  Extended Lagrangian Born-Oppenheimer molecular dynamics [{\em Phys.\ Rev.\
Lett.\ } {\bf 2008}, {\em 100}, 123004] is presented for Hartree-Fock theory,
where the extended electronic degrees of freedom are represented by a density
matrix, including fractional occupation numbers at elevated electronic
temperatures. In contrast to regular direct Born-Oppenheimer molecular dynamics
simulations, no iterative self-consistent field optimization is required prior
to the force evaluations. To sample regions of the potential energy landscape
where the gap is small or vanishing, which leads to particular convergence
problems in regular direct Born-Oppenheimer molecular dynamics simulations, an
adaptive integration scheme for the extended electronic degrees of freedom is
presented. The integration scheme is based on a tunable, low-rank approximation
of a fourth-order kernel, ${\cal K}$, that determines the metric tensor, ${\cal
T}\equiv {\cal K}^T{\cal K}$, used in the extended harmonic oscillator of the
Lagrangian that generates the dynamics of the electronic degrees of freedom.
The formulation and algorithms provide a general guide to implement extended
Lagrangian Born-Oppenheimer molecular dynamics for quantum chemistry, density
functional theory, and semiempirical methods using a density matrix formalism.
