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$$ P_n\alpha(\mathbfk) = \langle \chi_\alpha | \psi_n,\mathbfk \rangle = \int \chi_\alpha^*(\mathbfr) \psi_n,\mathbfk(\mathbfr) , d\mathbfr $$

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In modern computational materials science, the accurate and efficient representation of electronic wavefunctions is paramount. This paper examines the concept designated by the term "prxyorb" (Projected/Proxy Orbital), a mechanism utilized in plane-wave density functional theory (DFT) codes. We explore how the projection of plane-wave states onto localized atomic orbital bases facilitates the analysis of chemical bonding, the calculation of band decomposed charge densities, and the implementation of order-N scaling methods. This technical review highlights the mathematical formulation of orbital projection and its critical role in bridging the delocalized plane-wave representation with the chemically intuitive localized orbital picture. There is no need to download executable files

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The magnitude of these coefficients serves as a for the contribution of a specific atomic orbital $\alpha$ to the electronic band state $n$. In computational codebases, the data structure or file storing these coefficients is frequently labeled prxyorb or similar variations (e.g., PROCAR output derives from these calculations).