UNEX Tutorial

In the C_2v symmetry SCl₂ molecule has only two independent geometrical parameters and we can refine them from three experimental rotational constants of only single parent isotopologue. Lets create UNEX input file and as the very first lines we add the following minimal set of commands:

UNEX will interpret these commands line by line:

Now we need to provide the field of data expected by the BASE:READ command. Add to the input file the block:

which contains the only single keyword Molecules. We use it to define the identifier mol for our molecule. When interpreting this keyword UNEX expects a molecule-specific field between tags constructed on the basis of the molecule identifier as <mol> and </mol>. Lets create and fill in this field with required for our purposes data:

Here we define:

Now for the ZMATRIX:READ command we need to provide the actual Z-matrix of our molecule. Add to the file the following block:

This Z-matrix defines two internal geometrical parameters, the bond length Rscl (one parameter for both bonds) and the angle Aclcl, with respective initial values. We want the refine both parameters, so next to their values we also define refinement group numbers as 1 and 2. Note, here we use two different groups, thus the parameters will be refined independently from each other.

Now your input file is ready, its content must be similar to that of the file SCl2_1.inp coming with this tutorial. We can run UNEX with this file and inspect the produced output file. The part of the output from the refinement goes after the line LSQ functional processing …​. You can see some information on experimental data, parameters to be refined, the convergence of the iterative procedure and different statistics. In the statistics we may be particularly interested in line Rotational constants: ESD=3.279e+00 MaxD=2.29e+00 RMSD=1.89e+00 WRMSD=2.09e+00 MHz, which gives us maximal, root-mean-square and weighted root-mean-square deviations of rotational constants in MHz. In particular the weighted root-mean-square deviation is equal to 2.09 MHz. Also after the convergence a table with parameters is printed, for example:

The meaning of the table content is essentially self-explaining. The refined values of the parameters are now r(S—​Cl) = 2.0129(6) Å and α(Cl—​S—​Cl) = 102.82(4) degrees, with least-squares standard deviations in parentheses. Note, we used rotational constants A₀, B₀ and C₀ without any vibrational corrections. Thus, the refined structure is in fact of the r₀ type.

After the table with parameters you can find the matrix of correlations printed as

Fortunately, in our example this matrix is very compact. It shows that the refined parameters have correlation coefficient equal to 15.5 %.

Bizzocchi et al. [1] determined rotational constants not only for the parent molecule of sulfur dichloride but also for its three isotopologues. We can utilize these data in our structural refinement. First of all, in the input UNEX file we need to define additional molecules corresponding to the three additional isotopologues. For this, we use the keyword IsotopMols in the field of the parent molecule:

Accordingly, we add three molecule-specific fields with data for iso1, iso2 and iso3:

Just like for the parent molecule, we must define Z-matrices also for the additional isotopologues using the ZMATRIX:READ command:

and by providing the respective data fields:

Notice how the atomic masses for the non-standard isotopes are declared right in Z-matrix bodies, next to atomic symbols. In this way we indicate only non-standard isotopes. For the other atoms UNEX automatically assigns built-in atomic masses of the most abundant isotopes, in this case ³²S and ³⁵Cl. Also note that the initial values of bond lengths and angles in all isotopologues are the same and constitute two refinement groups, 1 and 2. Thus, we are going to refine two parameters under assumption that all isotopologues have the same r₀ structure.

Now your input file must be similar to the file SCl2_2.inp of this tutorial and it is ready to be processed by UNEX. The least squares refinement converges very quickly and takes less than a second on modern hardware. Note that we have now in total 12 experimental constants:

Interestingly, the overall agreement of model with data is now even slightly better, WRMSD=2.03e+00 MHz, than in the previous case of only one isotopologue with three experimental constants in total. Due to the increased number of statistically independent experimental data the refined parameters are now more precise, r(S—​Cl) = 2.0129(2) Å and α(Cl—​S—​Cl) = 102.82(1) degrees, and the correlation coefficient has lowered to 10 %.

We can also print Cartesian coordinates of atoms in all or selected isotopologues. After the LSQFUC command add the line

This will give you the coordinates of atoms in mol. In the default format you can also see the atomic masses actually used by UNEX. In context of rotational spectroscopy it makes sense to print Cartesian coordinates in the system of principal axes of inertia tensor. For this, add the keyword PrintXYZPAS=true to the field of basic information. On the output you should get

The coordinates can be visualized producing a structure like in the figure below.

