted averaging in common ranges of argument.
Next, there are three general modes for calculation of radial distribution functions, which directly influence the minimal and maximal values of integration argument s.
The modes, controlled by the RdfType
keyword in BASE, are as follows:

old
, the most simple method, takes (combined) experimental sM(s) curve and performes integration in the range of svalues in which this curve is defined. The resulting radial distribution function usually looks not nice since integration starts from s_{min} not equal to zero. 
classic
, a more advanced method, in which experimental sM(s) curve is supplemented with respective model curve on the left side before integration. This is done in order to get an "experimental" curve, which starts from s=0. This, in turn, stabilizes integration and improves the overall appearance of the radial distribution function. Note, this makes sense if model sM(s) function fits well the experimental data. On the right side the experimental curve remains unchanged and integration is done till maximal s value for which the curve is defined. This can lead to problems with integration since at maximal experimentally achievable svalues of 3040 Å^{1} sM(s) functions are numerically not enough converged to zero. To overcome this problem the experimental sM(s) can be multiplied by an exponential function for damping, see keywordRdfDamp
in BASE. Note, this procedure leads to broadening of peaks on the radial distribution function and to reduction of its resolution. 
modern
, the most advanced method, which supplements experimental sM(s) with model data not only on the left side as in theclassic
variant but also on the right side, so that s_{max} is as large as possible (60 Å^{1} in current implementation). This allows to avoid usage of damping exponential function and, as a result, leads to improved resolution of F(r). However, to obtain good results the model function must fit experimental data well.
Note, the concatenation of model and experimental sM(s) curves in classic
and modern
types of RDF requires some overlap of these functions.
The extent of the overlapping can be adjusted using keyword RdfNconcat
.
Below are some examples of RDFs for benzene using different options of RdfType
:

Below is the graphical representation of the influence of damping function on the RDF in case of benzene when RdfType=classic
and experimental data available only up to 30 Å^{1}.
If the damping is turned off, i.e. RdfDamp=0.0
, the RDF has multiple false peaks. An optimal value of damping factor removes these peaks.
Too large value of the damping factor increases widths of true peaks so that the resolution of RDF is too low.
The next issue in calculation of RDF is connected with representation of contributions of different terms in sM(s) functions. In a crude approximation the contribution of a pair of atoms to diffraction pattern is proportional to the product of their atomic numbers. The respective RDF in this case can be easily analysed.
In reality, however, contributions of atomic pairs to diffractions patterns are not constant on the sscale and are even not linear (this property is characterized by gfunctions). As a consequence, the calculated RDF is difficult to analyze. Fortunately, ratios of many gfunctions for different types of atoms are much closer to constants than gfunctions themselves. Therefore UNEX provides a possibility to divide the integrated sM(s) data by a gfunction, see RdfDivGf
keyword in BASE. This can improve the appearance of the obtained RDFs. By default for this purpose UNEX chooses gfunction for the pair of atoms with maximal product of atomic numbers. Note, however, that this logic can fail in molecules containing atoms with very different atomic numbers. In this case some gfunctions can go through zero and change sign. Accordingly, RDF cannot be obtained by integration of sM(s) modified by such a gfunction. In this case an optimal gfunction can be chosen manually by using RdfDivGfAtoms
keyword.
In case of benzene the influence of RdfDivGf
is as follows:
For obtaining RDFs integration is done numerically. For this UNEX implements two methods, simple trapezoidal and more accurate but slower Romberg method. In most cases the first method is accurate enough and is used by default. Switching between integration methods is done by RdfIntegMethod
keyword in BASE.
Regarding rvalues, for which RDFs are calculated, two schemes are implemented in UNEX:

If
RdfAdaptiveR=1
(by default it is=0
, i.e. turned off) the socalled adaptive step size is used depending on the local curvature of the RDF in each point so that obtained function is accurate enough for numerical analysis. 
For purposes of visual analysis RDF is calculated on a fixed grid on the rscale, where step is determined by the
RdfRdr
keyword. However, for better appearance some points can be skipped, so that in general they are arranged nearly equally dense along the RDF line and not along the rscale. This is default and controlled by theRdfPruneRlen
keyword. To turn off the prunning of points setRdfPruneRlen=0.0
.
The RDF defined above as integral F(r) is essentially the distribution P(r)/r, where r is the distance between atoms and P(r) is its distribution function. The first moment of the function P_{ij}(r)/r for a particular pair of atoms ij is the r_{a} type of thermally averaged distance between these atoms. Thus, RDFs calculated as described above show peaks centered at r_{a} distances. There is, however, a possibility to obtain RDF defined as P(r), which is more natural. For this, UNEX can multiply the integral by r, see RdfMultR
keyword. In this case the peaks are centered at r_{g} distances between atoms. Note also that this procedure naturally increases the difference curve (difference between model and experimental RDFs) proportionally, so this should not be misinterpreted as a worsening in the model fit.
Below is such an example using data for PhCH_{2}CH_{2}CH_{2}SeCF_{3} molecule. Note again, both RDFs were obtained for exactly the same experimental data and model.
The advantages of the P(r) function are clearer physical meaning and more distinct visibility of contributions from terms with large interatomic distances.
In contrast, the P(r)/r function effectively hides discrepancies between data and model, which hinders analysis.
Therefore in UNEX the default setting is RdfMultR=1
so that P(r) RDFs are generated.
UNEX can also calculate pure model radial distribution functions if the 
If your experimental molecular intensities have meaningful standard deviations it is possible
to calculate errors of experimental RDFs by switching the RdfCalcStdevs
keyword on.
Standard deviations for sM(s) can be defined in input file as absolute values,
calculated in averaging procedure or in background procedure from respective errors of total intensity functions.
Standard deviations for sM(s) are also estimated in MINIMIZE
but should be used with care in case of large Rfactors.
Note that the calculated values of standard deviations for RDF’s only represent random errors
propagated from respective errors of sM(s).
The default method for calculation of standard deviations uses error propagation formula and numerical
differentiation of sM(s) functions. For large models the calculations can be (very) time consuming.
There is an alternative procedure which uses the MonteCarlo method. This can be activated by setting the keyword
RdfMCEsdIter
to some positive integer value, indicating the number of iterations.
It is recommended to investigate the convergence of the calculated results with respect to the number of iterations.
In some cases the optimal value of RdfMCEsdIter
can be from several hundreds to several thousands.
Below in Figure simulated molecular intensity functions for 1,2dichloroethane (1:1 mixture of anti and gauche conformers) are shown. Random Gaussian noise, with standard deviations 0.03 for the curve above and 0.025 for the other curve, was added to the simulated experimental data.
These data were used to calculate model and experimental RDFs with respective standard deviations. The obtained data are plotted on the Figure below. Note, the calculation was done with RdfMultR=1
, that is RDFs corresponding to P(r) were calculated. Therefore oscillations of the difference curve and the standard deviations increase when r increases.
Alternatively UNEX can calculate more traditional RDFs of type P(r)/r by switching the RdfMultR
keyword off. Usually in this case standard deviations are distributed approximately equally along r scale. The Figure below demonstrates P(r)/r for 1,2dichloroethane calculated from the simulated sM(s) data shown above.
For graphical interpretation of RDFs it is useful to print also interatomic terms and their contributions with 
Refinement of molecular parameters
Refinement of all kinds of parameters in UNEX is closely associated with the term group.
Group is a list of parameters tied by particular constraints.
Most often the constraints are fixed differences between values of parameters within each group.
In case of vibrational amplitudes for interatomic distances there is also a possibility to fix their ratios within each group instead of differences (see GedVarAmplScale
keyword).
The number of parameters in a group is not limited. However, only particular kinds of parameters can be grouped together.
Formally a group can also consist of only one parameter.
In this case the respective parameter is not tied to any other parameter in refinement procedures.
Groups are defined by unique integer numbers. Each parameter can be defined with its respective group number.
Several parameters with the same group number are combined together to a single group and
refined with fixed constraints.
By default parameters are defined without group numbers, which is the same as to define the group number to 0.
If you want to refine parameters, you have to define their group numbers larger than zero explicitly.
For refinement procedures the amount of variables is equal to the number of active groups.
Thus, multiple parameters in a group in fact act as a single parameter in leastsquares refinement.
Consequently they share the same estimated standard deviation as a group,
not as they had individual but equal standard deviations!
Below is the list of parameter types, which can be refined in UNEX.

