Quasicontinuum-like Reduction of Dft Calculations of Nanostructures

Density functional theory (DFT) can accurately predict chemical and mechanical properties of nanostructures, although at a high computational cost. A quasicontinuum-like framework is proposed to substantially increase the size of the nanostructures that can be simulated ab initio. This increase stems from two facts. First, in order to find approximate ground state electron density (the electronic problem), expensive DFT calculations are limited to a small number of subdomains, and the solution is interpolated everywhere else. The electronic problem embeds interpolation and coupled cross-domain optimization techniques through a process called electronic reconstruction. Second, for the optimization of nuclei positions (the ionic problem), computational gains result from explicit consideration of a reduced number of representative nuclei interpolating the positions of the rest of nuclei following the quasicontinuum paradigm. In the proposed approach, the new ionic configuration is the solution of a nonlinear system obtained as the first-order optimality condition for the minimization of the total energy associated with the nanostructure. This is an optimization problem with equilibrium constraints because the electronic density is itself the solution of a minimization problem. Numerical tests using the Thomas-Fermi-Dirac functional demonstrate the validity of the proposed framework within the orbital-free density functional theory. INTRODUCTION Nanostructures have dimensions in the range of 1 ∼ 100 nm and typically contain 102 ∼ 108 atoms. Density functional methods within the Kohn-Sham approach1 are typically applied to systems with fewer than 100 atoms. Contemporary implementations of order-N methods2 such as SIESTA3, ONETEP4, and CONQUEST5 that exhibit linear scaling of computation time with system size enable an increase in the number of atoms by one to two orders ∗Address all correspondence to this author. 1 of magnitude on massively parallel computers. Larger system sizes are accessible to classical interatomic potential methods but these methods cannot account for spin and charge relaxation and conjugation effects, which are important in modeling reactions, electronic excitation, and bond breaking processes. Therefore, new computational paradigms are needed that enable larger scale electronic structure calculations. A combination of methods with different fidelity is often used to reduce computational effort if only local information is needed with high accuracy. An example of such an approach is the ONIOM method6 for computations of chemical properties. However, such schemes have an inherent problem with conditions at the boundaries of different fidelity regions. Another approach to reduce computational effort, called the quasicontinuum method, is based on explicit treatment of only representative atoms and on interpolations for the rest. It has been successfully used in atomistic studies of mechanical properties with classical potentials and has been under development to include electronic-level calculations. This type of approach is particularly suitable for many nanostructures because large regions of the structures are perturbed relatively little as compared to periodic structures and, therefore, can be treated by using interpolation schemes. The present work proposes a quasicontinuum-like technique that, by closing the spatial scale gap, renders electronic structure information at the nanoscale. The proposed methodology follows in the steps of the quasicontinuum approach discussed in7;8;9 for mechanical analysis at the nanoscale. Specifically, this is an extension of the work in8;9, because, rather than considering a potential-based interatomic interaction that has a limited range of validity and is difficult to generalize to inhomogeneous materials, the methodology proposed uses ab initio methods to provide for particle interaction. At the same time it is a generalization of the method proposed in7 because, rather than considering electron density within each mesh discretization element separately, the proposed method treats the electronic density distribution in all elements in a generic optimization framework. Our approach does not rely on a strict periodicity assumption; it merely assumes that the material displays a nearly periodic structure in certain regions of the nanostructure. However, in order to bridge the gap between subatomic scale associated with the electron density and the nanoscale associated with the structures investigated, we have assumed that almost everywhere in the nanostructure the optimized structure results in only small deformations of periodic structure. This assumption is referred to as near-periodicity, because the nonperiodic part of the state variables is approximated as a macroscopic smoothly varying field. As explained later, the near-periodicity assumption enables the use of interpolation for electronic structure reconstruction. Here we define the small deformations that are present in most of the material. The nanostructure is considered to have an initial reference configuration D0 ⊂ R3. The structure undergoes a deformation described by a deformation mapping Φ(r0, t) ∈ R3, which gives the location r in the global Cartesian reference frame of each point r0 represented in the undeformed material frame. As indicated, the mapping might depend on time t. The variable t does not necessarily represent the time contemporary with the structure under consideration. In a static simulation framework this variable might be an iteration index of an optimization algorithm that solves for the system ground state. The components of the deformation gradient are introduced as FiJ = ∂Φi ∂r0 J , where upper-case indices refer to the material frame, and lower-case indices to the Cartesian global frame. Thus, F = ∇0 Φ, where ∇0 represents the material gradient operator. Using the repeated index summation rule, we express the deformation of an infinitesimal material neighborhood dr0 about a point r0 of D0 as dri = FiJ dr 0 J . If u = r − r0, the concept of small distortion is equivalent to requiring that the spectral radius of F̄ = ∇ u be sufficiently small; that is, ||∇0 u||2 < K is expected to hold almost everywhere in the domain D0, for a suitable chosen value of K.