UNEX User Manual

UNEX is distributed for free. Conditions of its distribution are of "AS IS" type. You use this program on your own risk. Before downloading and using UNEX you must accept the license agreement, see files license.html, license.pdf or license.txt.

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 UNEX. If you still cannot find the source of the problem it is possible to write an e-mail 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.

For questions, comments or bug reports you can use one the following E-mail addresses of the main UNEX developer, Dr. Yury V. Vishnevskiy

Reading this manual you can meet numbers expressed using scientific notation and symbol e, for example 5.5e13. This corresponds to base-10 exponentiation, so the number above is equivalent to 5.5×10¹³. 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.

UNEX program is distributed together with some supplementary programs, testing files, documentation and other parts. For the installation there is no need to do any special actions, simply copy all files from the distribution to any suitable directory. It is recommended to place them to one dedicated directory listed in the environment variable PATH so that the executables can be called from any directory in the system. If all UNEX files are in one place then it is also easier to update them by replacing old files in this particular directory. Checking for new versions and updating can be also performed automatically by starting the special script update.sh (in Linux and macOS) or update.cmd (for Windows). Note, automatic update may not work for different reasons. First, it requires access to internet for checking the availability of new versions. Second, the scripts are used some system utilities, which must be already installed. Finally, the automatic update may possibly not work due to some major changes in the procedure. In this case you need to download the newest version of UNEX and install it manually.

UNEX is a command line program. To use it an input file should be prepared first (for details see below). Starting UNEX without any input file prints general information about its usage. In the 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, which contains all results in text form. If you do not indicate the output file name explicitly in command line, then UNEX automatically creates one named similar to the input file with added underline symbol together with 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, in this mode output files are never overwritten. Alternatively, you can define in command line the name for the output file explicitly.

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 so-called 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

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

In most cases UNEX input files begin with introduction of basic information. For this purpose the BASE command is used:

The first word BASE is the name of the command. After the = symbol goes the mode or input field format type. Here the only available mode is READ. The other two words are tags pointing to the start and the end lines of the field containing the basic information. Thus, UNEX will try to find the field and read the corresponding information from the input file between the following tags

BASE field can contain control keywords set to particular values. Depending on the job type different keywords can be used. The keyword molecules is used most often whenever models of molecules are created and manipulated, for example in structural analyses. See section Molecules on how to read in molecule-specific keywords.

Generally lines in a BASE field look like following: keyword name, whitespace(s) and/or = character, keyword value. The values can be strings, integers or floating point numbers. Usually keywords 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 the job. This can be achieved by calling BASE command several times, for example

Below is the list of keywords valid in BASE field:

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

Possible keywords in field of molecule:

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

Valid keywords in image fields are:

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:

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

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

Several important points can be mentioned for this example.

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.

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 V₂, Δ₂ and w₂ (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.

A term in GED is a pair of atoms with associated parameters like distance, vibrational amplitude, correction and asymmetry (anharmonic, phase-shift) 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:

where format can be either FREEU, SHRINKU or ELDIFF.

An example of data field in FREEU format:

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 ų. 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 a-parameters with the local keyword a-Units as in the example:

The setting kA3 for a-Units indicates here that the input a-values are in fact asymmetry constants κ in ų. 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 pm³. Yet another option is a-Units=c3pm3, indicating that the input values are c₃ parameters from the cumulant approximation [17]. They are converted internally to asymmetry parameters as κ = 10^-6c₃/6. 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 a-Units=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

If a-parameters are zero, they also can be omitted. Note, group numbers may still be defined

If a correction is zero and the respective a-parameter is also zero, then both numbers may be omitted. But if the correction is zero and the a-parameter is not zero, then both numbers must be defined explicitly.

Another format option is SHRINKU. It is implemented for reading data produced by the Shrink [18] program. Specifically UNEX expects data as it is printed by Shrink in tables for the second approximation:

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_e-r_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.

The third available format ELDIFF is for introducing data produced by the ElDiff [19] program. Here is a shortened example of such data field:

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 pm³ are converted internally to asymmetry parameters κ in ų. 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 Z-matrix), the RENUM command can be used:

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 Z-matrices or Cartesian coordinates.

There is, however, an alternative command GEDTERMS for defining all required parameters for calculation of ED molecular intensity curves

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

Note, the definition of a-parameters follows here the same rules as for the AMPLITUDES command, likewise it is possible to use the local RENUM command and the a-Units keyword. See above for details.

