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Part 1
Improving Self-consistent Field Convergence
any initial guess D 0 , there exists a level shift µ 0 > 0 such that for level shift parameters µ > µ 0 , the
energy decreases at each step and converges towards a stationary value.
The level shift technique is still routinely used for cases where the DIIS scheme has problems. The
level shifts are typically found on a trial and error basis. Recently, we advocated the use of a level
shift to control the changes introduced in the Roothaan-Hall step 38 , and we suggested a way of
optimizing the level shift at each iteration based on physical arguments and without guesswork. The
algorithm is based on the trust region philosophy in which a model energy function is optimized,
but restricted with respect to the step length. We thus named the algorithm trust region Roothaan-
Hall (TRRH), even though it is not a true trust region optimization scheme like e.g. the energy
minimization of Francisco, Martínez, and Martínez 26 or our TRDSM scheme 38 .
Level shifting can be combined with a damping or extrapolation scheme. When the TRRH approach
is combined with the subspace minimization method TRDSM it seems to outperform DIIS in
stability and to have a better or similar convergence rate, as will be illustrated in the following
sections. Combining level shifting with DIIS can occasionally be a benefit, but typically DIIS and
level-shifting does not work well together, and in Section 1.4.1.3 we will try to justify this.
1.4 Development of SCF Optimization Algorithms
The SCF scheme as it typically looks today is sketched in Fig. 1.2. Compared to Fig. 1.1, the step
is inserted, illustrating a density subspace minimization, where
some function f is minimized with respect to the coefficients c i
which expand the previous densities D i . The function f could
be the gradient norm as in DIIS or some energy model
D 0
F(D n
)
n
approximating the SCF energy in the subspace of the previous
D = ∑ciDi,minf
( c)
densities as in EDIIS and TRDSM. In the Roothaan-Hall step
i=
1
, the averaged Fock matrix F found from the optimization in
n
n = n+1 F =
is then used instead of the most recent Fock matrix F(D n ) to
∑ciF( Di)
i=
1
find a new trial density D n+1 . In general, the averaged density
matrix D is not idempotent and therefore does not represent a
valid density matrix; moreover, since the Kohn-Sham matrix
F D n+1
(unlike the Fock matrix) is nonlinear in the density matrix, the
averaged Kohn-Sham matrix F is different from FD. ( ) For
these reasons, the averaged Fock matrix F cannot be
no
D n+1
≈ D n
yes
D conv
associated uniquely with a valid Fock matrix. Usually, this
Fig. 1.2 Flow diagram of the SCF
does not matter much since the subsequent diagonalization of scheme including the density
the Fock matrix nevertheless produces a valid density matrix subspace minimization step.
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