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Register a new userBeyond Wavelets: Exactness theorems and algorithms for physical calculations
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This paper develops the use of wavelets as a basis set for the solution of physical problems exhibiting behavior over wide-ranges in length scale. In a simple diagrammatic language, this article reviews both the mathematical underpinnings of wavelet theory and the algorithms behind the fast wavelet transform. This article underscores the fact that traditional wavelet bases are fundamentally ill-suited for physical calculations and shows how to go beyond these limitations by the introduction of the new concept of semicardinality, which leads to the profound, new result that basic physical couplings may be computed {em without approximatation} from very sparse information, thereby overcoming the limitations of traditional wavelet bases in the treatment of physical problems. The paper then explores the convergence rate of conjugate gradient solution of the Poisson equation in both semicardinal and lifted wavelet bases and shows the first solution of the Kohn-Sham equations using a novel variational principle.
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Daubechies wavelets are a powerful systematic basis set for electronic structure calculations because they are orthogonal and localized both in real and Fourier space. We describe in detail how this basis set can be used to obtain a highly efficient and accurate method for density functional electronic structure calculations. An implementation of this method is available in the ABINIT free software package. This code shows high systematic convergence properties, very good performances and an excellent efficiency for parallel calculations.
Very recently we developed an efficient method to calculate the free energy of 2D materials on substrates and achieved high calculation precision for graphene or $gamma$-graphyne on copper substrates. In the present work, the method was further confirmed to be accurate by molecular dynamic simulations of silicene on Ag substrate using empirical potential and was applied to predict the optimum conditions based on emph{ab initio} calculations for silicene growth on Ag (110) and Ag (111) surface, which are in good agreement with previous experimental observations.
We introduce a simple but efficient method for grand-canonical twist averaging in quantum Monte Carlo calculations. By evaluating the thermodynamic grand potential instead of the ground state total energy, we greatly reduce the sampling errors caused by twist-dependent fluctuations in the particle number. We apply this method to the electron gas and to metallic lithium, aluminum, and solid atomic hydrogen. We show that, even when using a small number of twists, grand-canonical twist averaging of the grand potential produces better estimates of ground state energies than the widely used canonical twist-averaging approach.
Localized basis sets in the projector augmented wave formalism allow for computationally efficient calculations within density functional theory (DFT). However, achieving high numerical accuracy requires an extensive basis set, which also poses a fundamental problem for the interpretation of the results. We present a way to obtain a reduced basis set of atomic orbitals through the subdiagonalization of each atomic block of the Hamiltonian. The resulting local orbitals (LOs) inherit the information of the local crystal field. In the LO basis, it becomes apparent that the Hamiltonian is nearly block-diagonal, and we demonstrate that it is possible to keep only a subset of relevant LOs which provide an accurate description of the physics around the Fermi level. This reduces to some extent the redundancy of the original basis set, and at the same time it allows one to perform post-processing of DFT calculations, ranging from the interpretation of electron transport to extracting effective tight-binding Hamiltonians, very efficiently and without sacrificing the accuracy of the results.
The Hugoniot curves for shock-compressed molybdenum with initial porosities of 1.0, 1.26, 1.83, and 2.31 are theoretically investigated. The method of calculations combines the first-principles treatment for zero- and finite-temperature electronic contribution and the mean-field-potential approach for the ion-thermal contribution to the total free energy. Our calculated results reproduce the Hugoniot properties of porous molybdenum quite well. At low porosity, in particular, the calculations show a complete agreement with the experimental measurements over the full range of data. For the two large porosity values of 1.83 and 2.31, our results are well in accord with the experimental data points up to the particle velocity of 3.5 km/s, and tend to overestimate the shock-wave velocity and Hugoniot pressure when further increasing the particle velocity. In addition, the temperature along the principal Hugoniot is also extensively investigated for porous molybdenum.
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