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We extend our linear-scaling approach for the calculation of Hartree–Fock exchange energy using localized in situ optimized orbitals [Dziedzic et al., J. Chem. Phys. 139, 214103 (2013)] to leverage massive parallelism. Our approach has been implemented in the onetep (Order-N Electronic Total Energy Package) density functional theory framework, which employs a basis of non-orthogonal generalized Wannier functions (NGWFs) to achieve linear scaling with system size while retaining controllable near-complete-basis-set accuracy. For the calculation of Hartree–Fock exchange, we use a resolution-of-identity approach, where an auxiliary basis set of truncated spherical waves is used to fit products of NGWFs. The fact that the electrostatic potential of spherical waves (SWs) is known analytically, combined with the use of a distance-based cutoff for exchange interactions, leads to a calculation cost that scales linearly with the system size. Our new implementation, which we describe in detail, combines distributed memory parallelism (using the message passing interface) with shared memory parallelism (OpenMP threads) to efficiently utilize numbers of central processing unit cores comparable to, or exceeding, the number of atoms in the system. We show how the use of multiple time-memory trade-offs substantially increases performance, enabling our approach to achieve superlinear strong parallel scaling in many cases and excellent, although sublinear, parallel scaling otherwise. We demonstrate that in scenarios with low available memory, which preclude or limit the use of time-memory trade-offs, the performance degradation of our algorithm is graceful. We show that, crucially, linear scaling with system size is maintained in all cases. We demonstrate the practicability of our approach by performing a set of fully converged production calculations with a hybrid functional on large imogolite nanotubes up to over 1400 atoms. We finish with a brief study of how the employed approximations (exchange cutoff and the quality of the SW basis) affect the calculation walltime and the accuracy of the obtained results. [ABSTRACT FROM AUTHOR]

Progress in electrochemical technologies, such as automotive batteries, supercapacitors, and fuel cells, depends greatly on developing improved charged interfaces between electrodes and electrolytes. The rational development of such interfaces can benefit from the atomistic understanding of the materials involved by first-principles quantum mechanical simulations with Density Functional Theory (DFT). However, such simulations are typically performed on the electrode surface in the absence of its electrolyte environment and at constant charge. We have developed a new hybrid computational method combining DFT and the Poisson–Boltzmann equation (P–BE) capable of simulating experimental electrochemistry under potential control in the presence of a solvent and an electrolyte. The charged electrode is represented quantum-mechanically via linear-scaling DFT, which can model nanoscale systems with thousands of atoms and is neutralized by a counter electrolyte charge via the solution of a modified P–BE. Our approach works with the total free energy of the combined multiscale system in a grand canonical ensemble of electrons subject to a constant electrochemical potential. It is calibrated with respect to the reduction potential of common reference electrodes, such as the standard hydrogen electrode and the Li metal electrode, which is used as a reference electrode in Li-ion batteries. Our new method can be used to predict electrochemical properties under constant potential, and we demonstrate this in exemplar simulations of the differential capacitance of few-layer graphene electrodes and the charging of a graphene electrode coupled to a Li metal electrode at different voltages. [ABSTRACT FROM AUTHOR]

Density functional theory (DFT) is often used for simulating extended materials such as infinite crystals or surfaces, under periodic boundary conditions (PBCs). In such calculations, when the simulation cell has non-zero charge, electrical neutrality has to be imposed, and this is often done via a uniform background charge of opposite sign ("jellium"). This artificial neutralization does not occur in reality, where a different mechanism is followed as in the example of a charged electrode in electrolyte solution, where the surrounding electrolyte screens the local charge at the interface. The neutralizing effect of the surrounding electrolyte can be incorporated within a hybrid quantum–continuum model based on a modified Poisson–Boltzmann equation, where the concentrations of electrolyte ions are modified to achieve electroneutrality. Among the infinite possible ways of modifying the electrolyte charge, we propose here a physically optimal solution, which minimizes the deviation of concentrations of electrolyte ions from those in open boundary conditions (OBCs). This principle of correspondence of PBCs with OBCs leads to the correct concentration profiles of electrolyte ions, and electroneutrality within the simulation cell and in the bulk electrolyte is maintained simultaneously, as observed in experiments. This approach, which we call the Neutralization by Electrolyte Concentration Shift (NECS), is implemented in our electrolyte model in the Order-N Electronic Total Energy Package (ONETEP) linear-scaling DFT code, which makes use of a bespoke highly parallel Poisson–Boltzmann solver, DL_MG. We further propose another neutralization scheme ("accessible jellium"), which is a simplification of NECS. We demonstrate and compare the different neutralization schemes on several examples. [ABSTRACT FROM AUTHOR]

