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Accuracy and precision for molecular solids

Periodic Reporting for period 4 - APES (Accuracy and precision for molecular solids)

Reporting period: 2022-07-01 to 2023-12-31

The problem that we want to solve in this project is reliable and accurate calculation
of binding energies of molecular solids.
Molecular solids are materials important both in nature and in industries.
An example of molecular solid occurring naturally is water ice, solid forms of pharmaceuticals
are examples of molecular solids produced by industry.
Depending on the temperature and pressure, different structures of water ice can form,
called phases.
Similarly, molecules used as pharmaceuticals can form crystals with different structure,
called polymorphs, depending on the conditions used for crystallisation.
The quantity deciding the stability of these different crystals is their
energy and this is the quantity we are interested in within the project.

The main goal of this project is to develop theoretical methods that would allow reliable
calculations of energies of molecular solids and, as a consequence, would allow to assess
the stability of the different forms.
The challenge of this goal lies in the fact that to obtain the main component of the energy
the equations of quantum mechanics need to be solved.
Unfortunately, this is only possible exactly for small molecules.
To obtain energies of molecular solids we need to introduce approximations that reduce
the accuracy of the predicted energies.
In the project we want to combine two approximations to obtain a scheme capable of providing
very accurate binding energies of molecular solids.
The first one is currently the most accurate scheme for calculating the properties of the solids,
the second is a rather mature approach used to obtain reference quality binding energies of molecules.
Apart from that, we also want to develop and implement methods that would allow us to understand
the precision of our predicted binding energies.
That is, we would like to be able to say how much one can trust the result.
The ability to obtain reliably energies of different crystal forms of a given compound will increase
the reliability of predicted phase diagrams or lead to reference data of energy stability of different polymorphs.

Computational modelling plays an important role in our daily lives and in solving our current
and future problems, such as energy production or development of novel drugs.
In any area, modelling is the more useful the more reliable predictions it can make.
In this project we want to increase the reliability of methods used to predict properties
of molecular solids and other solid materials.
Having more reliable methods at hand would reduce the cost and time needed for the development
of materials useful for solving our current and future problems.
The goal of the project is to develop methods for reliable calculation of binding energies of molecular solids.
To reach this goal we worked on three main topics: i) accuracy, ii) precision, and iii) reference data.

In the first part, we study accuracy of methods that are used to calculate binding energies or similar properties
for molecular solids or clusters. We focus on density functional theory (DFT) approximations and random
phase approximation (RPA). As a reference, we use a coupled clusters scheme CCSD(T).
To understand the accuracy for molecular systems we divide them into smaller fragments, such as molecular
dimers, trimers, or tetramers. For the clusters we calculate binding energies with the reference method
and with the tested schemes. We then calculate the errors of the simpler methods and analyze how they
depend on different parameters or how they behave for different systems. Within the project we performed
this analysis for methane clathrate cluster (methane molecule in a cage of twenty water molecules), for crystals
of short hydrocarbons, and for other systems. We found that the simple DFT functionals lead to substantial errors,
especially for errors in clusters with three or four molecules. This is quite a problem as there is no simple and
computationally cheap way to correct for these errors. We also found that the errors are largely inherited in
RPA calculations which are based on the DFT input. We found that the errors are reduced when Hartree-Fock
exchange is included, even though this can lead to substantial increase of computational cost. The work was
published in three articles so far with two more articles close to being submitted. Furthermore, we used the results
and understanding to publish other studies and also to take part in a test set of methods for predicting polymorphism,
which will lead to other publication. Moreover, we will use the developed methods to in a COST project focused
on polymorphism and in further studies of properties of molecular solids.

Concerning precision we assessed the effect of different approximations on the results. For example, we compared
binding energies calculated within periodic boundary conditions to those obtained from the smaller fragments.
We were able to obtain a close agreement between both approaches and discovered several unexpected problems
that can deteriorate the precision of the result. We also found several ways how to substantially speed up the calculations
without affecting the precision. In another work we analyzed how approximate treatment of core electron affects
the binding energies. We were able to identify the cause and develop a correction for this error. This code to
calculate the correction is available and easy to be implemented in computational packages. We hope that it will
be used to reduce the errors which can much larger than the errors of theoretical methods.

Finally, we have generated a large amount of data and we made them available, together with various scripts
for their set-up and analysis. The reference data can be used by other researchers to test their methods and should
be helpful in developing improved methods.
We demonstrated the good performance of the correction scheme based on RPA with CCSD(T) corrections for
the case of methane clathrate. The performance for bulk materials was worse due to large errors of RPA based
on standard DFT approximations. We found that the errors are much smaller when RPA is evaluated on Hartree-Fock
states. This approach is therefore much more reliable to be used within the correction scheme. A significant
outcome of this part are large datasets with reference many-body energies for different molecular solids and scripts
to obtain them. We used these datasets to illustrate the large errors of DFT approximations that are widely used
to calculate properties of molecular solids. The data can be used to improve upon the existing methods.

We have analysed the error due to neglect of core electrons in the calculations employing periodic boundary conditions
by utilising interaction energies of molecular dimers. We have been able to devise a simple physics based model
for the errors and made it available for others to use. The work gives a new view and possibilities to test the precision
of the approximations related to neglect of core electrons.
Tetramer of oxalic acid taken from the crystal structure of polymorph alpha.