Until now we refined r₀ structure directly from experimental rotational constants A₀, B₀ and C₀. Omitting a discussion of disadvantages related to this type of structure, we can proceed directly to the example on how to refine structures corrected for vibrational effects. For this, next to experimental rotational constants we need to define respective vibrational corrections. Nowadays in many cases they can be routinely calculated theoretically with reasonable confidence. Even better is when you have them refined from experimental ro-vibrational spectra. In any case they must be introduced in UNEX using the keywords RotAVibCorVal, RotBVibCorVal and RotCVibCorVal. Taking your previous input file you can add the following data to the field of the parent molecule:

to the first isotopologue iso1:

to iso2:

and to iso3:

All these values are quantum-chemically calculated corrections to equilibrium structure. As a reference this tutorial provides a file SCl2_3.inp, which should also contain all the required information. Running UNEX with the prepared input leads to a much better fit with WRMSD equal to 3.50e-02 MHz. This may be compared with the previous value (2.03e+00 MHz) from the example above without any vibrational corrections. Thus we are getting here more consistent model. The refined parameters, r_e^se(S—​Cl) = 2.011023(3) Å and α_e^se(Cl—​S—​Cl) = 102.6808(2) degrees, are now much more precise, at least nominally. Also, in comparison to the previous values, r₀(S—​Cl) = 2.0129(2) Å and α₀(Cl—​S—​Cl) = 102.82(1), we see significant deviations due to the different types of structures. Note, we use the r_e^se designation to underline that the structure is in fact semi-experimental (also known as semi-empirical) due to using theoretical vibrational corrections. Please keep in mind that obtained in this example very small standard deviations manifest only random noise in experimental data. Of course, the total errors must be larger at least due to uncertainties in vibrational corrections. This is, however, a different topic.

What we can also do in this example is to print details about rotational constants. For this, you can add the following line after the LSQFUNC command:

On the output you get the following data for the parent molecule mol:

Here in the first table you can see the particular components of the model values, including the vibrational corrections. Note, in sulfur dichloride these corrections constitute about 0.1 — 0.5 % of the total rotational constants. The default model rrpatm-vibc-elc1 has also electronic correction, which will be discussed in the next example. In the second table are provided experimental and model values and respective deviations for the rotational constants separately.

In addition to vibrational correction our model for rotational constants contains also an electronic correction (consult UNEX manual for details). Usually they are negligibly small but in investigations of some molecules can play significant role. In UNEX the corrections are internally calculated from diagonal aa, bb and cc components of rotational g-tensors. The latter can be determined experimentally or, most usually, calculated theoretically. In the input file we define them with the keywords RotGTaaVal, RotGTbbVal and RotGTccVal for each molecule. In this example we add the following values for the parent isotopologue mol:

for iso1:

for iso2:

and for iso3:

Now the file is ready for running and must be similar to SCl2_4.inp from this tutorial. After the least-squares refinement with LSQFUNC:MINIMIZE you can see that the overall agreement is now even better, WRMSD=3.13e-02 MHz, although to a very small extent. Thus, the vibrational corrections do the major job and the electronic corrections lead to only slight improvement. The latter can sometimes even slightly worsen the agreement due to inaccuracy and inconsistency with other data. Looking at the refined parameters, r_e^se(S—​Cl) = 2.011010(3) Å and α_e^se(Cl—​S—​Cl) = 102.6811(2) degrees, we can see some notable shift in the bond length compared to the previous refinement without electronic corrections, i.e. r_e^se(S—​Cl) = 2.011023(3) Å and α_e^se(Cl—​S—​Cl) = 102.6808(2) degrees. Formally, if we consider only the provided in parentheses standard deviations, this effect could be accepted as significant. However, we keep in mind that the other sources of uncertainties are not taken into account and thus the seeming effect is rather unreliable in this particular case. After the refinement we can get lots of interesting information by executing PRINT: ROTCON all. This variant of the command prints rotational constants for all defined molecules, so that you do not need to call PRINT for each molecule individually.

In particular, for mol we get

Now you can see how large are the absolute values of electronic corrections in sulfur dichloride. For the parent molecule, as well as for other isotopologues, they are of the order 3×10^-1 MHz for A and 3×10^-2 HMz for B and C, which is 0.5 — 1.5 % of the corresponding vibrational correction or 0.001 — 0.002 % of the total rotational constant.

At this point we can consider our refinement as complete.

As an example we take a portion of experimental diffraction data measured for carbon tetrachloride CCl₄ in the work of Vishnevskiy et al. [2]. The molecules of CCl₄ are small and highly symmetric (T_d point group) and thus are very convenient for a GED structural analysis. It is also handy from the experimental point of view, since diffraction patterns can be easily measured for this compound.

In this simple example we assume that the data reduction and electron wavelength calibration have already been done and we take for the analysis only one experimental total intensity curve. Our next steps are briefly as follows:

So lets start from the very beginning. Below are described commands, which should be added line-by-line to a newly created UNEX input file.

