  Reducing the number of candidate structures is crucial to improve the
efficiency of global optimization. Herein, we demonstrate that the generalized
Hamiltonian can be described by the atom classification model (ACM) based on
symmetry, generating competent candidates for the first-principles calculations
to determine ground states of alloy directly. The candidates can be obtained in
advance through solving the convex hull step by step, because the correlation
functions of ACM can be divided into various subspace according to the defined
index $l$. As an important inference, this index can be converted to the number
of Wyckoff positions, revealing the dominant effect of geometry symmetry on
structural stability. Taking Ni-Pt, Ag-Pd, Os-Ru, Ir-Ru and Mo-Ru as examples,
we not only identify the stable structures in previous theoretical and
experimental results, but also predict a dozen of configurations with lower
formation energies, such as Ag$_{0.5}$Pd$_{0.5}$ ($Fd$-$3m$),
Os$_{0.5}$Ru$_{0.5}$ ($Pnma$), Ir$_{1/3}$Ru$_{2/3}$ ($P6_{3}/mmc$), and
Mo$_{0.25}$Ru$_{0.75}$ ($Cmcm$).
