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I am rather new to VASP and I have seen people using different methods to compute crystal structure:

  1. Using Birch-Murnaghan equation to find optimal lattice volume, and then adjust the structure inside the lattice by tuning the structural parameter.
  2. Using ISIF=3 to relax the ions directly.

What are their advantages respectively and which one should I use?

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The whole idea is to find minimum energy structure, one can search entire combination of lattice parameters to get the minimum energy configuration. However the complexity of problem increases very fast specially for tetragonal, monoclinic, triclinic structures. In case of cubic only one degree of freedom is sufficient to find energy minima. If one have energy-volume curve. Fit into The Birch-Murnaghan equation which incorporate pressure dependent bulk-modulus in to the fitting equations.

ISIF=3 allows minimization algorithm to find minima in convex surface without prior knowledge of landscape. You only need to provide initial configuration. Ideally both method should end up at same value, however, numerical method always provide slightly different value. One also can see different IBIRION tag to choose minimization algorithm may result slightly different value.

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