Generalized Pseudopotential Theory Research Articles

A new form of pseudopotential for the study of lattice mechanical properties of transition metals

ABSTRACT The study of lattice mechanical properties of transition metals using pseudopotential is burdensome as the presence of d-band in these metals complicates their electronic structure. The use of pseudopotential for the study of physical properties can be justified if it accounts for the effect of d-band. The local form of the pseudopotentials is widely used to study the physical properties of materials due to their computational simplicity, physical transparency, and transferability to other environments. It is a well-established fact that non-local pseudopotentials should be preferred to the local ones. But, non-local pseudopotentials are proven to be at a higher computational cost. Inspired by such requirements, we, in the present communication, propose a pseudopotential comprising of one non-local (energy-dependent) and another local term in q-space. The non-local (energy-dependent) term accounts for s-p interaction, while the other term is used to account for the effect of s-d hybridisation. The Generalized Pseudopotential Theory (GPT) has been used to formulate the pseudopotential. Further, the pseudopotential has been used to study static binding energy, lattice dynamics, lattice mechanical properties and second-order elastic constants in the long-wavelength limit (q→0) for transition metals γ-Fe (fcc), Co and α-Fe (bcc). Computed results are found to be in good agreement with experimental and other theoretical results, which proves that the proposed pseudopotential itself accounts for the effect of d-band without using additional short-ranged Born–Mayer type potential and effective valency.

Ising-like models for stacking faults in a free electron metal.

We propose an extension of the axial next nearest neighbour Ising (ANNNI) model to a general number of interactions between spins. We apply this to the calculation of stacking fault energies in magnesium—particularly challenging due to the long-ranged screening of the pseudopotential by the free electron gas. We employ both density functional theory (DFT) using highest possible precision, and generalized pseudopotential theory (GPT) in the form of an analytic, long ranged, oscillating pair potential. At the level of first neighbours, the Ising model is reasonably accurate, but higher order terms are required. In fact, our ‘ ANNNI model’ is slow to converge—an inevitable feature of the free electron-like electronic structure. In consequence, the convergence and internal consistency of the ANNNI model is problematic within the most precise implementation of DFT. The GPT shows the convergence and internal consistency of the DFT bandstructure approach with electron temperature, but does not lead to loss of precision. The GPT is as accurate as a full implementation of DFT but carries the additional benefit that damping of the oscillations in the ANNNI model parameters are achieved without entailing error in stacking fault energies. We trace this to the logarithmic singularity of the Lindhard function.

Local volume effects in the generalized pseudopotential theory

The generalized pseudopotential theory (GPT) is a powerful method for deriving real-space transferable interatomic potentials. Using a coarse-grained electronic structure, one can explicitly calculate the pair ion-ion and multi-ion interactions in simple and transition metals. Whilst successful in determining bulk properties, in central force metals the GPT fails to describe crystal defects for which there is a significant local volume change. A previous paper [PhysRevLett.66.3036 (1991)] found that by allowing the GPT total energy to depend upon some spatially-averaged local electron density, the energetics of vacancies and surfaces could be calculated within experimental ranges. In this paper, we develop the formalism further by explicitly calculating the forces and stress tensor associated with this total energy. We call this scheme the adaptive GPT (aGPT) and it is capable of both molecular dynamics and molecular statics. We apply the aGPT to vacancy formation and divacancy binding in hcp Mg and also calculate the local electron density corrections to the bulk elastic constants and phonon dispersion for which there is refinement over the baseline GPT treatment.

Generalized pseudopotential theory of d-band metals

The generalized pseudopotential theory (GPT) of metals is reviewed with emphasis on recent developments. This theory, which attempts to rigorously extend to d-band metals the spirit of conventional simple-metal pseudopotential perturbation theory, has now been optimized and fully integrated with the Kohn-Sham local-density-functional formalism, allowing for systematic first-principles calculations. Recent work on the problems of cohesion, lattice dynamics, structural phase stability, pressure- and temperature-induced phase transitions, and melting is discussed.

