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We analyse three related preconditioned steepest descent algorithms, whichare partially popular in Hartree-Fock and Kohn-Sham theory as well as invariantsubspace computations, from the viewpoint of minimization of the correspondingfunctionals, constrained by orthogonality conditions. We exploit the geometryof the of the admissible manifold, i.e. the invariance with respect to unitarytransformations, to reformulate the problem on the Grassmann manifold as theadmissible set. We then prove asymptotical linear convergence of the algorithmsunder the condition that the Hessian of the corresponding Lagrangian iselliptic on the tangent space of the Grassmann manifold at the minimizer.

Direct minimization for calculating invariant subspaces in density functional computations of the electronic structure