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Method of increments - a wavefunction-based ab initio correlation method for solids

The field of many-body theory has been developing at high speed in recent years. A significant amount of effort is directed toward affordable descriptions of electronic correlation in infinite periodic systems. The most widely used approach here is density-functional theory (DFT). However, the present DFT-based approaches have in practice many shortcomings. When applying quantum-chemical correlation methods to extended systems one can make use of the property that the correlation hole is fairly local. For a solid it is useful, therefore, to switch from the description with delocalized Bloch states to localized Wannier orbitals. This is the starting-point for the so-called local correlation methods. One method of this type, the method of increments [1], combines HF calculations for periodic systems with correlation calculations for finite embedded clusters, and the total correlation energy per unit cell of a solid is written as a cumulant expansion in terms of contributions from localized orbital groups of increasing size. A schematic overview of the procedure is given in Fig. 1.

Since 1992, when H. Stoll has applied the idea of local excitations in the electronic theory of solids for considering correlation energy of diamond [1], silicon [2] and graphite [3] firstly, the method of increments has been successfully applied for various insulators and semiconductors ranging from strongly bound covalent compounds to weakly bound van der Waals rare-gas crystals (for a review see [4]).

The problem to extend the method to metallic systems is twofold. The first problem consists in the modeling of a metallic solid by a finite cluster and connected with this the question about the localisation of orbitals in a metal arises. As a solution we suggest an embedding scheme [5] which has itself no metallic character but can mimic the metal in the internal region, where the atoms are correlated. Within the incremental scheme we allow for a delocalisation of the orbitals and therefore account for the metallic character of the systems. With this approach the convergence rate, especially with the order of increments, is not as good as for insulators, but still three- and higher-body terms account for only about 15 % of the correlation contribution to the binding. Up to now we have carefully described closed shell metals [6-13]. However, it is possible to extend this approach also to opened-shell cases. So, the correlation energy of a metallic Li-ring has been considered applying a projection technique for non-orthogonal occupied and virtual orbitals within the incremental scheme [14,15]. Even an embedding scheme was suggested for 3-dimensional lithium relying on pairs of atoms [16]. In the case of lithium the s band is crossing the Fermi level and is half occupied. In this regard the second problem occurring in metals has to be mentioned, namely the quasi-degeneracy of the orbitals around the Fermi surface. Then a single-reference correlation method can fail and the incremental scheme has to be extended to the use of multi-reference correlation treatments. This can be the case not only in opened-shell systems, but also, for example, in barium [17], where the d-bands are crossing the Fermi level.

Thus, with the method of increments we have a tool, which allows the evaluation the ground-state properties of metals. Starting from the HF solution for the extended systems, which is a routine calculation nowadays, we can localize the orbitals and perform an incremental expansion for the correlation energy. An asset of the incremental scheme is that applying it one gets a possibility not only to obtain systematically improvable result, but also to investigate influence of individual contributions on the correlation energy. This opens up a large field of applications. With such a method at hand we can try to understand unusual structures of some metals. Preliminary work has been done in our laboratory for gallium [18], which forms Ga-dimers within the metallic phase. One more class of the object affecting the great interest is intermetallic compounds, especially Laves phases [19]. Here our approach could be useful for understanding their stability and phase formation as a function of components.

An advantage of the method of increments in comparison with other local correlation methods is that the translational symmetry of the system can be used to reduce the effort of the calculation. Translational symmetry is, however, not a prerequisite of the method. Therefore it is possible to extend the method to systems with reduced symmetry like surfaces and molecules on the surface [20]. Impurities and functional groups in large biological molecules can be described as well. If the description for metals and the one for impurities are combined, the whole field of Kondo physics opens up. But one has to be aware of the fact that for the small energy scales involved in the Kondo effect require highly accurate quantum-chemical methods, and very extended basis sets are necessary to achieve reliable results.

[1] H. Stoll, Phys. Rev. B 46, 6700 (1992).
[2] H. Stoll, Chem. Phys. Lett. 191, 548 (1992).
[3] H. Stoll, J. Chem. Phys. 97, 8449 (1992).
[4] B. Paulus, Phys. Rep. 428, 1 (2006).
[5] E. Voloshina, N. Gaston, and B. Paulus, J. Chem. Phys. 126, 134115 (2007).
[6] B. Paulus and K. Rosciszewski, Chem. Phys. Lett. 394, 96 (2004).
[7] B. Paulus, K. Rosciszewski, N. Gaston, P. Schwerdtfeger, and H. Stoll, Phys. Rev. B 70, 165106 (2004).
[8] N. Gaston, B. Paulus, K. Rosciszewski, P. Schwerdtfeger, and H. Stoll, Phys. Rev. B 74, 094102 (2006).
[9] E. Voloshina and B. Paulus, Phys. Rev. B 75, 245117 (2007).
[10] E. Voloshina and B. Paulus, Mol. Phys. 105, 2849 (2007).
[11] N. Gaston and B. Paulus, Phys. Rev. B 76, 214116 , (2007).
[12] N. Gaston, B. Paulus, U. Wedig, and M. Jansen, Phys. Rev. Lett. 100, 226404 (2008).
[13] E. Voloshina, B. Paulus, and H. Stoll, J. Phys.: Conf. Ser. 117, 012029 (2008).
[14] H. Stoll, Ann. Physik 5, 355 (1997).
[15] B. Paulus, Chem. Phys. Lett. 371, 7 (2003).
[16] H. Stoll, B. Paulus, and P. Fulde, Chem. Phys. Lett. 469, 90 (2009).
[17] B. Paulus and A. Mitin, Lecture series on computer and computational science 7, 935 (2006).
[18] E. Voloshina, K. Rosciszewski, and B. Paulus, Phys. Rev. B 79, 045113 (2009).
[19] F. Laves, H. Witte, Metallwirtschaft 14, 645 (1935).
[20] C. Müller, B. Herschend, K. Hermansson, and B. Paulus, J. Chem. Phys. 128, 214701 (2008).

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