Robust quantum-based interatomic potentials for multiscale modeling in transition metals

First-principles generalized pseudopotential theory (GPT) provides a fundamental basis for transferable multi-ion interatomic potentials in transition metals and alloys within density-functional quantum mechanics. In the central body-centered cubic (bcc) metals, where multi-ion angular forces are important to materials properties, simplified model GPT (MGPT) potentials have been developed based on canonical d bands to allow analytic forms and large-scale atomistic simulations. Robust, advanced-generation MGPT potentials have now been obtained for Ta and Mo and successfully applied to a wide range of structural, thermodynamic, defect, and mechanical properties at both ambient and extreme conditions. Selected applications to multiscale modeling discussed here include dislocation core structure and mobility, atomistically informed dislocation dynamics simulations of plasticity, and thermoelasticity and high-pressure strength modeling. Recent algorithm improvements have provided a more general matrix representation of MGPT beyond canonical bands, allowing improved accuracy and extension to f-electron actinide metals, an order of magnitude increase in computational speed for dynamic simulations, and the development of temperature-dependent potentials.

Transition-metal interactions in aluminum-rich intermetallics

The extension of the first-principles generalized pseudopotential theory (GPT) to transition-metal (TM) aluminides produces pair and many-body interactions that allow efficient calculations of total energies. In aluminum-rich systems treated at the pair-potential level, one practical limitation is a transition-metal overbinding that creates an unrealistic TM-TM attraction at short separations in the absence of balancing many-body contributions. Even with this limitation, the GPT pair potentials have been used effectively in total-energy calculations for Al-TM systems with TM atoms at separations greater than 4 \AA{}. An additional potential term may be added for systems with shorter TM atom separations, formally folding repulsive contributions of the three- and higher-body interactions into the pair potentials, resulting in structure-dependent TM-TM potentials. Towards this end, we have performed numerical ab initio total-energy calculations using the Vienna ab initio simulation package for an Al-Co-Ni compound in a particular quasicrystalline approximant structure. The results allow us to fit a short-ranged, many-body correction of the form ${a(r}_{0}{/r)}^{b}$ to the GPT pair potentials for Co-Co, Co-Ni, and Ni-Ni interactions.

First-principles interatomic potentials for transition-metal aluminides. III. Extension to ternary phase diagrams

Modeling structural and mechanical properties of intermetallic compounds and alloys requires detailed knowledge of their interatomic interactions. The first two papers of this series [Phys. Rev. B $56,$ 7905 (1997); $58,$ 8967 (1998)] derived first-principles interatomic potentials for transition-metal (TM) aluminides using generalized pseudopotential theory (GPT). Those papers focused on binary alloys of aluminum with first-row transition metals and assessed the ability of GPT potentials to reproduce and elucidate the alloy phase diagrams of Al-Co and Al-Ni. This paper addresses the phase diagrams of the binary alloy Al-Cu and the ternary systems Al-Co-Cu and Al-Co-Ni, using GPT pair potentials calculated in the limit of vanishing transition-metal concentration. Despite this highly simplifying approximation, we find rough agreement with the known low-temperature phase diagrams, up to 50% total TM concentration provided the Co fraction is below 25%. Full composition-dependent potentials and many-body interactions would be required to correct deficiencies at higher Co concentration. Outside this troublesome region, the experimentally determined stable and metastable phases all lie on or near the convex hull of a scatter plot of energy versus composition. We verify, qualitatively, reported solubility ranges extending binary alloys into the ternary diagram in both Al-Co-Cu and Al-Co-Ni. Finally, we reproduce previously conjectured transition-metal positions in the decagonal quasicrystal phase.