For investigation purposes with UNEX it is possible to refine r_a distances independently, resulting in a so-called geometrically inconsistent structure. The corresponding group numbers for r_a distances can be defined only with the GEDTERMS command.

It is possible to set all distance corrections of a molecule to a single value at once. For this the SET command can be used as SET=ALLRCORR,mol,value. For example,

will zero out all distance corrections in the mymol molecule.

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 f-functions, which characterize scattering ability of atom pairs. Also in some cases it is convenient to operate with so called g-functions defined as the 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 the GF command:

where <electron_energy> is the floating point number defining the energy of electrons. There are two possibilities to provide this quantity, in form of the electron wavelength in (in Å) or using accelerating voltage (in kV). This is controlled by the optional keyword Units, which accepts options A for Angstroms and kV for kilovolts. The electron wavelength and accelerating voltage are recomputed in each other internally in UNEX using the formula (4.3.1.33) in [8]. By default the electron wavelength is expected, so for example the command

calculates scattering factors for mol and the electron wavelength 0.05 Å. This is equivalent to the command

The methods for calculation of elastic and inelastic scattering factors are globally defined using the keywords EDElScatFacMethod and EDInelScatFacMethod, respectively. For electrons with energy 10-100 kV the options PwTab2 and MorseTab2 are recommended. In this case UNEX uses built-in tables [8] of inelastic factors and scattering aplitudes and phases defined for the accelerating voltages 10, 40, 60, 90 kV and for s-values up to 60 Å^-1. Two-dimensional cubic splines are used for calculation of required parameters from tabulated values. Note, for accelerating voltages outside the 10-90 kV range the scattering aplitudes and phases are calculated by cubic extrapolation. This may be inaccurate for energies significantly deviating from the stated range. For MeV electrons the recommended options are Born1Tab1C1 and MorseTab2, respectively. Note, the GF command can be called multiple times for the same molecule. The corresponding functions will be recalculated with defined parameters.

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 1-1 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.

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 so-called 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

Here format can be READTOTAL or READREDUCED.

The READTOTAL format is used to input total sector function

The data field in the example above contains distances in mm from the center 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

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:

or

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:

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 B-splines. This is done by command

Analogous command exists for smoothing regularizing sector function:

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 16-bit 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:

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.

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 16-bit grayscale TIFF format with little-endian byte order. For introduction of images in UNEX the imgfiles keyword in BASE must be used. The major procedure for data reduction [21] is started by command

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 [21] 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:

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:

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.

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

where jobtype is the type of modification, int1 and possibly int2 and so on are intensity identificators. The types of the modification are

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

The available variants for jobtype are

In independent atom approximation total electron diffraction intensity can defined as [22]:

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 t-factor 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 thermal-average 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 c₃ constants from the cumulant approximation [17] (as printed, for example, by the ElDiff program [19]) are related to κ parameters as κ = c₃/6.

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.

In UNEX there is a possibility for averaging ED intensity curves with an AVERAGE command. The general syntax of the command is as follows:

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:

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 R-factors [30]. For each of the curve, participating in averaging procedure, individual experimental R-factors are calculated as

where is the i-th point of the intensity curve, for which the R-factor 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 R-factors 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 R-factors:

Note, in INT and SMS modes standard deviations are not calculated so both types of experimental R-factors are equal. Weighting works in SMSS and INTS modes, when standard deviations for the average curve are calculated.

Next UNEX calculates total experimental R-factor 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 R-factor allows to represent numerically the overall reproducibility of experimental data and their general quality. Weighted and non-weighted (setting all w_i = 1) total and average experimental R-factors are calculated. Note, the weighted experimental R-factors 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 R-factors.

The advantage of experimental R-factors 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 R-factors on the basis of sM(s) curves are directly comparable with structural R-factors:

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:

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 non-weighted R_str should be compared with non-weighted R_exp.

Below are several examples of averaging commands.

Another possibility of converting multiple ED curves into a single curve provides COMBINE command. It has exactly the same syntax as the AVERAGE

but in general case does a different job creating a curve with s-values present in all input curves (in contrast, AVERAGE takes points s only from the first input curve). Thus, curves with different s-ranges 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 nozzle-to-detector distances. The respective command was

Note, in COMBINE experimental R-factors are calculated in a similar manner is in AVERAGE (see above). There are, however, some peculiarities. The weighted R-factors 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.

