The Atomic Simulation Environment (ASE) is a software package written in the Python programming language with the aim of setting up, steering, and analyzing atomistic simula- tions. In ASE, tasks are fully scripted in Python. The powerful syntax of Python combined with the NumPy array library make it possible to perform very complex simulation tasks. For example, a sequence of calculations may be performed with the use of a simple "for-loop" construction. Calculations of energy, forces, stresses and other quantities are performed through interfaces to many external electronic structure codes or force fields using a uniform interface. On top of this calculator interface, ASE provides modules for performing many standard simulation tasks such as structure optimization, molecular dynamics, handling of constraints and performing nudged elastic band calculations.

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The Atomic Simulation Environment - A Python library for working with atoms

A methodology for semiempirical density functional optimization, using regularization and cross-validation methods from machine learning, is developed. We demonstrate that such methods enable well-behaved exchange-correlation approximations in very flexible model spaces, thus avoiding the overfitting found when standard least-squares methods are applied to high-order polynomial expansions. A general-purpose density functional for surface science and catalysis studies should accurately describe bond breaking and formation in chemistry, solid state physics, and surface chemistry, and should preferably also include van der Waals dispersion interactions. Such a functional necessarily compromises between describing fundamentally different types of interactions, making transferability of the density functional approximation a key issue. We investigate this trade-off between describing the energetics of intramolecular and intermolecular, bulk solid, and surface chemical bonding, and the developed optimization method explicitly handles making the compromise based on the directions in model space favored by different materials properties. The approach is applied to designing the Bayesian error estimation functional with van der Waals correlation (BEEF--vdW), a semilocal approximation with an additional nonlocal correlation term. Furthermore, an ensemble of functionals around BEEF--vdW comes out naturally, offering an estimate of the computational error. An extensive assessment on a range of data sets validates the applicability of BEEF--vdW to studies in chemistry and condensed matter physics. Applications of the approximation and its Bayesian ensemble error estimate to two intricate surface science problems support this.

Density functionals for surface science: Exchange-correlation model development with Bayesian error estimation

The exchange-correlation functionals of the generalized gradient approximation (GGA) are still the most used for the calculations of the geometry and electronic structure of solids. The PBE functional [J. P. Perdew et al., Phys. Rev. Lett. 77, 3865 (1996)], the most common of them, provides excellent results in many cases. However, very recently other GGA functionals have been proposed and compete in accuracy with the PBE functional, in particular for the structure of solids. We have tested these GGA functionals, as well as the local-density approximation (LDA) and TPSS (meta-GGA approximation) functionals, on a large set of solids using an accurate implementation of the Kohn-Sham equations, namely, the full-potential linearized augmented plane-wave and local orbitals method. Often these recently proposed GGA functionals lead to improvement over LDA and PBE, but unfortunately none of them can be considered as good for all investigated solids.

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Calculation of the lattice constant of solids with semilocal functionals

Predictions of observable properties by density-functional theory calculations (DFT) are used increasingly often by experimental condensed-matter physicists and materials engineers as data. These predictions are used to analyze recent measurements, or to plan future experiments in a rational way. Increasingly more experimental scientists in these fields therefore face the natural question: what is the expected error for such a first-principles prediction? Information and experience about this question is implicitly available in the computational community, scattered over two decades of literature. The present review aims to summarize and quantify this implicit knowledge. This eventually leads to a practical protocol that allows any scientist—experimental or theoretical—to determine justifiable error estimates for many basic property predictions, without having to perform additional DFT calculations. A central role is played by a large and diverse test set of crystalline solids, containing all ground-state...