First-principles interatomic potentials for transition-metal aluminides. II. Application to Al-Co and Al-Ni phase diagrams

Aluminum-rich intermetallic compounds and alloys are important for their technological applications and scientifically interesting for their complex structures such as quasicrystals. Detailed knowledge of interatomic interactions can help explain structural and mechanical properties of these systems. The first paper of this series [Phys. Rev. B 56, 7905 (1997)] derived first-principles interatomic potentials for alloys of aluminum with first row transition metals from generalized pseudopotential theory (GPT). This paper assesses the ability of those potentials to reproduce and elucidate the binary alloy phase diagrams of ${\mathrm{Al}}_{1\ensuremath{-}x}{\mathrm{Co}}_{x}$ and ${\mathrm{Al}}_{1\ensuremath{-}x}{\mathrm{Ni}}_{x}$. When the full theory is taken into account, we successfully reproduce the phase diagrams up to $x=0.3.$ While many of the general features of the phase diagrams can be obtained with $x=0$ GPT pair potentials alone, the volume and composition dependence of the total energy become important by $x=0.25.$ In addition, for certain complex structures, we introduce partial aluminum occupancy and demonstrate its importance. At high transition-metal concentration $xg0.25,$ we must further include three- and four-body transition-metal interactions to account for the stability of ${\mathrm{Al}}_{5}{\mathrm{Co}}_{2}$ in the ${\mathrm{Al}}_{1\ensuremath{-}x}{\mathrm{Co}}_{x}$ phase diagram.

First-principles interatomic potentials for transition-metal aluminides: Theory and trends across the3dseries

In this paper the first-principles generalized pseudopotential theory (GPT) of transition-metal interatomic potentials [J. A. Moriarty, Phys. Rev. B 38, 3199 (1988)] is extended to $\mathrm{AB}$ binary compounds and alloys. For general transition-metal (TM) systems, the GPT total-energy functional involves a volume term, central-force pair potentials, and angular-force multi-ion potentials, which are both volume $(\ensuremath{\Omega})$ and concentration $(x)$ dependent and include all $sp,$ $sp\ensuremath{-}d,$ and $d\ensuremath{-}d$ interactions within local density-functional quantum mechanics. The formalism is developed here in detail for intermetallic systems where $A$ is a simple metal and $B$ is a transition metal and applied to the prominent special case of the transition-metal aluminides ${\mathrm{TM}}_{x}{\mathrm{Al}}_{1\ensuremath{-}x},$ where $sp\ensuremath{-}d$ hybridization is especially important. Emphasis is given to the aluminum-rich $3d$ binary systems for $x<0.30,$ which appear to be well described at the pair-potential level without angular forces and for which the present GPT potentials can be used directly in atomistic simulations. Volume terms and pair potentials for all of the $3d$ aluminides have been calculated and their behavior with atomic number, $\ensuremath{\Omega},$ and $x$ is elaborated through illustrative applications to the cohesive and structural trends across the $3d$ series. More extensive applications to the Co-Al and Ni-Al phase diagrams will be given elsewhere.

Angular forces and melting in bcc transition metals: A case study of molybdenum.

