The tight-binding bond model

Efficient Parameterization of Density Functional Tight-Binding for 5f-Elements: A Th-O Case Study

Density functional tight binding (DFTB) models for f-element species are challenging to parametrize owing to the large number of adjustable parameters. The explicit optimization of the terms entering the semiempirical DFTB Hamiltonian related to f orbitals is crucial to generating a reliable parametrization for f-block elements, because they play import roles in bonding interactions. However, since the number of parameters grows quadratically with the number of orbitals, the computational cost for parameter optimization is much more expensive for the f-elements than for the main group elements. In this work we present a set of efficient approaches for mitigating the hurdle imposed by the large size of the parameter space. A novel group-by-orbital correction functions for two-center bond integrals was developed. With this approach the number of parameters is reduced, and it grows linearly with the number of elements, maintaining the accuracy and the number of parameters, in the case of f elements, by more than 40%. The parameter optimization step was accelerated by means of the mini-batch BFGS method. This method allows parameter optimizations with much larger training sets than other single batch methods. A stochastic optimizer was employed that helped overcome shallow local minima in the objective function. The proposed algorithm was used to parametrize the DFTB Hamiltonian for the Th-O system, which was subsequently applied to the study of ThO2 nanoparticles. The training set consisted of 6322 unique structures, which is barely feasible with conventional optimization methods. The optimized parameter set, LANL-ThO, displays good agreement with DFT-calculated properties such as energies, forces, and structures for both clusters and bulk ThO2. Benefiting from the fewer number of parameters and lower computational costs for objective function evaluations, this new approach shows its potential applications in DFTB parametrization for elements with high angular momentum, which present a challenge to conventional methods.

Understanding metal bonding

This theoretical discussion covers several effects of metallic bonding based on a simple formula. It comes from the first steps in the Moment Method for calculating the local electronic structure of a solid (such as at a surface or in a random alloy), and depends on the square root of the total coordination numberCof near neighbours. Each atom is covalently bonded to its cluster of near neighboursas a whole. The properties of metals touched on include malleability, crystal structure and phase transitions, vacancy formation energy, surface catalysis, surface reconstruction, graphite stability, and some aspects of the benzene molecule seen as an atomic metal ring. In most of these, the 'saturation' type of curvature of the square root function plays a crucial role. A short historical survey indicates the development of the ideas from Bloch (1929Z. Phys.52555) to recent times.

Analytical and Parameter-Free Hückel Theory Made Possible for Symmetric Hx Clusters

It is widely accepted that energetics of chemical bond breaking and formation can be described with simple mathematical forms only at the expense of extensive parameterization. In this work, the discovery of a simple tight-binding-type mathematical framework that can accurately predict the relative energetics of regular Hx polygons (2 ≤ x ≤ 15) in the ground states with their respective spin multiplicities using no parameters has been reported. The framework recasts Hückel theory in a density functional theory form by making use of Anderson and Adams-Gilbert theories of localized orbitals. For the systems examined, the method exhibits mean absolute errors of ∼0.02 Å (edge lengths) and ∼0.15 eV/atom (energy minima) relative to correlated-electron quantum chemistry calculations. Its accuracy is found to be comparable to the generalized gradient approximation and superior to standard parameterized tight binding and reactive potentials applied to Hx structures. Generalization of the theoretical framework to systems of many-electron atoms is presented, along with the comparison of the method to existing semiempirical tight binding and bond order potential approaches.

