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## van der Waals forces in DFT, RPA and beyond

#### CECAM-HQ-EPFL, Lausanne, Switzerland

#### Organisers

Density functional theory (DFT) is usually the preferred method for modeling the electronic energy of large multi-nuclear systems where quantum chemical methods are too computationally intensive. However in many nanostructures the dispersion force is a very significant component of the energy, and therefore forms a vital consideration for self-assembly and other technologically important processes. (This applies, e.g., in graphitics such as graphene and carbon nanotubes). Unfortunately conventional local or semi-local DFT misses the attractive outer part of the van der Waals (vdW) force and can severely misrepresent it at smaller distances. Presently many calculations of van der Waals energetics in nanostructures are carried out by force field modelling using some variant of the Lennard-Jones potential, or by adding a suitably damped version of the attractive 1/R^{6} Lennard-Jones component to a conventional density functional calculation (Refs 1, 2 and references therein). In either case the vdW component is thus modelled by a sum of pairwise-additive 1/R^{6} terms.

While this procedure may work fairly well for compact, insulating systems with a low dielectric polarizability, it has become clear, since the last CECAM meeting on the subject in 2005, that it can be quite inaccurate for anisotropic systems with a high polarizabilty, where non-additive cross-polarization effects yield a substantially altered vdW attraction strength.[3, 4]. It has also become clear that such non-additive effects can even cause a change in the functional form of the interaction at large separations, when the interacting nanostructures are metallic or have a small electronic energy gap.[5, 6]

Recently there has been considerable development of applications based on a "vdW Density Functional".[7-11] This is the first practically-implementable, nonlocal, non-empirical density functional to include a seamless description of vdW forces at all separations. While it has made predictions for both molecular and graphitic systems, its application to these latter anisotropic highly polarizable systems requires further investigation because its vdW component takes the form of a pairwise additive contribution with 1/R^{6} asymptotics.

Unfortunately experimental data for these systems is sparse, imprecise and barely mutually compatible. There is therefore an urgent need for further experiment, and especially for non-empirical theories that do not make the pairwise-additive assumption for the vdW energy. The DFT-SAPT theory [12] is available but is workable only for relatively small systems in practice . Two important microscopic classes of theory that are promising for benchmark calculations in large nanostructures are (i) the Random Phase Approximation (RPA) [13-20] and related methods e.g. in the coupled-cluster class,[21]; (ii) the Variational and diffusion Monte Carlo methods applied to the many-electron problem [22-25]. Preliminary results on graphite and bi-graphene from both (i) and (ii) are currently circulating but still have outstanding convergence issues. One may hope definitive benchmarks for these simple nanostructures will be settled by the date of the proposed CECAM meeting. It seems clear however that Methods (i) and (ii) will not be practicable for more complex nanostructures in the foreseeable future, so much development is needed on less demanding approximate methods. The discussion of benchmarks and the development of reliable simplified theories are therefore key activities for the proposed Workshop.

A further important technique under recent development is that of Range Separation.[20, 21, 26-31] This allows a non-arbitrary combination of different theories (e.g. DFT and RPA) to be used for the long and short ranged components of the Coulomb interaction. This technique is important in allowing theories of the RPA class to deal with the short-ranged cusp in the electronic pair distribution, without the need for massive basis sets. It also permits inclusion of some short-ranged correlation effects missed by the simpler theories of the RPA class, regardless of basis set. There has been encouraging recent progress [20] with this type of theory for the binding energy curves of rare gas dimers and alkali metal dimers where, amazingly, state-of-the-art quantum chemical methods are at the limit of their adequacy. The Range Separation approach is likely to be very significant in developing methods for nanostructures and for larger molecules (e.g. coronene-coronene interactions where SAPT may be able to provide benchmarks for the lower members of the series). Range separation is currently being added to popular packages such as VASP.

## References

**Australia**

John F. Dobson (Griffith University, Nathan) - Organiser & speaker

**France**

Sebastien Lebegue (University of Nancy) - Organiser & speaker

Andreas Savin (CNRS and Sorbonne University) - Organiser

Janos Ángyán (CNRS and Université de Lorraine) - Organiser & speaker