A linear-scaling implementation of Hartree-Fock and Kohn-Sham self-consistent field SCF
theories is presented and illustrated with applications to molecules consisting of more than 1000
atoms. The diagonalization bottleneck of traditional SCF methods is avoided by carrying out a
minimization of the Roothaan-Hall RH energy function and solving the Newton equations using
the preconditioned conjugate-gradient PCG method. For rapid PCG convergence, the Löwdin
orthogonal atomic orbital basis is used. The resulting linear-scaling trust-region Roothaan-Hall
LS-TRRH method works by the introduction of a level-shift parameter in the RH Newton
equations. A great advantage of the LS-TRRH method is that the optimal level shift can be
determined at no extra cost, ensuring fast and robust convergence of both the SCF iterations and the
level-shifted Newton equations. For density averaging, the authors use the trust-region
density-subspace minimization TRDSM method, which, unlike the traditional direct inversion in
the iterative subspace DIIS scheme, is firmly based on the principle of energy minimization. When
combined with a linear-scaling evaluation of the Fock/Kohn-Sham matrix including a boxed fitting
of the electron density, LS-TRRH and TRDSM methods constitute the linear-scaling trust-region
SCF LS-TRSCF method. The LS-TRSCF method compares favorably with the traditional SCF/
DIIS scheme, converging smoothly and reliably in cases where the latter method fails. In one case
where the LS-TRSCF method converges smoothly to a minimum, the SCF/DIIS method converges
to a saddle point.