Carbon Nanodots from an In Silico Perspective

Carbon nanodots (CNDs) are the latest and most shining rising stars among photoluminescent (PL) nanomaterials. These carbon-based surface-passivated nanostructures compete with other related PL materials, including traditional semiconductor quantum dots and organic dyes, with a long list of benefits and emerging applications. Advantages of CNDs include tunable inherent optical properties and high photostability, rich possibilities for surface functionalization and doping, dispersibility, low toxicity, and viable synthesis (top-down and bottom-up) from organic materials. CNDs can be applied to biomedicine including imaging and sensing, drug-delivery, photodynamic therapy, photocatalysis but also to energy harvesting in solar cells and as LEDs. More applications are reported continuously, making this already a research field of its own. Understanding of the properties of CNDs requires one to go to the levels of electrons, atoms, molecules, and nanostructures at different scales using modern molecular modeling and to correlate it tightly with experiments. This review highlights different in silico techniques and studies, from quantum chemistry to the mesoscale, with particular reference to carbon nanodots, carbonaceous nanoparticles whose structural and photophysical properties are not fully elucidated. The role of experimental investigation is also presented. Hereby, we hope to encourage the reader to investigate CNDs and to apply virtual chemistry to obtain further insights needed to customize these amazing systems for novel prospective applications.

Equivariant representations for molecular Hamiltonians and N-center atomic-scale properties

Symmetry considerations are at the core of the major frameworks used to provide an effective mathematical representation of atomic configurations that is then used in machine-learning models to predict the properties associated with each structure. In most cases, the models rely on a description of atom-centered environments and are suitable to learn atomic properties or global observables that can be decomposed into atomic contributions. Many quantities that are relevant for quantum mechanical calculations, however-most notably the single-particle Hamiltonian matrix when written in an atomic orbital basis-are not associated with a single center, but with two (or more) atoms in the structure. We discuss a family of structural descriptors that generalize the very successful atom-centered density correlation features to the N-center case and show, in particular, how this construction can be applied to efficiently learn the matrix elements of the (effective) single-particle Hamiltonian written in an atom-centered orbital basis. These N-center features are fully equivariant-not only in terms of translations and rotations but also in terms of permutations of the indices associated with the atoms-and are suitable to construct symmetry-adapted machine-learning models of new classes of properties of molecules and materials.

Coarse-grained tight-binding models

Calculating the electronic structure of systems involving very different length scales presents a challenge. Empirical atomistic descriptions such as pseudopotentials or tight-binding models allow one to calculate the effects of atomic placements, but the computational burden increases rapidly with the size of the system, limiting the ability to treat weakly bound extended electronic states. Here we propose a new method to connect atomistic and quasi-continuous models, thus speeding up tight-binding calculations for large systems. We divide a structure into blocks consisting of several unit cells which we diagonalize individually. We then construct a tight-binding Hamiltonian for the full structure using a truncated basis for the blocks, ignoring states having large energy eigenvalues and retaining states with energies close to the band edge energies. A numerical test using a GaAs/AlAs quantum well shows the computation time can be decreased to less than 5% of the full calculation with errors of less than 1%. We give data for the trade-offs between computing time and loss of accuracy. We also tested calculations of the density of states for a GaAs/AlAs quantum well and find a ten times speedup without much loss in accuracy.

The partition principles for atomic-scale structures and their physical properties: application to the nonlinear optical crystal material KBe2BO3F2

In implementing the materials genome approach to search for new materials with interesting properties or functions, it is necessary to find the correct functional motif. To this end, it is common to partition an extended structure into various building units and then partition its properties to find the appropriate functional motif. We have developed the general principles for partitioning a structure and its properties in terms of a set of atoms and bonds by analyzing the differential cross-sections of neutron and X-ray scattering phenomena and proposed the procedures with which to partition an extended structure and its properties. We demonstrate how these procedures work by analyzing the nonlinear optical crystal KBe2BO3F2. Our partitioning analysis of KBe2BO3F2 leads to the conclusion that the second harmonic generation response of KBe2BO3F2 is dominated by the ionically bonded metal-centered groups.

Current-constrained one-electron reduced density-matrix theory for non-equilibrium steady-state molecular conductivity

In the effort to create ever smaller electronic devices, the idea of single molecule circuit elements has sparked the imagination of scientists for nearly fifty years.

