orbitalprojected

Orbital-projected refers to a set of techniques and analyses that express or analyze electronic structure quantities in terms of contributions from a chosen set of atomic or molecular orbitals. The core idea is to project a quantum state, wavefunction, or operator onto the subspace spanned by selected orbitals to quantify how much each orbital character contributes to the property of interest.

Mathematically, if {|phi_i>} denotes a selected set of normalized orbitals, the projection operator is P = sum_i |phi_i><phi_i| (with appropriate care for non-orthogonality in some bases). A state psi can be projected as psi_proj = P psi, and the projection coefficients are c_i = <phi_i|psi>. In practice, quantities are often described by orbital-resolved weights, such as the partial density of states g_i(E) = sum_k |<phi_i|psi_k>|^2 delta(E - E_k), which reveals how specific orbitals contribute at different energies.

Applications abound in quantum chemistry and condensed matter physics. Orbital projection is used to obtain partial density of states, to construct orbital-resolved band structures, and to analyze bonding, charge transfer, or magnetic moments by attributing features to particular orbitals. It also underpins methods that build localized representations, such as maximally localized Wannier functions, where Bloch states are projected onto a chosen orbital subspace to generate initial guesses. In practice, orbital projection depends on the choice of orbital set and can be influenced by basis completeness and overlap between orbitals, requiring careful interpretation.

Related concepts include projection operators, Mulliken or Löwdin population analyses, and Wannier function construction. Orbital-projected approaches provide a useful lens for dissecting complex electronic structures into chemically or physically meaningful contributions.