Using multi-ion interatomic potentials derived from first-principles generalized pseudopotential theory (GPT) together with molecular-dynamics (MD) simulation, a detailed study of melting and related high-temperature solid and liquid properties in molybdenum has been performed. The energetics in such bcc transition metals are dominated by d-state interactions that give rise to both many-body angular forces and enhanced electron-thermal contributions. The angular forces are accounted for in the GPT through explicit three- and four-ion potentials, ${\mathit{v}}_{3}$ and ${\mathit{v}}_{4}$, which in the present work are applied in analytic model-GPT (MGPT) form. With the MGPT potentials, ion-thermal melting in Mo has been investigated both mechanically, by cycling up and down through the observed MD melting point at constant volume, and thermodynamically, by calculating solid and liquid free energies. In the former approach, parallel MD simulations have also been done with a corresponding effective-pair potential ${\mathit{v}}_{2}^{\mathrm{eff}}$ in which the angular dependence of ${\mathit{v}}_{3}$ and ${\mathit{v}}_{4}$ has been averaged. The multi-ion angular forces, which are essential to an accurate description of the bcc solid, are found to lower the dynamically observed melting point by about 1000 K. Above the melting transition, however, ${\mathit{v}}_{2}^{\mathrm{eff}}$ gives a reasonbly good account of the structure and thermal energy of the liquid and the accuracy of this description improves with increasing temperature.Both the multi-ion and effective pair potentials also permit a large amount of supercooling of the liquid before the onset of freezing. With ${\mathit{v}}_{2}^{\mathrm{eff}}$ a bcc structure is nucleated at freezing, while with the multi-ion potentials an amorphous glasslike structure is obtained, which appears to be related to the energetically competitive A15 structure. In our second approach to melting, the multi-ion potentials have been used to obtain accurate solid and liquid free energies from quasiharmonic lattice dynamics and MD calculations of thermal energies and pressures. The resulting ion-thermal melting curve exactly overlaps the dynamically observed melting point, indicating that no superheating of the solid occurred in our MD simulations. To obtain a full melting curve, electron-thermal contributions to the solid and liquid free energies are added in terms of the density of electronic states at the Fermi level, \ensuremath{\rho}(${\mathit{E}}_{\mathit{F}}$). Here the density of states for the solid has been calculated with the linear-muffin-tin-orbital method, while for the liquid tight-binding calculations have been used to justify a simple model. In the liquid \ensuremath{\rho}(${\mathit{E}}_{\mathit{F}}$) is increased dramatically over the bcc solid, and the net effect of the electron-thermal contributions is to lower the calculated melting temperatures by about a factor of 2. A full melting curve to 2 Mbar has thereby been obtained and the calculated melting properties near zero pressure are in generally good agreement with experiment.

Analytic representation of multi-ion interatomic potentials in transition metals.

The first-principles, density-functional version of the generalized pseudopotential theory (GPT), previously developed for empty- and filled-d-band metals, recently has been extended to pure transition metals with partially filled d bands [Phys. Rev. B 38, 3199 (1988)]. Within this formalism, a rigorous real-space expansion of the bulk total energy has been obtained in terms of widely transferable, structure-independent interatomic potentials, including both central-force pair interactions and angular-force triplet and quadruplet interactions. In the central transition metals, the three- and four-ion potentials, ${\mathit{v}}_{3}$ and ${\mathit{v}}_{4}$, are essential to a proper description of materials properties, but are necessarily multidimensional functions which cannot be easily tabulated for application purposes.We develop here a simplified version of the theory, the model GPT, in which these potentials can be expressed analytically while retaining the most important physics of the full first-principles treatment. The analytic treatment of ${\mathit{v}}_{3}$ and ${\mathit{v}}_{4}$ is made possible because of three simplifying features in the central transition metals. First, due to the nonspherical nature of the Fermi surface in such metals, the long-range sp-d hybridization tails of the first-principles potentials destructively interfere in total-energy calculations and thus can be dropped at the outset without major consequences. Second, the direct d-d contributions to the potentials are short ranged and need only be retained to fourth order in interatomic d-state matrix elements to obtain a good representation of the d-band-structure energy. Third, the d bands are canonical in nature, with the interatomic matrix elements well approximated by simple forms, so that all remaining low-order d-state contributions can be evaluated analytically.This leads to a description of ${\mathit{v}}_{3}$ and ${\mathit{v}}_{4}$ in terms of universal short-range radial and angular functions. The model GPT is made quantitatively accurate for real materials by allowing the coefficient of each d-state contribution to be adjusted to match first-principles calculations and/or experimental data. In this manner, one can achieve a set of potentials which simultaneously yield a good description of cohesion, vacancy formation, structural phase stability, elastic constants, and phonons, as is demonstrated for the representative case of molybdenum. More generally, the analytic potentials are suitable for widescale applications and permit for the first time the use of the transition-metal GPT in molecular-dynamics and Monte Carlo simulations.

