Flexibilities of wavelets as a computational basis set for large-scale electronic structure calculations.

The BigDFT project was started in 2005 with the aim of testing the advantages of using a Daubechies wavelet basis set for Kohn-Sham (KS) density functional theory (DFT) with pseudopotentials. This project led to the creation of the BigDFT code, which employs a computational approach with optimal features of flexibility, performance, and precision of the results. In particular, the employed formalism has enabled the implementation of an algorithm able to tackle DFT calculations of large systems, up to many thousands of atoms, with a computational effort that scales linearly with the number of atoms. In this work, we recall some of the features that have been made possible by the peculiar properties of Daubechies wavelets. In particular, we focus our attention on the usage of DFT for large-scale systems. We show how the localized description of the KS problem, emerging from the features of the basis set, is helpful in providing a simplified description of large-scale electronic structure calculations. We provide some examples on how such a simplified description can be employed, and we consider, among the case-studies, the SARS-CoV-2 main protease.

Reproducibility in density functional theory calculations of solids.

The widespread popularity of density functional theory has given rise to an extensive range of dedicated codes for predicting molecular and crystalline properties. However, each code implements the formalism in a different way, raising questions about the reproducibility of such predictions. We report the results of a community-wide effort that compared 15 solid-state codes, using 40 different potentials or basis set types, to assess the quality of the Perdew-Burke-Ernzerhof equations of state for 71 elemental crystals. We conclude that predictions from recent codes and pseudopotentials agree very well, with pairwise differences that are comparable to those between different high-precision experiments. Older methods, however, have less precise agreement. Our benchmark provides a framework for users and developers to document the precision of new applications and methodological improvements.

Toward Fast and Accurate Evaluation of Charge On-Site Energies and Transfer Integrals in Supramolecular Architectures Using Linear Constrained Density Functional Theory (CDFT)-Based Methods.

A fast and accurate scheme has been developed to evaluate two key molecular parameters (on-site energies and transfer integrals) that govern charge transport in organic supramolecular architecture devices. The scheme is based on a constrained density functional theory (CDFT) approach implemented in the linear-scaling BigDFT code that exploits a wavelet basis set. The method has been applied to model disordered structures generated by force-field simulations. The role of the environment on the transport parameters has been taken into account by building large clusters around the active molecules involved in the charge transfer.

Multipole-preserving quadratures for the 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, which 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 up the path towards reliable low-accuracy DFT calculations with a reduced number of degrees of freedom.

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 are 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.

Accurate and efficient linear scaling DFT calculations with universal applicability.

Density functional theory calculations are computationally extremely expensive for systems containing many atoms due to their intrinsic cubic scaling. This fact has led to the development of so-called linear scaling algorithms during the last few decades. In this way it becomes possible to perform ab initio calculations for several tens of thousands of atoms within reasonable walltimes. 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 system with linear scaling walltimes requiring only a moderate demand of computing resources. We prove the effectiveness of our method on a wide variety of systems with different boundary conditions, for single-point calculations as well as for geometry optimizations and molecular dynamics.

Daubechies wavelets for linear scaling density functional theory.

We demonstrate that Daubechies wavelets can be used to construct a minimal set of optimized localized adaptively 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 adaptively 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 density functional theory 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 adaptively contracted basis functions for closely related environments, e.g., in geometry optimizations or combined calculations of neutral and charged systems.

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 Schrödinger 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.

Norm-conserving pseudopotentials with chemical accuracy compared to all-electron calculations.

By adding a nonlinear core correction to the well established dual space Gaussian type pseudopotentials for the chemical elements up to the third period, we construct improved pseudopotentials for the Perdew-Burke-Ernzerhof [J. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996)] functional and demonstrate that they exhibit excellent accuracy. Our benchmarks for the G2-1 test set show average atomization energy errors of only half a kcal/mol. The pseudopotentials also remain highly reliable for high pressure phases of crystalline solids. When supplemented by empirical dispersion corrections [S. Grimme, J. Comput. Chem. 27, 1787 (2006); S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, J. Chem. Phys. 132, 154104 (2010)] the average error in the interaction energy between molecules is also about half a kcal/mol. The accuracy that can be obtained by these pseudopotentials in combination with a systematic basis set is well superior to the accuracy that can be obtained by commonly used medium size Gaussian basis sets in all-electron calculations.

Optimized energy landscape exploration using the ab initio based activation-relaxation technique.