Radial distribution functions in UNEX are calculated and printed by the RDF command:

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 sine-Fourier 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

Note, the curves must have common range of s-values. The concatenation procedure includes relative scaling of curves, re-interpolation to common argument values (if required) and weighted 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:

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 diffraction patterns are not constant on the s-scale and are even not linear (this property is characterized by g-functions). As a consequence, the calculated RDF is difficult to analyze. Fortunately, ratios of many g-functions for different types of atoms are much closer to constants than g-functions themselves. Therefore UNEX provides a possibility to divide the integrated sM(s) data by a g-function, see RdfDivGf keyword in BASE. This can improve the appearance of the obtained RDFs. By default for this purpose UNEX chooses g-function 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 g-functions can go through zero and change sign. Accordingly, RDF cannot be obtained by integration of sM(s) modified by such a g-function. In this case an optimal g-function 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 r-values, for which RDFs are calculated, two schemes are implemented in UNEX:

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 Ph-CH₂-CH₂-CH₂-Se-CF₃ 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.

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 R-factors. 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 Monte-Carlo 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 1,2-dichloroethane (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 1,2-dichloroethane calculated from the simulated sM(s) data shown above.

In the simplest case, model rotational constants are computed by diagonalizing inertia tensors, which are calculated for the current geometrical structure and using atomic masses located in infinitesimal points. The type of the structure is determined by user, for example it may be equilibrium. This model is called "rigid rotor - point atomic masses" and corresponds to the setting RotConstModel=rrpatm:

The other option, RotConstModel=rrpatm-vibc, in addition utilizes input vibrational corrections, defined by the keywords RotA_vibc_value, RotB_vibc_value and RotC_vibc_value. First, the "rrpatm" values are calculated, from which the respective vibrational corrections are subtracted. Note, the correction is defined as

typically

Thus, the model rotational constant is calculated as

The third option, RotConstModel=rrpatm-elc1-vibc, adds an electronic correction in addition to the vibrational correction. The electronic correction is calculated as (for details see the book of W. Gordy and R. L. Cook [31] page 548)

where is the axis in the principal system, is the effective model rotational constant, and are the rotational constants calculated using atomic (the "rrpatm" definition) and nuclear masses, respectively, m and M are masses of the electron and proton, respectively and is the corresponding diagonal element of the rotational g tensor. Thus, using UNEX designations (and omitting the axis symbols for simplicity) the total model rotational constant is calculated as

The diagonal components of the rotational g tensor are defined using the keywords RotG_gaa_value, RotG_gbb_value and RotG_gcc_value. Note, in the actual calculation automatically corrected g values (g_corrected) can be used, if respective relative shifts were also defined, RotG_gaa_rshift, RotG_gbb_rshift and RotG_gcc_rshift. The definition for these shifts is

Accordingly, the corrected g value is calculated as

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 least-squares 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.

Semi-automatic grouping of amplitudes can be done using AMPLGROUP command.

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

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:

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 amplitudes which had previously group numbers equal to zero.

In inverse problems parameters are refined with uncertainties. After each least-squares 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 Monte-Carlo simulation approach [52]. This method is implemented in UNEX and can be started by calling the command MCMIN:

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 Monte-Carlo procedure is started. Parameters and/or data are sampled randomly from respective distributions and for the newly generated model and data a least-squares 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:

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

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 keyword RotA_exp_pdf_p2 (analogously for constants B and C). 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 the keyword MCRandDataSeed. Set this parameter to a particular integer value if you want to obtain reproducible results. Otherwise it is initialized automatically in a pseudo-random 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 non-zero group numbers. The rules for combination of parameters in groups are the same as for refinements.

For grouping of Z-matrix parameters ZMATMC command is used:

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 Z-matrix of the current molecule, randomization group number for this parameter and two floating-point numbers defining limits for randomization. If type is ABS then absolute minimal and maximal values should be defined, like in the example:

Here F1 and F2 are names of torsion angles, which have been defined earlier in a Z-matrix 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:

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:

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 Z-matrix parameters are listed in the respective input field. It is enough to indicate only those parameters, which should be randomized.