Geometrical parameters of Zmatrices. See respective chapter on how to assign group numbers to these parameters. Parameters of the same type can be combined together in a signle group, for example a distance can be combined only with other distances. It is generally impossible to combine different types of geometrical parameters into a group. However, it is allowed to combine angles and torsional angles. Note, geometrical parameters of different molecules can be tied together in one group.

Interatomic distances r_{a} can be refined independently. See
GEDTERMS
command on how to define respective groups. 
Vibrational amplitudes of interatomic pairs. Respective groups can be defined in the same fields as the values for amplitudes. Alternatively, there are possibilities to group amplitudes (semi) automatically after their definition (see below). Amplitudes in different molecules can be grouped together.

Mole fractions for molecules in mixtures, see parameters
molx
andvarx
in molecular fields. 
Parameters of potential functions in dynamic GED models of molecules.

Scale factors for GED molecular intensity functions, see the
VarSc
keyword in field of intensity curves.
Semiautomatic grouping of amplitudes can be done using AMPLGROUP
command.
AMPLGROUP=mol,gstart, r1s,r1e, r2s,r2e, ...
Here the grouping is done for the molecule mol
. The numbering of groups starts from the number gstart
.
The parameters r1s
and r1e
should be floating point numbers, defining range of distances in the molecules.
Amplitudes of terms, whose r_{a} distances are within this range, are grouped together.
Multiple ranges can be defined, r2s
and r2e
above indicate the second range.
In the following example
AMPLGROUP=mol,200 1.4,1.6 2.0,3.0
amplitudes of terms in mol
with r_{a} distances between 1.4 and 1.6 Å are assigned to a group with the number 200
.
Next, amplitudes of terms with r_{a} distances in the range from 2.0 to 3.0 Å are assigned to a group with the number 201.
Another possibility is the fully automatic grouping of amplitudes for all molecules
in the model with AUTOGROUP
command:
AUTOGROUP=AMPLITUDES
This procedure calculates internally a radial distribution function (RDF) for the current ED model
and assigns equal group numbers for amplitudes of the terms, which reside within single particular peak of the RDF.
Accordingly, this method requires completely initialized ED model.
In real practice it is strongly advised to analyse the performed by this procedure grouping and adjust it manually if need.
Also note, both AUTOGROUP
and AMPLGROUP
procedures perform grouping only for those aplitudes
which had previously group numbers equal to zero.
MINIMIZE
UNEX provides several refinement procedures. The most important one is the leastsquares (LS) method.
In UNEX minimization of LS functionals can be done with MINIMIZE
command.
Its general syntax is
MINIMIZE=functional(s) [data] [groups]
The type of functional is the required parameter of the command, data should be defined in case of ED, groups can be defined or prohibited explicitly but this is optional (see below). There are several types of functionals:

GEDSMS
— molecular part of electron diffraction intensity in form of sM(s). 
ROTCONST
— rotational constants. 
REGPRM
— regularization parameters, also known as flexible constraints or restraints. 
RESTRGEOM
— restraining geometrical parameters.
In most general form the complete functional is represented as
Here the first, second, third and fourth terms correspond to GEDSMS
, ROTCONST
, REGPRM
and RESTRGEOM
types of functional, respectively.
The relative global weight factors α_{rot}, α_{reg} and α_{rgp} are defined by the keywords RotConstAlpha
, RegAlpha
and RestrGeomAlpha
, respectively.
If GEDSMS
is given, particular ED intensity curves must be indicated explicitly for constructing functional, for example
MINIMIZE=GEDSMS,11,21
Individual weights w_{i} for the sM(s) data points are calculated automatically from their standard deviations
as w_{i}=σ_{i}^{2}.
The other types of functional do not require explicit indication of their data. ROTCONST
automatically includes all defined
rotational constants for all molecules and for their isotopologues, see keywords ExpRotA
, ExpRotB
and ExpRotC
.
The weights are calculated in the same manner as for ED data from respective standard deviations defined with keywords
SigRotA
, SigRotB
and SigRotC
.
Functional REGPRM
uses data, which can be read in using the respective
command REGPRM
. The syntax of the command is:
REGPRM=mode,otag,ctag
The only available mode of the command is INPUNEX
, which means reading of the data as in the following example:
REGPRM=INPUNEX,<myregparams>,</myregparams> <myregparams> 1 1.490 0.001 2 110.0 0.1 3 0.05 0.01 </myregparams>
Here in the first column group numbers are indicated. In the second and third columns regularization values and their individual standard deviations are defined. In the equation above the regularization values correspond to p_{k}^{reg}, while the weights w_{k} are calculated from the introduced standard deviations as w_{k}=σ_{k}^{2}.
In UNEX versions before 1.61273 the values in the third column are interpreted as weights, not as standard deviations! 
Note, the regularization is applied to first parameters in groups, so the regularization values must be appropriate.
The other parameters in the groups are regularized automatically to the same extent due to rigid constraints.
Regularization must not necessarily be used for all refined parameters, restraints can be applied to some selected groups.
With REGPRM
it is possible to define regularization parameters for groups, whose parameters are not refined.
In this case the respective restraints are ingnored and not included in least squares functional.
This is because in different MINIMIZE
(or in SEARCH
and others) procedures different groups of parameters can be refined,
whereas the list of regularization parameters is global.
The units of the regularization values must be the same as the units used for introduction of respective model parameters.
For parameters of potential functions internal UNEX units must be used. They can be checked by printing data using PRINT=POTENTIAL
.
RESTRGEOM
functionals are built on the basis of geometrical parameters of molecules.
The allowed types of parameters are interatomic distances, angles, torsion angles and outofplane angles.
For definition of parameters and numeration of atoms see section for geometrical parameters in chapter Data printing.
The restraining data are read in for particular molecules with RESTRGEOM
command as in the example:
RESTRGEOM=mol,INPUNEX,<rgeom>,</rgeom> <rgeom> bond 1 2 0.970 0.010 bond 2 3 1.385 0.005 bond 3 4 1.200 0.002 bond 3 5 1.185 0.002 angle 1 2 3 103.1 0.5 angle 2 3 4 115.6 0.5 angle 2 3 5 114.1 0.5 angle 4 3 5 130.1 0.5 torsion 1 2 3 4 0.0 0.00001 oop 5 3 2 4 0.0 0.00001 </rgeom>
In each line first a string with type of parameter is given. The possible types are bond
(dist
and distance
can also be used as synonyms), angle
, torsion
and oop
(for outofplane angles).
Next, numbers of atoms must be defined. Finally, values and respective standard deviations for the defined parameters are given.
Note, the standard deviations are optional but it is recommended to indicate them explicitly. Otherwise they are assumed to be 1.0.
The units for distances and angles are Angstroms and degrees, respectively. In LS functional the weights are calculated from the defined here
standard deviations σ as w_{l}=σ_{l}^{2}. Note, for models of mixtures each molecule can have
an individual set of restraining geometrical parameters. All of them are participating in the same respective sum in the LS
functional given above. The prefactor α_{rgp} is thus the same for all sets.
In should be noted that restraining geometrical parameters should not necessarily be equal to the parameters in
definition of molecular geometry, which can be refined in MINIMIZE
(or in other procedure).
Also, distances can be given for chemically nonbonded atoms.
In the beginning of the MINIMIZE
procedure the information on data for constructing the LS functional is printed.
This include number of data, α values, etc. If electron diffraction intensities are used, then
the structural resolution and maximal structural distance are estimated. The former is calculated as
where s_{max} and s_{min} are the maximal and minimal svalues for which experimental ED intensity values are available. The maximal structural distance is calculated as Nyquist frequency due to the Whittaker–Nyquist–Kotelnikov–Shannon sampling theorem [29, 30]:
where Δs is the average spacing between adjacent intensity data points in Å^{1}. The r_{max} value shows the largest interatomic distance, which can possibly be determined from the current electron diffraction data set.
Minimization of the LS functional can be done using two methods:
A particular method to use (or a combination of methods) can be chosen with the MinMethod
keyword.
All methods are iterative, the maximal allowed number of iterations is defined by keyword MaxIter
.
Iterations stop in several cases:

Relative change in functional is less than threshold value (see keyword
LsqFunTol
). 
Maximal relative parameter addition is less than threshold value (see keyword
LsqAddTol
). 
Maximal relative gradient (partial derivative of functional with respect to some refined parameter) is less than threshold value (see keyword
LsqGrdTol
). 
Parameter
Lambda
in LevenbergMarquardt method increased too many iterations in a row (see keywordLsqLamMaxInc
). 
Maximal allowed number of iterations performed.
In UNEX there are two different weighting schemes in LS analysis: relative and absolute.
They influence calculation of standard deviations of refined parameters.
With absolute weights the matrix of covariances (with squares of standard deviations of parameters as diagonal elements)
is obtained directly as inverse of normal matrix.
In case of relative weighting the calculated cofactor matrix (the inverse of the normal matrix) is multiplied by the factor
χ^{2}/v, where v is the number of degress of freedom, calculated as the number of data points minus the number of
refined parameters v = N_{data}  N_{prm}.
The switching between absolute and relative weighting can be done using the MinAbsWeighting
keyword.
Depending on settings MINIMIZE
can print different types of data for information on minimization status, properties,
convergence and so on:

Total absolute and relative values of functional χ^{2} designated in output as
X^2
. The relative values are printed during iterations of solving LS problem. In this case the initial value of the functional is scaled to be 1.0 and the following values are relative to this initial unity. 
Lambda
is the parameter in LevenbergMarquardt method [31, 32] for improving convergence. Stable minimization is accompanied with decreasing of this parameter. 
Rf
andwRd
are printed in case of using electron diffraction data. The first one is the regular Rfactor calculated without weights:The other,
wRd
, is the Rfactor with diagonal weighting:Note, on iterations total wR_{d} values are printed, i.e. the summations are performed for all data points of all intensity curves, if several are used. Individual different types of Rfactors for particular intensity curves are printed after minimization.

RMSD
andWRMSD
are rootmeansquare and weighted rootmeansquare deviations, respectively. For rotational constants they are calculated aswhere N is the total number of rotational constants, and
Analogous formulae are used for other types of data. During iterations WRMSD for rotational constants and all restraining geometrical parameters are printed in columns
RotWRMSD
andRgpWRMSD
, respectively. 
When regularization data are used, the respective part of the total functional (see equation for χ^{2} above) is printed as
RegQ
.
After minimization several blocks of data are printed:

Information on the convergence of the procedure.

Statistics for the data and model:

Number of degrees of freedom (number of data points minus number of refined parameter groups).

Condition number (ratio of maximal and minimal singular values of normal matrix). Large values can indicate numerical instability.

Rank and nullity of the design matrix in LSQ method.

Goodnessoffit value printed in case of absolute weighting. It is defined as 1Q, where Q is the probability that the functional χ^{2} should exceed its refined minimal value by chance. Small values (close to zero) of goodnessoffit can indicate that (i) model is not adequate, (ii) standard deviations for data points are probably larger than stated, (iii) measurement errors are not normally distributed. On the other hand values of goodnessoffit close to or equal 1 can indicate that defined standard deviations of data points are too large/pessimistic.

Values of functional parts.

For sM(s) data different Rfactors (diagonalweighted
wRd
and without weightsRf
), RMSD/WRMSD, estimated standard deviations for the data (seeESD
in output), etc. 
For electron diffraction sM(s) data the DurbinWatson statistics [34, 35] are printed (indicated as
DW
). 
Also, unweighted Rfactors (for each curve separately and total) are calculated and printed for M(s). However, they are provided mostly for the completeness and for a possible comparison with results from other programs. Normally, for the assessment of the data fit, it is recommended to use the Rfactors calculated for sM(s) data.


Table with refined parameter values, their absolute and relative errors and partial derivatives of total functional with respect to these parameters. Errors of parameters are leastsquares standard deviations multiplied by factor
PrintEsdFactor
. 
Optional tables with contributions of functionals to refined parameters (see
CalcFuncProportion
keyword). Note, here errors (standard deviations possibly multiplied by a factor) and respective socalled experimental errors of the refined parameters are also printed. The experimental errors are defined and calculated as described in [4]. KeywordMinSigmaExcludeFunc
can be used to control this method. 
Matrix of correlations.

Table with correlations above 0.5, if there are any.

