Euler–Lagrange Equations: Kohn–Sham Energy and X‐Ray Density Matrix

ABSTRACT The Kohn–Sham equations and the one‐body equations associated with X‐ray density‐matrix refinement arise from different constrained variational problems but have parallel Euler–Lagrange structure. On the Kohn–Sham side, stationarity of the electronic energy gives the one‐electron Hamiltonian eigenvalue equations. On the X‐ray side, stationarity of a constrained residual functional gives a one‐body eigenvalue equation involving a constrained Euler–Lagrange operator, denoted here by D . The practical construction of the X‐ray projector, however, is carried out by the Clinton equations, which iteratively enforce projector admissibility and experimental constraints. We make explicit the mathematical relationship between these two descriptions: the Clinton iteration constructs the projector , while the constrained Euler–Lagrange equation characterizes the same stationary projector through the condition D p = 0, or equivalently, for Hermitian D , = 0. Consequently, the occupied subspace of the converged Clinton projector can be represented by orbitals satisfying D = . For exact data, both the X‐ray and Kohn–Sham developments yield the same exact density; under the stated finite‐basis uniqueness conditions they yield the same occupied projector, although D and the Kohn–Sham Hamiltonian remain physically distinct operators.