Information required for randomization of ED vibrational amplitudes and corrections is introduced with the TERMSMC command:

Here type can be one of AMPLPERC, CORRPERC, AMPLABS, CORRABS, AMPLABSDEV and CORRABSDEV. Prefixes AMPL and CORR indicate amplitudes and corrections. Suffixes ABS, PERC and ABSDEV indicate that the data are introduced as absolute values, as percent deviations from current active values or as absolute deviations from current values. This scheme works in the same way as in the ZMATMC command. Below is an example for CO₂ molecule:

UNEX provides a (semi)automatic procedure for collecting statistics on minimal and maximal values of parameters (Z-matrix parameters, vibrational amplitudes and corrections), which can be printed in a way suitable for later introducing in Monte-Carlo procedure. This is demonstrated in the example below:

The idea consists in reading different sets of geometrical parameters (here in form of Cartesian coordinates, which update Z-matrix parameters) and vibrational amplitudes and corrections. The first command RESTERMSTATS is used for resetting of statistics. Next new parameters are introduced in a usual way. After reading of data the command UPDPRMSTATS is used for updating of parameter statistics taking into account the freshly introduced values. This can be repeated multiple times. In the end the collected minimal and maximal values are printed by PRINT=ZMATMCTMPL,mol, =AMPLMCTMPL and =CORRMCTMPL. Note, by default the printed ranges of parameters are extended by particular factors, see documentation for these printing commands. With slight modifications these data can be used as input for MC procedure.

During the MC procedure in intermediate and in final calculations of parameter statistics two methods can be used depending on the setting of the MCWeightedStats keyword, i.e. the statistics can be calculated with and without weighting factors. In the former case the weights are calculated as

where χ²_i is the absolute minimized value of the least-squares functional on the i-th iteration of the Monte-Carlo procedure.

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:

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 StdDefType 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 t-factors can be refined if StdBglRefScaleMaxIter is greater than zero.

The other option for the method is LSQ [53]. In this case the least-squares 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 non-zero 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 non-zero 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 non-zero 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 r-values at the sector plane. The particular set of r-values 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.

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

It is also possible to define a custom set of standard parameters for each standard molecule with the STDPARAMS command:

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:

In UNEX it is possible to do a simple vibrational analysis. Direct vibrational problem — calulation of harmonic vibrational modes and frequencies can be solved using the command VMOD as

This runs a procedure, which diagonalizes the matrix of harmonic force constants in mass-weighted Cartesian coordinates. Note, the force constants, as well as the respectively oriented Cartesian coordinates of atoms (and their masses!) must be already available in UNEX. For this, for example, the commands F2C and MOLXYZ can be used. Results of the procedure can be printed using the PRINT=VMODMC command (see below).

UNEX can calculate thermodynamic functions using statistical thermodynamics theory [57]. For this the THERMO command should be used:

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 (see Vibrational modes). Depending on the ThermoModel setting of the molecule the calculation can be performed in different ways. In the simplest case, ThermoModel=sRRHO, UNEX uses the model of ideal gas, assumption on uncoupled translational, rotational and vibrational motions, rigid rotator and harmonic oscillator (RRHO) approximation [58]. Note, the frequencies may be scaled by using the ThermoFreqScale keyword, which turns the approximation effectively into the anharmonic oscillator, respectively. The other option, ThermoModel=msRRHO-1, utilizes the so called modified scaled RRHO method by S. Grimme [12], with a modified procedure for the calculation of the vibrational entropy. Two keywords are related to this method, ThermoMSRRHOWcutoff1 and ThermoMSRRHOWalpha1. Note, this method was also known as quasi-RRHO [11]. The last option, ThermoModel=msRRHO-2, in addition to the entropy correction (as in msRRHO-1) uses also similar correction to enthalpy as described in [13]. The respective control keywords are ThermoMSRRHOWcutoff2 and ThermoMSRRHOWalpha2.

Thermodynamic functions are calculated for conditions (temperature and pressure) defined by optional parameters to the command (see below) or by global parameters. 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 the SpinMult keyword in the field of the respective molecule.

The possible optional parameters of the THERMO command are as follows

For example, the command

calculates thermodynamic functions for the h2o molecule at 500 K and default (standard) pressure. All types of motions will be taken into account, except for translations.

On the output are printed inner energy U, enthalpy [H(T)-H(0)], entropy S, Gibbs free energy G, constant volume heat capacity Cv and constant pressure heat capacity Cp. The particular contributions to the thermodynamic functions and to the molecular partition function Q from different types of motions are also printed. If the electronic energy has been defined for the molecule (see the ElEnergy keyword) then total values for the thermal energy, enthalpy and Gibbs free energy are printed.

Note, small vibrational frequencies lead to large errors in calculated thermodynamic functions within standard RRHO approximation. Depending on the application, it may be reasonable to ignore such frequencies by using the keyword ThermoFreqCutoff. However, a more universal way for solving this problem is to use the msRRHO methods.