A new embedded-atom method approach based on the pth moment approximation

Large scale atomistic simulations with suitable interatomic potentials are widely employed by scientists or engineers of different areas. The quick generation of high-quality interatomic potentials is urgently needed. This largely relies on the developments of potential construction methods and algorithms in this area. Quantities of interatomic potential models have been proposed and parameterized with various methods, such as the analytic method, the force-matching approach and multi-object optimization method, in order to make the potentials more transferable. Without apparently lowering the precision for describing the target system, potentials of fewer fitting parameters (FPs) are somewhat more physically reasonable. Thus, studying methods to reduce the FP number is helpful in understanding the underlying physics of simulated systems and improving the precision of potential models. In this work, we propose an embedded-atom method (EAM) potential model consisting of a new manybody term based on the pth moment approximation to the tight binding theory and the general transformation invariance of EAM potentials, and an energy modification term represented by pairwise interactions. The pairwise interactions are evaluated by an analytic-numerical scheme without the need to know their functional forms a priori. By constructing three potentials of aluminum and comparing them with a commonly used EAM potential model, several wonderful results are obtained. First, without losing the precision of potentials, our potential of aluminum has fewer potential parameters and a smaller cutoff distance when compared with some constantly-used potentials of aluminum. This is because several physical quantities, usually serving as target quantities to match in other potentials, seem to be uniquely dependent on quantities contained in our basic reference database within the new potential model. Second, a key empirical parameter in the embedding term of the commonly used EAM model is found to be related to the effective order of moments of local density of states. This may provide a way to improve the precision of EAM potentials further through more precise approximations to tight binding theory. In addition, some critical details about construction procedures are discussed.

Incorporating non-adiabatic effects in embedded atom potentials for radiation damage cascade simulations

In radiation damage cascade displacement spikes ions and electrons can reach very high temperatures and be out of thermal equilibrium. Correct modelling of cascades with molecular dynamics should allow for the non-adiabatic exchange of energy between ions and electrons using a consistent model for the electronic stopping, electronic temperature rise, and thermal conduction by the electrons. We present a scheme for correcting embedded atom potentials for these non-adiabatic properties at the level of the second-moment approximation, and parameterize for the bcc transition metals above the Debye temperature. We use here the Finnis-Sinclair and Derlet-Nguyen-Manh-Dudarev potentials as models for the bonding, but the corrections derived from them can be applied to any suitable empirical potential. We show with two-temperature MD simulations that computing the electronic thermal conductivity during the cascade evolution has a significant impact on the heat exchange between ions and electrons.

Non-collinear magnetism with analytic Bond-Order Potentials

The theory of analytic Bond-Order Potentials as applied to non-collinear magnetic structures of transition metals is extended to take into account explicit rotations of Hamiltonian and local moment matrix elements between locally and globally defined spin-coordinate systems. Expressions for the gradients of the energy with respect to the Hamiltonian matrix elements, the interatomic forces and the magnetic torques are derived. The method is applied to simulations of the rotation of magnetic moments in α iron, as well as α and β manganese, based on d-valent orthogonal tight-binding parametrizations of the electronic structure. A new weighted-average terminator is introduced to improve the convergence of the Bond-Order Potential energies and torques with respect to tight-binding reference values, although the general behavior is qualitatively correct for low-moment expansions.

Analytic bond-order potentials for the bcc refractory metals Nb, Ta, Mo and W

Bond-order potentials (BOPs) are based on the tight-binding approximation for determining the energy of a system of interacting atoms. The bond energy and forces are computed analytically within the formalism of the analytic BOPs. Here we present parametrizations of the analytic BOPs for the bcc refractory metals Nb, Ta, Mo and W. The parametrizations are optimized for the equilibrium bcc structure and tested for atomic environments far from equilibrium that had not been included in the fitting procedure. These tests include structural energy differences for competing crystal structures; tetragonal, trigonal, hexagonal and orthorhombic deformation paths; formation energies of point defects as well as phonon dispersion relations. Our tests show good agreement with available experimental and theoretical data. In practice, we obtain the energetic ordering of vacancy, [1 1 1], [1 1 0], and [1 0 0] self-interstitial atom in agreement with density functional theory calculations.

