Extended Lagrangian Born–Oppenheimer molecular dynamics: from density functional theory to charge relaxation models

We present a review of extended Lagrangian Born–Oppenheimer molecular dynamics and its most recent development. The framework is first derived for general Hohenberg–Kohn density functional theory it then presented in explicit forms thermal Hartree–Fock using matrix formalism, self-consistent charge tight-binding theory, non-linear relaxation models that can be designed optimized modern machine learning methods. Our intention to give self-contained but brief hopefully pedagogical presentation.