## Final Report Summary - WFTINDFTCE (Wave-function theory embedded in density-functional theory with coupled excitations)

Wave-function theory embedded in density-functional theory with coupled excitations: ''WFTinDFTce''

Background

Electronic excitations play an important role in many biological processes, such as photosynthesis and vision, as well as in technological applications like sensors and lighting materials. Computational chemistry is widely used in the interpretation of complex experimental data and to assist in designing new materials.

In the field of theoretical chemistry, various and complementary approaches have been developed to describe response properties and excited states of molecular systems. On one side of the methodological spectrum, wave-function theory (WFT) based methods to compute accurate excitation energies are available, but these techniques remain computationally very expensive. On the other side, (time-dependent) density-functional theory (DFT) has emerged as the standard first-principles approach for treating large-scale systems. Based on the electron density, the latter approach is significantly faster to calculate, but in practice the accuracy is more difficult to guarantee.

Also DFT methods are, however, limited to up to a few hundred atoms, prohibiting the treatment of biological systems. To make the computations on such systems feasible, subsystem methods need to be employed, in which a complex system is split into smaller subunits that can be computed separately. This overcomes the scaling problem and also allows for a convenient definition of subsystem properties and better analysis of the results. The most common approach (QM/MM) is to treat only the molecule of interest quantum mechanically (QM), and use a parametrized molecular mechanics (MM) description of the environment. While very efficient, the method inherently introduces approximations and needs careful parameterization and testing. An alternative is to use so-called QM/QM schemes in which the subsystem approach is retained, but where (part of) the environment is also treated quantum mechanically.

A particularly efficient QM/QM approach is the frozen-density embedding (FDE) scheme, which is an exact subsystem formulation of DFT in which an "active" subsystem is computed

in the presence of a frozen environment density. While FDE has been quite successful in a DFT-only formulation, applications are restricted to cases in which a DFT (response) treatment of the subsystems suffices. For charge-transfer (CT) excitations, that are of particular importance in many biological systems, DFT is known to have an intrinsic problem due to the too approximate nature of the exchange-correlation functionals that are available. An alternative is found in WFT methods, that can handle such difficult cases reliably. In order to be able to employ such methods, it was necessary

to generalise the subsystem DFT theory to be able to include subsystems treated by WFT. This was the main aim of the project.

Theory

We were able to derive a consistent formulation of response in the FDE framework, for both DFT as well as coupled-cluster methods.

The procedure can be carried out to calculate arbitrary (response) properties and is based on a set of linear-response equations in which the coupling between

the different subsystems is described.

Individual molecular response properties are obtained in a series of calculations on the subsystems, which is followed by a final step in which all subsystems

are coupled and the response of the full system to a perturbation with a specific frequency is calculated.

Naturally, this separation allows to set up a hierarchy of approximations, ranging from very efficient but approximate to very accurate.

For instance, for the fastest scheme not only the coupling can be neglected but also explicit excited-state contributions of the interaction with the environment.

Furthermore, the newly developed formulation is very generic and thus does not only allow for DFT-in-DFT or WFT-in-DFT treatments, but holds also for WFT-in-WFT frozen-density embedding.

Implementation and Applications

In order to to assess the proposed methods and evaluate its performance for charge-transfer (CT) excitations, we have implemented the new equations in computer programs.

The implementations were carried out at coupled cluster level of theory with singles and doubles excitations (CCSD) as well as an approximated and more efficient scheme (denoted CC2).

Among the investigated proof-of-principle applications are small compounds such as an water..ammonia complex as well as the DNA base pair guanine..cytosine (GC), where we have used the FDE approximation to separate the two molecules in vacuum.

Beyond these, we have also studied exemplary water surrounded by 127 water molecules at one geometry snapshot and also uracil surrounded by a larger solvation shells.

These calculations are only possible due to the FDE approximations and supermolecule calculations are not feasible any more.

A WFT-in-WFT realisation of the subsystem approach was also studied and presents a step beyond the original proposal.

The sample calculations presented show that the main contribution to the solvatochromic shift is accounted for using uncoupled WFT-in-WFT embedding, which may lead to an efficient treatment for larger systems.

Conclusions

With this project, key steps have been achieved towards a more general use of the FDE approximation in several theoretical approaches, applicable in many computer programs and for general molecular properties.

The approximations introduced reduce the numerical accuracy compared to the supermolecule approach, but offer the possibility to study realistic models of complex systems which are out of reach in conventional calculations.

