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074103-12 Thøgersen et al. J. Chem. Phys. 123, 074103 2005
and accept a favorable configuration shift. A configuration
shift may be recognized when an a orb min profile has an
abrupt change where on the right-hand side a orb min is close to 1
and on the left-hand side a orb min is close to 0. To maintain the
high degree of control characteristic of the TRSCF method,
the energy of the new configuration is checked before the
shift is accepted, at the cost of an additional Kohn–Sham
matrix build. As seen from Fig. 4a, this check is well worth
the effort, saving more than ten iterations, and thus it is made
an integrated part of our TRSCF implementation.
VI. THE DIIS METHOD VIEWED
AS A QUASI-NEWTON METHOD
Since its introduction by Pulay in 1980, the DIIS method
has been extensively and successfully used to accelerate the
convergence of SCF optimizations. We here present a rederivation
of the DIIS method to demonstrate that, in the iterative
subspace of density matrices, it is equivalent to a quasi-
Newton method. From this observation, we conclude that, in
the local region of the SCF optimization, the DIIS steps can
be used safely and will lead to fast convergence. The convergence
of the DIIS algorithm in the global region is also
discussed and is much more unpredictable.
We assume that, in the course of the SCF optimization,
we have determined a set of n+1 AO density matrices
D 0 ,D 1 ,D 2 ,...,D n and the associated Kohn–Sham or Fock
matrices FD 0 ,FD 1 ,FD 2 ,...,FD n . Since the electronic
gradient gD is given by 11
gD =4SDFD − FDDS,
79
we also have available the corresponding gradients
gD 0 ,gD 1 ,gD 2 ,...,gD n . We now wish to determine a
corrected density matrix,
n
D¯ = D 0 + c i D i0 , D i0 = D i − D 0 , 80
i=1
that minimizes the norm of the gradient gD¯ . For this purpose,
we parameterize the density matrix in terms of an antisymmetric
matrix X=−X T and the current density matrix
D 0 as 11
DX = exp− XSD 0 expSX.
81
With each old density matrix D i , we now associate an antisymmetric
matrix X i such that
D i = exp− X i SD 0 expSX i = D 0 + D 0 ,X i S + OX 2 i .
82
Introducing the averaged antisymmetric matrix,
n
X¯ = c i X i ,
i=1
we obtain
83
n
DX¯ = D 0 + c i D 0 ,X i S + OX¯ 2 ,
i=1
84
where we have used the S-commutator expansion of DX¯
analogeous to Eq. 82. Our task is hence to determine X¯ in
Eq. 83 such that DX¯ minimizes the gradient norm
gDX¯ . In passing, we note that, whereas D¯ is not in
general idempotent and therefore not a valid density matrix,
DX¯ is a valid, idempotent density matrix for all choices of
c i .
Expanding the gradient in Eq. 79 about the currentdensity
matrix D 0 , we obtain
gDX¯ = gD 0 + HD 0 X¯ + OX¯ 2 ,
85
where HD is the Jacobian matrix. Neglecting the higherorder
terms, our task is therefore to minimize the norm of the
gradient,
n
gc = gD 0 + c i HD 0 X i ,
86
i=1
with respect to the elements of c. For an estimate of
HD 0 X i , we truncate the expansion,
gD i = gD 0 + HD 0 X i + OX i 2 ,
and obtain the quasi-Newton condition,
gD i − gD 0 = HD 0 X i .
Inserting this condition into Eq. 86, we obtain
n
gc = gD 0 +
i=1
n
c i gD i − gD 0 = c i gD i ,
i=0
87
88
89
where we have introduced the parameter c 0 =1− n
i=1 c i . The
minimization of gc=gc may therefore be carried out as
a least-squares minimization of gc in Eq. 89 subject to the
constraint
n
c i =1.
90
i=0
If we consider gD i as an error vector for the density matrix
D i , this procedure becomes identical to the DIIS method.
From Eq. 86 we also see that DIIS may be viewed as a
minimization of the residual for the Newton equation in the
subspace of the density matrix differences D i −D 0 , i=1, n,
where the quasi-Newton condition is used to set up the subspace
equations. Since the quasi-Newton steps are reliable
only in the local region of the optimization, we conclude that
the DIIS method can be used safely only in this region, when
the electronic Hessian is positive definite.
The optimal combination of the density matrices is obtained
in the DIIS method, by carrying out a least-squares
minimization of the gradient norm subject to the constraint in
Eq. 90. However, since a small gradient norm in the global
region does not necessarily imply a low Kohn–Sham energy,
the DIIS convergence may be unpredictable. Furthermore,
we may encounter regions where the gradient norms are
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