arXiv subjects

Anders M. N. Niklasson

Publications and source records attributed to Anders M. N. Niklasson.

At least 37 records · Page 2Linked to original sources

arXiv2023

Shadow Energy Functionals and Potentials in Born-Oppenheimer Molecular Dynamics

In Born-Oppenheimer molecular dynamics (BOMD) simulations based on density functional theory (DFT), the potential energy and the interatomic forces are calculated from an electronic ground state density that is determined by an iterative self-consistent field optimization procedure, which in practice never is fully converged. The calculated energies and the forces are therefore only approximate, which may lead to an unphysical energy drift and instabilities. Here we discuss an alternative shadow BOMD approach that is based on a backward error analysis. Instead of calculating approximate solutions for an underlying exact regular BO potential, we do the opposite. Instead, we calculate the exact electron density, energies, and forces, but for an underlying approximate shadow BO potential. In this way the calculated forces are conservative with respect to the shadow potential and generate accurate molecular trajectories with long-term energy stability. We show how such shadow BO potentials can be constructed at different levels of accuracy as a function of the integration time step, dt, from the minimization of a sequence of systematically improvable, but approximate, shadow energy density functionals. For each functional there is a corresponding ground state BO potential. These pairs of shadow energy functionals and potentials are higher-level generalizations of the original "0th-level" shadow energy functionals and potentials used in extended Lagrangian BOMD [Eur. Phys. J. B vol. 94, 164 (2021)]. The proposed shadow energy functionals and potentials are useful only within this dynamical framework, where also the electronic degrees of freedom are propagated together with the atomic positions and velocities. The theory is general and can be applied to MD simulations using approximate DFT, Hartree-Fock or semi-empirical methods, as well as to coarse-grained flexible charge models.

physics.chem-ph↗

arXiv2022

Graph-based Quantum Response Theory and Shadow Born-Oppenheimer Molecular Dynamics

Graph-based linear scaling electronic structure theory for quantum-mechanical molecular dynamics simulations is adapted to the most recent shadow potential formulations of extended Lagrangian Born-Oppenheimer molecular dynamics, including fractional molecular-orbital occupation numbers, which enables stable simulations of sensitive complex chemical systems with unsteady charge solutions. The proposed formulation includes a preconditioned Krylov subspace approximation for the integration of the extended electronic degrees of freedom, which requires quantum response calculations for electronic states with fractional occupation numbers. For the response calculations we introduce a graph-based canonical quantum perturbation theory that can be performed with the same natural parallelism and linear scaling complexity as the graph-based electronic structure calculations for the unperturbed ground state. The proposed techniques are particularly well-suited for semi-empirical electronic structure theory and the methods are demonstrated using self-consistent charge density-functional tight-binding (SCC-DFTB) theory, both for the acceleration of self-consistent field calculations and for quantum molecular dynamics simulations. The graph-based techniques combined with the semi-empirical theory enable stable simulations of large, complex chemical systems, including tens-of-thousands of atoms.

physics.chem-ph↗

arXiv2022

Quantum perturbation theory using Tensor cores and a deep neural network

Time-independent quantum response calculations are performed using Tensor cores. This is achieved by mapping density matrix perturbation theory onto the computational structure of a deep neural network. The main computational cost of each deep layer is dominated by tensor contractions, i.e. dense matrix-matrix multiplications, in mixed precision arithmetics which achieves close to peak performance. Quantum response calculations are demonstrated and analyzed using self-consistent charge density-functional tight-binding theory as well as coupled-perturbed Hartree-Fock theory. For linear response calculations, a novel parameter-free convergence criterion is presented that is well-suited for numerically noisy low precision floating point operations and we demonstrate a peak performance of almost 200 Tflops using the Tensor cores of two Nvidia A100 GPUs.

physics.comp-ph↗

arXiv2021

Quantum-based Molecular Dynamics Simulations Using Tensor Cores

Tensor cores, along with tensor processing units, represent a new form of hardware acceleration specifically designed for deep neural network calculations in artificial intelligence applications. Tensor cores provide extraordinary computational speed and energy efficiency, but with the caveat that they were designed for tensor contractions (matrix-matrix multiplications) using only low-precision floating point operations. In spite of this, we demonstrate how tensor cores can be applied with high efficiency to the challenging and numerically sensitive problem of quantum-based Born-Oppenheimer molecular dynamics, which requires highly accurate electronic structure optimizations and conservative force evaluations. The interatomic forces are calculated on-the-fly from an electronic structure that is obtained from a generalized deep neural network, where the computational structure naturally takes advantage of the exceptional processing power of the tensor cores and allows for high performance in excess of 100 Tflops on the tensor cores of a single Nvidia A100 GPU. Stable molecular dynamics trajectories are generated using the framework of extended Lagrangian Born-Oppenheimer molecular dynamics, which combines computational efficiency with long-term stability, even when using approximate charge relaxations and force evaluations that are limited in accuracy by the numerically noisy conditions caused by the low precision tensor core floating-point operations. A canonical ensemble simulation scheme is also presented, where the additional numerical noise in the calculated forces is absorbed into a Langevin-like dynamics.

