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Methods for Large-Scale Density Functional Calculations on Metallic Systems Jolyon Aarons, Misbah Sarwar, David Thompsett, and Chris-Kriton Skylaris

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Methods for Large-Scale Density Functional Calculations on Metallic Systems Jolyon Aarons, Misbah Sarwar, David Thompsett, and Chris-Kriton Skylaris Perspective: Methods for large-scale density functional calculations on metallic systems Jolyon Aarons, Misbah Sarwar, David Thompsett, and Chris-Kriton Skylaris Citation: J. Chem. Phys. 145, 220901 (2016); doi: 10.1063/1.4972007 View online: http://dx.doi.org/10.1063/1.4972007 View Table of Contents: http://aip.scitation.org/toc/jcp/145/22 Published by the American Institute of Physics THE JOURNAL OF CHEMICAL PHYSICS 145, 220901 (2016) Perspective: Methods for large-scale density functional calculations on metallic systems Jolyon Aarons,1 Misbah Sarwar,2 David Thompsett,2 and Chris-Kriton Skylaris1,a) 1School of Chemistry, University of Southampton, Southampton SO17 1BJ, United Kingdom 2Johnson Matthey Technology Centre, Sonning Common, Reading, United Kingdom (Received 14 July 2016; accepted 28 November 2016; published online 15 December 2016) Current research challenges in areas such as energy and bioscience have created a strong need for Density Functional Theory (DFT) calculations on metallic nanostructures of hundreds to thousands of atoms to provide understanding at the atomic level in technologically important processes such as catalysis and magnetic materials. Linear-scaling DFT methods for calculations with thousands of atoms on insulators are now reaching a level of maturity. However such methods are not applicable to metals, where the continuum of states through the chemical potential and their partial occupancies provide significant hurdles which have yet to be fully overcome. Within this perspective we outline the theory of DFT calculations on metallic systems with a focus on methods for large-scale calculations, as required for the study of metallic nanoparticles. We present early approaches for electronic energy minimization in metallic systems as well as approaches which can impose partial state occupancies from a thermal distribution without access to the electronic Hamiltonian eigenvalues, such as the classes of Fermi operator expansions and integral expansions. We then focus on the significant progress which has been made in the last decade with developments which promise to better tackle the length- scale problem in metals. We discuss the challenges presented by each method, the likely future directions that could be followed and whether an accurate linear-scaling DFT method for metals is in sight. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4972007] I. INTRODUCTION control the emissions released, and the production of certain foodstuffs also relies on catalytic processes, all of which can Electronic structure theory calculations using the Density use metallic nanoparticles. Catalysis also has a significant role Functional Theory (DFT) approach are widely used to com- to play in Proton Exchange Membrane (PEM) fuel cells, for pute and understand the chemical and physical properties of example, offering a promising source of clean energy, produc- molecules and materials. The study of metallic systems, in ing electricity by the electrochemical conversion of hydrogen particular, is an important area for the employment of DFT and oxygen to water. Either monometallic or alloyed, metal- simulations as there is a broad range of practical applica- lic nanoparticles are used as catalysts in these processes and tions. These applications range from the study of bulk metals anchored to a support such as oxide or carbon. The size, shape, and surfaces to the study of metallic nanoparticles, which is and composition (e.g., core-shell, bulk alloy, and segregated a rapidly growing area of research due to its technological structure)7 of these nanoparticles influence their chemical relevance.1 properties, and catalytically important sizes of nanoparticles For example, metallic nanoparticles have optical proper- (diameters of 2-10 nm) can consist of hundreds to thousands ties that are tunable and entirely different from those of the of atoms. bulk material as their interaction with light is determined by Even though extended infinite surface slabs of one type of their quantization of energy levels and surface plasmons. Due crystal plane are often used as models,8 which correspond to to their tunable optical properties, metal nanoparticles have the limit of very large nanoparticles (&5 nm), we can see from found numerous applications in biodiagnostics2,3 as sensitive Figure1 that this limit is not reached before nanoparticles with markers, such as for the detection of DNA by Au nanoparticle thousands of atoms are considered. The slab model has been markers. Another very promising area of metallic nanoparti- applied successfully in screening metal and metal alloys for cle usage concerns their magnetic properties which intricately a variety of reactions, providing guidance on how to improve depend not only on their size but also on their geometry,4 and current catalysts or identifying novel materials or composi- can