This section contains reference and verification datasets of equations of state (EOS) calculated with different density functional theory (DFT) codes using the AiiDA common workflows (ACWF) infrastructure. The data is published and discussed in the article:
E. Bosoni et al., How to verify the precision of density-functional-theory implementations via reproducible and universal workflows, Nat. Rev. Phys. 6, 45-58 (2024)
Select an element to show the equation of state (EOS) curves for various reference structures containing it (unaries and oxides) and to compare results among multiple codes and computational approaches. Note: the PBE functional is used for the current data.
H
He
Li
Be
B
C
N
O
F
Ne
Na
Mg
Al
Si
P
S
Cl
Ar
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
Ga
Ge
As
Se
Br
Kr
Rb
Sr
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
Cd
In
Sn
Sb
Te
I
Xe
Cs
Ba
Hf
Ta
W
Re
Os
Ir
Pt
Au
Hg
Tl
Pb
Bi
Po
At
Rn
Fr
Ra
Rf
Db
Sg
Bh
Hs
Mt
Ds
Rg
Cn
Nh
Fl
Mc
Lv
Ts
Og
La
Ce
Pr
Nd
Pm
Sm
Eu
Gd
Tb
Dy
Ho
Er
Tm
Yb
Lu
Ac
Th
Pa
U
Np
Pu
Am
Cm
Bk
Cf
Es
Fm
Md
No
Lr
Density functional theory (DFT) is extensively used in condensed matter physics and materials science. Many different software codes and computational approaches have been developed to perform DFT calculations. This website is part of an effort to systematically assess the precision and evaluate the reliability of many of these computational approaches. It supplements the accompanying publication[1] and interactively visualizes the relevant data. The raw data is published on the Materials Cloud Archive[2].
This effort is a continuation of the the systematic assessment published in 2016, that compared the performance of the DFT codes by calculating the equation of state (EOS) curves for 71 elemental crystals[3]. The website illustrating the results of that project is available at [4].
The performance of the computational approaches to solve the DFT equations is assessed, similarly to the previous study, by calculating the EOS curves. For every element in range Z=1 to 96, ten different crystals are calculated using the PBE functional[5]. They include 4 unaries (simple cubic, BCC, diamond, FCC) and 6 oxides (X2O, XO, X2O3, XO2, X2O5, XO3). The datasets produced by calculating the EOS curves for these 960 crystals can be divided into
  1. A curated reference set of highly converged results using two independent all-electron DFT codes (FLEUR and WIEN2k).
  2. Other datasets obtained with various pseudopotential codes.
These datasets were produced in an automated fashion using the AiiDA common workflows (ACWF) infrastructure[6][7] and the relevant scripts are available on GitHub[8].
The EOS curves are calculated by fitting a variation of the DFT total energy \(E\) (or, more precisely, of the free energy \(E - TS\) including the entropic contribution due to the electronic smearing) versus cell volume \(V\) to the Birch-Murnaghan EOS, given by$$ E(V) = E_0 + \frac{9V_0B_0}{16} \left\{ \left[ \left( \frac{V_0}{V} \right)^{\frac{2}{3}} - 1 \right]^3 B_1 + \left[ \left( \frac{V_0}{V} \right)^{\frac{2}{3}} - 1 \right]^2 \left[ 6 - 4 \left( \frac{V_0}{V} \right)^{\frac{2}{3}} \right] \right\}, \tag{1}$$where the equilibrium volume \(V_0\), the bulk modulus \(B_0\), and its derivative with respect to pressure \(B_1\) can be extracted from a fitting procedure.
Comparison metrics
In order to conveniently compare the performance of two different codes or computational approaches, we need a single-valued metric to describe the difference between two EOS curves. In this application we consider the following comparison metrics:
  1. ε (epsilon) - a metric that represents the area between the two EOS curves normalized by the average variance of the two curves, and is given by$$ \varepsilon(a,b) = \sqrt{\frac{ \langle[E_{a}(V) - E_{b}(V)]^2 \rangle} {\sqrt{\langle [E_{a}(V) - \langle E_{a} \rangle]^2 \rangle \langle [ E_{b}(V) - \langle E_{b} \rangle]^2 \rangle}} }, \tag{3}$$where$$ \langle f \rangle = \frac{1}{V_{M}-V_{m}}\int_{V_{m}}^{V_{M}} f(V) ~ dV, \tag{4}$$and \(V_M\), \(V_m\) define the considered volume range (\(\pm 6\%\) around a central reference volume for our case)[1]. As reported in the publication[1], an excellent agreement is \(\varepsilon<0.06\), while a good agreement is \(\varepsilon<0.2\).
  2. ν (nu) - a metric that captures the relative difference of the equilibrium volume (\(V_0\)), bulk modulus (\(B_0\)) and its derivative w.r.t. pressure (\(B_1\)) with specified weights. The metric is calculated by$$ \nu_{w_{V_0},w_{B_0},w_{B_1}}(a,b) = 100 \sqrt{ \sum_{Y=V_0,B_0,B_1} \left[ w_{Y} \cdot \frac{Y_{a}-Y_{b}}{(Y_{a}+Y_{b})/2} \right] ^2}, \tag{2}$$where \((V_0)_a\) indicates the value of \(V_0\) obtained by fitting the data of method \(a\), and so on. The values of the weights were determined as \(w_{V_0} = 1\), \(w_{B_0} = \frac{1}{20}\) and \(w_{B_1} = \frac{1}{400}\) based the sensitivity of each parameter to numerical noise in the fitting procedure[1]. As reported in the publication[1], an excellent agreement is \(\nu<0.1\), while a good agreement is \(\nu<0.33\).
  3. Δ (delta) - the metric used in the previous study[3], representing the area between the two EOS curves. It is given by$$ \Delta(a,b) = \sqrt{\langle [E_{a}(V) - E_{b}(V)]^2 \rangle}. \tag{5}$$This metric has the shortcoming of being too sensitive to the value of the bulk modulus of the material. In this website, the values are normalized by the number of atoms. As mentioned in the publication[1], an excellent agreement (when the bulk modulus is not too small) could be considered for \(\Delta<0.3\ \text{[meV/atom]}\), while a good agreement for \(\Delta<0.95\ \text{[meV/atom]}\).
References
[1] E. Bosoni et al. How to verify the precision of density-functional-theory implementations via reproducible and universal workflows. Nat. Rev. Phys. 6, 45-58 (2024), https://doi.org/10.1038/s42254-023-00655-3. [open-access arXiv version]
[2] E. Bosoni et al. How to verify the precision of density-functional-theory implementations via reproducible and universal workflows. Materials Cloud Archive 2023.81 (2023), https://doi.org/10.24435/materialscloud:s4-3h.
[3] K. Lejaeghere et al. Reproducibility in density functional theory calculations of solids. Science 351, aad3000-aad3000 (2016), https://doi.org/10.1126/science.aad3000.
[5] J. P. Perdew, K. Burke, M. Ernzerhof, Generalized Gradient Approximation Made Simple. PRL 77, 18 (1996), https://doi.org/10.1103/PhysRevLett.77.3865.
[6] S. P. Huber et al. Common workflows for computing material properties using different quantum engines. npj Comput. Mater. 7, 136 (2021), https://doi.org/10.1038/s41524-021-00594-6.