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Part 1
Improving Self-consistent Field Convergence
1.9 Conclusion
In this part of the thesis the trust region SCF (TRSCF) algorithm is presented as a means to improve
SCF convergence compared to methods typically used today e.g. DIIS. In the TRSCF method, both
the Roothaan-Hall (RH) step and the density-subspace minimization (DSM) steps are replaced by
optimizations of local energy models of the Hartree-Fock/Kohn-Sham energy E SCF . These local
models have the same gradient as the energy E SCF , but an approximate Hessian. Restricting the steps
of the TRSCF algorithm to the trust region of these local models, that is, to the region where the
local models approximate E SCF well, smooth and fast convergence may be obtained.
The developments through the years in SCF optimization algorithms are reviewed, and it is found
that the fundamental schemes used in TRSCF to improve convergence have been around for several
years; DIIS is actually a subspace minimization in the gradient norm, and level shifts have been
used to improve or force convergence since 1973. Anyhow, the level shifts have previously been
found on a trial and error basis as a constant parameter, whereas we advocate a dynamic level shift
scheme in which the level shift is used to control the density change in the RH step. As such the
level shift is optimized in each iteration to allow the density to change to the trust radius of the RH
energy model, hence the name trust region Roothaan-Hall (TRRH) for our RH scheme. Also, the
density subspace minimization has been improved compared to previous methods. An accurate
energy model is constructed in the iterative subspace, where only minor approximations are made
compared to the SCF energy. The trust region minimization of this energy model thus corresponds
well to a minimization of E SCF in the iterative subspace, thus resulting in an energy decrease in each
trust region DSM (TRDSM) step. The TRRH and TRDSM steps in combination make up a
successful scheme with a high convergence rate without compromising the control of the density
changes in each step.
Compared to ref. 38 and 39 , an alternative level shift scheme (d orth -shift) for the TRRH step is
presented which does not control the density change through the overlap of the individual orbitals,
but instead controls the amount of new information added to the density subspace. Thus the d orth -
shift scheme does not contain any reference to the MO basis and can be used in connection with
alternatives to diagonalization. Also, it is found that the d orth -shift scheme leads to a faster
convergence since the former level shift scheme is too restrictive, ignoring the well known changes
contained in the density subspace.
For TRDSM, an improvement of the energy model is developed, in which a part of the term
neglected in the DSM energy model compared to the SCF energy is recovered. However, the effects
of the improvement are found rather small compared to the extra complexity added to the algorithm.
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