Among the key features of this approach is the separation of e.g. [~]solute[/~] and solvent, thus serving the interpretation of spectra of molecules in complex environments in terms of chemical subunits with accurate wave-function methods.

Background

Electronic excitations play an important role in many biological processes, such as photosynthesis and vision, as well as in technological applications like sensors and lighting materials. Computational chemistry is widely used in the interpretation of complex experimental data and to assist in designing new materials.

In the field of theoretical chemistry, various and complementary approaches have been developed to describe response properties and excited states of molecular systems. On one side of the methodological spectrum, wave-function theory (WFT) based methods to compute accurate excitation energies are available, but these techniques remain computationally very expensive. On the other side, (time-dependent) density-functional theory (DFT) has emerged as the standard first-principles approach for treating large-scale systems. Based on the electron density, the latter approach is significantly faster to calculate, but in practice the accuracy is more difficult to guarantee.

Also DFT methods are, however, limited to up to a few hundred atoms, prohibiting the treatment of biological systems. To make the computations on such systems feasible, subsystem methods need to be employed, in which a complex system is split into smaller subunits that can be computed separately. This overcomes the scaling problem and also allows for a convenient definition of subsystem properties and better analysis of the results. The most common approach (QM/MM) is to treat only the molecule of interest quantum mechanically (QM), and use a parametrized molecular mechanics (MM) description of the environment. While very efficient, the method inherently introduces approximations and needs careful parameterization and testing. An alternative is to use so-called QM/QM schemes in which the subsystem approach is retained, but where (part of) the environment is also treated quantum mechanically.

A particularly efficient QM/QM approach is the frozen-density embedding (FDE) scheme, which is an exact subsystem formulation of DFT in which an "active" subsystem is computed

in the presence of a frozen environment density. While FDE has been quite successful in a DFT-only formulation, applications are restricted to cases in which a DFT (response) treatment of the subsystems suffices. For charge-transfer (CT) excitations, that are of particular importance in many biological systems, DFT is known to have an intrinsic problem due to the too approximate nature of the exchange-correlation functionals that are available. An alternative is found in WFT methods, that can handle such difficult cases reliably. In order to be able to employ such methods, it was necessary

to generalise the subsystem DFT theory to be able to include subsystems treated by WFT. This was the main aim of the project.

Theory

We were able to derive a consistent formulation of response in the FDE framework, for both DFT as well as coupled-cluster methods.

The procedure can be carried out to calculate arbitrary (response) properties and is based on a set of linear-response equations in which the coupling between

the different subsystems is described.

Individual molecular response properties are obtained in a series of calculations on the subsystems, which is followed by a final step in which all subsystems

are coupled and the response of the full system to a perturbation with a specific frequency is calculated.

Naturally, this separation allows to set up a hierarchy of approximations, ranging from very efficient but approximate to very accurate.

For instance, for the fastest scheme not only the coupling can be neglected but also explicit excited-state contributions of the interaction with the environment.

Furthermore, the newly developed formulation is very generic and thus does not only allow for DFT-in-DFT or WFT-in-DFT treatments, but holds also for WFT-in-WFT frozen-density embedding.

Implementation and Applications

In order to to assess the proposed methods and evaluate its performance for charge-transfer (CT) excitations, we have implemented the new equations in computer programs.

The implementations were carried out at coupled cluster level of theory with singles and doubles excitations (CCSD) as well as an approximated and more efficient scheme (denoted CC2).

Among the investigated proof-of-principle applications are small compounds such as an water..ammonia complex as well as the DNA base pair guanine..cytosine (GC), where we have used the FDE approximation to separate the two molecules in vacuum.

Beyond these, we have also studied exemplary water surrounded by 127 water molecules at one geometry snapshot and also uracil surrounded by a larger solvation shells.

These calculations are only possible due to the FDE approximations and supermolecule calculations are not feasible any more.

A WFT-in-WFT realisation of the subsystem approach was also studied and presents a step beyond the original proposal.

The sample calculations presented show that the main contribution to the solvatochromic shift is accounted for using uncoupled WFT-in-WFT embedding, which may lead to an efficient treatment for larger systems.

Conclusions

With this project, key steps have been achieved towards a more general use of the FDE approximation in several theoretical approaches, applicable in many computer programs and for general molecular properties.

The approximations introduced reduce the numerical accuracy compared to the supermolecule approach, but offer the possibility to study realistic models of complex systems which are out of reach in conventional calculations.

Among the key features of this approach is the separation of e.g. [~]solute[/~] and solvent, thus serving the interpretation of spectra of molecules in complex environments in terms of chemical subunits with accurate wave-function methods.