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Mooraj et al. J Mater Inf 2023;3:4 https://dx.doi.org/10.20517/jmi.2022.41 Page 11 of 45

First-principles calculations
First-principles calculations (also called ab-initio) are computational methods that rely purely on
[93]
fundamental quantum physical laws without additional assumptions . The literature on first-principles
methods is vast and presents a comprehensive overview beyond the scope of this work. This section will
provide a sufficiently broad outline of the general concepts, advantages and disadvantages of these
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techniques pertinent to combinatorial studies of HEAs . The main strength of these methods is that they
do not require previous empirical observations of the predicted properties, and thus very little prior work is
[94]
needed to implement them . The most common practical implementation of ab-initio calculations is
[94,95]
density functional theory (DFT) in the Khon-Sham formalism . This method maps the quantum-
mechanical many-electron Schrodinger equation onto an effective one-electron problem using electron
density as a key variable. This mapping also requires the use of the exchange-correlation functional of the
electron density which is not known for most systems and must be approximated either with the local
density approximation (LDA) [96,97] or the generalized gradient approximation (GGA) [98-100] . Once these
fundamental functions are calculated, the overall energy of the system can be calculated and used to
determine the energy of formation for the possible phases of the system. This result can help researchers
determine the stability of different phases to determine which phases are likely to form. It should be noted
that in its initial state, DFT is a ground state theory and thus only provides the ground state energy at 0 K
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for a given configuration of atoms . These results can be combined with thermodynamic concepts and
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statistical sampling techniques to bridge the gap between 0 K to a finite temperature .

Despite the strong predictive power of ab-initio calculations, they often suffer from high computational
costs, which can significantly decrease the ability of researchers to explore the vast design space that is
necessary to build accurate property maps for HEAs. To overcome this challenge, many researchers either
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combine first-principle calculations with more high-throughput methods like ML or use new algorithms
and models to improve the computational efficiency to the point where first-principles calculations can be
used to explore hundreds to thousands of compositions in relatively short periods. Examples of such works
will be discussed in this section.

One approach that is considered highly promising toward high fidelity and high throughput computations
[92]
of HEAs is based on the small set of ordered structures (SSOS) containing several atoms . This method
works well to predict properties of equiatomic configurations of HEAs but loses computational efficiency
when employed for non-equiatomic compositions. Sorkin et al. implemented a preselected set of small
ordered structures (PSSOS) approach to address the issue of computational efficiency and used it to
[92]
estimate the stability of BCC and FCC phases within the Al-Co-Cr-Fe-Ni system . Traditionally the SSOS
method uses a set of small, ordered structures (SOS) to model a HEA with a given composition. First,
symmetry-unique SOS are constructed using non-conventional, non-primitive unit cells of cubic lattices.
Each SOS has a unique pair correlation function. The complete set of possible SOS solutions is constructed
and optimized using DFT. Then a small subset of SOS is selected by matching the pair correlation function
of the target composition as a linear combination of the pair correlation functions of the selected SOS. This
small set of SOS constitutes the solution of the SSOS. Screening the entire SOS solution space is impractical
when studying HEAs, so the authors restricted their SOS space to those containing 5, 6, or 7 atoms. They
selected the most frequent SOS structures in the solution set to further reduce the SOS space and only
optimized those using DFT. This selection decreases the original SOS from over 50,000 sets to 1,500.

Through the above-mentioned process, the authors can predict the formation energy and density of the
alloy system’s BCC and FCC phases of 8,801 compositions. This result is exemplified in Figure 5A, which
shows a plot of the formation energy and density of the BCC phase with varying Al and Cr compositions.
Here the marker color represents the Ni content, and the marker size represents the Co content. It can be