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Multiresolution analyses based upon interpolets, interpolating scaling functions introduced by Deslauriers and Dubuc, are particularly well-suited to physical applications because they allow exact recovery of the multiresolution representation of a function from its sample values on a finite set of points in space. We present a detailed study of the application of wavelet concepts to physical problems expressed in such bases. The manuscript describes algorithms for the associated transforms which, for properly constructed grids of variable resolution, compute correctly without having to introduce extra grid points. We demonstrate that for the application of local homogeneous operators in such bases, the non-standard multiply of Beylkin, Coifman and Rokhlin also proceeds exactly for inhomogeneous grids of appropriate form. To obtain less stringent conditions on the grids, we generalize the non-standard multiply so that communication may proceed between non-adjacent levels. The manuscript concludes with timing comparisons against naive algorithms and an illustration of the scale-independence of the convergence rate of the conjugate gradient solution of Poissons equation using a simple preconditioning, suggesting that this approach leads to an O(n) solution of this equation.
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We demonstrate that Daubechies wavelets can be used to construct a minimal set of optimized localized contracted basis functions in which the Kohn-Sham orbitals can be represented with an arbitrarily high, controllable precision. Ground state energies and the forces acting on the ions can be calculated in this basis with the same accuracy as if they were calculated directly in a Daubechies wavelets basis, provided that the amplitude of these contracted basis functions is sufficiently small on the surface of the localization region, which is guaranteed by the optimization procedure described in this work. This approach reduces the computational costs of DFT calculations, and can be combined with sparse matrix algebra to obtain linear scaling with respect to the number of electrons in the system. Calculations on systems of 10,000 atoms or more thus become feasible in a systematic basis set with moderate computational resources. Further computational savings can be achieved by exploiting the similarity of the contracted basis functions for closely related environments, e.g. in geometry optimizations or combined calculations of neutral and charged systems.
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.
A proposal for a magnetic quantum processor that consists of individual molecular spins coupled to superconducting coplanar resonators and transmission lines is carefully examined. We derive a simple magnetic quantum electrodynamics Hamiltonian to describe the underlying physics. It is shown that these hybrid devices can perform arbitrary operations on each spin qubit and induce tunable interactions between any pair of them. The combination of these two operations ensures that the processor can perform universal quantum computations. The feasibility of this proposal is critically discussed using the results of realistic calculations, based on parameters of existing devices and molecular qubits. These results show that the proposal is feasible, provided that molecules with sufficiently long coherence times can be developed and accurately integrated into specific areas of the device. This architecture has an enormous potential for scaling up quantum computation thanks to the microscopic nature of the individual constituents, the molecules, and the possibility of using their internal spin degrees of freedom.
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.
The emergence of ultra-fast X-ray free-electron lasers opens the possibility of imaging single molecules in the gas phase at atomic resolution. The main disadvantage of this imaging technique is the unknown orientation of the sample exposed to the X-ray beam, making the three dimensional reconstruction not trivial. Induced orientation of molecules prior to X-ray exposure can be highly beneficial, as it significantly reduces the number of collected diffraction patterns whilst improving the quality of the reconstructed structure. We present here the possibility of protein orientation using a time-dependent external electric field. We used ab initio simulations on Trp-cage protein to provide a qualitative estimation of the field strength required to break protein bonds, with 45 V/nm as a breaking point value. Furthermore, we simulated, in a classical molecular dynamics approach, the orientation of ubiquitin protein by exposing it to different time-dependent electric fields. The protein structure was preserved for all samples at the moment orientation was achieved, which we denote `orientation before destruction. Moreover, we find that the minimal field strength required to induce orientation within ten ns of electric field exposure, was of the order of 0.5 V/nm. Our results help explain the process of field orientation of proteins and can support the design of instruments for protein orientation.
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