Representing potential energy surfaces by high-dimensional neural network potentials

The development of interatomic potentials employing artificial neural networks has seen tremendous progress in recent years. While until recently the applicability of neural network potentials (NNPs) has been restricted to low-dimensional systems, this limitation has now been overcome and high-dimensional NNPs can be used in large-scale molecular dynamics simulations of thousands of atoms. NNPs are constructed by adjusting a set of parameters using data from electronic structure calculations, and in many cases energies and forces can be obtained with very high accuracy. Therefore, NNP-based simulation results are often very close to those gained by a direct application of first-principles methods. In this review, the basic methodology of high-dimensional NNPs will be presented with a special focus on the scope and the remaining limitations of this approach. The development of NNPs requires substantial computational effort as typically thousands of reference calculations are required. Still, if the problem to be studied involves very large systems or long simulation times this overhead is regained quickly. Further, the method is still limited to systems containing about three or four chemical elements due to the rapidly increasing complexity of the configuration space, although many atoms of each species can be present. Due to the ability of NNPs to describe even extremely complex atomic configurations with excellent accuracy irrespective of the nature of the atomic interactions, they represent a general and therefore widely applicable technique, e.g. for addressing problems in materials science, for investigating properties of interfaces, and for studying solvation processes.

Comparison of analytic and numerical bond-order potentials for W and Mo

Bond-order potentials (BOPs) are derived from the tight-binding approximation and provide a linearly-scaling computation of the energy and forces for a system of interacting atoms. While the numerical BOPs involve the numerical integration of the response (Green's) function, the expressions for the energy and interatomic forces are analytical within the formalism of the analytic BOPs. In this paper we present a detailed comparison of numerical and analytic BOPs. We use established parametrizations for the bcc refractory metals W and Mo and test structural energy differences; tetragonal, trigonal, hexagonal and orthorhombic deformation paths; formation energies of point defects as well as phonon dispersion relations. We find that the numerical and analytic BOPs generally are in very good agreement for the calculation of energies. Different from the numerical BOPs, the forces in the analytic BOPs correspond exactly to the negative gradients of the energy. This makes it possible to use the analytic BOPs in dynamical simulations and leads to improved predictions of defect energies and phonons as compared to the numerical BOPs.

Quantum-classical simulations of the electronic stopping force and charge on slow heavy channelling ions in metals

By simulating the passage of heavy ions along open channels in a model crystalline metal using semi-classical Ehrenfest dynamics we directly investigate the nature of non-adiabatic electronic effects. Our time-dependent tight-binding approach incorporates both an explicit quantum mechanical electronic system and an explicit representation of a set of classical ions. The coupled evolution of the ions and electrons allows us to explore phenomena that lie beyond the approximations made in classical molecular dynamics simulations and in theories of electronic stopping. We report a velocity-dependent charge-localization phenomenon not predicted by previous theoretical treatments of channelling. This charge localization can be attributed to the excitation of electrons into defect states highly localized on the channelling ion. These modes of excitation only become active when the frequency at which the channelling ion moves from interstitial point to equivalent interstitial point matches the frequency corresponding to excitations from the Fermi level into the localized states. Examining the stopping force exerted on the channelling ion by the electronic system, we find broad agreement with theories of slow ion stopping (a stopping force proportional to velocity) for a low velocity channelling ion (up to about 0.5 nm fs(-1) from our calculations), and a reduction in stopping power attributable to the charge localization effect at higher velocities. By exploiting the simplicity of our electronic structure model we are able to illuminate the physics behind the excitation processes that we observe and present an intuitive picture of electronic stopping from a real-space, chemical perspective.

Environmental tight-binding modeling of nickel and cobalt clusters

Tight-binding models derived from density functional theory potentially provide a systematic approach to the development of accurate and transferable models of multicomponent systems. We introduce a systematic methodology for environmental tight binding in which both the overlap and environmental contributions to the electronic structure are included. The parameters of the model are determined directly from ab initio considerations, thus providing a formal conceptual link to density functional approaches. In order to test the validity of the approach, the model is applied to small clusters of Ni and Co, whose electronic structure is largely determined by the interplay of tightly bound d-valent states and the disperse s-states. We numerically illustrate that it is essential to include environmental contributions in the tight-binding approach in order to reliably reproduce the electronic structure of such clusters.

