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Part 1
Improving Self-consistent Field Convergence
and furthermore the MO coefficient matrices C are inherently non-sparse. Several linear or nearlinear
scaling alternatives to diagonalization have been suggested in the literature 18-20 . These
methods could be reformulated with a dynamical level shift scheme like ours if the scheme could do
without the MO information, but it is not an easy task to find a good dynamic level shift scheme
with a high level of control without the knowledge of the developments in the individual MOs. The
search used to find the level shift in TRRH-LS is directly applicable since it is not dependent on the
MO information; the problem is only the number of Fock evaluations. The Fock evaluation is still
expensive even though algorithms which make the evaluation of the Fock matrix cheaper are
continually developed.
This section describes a very recently developed approach to find the optimal level shift in the
TRRH step without the use of individual MOs or knowledge of the HOMO-LUMO gap. So far it
has proven to be the most successful level shift scheme we have studied. The scheme is build on the
assumption that the TRRH step is taken in connection with a TRDSM step (or some other density
subspace minimization method). In this case it can be exploited that TRDSM is a very good energy
model (see Section 1.4.2.2) and can be trusted with the responsibility to find the best direction as
long as not too much new information is introduced to the density subspace in each step.
A new density, found by diagonalization of a level shifted Fock matrix or by some alternative, can
be split in a part D ⊥
that can be described in the previous densities and a part D with new
information orthogonal to the existing subspace
D can be expanded in the previous densities as
⊥
D( µ ) = D + D . (1.35)
n
D = ∑ωiDi
, (1.36)
i=
1
where n is the number of previously stored densities D i and the expansion coefficients ω i are
dependent on µ and determined in a least-squares manner
n
−1
ω i ( µ ) = ∑ ⎡⎣M ⎤⎦
Tr D jSD( µ ) S, Mij = Tr DiSD jS . (1.37)
j=
1
ij
⊥
It is obvious that when µ → ∞ then D → 0 since the new density then approaches the initial
density D 0 , see Eq. (1.32) and (1.33), which belongs to the set of previous densities. Thus, there is a
⊥
connection between D and µ which we can exploit. If the ratio d orth ⊥ 2
of the square norm D
S
2
relative to D
S
is small, only small changes to the density subspace are introduced;
20