First, we must introduce some basic information

Next we define a geometrical model for the molecule. In this example we use geometrically-consistent GED model, so the structure must be defined as a Z-matrix with the command (right after BASE:)

and providing the required field

Note, to the R1 parameter we already assigned the group number 2. This is not required now, but will be used below in the actual refinement procedure. When introducing Z-matrices it is advised to check their correctness by printing calculated Cartesian coordinates. This can be done with the command

and visualizing the printed coordinates by any suitable software. Note, you can print data in different formats, see manual for details. You may also want to check the symmetry of your input structure by using

For our example this command must produce some information on symmetry elements and also the line

which is what we expect for CCl₄.

Next we introduce information on GED terms (interatomic pairs) for the molecule with the EDTERMS command:

Add the corresponding data field to the input file with precalculated values:

Here we are tying together amplitudes for the bonded distances C—​Cl in the refinement group number 3, and the amplitudes for the non-bonded distances Cl…​Cl in the group 4.

At this point the molecular model is defined completely and we can introduce the diffraction data. With the command EDDATA UNEX will read-in the experimental total diffraction intensity:

The initial part of the respective data field looks like:

For the complete data see the file CCl4_1.inp supplemented to this tutorial. For the refinement we assign the group number 1 to the scale factor of the molecular intensity. The initial value for this factor is also explicitly defined as ImolSf=1.0. Now the ED data set ed1 is defined and we can use this identifier later in other commands.

For the calculation of model intensities we must introduce ED scattering factors. In UNEX they can be calculated simply by calling the EDSCATFAC command. Add the following line after the EDDATA:READ call:

With the keyword Lambda we indicate the calibrated electron wavelength.

For the least-squares refinement we need experimental molecular intensity function. It must be extracted from the total intensity, which is in the ED data set ed1. For this use the command EDIMOL:

In this procedure the total intensity is decomposed into molecular and background components. As the model we choose mbgr, i.e. containing multiplicative background. The background will be approximated with a cubic spline and as a smoothness criterion we allow no more than 3 inflection points. Note, the actual number of inflection points may be smaller. UNEX prints this information on output. In this example you should get

Finally we have both model and data ready for the refinement. For the least-squares determination of parameters we must use the LSQFUNC command. Add the following line for minimization of the LSQ functional built on ED sM(s) function from the ed1 data set:

From this command UNEX prints lots of important information to the output file, including the initial values of parameters and LSQ functional, convergence during the iterations and at the end, statistics of the refinement, a table with refined parameters and correlation matrix. After the convergence we have the best (minimal) achieved R-factor

Initial and refined parameters, as well as corresponding least-squares standard deviations, are printed in the table:

Next are printed the correlation coefficients:

As you can see, the refinement procedure has changed the values of parameters substantially. We need to keep in mind that the EDIMOL:GETEXP procedure used model-dependent background (see UNEX manual for details). Thus, if the model has significantly changed, it makes sense to repeat again the combination of molecular intensity extraction and parameter refinement. Add these two lines (same as before) to the input file:

After the second LSQ functional minimization you should get a better (lower) R-factor

and the refined parameters

The changes between the old (before this last refinement) and new (after the refinement) values are now much smaller. The correlation matrix remained nearly the same

and shows that the largest correlation (73 %) is between the molecular intensity scale factor EDImolSf (group 1) and the vibrational amplitudes l(Cl…​Cl) tied together in group 4.

We may want to check the electron diffraction data and plot various functions for better analysis. Add the following command after the last LSQ minimization:

With the keyword Data is indicated that the following data types from the set ed1 are printed together:

These data can be visualized by any suitable software. We can use the UIntPlot program bundled together with UNEX. Starting it in the command line as

will produce a plot of the experimental total intensity with the respective background line. Note, UIntPlot requires that the Gnuplot program [3] is already installed and can be started simply as gnuplot from you working directory. Also make sure that last command line argument to uintplot is the actual name of your UNEX output file.

Visual inspection of molecular intensity functions is also very important. We plot them running uintplot as

which produces the following image.

As we can see, the difference curve is rather small and shows mostly random deviations between experimental data and model. Thus the fit is acceptable.

Usually electron diffraction data are also represented in the form of radial distribution functions (RDF). UNEX will calculate them, if you add the following command:

Also, for the visualization it is convenient to print the ED terms as

Now after processing of all commands your UNEX output file must contain all the data required by the URDFPlot program (also bundled together with UNEX), which is useful for plotting RDFs. Start it simply as (provide the actual name of your output file as the argument):

and you will get the following image.

As we can see, in the RDF representation the difference between the experimental data and model is also small.

On the basis of the R-factor value of 5.05 % and visual inspection of deviations between experimental data and best refined model, we may now summarize the main results of our investigation of CCl₄ as r_e^se(C—​Cl) = 1.7601(2) Å, l(C—​Cl) = 0.0464(4) Å and l(Cl…​Cl) = 0.0693(4) Å, where values in parentheses are least-squares standard deviations. Note, because of the used theoretical vibrational corrections (we introduced them with EDTERMS:READ) to the equilibrium geometry, the determined structural parameter is semi-experimental r_e^se.