First-Principles Interatomic Potentials for Transition Metals and Their Surfaces

ABSTRACTFor bulk transition metals, a first-principles generalized pseudopotential theory (GPT) of interatomic potentials has been developed in which the cohesive-energy unctional takes the form of a volume term plus sums over widely transferable two-, three-, and four-ion potentials. The GPT has been further extended to surfaces by making an internal transformation of this functional to an embedded-atom-like format in which the embedding function is identified as the bulk volume term and the atomic volume is replaced by an average electron density. Applications of the bulk and surface GPT to the calculation of structural, vacancy-formation, and surface energies in Cu and Mo, and to the investigation of surface relaxation and reconstruction in Mo are discussed.

Density-functional formulation of the generalized pseudopotential theory. III. Transition-metal interatomic potentials.

The first-principles, density-functional version of the generalized pseudopotential theory (GPT) developed in papers I and II of this series [Phys. Rev. B 16, 2537 (1977); 26, 1754 (1982)] for empty- and filled-d-band metals is here extended to pure transition metals with partially filled d bands. The present focus is on a rigorous, real-space expansion of the bulk total energy in terms of widely transferable, structure-independent interatomic potentials, including both central-force pair interactions and angular-force triplet and quadruplet interactions. To accomplish this expansion, a specialized set of starting equations is derived from the basic local-density formalism for a pure metal, including refined expansions for the exchange-correlation terms and a simplified yet accurate representation of the cohesive energy. The parent pseudo-Green's-function formalism of the GPT is then used to develop these equations in a plane-wave, localized-d-state basis. In this basis, the cohesive energy divides quite naturally into a large volume component and a smaller structural component.The volume component,which includes all one-ion intra-atomic energy contributions, already gives a good description of the cohesion in lowest order. The structural component is expanded in terms of weak interatomic matrix elements and gives rise to a multi-ion series which establishes the interatomic potentials. Special attention is focused on the dominant d-electron contributions to this series and complete formal results for the two-ion, three-ion, and four-ion d-state potentials (${v}_{2}^{d}$, ${v}_{3}^{d}$, and ${v}_{4}^{d}$) are derived. In addition, a simplified model is used to demonstrate that while ${v}_{3}^{d}$ can be of comparable importance to ${v}_{2}^{d}$, ${v}_{4}^{d}$ is inherently small and the series is rapidly convergent beyond three-ion interactions. Analytic model forms are also derived for ${v}_{2}^{d}$ and ${v}_{3}^{d}$ in the case of canonical d bands. In this limit, ${v}_{2}^{d}$ is purely attractive and varies with interatomic distance as ${r}^{\mathrm{\ensuremath{-}}10}$, while ${v}_{3}^{d}$ is weak and attractive for almost empty or filled d bands and maximum in strength and repulsive for half-filled d bands. Full first-principles expressions are then developed for the total two-ion and three-ion potentials and implemented for all 20 3d and 4d transition metals. The first-principles potentials qualitatively display all of the trends predicted by the model results, but they also reflect additional effects, including long-range hybridization tails which must be suitably screened in real-space calculations. Finally, illustrative application of the first-principles potentials is made to the calculation of the [100] phonon spectrum for V and Cr, where the importance of three-ion angular forces is explicitly demonstrated.

Parametrized version of the generalized pseudopotential theory for noble metals: Application to copper.

A parametrized version of the generalized pseudopotential theory (GPT) of Moriarty is proposed for the special case of the noble metals. Copper is taken as an example. To define the parametrization, we consider separately the contributions which in the GPT are defined as the simple-metal limit (SML), the s-d coupling, and the overlap. The perturbation expansion is done from the basis of the atomic d states, and the d bands of the noble metal are considered to be completely filled. The SML is represented as an empty-core potential of radius ${r}_{c}$, and the s-d--coupling calculation is done exactly, but the mean energy ${E}_{d}$ of the d band relative to the conduction band is considered to be adjustable. The two parameters ${r}_{c}$ and ${E}_{d}$ are determined from particular points of the band structure. The results are found to be consistent with the large-L band-gap value and the width of the resonance of copper. The SML of the form factor, and the total form factor obtained, are in good agreement with those of the ab initio GPT. By means of the optimized random-phase approximation, we analyze the influence of the overlap on the liquid structure. We find that the expression for the overlap pair potential of Moriarty gives a good estimation of the liquid structure factor. Conversely, this quantity is used to adjust the overlap pair potential. Further concluding tests of the parametrization are done by calculating the entropy ${C}_{V}$, the compressibility, and the resistivity of the liquid. It is found that the parameters ${r}_{c}$, ${E}_{d}$, and those of the overlap are almost insensitive to volume changes with temperature in the solid and liquid under normal pressure. We conclude that the parametrized approach gives a good picture of the electron-ion and interionic interactions without losing the essence of the GPT. The method cannot replace the ab initio calculation, but could be useful in the investigation of noble-metal properties in systems which are not easily tractable in a detailed theory.

