Assessment of density matrix methods for linear scaling electronic structure calculations

Abstract

Purification and minimization methods for linear scaling computation of the one-particledensity matrix for a fixed Hamiltonian matrix are compared. This is done by consideringthe work needed by each method to achieve a given accuracy in terms of the difference fromthe exact solution. Numerical tests employing orthogonal as well as non-orthogonal versionsof the methods are performed using both element magnitude and cutoff radius basedtruncation approaches. It is investigated how the convergence speed for the differentmethods depends on the eigenvalue distribution in the Hamiltonian matrix. An expressionfor the number of iterations required for the minimization methods studied is derived,taking into account the dependence on both the band gap and the chemical potential. Thisexpression is confirmed by numerical tests. The minimization methods are found toperform at their best when the chemical potential is located near the center ofthe eigenspectrum. The results indicate that purification is considerably moreefficient than the minimization methods studied even when a good starting guessfor the minimization is available. In test calculations without a starting guess,purification is more than an order of magnitude more efficient than minimization.