Failure of Density-matrix Minimization Methods for Linear-scaling Density-functional Theory Using the Kohn Penalty-functional

We examine the recently-proposed scheme W. Kohn, Phys. Rev. Lett. 76, 3168 (1996)] for performing linear-scaling calculations within density-functional theory by direct minimization with respect to the single-particle density-matrix using a penalty-functional to exactly enforce the idempotency constraint. We show that such methods are incompatible with standard minimization algorithms (using conjugate gradients as an example) and demonstrate that this is a direct result of the non-analytic form of penalty-functional which must be chosen to obtain a variational principle for the total energy. The traditional formulation of density-functional theory 1] (DFT) in terms of a set of extended single-particle wave functions has led to the development of schemes for performing total-energy calculations which require a computational eeort which scales as the cube of the system-size (i.e. the number of atoms, electrons or volume of the system). This scaling results from the cost of diagonalizing the Hamiltonian or orthogonalizing the wave functions. Methods based upon the single-particle density-matrix (DM), which is free from orthogonality constraints and short-ranged in real-space, ooer the prospect of electronic structure calculations at a cost which scales only linearly with system-size. We investigate one scheme that has been proposed for achieving this goal, which uses a penalty-functional to impose the idempotency constraint on the DM. We show that the form of penalty-functional which must be chosen to obtain a variational principle for the total energy precludes the use of eecient minimization algorithms commonly used in electronic structure calculations. We apply the method to crystalline silicon and show that the desired minimum cannot be found and therefore that the variational principle cannot be used in practice.