Technically the term molecular trajectory is here defined as a set of geometries for a molecule. For UNEX it must be represented as a file in the standard XYZ format, in which sets of Cartesian coordinates follow one after another. A single set of Cartesian coordinates within trajectory is called frame. Trajectories can be obtained, for example, from molecular dynamics and Monte Carlo simulations. For their processing UNEX provides different functions, invoked by the TRJXYZ command.

For getting current settings use

Brief information about hardware and operating system is printed by

For information about loaded images you can use

The command below prints symmetry elements for mol and respective point group. Geometry for mol must be already defined.

The command prints experimental and model rotational constants for mol. Model values are calculated for the current geometrically consistent structure of mol. 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 root-mean-square 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.

Force constants can be printed by

These variants print harmonic force constants in Cartesian coordinates for mol. The difference is in the format of printing. Below is the command for printing cubic force constants in Cartesian coordinates. The number of columns in each block is controlled by the F3cBlockCols keyword.

Cubic force constants in normal coordinates can be printed using the command

By default the constants are printed in internal units Hartree amu^-3/2 Bohr^-3. However, it is possible to get the values in cm^-1 by using the appropriate Units keyword:

Vibrational modes in mass-weighted Cartesian coordinates and respective frequencies can be printed as

Each command also prints the particular units of the output data.

UNEX can prepare input data for DISP and ElDiff, the programs for vibrational spectroscopy and electron diffraction written by Igor Kochikov [19].

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.

Particular geometrical parameters of molecules are printed by calling PRINT command with options DISTANCE (aliases are DIST and BOND), ANGLE (short variant is ANG), TORSION (short variant TORS) and OOP for out-of-plane 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:

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 and r_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 B₀ are used without any correction then the refined r_c are in fact r₀. 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 approximation

For 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

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 [62] plus some fraction of this sum (see GeomBondTol keyword). Out-of-plane angles are generated and printed if their values are below 10 degrees. Note, the procedure ignores dummy atoms. It is possible to force inclusion of particular atom pairs in the list of bonds and hence to influence the generation of angles. For this the keyword GenBondsInclude in the information field of the particular molecule can be used. For example

forces inclusion of bonds between atoms in pairs 1—​3 and 4—​6 of the molecule mol irrespective of the distances. Note, the numeration of atoms starts from 1 and includes also dummy atoms, although they are skipped in the procedure. The autogeneration of internal parameters may be switched off by defining explicit set of parameters for printing. For this, the PRTGEOM command must be used as in the example below

If geometrical structure was refined by minimizing a combined least-squares functional then it makes sense to print geometrical parameters with respective contributions of different LSQ functional parts into these parameters. For this two commands can be used:

This will print contributions of different parts of least squares functional from latest MINIMIZE procedure to geometrical parameters of mol. The values are calculated according to the methods W1 [6] and W2 [7]. 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 [6] (UNEX uses a generalized form of Eq. 6 from the cited paper). The keyword MinSigmaExcludeFunc 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 the PRTGEOM 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.

This command prints restraining geometrical parameters for the molecule mol in comparison to respective model values. Note, the standard deviations are those from input and related to restraining values, not to the refined model values! In the end of the table some statistics are printed: maximal absolute deviation between model and restraining values MaxD, root-mean-square deviation RMSD and weighted (using input standard deviations of restraining values) root-mean-square deviation WRMSD, number of parameters used in calculation of statistics Nprm and index of parameter with the largest deviation ImaxD. The statistics are printed for all parameters and for groups of parameters of different types.

Z-matrices of molecules can be printed by calling

Note, PRINT=UNEXZM and PRINT=PARAMS are obsolete starting from 19 March 2022. To print only parameters of Z-matrices use

There is a possibility to generate and print Z-matrices which include only Cartesian coordinates as parameters. Such a Z-matrix can be used as a template for further work. The command is

Note, with the keyword CzmTmplStartGroup it is possible to set the starting value for the generated group numbers. Printing of collected statistics (minimal and maximal values) for Z-matrix parameters is done by the command

The output data can be used as template for introducing data for Monte-Carlo (MC) procedure. Note, the output of the command is also supplemented with data on differences between minimal and maximal parameter values (d=) and group numbers (RG=). In case of large differences a warning message is also printed. The automatic grouping of parameters for MC procedure is done on the basis of their numerical similarity. This should be manually checked and corrected, if need.

The most general command for printing Cartesian coordinates of atoms in a molecule is

The output format of this command is suitable for visualization with UMV program. Two other options MOL and XMOL print Cartesian coordinates in XYZ format [63].

