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Part 1
Improving Self-consistent Field Convergence
D
+
=
n
∑
i=
1
c D
i0
D = D −D
i0 i 0 .
i
(1.46)
Unlike the symmetry and trace conditions in Eq. (1.2), the idempotency condition is in general not
fulfilled for linear combinations of D i . Still, for any averaged density matrix D in Eq. (1.45) that
does not fulfill the idempotency condition, we may generate a purified density matrix with a smaller
idempotency error by the transformation 8
D = 3DSD−2DSDSD. (1.47)
Introducing the idempotency correction
Dδ = D − D, (1.48)
we may then write the purified averaged density matrix in the form
D = D + D + D . (1.49)
0 + δ
1.4.2.2 The Trust Region DSM Energy Function
Having established a useful parameterization of the averaged density matrix Eq. (1.45) and having
considered its purification Eq. (1.47), let us now consider how to determine the best set of
coefficients c i . Expanding the energy in the purified averaged density matrix, Eq. (1.49), around the
reference density matrix D 0 , we obtain to second order
T
( ) ( ) ( ) (1) 1
T
D = D + D+ + D E + ( D+ + D ) E (2) ( D+
+ D )
E E δ δ δ . (1.50)
SCF(2) SCF 0 0 2
0
To evaluate the terms containing
(1)
E
0
and
(2)
E
0
we make the identifications
(1)
0
= 2 0
2 2
0 + = 2 + + +
E F (1.51)
( )
( )
E D F O D , (1.52)
which follow from Eq. (1.4) and from the second-order Taylor expansion of about D 0 . The
n
notation Eq. (1.46) has now been generalized to the Fock matrix F+ = ∑ c
i=
1 iF i0
. Ignoring the
terms quadratic in D δ in Eq. (1.50) and quadratic in D + in Eq. (1.52), we then obtain the DSM
energy
DSM
E () = ESCF ( 0 ) + 2Tr + 0 + Tr + + + 2Tr δ 0 + 2Tr δ +
(1)
E0
c D DF DF DF DF. (1.53)
Finally, for a more compact notation, we introduce the weighted Fock matrix
n
0 + 0 ci
i0
i=
1
and find that the DSM energy may be written in the form
F = F + F = F +∑ F , (1.54)
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