CECAM/Psi-k Workshop

Local orbitals and linear-scaling ab initio calculations

CECAM, Lyon, 3-7 September 2001

Abstracts for Friday, 7 September 2001

CoSi₂ is already an important material for metalization in VLSI devices. This industrial interest provides a strong incentive to understand in detail the behavior of Co during deposition and annealing. In this talk I will discuss two sets of calculations using Plato: 1) those performed to study the relative stabilities of a number of surface and subsurface sites; 2) those performed to study the diffusion of Co in Si.

A method for obtaining spatially localized crystalline orbitals starting from delocalized Bloch Functions has been implemented in the periodic LCAO CRYSTAL code [1]; it provides a set of well localized Wannier functions (WF) through an iterative mixed Wannier-Boys scheme [2].

The WFs of seven oxygen containing compounds with increasing degree of covalent character (MgO, MnO, ZnO, Al₂O₃, SiO₂, AlPO₄ and CaSO₄) [3] and of six semiconductors (Si, C, BP, AlP, SiC and BN) [4] are analyzed in terms of various indices (centroids positions, second order central moment tensor, its eigenvalues and principal axes, Mulliken population analysis and atomic localization indices) and through their graphical representations. Systematic trends are observed along the series.

As an example, Fig. 1 gives the atomic delocalization index [5] (top) defined as:

(1)

(2)

Figure 1: Atomic delocalization index (eq. 1; in ) and "normalized" standard deviation (eq. 2) of the considered oxygen containing compounds. Full and open circles refer to ionic/covalent (io, cv) and lone pair (lp) WFs, respectively.

An example of graphical representation is given in Fig. 2 for the diamond valence WF; the electronic density map and profile (see the caption for more details) are reported.

Figure 2: Electron density map in the (1 1 1) plane (top) and profile along the bond axis (down), of one of the four diamond valence WFs. In the map (4 Å wide), iso-lines differing by 0.015 in the range 0.015-0.3 are represented. Gray dots and lines indicate nuclei positions and the bond axes, respectively. Density in and distances in Å.

The WFs have also been used for the calculation of dielectric properties of various compounds, in alternative to the Berry phase scheme based on the Bloch functions formalism [6,7]. The effective Born charges, spontaneous polarization and dielectric constants of various compounds are easily derived simply from the coordinates of the WF centroids [8,9].

Table 1 provides an example referring to KNbO₃; Fig. 3 shows the dependence of the centroid position of a valence and a core WFs of BeO, on the strain .

Method
BP	8.073	1.001	-5.964	-1.556	-0.0003	0.347
LWF	8.089	1.000	-5.985	-1.552	0.0004	0.348

Table 1: Born effective charges (in ), acoustic sum rule (in ) and (in C/m²) obtained for KNbO₃, by using the BP and LWF schemes.

Figure 3: Dependence of the centroid z fractional coordinate for a core (left scale, continuous line and open circles) and a valence (right scale, dashed line and full circles) LWF of BeO centered on oxygen, as a function of the cell parameter c (in Å). and Å^-1 for valence and core centroids, respectively.

1. V R Saunders, R Dovesi, C Roetii, R Orlando, M Causa, N M Harrison and C M Zicovich-Wilson, CRYSTAL98 User's Manual, Università di Torino, Torino, 1998.
2. C M Zicovich-Wilson, R Dovesi and V R Saunders, submitted to J. Chem. Phys. (2001).
3. C Zicovich-Wilson, A Bert, C Roetti, R Dovesi and V Saunders, in preparation.
4. A Bert, C M Zicovich-Wilson, Roberto Dovesi and M Llunell, in preparation.
5. J Pipek and P G Mezey, J. Chem. Phys. 90, 4916 (1989).
6. R D King-Smith and D Vanderbilt, Phys. Rev. B 47, 1651 (1993).
7. R Resta, Rev. Mod. Phys. 66, 809 (1994).
8. Ph Baranek, C M Zicovich-Wilson, R Orlando and R Dovesi, Phys. Rev. B, in press (2001).
9. Y Noel, B Civalleri, C M Zicovich-Wilson, Ph D'Arco and R Dovesi, submitted to Phys. Rev. B (2001).

We present a method for obtaining well-localized Wannier-like functions (WFs) for energy bands that are attached to or mixed with other bands. The present scheme removes the limitation of the usual maximally-localized WF method [1] that the bands of interest should form an isolated group, separated by gaps from higher and lower bands everywhere in the Brillouin zone. An energy window encompassing N bands of interest is specified by the user, and the algorithm then proceeds to disentangle these from the remaining bands inside the window by filtering out an optimally connected N-dimensional subspace. This is achieved by minimizing a functional that measures the subspace dispersion across the Brillouin zone. The maximally-localized WFs for the optimal subspace are then obtained via the algorithm of Marzari and Vanderbilt [1]. The method, which functions as a postprocessing step using the output of conventional electronic-structure codes, is applied to the s and d bands of copper, and to the valence and low-lying conduction bands of silicon. For the low-lying nearly-free-electron bands of copper we find WFs which are centered at the tetrahedral interstitial sites (see right panel of the figure), suggesting an alternative tight-binding parametrization.

1. N Marzari and D Vanderbilt, Phys. Rev. B 56, 12847 (1997).

Left Panel: Solid lines: Calculated band structure of copper. Dotted lines: Seven Wannier-interpolated bands obtained from our procedure, by choosing an optimal seven-dimensional space from the states inside the outer energy window, while constraining the states inside the inner window to be included in that subspace. The maximally-localized WFs consist of five atom-centered d-like orbitals plus two equivalent s-like orbitals, each centered at one of the two tetrahedral interstitial sites. The contour-surface plot of one of the latter is shown in the right panel. The amplitudes are (light gray) and (dark gray), where is the volume of the primitive cell.