Structural phase stability in third-period simple metals

Extensive phase-stability calculations for the fcc, hcp, and bcc structures of Na, Mg, Al, and Si are reported both near 1 atm and under high pressure, where Si becomes a natural member of this set of third-period simple metals Calculations for each element have been carried out by two entirely different first-principles techniques: the generalized pseudopotential theory (GPT) and the linear-muffin-tin orbitals (LMTO) method. The two techniques give results in good qualitative agreement for Mg, Al, and Si, and predict sequences of high-pressure structural phase transitions for these elements which arise from the lowering and partial filling of the initially empty $3d$ band under compression. Detailed analysis shows that major trends in the phase stability of the third-period metals are correlated with specific features both in the LMTO one-electron densities of states and in the GPT interatomic pair potentials. Quantitative comparisons between the two theoretical techniques, as well as with available experimental data and other recent calculations, are further used to assess the accuracy of the approximations employed by these methods in calculating structural energy differences.

Density-functional formulation of the generalized pseudopotential theory. II

The density-functional formulation of the generalized pseudopotential theory (GPT) set forth in paper I of this set is recast in an optimum representation and more widely applied to empty- and filled-$d$-band metals. Optimization is achieved by making the most advantageous separation possible of (i) the hybridization potential into volume-dependent (${\ensuremath{\Delta}}_{\mathrm{vol}}$) and structure-dependent (${\ensuremath{\Delta}}_{\mathrm{struc}}$) parts, and (ii) the nonuniform component of the valence-electron density into screening and orthogonalization-hole contributions. The resulting new definitions of these quantities permit the entire contribution of ${\ensuremath{\Delta}}_{\mathrm{struc}}$ to the total energy to be folded into the original framework of the theory, where ${\ensuremath{\Delta}}_{\mathrm{struc}}$ was n\'eglected. The energy-wave-number characteristic $F(q)$ and overlap potential ${v}_{\mathrm{o}1}(r)$ then assume simpler and more computationally efficient forms, in which certain large numerical cancellations otherwise inherent in the calculation of physical properties are eliminated. The new representation also makes clearer the ranges of applicability of the empty-and filled-$d$-band limits of the theory. The optimized GPT is shown to provide an excellent description of the group-$\mathrm{II}A$ metals Ca and Sr in the empty-$d$-band limit and of the group-$\mathrm{II}B$ metals Zn and Cd in the filled-$d$-band limit. Somewhat surprisingly, however, a filled-$d$-band treatment is found not to be adequate in the noble metals because the number of electrons effectively emptied out of the $d$ states through hybridization (\ensuremath{\sim} \textonehalf{} electron/atom) is not small relative to the nominal valence (1 electron/atom). It is further shown that a much more accurate description of the noble metals can be expected by allowing the $d$ states to unfill and a self-consistent valence to be achieved in zero order, and the first steps towards implementing the partially-filled-$d$-band limit of the GPT are considered here. Finally, extensive applications of the optimized GPT that we have made on the band structure, cohesion, liquid-metal transport, lattice dynamics, and structural phase stability in 22 simple and -$d$-band metals are summarized and compared with both experiment and the density-functional calculations of Moruzzi, Janak, and Williams.