Tight-binding simulation of transition-metal alloys

In order to perform atomistic simulations of steel, it is necessary to have a detailed understanding of the complex interatomic interactions in transition metals and their alloys. The tight-binding approximation provides a computationally efficient, yet accurate, method to investigate such interactions. In the present work, an orthogonal tight-binding model for Fe, Mn and Cr, with the explicit inclusion of magnetism, has been parameterized from ab initio density-functional calculations.

Negative Cauchy Pressure within the Tight-Binding Approximation

It is well-known that the Embedded Atom Method (EAM) predicts positive Cauchy pressures for cubic metals if physically-motivated embedding functions are used. Supris-ingly, even if the angular character of the covalent bonding is included within an orthorgonal or non-orthorgonal Tight-Binding (TB) description, the Cauchy pressure for most elemental and binary metallic systems remains positive. We describe the results of a detailed breakdown of the different contributions to the Cauchy pressure within the Harris-Foulkes approximation (HFA) to density functional theory. We show that negative values of the Cauchy pressure for both elemental transition metals such as Ir and binary intermetallics such as Ti3Al, TiAl and TiAl3 are well reproduced by the HFA. We argue that the negative Cauchy pressure (NCP) arises namely from the environment dependence of the local TB orbitals which leads to both environment-dependent bonding integrals and overlap repulsion. We discuss a particular functional form for overlap repulsion which leads to NCP and compare it with different fitting schemes proposed recently in TB theory.

A Selfconsistent-Charge Density-Functional Tight-Binding Scheme

We present an extension to the tight-binding (TB) approach to improve total energies, forces and transferability in the presence of considerable long-range Coulomb interactions. We derive an approximate energy expression in terms of charge density fluctuations δn at a reference (input) density n0, which is a second order approximation to the total energy expression in density functional theory (DFT). With the choice of n0 as a superposition of densities of neutral atomic fragments, we can define a repulsive potential as in standard TB theory, which is pairwise, short ranged and transferable. The zero order terms in the total energy expression are recoverd as the standard terms of our density-functional based tight-binding (DF-TB). For the second order terms, the charge density fluctuations δn are approximated by the total charge fluctuation Δqα at atom α, which is qualitatively estimated by employing the Mullikan charge analysis. Within this approximations the total energy expression contains new parameters, which are related to ab-intio DFT calculations. Finally, by introducing localized basis functions and applying the variational principle we arrive at the Hamilton matrix elements, wich themselves depend on the charge fluctuations and, therefore, the general eigenvalue problem has to be solved self-consistently. To obtain forces for efficient geometry relaxation and molecular-dyamics, we calculated analytical derivatives of the total energy with respect to the atomic sites. In order to demonstrate the strenghts of our self-consistent-charge tight-binding (SCC-TB), we calculated reaction energies, geometries and vibrational frequencies for a large set of molecules and compare the results to semi-empirical methods, density-functional calculations and experiment.

Interface structure and mechanics between graphene and metal substrates: a first-principles study

Graphene is a fascinating material not only for technological applications, but also as a test bed for fundamental insights into condensed matter physics due to its unique two-dimensional structure. One of the most intriguing issues is the understanding of the properties of graphene and various substrate materials. In particular, the interfaces between graphene and metal substrates are of critical importance in applications of graphene in integrated electronics, as thermal materials, and in electromechanical devices. Here we investigate the structure and mechanical interactions at a graphene-metal interface through density functional theory (DFT)-based calculations. We focus on copper (111) and nickel (111) surfaces adhered to a monolayer of graphene, and find that their cohesive energy, strength and electronic structure correlate directly with their atomic geometry. Due to the strong coupling between open d-orbitals, the nickel-graphene interface has a much stronger cohesive energy with graphene than copper. We also find that the interface cohesive energy profile features a well-and-shoulder shape that cannot be captured by simple pair-wise models such as the Lennard-Jones potential. Our results provide a detailed understanding of the interfacial properties of graphene-metal systems, and help to predict the performance of graphene-based nanoelectronics and nanocomposites. The availability of structural and energetic data of graphene-metal interfaces could also be useful for the development of empirical force fields for molecular dynamics simulations.