physics.comp-ph↗

arXiv2021

Performance Optimizations of Recursive Electronic Structure Solvers targeting Multi-Core Architectures (LA-UR-20-26665)

As we rapidly approach the frontiers of ultra large computing resources, software optimization is becoming of paramount interest to scientific application developers interested in efficiently leveraging all available on-Node computing capabilities and thereby improving a requisite science per watt metric. The scientific application of interest here is the Basic Math Library (BML) that provides a singular interface for linear algebra operation frequently used in the Quantum Molecular Dynamics (QMD) community. The provisioning of a singular interface indicates the presence of an abstraction layer which in-turn suggests commonalities in the code-base and therefore any optimization or tuning introduced in the core of code-base has the ability to positively affect the performance of the aforementioned library as a whole. With that in mind, we proceed with this investigation by performing a survey of the entirety of the BML code-base, and extract, in form of micro-kernels, common snippets of code. We introduce several optimization strategies into these micro-kernels including 1.) Strength Reduction 2.) Memory Alignment for large arrays 3.) Non Uniform Memory Access (NUMA) aware allocations to enforce data locality and 4.) appropriate thread affinity and bindings to enhance the overall multi-threaded performance. After introducing these optimizations, we benchmark the micro-kernels and compare the run-time before and after optimization for several target architectures. Finally we use the results as a guide to propagating the optimization strategies into the BML code-base. As a demonstration, herein, we test the efficacy of these optimization strategies by comparing the benchmark and optimized versions of the code.

cs.PF↗

arXiv2021

Mixed Precision Fermi-Operator Expansion on Tensor Cores From a Machine Learning Perspective

We present a second-order recursive Fermi-operator expansion scheme using mixed precision floating point operations to perform electronic structure calculations using tensor core units. A performance of over 100 teraFLOPs is achieved for half-precision floating point operations on Nvidia's A100 tensor core units. The second-order recursive Fermi-operator scheme is formulated in terms of a generalized, differentiable deep neural network structure, which solves the quantum mechanical electronic structure problem. We demonstrate how this network can be accelerated by optimizing the weight and bias values to substantially reduce the number of layers required for convergence. We also show how this machine learning approach can be used to optimize the coefficients of the recursive Fermi-operator expansion to accurately represent fractional occupation numbers of the electronic states at finite temperatures.

physics.comp-ph↗

arXiv2021

Shadow Lagrangian dynamics for superfluidity

Motivated by a similar approach for Born-Oppenheimer molecular dynamics, this paper proposes an extended "shadow" Lagrangian density for quantum states of superfluids. The extended Lagrangian contains an additional field variable that is forced to follow the wave function of the quantum state through a rapidly oscillating extended harmonic oscillator. By considering the adiabatic limit for large frequencies of the harmonic oscillator, we can derive the two equations of motions, a Schrödinger-type equation for the quantum state and a wave equation for the extended field variable. The equations are coupled in a nonlinear way, but each equation individually is linear with respect to the variable that it defines. The computational advantage of this new system is that it can be easily discretized using linear time stepping methods, where we propose to use a Crank-Nicolson-type approach for the Schrödinger equation and an extended leapfrog scheme for the wave equation. Furthermore, the difference between the quantum state and the extended field variable defines a consistency error that should go to zero if the frequency tends to infinity. By coupling the time-step size in our discretization to the frequency of the harmonic oscillator we can extract an easily computable consistency error indicator that can be used to estimate the numerical error without any additional costs. The findings are illustrated in numerical experiments.

math.NA↗

arXiv2020

Modeling solid-liquid interface reactions with next generation extended Lagrangian quantum-based molecular dynamics

We demonstrate the applicability of extended Lagrangian Born-Oppenheimer quantum-based molecular dynamics (XL-BOMD) to model electron transfer reactions occurring on solid-liquid interfaces. Specifically, we consider the reduction of O$_2$ as catalyzed at the interface of an N-doped graphene sheet and H$_2$O at fuel cell cathodes. This system is a good testbed for next-generation computational chemistry methods since the electrochemical functionalities strongly depend on atomic-scale quantum mechanics. As opposed to prior iterations of first principles molecular dynamics, XL-BOMD only requires a full self-consistent-charge relaxation during the initial time step. The electronic ground state and total energy are stabilized thereafter through nuclear and electronic equations of motion assisted by an inner-product kernel updated with low-rank approximations. A species charge analysis reveals that the kernel-based XL-BOMD simulation can capture an electron transfer between the PGM-free catalyst and a solvated O$_2$ molecule mediated by H$_2$O, which results in the molecular dissociation of O$_2$.

