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## Theoretical Chemistry for Periodic Systems: Systematically Improvable Electronic Structure Methods

#### Ax-Les-Thermes -- Midi-Pyrenees -- France (CF-CAM Sud-Ouest)

#### Organisers

The development of electronic structure methods and rapid advances in computing technology have transformed the role of quantitative numerical calculations in both theoretical and experimental studies in chemistry, chemical physics, condensed-phase and solid-state sciences. The last five decades, in which this progress occurred, have seen enormous developments. During the 1980s, density-functional theory (DFT) -- the computational workhorse of solid-state physics -- was imported and established in chemistry, while, during the same period, the development of the CRYSTAL code by the Pisani group at Turin matured and made the Hartree-Fock (HF) method available for periodic systems. The following two decades have seen a huge amount of further development in practical and robust software implementations, also adapted to novel parallel processor hardware, such that electronic structure calculations are now a routine or even an essential tool in many research fields.

In the past several years, there has been a dramatic shift toward using accurate, systematic improvable electronic structure methods for large systems, including systems showing periodicity in one, two, or three dimensions, see, e.g., [Paulus_2006, Schmitt_2009, Manby_2010, Beran_2012]. While conventional Kohn-Sham DFT remains a widely used method for electronic structure calculations on any kind of systems, the newer, systematically improvable wavefnction-based methods go beyond it in the range of problems they can address, for example excited states, dispersion and other weak intermolecular interactions, and strong correlation. Electronic structure methods that have traditionally been developed, refined, and used by chemists can now be (and are already) exported to materials science and solid-state physics. The introduction of these new tools will impact those fields in much the same way as DFT impacted chemistry in the 1980s. It is thus of importance to disseminate the new developed knowledge about these methods, and about the handling of the tools into which they have been implemented, as quickly and widely as possible. The workshop scheduled for February 2013 will contribute significantly to this effort.

Historically, the first of the wavefunction-based systematically improvable electron correlation methods was the so-called Local Ansatz by Stollhoff and Fulde [Stollhoff_1977], where in addition to periodic Hartree-Fock results local excitations were selected in a way comparable to a selection as in a CEPA approach. Two closely related methods, which were much development in the recent years, are the periodic local MP2 method (LMP2) by Pisani et al. [Pisani_2005] and the plane-wave based canonical MP2 method by Kresse et al. [Marsman_2009, Grüneis_2010]. The former method employs orthonormal localized orbitals for the occupied space (Wannier functions) and mutually non-orthogonal, redundant projected atomic orbitals (PAOs) for the virtual space, and has been implemented now in the CRYSCOR program, see [Pisani_2012] and references therein. Considering the relatively short time the LMP2 method has been available, it has been tested already extensively for a large variety of systems, ranging from weakly bound rare-gas solids and molecular crystals to semiconductors, ionic solids, covalently bound solids. Plane-wave based canonical MP2 was tested for a couple of systems with band gaps from 2 to 20 eV [Grüneis_2010] and is implemented in the VASP program [Kresse_1999]. Very recently, implementations of periodic canonical coupled-cluster singles and doubles (CCSD) have been reported by Grüneis et al. [Grüneis_2011], based on the plane-wave canonical MP2 method by Kresse et al. mentioned above. Here, the use of approximated natural orbitals improves the computational time of the canonical MP2 and allows even for the more expensive CCSD method within this plane wave framework.

A conceptually rather different approach, the method of increments (MI), was invented by Stoll in 1992 for calculating electron correlation effects in solids [Stoll_1992]. MI relies on localized molecular orbitals and expands the total electron correlation energy in terms of contributions from groups of these orbitals. One of these groups contains localized orbitals sharing a common center, which is in the most common case an atom but in some special cases the center of a bond or an entire molecule. The correlation energy, calculated for the electrons in orbitals of one group, is called the one-center increment. The non-additive part of the correlation energy calculated for the electrons in two groups of orbitals is called the two-center increment. Analogously, for three groups of orbitals one obtains three-center increments, and so on. Any size-extensive method can yield the exact electron correlation energy as a sequence in terms of these increments, weighted with appropriate weight factors. The correlation energy thus obtained has to be combined then with the energy from a periodic Hartree-Fock calculation, in order to obtain the total energy per unit cell. The total number of increments increases rapidly with the size of a system, so the success of the method of increments relies on the fast convergence of the incremental expansion with respect to the order of the increments and with respect to the distance between orbital groups. Different approaches exist for the actual calculation of the increments. For materials with a band gap these approaches are ranging from non-embedded clusters, via electrostatically embedded clusters or clusters surrounded by ghost atoms to clusters with correlated atoms surrounded by frozen Hartree-Fock orbitals, which can be described by a smaller basis set. Special embeddings were invented for covalent systems, metals, surfaces, ans molecules adsorbed on surfaces. However, all these embeddings were designed in a way that allow to use standard quantum chemical program packages for the size-extensive correlated calculations. For further details see [Müller_2012] and references cited therein. Alternatively, Friedrich et al. develop a fully automated implementation of the incremental scheme [Friedrich_2007] which has, however, not been applied to any periodic system yet.

For completeness, we also mention that the full configuration interaction quantum Monte Carlo (FCI-QMC) method has also been extended recently to periodic systems [Alavi_2012].

## References

**France**

Stefano Evangelisti (Laboratoire de Chimie et Physique Quantiques) - Organiser

Thierry Leininger (Laboratoire de Chimie et Physique Quantiques UMR5626 CNRS Université Toulouse 3) - Organiser

**Germany**

Dirk Andrae (Physikalische und Theoretische Chemie - Institut für Chemie und Biochemie - Freie Universität Berlin) - Organiser

Beate Paulus (Physikalische und Theoretische Chemie - Institut für Chemie und Biochemie - Freie Universität Berlin) - Organiser