Relaxation of Incommensurate Structures via Quantum Models

Accurately modeling structural relaxation in incommensurate systems is intrinsically challenging due to the absence of global translational symmetry. In this work, we develop a variational quantum framework for structural relaxation in incommensurate Schr\"{o}dinger models, where displacement fields are formulated on the configuration space and the electronic Hamiltonian is represented in reciprocal space. This yields well-defined relaxed energy, local density of states, and forces through thermodynamic limits. We propose an anisotropic scattering-channel approximation, and prove exponential convergence of the approximate equilibria. Numerical experiments are performed to support the analysis and show that the model captures domain-wall formation and its impact on the electronic spectrum.