physics.chem-ph↗

arXiv2020

Density-matrix based Extended Lagrangian Born-Oppenheimer Molecular Dynamics

Extended Lagrangian Born-Oppenheimer molecular dynamics [{\em Phys.\ Rev.\ Lett.\ } {\bf 2008}, {\em 100}, 123004] is presented for Hartree-Fock theory, where the extended electronic degrees of freedom are represented by a density matrix, including fractional occupation numbers at elevated electronic temperatures. In contrast to regular direct Born-Oppenheimer molecular dynamics simulations, no iterative self-consistent field optimization is required prior to the force evaluations. To sample regions of the potential energy landscape where the gap is small or vanishing, which leads to particular convergence problems in regular direct Born-Oppenheimer molecular dynamics simulations, an adaptive integration scheme for the extended electronic degrees of freedom is presented. The integration scheme is based on a tunable, low-rank approximation of a fourth-order kernel, ${\cal K}$, that determines the metric tensor, ${\cal T}\equiv {\cal K}^T{\cal K}$, used in the extended harmonic oscillator of the Lagrangian that generates the dynamics of the electronic degrees of freedom. The formulation and algorithms provide a general guide to implement extended Lagrangian Born-Oppenheimer molecular dynamics for quantum chemistry, density functional theory, and semiempirical methods using a density matrix formalism.

physics.comp-ph↗

arXiv2019

Using Graph Partitioning for Scalable Distributed Quantum Molecular Dynamics

The simulation of the physical movement of multi-body systems at an atomistic level, with forces calculated from a quantum mechanical description of the electrons, motivates a graph partitioning problem studied in this article. Several advanced algorithms relying on evaluations of matrix polynomials have been published in the literature for such simulations. We aim to use a special type of graph partitioning in order to efficiently parallelize these computations. For this, we create a graph representing the zero-nonzero structure of a thresholded density matrix, and partition that graph into several components. Each separate submatrix (corresponding to each subgraph) is then substituted into the matrix polynomial, and the result for the full matrix polynomial is reassembled at the end from the individual polynomials. This paper starts by introducing a rigorous definition as well as a mathematical justification of this partitioning problem. We assess the performance of several methods to compute graph partitions with respect to both the quality of the partitioning and their runtime.

physics.comp-ph↗

arXiv2018

Graph Partitioning Methods for Fast Parallel Quantum Molecular Dynamics

We study a graph partitioning problem motivated by the simulation of the physical movement of multi-body systems on an atomistic level, where the forces are calculated from a quantum mechanical description of the electrons. Several advanced algorithms have been published in the literature for such simulations that are based on evaluations of matrix polynomials. We aim at efficiently parallelizing these computations by using a special type of graph partitioning. For this, we represent the zero-nonzero structure of a thresholded matrix as a graph and partition that graph into several components. The matrix polynomial is then evaluated for each separate submatrix corresponding to the subgraphs and the evaluated submatrix polynomials are used to assemble the final result for the full matrix polynomial. The paper provides a rigorous definition as well as a mathematical justification of this partitioning problem. We use several algorithms to compute graph partitions and experimentally evaluate their performance with respect to the quality of the partition obtained with each method and the time needed to produce it.

quant-ph↗

arXiv2017

Next generation extended Lagrangian first principles molecular dynamics

Extended Lagrangian Born-Oppenheimer molecular dynamics [Phys. Rev. Lett., ${\bf 100}$, 123004 (2008)] is formulated for general Hohenberg-Kohn density functional theory and compared to the extended Lagrangian framework of first principles molecular dynamics by Car and Parrinello [Phys. Rev. Lett. ${\bf 55}$, 2471 (1985)]. It is shown how extended Lagrangian Born-Oppenheimer molecular dynamics overcomes several shortcomings of regular, direct Born-Oppenheimer molecular dynamics, while improving or maintaining important features of Car-Parrinello simulations. The accuracy of the electronic degrees of freedom in extended Lagrangian Born-Oppenheimer molecular dynamics, with respect to the exact Born-Oppenheimer solution, is of second order in the size of the integration time step and of fourth order in the potential energy surface. Improved stability over recent formulations of extended Lagrangian Born-Oppenheimer molecular dynamics is achieved by generalizing the theory to finite temperature ensembles, using fractional occupation numbers in the calculation of the inner-product kernel of the extended harmonic oscillator that appears as a preconditioner in the electronic equations of motion. Materials systems that normally exhibit slow self-consistent field convergence can be simulated using integration time steps of the same order as in direct Born-Oppenheimer molecular dynamics, but without the requirement of an iterative, non-linear electronic ground state optimization prior to the force evaluations and without a systematic drift in the total energy. In combination with proposed low-rank and on-the-fly updates of the kernel, this formulation provides an efficient and general framework for quantum based Born-Oppenheimer molecular dynamics simulations.