Metal oxide supports often play an active part in heterogeneous catalysis by moderating both the structure and the electronic properties of the metallic catalyst particle. In order to provide some fundamental understanding on these effects, we present here a density functional theory (DFT) investigation of the binding of O and CO on Pt nanoparticles supported on titania (anatase) surfaces. These systems are complex, and in order to develop realistic models, here, we needed to perform DFT calculations with up to ∼1000 atoms. By performing full geometry relaxations at each stage, we avoid any effects of "frozen geometry" approximations. In terms of the interaction of the Pt nanoparticles with the support, we find that the surface deformation of the anatase support contributes greatly to the adsorption of each nanoparticle, especially for the anatase (001) facet. We attempt to separate geometric and electronic effects and find a larger contribution to ligand binding energy arising from the former. Overall, we show an average weakening (compared to the isolated nanoparticle) of ∼0.1 eV across atop, bridge and hollow binding sites on supported Pt55 for O and CO, and a preservation of site preference. Stronger effects are seen for O on Pt13, which is heavily deformed by anatase supports. In order to rationalize our results and examine methods for faster characterization of metal catalysts, we make use of electronic descriptors, including the d-band center and an electronic density based descriptor. We expect that the approach followed in this study could be applied to study other supported metal catalysts. [ABSTRACT FROM AUTHOR]

We extend our recently developed quantum-mechanical/molecular mechanics (QM/MM) approach [Dziedzic et al., J. Chem. Phys. 145, 124106 (2016)] to enable in situ optimization of the localized orbitals. The quantum subsystem is described with onetep linear-scaling density functional theory and the classical subsystem – with the AMOEBA polarizable force field. The two subsystems interact via multipolar electrostatics and are fully mutually polarizable. A total energy minimization scheme is employed for the Hamiltonian of the coupled QM/MM system. We demonstrate that, compared to simpler models using fixed basis sets, the additional flexibility offered by in situ optimized basis functions improves the accuracy of the QM/MM interface, but also poses new challenges, making the QM subsystem more prone to overpolarization and unphysical charge transfer due to increased charge penetration. We show how these issues can be efficiently solved by replacing the classical repulsive van der Waals term for QM/MM interactions with an interaction of the electronic density with a fixed, repulsive MM potential that mimics Pauli repulsion, together with a modest increase in the damping of QM/MM polarization. We validate our method, with particular attention paid to the hydrogen bond, in tests on water-ion pairs, the water dimer, first solvation shells of neutral and charged species, and solute-solvent interaction energies. As a proof of principle, we determine suitable repulsive potential parameters for water, K+, and Cl−. The mechanisms we employed to counteract the unphysical overpolarization of the QM subsystem are demonstrated to be adequate, and our approach is robust. We find that the inclusion of explicit polarization in the MM part of QM/MM improves agreement with fully QM calculations. Our model permits the use of minimal size QM regions and, remarkably, yields good energetics across the well-balanced QM/MM interface. [ABSTRACT FROM AUTHOR]

The intrinsic sampling method (ISM) is a powerful tool that allows the exploration of interfacial properties from molecular simulations by fitting a function that represents the local boundary between two phases. However, owing to the non-physical nature of an "intrinsic" surface, there remains an ambiguity surrounding the comparison of theoretical properties with the physical world. It is therefore important that the ISM remains internally consistent when reproducing simulated properties which match experiments, such as the surface tension or interfacial density distribution. We show that the current ISM procedure causes an over-fitting of the surface to molecules in the interface region, leading to a biased distribution of curvature at these molecular coordinates. We assert that this biased distribution is a cause of the disparity between predicted interfacial densities upon convolution to a laboratory frame, an artefact which has been known to exist since the development of the ISM. We present an improvement to the fitting procedure of the ISM in an attempt to alleviate the ambiguity surrounding the true nature of an intrinsic surface. Our "surface reconstruction" method is able to amend the shape of the interface so as to reproduce the global curvature distribution at all sampled molecular coordinates. We present the effects that this method has on the ISM predicted structure of a simulated Lennard-Jones fluid air-liquid interface. Additionally, we report an unexpected relationship between surface thermodynamic predictions of our reconstructed ISM surfaces and those of extended capillary wave theory, which is of current interest. [ABSTRACT FROM AUTHOR]