Optional table with χ^{2} (hyper)ellipsoid (see
MinPrintEllipsoid
keyword).
Several types of functional can be combined in MINIMIZE
. For example, the following command
MINIMIZE=GEDSMS+ROTCONST,11,21
refines parameters from rotational constants and indicated ED data simultaneously. In the same manner three types of data can be used
MINIMIZE=GEDSMS+ROTCONST+REGPRM,11,21
or with restraining geometrical parameters
MINIMIZE=GEDSMS+ROTCONST+RESTRGEOM,11,21
Other combinations are possible.
As already stated refinement of particular parameter groups can be (dis)allowed. By default all parameters with group numbers greater than zero are refined. If group numbers are defined explicitly in MINIMIZE
then only parameters in these groups will be refined. The following example demonstrates how to refine parameters only in groups 1
and 2
.
MINIMIZE=GEDSMS,11 1,2
The other possibility is to prohibit refinement of parameters from particular groups. In the following example parameters from all groups except group 5
are refined:
MINIMIZE=GEDSMS,11 !5
Several groups can also be excluded from refinement:
MINIMIZE=GEDSMS,11 !5,!6,!26
Combinations of permisive and prohibitive definitions of groups are meaningless. The command
MINIMIZE=GEDSMS,11 1,!2
is equivalent to
MINIMIZE=GEDSMS,11 1
i.e. refines parameters only from the first group.
OPTALPHA
In case of relative weighting of data in leastsquares functional the factors
α_{rot}, α_{reg} and α_{rgp} should have appropriate values.
The problem is that there is no single clearly defined criterion for their optimal values.
Usually these factors are adjusted according to specific requirements of a particular investigation
(see discussion of the problem in [36]).
There are, however, some heuristic criteria. One of them is implemented in UNEX
[37] (detailed description is given in [4]).
The respective command is OPTALPHA
:
OPTALPHA=functional(s) [data] [groups]
The syntax above is the same as for MINIMIZE
. In fact this is an iterative procedure,
which internally starts MINIMIZE
on each iteration.
Note, the procedure is implemented only for some cases when two types of functional combined together.
Always analyse result(s) of this command. Check whether the obtained alpha parameter fits your needs. In many cases it can be used only as a starting point for further search of optimal values. 
This procedure does not refine any parameters except respecive α. That is, any other types of parameters,
including molecular geometry, remain unchanged after this command. Also, standard deviations and correlations are not determined.
Thus, if you run only 
ROBUSTM
The described above command for minimization of LS functionals implements the main procedure in UNEX for these purposes.
A more complex method is provided by the ROBUSTM
command. In contrast to MINIMIZE
it modifies weights of
experimental data using bisquare weights scheme of Tukey [38, 39]. Accordingly, standard deviations of data points are adjusted.
The weights are calculated iteratively from respective residuals. In fact ROBUSTM
internally starts MINIMIZE
procedure on each iteration,
refines model and updates weights of all data points. This procedure is repeated until weights are converged or the limit for number of
iterations is achieved (see MinRobMaxIter
keyword). As the last step a normal minimization is performed and results are printed as
usually from MINIMIZE
. The syntax for ROBUSTM
is the same as for MINIMIZE
:
ROBUSTM=functional(s) [data] [groups]
By design 
SEARCH
MINIMIZE
and ROBUSTM
are local methods for optimization of models.
This means that they typically converge to local minima depending on starting approximation.
With these methods there is no possibility to know exactly whether obtained solution corresponds to global minimum on the functional (hyper)surface.
For this purpose in UNEX there is a special command SEARCH
, which provides two methods.
The first one implements a systematic scanning of functional by testing different values of parameters in defined ranges.
The syntax of the command is as follows:
SEARCH=SCAN,functional(s),otag,ctag,[data]
Here functional and data are defined exactly in the same manner as for MINIMIZE
command.
The opening and closing tags otag
and ctag
are for data field with information on which parameters should be used in scanning procedure.
The format of the data field can be best shown by example:
<scan> 2 1.76 100 1.78 4 0.04 100 0.06 </scan>
This is definition of a twodimensional scan. Each line defines a parameter or a group of parameters, which should be scanned.
Here the first line is for group number 2
. The first parameter in this group should be scanned in range from 1.76
to 1.78
(other parameters in this group will be automatically adjusted using fixed constraints) with 100
steps.
In the same manner scanning for the second group (in the example group number 4
) is defined.
In total, the twodimensional scan will do 100x100=10000 calculations of functional for different combinations of parameters.
After this UNEX reports statistics for parameters:

Values of paramaters (first parameters in respective groups) which correspond to minimal value of total functional.

Weighted statistics for each parameter:

Mean value.

Standard deviation.

Skewness.

Kurtosis (but not excess kurtosis).


Minimal tested value.

Maximal tested value.
In addition UNEX prints weighted correlations between parameters.
Accuracy of weighted statistics (including correlations) can be sensitive to the size of scanned area and to the number of tested points. In particular, area in the vicinity of minimum of functional must be sampled dense enough in oder to get well converged values of statistics. Otherwise values of skewness and kurtosis are printed as zeros if they cannot be calculated accurately. The sign for this is a very small value of respective standard deviation. In case of numerical problems with calculation of correlations respective rows and columns and diagonal elements are printed as zeros. Otherwise diagonal elements are exactly 1. In many cases to get reasonable values of statistics at least several tens of thousands points are required. To test for convergence you need to recalculate the values for at least twice as large number of scanned points. 
The procedure in SEARCH=SCAN
finds global minimum of functional within defined limits for values of parameters and with accuracy determined by scanning step size(s).
The problem is, however, that the required number of scanning points scales as S^{N}, where S is the number of steps for each parameter and N is the number of parameters or groups of parameters.
Thus, the total number of points increases very quickly with the number of scanning dimensions.
To avoid this problem UNEX implements MonteCarlo method for searching of global minimum of functional.
The syntax of the respective command is very similar to that shown above. The only difference is that RAND
should be indicated:
SEARCH=RAND,functional(s),otag,ctag,[data]
The definition of search parameters is also the same as in the example above.
For compatibility purposes the integer values (number of steps in case of SCAN
) between minimal and maximal allowed parameter values
are also indicated but do not play any role and simply ignored by UNEX.
In case of RAND
the amount of sampled points is controlled by time allowed for SEARCH
, see SearchTime
keyword.
Uncertainties of parameters
In inverse problems parameters are refined with uncertainties. After each leastsquares refinement standard deviations and correlation factors are calculated for parameters. They can be accurate in a simplest case, when the only source of uncertainty is the noize in experimental data and under condition that an appropriate weighting for experimental data is used. However, in real practice uncertainties in other parts of the refined model can propagate into uncertainties of the refined parameters.
A possible way to take this into account is to use the direct MonteCarlo simulation approach [40]. This method is implemented in UNEX and can be started by calling the command MCMIN
:
MCMIN=functional(s) [data] [groups]
This command has the same syntax as MINIMIZE
described above.
In fact, the MCMIN
procedure calls MINIMIZE
internally as a first step. After its convergence the MonteCarlo procedure is started. Parameters and/or data are sampled randomly from respective distributions and for the newly generated model and data a leastsquares method is started for minimization of defined in MCMIN
command functional. The refined values of parameters are saved. These steps are performed multiple times in a loop. Thus statistics for refined parameters are collected and the procedure is repeated until convergence or until the maximal allowed number of iterations is reached.
The following data can be randomized:

Experimental data (if
MCRandData=1
is defined inBASE
).
Experimental molecular scattering intensity functions. For details see below.

