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Part 1
Improving Self-consistent Field Convergence
(1) ∂ESCF
( κ )
⎡ ⎤
⎣
E
SCF ⎦
= = −4F
ai . (1.99)
ai ∂κ ai κ=
0
The expressions are found replacing D in Eqs. (1.96) and (1.97) with D MO in Eq. (1.94) and
differentiating with respect to κ ai .
All higher order terms in κ arising from 2TrF(D 0 )(D - D 0 ) are consequently also shared for the SCF
and RH energies whereas terms of second and higher order arising from the last term(s) in Eq. 1.94
are neglected in the RH energy model. To study the differences, the second order derivatives in κ
are found in the same way as the first derivatives
2 RH
(2) ∂ E ( κ)
⎡ ⎤
⎣
E
RH ⎦
= = 4δ ij δ ab ( ε a −ε
i )
(1.100)
aibj ∂κ
∂κ
2
ai
bj
κ=
0
(2) ∂ ESCF
( κ)
⎡ ⎤
⎣
E
SCF ⎦
= = 4δδ ij ab ( ε a − ε i ) + W aibj , (1.101)
aibj ∂κ
∂κ
ai
bj
κ=
0
where
HF
16 4( )
W = g − g + g
(1.102)
aibj
aibj abij ajib
( )
DFT
Waibj
= 16gaibj − 4 γ gabij + gajib
+ ⎡ ( ) ⎤ ⎣
E κ ⎦
. (1.103)
(2)
XC
aibj
(2)
E XC ( κ ) is the second derivative of the term E XC expanded in the orbital rotation parameters κ. The
error in the RH energy model can then be said to depend partly on the size of W and partly on the
size of the third and higher order contributions from the nonlinear terms in Eq. (1.97) which are not
included in Eq. (1.96). This general consideration goes for DFT as well as HF, but with different
impact. As seen in Eq. (1.102) and (1.103), the definition of W differs in the two approaches and
even differs depending on which DFT functional is chosen. Furthermore, since the size of the
HOMO-LUMO gap ∆ε ai = ε a - ε i is typically smaller in DFT, the term 4δ ij δ ab (ε a – ε i ) will have
different weights in Eq. (1.101) depending on the method. Also the size of the third and higher
order contributions in Eq. (1.97) would be expected to differ for HF and DFT, since for DFT both
the terms Tr(D - D 0 )G(D - D 0 ) and E XC (D) contribute whereas HF only contains the Tr(D - D 0 )G(D
- D 0 ) term. In the beginning of the optimization, where large steps are taken, the size of the third
and higher order contributions is the potential source of error. Near convergence this should be less
of an issue, and in this region the size of the lowest Hessian eigenvalues should be the decisive error
source.
HF and LDA calculations have been carried out and the part of the SCF energy change arising from
RH
the RH step ∆ E SCF
has been found as well as the change in the RH energy model ∆E RH in each
iteration.
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