Solid Solutions and Disordered Crystalline Systems:

II-Symmetry and Random Sampling of the Configurational Space

Philippe D’Arco

Laboratoire de Pétrologie, Case 110, UPMC Tour 26, 4, place Jussieu, 75252 Paris Cedex 05, France

The description of the properties of disordered crystalline systems is based on weighted averages of ordered configurations. Very efficient algorithms taking advantage of crystal symmetry have been proposed to count and enumerate symmetry independent configurations (SICs) in order to reduce the number of configurations to be calculated [1]. However, as the number of SICs increases, average properties are usually calculated using randomly chosen configurations [2]. Whatever the energy calculation is, such a strategy suffers several drawbacks: symmetry equivalent configurations are computed, high symmetry configurations have a low probability to be found and convergence of averages as a function of draws can hardly be estimated. Random search enhances the visibility of low symmetry SICs due to their large multiplicity [3]. In "modest” configurational spaces, the most symmetric configurations can be missed in favor of the asymmetric ones. If the proposed relation between symmetry and stablest configuration hold, the calculated average properties can then be biased [4].

In 2007, Catlow and co-workers called for Monte Carlo sampling techniques combining symmetry arguments in order to more efficiently sample the configurational space [5].

Starting from the algorithm originally developed for the random generation of unlabelled graphs on n vertices [6], we present a symmetry-adapted algorithm which circumvents this problem and produces uniformly at random the set of SICs. In this Monte Carlo (at variance with respect to Metropolis) scheme, high and low symmetry SICs have the same draw probability. Starting from Pòlya's formula and the conceptual action matrix [7], both introduced in the previous talk, the role of the conjugacy classes of the symmetry group in uniform sampling is shown.

The probability of finding a given SIC or a subset of SICs is discussed as a function of the number of draws and their precise estimate is given. The algorithm is flexible: SICs can be obtained for all the possible compositions or for a chosen one, and the probability distribution can be unbalanced in order to favorize some symmetric SICS. The present low-memory demanding implementation in the CRYSTAL code is briefly sketched.

Its yields the multiplicity of the SICs, allows to operate configurational statistics in the reduced space of the SICs and avoids calculations for symmetry equivalent configurations. The reliability of the method will be illustrated applying it to a binary series of carbonates.

The binary normal spinel solid solution Mg(Al,Fe)2O4 will be considered as a test case to calculate average properties.

This work is under press : J. Phys. : Condens. Matter, 2013.

[1] Hart, G.L.W. and Forcade, R.W. 2008 Phys. Rev. B, 77, 224115 ; Hart, G.L.W. and Forcade, R.W. 2009 Phys. Rev. B, 80, 014120 ; Hart, G.L.W., Nelson, L.J. and Forcade, R.W. 2012 Comput. Mater. Sci., 59, 101-7 ; Mustapha, S., D'Arco, Ph., de la Pierre, M., Noel, Y., Ferrabone, M. and Dovesi, R. 2013, J. Phys. : Condens. Matter, 25, 105401.

[2] Todorov, I.T., Allan, N.L., Laurentiev, M.Y, Freeman, C.L., Mohn, C.E. and Purton, J.A. 2004 J. Phys.: Condens. Matter, 16 S2751.

[3] Wang, Q., Grau-Crespo, R. and de Leew, N.H. 2011 J. Phys. Chem. B, 115,13854.

[4] Pickard, C.J. and Needs, R.J. 2013 J. Phys.: Condens. Matter, 23:53201.

[5] Grau-Crespo, R., Hamad, S., Catlow, C.R.A. and de Leeuw N.H. 2007 J. Phys. Condens. Matter, 19, 256201.

[6] Dixon, J.D. and Wilf, H.S. 1983 J. Algorithms, 4, 205213 ; Kerber, A., Laue, R., Hager, R. and Weber, W., 1990 J. Graph Theory, 14, 559563.

[7] Williamson, S.G. 1985 Combinatorics for computer science (NY : Computer Science Press)