Experimental rotational constants, see keywords
MCStdRotA
,MCStdRotB
andMCStdRotC
in molecular fields. 
Regularization parameters, which indeed can be experimental from some other source or they can be assumed values with some defined absolute standard deviation.


Model parameters (if
MCRandParams=1
is defined inBASE
).
Parameters defining background spline smoothness, see keywords
MCBglQ
,MCBglQMin
andMCBglQMax
in field for scattering intensity. 
Electron wavelengths, see keywords
MCLam
andMCLamS
in input field for scattering intensity. 
Geometrical constraints (fixed values of parameters and their differences), see below command
ZMATMC
. 
Fixed amplitudes of interatomic vibrations or their differences and ratios, see command
TERMSMC
. 
Vibrational corrections for interatomic distances, see command
TERMSMC
. 
Relative abundances of molecules in mixtures, see keywords
MCvarx
,MCxMin
andMCxMax
.

There are several possibilities on how electron diffraction intensities are randomized.

In case of absolute weighting (
MinAbsWeighting
) each point is sampled from a gaussian distribution with respective standard deviation and centered on original experimental value of this point. 
If standard deviation is defined for sM(s) globally with the keyword
MCsMsSpread
then this value is used for all points. 
Otherwise are used estimated standard deviations for each curve independently from the initial leastsquares run. Note, within each curve all points have the same value of standard deviation.
The first variant is the best if accurate standard deviations are known for molecular intensities.
For rotational constants the only option is to define directly standard deviations in molecular field(s) using keywords MCStdRotA
, MCStdRotB
and MCStdRotC
. The random values are sampled from respective gaussian distributions centered on original experimental values. The initial state of used internally random number generator can be controlled using keyword MCRandDataSeed
. Set this parameter to a particular integer value if you want to obtain reproducible results. Otherwise it is initialized automatically in a pseudorandom manner so that you get numerically different results in each run. However, if the procedure is well converged the results must be very similar.
For randomization of parameters group numbers are used.
If a parameter should be randomized an integer group number should be assigned to this parameter using special keywords (MCBglQ
, MCLam
, MCvarx
) and/or commands (ZMATMC
, TERMSMC
).
Parameters in the same groups (with equal group numbers) are randomized so that differences between their values are kept fixed.
This is similar to refinement of parameters, when only those parameters are refined, which have nonzero group numbers.
The rules for combination of parameters in groups are the same as for refinements.
The group numbering systems for refinement and for randomization are completely separated, so you can use the same numbers for these two purposes. However, it makes no sense to randomize parameters which are subject for refinement. 
For grouping of Zmatrix parameters ZMATMC
command is used:
ZMATMC=mol,type,otag,ctag
where type
defines type of limits for minimal and maximal parameter values, which is one of ABS
, ABSDEV
or PERC
.
The input field for this command should consist of four columns: name of parameter from Zmatrix of the current molecule, randomization group number for this parameter and two floatingpoint numbers defining limits for randomization. If type
is ABS
then absolute minimal and maximal values should be defined, like in the example:
<ZM_mol_MC> F1 1 150.0 180.0 F2 2 0.0 60.0 </ZM_mol_MC>
Here F1
and F2
are names of torsion angles, which have been defined earlier in a Zmatrix for a molecule. These two angles will be randomized independently in two separate groups with group numbers 1
and 2
. The first angle will be randomized in the range from 150.0 to 180.0 degrees, the other angle will be randomized in the range [0,60]. Note, parameters are sampled from uniform distributions with defined minimal and maximal values.
If type
is ABSDEV
then deviations from current values of parameters are defined in the input field, for example:
<ZM_mol_MC> F1 1 10.0 15.0 F2 2 5.0 10.0 </ZM_mol_MC>
After reading of this data the limits for randomization of F1
are [V_{F1}10,V_{F1}+15], where V_{F1} is the current value of F1
. The limits for randomization of F2
are [V_{F2}5,V_{F2}+10].
If type
is PERC
then deviations are defined in percents from current values of parameters, for example:
<ZM_mol_MC> F1 1 0.5 0.5 F2 2 1.0 1.0 </ZM_mol_MC>
Here the limits for F1
are in fact [V_{F1}0.5*V_{F1}/100,V_{F1}+0.5*V_{F1}/100], where V_{F1} is the current value of F1
. Similarly for F2
the limits will be [V_{F2}1.0*V_{F2}/100,V_{F2}+1.0*V_{F2}/100].
Note, command ZMATMC
does not require that all Zmatrix parameters are listed in the respective input field. It is enough to indicate only those parameters, which should be randomized.
Note, for correct calculation of biases the refined parameters must be already optimal for the given functional before starting the 
ED standards
UNEX can process ED intensities of gas standards. In most cases this is done in order to refine electron wavelength.
The respective command is STANDARD
:
STANDARD=method,int1,...
Here method
can be SCANLAM
, REFINELAM
or LSQ
. Also in the command there must be defined one or more identificators of intensities,
which should be processed. For each intensity the type of standard can be defined with the keyword Std
,
otherwise default value from the global keyword DefStdType
will be used.
In most cases initial value of the electron wavelength (keyword Lam
) and distance from nozzle to detector (keyword NtoP
) must be defined.
It is also recommended to set sector to detector distance (keyword StoP
) to an appropriate value
if sector device was used for obtaining experimental data and sector function is used in the procedure.
The first method SCANLAM
does searching of the best electron wavelength by scanning.
The control keywords for this method are StdScanIter
, StdScanLamMin
and StdScanLamMax
.
The other method REFINELAM
searches the best electron wavelength using golden section method.
In contrast to the simple scanning it recalculates background on each iteration, which increases the overall accuracy of the obtained electron wavelength.
The most important keywords for this method are StdRefLamMaxIter
and StdRefLamTol
.
Note, the type of model for total intensities in these both methods can be chosen using individual for each intensity curve
keyword IModel
. In fact this determines the type of background calculated for the intensities.
The valid options are mbgl
, a1bgl
, a2bgl
and none
.
The last option none
explicitly indicates that no background should be calculated.
The background and respective experimental sM(s) are also not calculated if IModel
is not defined.
In this case the experimental sM(s) must be defined and available from some other source, for example from input file.
The approximation for background lines can be defined using global keyword BglApproxType
.
The flexibility of the lines can be controlled by defining the maximal number of inflection points with
the keyword BglNinflThr
for each curve individually, or with the global keyword BglNinflThr
.
If polynoms are used for background approximations, then the relevant local and global keyword is BglPolPow
.
Depending on the type of background scale or tfactors can be refined if StdBglRefScaleMaxIter