The difference between MOL and XMOL is that the former does not print dummy atoms. All variants can print coordinates in Z-matrix (or input) orientation or in the system of principal axes of rotation. This depends on the setting of the PrintMainInertXYZ keyword. In the latter case, rotational constants are also printed. They are calculated for the current geometry in the RRPATM approximation: rigid rotor — point atomic masses.

ED scattering factors in form of g-functions (scattering factor of atomic pair divided by atomic intensity) for all types of atomic pairs in the molecule can be printed using the command

This outputs data, produced by the respective GF command (see ED scattering factors). Note, in the case of models with mixtures of molecules the printed g-functions are calculated by dividing scattering factors by atomic scattering function of the respective molecule only. The bare scattering factors (not divided by atomic intensity) of atom pairs can be printed with the command

It is also possible to print atomic scattering functions for a specific molecule with the command

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:

Note, theoretical (model) functions are (re)calculated before printing.

In the most complete mode all types of data can be printed by calling

However, in most cases only several types of data are required for inspection/analysis and thus only particular arguments can be given, for example

The order of the arguments does not play any role, for example the command below prints the same set of data as in the example above

By default all defined sets of ED functions are printed. However, it is possible to print only particular set(s) of data, indicating respective identificators in the PRINT command. The following command prints experimental sM(s) only from the 1-1 data set.

The levelled versions of total intensity and background, LINT and LBGL, 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 by LvlInfl keyword) or a polynomial (if keyword LvlPow > 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 and RBGL. 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 self-explanatory and correspond to keywords described in ED intensities. Some others are explained here:

Delta sM(s) (difference between experimental and model) can be printed in a special form suitable for producing Poincaré plots as

Data from selected set(s) can also be printed just as in other modes above. This example will print only data from the set 1-1:

There is a special command to print results of comparison of model and experimental ED molecular intensities:

This will print different types of R-factors (unweighted Rf und weighted wRd) in percent units for each set of ED data and total values for all data sets together. The definition of R-factors for sM(s) was given above in MINIMIZE chapter. In addition here are printed also R-factors, calculated for M(s) and s⁴I_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 case wRd are equal for all types of molecular intensity, sM(s), M(s) and s⁴I_mol(s), by definition. Total wRd are meaningful only if standard deviations were defined or calculated for each data set. It is possible to print individual and total R-factors calculated only for a particular set of ED curves by giving in the command identificators of respective ED data sets, for example

For sM(s) in addition to R-factors are also printed Durbin–Watson statistics (see above), mean and maximal absolute values, mean and maximal absolute differences between model and experimental values.

Several modes exist for printing parameters of pairs of atoms in molecules. The most general command is

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 calling PRINT=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:

A special possibility exists for printing terms with their contributions to radial distribution functions:

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 and RdfTermDivAmpl. If RdfMultR=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. For RdfMultR=0 the contributions are additionally divided by respective r_a distances. If RdfTermDif 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 of RdfTermDivAmpl=1 the contributions are divided by respective amplitude values. Collected statistics (minimal and maximal values) for amplitudes and corrections are printed by commands

These data can be used as templates for introducing data required in Monte-Carlo (MC) procedure. Note, on output the ranges (minimal - maximal values) are increased by particular percentages, which can be explicitly defined using MCAmplTmplExr and MCCorrTmplExr keywords. Additionally printed are differences between minimal and maximal values (d=) and group numbers (RG= in case of amplitudes). The automatic grouping for MC is performed using the principal of numerical similarity. The terms are combined in one group if their distances, amplitudes and corrections are equal with 0.0001 Å. However, the correctness of such grouping should be checked manually! Warning messages are printed if the difference between the minimal and maximal values within the single term is too large (WarnD!) or if differences between minimal or maximal values in the current term and in the first term of the group are too large (WarnG!).

In dynamic GED models potential energy functions are used. The command to print data on potential function is

Note, the units of parameters are so that the potential energy is in kJ mol^-1 for the dynamic coordinate in radians.

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

Total and reduced sector functions are printed by

Regularization sector function can be printed by

Currently active (calculated, refined or introduced from the input file) response function for ED detector can be printed using the command

Several types of data can also be automatically plotted in output files by using pseudo-graphics. The corresponding command is PLOT. Width and height of pseudographics are determined by the keywords Wplot and Hplot, respectively. The available currently options are described below.

This command plots input (introduced by POTENTIAL=mol,PTL1) and current model potential functions for the molecule mol. Note, this will only work if potential function in numerical form has been already defined.

The other plotting posibility is implemented for ED intensity functions.

This command plots all available experimental molecular intensity functions sM(s).