Self-consistent field tight-binding model for neutral and (multi-) charged carbon clusters

A semiempirical model for carbon clusters modeling is presented, along with structural and dynamical applications. The model is a tight-binding scheme with additional one- and two-center distance-dependent electrostatic interactions treated self-consistently. This approach, which explicitly accounts for charge relaxation, allows us to treat neutral and (multi-) charged clusters not only at equilibrium but also in dissociative regions. The equilibrium properties, geometries, harmonic spectra, and relative stabilities of the stable isomers of neutral and singly charged clusters in the range n=1-14, for C(20) and C(60), are found to reproduce the results of ab initio calculations. The model is also shown to be successful in describing the stability and fragmentation energies of dictations in the range n=2-10 and allows the determination of their Coulomb barriers, as examplified for the smallest sizes (C(2) (2+),C(3) (2+),C(4) (2+)). We also present time-dependent mean-field and linear response optical spectra for the C(8) and C(60) clusters and discuss their relevance with respect to existing calculations.

Surface passivation for tight-binding calculations of covalent solids

Simulation of a cluster representing a finite portion of a larger covalently bonded system requires the passivation of the cluster surface. We compute the effects of an explicit hybrid orbital passivation (EHOP) on the atomic structure in a model bulk, three-dimensional, narrow gap semiconductor, which is very different from the wide gap, quasi-one-dimensional organic molecules where most passivation schemes have been studied in detail. The EHOP approach is directly applicable to minimal atomic orbital basis methods such as tight-binding. Each broken bond is passivated by a hybrid created from an explicitly expressed linear combination of basis orbitals, chosen to represent the contribution of the missing neighbour, e.g. a sp(3) hybrid for a single bond. The method is tested by computing the forces on atoms near a point defect as a function of cluster geometry. We show that, compared to alternatives such as pseudo-hydrogen passivation, the force on an atom converges to the correct bulk limit more quickly as a function of cluster radius, and that the force is more stable with respect to perturbations in the position of the cluster centre. The EHOP method also obviates the need for parameterizing the interactions between the system atoms and the passivating atoms. The method is useful for cluster calculations of non-periodic defects in large systems and for hybrid schemes that simulate large systems by treating finite regions with a quantum-mechanical model, coupled to an interatomic potential description of the rest of the system.

Topological frustrations in Mn films on Fe(001)

The high lateral resolution of spin-polarized scanning tunneling microscopy allows new insights into the spin structure of antiferromagnets on the nanometer range. We demonstrate the capability to image a well-defined in-plane component of the sample spin polarization and discuss the spin structure of antiferromagnetic bct Mn in contact with the ferromagnetic Fe(001) substrate. Mn atoms couple ferromagnetically within a Mn atomic plane, while normal to the surface a layer-wise antiferromagnetic order was found. Magnetic frustrations arise in this system at Fe substrate steps at the interface, where topologically induced 180 degrees domain walls are created in the Mn film. A clear widening of the enforced domain walls with increasing Mn thickness was found. The measured widths could be fitted with a linear function and are explained on the basis of a Heisenberg model.

Atomic force algorithms in density functional theory electronic-structure techniques based on local orbitals

Electronic structure methods based on density-functional theory, pseudopotentials, and local-orbital basis sets offer a hierarchy of techniques for modeling complex condensed-matter systems with a wide range of precisions and computational speeds. We analyze the relationships between the algorithms for atomic forces in this hierarchy of techniques, going from empirical tight-binding through ab initio tight-binding to full ab initio. The analysis gives a unified overview of the force algorithms as applied within techniques based either on diagonalization or on linear-scaling approaches. The use of these force algorithms is illustrated by practical calculations with the CONQUEST code, in which different techniques in the hierarchy are applied in a concerted manner.