is greater than zero.
The other option for the method
is LSQ
.
In this case the leastsquares method is used.
It refines parameters of total intensities by minimizing a functional, which is generally defined as
The summation in the first term is performed for all points of all intensity curves processed simultaneously.
The second term is for regularization of the refined sector function,
if the keyword StdRegSecAlpha
is set to nonzero value and a regularizing sector function is defined with the REGSEC
command.
The third term is for regularization of refined background functions.
For this the StdRegBglAlpha
keyword must be set to some nonzero value.
The regularizing value itself is defined by the keyword StdRegBglValue
.
The fourth term is for the additional control of the background flexibility,
if the keyword StdRegDBglAlpha
is set to nonzero value.
In fact this is a sum of second derivatives of backgound values of all curves in all points.
Again, the used model for the total intensity depends on the setting of the IModel
keyword for each intensity curve.
In the least squares method the valid options are a1bgl
and a2bgl
. For details see Models for ED intensity.
In the first model background β is refined, in the second case background B is refined.
The reduced sector function is modelled numerically as a set of its values at particular rvalues at the sector plane.
The particular set of rvalues with explicit initial approximation for the reduced sector function can be defined with the SECTOR
command.
Alternatively UNEX can automatically initialize initial reduced sector function, see keyword StdInitSecStep
.
Note, in any case a model sector function must be defined, see SecModelType
and SecPrm*
keywords.
Backgrounds β or B are modelled as Chebyshev polynomials.
The order of polynomials can be defined for each curve individually using the StdModBglPow
keyword.
Normally an initial approximation for the background is obtained by fitting the polynomial to a background,
which is obtained by running internally the BGL
procedure.
This can be turned off by setting StdInitRefBgl=0
.
The types of parameters to be refined is determined by the StdVar*
set of keywords.
Refinement of background, sector function and other parameters in 
Diffraction intensities of the following molecules can be processed in UNEX as gas standards (default standard values are taken from the cited papers):
Currently used standard values of parameters for these molecules can be printed with the command
PRINT=STDPARAMS
It is also possible to define a custom set of standard parameters for each standard molecule with the STDPARAMS
command:
STDPARAMS=stdtype,STDTERMS,otag,ctag
Here stdtype
is the type of standard, one of the following: CCl4
, C6H6
, CO2
, CS2
.
In input file between tags otag
and ctag
must be defined types of terms and their parameters:
r_{a} distance, amplitude l and asymmetry (or Morse) constant.
Note, the last parameter should correspond to your setting of the ImolAnhTermModel
keyword.
By default it is ImolAnhTermModel=Asym
, so asymmetry parameters are expected.
Model molecular intensity functions are calculated from these parameters using the respective approximation as described in chapter Models for ED intensity.
In these calculations multiplicity factors (number of equal terms of a given type in a molecule) are used automatically.
They are predefined for each term in each type of standard molecules in UNEX.
Here are examples of STDPARAMS
command:
STDPARAMS=CCl4,STDTERMS,<ccl4terms>,</ccl4terms> <ccl4terms> CCl 1.7667 0.0496 5.0e7 Cl..Cl 2.8892 0.0712 6.0e7 </ccl4terms>
STDPARAMS=C6H6,STDTERMS,<c6h6terms>,</c6h6terms> <c6h6terms> CC 1.397760 0.046360 5.5e7 C..C 2.418800 0.055180 8.1e7 C..C 2.792300 0.058980 0.0 CH 1.095600 0.077060 0.0 C..H 2.158700 0.099850 0.0 C..H 3.404100 0.096880 0.0 C..H 3.879200 0.093380 7.0e8 H..H 2.483300 0.157990 0.0 H..H 4.300200 0.133170 0.0 H..H 4.964400 0.118180 0.0 </c6h6terms>
STDPARAMS=CO2,STDTERMS,<co2terms>,</co2terms> <co2terms> CO 1.16419 0.0327 3.0e7 O..O 2.32427 0.0393 2.9e7 </co2terms>
STDPARAMS=CS2,STDTERMS,<cs2terms>,</cs2terms> <cs2terms> CS 1.559 0.040 7.7e7 SS 3.112 0.052 8.8e7 </cs2terms>
The input order of terms is important and should be as in the examples above. 
The values in the examples are only for the demonstration of the input syntax. Do not use them in real investigations! 
Thermodynamics
UNEX can calculate thermodynamic functions using statistical thermodynamics theory [43].
For this the THERMO
command should be used:
THERMO=STAT,mol
Geometry and vibrational frequencies should be defined for the respective molecule before running this command.
The frequencies can be calculated or introduced with the VMOD
command.
The calculation of thermodynamic parameters is performed using the model of ideal gas
and assumption on uncoupled translational, rotational and vibrational motions.
The values of pressure and temperature, for which the calculation is done, can be defined in BASE
with the respective keywords Pressure
and Temperature
.
In calculations it is also assumed that only the ground electronic state is populated and other states are not achievable.
The contribution of the ground electronic state is determined by the spin multiplicity and
can be defined using keyword SpinMult
in the information field of the respective molecule.
Data printing
Normally UNEX by default prints status of executed commands and some summarized results of these commands.
There is, however, a special command PRINT
for outputting different kinds of data.
This command can be executed at any stage of data processing.
The only requirement is that the respective data must be already initialized at the time of requesting printing.
In many cases UNEX prints values of parameters with respective errors from leastsquares refinement or other procedures. Some parameters are not directly refined but rather calculated from values of other parameters. Such parameters are called dependent. Standard deviations for dependent parameters are calculated using the formula for error propagation
where f is the dependent parameter represented here as a function of independent parameters p_{i} with standard deviations s_{i}
and covariations ρ_{ij}, N is the number of groups of parameters.
By default UNEX uses covariations if they are available from latest leastsquares refinement. This can be turned off with DepSigmaCovar
keyword.
Standard deviations for independent parameters s_{i} are normally those from latest leastsquares refinement or MonteCarlo simulation.
However, in some cases they can be defined directly in input file. For example, values of parameters of Zmatrices can be introduced together with respective standard deviations.
Below is the list of data types for printing and corresponding variants of the PRINT
command.

General information
For getting current settings:
PRINT=INFO
Brief information about hardware and operating system:
PRINT=COMPINFO
Information about loaded images:
PRINT=IMGINFO

Data for DISP/ElDiff
DISP and ElDiff are programs for vibrational spectroscopy and electron diffraction written by Igor Kochikov.
PRINT=DISPINPXYZ,mol PRINT=DISPINPINT,mol
The first example prints data used as input for calculations in ElDiff using Cartesian coordinate system. The second command prints data for calculations in internal coordinates. Note, to be able to print such data there must be introduced harmonic and optionally cubic force field(s) and Cartesian coordinates for the respective molecule.

