Get my PhD Thesis

Part 1
Improving Self-consistent Field Convergence
tot
(1)
∆c = α ⋅ c (1.67)
DSM
and a search is made in ∆ E SCF
to find the step ∆c tot that leads to the largest decrease in energy.
E SCF (α) is found by evaluating the averaged density of Eq. (1.45) for the coefficients (c 0 + ∆c tot ),
purifying it as in Eq. (1.32)–(1.33) and inserting it in the energy expression of Eq. (1.1). Then
DSM
∆ E SCF ( α)
can be found as DSM
∆ E ( α ) = E ( α ) − E ( D ). (1.68)
SCF
SCF SCF 0
Fig. 1.20 and Fig. 1.21 illustrate the search in α, again for iteration seven of the HF and LDA
calculations on the zinc complex in Fig. 1.3. For α = 0, no step is taken and hence no energy
decrease is seen. For the marked choice of α, the optimal step length is obtained.
0
-5
-10
-15
-20
-25
-30
-35
0 4 8 12 16 20
α
Fig. 1.20 Decrease in HF energy as a function of
the step length α.
0
-5
-10
-15
-20
-25
0 4 8 12 16 20
α
Fig. 1.21 Decrease in LDA energy as a function of
the step length α.
1.4.2.5 The Missing Term
In the construction of the TRDSM energy model Eq. (1.55), the term of second order in the
idempotency correction D δ was neglected from Eq. (1.50), since this term required a new Fock
evaluation F(D δ ), which would increase the expenses of the scheme considerably. This section will
be concerned with this neglected term and how a part of it can be described without the evaluation
of a new Fock matrix, leading to an improved energy model for TRDSM at no considerable extra
cost. The actual effect of this improvement to the energy model will then be discussed through a
case study. This section will only be concerned with Hartree-Fock theory and examples, but it might
equally well be done for DFT even though the improvement should be less significant since for
DFT, also terms of order ||D + || 3 are neglected. These are of the same size as the neglected term
quadratic in D δ . In Section 1.5.2 these errors are discussed.
Since the only neglect in the DSM energy model Eq. (1.55) for Hartree-Fock is the term quadratic
in D δ , and since the only term quadratic in the density is TrDG(D), the HF energy for the density D
can be written as
30