Multipole-Preserving Quadratures for Discretization of Functions in Real-Space Electronic Structure Calculations

Discretizing an analytic function on a uniform real-space grid is often done via a straightforward collocation method. This is ubiquitous in all areas of computational physics and quantum chemistry. An example in Density Functional Theory (DFT) is given by the external potential or the pseudo-potential describing the interaction between ions and electrons. The accuracy of the collocation method used is therefore very important for the reliability of subsequent treatments like self-consistent field solutions of the electronic structure problems. By construction, the collocation method introduces numerical artifacts typical of real-space treatments, like the so-called egg-box error, that may spoil the numerical stability of the description when the real-space grid is too coarse. As the external potential is an input of the problem, even a highly precise computational treatment cannot cope this inconvenience. We present in this paper a new quadrature scheme that is able to exactly preserve the moments of a given analytic function even for large grid spacings, while reconciling with the traditional collocation method when the grid spacing is small enough. In the context of real-space electronic structure calculations, we show that this method improves considerably the stability of the results for large grid spacings, opening the path towards reliable low-accuracy DFT calculations with reduced number of degrees of freedom.Comment: 20 pages, 7 figure

Exact solution of the many-body problem with a O(n6)\mathcal{O}\left(n^6\right) complexity

In this article, we define a new mathematical object, called a pair $D=\left(A,C\right)$ of anti-commutation matrices (ACMP) based on the anti-commutation relation $a^\dag_{i}a_{j} + a_{j}a^\dag_{i} = \delta_{ij}$ applied to the scalar product between the many-body wavefunctions. This ACMP explicitly separates the different levels of correlation. The one-body correlations are defined by a ACMP $D^0=\left(A^0,C^0\right)$ and the two-body ones by a set of $n$ ACMPs $D^i=\left(A^i,C^i\right)$ where $n$ is the number of states. We show that we can have a compact and exact parametrization with $n^4$ parameters of the two-body reduced density matrix (\TRDM) of any pure or mixed $N$-body state to determine the ground state energy with a $\mathcal{O}\left(n^6\right)$ complexity

Accurate Complex Scaling of Three Dimensional Numerical Potentials

The complex scaling method, which consists in continuing spatial coordinates into the complex plane, is a well-established method that allows to compute resonant eigenfunctions of the time-independent Schroedinger operator. Whenever it is desirable to apply the complex scaling to investigate resonances in physical systems defined on numerical discrete grids, the most direct approach relies on the application of a similarity transformation to the original, unscaled Hamiltonian. We show that such an approach can be conveniently implemented in the Daubechies wavelet basis set, featuring a very promising level of generality, high accuracy, and no need for artificial convergence parameters. Complex scaling of three dimensional numerical potentials can be efficiently and accurately performed. By carrying out an illustrative resonant state computation in the case of a one-dimensional model potential, we then show that our wavelet-based approach may disclose new exciting opportunities in the field of computational non-Hermitian quantum mechanics.Comment: 11 pages, 8 figure

Fragment Approach to Constrained Density Functional Theory Calculations using Daubechies Wavelets

In a recent paper we presented a linear scaling Kohn-Sham density functional theory (DFT) code based on Daubechies wavelets, where a minimal set of localized support functions is optimized in situ and therefore adapted to the chemical properties of the molecular system. Thanks to the systematically controllable accuracy of the underlying basis set, this approach is able to provide an optimal contracted basis for a given system: accuracies for ground state energies and atomic forces are of the same quality as an uncontracted, cubic scaling approach. This basis set offers, by construction, a natural subset where the density matrix of the system can be projected. In this paper we demonstrate the flexibility of this minimal basis formalism in providing a basis set that can be reused as-is, i.e. without reoptimization, for charge-constrained DFT calculations within a fragment approach. Support functions, represented in the underlying wavelet grid, of the template fragments are roto-translated with high numerical precision to the required positions and used as projectors for the charge weight function. We demonstrate the interest of this approach to express highly precise and efficient calculations for preparing diabatic states and for the computational setup of systems in complex environments

Density Functional Theory calculation on many-cores hybrid CPU-GPU architectures

The implementation of a full electronic structure calculation code on a hybrid parallel architecture with Graphic Processing Units (GPU) is presented. The code which is on the basis of our implementation is a GNU-GPL code based on Daubechies wavelets. It shows very good performances, systematic convergence properties and an excellent efficiency on parallel computers. Our GPU-based acceleration fully preserves all these properties. In particular, the code is able to run on many cores which may or may not have a GPU associated. It is thus able to run on parallel and massive parallel hybrid environment, also with a non-homogeneous ratio CPU/GPU. With double precision calculations, we may achieve considerable speedup, between a factor of 20 for some operations and a factor of 6 for the whole DFT code.Comment: 14 pages, 8 figure

Accurate and efficient linear scaling DFT calculations with universal applicability

Density Functional Theory calculations traditionally suffer from an inherent cubic scaling with respect to the size of the system, making big calculations extremely expensive. This cubic scaling can be avoided by the use of so-called linear scaling algorithms, which have been developed during the last few decades. In this way it becomes possible to perform ab-initio calculations for several tens of thousands of atoms or even more within a reasonable time frame. However, even though the use of linear scaling algorithms is physically well justified, their implementation often introduces some small errors. Consequently most implementations offering such a linear complexity either yield only a limited accuracy or, if one wants to go beyond this restriction, require a tedious fine tuning of many parameters. In our linear scaling approach within the BigDFT package, we were able to overcome this restriction. Using an ansatz based on localized support functions expressed in an underlying Daubechies wavelet basis -- which offers ideal properties for accurate linear scaling calculations -- we obtain an amazingly high accuracy and a universal applicability while still keeping the possibility of simulating large systems with only a moderate demand of computing resources

Efficient and accurate three dimensional Poisson solver for surface problems

We present a method that gives highly accurate electrostatic potentials for systems where we have periodic boundary conditions in two spatial directions but free boundary conditions in the third direction. These boundary conditions are needed for all kind of surface problems. Our method has an O(N log N) computational cost, where N is the number of grid points, with a very small prefactor. This Poisson solver is primarily intended for real space methods where the charge density and the potential are given on a uniform grid.Comment: 6 pages, 2 figure

Structural Metastability of Endohedral Silicon Fullerenes

Endohedrally doped Si20 fullerenes appear as appealing building blocks for nanoscale materials. We investigate their structural stability with an unbiased and systematic global geometry optimization method within density-functional theory. For a wide range of metal doping atoms, it was sufficient to explore the Born Oppenheimer surface for only a moderate number of local minima to find structures that clearly differ from the initial endohedral cages, but are considerably more favorable in terms of energy. Previously proposed structures are thus all metastable.Comment: 4 pages, 1 figur