Jonathan R. Yates

Reproducibility in density functional theory calculations of solids

A comparison of DFT methods Density functional theory (DFT) is now routinely used for simulating material properties. Many software packages are available, which makes it challenging to know which are the best to use for a specific calculation. Lejaeghere et al. compared the calculated values for the equation of states for 71 elemental crystals from 15 different widely used DFT codes employing 40 different potentials (see the Perspective by Skylaris). Although there were variations in the calculated values, most recent codes and methods converged toward a single value, with errors comparable to those of experiment. Science, this issue p. 10.1126/science.aad3000; see also p. 1394 A survey of recent density functional theory methods shows a convergence to more accurate property calculations. [Also see Perspective by Skylaris] INTRODUCTION The reproducibility of results is one of the underlying principles of science. An observation can only be accepted by the scientific community when it can be confirmed by independent studies. However, reproducibility does not come easily. Recent works have painfully exposed cases where previous conclusions were not upheld. The scrutiny of the scientific community has also turned to research involving computer programs, finding that reproducibility depends more strongly on implementation than commonly thought. These problems are especially relevant for property predictions of crystals and molecules, which hinge on precise computer implementations of the governing equation of quantum physics. RATIONALE This work focuses on density functional theory (DFT), a particularly popular quantum method for both academic and industrial applications. More than 15,000 DFT papers are published each year, and DFT is now increasingly used in an automated fashion to build large databases or apply multiscale techniques with limited human supervision. Therefore, the reproducibility of DFT results underlies the scientific credibility of a substantial fraction of current work in the natural and engineering sciences. A plethora of DFT computer codes are available, many of them differing considerably in their details of implementation, and each yielding a certain “precision” relative to other codes. How is one to decide for more than a few simple cases which code predicts the correct result, and which does not? We devised a procedure to assess the precision of DFT methods and used this to demonstrate reproducibility among many of the most widely used DFT codes. The essential part of this assessment is a pairwise comparison of a wide range of methods with respect to their predictions of the equations of state of the elemental crystals. This effort required the combined expertise of a large group of code developers and expert users. RESULTS We calculated equation-of-state data for four classes of DFT implementations, totaling 40 methods. Most codes agree very well, with pairwise differences that are comparable to those between different high-precision experiments. Even in the case of pseudization approaches, which largely depend on the atomic potentials used, a similar precision can be obtained as when using the full potential. The remaining deviations are due to subtle effects, such as specific numerical implementations or the treatment of relativistic terms. CONCLUSION Our work demonstrates that the precision of DFT implementations can be determined, even in the absence of one absolute reference code. Although this was not the case 5 to 10 years ago, most of the commonly used codes and methods are now found to predict essentially identical results. The established precision of DFT codes not only ensures the reproducibility of DFT predictions but also puts several past and future developments on a firmer footing. Any newly developed methodology can now be tested against the benchmark to verify whether it reaches the same level of precision. New DFT applications can be shown to have used a sufficiently precise method. Moreover, high-precision DFT calculations are essential for developing improvements to DFT methodology, such as new density functionals, which may further increase the predictive power of the simulations. Recent DFT methods yield reproducible results. Whereas older DFT implementations predict different values (red darts), codes have now evolved to mutual agreement (green darts). The scoreboard illustrates the good pairwise agreement of four classes of DFT implementations (horizontal direction) with all-electron results (vertical direction). Each number reflects the average difference between the equations of state for a given pair of methods, with the green-to-red color scheme showing the range from the best to the poorest agreement. The widespread popularity of density functional theory has given rise to an extensive range of dedicated codes for predicting molecular and crystalline properties. However, each code implements the formalism in a different way, raising questions about the reproducibility of such predictions. We report the results of a community-wide effort that compared 15 solid-state codes, using 40 different potentials or basis set types, to assess the quality of the Perdew-Burke-Ernzerhof equations of state for 71 elemental crystals. We conclude that predictions from recent codes and pseudopotentials agree very well, with pairwise differences that are comparable to those between different high-precision experiments. Older methods, however, have less precise agreement. Our benchmark provides a framework for users and developers to document the precision of new applications and methodological improvements.