Unbiased open-ended methods for finding transition states are powerful tools to understand diffusion and relaxation mechanisms associated with defect diffusion, growth processes, and catalysis. They have been little used, however, in conjunction with ab initio packages as these algorithms demanded large computational effort to generate even a single event. Here, we revisit the activation-relaxation technique (ART nouveau) and introduce a two-step convergence to the saddle point, combining the previously used Lanczós algorithm with the direct inversion in interactive subspace scheme. This combination makes it possible to generate events (from an initial minimum through a saddle point up to a final minimum) in a systematic fashion with a net 300-700 force evaluations per successful event. ART nouveau is coupled with BigDFT, a Kohn-Sham density functional theory (DFT) electronic structure code using a wavelet basis set with excellent efficiency on parallel computation, and applied to study the potential energy surface of C(20) clusters, vacancy diffusion in bulk silicon, and reconstruction of the 4H-SiC surface.

Assessment of noncollinear spin-flip Tamm-Dancoff approximation time-dependent density-functional theory for the photochemical ring-opening of oxirane.

Under the usual assumption of noninteracting v-representability, density-functional theory (DFT) together with time-dependent DFT (TDDFT) provide a formally exact single-reference method suitable for the theoretical description of the electronic excited-states of large molecules, and hence for the description of excited-state potential energy surfaces important for photochemistry. The quality of this single-reference description is limited in practice by the need to use approximate exchange-correlation functionals. In particular it is far from clear how well approximations used in contemporary practical TDDFT calculations can describe funnel regions such as avoided crossings and conical intersections. These regions typically involve biradical-like structures associated with bond breaking and conventional wisdom would seem to suggest the need to introduce explicit double excitation character to describe these structures. Although this is lacking in ordinary spin-preserving (SP) TDDFT, it is present to some extent in spin-flip (SF) TDDFT. We report our tests of Wang-Ziegler noncollinear SF-TDDFT within the Tamm-Dancoff approximation for describing the avoided crossing in the C(2v) CC ring-opening reaction of oxirane and for describing the conical intersection relevant for the more physical asymmetric CO ring-opening reaction of oxirane. Comparisons are made with complete active space self-consistent field and quantum Monte Carlo benchmark results from two previous papers on the subject [J. Chem. Phys., 2007, 127, 164111; ibid 129, 2008, 124108]. While the avoided crossing in the C(2v) pathway is found to be reasonably well described, the method was found to be only partially successful for the conical intersection (CX) associated with the physically more important asymmetric pathway. The origin of the difficulties preventing the noncollinear SF-TDDFT method from giving a completely satisfactory description of the CX was traced back to the inability of SF-TDDFT based upon a single triplet reference state to correlate all potentially relevant configurations involving not just two but three nearly degenerate orbitals (n, σ(CO), and σ(CO)(*)). This article is also the first report of our implementation of SF-TDDFT within the deMon2k program.

Metallofullerenes as fuel cell electrocatalysts: a theoretical investigation of adsorbates on C59Pt.

Nano-structured electrode degradation in state-of-the-art polymer electrolyte membrane fuel cells (PEMFCs) is one of the main shortcomings that limit the large-scale development and commercialization of this technology. During normal operating conditions of the fuel cell, the PEMFC lifetime tends to be limited by coarsening of the cathode's Pt-based catalyst and by corrosion of the cathode's carbon black support. Because of their chemical properties, metallofullerenes such as C(59)Pt may be more electrochemically stable than the Pt/C mixture. In this paper we investigate, by theoretical methods, the stability of oxygen reduction reaction (ORR) adsorbates on the metallofullerene C(59)Pt and evaluate its potential as a PEMFC fuel cell catalyst.

Density functional theory calculation on many-cores hybrid central processing unit-graphic processing unit architectures.

We present the implementation of a full electronic structure calculation code on a hybrid parallel architecture with graphic processing units (GPUs). This implementation is performed on a free software code based on Daubechies wavelets. Such code 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, and thus on parallel and massive parallel hybrid machines. 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 density functional theory code.

Daubechies wavelets as a basis set for density functional pseudopotential calculations.

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.

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 kinds 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.

Efficient solution of Poisson's equation with free boundary conditions.

Interpolating scaling functions give a faithful representation of a localized charge distribution by its values on a grid. For such charge distributions, using a fast Fourier method, we obtain highly accurate electrostatic potentials for free boundary conditions at the cost of O(N log N) operations, where N is the number of grid points. Thus, with our approach, free boundary conditions are treated as efficiently as the periodic conditions via plane wave methods.

A fourfold coordinated point defect in silicon.

Vacancies, interstitials, and Frenkel pairs are considered to be the basic point defects in silicon. We challenge this point of view by presenting density functional calculations that show that there is a stable point defect in silicon that has fourfold coordination and is lower in energy than the traditional defects.