We present an unexpected finite size effect affecting interfacial molecular simulations that is proportional to the width-to-surface-area ratio of the bulk phase Ll/A. This finite size effect has a significant impact on the variance of surface tension values calculated using the virial summation method. A theoretical derivation of the origin of the effect is proposed, giving a new insight into the importance of optimising system dimensions in interfacial simulations. We demonstrate the consequences of this finite size effect via a new way to estimate the surface energetic and entropic properties of simulated air-liquid interfaces. Our method is based on macroscopic thermodynamic theory and involves comparing the internal energies of systems with varying dimensions. We present the testing of these methods using simulations of the TIP4P/2005 water forcefield and a Lennard-Jones fluid model of argon. Finally, we provide suggestions of additional situations, in which this finite size effect is expected to be significant, as well as possible ways to avoid its impact. [ABSTRACT FROM AUTHOR]

Accurate and computationally efficient exchange-correlation functionals are critical to the successful application of linear-scaling density functional theory (DFT). Local and semi-local functionals of the density are naturally compatible with linear-scaling approaches, having a general form which assumes the locality of electronic interactions and which can be efficiently evaluated by numerical quadrature. Presently, the most sophisticated and flexible semi-local functionals are members of the meta-generalized-gradient approximation (meta-GGA) family, and depend upon the kinetic energy density, τ, in addition to the charge density and its gradient. In order to extend the theoretical and computational advantages of τ-dependent meta-GGA functionals to large-scale DFT calculations on thousands of atoms, we have implemented support for τ-dependent meta-GGA functionals in the ONETEP program. In this paper we lay out the theoretical innovations necessary to implement τ-dependent meta-GGA functionals within ONETEP's linear-scaling formalism. We present expressions for the gradient of the τ-dependent exchange-correlation energy, necessary for direct energy minimization. We also derive the forms of the τ-dependent exchange-correlation potential and kinetic energy density in terms of the strictly localized, self-consistently optimized orbitals used by ONETEP. To validate the numerical accuracy of our self-consistent meta-GGA implementation, we performed calculations using the B97M-V and PKZB meta-GGAs on a variety of small molecules. Using only a minimal basis set of self-consistently optimized local orbitals, we obtain energies in excellent agreement with large basis set calculations performed using other codes. Finally, to establish the linear-scaling computational cost and applicability of our approach to large-scale calculations, we present the outcome of self-consistent meta-GGA calculations on amyloid fibrils of increasing size, up to tens of thousands of atoms. [ABSTRACT FROM AUTHOR]

We present a novel quantum mechanical/molecular mechanics (QM/MM) approach in which a quantum subsystem is coupled to a classical subsystem described by the AMOEBA polarizable force field. Our approach permits mutual polarization between the QM and MM subsystems, effected through multipolar electrostatics. Self-consistency is achieved for both the QM and MM subsystems through a total energy minimization scheme. We provide an expression for the Hamiltonian of the coupled QM/MM system, which we minimize using gradient methods. The QM subsystem is described by the ONETEP linear-scaling DFT approach, which makes use of strictly localized orbitals expressed in a set of periodic sinc basis functions equivalent to plane waves. The MM subsystem is described by the multipolar, polarizable force field AMOEBA, as implemented in TINKER. Distributed multipole analysis is used to obtain, on the fly, a classical representation of theQMsubsystem in terms of atom-centered multipoles. This auxiliary representation is used for all polarization interactions between QM and MM, allowing us to treat them on the same footing as in AMOEBA.We validate our method in tests of solute-solvent interaction energies, for neutral and charged molecules, demonstrating the simultaneous optimization of the quantum and classical degrees of freedom. Encouragingly, we find that the inclusion of explicit polarization in the MM part of QM/MM improves the agreement with fully QM calculations. [ABSTRACT FROM AUTHOR]