Molecular symmetry
PRINT=SYMMETRY,mol
This command prints symmetry elements for
mol
and respective point group. Geometry formol
must be already defined. 
Rotational constants
PRINT=ROTCONSTS,mol
The command prints experimental and model rotational constants for
mol
. Model values are calculated for the current geometrically consistent structure ofmol
. Experimental values are printed as defined in the input. Additionally, for each rotational constant respective corrections (defined earlier in the field of the molecule), experimental standard deviations, differences between experimental and corrected (using input corrections) model values and errors are printed. The errors are calculated on the basis of standard deviations of refined molecular parameters using error propagation formula. In the end the values of rootmeansquare deviation (RMSD) and weighted RMSD (WRMSD, taking into account experimental standard deviations) are calculated and printed. Note, (W)RMSD are printed only if at least one experimental value is not zero. 
Vibrational force constants
PRINT=F2CMATRIX,mol PRINT=F2CSHRINK,mol PRINT=F2CGAMESS,mol
This three variants print harmonic force constants in Cartesian coordinates for
mol
. The difference is in the format of printing.PRINT=F3CBLOCKS,mol
This is for printing cubic force constants in Cartesian coordinates. The number of columns in each block is controlled by the
F3cBlockCols
keyword. 
Geometrical parameters
Particular geometrical parameters of molecules are printed by calling
PRINT
command with optionsDISTANCE
(aliases areDIST
andBOND
),ANGLE
(short variant isANG
),TORSION
(short variantTORS
) andOOP
for outofplane angles. Syntax of the command includes indication of parameter type, name of molecule and numbers of atoms from two to four, depending on the type of parameter:PRINT=DISTANCE,mol,1,2 PRINT=ANGLE,mol,1,2,3 PRINT=TORSION,mol,1,2,3,4 PRINT=OOP,mol,1,2,3,4
The numeration scheme is as in the image below
UNEX prints three types of distances, r_{c}, r_{a} and r_{g}. In UNEX output they are indicated as
r_c
,r_a
andr_g
, respectively. r_{c} is the geometrically consistent distance as calculated from Cartesian coordinates. In ED structural refinements its definition is closely related to the type of vibrational corrections. If the corrections are (r_{e}  r_{a}) then the r_{c} distances are in fact r_{e}. In refinements from rotational constants the definition of r_{c} depends on the type of corrections for these constants. If B_{0} are used without any correction then the refined r_{c} are in fact r_{0}. There can be also other possibilities, depending on details of structural analysis. r_{a} and r_{g} are two kinds of vibrationally averaged distances. The r_{a} can be calculated from r_{c} internally by subtraction respective vibrational corrections. The r_{g} distances are calculated from r_{a} and respective vibrational amplitudes l using approximationFor all types of geometrical parameters errors are calculated (see explanation above) and printed. By default they correspond to estimated standard deviations, but they also can be modified by a factor defined by
PrintEsdFactor
keyword.There is also a possibility to generate automatically a complete set of internal geometrical parameters and print their values with respective errors. This can be done by calling
PRINT=ALLGEOM,mol
In this command UNEX tries first to identify bonds and then to generate all other parameters based on the connectivity information. Two atoms are assumed to be connected with a bond if distance between them is less than the sum of their covalent radii [44] plus some fraction of this sum (see
GeomBondTol
keyword). Outofplane angles are generated and printed if their values are below 10 degrees. Note, the procedure ignores dummy atoms. It is possible to switch off this procedure and to define explicitly another particular set of parameters, which must be printed. For this, the commandPRTGEOM
should be used as in the example belowPRTGEOM=mol,INPUNEX,<pgprm>,</pgprm> <pgprm> distance 1 2 angle 1 2 3 torsion 1 2 3 4 oop 1 2 3 4 </pgprm>
If geometrical structure was refined by minimizing a combined leastsquares functional then it makes sense to print geometrical parameters with respective contributions of different LSQ functional pars into these parameters. For this two commands can be used:
PRINT=GEOMFUNCW2,mol PRINT=GEOMFUNCW12,mol
This will print contributions of different parts of least squares functional from latest
MINIMIZE
procedure to geometrical parameters ofmol
. The values are calculated according to the methods W2 [5] and W12 [4]. Together with contributions are printed values of parameters, their errors from the latest refinement procedure and experimental errors. The latter are calculated according to the procedure described in [4] (UNEX uses a generalized from of the Eq. 6 from the cited paper). The keywordMinSigmaExcludeFunc
can be used to control this procedure. Note, the set of internal geometrical parameters is automatically generated by default. It is, however, possible to define explicitly another custom set using thePRTGEOM
command as described above. Note, however, that the calculated values of contributions can depend to some extent on the composition of the complete set of internal parameters. This is intrinsic property of the procedure.PRINT=RESTRGEOM,mol
This command prints restraining geometrical parameters for the molecule
mol
in comparison to respective model values. 
Zmatrices
Zmatrices of molecules can be printed by calling
PRINT=UNEXZM,mol
The other option prints only parameters of the Zmatrix
PRINT=PARAMS,mol
There is a possibility to generate and print Zmatrices which include only Cartesian coordinates as parameters. Such a Zmatrix can be used as a template for further work. The command is
PRINT=CZMTMPL,mol
Note, with the keyword
CzmTmplStartGroup
it is possible to set the starting value for the generated group numbers. 
Cartesian coordinates
The most general command for printing Cartesian coordinates of atoms in a molecule is
PRINT=XYZ,mol
The output format of this command is suitable for visualization with UMV program. Two other options
MOL
andXMOL
print Cartesian coordinates in XYZ format [45].PRINT=MOL,mol PRINT=XMOL,mol
The difference between
MOL
andXMOL
is that the former does not print dummy atoms. 
ED scattering factors
ED scattering factors in form of gfuntions for all types of atomic pairs in a molecule can be printed using the command
PRINT=GF,mol
This outputs data, produced by the respective
GF
command (see ED scattering factors). Note, in case of models of mixtures the printed gfunctions are calculated by dividing scattering factors by atomic scattering function of the respective molecule only. In the same manner it is also possible to print atomic scattering functions for a specific molecule with the commandPRINT=FULLIAT,mol

