Krylov Subspace Method for Molecular Dynamics Simulation Based on Large-Scale Electronic Structure Theory

Takayama, R.; Hoshi, Takeo; Fujiwara, Takeo

doi:10.1143/jpsj.73.1519

Cited by 46 publications

(73 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For instance, a Lanczos procedure generates basis vectors in a Krylov subspace which are orthogonal. It has been proposed [317] that diagonalising within the subspace (similar, in effect, to the method of Ozaki [254] described above) will give an efficient linear scaling method; they find that around 30 vectors is sufficient. A more efficient variant of this method [318] solves linear equations:…”

Section: Recursive and Stochastic Approachesmentioning

confidence: 99%

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

2012

View full text Add to dashboard Cite

Abstract. Linear scaling methods, or O(N ) methods, have computational and memory requirements which scale linearly with the number of atoms in the system, N , in contrast to standard approaches which scale with the cube of the number of atoms. These methods, which rely on the short-ranged nature of electronic structure, will allow accurate, ab initio simulations of systems of unprecedented size. The theory behind the locality of electronic structure is described and related to physical properties of systems to be modelled, along with a survey of recent developments in real-space methods which are important for efficient use of high performance computers. The linear scaling methods proposed to date can be divided into seven different areas, and the applicability, efficiency and advantages of the methods proposed in these areas is then discussed. The applications of linear scaling methods, as well as the implementations available as computer programs, are considered. Finally, the prospects for and the challenges facing linear scaling methods are discussed.

show abstract

Section: Recursive and Stochastic Approachesmentioning

confidence: 99%

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

2012

View full text Add to dashboard Cite

show abstract

“…We developed the subspace diagonalization method and the shifted conjugate orthogonal conjugate gradient (COCG) method. [13][14][15][16] Then, the methods were applied to the fracture propagation and surface formation in Si crystals with the tight-binding Hamiltonian based on an orthogonal basis set. 1,2 On the other hand, since its Hamiltonian is described by the tight-binding Hamiltonian based on a nonorthogonal basis set, the problem of the formation of Au multishell helical nanowires was solved by the exact diagonalization method.…”

Section: Introductionmentioning

confidence: 99%

“…3,4 Development of efficient linear algebraic methods has been, so far, mainly based on the orthogonal basis sets. 13,14,[17][18][19] However, localized basis wave functions are generally nonorthogonal and it is much more desirable to generalize the methods to the case of a nonorthogonal basis set. The most popular strategy of the generalized eigenvalue problem (represented by the nonorthogonal basis set) would be the transformation to the standard eigenvalue problem.…”

Section: Introductionmentioning

confidence: 99%

Efficient and accurate linear algebraic methods for large-scale electronic structure calculations with nonorthogonal atomic orbitals

et al. 2011

View full text Add to dashboard Cite

The need for large-scale electronic structure calculations arises recently in the field of material physics, and efficient and accurate algebraic methods for large simultaneous linear equations become greatly important. We investigate the generalized shifted conjugate orthogonal conjugate gradient method, the generalized Lanczos method, and the generalized Arnoldi method. They are the solver methods of large simultaneous linear equations of the one-electron Schrödinger equation and map the whole Hilbert space to a small subspace called the Krylov subspace. These methods are applied to systems of fcc Au with the NRL tight-binding Hamiltonian [F. Kirchhoff et al., Phys. Rev. B 63, 195101 (2001)]. We compare results by these methods and the exact calculation and show them to be equally accurate. The system size dependence of the CPU time is also discussed. The generalized Lanczos method and the generalized Arnoldi method are the most suitable for the large-scale molecular dynamics simulations from the viewpoint of CPU time and memory size.

show abstract

“…Although such simulations have been carried out thus far, 8,9,13 the investigation is still limited, owing to the system size of 10 2 atoms. In this paper, the cleavage of silicon is studied with quantum mechanical calculations for large-scale electronic structures 16,17,18,19,20,21 and we use a transferable Hamiltonian 22 in the Slater-Koster (tight-binding) form. The methodology is reviewed briefly in Appendix A.…”

Section: Introductionmentioning

confidence: 99%

Nanoscale structures formed in silicon cleavage studied with large-scale electronic structure calculations: Surface reconstruction, steps, and bending

2005

Self Cite

View full text Add to dashboard Cite

The 10-nm-scale structure in silicon cleavage is studied by the quantum mechanical calculations for large-scale electronic structure. The cleavage process on the order of 10 ps shows surface reconstruction and step formation. These processes are studied by analyzing electronic freedom and compared with STM experiments. The discussion presents the stability mechanism of the experimentally observed mode, the (111)-(2 × 1) mode, beyond the traditional approach with surface energy. Moreover, in several results, the cleavage path is bent into the experimentally observed planes, owing to the relative stability among different cleavage modes. Finally, several common aspects between cleavage and other phenomena are discussed from the viewpoints of the nonequilibrium process and the 10-nm-scale structure.

show abstract

Krylov Subspace Method for Molecular Dynamics Simulation Based on Large-Scale Electronic Structure Theory

Cited by 46 publications

References 22 publications

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

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

Efficient and accurate linear algebraic methods for large-scale electronic structure calculations with nonorthogonal atomic orbitals

Nanoscale structures formed in silicon cleavage studied with large-scale electronic structure calculations: Surface reconstruction, steps, and bending

Contact Info

Product

Resources

About