Recently developed density functionals have good accuracy for both thermochemistry (TC) and non-covalent interactions (NC) if very large atomic orbital basis sets are used. To approach the basis set limit with potentially lower computational cost, a new self-consistent field (SCF) scheme is presented that employs minimal adaptive basis (MAB) functions. The MAB functions are optimized on each atomic site by minimizing a surrogate function. High accuracy is obtained by applying a perturbative correction (PC) to the MAB calculation, similar to dual basis approaches. Compared to exact SCF results, using this MAB-SCF (PC) approach with the same large target basis set produces <0.15 kcal/mol root-mean-square deviations for most of the tested TC datasets, and <0.1 kcal/mol for most of the NC datasets. The performance of density functionals near the basis set limit can be even better reproduced. With further improvement to its implementation, MAB-SCF (PC) is a promising lower-cost substitute for conventional large-basis calculations as a method to approach the basis set limit of modern density functionals. [ABSTRACT FROM AUTHOR]

A new method for finite-temperature density functional theory calculations which significantly increases the number of atoms that can be simulated in metallic systems is presented. A self-consistent, direct minimization technique is used to obtain the Helmholtz free energy of the electronic system, described in terms of a set of non-orthogonal, localized functions which are optimized in situ using a periodic-sinc basis set, equivalent to plane waves. Most parts of the calculation, including the demanding operation of building the Hamiltonian matrix, have a computational cost that scales linearly with the number of atoms in the system. Also, this approach ensures that the Hamiltonian matrix has a minimal size, which reduces the computational overhead due to diagonalization, a cubic-scaling operation that is still required. Large basis set accuracy is retained via the optimization of the localized functions. This method allows accurate simulations of entire metallic nanostructures, demonstrated with calculations on a supercell of bulk copper with 500 atoms and on gold nanoparticles with up to 2057 atoms. [ABSTRACT FROM AUTHOR]

In situ optimization of a set of localized orbitals with respect to a systematically improvable basis set independent of the position of the atoms, such as psinc functions, would theoretically eliminate the correction due to Pulay forces from the total ionic forces. We demonstrate that for strict localization constraints, especially with small localization regions, there can be non-negligible Pulay forces that must be calculated as a correction to the Hellmann-Feynman forces in the ground state. Geometry optimization calculations, which rely heavily upon accurate evaluation of the total ionic forces, show much better convergence when Pulay forces are included. The more conventional case, where the local orbitals remain fixed to pseudo-atomic orbital multiple-ζ basis sets, also benefits from this implementation. We have validated the method on several test cases, including a DNA fragment with 1045 atoms. [ABSTRACT FROM AUTHOR]

Biomolecular simulations with atomistic detail are often required to describe interactions with chemical accuracy for applications such as the calculation of free energies of binding or chemical reactions in enzymes. Force fields are typically used for this task but these rely on extensive parameterisation which in cases can lead to limited accuracy and transferability, for example for ligands with unusual functional groups. These limitations can be overcome with first principles calculations with methods such as density functional theory (DFT) but at a much higher computational cost. The use of electrostatic embedding can significantly reduce this cost by representing a portion of the simulated system in terms of highly localised charge distributions. These classical charge distributions are electrostatically coupled with the quantum system and represent the effect of the environment in which the quantum system is embedded. In this paper we describe and evaluate such an embedding scheme in which the polarisation of the electronic density by the embedding charges occurs self-consistently during the calculation of the density. We have implemented this scheme in a linear-scaling DFT program as our aim is to treat with DFT entire biomolecules (such as proteins) and large portions of the solvent. We test this approach in the calculation of interaction energies of ligands with biomolecules and solvent and investigate under what conditions these can be obtained with the same level of accuracy as when the entire system is described by DFT, for a variety of neutral and charged species. [ABSTRACT FROM AUTHOR]

We present a comparison of methods for treating the electrostatic interactions of finite, isolated systems within periodic boundary conditions (PBCs), within density functional theory (DFT), with particular emphasis on linear-scaling (LS) DFT. Often, PBCs are not physically realistic but are an unavoidable consequence of the choice of basis set and the efficacy of using Fourier transforms to compute the Hartree potential. In such cases the effects of PBCs on the calculations need to be avoided, so that the results obtained represent the open rather than the periodic boundary. The very large systems encountered in LS-DFT make the demands of the supercell approximation for isolated systems more difficult to manage, and we show cases where the open boundary (infinite cell) result cannot be obtained from extrapolation of calculations from periodic cells of increasing size. We discuss, implement, and test three very different approaches for overcoming or circumventing the effects of PBCs: truncation of the Coulomb interaction combined with padding of the simulation cell, approaches based on the minimum image convention, and the explicit use of open boundary conditions (OBCs). We have implemented these approaches in the ONETEP LS-DFT program and applied them to a range of systems, including a polar nanorod and a protein. We compare their accuracy, complexity, and rate of convergence with simulation cell size. We demonstrate that corrective approaches within PBCs can achieve the OBC result more efficiently and accurately than pure OBC approaches. [ABSTRACT FROM AUTHOR]

