Large-scale electronic structure theory for simulating nanostructure processes

Abstract

Fundamental theories and practical methods for large-scale electronic structure calculations are given, in which the computational cost is proportional to the system size. Accuracy controlling methods for microscopic freedoms are focused on two practical solver methods, Krylov-subspace method and generalized-Wannierstate method. A general theory called the 'multi-solver' scheme is also formulated, as a hybrid between different solver methods. Practical examples are carried out in several insulating and metallic systems with 10 3 -10 5 atoms. All the theories provide general guiding principles of constructing an optimal calculation for simulating nanostructure processes, since a nanostructured system consists of several competitive regions, such as bulk and surface regions, and the simulation is designed to reproduce the competition with an optimal computational cost.

\mathcal{O}(N) methods in electronic structure calculations

“…Once the vectors |x j have been generated for one energy, further energies are almost trivial, and certainly scale as O(1). An overview and summary of applications using tight binding have been given [319,320], and the method has been extended to non-orthogonal orbitals [321]. The recursion methods were the first set of linear scaling methods proposed, and Lanczos approaches are widely used in many areas of physics and mathematics.…”

Large scale simulation of quantum-mechanical molecular dynamics for nano-polycrystalline diamond

“…[5,6,7] In the Krylov subspace theory, the Green's function is calculated, instead of eigenstates. The methodology has a rigorous mathematical foundation as iterative linear-algebraic algorithms and is applicable to both insulators and metals.…”

“…The methodological details of the Krylov subspace theory in the present simulation are the same as those for liquid carbon. [7] In the first stage of our research, the simulations were carried out by expanding a diamond structure with initial structural defects, a small fraction of threefold-coordinated atoms and deformed fourfold-coordinated atoms, not more than 10 %. In this way, we aim to obtain the intrinsic metastable structures of nano domains within the limit of the computational time scales.…”

Development of the simulation package ‘ELSES’ for extra-large-scale electronic structure calculation

“…For years, we have developed a set of theories and program code for such nanoscience researches. [1,2,3,4,5,6,7,8,9] One crucial point is that large-scale quantum-mechanical calculation can be realized, in principle, by calculating the one-body density matrix, instead of one-electron eigen states, since the computational cost can be drastically reduced. [10] An overview of these theories can be found in the introduction part of Ref.…”

“…Electronic property, such as density of states, is also calculated. [7] Now the code has named Extra Large Scale Electronic Structure calculation (ELSES) code (www.elses.jp).…”