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The Broyden (Broyden—Fletcher—Goldfarb—Shanno or BFGS) algorithm can be known in the matter modeling to optimize a crystal lattice geometry — to clarify positions of nodes of a crystal lattice in calculation process of eigenvalues and eigenfunctions with a Hartree-Fock (or DFT) method. How much will the Broyden algorithm be clarifying right (energetically beneficial) positions of atomic nuclei if a crystal lattice of a substance will intentionally be significantly distorted? But what things can be done if a hypothetical alloy (some mix of metals and non-metals in some proportion, which is raised to the monocrystalline solid level) is never explored before, including analyzing in an X-ray diffractometer to define positions of nuclei of lattice of a monocrystalline solid? Whether can the Broyden algorithm correctly upbuild a crystalline structure of some alloy? If it cannot, so is there a way to roughly calculate this thing maybe before a Hartree-Fock (or DFT) calculation and free software implementing such a calculation?

Or to put it more simply

is it possible to calculate an "a priori" alloy?

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BFGS is a local minimization method - it will find you the closest minimum to your starting structure - whereas finding the crystal structure is a global optimization problem: what is the structure that yields the lowest possible energy with the given atoms?

BFGS may yield you the correct crystal structure, if you already know the right crystal space group and the structure is simple enough so that you can make a good guess for the structure. However, if you do not know the symmetry of the crystal, your structure search needs to consider many alternatives.

There is already a code dedicated to solving these kinds of structure search problems called USPEX; I recommend you to study the related literature which should answer all of your questions in detail.

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  • $\begingroup$ Thanks to the answer! What is a theory or method on which the USPEX is based? $\endgroup$
    – SFriendly
    May 4, 2023 at 12:50
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    $\begingroup$ @SFriendly since I answered the question, please accept it. USPEX is documented on its web page. $\endgroup$ May 5, 2023 at 13:22

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