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Part 1
Improving Self-consistent Field Convergence
⊥ ⊥
Inserting Eq. (1.77) in the last term of Eq. (1.76) and neglecting the term Tr ∆ G ( ∆ ) , the
augmented Roothaan-Hall energy model can be written as
( ) ( ) ( )
ARH ( ) RH ( ) ( ) RH
E D = E D + EHF D0 − E ( D0 ) + Tr 2∆−∆ G ∆ , (1.80)
where G ( ∆ ) is evaluated as a linear combination of previous Fock matrices
n
( ) ∑ωi ( i ) ∑ωi ( i )
G ∆ = G D = ( F D − h ). (1.81)
i= 1 i=
1
The energy model E ARH has no intrinsic restrictions with respect to how different the densities
spanning the subspace are allowed to be, and this is one of the benefits compared to the TRSCF
scheme. For the TRDSM energy model, the purification implicit in the DSM energy makes no sense
if the densities are too different, in particular if they have different electron configurations. In ARH,
configuration shifts can be handled without problems, and whereas old, obsolete densities pollute
the DSM energy model, they simply disappear from the ARH energy model, since their weights ω i
diminish.
We expect a faster convergence rate for ARH compared to TRSCF, mainly because the RH and
DSM steps are merged to an energy model with correct gradient (not just in the subspace) and an
approximate Hessian, which is improved in each iteration using the information from the previous
density and Fock matrices.
1.4.3.2 The Augmented RH Optimization
The density for which the ARH energy model should be optimized can be expanded in the antisymmetric
matrix X
n
D ( X () () () ()
) = exp 1
( − XS ) D i 0 exp ( SX ) = D i ⎡ i 0
+
0 , ⎤ + ⎡⎡ i
2 0
, ⎤ , ⎤
⎣ D X ⎦ ⎣ ⎦
+
⎣
D X X ⎦
, (1.82)
() i
S S S
where D
0
is the reference density from which the step X is taken. Optimizing the ARH energy is
thus a nonlinear problem and an iterative scheme should be applied.
A Newton-Raphson (NR) optimization of the ARH energy is therefore carried out, and the steps are
ARH
found minimizing a second order approximation of the ARH energy E
(2)
by the preconditioned
conjugate gradient (PCG) method. The second order approximation of the ARH energy, where the
constant terms are excluded, can be written as
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