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 molecular dynamics framework is first derived for general Hohenberg–Kohn density functional theory and it is then presented in explicit forms for thermal Hartree–Fock theory using a density matrix formalism, for self-consistent charge density functional tight-binding theory, and for general non-linear charge relaxation models that can be designed and optimized using modern machine learning methods. Our intention is to give a self-contained but brief and hopefully pedagogical presentation. Graphic abstract