  Extended Lagrangian Born-Oppenheimer molecular dynamics [Phys. Rev. Lett.,
${\bf 100}$, 123004 (2008)] is formulated for general Hohenberg-Kohn density
functional theory and compared to the extended Lagrangian framework of first
principles molecular dynamics by Car and Parrinello [Phys. Rev. Lett. ${\bf
55}$, 2471 (1985)]. It is shown how extended Lagrangian Born-Oppenheimer
molecular dynamics overcomes several shortcomings of regular, direct
Born-Oppenheimer molecular dynamics, while improving or maintaining important
features of Car-Parrinello simulations. The accuracy of the electronic degrees
of freedom in extended Lagrangian Born-Oppenheimer molecular dynamics, with
respect to the exact Born-Oppenheimer solution, is of second order in the size
of the integration time step and of fourth order in the potential energy
surface. Improved stability over recent formulations of extended Lagrangian
Born-Oppenheimer molecular dynamics is achieved by generalizing the theory to
finite temperature ensembles, using fractional occupation numbers in the
calculation of the inner-product kernel of the extended harmonic oscillator
that appears as a preconditioner in the electronic equations of motion.
Materials systems that normally exhibit slow self-consistent field convergence
can be simulated using integration time steps of the same order as in direct
Born-Oppenheimer molecular dynamics, but without the requirement of an
iterative, non-linear electronic ground state optimization prior to the force
evaluations and without a systematic drift in the total energy. In combination
with proposed low-rank and on-the-fly updates of the kernel, this formulation
provides an efficient and general framework for quantum based Born-Oppenheimer
molecular dynamics simulations.
