## Periodic Reporting for period 2 - APES (Accuracy and precision for molecular solids)

Reporting period: 2019-07-01 to 2020-12-31

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.

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 main approximation that we are using is the so-called random-phase approximation (RPA)

which is currently the most accurate scheme for obtaining binding energies of molecular solids.

The work so-far can be divided into three main parts: i) accuracy, ii) precision, and iii) reference data.

In the first part we focused on understanding the accuracy of RPA for binding energies

of molecular dimers, trimers, and tetramers.

To this end, we used simple systems of noble gas trimers, a standard test set of molecular trimers,

and a methane clathrate model.

This range of structures and compounds allowed us to understand the accuracy of RPA and improve it in a simple way.

Specifically, the RPA error is tightly bound to the errors of the underlying density functional theory approximation which

is used to provide the eigenstates used to evaluate the RPA energy.

This part has lead to two publications, first one about molecular trimers and the second one about methane clathrate.

To be able to perform this study we developed an efficient and scalable computer code for calculating RPA binding

energies of molecular clusters.

The programme can obtain highly precise RPA energies for medium sized systems which has not been possible before.

One observation from the work on clathrate is that RPA errors are the largest for compact clusters.

Therefore, this allows one to focus on these energies and replace them with energies from a higher-level scheme,

as we have proposed in the project.

In the case of clathrate, only two-body energies and three-body energies of compact clusters need to be replaced,

much reducing the computational cost.

Concerning the precision we are assessing the effect of different approximations made in the calculations,

either those performed within periodic boundary conditions or those done using finite clusters of molecules.

In the first case we have analysed the precision of different projector-augmented wave data sets, identified

possible cause for the errors and a simple physics-based correction to reduce the error.

For many-body expansion which employs the finite cluster calculations we have identified issues due to

finite precision of the results and are testing ways how to avoid them.

To gain more insight into the various errors we have obtained binding energies of simple molecular solids

within periodic calculations and using the many-body expansion and analysed how large are the differences

and what can be done to reduce them.

The high quality energies of molecular clusters obtained from clathrate work or from solids allow us to assess

the accuracy of simpler schemes, such as density functional theory.

One interesting outcome of this is a high error of even recent dispersion corrected density functional theory

approximations for the methane clathrate.

which is currently the most accurate scheme for obtaining binding energies of molecular solids.

The work so-far can be divided into three main parts: i) accuracy, ii) precision, and iii) reference data.

In the first part we focused on understanding the accuracy of RPA for binding energies

of molecular dimers, trimers, and tetramers.

To this end, we used simple systems of noble gas trimers, a standard test set of molecular trimers,

and a methane clathrate model.

This range of structures and compounds allowed us to understand the accuracy of RPA and improve it in a simple way.

Specifically, the RPA error is tightly bound to the errors of the underlying density functional theory approximation which

is used to provide the eigenstates used to evaluate the RPA energy.

This part has lead to two publications, first one about molecular trimers and the second one about methane clathrate.

To be able to perform this study we developed an efficient and scalable computer code for calculating RPA binding

energies of molecular clusters.

The programme can obtain highly precise RPA energies for medium sized systems which has not been possible before.

One observation from the work on clathrate is that RPA errors are the largest for compact clusters.

Therefore, this allows one to focus on these energies and replace them with energies from a higher-level scheme,

as we have proposed in the project.

In the case of clathrate, only two-body energies and three-body energies of compact clusters need to be replaced,

much reducing the computational cost.

Concerning the precision we are assessing the effect of different approximations made in the calculations,

either those performed within periodic boundary conditions or those done using finite clusters of molecules.

In the first case we have analysed the precision of different projector-augmented wave data sets, identified

possible cause for the errors and a simple physics-based correction to reduce the error.

For many-body expansion which employs the finite cluster calculations we have identified issues due to

finite precision of the results and are testing ways how to avoid them.

To gain more insight into the various errors we have obtained binding energies of simple molecular solids

within periodic calculations and using the many-body expansion and analysed how large are the differences

and what can be done to reduce them.

The high quality energies of molecular clusters obtained from clathrate work or from solids allow us to assess

the accuracy of simpler schemes, such as density functional theory.

One interesting outcome of this is a high error of even recent dispersion corrected density functional theory

approximations for the methane clathrate.

The RPA approach is promising as it performs well even for systems where the latest DFT approximations fail

or show unreliable errors. One of the goals of our project is to understand if the accuracy of RPA can be further

improved and what would be the physical origin of that. We have made substantial progress along this direction

and we are exploring how to improve the accuracy further and what are the consequences of our observations

for different systems.

We have combined RPA with the more accurate coupled clusters approach to simplify the evaluation of interaction

energy of methane clathrate and observed very promising results. We have now data to assess the scheme

for solids, we still need to obtain energies of the RPA variants with higher accuracy to see the highest benefit.

One of the issues we have to deal with is a large number of calculations which also leads to accumulation

of numerical errors due to finite arithmetic. We are exploring ways how to avoid these errors or reduce them.

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. We don't think this has been done before as it's usually considered there is not much to be done about this issue.

However, we hope to be able to push the model (using machine learning or similar techniques) to be able

to much improve the binding energies. This would reduce the computational cost of the calculations leading

to capability to increase the system size or increase the throughput of the calculations, which is relevant

for crystal structure prediction and other fields.

or show unreliable errors. One of the goals of our project is to understand if the accuracy of RPA can be further

improved and what would be the physical origin of that. We have made substantial progress along this direction

and we are exploring how to improve the accuracy further and what are the consequences of our observations

for different systems.

We have combined RPA with the more accurate coupled clusters approach to simplify the evaluation of interaction

energy of methane clathrate and observed very promising results. We have now data to assess the scheme

for solids, we still need to obtain energies of the RPA variants with higher accuracy to see the highest benefit.

One of the issues we have to deal with is a large number of calculations which also leads to accumulation

of numerical errors due to finite arithmetic. We are exploring ways how to avoid these errors or reduce them.

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. We don't think this has been done before as it's usually considered there is not much to be done about this issue.

However, we hope to be able to push the model (using machine learning or similar techniques) to be able

to much improve the binding energies. This would reduce the computational cost of the calculations leading

to capability to increase the system size or increase the throughput of the calculations, which is relevant

for crystal structure prediction and other fields.