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The Quality of the Energy Models for HF and DFT
Let us take a closer look at the errors one by one. In ref. 39 a general order analysis of the purified
density D used in the parameterization of the DSM energy is given, and the results are summarized
in Table 1-5.
Table 1-5. Comparison of the properties of the unpurified density D and the purified
density D . c is the density expansion coefficients and κ is the orbital rotation parameters
that change D 0 to another density in the subspace D i .
D
Differences D+ = D− D0 = ( c κ )
O
2
Dδ = D
− D = O ( c κ )
Idempotency error
2
4
DSD − D = O ( c κ ) DSD − D
= O ( c 2 κ )
Trace error Tr DS − N / 2 = 0
2 4
Tr DS − N / 2 = O ( c κ )
In the D column, the order of the idempotency correction D δ and the idempotency error for D are
found. These are the same for DFT and HF; the idempotency error is of order c 2 ||κ|| 4 , and since D δ
is of the order c||κ|| 2 , the error connected to the neglect of the term second order in D δ , will be of
order c 2 ||κ|| 4 as well.
The third possible source of errors is the truncation of the energy E( D ) after second order in the
density. Since the Hartree-Fock energy is quadratic in the density, this truncation leads to no errors
for HF, but for DFT there will be an error of order ||D + || 3 and from the first column in Table 1-5 it is
seen that it can be written as an error of order c 3 ||κ|| 3 , since D + is of the order c||κ||. Also since the
(3)
HF energy is quadratic in the density, no third derivative E
0
exists and thus the Taylor expansion
( 2 )
used to find E0 D+ = 2F + is terminated for HF, but for DFT terms of order ||D + || 2 are neglected.
( 2 )
Since E0 D + is multiplied by D + in the energy function Eq. (1.50), this gives an error for DFT of
the order ||D + || 3 or as before c 3 ||κ|| 3 . The sizes of the introduced errors are summarized in Table 1-6.
Table 1-6. Comparison of the errors introduced in the DSM energy model for
HF and DFT respectively.
D
1 Idempotency error DSD − D
2 Neglected term
3 Truncation of ( )
4 Approximation of
( )
error in HF
error in DFT
( 2 4
O c κ )
2 4
O ( c κ )
1 T [2]
D
2 δ
E0
D
2 4
2 4
δ O ( c κ ) O ( c κ )
E D 0 3 3
O ( c κ )
2
E0 D +
0 3 3
O ( c κ )
Depending on the sizes of c and ||κ|| respectively, the error for DFT will be of same or lower order
than the one for HF. To inspect whether or not the DSM energy is a poorer model for DFT than for
HF, a number of calculations have been carried out, and the sizes of ||D δ || and ||D + || for the DSM
step in each iteration are examined. Since D δ is of the order c||κ|| 2 and D + is of the order c||κ||, the
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