The theoretical description of the properties of complex molecular systems requires the use of multiscale methods, in which different approximations adequate for different parts of the whole system are applied. An important example are chemical problems that concern the properties of a subunit embedded in a complex environment. For instance, such a situation is encountered when studying solvent effects on molecular properties, when investigating chromophores embedded in protein environments, or when looking at catalytic reaction mechanisms in biological systems, in porous solids, in liquids, or on surfaces. Such applications require computational methods that can provide an accurate description for a small subsystem of interest, while using an efficient representation of its environment.

The proposed workshop concerns the class of multiscale simulation methods that are based on this idea of embedding a subsystem described by means of conventional quantum-mechanical descriptors (such as the wavefunction, a density matrix, or the charge density) in an environment represented by its density only. The Density Functional Theory (DFT) formulation of the many-electron problem provides the formal framework behind such methods, which has been developed in [1-3]. We will refer to this framework as Frozen-Density Embedding Theory (FDET). The key elements of FDET are:
(i) the application of a variational principle, which leads to a well-defined upper bound for the ground-state energy of the total system,
(ii) an embedding potential that is uniquely defined by the charge densities in the environment and in the embedded subsystem, and
(iii) self-consistent definitions of the density functionals for the energy and embedding potential.
This way, FDET provides a formally exact framework for quantum-chemical embedding methods. This makes it possible to control the approximations necessary when devising multiscale simulation methods and offers a possibility to go beyond a purely electrostatic embedding potential, such as the one applied in QM/MM schemes.

Recent years brought exciting new developments in multiscale simulation methods based on FDET. Many research groups are active in:

a) the development of improved approximations for the FDET embedding potential [4-6]
b) studies of mathematical properties of the FDET embedding potential [4,5,7] and of the relevant density functionals [8-10],
c) the development of algorithms for the reconstruction of the exact FDET embedding potential by inversion techniques [11-14],
d) the development of multi-level strategies that couple different quantum-chemical methods in a seamless fashion using such an FDET embedding potential [14-16],
d) the development of efficient numerical implementations (e.g., in ADF [17,18], deMon2k [19], MOLCAS [20-22], Molpro [16,23], Turbomole [24]) as well as scripting frameworks to couple different quantum-chemical program packages [25],
e) exploring new domains of applicability of multiscale simulation methods based on FDET, in particular for solving challenging problems in theoretical spectroscopy. Examples include EPR spectroscopy and the treatment of open-shell systems [26-28], NMR spectroscopy [29], electronic excited states [17,30-33], and CD spectroscopy [34].
f) the development of "beyond-FDET" embedding methods, i.e., methods in which the environment is described by means of descriptors other than the electron density [24,35-40]. The development of such methods is frequently inspired by difficulties simple FDET-based methods encountered in practice (e.g., for the description of covalent bonds between subsystems, of strongly correlated subsystems, or of the coupling between excitations).

United by the common formal background of such methods, each research group focusses on different specific issues in these areas. Mostly, the theoretical method development and the development of numerical implementations are driven by the need to employ multiscale simulations to tackle specific chemical problems (e.g., the prediction of properties of solvated molecules, unraveling the mechanism of excitation energy transfer in photosynthetic systems, or improving the catalytic activity of surfaces).

The experience gained in the community, especially in the last few years, now calls for an effort to gather the knowledge about strengths and weaknesses of various approximations within FDET, to establish the best numerical protocols ("best practices") for specific classes of problems, and to define future requirements and challenges.

Despite the impressive developments that have been happening within the last two years no dedicated scientific meeting that would unify scientists working on such theories and methods has taken place so far. The number of issues of common interest is great and the prompt endorsement we have received from many scientists active in the field justifies the need of such a meeting.

Starting from the formal framework of FDET, different multiscale simulation methods can be devised. Such methods differ in
(a) the way the embedding potential is approximated,
(b) the method used to generate the density of the environment, and
(c) the type of quantum mechanical descriptor employed for the embedded subsystem.
Since the introduction of FDET and the first numerical simulations based on it almost two decades ago [1], several key developments happened recently concerning all three aspects.

