We propose an adaptive planewave method for eigenvalue problems in electronic structure calculations. The method combines a priori convergence rates and accurate a posteriori error estimates into an effective way of updating the energy cut-off for planewave discretizations, for both linear and nonlinear eigenvalue problems. The method is error controllable for linear eigenvalue problems in the sense that for a given required accuracy, an energy cut-off for which the solution matches the target accuracy can be reached efficiently. Further, the method is particularly promising for nonlinear eigenvalue problems in electronic structure calculations as it shall reduce the cost of early iterations in self-consistent algorithms. We present some numerical experiments for both linear and nonlinear eigenvalue problems. In particular, we provide electronic structure calculations for some insulator and metallic systems simulated with Kohn--Sham density functional theory (DFT) and the projector augmented wave (PAW) method, illustrating the efficiency and potential of the algorithm.
We present phaseless auxiliary-field (AF) quantum Monte Carlo (QMC) calculations of the ground states of some hydrogen-bonded systems. These systems were selected to test and benchmark different aspects of the new phaseless AF QMC method. They include the transition state of H+H$_2$ near the equilibrium geometry and in the van der Walls limit, as well as H$_2$O, OH, and H$_2$O$_2$ molecules. Most of these systems present significant challenges for traditional independent-particle electronic structure approaches, and many also have exact results available. The phaseless AF QMC method is used ei…
The iterative diagonalization of a sequence of large ill-conditioned generalized eigenvalue problems is a computational bottleneck in quantum mechanical methods employing a nonorthogonal basis for {\em ab initio} electronic structure calculations. We propose a hybrid preconditioning scheme to effectively combine global and locally accelerated preconditioners for rapid iterative diagonalization of such eigenvalue problems. In partition-of-unity finite-element (PUFE) pseudopotential density-functional calculations, employing a nonorthogonal basis, we show that the hybrid preconditioned block ste…
The commutator direct inversion of the iterative subspace (commutator DIIS or C-DIIS) method developed by Pulay is an efficient and the most widely used scheme in quantum chemistry to accelerate the convergence of self consistent field (SCF) iterations in Hartree-Fock theory and Kohn-Sham density functional theory. The C-DIIS method requires the explicit storage of the density matrix, the Fock matrix and the commutator matrix. Hence the method can only be used for systems with a relatively small basis set, such as the Gaussian basis set. We develop a new method that enables the C-DIIS method t…
Routine applications of electronic structure theory to molecules and periodic systems need to compute the electron density from given Hamiltonian and, in case of non-orthogonal basis sets, overlap matrices. System sizes can range from few to thousands or, in some examples, millions of atoms. Different discretization schemes (basis sets) and different system geometries (finite non-periodic vs. infinite periodic boundary conditions) yield matrices with different structures. The ELectronic Structure Infrastructure (ELSI) project provides an open-source software interface to facilitate the impleme…
Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework II: Force, vibration, and molecular dynamics calculations
Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；University of California, Berkeley（加州大学伯克利分校）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）
An Adaptive Multiscale Approach for Electronic Structure Methods
Institut für Numerische Simulation, Universität Bonn（波恩大学数值模拟研究所）；Fraunhofer-Institut für Algorithmen und Wissenschaftliches Rechnen SCAI（弗劳恩霍夫算法与科学计算研究所）
In the present work, we investigate the computational efficiency afforded by higher-order finite-element discretization of the saddle-point formulation of orbital-free density functional theory. We first investigate the robustness of viable solution schemes by analyzing the solvability conditions of the discrete problem. We find that a staggered solution procedure where the potential fields are computed consistently for every trial electron-density is a robust solution procedure for higher-order finite-element discretizations. We next study the numerical convergence rates for various orders of…
We present an efficient computational approach to perform real-space electronic structure calculations using an adaptive higher-order finite-element discretization of Kohn-Sham density-functional theory (DFT). To this end, we develop an a-priori mesh adaption technique to construct a close to optimal finite-element discretization of the problem. We further propose an efficient solution strategy for solving the discrete eigenvalue problem by using spectral finite-elements in conjunction with Gauss-Lobatto quadrature, and a Chebyshev acceleration technique for computing the occupied eigenspace. …
A theoretical scheme for the treatment of an open molecular system with electrons and nuclei is proposed. The idea is based on the Grand Canonical description of a quantum region embedded in a classical reservoir of molecules. Electronic properties of the quantum region are calculated at constant electronic chemical potential equal to that of the corresponding (large) bulk system treated at full quantum level. Instead, the exchange of molecules between the quantum region and the classical environment occurs at the chemical potential of the macroscopic thermodynamic conditions. T he Grand Canon…
The calculation of potential energy surfaces for quantum dynamics can be a time consuming task -- especially when a high level of theory for the electronic structure calculation is required. We propose an adaptive interpolation algorithm based on polyharmonic splines combined with a partition of unity approach. The adaptive node refinement allows to greatly reduce the number of sample points by employing a local error estimate. The algorithm and its scaling behavior is evaluated for a model function in 2, 3 and 4 dimensions. The developed algorithm allows for a more rapid and reliable interpol…
The Discontinuous Galerkin (DG) electronic structure method employs an adaptive local basis (ALB) set to solve the Kohn-Sham equations of density functional theory (DFT) in a discontinuous Galerkin framework. The adaptive local basis is generated on-the-fly to capture the local material physics, and can systematically attain chemical accuracy with only a few tens of degrees of freedom per atom. A central issue for large-scale calculations, however, is the computation of the electron density (and subsequently, ground state properties) from the discretized Hamiltonian in an efficient and scalabl…
MRChem is a code for molecular electronic structure calculations, based on a multiwavelet adaptive basis representation. We provide a description of our implementation strategy and several benchmark calculations. Systems comprising more than a thousand orbitals are investigated at the HF level of theory, with an emphasis on scaling properties. With our design, terms which formally scale quadratically with the system size, in effect have a better scaling because of the implicit screening introduced by the inherent adaptivity of the method: all operations are performed to the requested precision…
A Precision Emulation Approach to the GPU Acceleration of Ab Initio Electronic Structure Calculations
Texas Advanced Computing Center, The University of Texas at Austin（德克萨斯大学奥斯汀分校高级计算中心）；Pittsburgh Computing Center, Carnegie Mellon University（卡内基梅隆大学匹兹堡计算中心）
In this paper we introduce a common problem in electronic measurements and electrical engineering: finding the first root from the left of an equation in the presence of some initial conditions. We present examples of electrotechnical devices (analog signal filtering), where it is necessary to solve it. Two new methods for solving this problem, based on global optimization ideas, are introduced. The first uses the exact a priori given global Lipschitz constant for the first derivative. The second method adaptively estimates local Lipschitz constants during the search. Both algorithms either fi…
