Formal Compatibility between Symmetry Adaption and Spatial Localization and their Use in the Computational Study of Crystalline Systems.

Claudio M. Zicovich-Wilson

Facultad de Ciencias, Universidad Autonóma del Estado de Morelos, Av. Universidad 1001, Col, Chamilpa, 62209,

Cuernavaca, (Morelos), Mexico

The combined exploitation of symmetry equivalences between basis set components and sparsity of the corresponding matrices has been one of the key points in the design of efficient computational methods for the quantum chemistry of periodic systems.1 In such a context, matrix sparsity is closely connected with spatial localization of the wave-functions that conform the basis sets and a clear linkage between this property and site symmetry in crystalline systems has been highlighted. [2]

A generalization of these concepts intended to formally establish the existence and effective computation of sets of functions that simultaneously feature optimal spatial localization properties and symmetry relationships,[3,4] is discussed in this talk. At variance with the usual symmetry-adapted basis sets, as bases of the irreducible representations (IRREP) of the symmetry group, these symmetrically-localized

orbitals (SLO) are in general bases of the induced representations (INDREP). Being the IRREPs a particular case of the INDREPs, the SLOs become the most general form of symmetry-adapted orbitals. It is shown that most of the powerful theorems concerning IRREPs can be somehow extended to INDREPs providing ways to maximally exploit symmetry equivalences together with matrix sparsity in the solution of several quantum mechanical problems connected to the electronic structure of materials.

As an illustrating example, it is shown that SLOs can be conveniently optimized -keeping their symmetry equivalence properties- with respect to functionals with well defined characteristics, for instance

the Foster-Boys one for spatial spread.[5] Similar strategies could be employed in variation methods for the electronic structure of large and symmetric systems.

Last but no least, the combination of symmetry equivalences and spatial localization is of particular utility in the study of crystalline systems as it allowed to extend periodic methods with atomic basis sets already implemented in public codes like Crystal[6] toward wave-function approximations that explicitly account for electronic correlation as, for instance, the periodic Local-MP2 method. [7]

[1] Pisani, C.; Dovesi, R.; Roetti, C., Hartree-Fock Ab Initio Treatment of Crystalline Solids, vol. 48 of Lecture Notes in Chemistry Series, Springer: Berlin, 1988.

[2] Evarestov, R. A.; Smirnov, V. P., Site symmetry in crystals: theory and applications, Springer series in Solid-state sciences, Springer-Verlag: Berlin Heidelberg, 1997, second enlarged ed., ISSN 0171-1873.

[3] Zicovich-Wilson, C. M.; Erba, A., Int. J. Quantum Chem. 2012, 112 , 3543-3551.

[4] Zicovich-Wilson, C. M.; Erba, A., Theor. Chem. Acc. 2010, 126 , 165.

[5] Foster, J. M.; Boys, S. F., Rev. Mod. Phys. 1960, 32 , 300-302.

[6] Dovesi, R.; Saunders, V. R.; Roetti, C.; Orlando, R.; Zicovich-Wilson, C. M.; Pascale, F.; Civalleri, B.; Doll, K.; Harrison, N. M.; Bush, I. J.; Arco, P. D.; Llunell, M., CRYSTAL09 Users Manual, University of Turin: Turin, 2009, see http://www.crystal.unito.it.

[7] Pisani, C.; Maschio, L.; Casassa, S.; Halo, M.; Schutz, M.; Usvyat, D., J. Comput. Chem. 2008, 29 , 2113, A. Erba and M. Halo, CRYSCOR09 Users manual, University of Torino, Torino, 2009 (www.cryscor.unito.it).