Linear-scaling methods for density functional theory promise to revolutionize the scope and scale of first-principles quantum mechanical calculations. Crystalline silicon has been the system of choice for exploratory tests of such methods in the literature, yet attempts at quantitative comparisons under linear-scaling conditions with traditional methods or experimental results have not been forthcoming. A detailed study using the ONETEP code is reported here, demonstrating for the first time that plane wave accuracy can be achieved in linear-scaling calculations on periodic systems. [ABSTRACT FROM AUTHOR]

We present ONETEP (order-N electronic total energy package), a density functional program for parallel computers whose computational cost scales linearly with the number of atoms and the number of processors. ONETEP is based on our reformulation of the plane wave pseudopotential method which exploits the electronic localization that is inherent in systems with a nonvanishing band gap. We summarize the theoretical developments that enable the direct optimization of strictly localized quantities expressed in terms of a delocalized plane wave basis. These same localized quantities lead us to a physical way of dividing the computational effort among many processors to allow calculations to be performed efficiently on parallel supercomputers. We show with examples that ONETEP achieves excellent speedups with increasing numbers of processors and confirm that the time taken by ONETEP as a function of increasing number of atoms for a given number of processors is indeed linear. What distinguishes our approach is that the localization is achieved in a controlled and mathematically consistent manner so that ONETEP obtains the same accuracy as conventional cubic-scaling plane wave approaches and offers fast and stable convergence. We expect that calculations with ONETEP have the potential to provide quantitative theoretical predictions for problems involving thousands of atoms such as those often encountered in nanoscience and biophysics.© 2005 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Linear-scaling electronic structure methods are essential for calculations on large systems. Some of these approaches use a systematic basis set, the completeness of which may be tuned with an adjustable parameter similar to the energy cut-off of plane-wave techniques. The search for the electronic ground state in such methods suffers from an ill-conditioning which is related to the kinetic contribution to the total energy and which results in unacceptably slow convergence. We present a general preconditioning scheme to overcome this ill-conditioning and implement it within our own first-principles linear-scaling density functional theory method. The scheme may be applied in either real space or reciprocal space with equal success. The rate of convergence is improved by an order of magnitude and is found to be almost independent of the size of the basis.© 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Density Functional Theory (DFT) calculations with computational effort which increases linearly with the number of atoms (linear-scaling DFT) have been successfully developed for insulators, taking advantage of the exponential decay of the one-particle density matrix. For metallic systems, the density matrix is also expected to decay exponentially at finite electronic temperature and linear-scaling DFT methods should be possible by taking advantage of this decay. Here we present a method for DFT calculations at finite electronic temperature for metallic systems which is effectively linear-scaling (O(N)). Our method generates the elements of the one-particle density matrix and also finds the required chemical potential and electronic entropy using polynomial expansions. A fixed expansion length is always employed to generate the density matrix, without any loss in accuracy by the application of a high electronic temperature followed by successive steps of temperature reduction until the desired (low) temperature density matrix is obtained. We have implemented this method in the ONETEP linear-scaling (for insulators) DFT code which employs local orbitals that are optimised in situ. By making use of the sparse matrix machinery of ONETEP, our method exploits the sparsity of Hamiltonian and density matrices to perform calculations on metallic systems with computational cost that increases asymptotically linearly with the number of atoms. We demonstrate the linear-scaling computational cost of our method with calculation times on palladium nanoparticles with up to ∼13 000 atoms. [ABSTRACT FROM AUTHOR]