Chemical shift computations on a crystallographic basis: some reflections and comments.

Computations for chemical shifts of molecular organic compounds using the gauge‐including projector augmented wave method and the NMR‐CASTEP code are reviewed. The methods are briefly introduced, and some general aspects involving the sources of uncertainty in the results are explored. The limitations are outlined. Successful applications of the computations to problems of interpretation of NMR results are discussed and the range of areas in which useful information is obtained is illustrated by examples. The particular value of the computations for comparing shifts between resonances where the same chemical site is involved is emphasised. Such cases arise for shifts between different crystallographically independent molecules of the same chemical species, between polymorphs and for shift anisotropies and asymmetries. Copyright

NMR crystallography of oxybuprocaine hydrochloride, Modification II°

The (13)C CPMAS spectrum is presented for the polymorph of oxybuprocaine hydrochloride which is stable at room temperature, i.e. Mod. II degrees . It shows crystallographic splittings arising from the fact that there are two molecules, with substantially different conformations, in the asymmetric unit. An INADEQUATE two-dimensional experiment was used to link signals for the same independent molecule. The chemical shifts are discussed in relation to the crystal structure. Of the four ethyl groups attached to NH(+) nitrogens, one gives rise to unusually low chemical shifts, very different from those of the other three ethyl groups. This is attributed empirically to gamma-gauche conformational effects, as is confirmed by shielding computations. These considerations allow (13)C signals to be assigned to specific carbons in the two crystallographically inequivalent molecules in the crystal structure. Indeed, information about the conformations is inherent in the NMR spectrum, which thus provides data of crystallographic significance. A (13)C/(1)H HETCOR experiment enabled resolution to be obtained in the (1)H dimension and allowed (1)H and (13)C signals for the same independent molecule to be linked.

Ab initio calculation of the anomalous Hall conductivity by Wannier interpolation

The intrinsic anomalous Hall conductivity in ferromagnets depends on subtle spin-orbit-induced effects in the electronic structure, and recent ab initio studies found that it was necessary to sample the Brillouin zone at millions of

Density functional theory in the solid state.

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Spectral and Fermi surface properties from Wannier interpolation

-points to converge the calculation. We present an efficient first-principles approach for computing this quantity. We start out by performing a conventional electronic-structure calculation including spin-orbit coupling on a uniform and relatively coarse

Complete 1H resonance assignment of β-maltose from 1H–1H DQ-SQ CRAMPS and 1H (DQ-DUMBO)–13C SQ refocused INEPT 2D solid-state NMR spectra and first principles GIPAW calculations

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Time Averaging of NMR Chemical Shifts in the MLF Peptide in the Solid State

-point mesh. From the resulting Bloch states, maximally localized Wannier functions are constructed which reproduce the ab initio states up to the Fermi level. The Hamiltonian and position-operator matrix elements, needed to represent the energy bands and Berry curvatures, are then set up between the Wannier orbitals. This completes the first stage of the calculation, whereby the low-energy ab initio problem is transformed into an effective tight-binding form. The second stage only involves Fourier transforms and unitary transformations of the small matrices setup in the first stage. With these inexpensive operations, the quantities of interest are interpolated onto a dense

Probing heteronuclear N-15-O-17 and C-13-O-17 connectivities and proximities by solid-state NMR spectroscopy

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Relativistic nuclear magnetic resonance chemical shifts of heavy nuclei with pseudopotentials and the zeroth-order regular approximation

-point mesh and used to evaluate the anomalous Hall conductivity as a Brillouin zone integral. The present scheme, which also avoids the cumbersome summation over all unoccupied states in the Kubo formula, is applied to bcc Fe, giving excellent agreement with conventional, less efficient first-principles calculations. Remarkably, we find that about 99% of the effect can be recovered by keeping a set of terms depending only on the Hamiltonian matrix elements, not on matrix elements of the position operator.