It is shown how the electronic equations of motion in extended Lagrangian Born-Oppenheimer molecular dynamics simulations [A. M. N. Niklasson, Phys. Rev. Lett. 100, 123004 (2008); J. Chem. Phys. 147, 054103 (2017)] can be integrated using low-rank approximations of the inverse Jacobian kernel. This kernel determines the metric tensor in the harmonic oscillator extension of the Lagrangian that drives the evolution of the electronic degrees of freedom. The proposed kernel approximation is derived from a pseudoinverse of a low-rank estimate of the Jacobian, which is expressed in terms of a generalized set of directional derivatives with directions that are given from a Krylov subspace approximation. The approach allows a tunable and adaptive approximation that can take advantage of efficient preconditioning techniques. The proposed kernel approximation for the integration of the electronic equations of motion makes it possible to apply extended Lagrangian first-principles molecular dynamics simulations to a broader range of problems, including reactive chemical systems with numerically sensitive and unsteady charge solutions. This can be achieved without requiring exact full calculations of the inverse Jacobian kernel in each time step or relying on iterative non-linear self-consistent field optimization of the electronic ground state prior to the force evaluations as in regular direct Born-Oppenheimer molecular dynamics. The low-rank approximation of the Jacobian is directly related to Broyden's class of quasi-Newton algorithms and Jacobian-free Newton-Krylov methods and provides a complementary formulation for the solution of nonlinear systems of equations.
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