2. Equation of State & Geometry Optimization

In density-functional theory (DFT), physical convergence refers to reaching a stationary state with respect to structural degrees of freedom, such as lattice constants or atomic positions. This is distinct from numerical convergence, where one checks stability of results with respect to discretization parameters (see Numerical Convergence Tutorial). [PAYNE1992]

Two key procedures are involved:

• Equation of State (EOS): Scan of total energy as a function of volume (lattice constant scaling) to locate the equilibrium lattice constant.

• Geometry Optimization: Relaxation of internal atomic positions to minimize residual forces at fixed lattice vectors.

• 2.1. Si_1−xGe_x alloy