Concerning (a): For setting up multiscale simulation methods based on FDET, it is necessary to calculate the contribution to the FDET embedding potential arising from the nonadditive kinetic energy. In general, this nonadditive kinetic potential is not easily available. Therefore, approximations based on kinetic-energy density functionals have been introduced. In particular, for weakly overlapping densities generalized gradient approximations (GGAs) have been very successful [41-43]. Recently, a number of novel GGA-type approximations have been introduced [4,6], which each address specific shortcomings of the existing approximations.
On the other hand, for large overlaps (i.e., for subsystems connected by covalent bonds) such semi-local approximations are known to fail [41,9,44-46]. This problem can be circumvented by introducing a more general partitioning [47,48]. Alternatively, new techniques based on numerical inversion were introduced into FDET-based methods recently. This has been a major breakthrough for the field, because it makes it possible to extend the range of applicability of the FDET embedding potential even to covalent interactions [14-16]. Moreover, accurate potentials obtained with inversion techniques (either numerical [12] or analytical [7,8,49]) can be used as guidelines for constructing better approximations for the non-additive kinetic potential.

Concerning (b), the choice of the frozen electron density (i.e., the density of the environment) represents a crucial approximation in FDET-based simulations. Usually, a density generated by some quantum-chemical method is used, but the available strategies differ significantly. There is a large body of scattered numerical results obtained in different groups working on a variety of systems. This calls for a systematic analysis at the proposed workshop. However, FDET allows for more general choices for the frozen density. For instance, in studies of solvent environments the frozen density can be generated from a classical-statistical theory of liquids, leading to a non-uniform continuum model [50]. On the other hand, the electron density for the environment might also be generated in fully variational calculations (subsystem DFT [51-53] or partition DFT [54,55]) in which it is optimized simultaneously with the electron density of the embedded subsystem [56]. This allows for a systematic refinement of the frozen density and leads to an efficient alternative to conventional KS-DFT calculations.

Concerning (c): Initially, FDET was formulated for embedding a subsystem described by means of non-interacting reference electrons (Kohn-Sham DFT). Subsequently, this has been extended to allow for other levels of approximation for the embedded subsystem. Della Salla and collaborators introduced the use of hybrid functionals within a generalized KS framework [24]. Carter and coworkers devised computational schemes [57,58], in which the FDET embedding potential was combined with wavefunction-based quantum-chemical methods. This makes it possible to overcome known failures of KS-DFT and enjoys great popularity nowadays [20,21,59-61]. A formal justification for the such a combination was given later using the constrained search definition of the relevant functionals [2]. Recently, such embedding methods employing a wavefunction-based description for the embedded subsystem have been combined with the use of accurate embedding potentials reconstructed using inversion techniques, allowing for a seamless coupling of different quantum-chemical methods [11,14-16].

In addition, there has been a significant amount of recent work on the extension of FDET-based methods to the treatment of excited states. In particular, subsystem DFT has been generalized to excited states using the linear response TDDFT framework [35]. Subsequently, the corresponding numerical methods were developed [17] and extended for studies of coupled chromophores [36,37]. Recently, the inclusion of wavefunction-based methods for treating excited states within the framework of FDET has also been (re-)considered [62,63].

Finally, certain limitations of FDET-based simulation methods have also become clear in recent years. These are in general not limitations of the theoretical framework, but hinge on both the restriction to a frozen-density description of the environment and the need to introduce approximations in FDE-based simulations. Therefore, several works address methods going beyond the formal framework of FDET by introducing additional quantum-mechanical descriptors for the environment, such as KS-orbitals [38] or wavefunctions [39,40]. This way, it becomes possible to treat systems dominated by long-range electronic correlations, i.e., situations in which the interaction between subsystems cannot be treated with DFT and the currently available exchange-correlation functionals. Similarly, to treat coupled electronic excitations it becomes necessary to introduced an orbital-based description of the environment [35-37,63]. The same applies if the polarization of the environment due to (excitations in) the embedded subsystem shall be included.