A novel Newton-Raphson geometry optimization method for density functional theory calculation is presented. Contrary to the standard Newton-Raphson procedure, our method does not require the calculation of the full Hessian matrix. We use instead density functional perturbation theory to evaluate the effect of the Hessian on a given atomic displacement and invert the Hessian using a conjugate gradient method. The method uses Cartesian coordinates and is very robust. In a variety of tested molecular systems, it has an efficiency comparable with that of more established methods. In difficult condensed matter systems, it appears to be far superior.