cond-mat.mtrl-sci↗

arXiv2016

Graph-based linear scaling electronic structure theory

We show how graph theory can be combined with quantum theory to calculate the electronic structure of large complex systems. The graph formalism is general and applicable to a broad range of electronic structure methods and materials, including challenging systems such as biomolecules. The methodology combines well-controlled accuracy, low computational cost, and natural low-communication parallelism. This combination addresses substantial shortcomings of linear scaling electronic structure theory, in particular with respect to quantum-based molecular dynamics simulations.

physics.comp-ph↗

arXiv2015

Canonical density matrix perturbation theory

Density matrix perturbation theory [Niklasson and Challacombe, Phys. Rev. Lett. 92, 193001 (2004)] is generalized to canonical (NVT) free energy ensembles in tight-binding, Hartree-Fock or Kohn-Sham density functional theory. The canonical density matrix perturbation theory can be used to calculate temperature dependent response properties from the coupled perturbed self-consistent field equations as in density functional perturbation theory. The method is well suited to take advantage of sparse matrix algebra to achieve linear scaling complexity in the computational cost as a function of system size for sufficiently large non-metallic materials and metals at high temperatures.

physics.chem-ph↗

arXiv2014

Generalized Extended Lagrangian Born-Oppenheimer Molecular Dynamics

Extended Lagrangian Born-Oppenheimer molecular dynamics based on Kohn-Sham density functional theory is generalized in the limit of vanishing self-consistent field optimization prior to the force evaluations. The equations of motion are derived directly from the extended Lagrangian under the condition of an adiabatic separation between the nuclear and the electronic degrees of freedom. We show how this separation is automatically fulfilled and system independent. The generalized equations of motion require only one diagonalization per time step and are applicable to a broader range of materials with improved accuracy and stability compared to previous formulations.

physics.chem-ph↗

arXiv2014

First principles molecular dynamics without self-consistent field optimization

We present a first principles molecular dynamics approach that is based on time-reversible ex- tended Lagrangian Born-Oppenheimer molecular dynamics [Phys. Rev. Lett. 100, 123004 (2008)] in the limit of vanishing self-consistent field optimization. The optimization-free dynamics keeps the computational cost to a minimum and typically provides molecular trajectories that closely follow the exact Born-Oppenheimer potential energy surface. Only one single diagonalization and Hamiltonian (or Fockian) costruction are required in each integration time step. The proposed dy- namics is derived for a general free-energy potential surface valid at finite electronic temperatures within hybrid density functional theory. Even in the event of irregular functional behavior that may cause a dynamical instability, the optimization-free limit represents an ideal starting guess for force calculations that may require a more elaborate iterative electronic ground state optimization. Our optimization-free dynamics thus represents a flexible theoretical framework for a broad and general class of ab initio molecular dynamics simulations.

physics.chem-ph↗

arXiv2013

Extended Lagrangian Born-Oppenheimer molecular dynamics in the limit of vanishing self-consistent field optimization

We present an efficient general approach to first principles molecular dynamics simulations based on extended Lagrangian Born-Oppenheimer molecular dynamics in the limit of vanishing self-consistent field optimization. The reduction of the optimization requirement reduces the computational cost to a minimum, but without causing any significant loss of accuracy or longterm energy drift. The optimization-free first principles molecular dynamics requires only one single diagonalization per time step and yields trajectories at the same level of accuracy as "exact", fully converged, Born-Oppenheimer molecular dynamics simulations. The optimization-free limit of extended Lagrangian Born-Oppenheimer molecular dynamics therefore represents an ideal starting point for a robust and efficient formulation of a new generation first principles quantum mechanical molecular dynamics simulation schemes.

physics.chem-ph↗

arXiv2013

Accelerated density matrix expansions for Born-Oppenheimer molecular dynamics

An accelerated polynomial expansion scheme to construct the density matrix in quantum mechanical molecular dynamics simulations is proposed. The scheme is based on recursive density matrix expansions, e.g. [Phys. Rev. B. 66 (2002), p. 155115], which are accelerated by a scale-and-fold technique [J. Chem. Theory Comput. 7 (2011), p. 1233]. The acceleration scheme requires interior eigenvalue estimates, which may be expensive and cumbersome to come by. Here we show how such eigenvalue estimates can be extracted from the recursive expansion by a simple and robust procedure at a negligible computational cost. Our method is illustrated with density functional tight-binding Born-Oppenheimer molecular dynamics simulations, where the computational effort is dominated by the density matrix construction. In our analysis we identify two different phases of the recursive polynomial expansion, the conditioning and purification phases, and we show that the acceleration represents an improvement of the conditioning phase, which typically gives a significant reduction of the computational cost.

physics.comp-ph↗