exhibit effects such as giant magnetoresistance.5,6 How- tions.9 However, these types of models, while providing useful ever, by far the domain in which metallic nanoparticles have insight, do not capture the complexity of the nanoparticle,10 found most application so far is the area of heterogeneous for example, the effect the particle size has on properties or catalysis. The catalytic cracking of hydrocarbons produces edge effects between different crystal planes. The influence of the fuels we use, the vehicles we drive contain catalysts to the support can modify the electronic structure and geometry of the nanoparticle as well as cause “spillover effects” which may, in turn, affect catalytic activity.11 Metallic nanoparticles a)Electronic address: [email protected] are also dynamic, and interaction with adsorbates can induce 0021-9606/2016/145(22)/220901/14/$30.00 145, 220901-1 Published by AIP Publishing. 220901-2 Aarons et al. J. Chem. Phys. 145, 220901 (2016) usually accurate for chemical properties. However, this is an area of ongoing developments18 and there have been appli- cations in specific cases such as the physical properties of liquid Li.19 The majority of DFT calculations are performed with the Kohn-Sham approach which describes the energy of the sys- tem as a functional of the electronic density n(r) and molecular orbitals f ig, X E[n] = f h j Tˆ j i + υ (r)n(r)dr i i i ext i + EH [n] + Exc[n], (1) where the terms on the right are the kinetic energy, the exter- nal potential energy, the Hartree energy, and the exchange- correlation energy of the electrons, respectively. The molecular FIG. 1. The first 6 “magic numbers” of platinum cuboctahedral nanoparticles orbitals f ig are the solutions of a Schrodinger¨ equation for a showing how the number of atoms scales cubically with the diameter; 5 nm fictitious system of non-interacting particles, is not reached until the nanoparticle has 2869 atoms. 1 2 − r + υˆKS i(r) = i i(r), (2) a change in their shape and composition, which can modify " 2 # their behaviour. A fundamental understanding of how catalytic where i are the energy levels and where the effective potential reactions occur on the surfaces of these catalysts is crucial for υˆKS has been constructed in such a way that the electronic improving their performance, and DFT simulations play a key density of the fictitious non-interacting system role.12–16 X n(r) = f j (r)j2 (3) The need to understand and control the rich and unique i i physical and chemical properties of metallic nanoparticles pro- i vides the motivation for the development of suitable DFT is the same as the electronic density of the system of interest methods for their study. Conventional DFT approaches used of interacting particles. to model metallic slabs are unsuited to modelling metallic We need to point out that the solution of the Kohn-Sham nanoparticles larger than ∼100 atoms as they are computation- equations (2) is not a trivial process as the Kohn-Sham poten- ally very costly with an increasing number of atoms. Therefore, tialυ ˆKS[n] is a functional of the density n which in turn depends the development of DFT methods for metals with reduced (ide- on the occupancies fi and the one-particle wavefunctions i via ally linear) scaling of computational effort with the number Equation (3). Thus the Kohn-Sham equations are non-linear of atoms is essential to modelling nanoparticles of appropri- and in practice they need to be solved iteratively until the wave- ate sizes for the applications mentioned above. Such methods functions, occupancies, and the density no longer change with also allow for the introduction of increased complexity into respect to each other, which is what is termed a self-consistent the models, such as the effect of the support or the influence solution to these equations. Typically this solution is obtained of the environment such as the solvent. by a Self-Consistent-Field (SCF) process whereυ ˆKS[n] is built In this perspective, we present an introduction to meth- from the current approximation to the density; then the Kohn- ods for large-scale DFT simulations of metallic systems with Sham equations are solved to obtain new wavefunctions and metallic nanoparticle applications in mind. We provide the occupancies to build a new density; from that new density a key ideas between the various classes of such methods and newυ ˆKS[n] is constructed and these iterations continue until discuss their computational demands. We conclude with some convergence. thoughts about likely future developments in this area. For a non-spin-polarized system, the occupancies fi are either 2 or 0, depending on whether the orbitals are occupied or not. The extension of the equations to spin polarization, II. DENSITY FUNCTIONAL THEORY FOR METALLIC with occupancies 1 or 0, is trivial. This formulation of DFT SYSTEMS is suitable for calculations on materials with a bandgap (or Density functional theory has become the method of HOMO-LUMO gap in molecules), and a wide range of algo- choice for electronic structure calculations of materials and rithms have been developed for the efficient numerical solution molecules due to its ability to provide sufficient accuracy for of these equations.
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