ED intensity functions
Different types of electron diffraction intensity and closely related functions can be printed by calling
PRINT
with one or more arguments from the list below:
IR
— Distances r (in mm) from center of diffraction pattern to detection points. These data are calculated from respective values of s if electron wavelength and nozzletodetector distance are defined. 
INT
— Experimental total intensity. 
TINT
— Theoretical (model) total intensity. 
RINT
— Experimental reduced total intensity, i.e. total intensity divided by tfactor, sector function and atomic component of total intensity. 
LINT
— Levelled experimental total intensity, see below. 
DINT
— Delta (difference) between experimental and theoretical total intensity. 
SMS
— Experimental reduced molecular intensity sM(s). 
TSMS
— Theoretical (model) reduced molecular intensity sM(s). 
DSMS
— Delta (difference) between experimental and theoretical (model) sM(s). 
BGL
— Background intensity. Type of background depends on the last used method for its calculation/estimation. 
RBGL
— Similar toRINT
background divided by tfactor, sector function and atomic component of total intensity. 
LBGL
— Levelled background, see below. 
SEC
— Sector function for each data set. 
IAT
— Atomic part of total electron diffraction intensity. 
INTS
— Standard deviations of experimental total intensity. 
SMSS
— Standard deviations of experimental sM(s).
Note, theoretical (model) functions are (re)calculated before printing.
In the most complete mode all types of data can be printed by calling
PRINT=IR,INT,TINT,RINT,LINT,DINT,SMS,TSMS,DSMS,BGL,RBGL,LBGL,SEC,IAT,INTS,SMSS
However, in most cases only several types of data are required for inspection/analysis and thus only particular arguments can be given, for example
PRINT=SMS,TSMS,DSMS
Note, the order of arguments is not important, for example the command below prints the same set of data as in the example above
PRINT=DSMS,SMS,TSMS
By default all defined sets of ED functions are printed. However, it is possible to print only particular sets of data, indicating respective identificators in the
PRINT
command. The following command prints experimental sM(s) only from the11
data set.PRINT=SMS,11
The levelled versions of total intensity and background,
LINT
andLBGL
, are curves obtained in the following manner. Total intensity function is approximated by a cubic spline (this is default, number of allowed inflection points is defined byLvlInfl
keyword) or a polynomial (if keywordLvlPow
> 0 is defined). The original total intensity and background are divided by this smooth function and printed. The idea is to output such curves which are easy to assess visually. It is important to use as less inflection points as possible (as low power for polynomial as possible) so that oscillations on the original intensity and background are not influenced. Note, with this requirement splines and polynomials are not always the best approximations, so manual levelling may be required.Another possibility for levelled total intensity and background are reduced variants
RINT
andRBGL
. For the description of calculation procedure of these functions see section ED background lines.Together with the data for each set of curves some parameters and current settings are printed. Most of them are selfexplanatory and correspond to keywords described in ED intensities. Some others are explained here:

sMesd
— estimated inMINIMIZE
standard deviation for sM(s). 
sMdDW
— Durbin–Watson statistic [34, 35] calculated in the latestMINIMIZE
(or any equivalent) procedure from delta sM(s) values. 
sMsigmaStatus
— status of individual standard deviations (which can be requested for printing asSMSS
) for sM(s) data points. If it isdefined
then the values were explicitly introduced in input or they were calculated in some procedure (for example averaging, combining, background procedures, etc.). Otherwise the status isundefined
. 
IsigmaStatus
— similar tosMsigmaStatus
but related to total experimental intensity functions and their standard deviations printed asINTS
. They have meaningful values if the status isdefined
.
Delta sM(s) (difference between experimental and model) can be printed in a special form suitable for producing Poincaré plots as
PRINT=DSMSPOINCARE
Data from selected set(s) can also be printed just as in other modes above. This example will print only data from the set
11
:PRINT=DSMSPOINCARE,11
There is a special command to print results of comparison of model and experimental ED molecular intensities:
PRINT=RFACTOR
This will print different types of Rfactors (unweighted
Rf
und weightedwRd
) in percent units for each set of ED data and total values for all data sets together. The definition of Rfactors for sM(s) was given above in MINIMIZE chapter. In addition here are printed also Rfactors, calculated for M(s) and s^{4}I_{mol}(s) functions in analogous manner. Note,wRd
values are meaningful only if standard deviations were defined (or calculated) for the respective sM(s). Also in this casewRd
are equal for all types of molecular intensity, sM(s), M(s) and s^{4}I_{mol}(s), by definition. TotalwRd
are meaningful only if standard deviations were defined or calculated for each data set. It is possible to print individual and total Rfactors calculated only for a particular set of ED curves by giving in the command identificators of respective ED data sets, for examplePRINT=RFACTOR,11,21
For sM(s) in addition to Rfactors are also printed Durbin–Watson statistics (see above), mean and maximal absolute values, mean and maximal absolute differences between model and experimental values.


ED terms of molecules
Several modes exist for printing parameters of pairs of atoms in molecules. The most general command is
PRINT=TERMS,mol
It prints distances of r_{a} type, vibrational amplitudes and corrections, asymmetry constants for all pairs of atoms. Together with amplitudes corresponding errors are printed, which are the LS standard deviations (possibly multiplied by the factor
PrintEsdFactor
) from the latest refinement. Note, it is possible that scale factors for amplitudes were refined. In this case their standard deviations are automatically recalculated into standard deviations for respective amplitudes. Additionally, experimental errors are calculated in the same manner as for geometrical parameters when callingPRINT=GEOMFUNCW2
. In the last two columns group numbers for r_{a} distances and for amplitudes (or for their scale factors) are printed. There is also a command, which prints terms sorted by values of r_{a} distances:PRINT=RSORTU,mol
A special possibility exists for printing terms with their contributions to radial distribution functions:
PRINT=GRAPHTERMS,mol
This command prints data which can be directly plotted. For example, the RdfPlot program can read and plot such data. Note, the output of this command depends on the keywords
RdfMultR
,RdfTermDif
andRdfTermDivAmpl
. IfRdfMultR=1
then the RDFs are approximations of P(r) and the printed distances are of r_{g} type, otherwise RDFs are P(r)/r and the printed distances are of r_{a} type, respectively. Basic values for contributions of terms are calculated as products of respective atomic charges. They can be automatically modified depending on settings. ForRdfMultR=0
the contributions are additionally divided by respective r_{a} distances. IfRdfTermDif
is active then the obtained values are multiplied by the respective multiplicity factors. In models of mixtures the contributions are also multiplied by respective mole fractions. In case ofRdfTermDivAmpl=1
the contributions are divided by respective amplitude values. 
Potential functions
In dynamic GED models potential energy functions are used. The command to print data on potential function is
PRINT=POTENTIAL,mol
Note, the units of parameters are so that the potential energy is in kJ/mol for the dynamic coordinate values in radians.

ED standards
Parameters of molecules, which can be used in UNEX as GED standards, are printed by
PRINT=STDPARAMS

Sector functions
Total and reduced sector functions are printed by
PRINT=SECTOR
Regularization sector function can be printed by
PRINT=REGSEC

Response functions
Currently active (calculated, refined or introduced from input file) response function for ED detector can be printed with command
PRINT=RESPFUNC
Several types of data can also be automatically plotted in output files by using pseudographics.
The corresponding command is PLOT
.
Width and height of pseudographics are determined by the keywords Wplot
and Hplot
, respectively.
The available currently options for plotting are:

Potential functions
PLOT=POTENTIAL,mol
This command plots input (introduced by
POTENTIAL=mol,PTL1
) and current model potential functions for the moleculemol
. Note, this will only work if potential function in numerical form has been already defined. 
ED intensity
PLOT=SMS,mol
This command plots all available experimental molecular intensity functions sM(s).
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