A new method for the constrained optimization of non-orthogonal, spatially localized orbitals using direct energy minimization techniques, in the context of electronic structure calculations, is presented. The variationally localized search direction (VLSD) method, as it was named, ensures that strict localization constraints are imposed upon the search direction vectors exactly, analytically and in a fully variational fashion. In contrast, the truncated search direction (TSD) method, of standard use in many electronic structure approaches with localization constraints, relies on the approximation that the truncated search direction vectors of the unconstrained problem resemble the exact search direction vectors of the constrained problem. With the TSD method, in order to maintain the localization constraints, a part of the pre-calculated information that is stored in the search direction vectors has to be deleted via an ad hoc, non-variational truncation step. The results on an extensive set of test molecules show that, in general, calculations with the VLSD method require less iterations to converge than with the TSD method for any size of the localization region. It was found that in calculations on certain systems where the TSD method is forced to delete a very large amount of information, the VLSD method is capable of achieving convergence in up to three times less iterations. Validation tests show that structural and electronic properties calculated with either method are accurate and in agreement with other electronic structure approaches. [ABSTRACT FROM AUTHOR]

We present an overview of the onetep program for linear-scaling density functional theory (DFT) calculations with large basis set (plane-wave) accuracy on parallel computers. The DFT energy is computed from the density matrix, which is constructed from spatially localized orbitals we call Non-orthogonal Generalized Wannier Functions (NGWFs), expressed in terms of periodic sinc (psinc) functions. During the calculation, both the density matrix and the NGWFs are optimized with localization constraints. By taking advantage of localization, onetep is able to perform calculations including thousands of atoms with computational effort, which scales linearly with the number or atoms. The code has a large and diverse range of capabilities, explored in this paper, including different boundary conditions, various exchange-correlation functionals (with and without exact exchange), finite electronic temperature methods for metallic systems, methods for strongly correlated systems, molecular dynamics, vibrational calculations, time-dependent DFT, electronic transport, core loss spectroscopy, implicit solvation, quantum mechanical (QM)/molecular mechanical and QM-in-QM embedding, density of states calculations, distributed multipole analysis, and methods for partitioning charges and interactions between fragments. Calculations with onetep provide unique insights into large and complex systems that require an accurate atomic-level description, ranging from biomolecular to chemical, to materials, and to physical problems, as we show with a small selection of illustrative examples. onetep has always aimed to be at the cutting edge of method and software developments, and it serves as a platform for developing new methods of electronic structure simulation. We therefore conclude by describing some of the challenges and directions for its future developments and applications.

Iterative energy minimization with the aim of achieving self-consistency is a common feature of Born-Oppenheimer molecular dynamics (BOMD) and classical molecular dynamics with polarizable force fields. In the former, the electronic degrees of freedom are optimized, while the latter often involves an iterative determination of induced point dipoles. The computational effort of the self-consistency procedure can be reduced by re-using converged solutions from previous time steps. However, this must be done carefully, as not to break time-reversal symmetry, which negatively impacts energy conservation. Self-consistent schemes based on the extended Lagrangian formalism, where the initial guesses for the optimized quantities are treated as auxiliary degrees of freedom, constitute one elegant solution. We report on the performance of two integration schemes with the same underlying extended Lagrangian structure, which we both employ in two radically distinct regimes-in classical molecular dynamics simulations with the AMOEBA polarizable force field and in BOMD simulations with the Onetep linear-scaling density functional theory (LS-DFT) approach. Both integration schemes are found to offer significant improvements over the standard (unpropagated) molecular dynamics formulation in both the classical and LS-DFT regimes.

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.

Recently developed density functionals have good accuracy for both thermochemistry (TC) and non-covalent interactions (NC) if very large atomic orbital basis sets are used. To approach the basis set limit with potentially lower computational cost, a new self-consistent field (SCF) scheme is presented that employs minimal adaptive basis (MAB) functions. The MAB functions are optimized on each atomic site by minimizing a surrogate function. High accuracy is obtained by applying a perturbative correction (PC) to the MAB calculation, similar to dual basis approaches. Compared to exact SCF results, using this MAB-SCF (PC) approach with the same large target basis set produces <0.15 kcal/mol root-mean-square deviations for most of the tested TC datasets, and <0.1 kcal/mol for most of the NC datasets. The performance of density functionals near the basis set limit can be even better reproduced. With further improvement to its implementation, MAB-SCF (PC) is a promising lower-cost substitute for conventional large-basis calculations as a method to approach the basis set limit of modern density functionals.