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Error Estimates for Solid-State Density-Functional Theory Predictions: An Overview by Means of the Ground-State Elemental Crystals

The ambient temperature equations of state (EoS) of iron, cobalt, nickel, zinc, molybdenum, and silver have been measured by x-ray diffraction. These transition metals were compressed using diamond anvil cells with a helium pressure transmitting medium. The maximum pressure reached during these experiments varied between 65 GPa (for cobalt) and 200 GPa (for iron). This work completes previous measurements on six other metals [Phys. Rev. B 70, 094112 (2004)] to quantify the differences between ab initio calculations and experiment on a large experimental set of transition metals. The compression curves of iron, cobalt, nickel, zinc, molybdenum, silver, platinum, and gold are also calculated ab initio within the density-functional theory (DFT) formalism using the projector augmented-wave (PAW) method and different exchange-correlation functionals (LDA, GGA-PBE, GGA-PBEsol). The difference between PAW and available all-electron calculations is found to be negligible up to very high pressures. The success of each exchange-correlation functional is correlated with the atomic number. For all metals, the bulk modulus becomes overestimated at high pressure. In addition, this extended data set of metals' EoS enables to reduce further, but marginally, the systematic uncertainty of the high-pressure metrology based on the ruby standard.

Compression curves of transition metals in the Mbar range: Experiments and projector augmented-wave calculations

Explicit functions are widely used to interpolate, extrapolate, and differentiate theoretical or experimental data on the equation of state (EOS) of a solid. We present two EOS functions which are theoretically motivated. The simplest realistic model for a simple metal, the stabilized jellium (SJ) or structureless pseudopotential model, is the paradigm for our SJEOS. A simple metal with exponentially overlapped ion cores is the paradigm for an augmented version (ASJEOS) of the SJEOS. For the three solids tested (Al, Li, Mo), the ASJEOS matches all-electron calculations better than prior equations of state. Like most of the prior EOS's, the ASJEOS predicts pressure P as a function of compressed volume $v$ from only a few equilibrium inputs: the volume ${v}_{0},$ the bulk modulus ${B}_{0},$ and its pressure derivative ${B}_{1}.$ Under expansion, the cohesive energy serves as another input. A further advantage of the new equation of state is that these equilibrium properties other than ${v}_{0}$ may be found by linear fitting methods. The SJEOS can be used to correct ${B}_{0}$ and the EOS found from an approximate density functional, if the corresponding error in ${v}_{0}$ is known. We also (a) estimate the typically small contribution of phonon zero-point vibration to the EOS, (b) find that the physical hardness $\mathrm{Bv}$ does not maximize at equilibrium, and (c) show that the ``ideal metal'' of Shore and Rose is the zero-valence limit of stabilized jellium.

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Energy and pressure versus volume: Equations of state motivated by the stabilized jellium model

Alchagirov et al. [Phys. Rev. B 63, 224115 (2001)] presented an augmented stabilized jellium equation of state (ASJEOS) related to an energy versus volume relation of an "augmented stabilized jellium" model with a higher-order discontinuity at the zero-pressure volume V 0 . While this ASJEOS has some resemblance at V/V 0 ≤1 to the recently proposed form APL [W. B. Holzapfel, High Press. Res. 16, 81 (1998)], which was not known to Alchagirov et al., the discontinuity and various other features, discussed in detail here, show that the ASJEOS has no advantage in comparison with the much simpler EOS form APL.

Comment on: Energy and pressure versus volume: Equations of state motivated by the stabilized jellium model. Authors' reply

In a recent paper [Phys. Rev. B $63,$ 224115 (2001)], we proposed two equations of state based upon microscopic insight into the cohesion of a solid: the stabilized jellium equation of state (SJEOS) and its augmented version (ASJEOS). In this Reply, we address the issues raised by Holzapfel in his Comment on that paper. We show that, according to the number of independent fitting parameters used, our ASJEOS is comparable with Holzapfel's third-order adapted polynomial (AP3) and that the ASJEOS performs slightly better than AP3 for the materials (Al, Li, and Mo) and pressure range $(P\ensuremath{\lesssim}15\mathrm{Mbar})$ discussed in our paper. We compare the advantages and the disadvantages of ASJEOS and AP3, and discuss the behavior of the equations of state as they approach the strong-compression and strong-expansion limits.

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Alim B. Alchagirov

Papers

Energy and pressure versus volume: Equations of state motivated by the stabilized jellium model

Comment on: Energy and pressure versus volume: Equations of state motivated by the stabilized jellium model. Authors' reply

Reply to ``Comment on `Energy and pressure versus volume: Equations of state motivated by the stabilized jellium model' ''