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Convergence for Problems with Several Stationary Points
no optimization method can make that a guarantee. Furthermore, it cannot be tested if the minimum
found is a local or the global minimum without knowledge of the whole surface. Depending on the
start guess and the optimization approach, an optimization can converge to different stationary
points. Further, it is necessary to decide in which subspace of orbital rotations the desired solution
should be found, since a solution representing a stable stationary point in one subspace is not
necessarily stable in another.
Orbital rotations can be divided in real and complex rotations and each of those can be further
divided in singlet and triplet rotations. Each of those can then again be divided in rotations within
the different point group symmetries. Generally, we do not consider the complex rotations, and we
only optimize in the real space. Further, when optimizing a closed shell wave function, only the
total-symmetric part of the singlet rotations is considered. A stationary point in the subspace of real,
total-symmetric, singlet rotations can be shown through elementary arguments to be a stationary
point for all types of rotations. However, a stationary point can both be a maximum, a saddle point
or a minimum. A way to realize if the stationary point also is a minimum is to evaluate the Hessian
eigenvalues. This is done within the subspace in which the solution should be stable. If a negative
Hessian eigenvalue is found in the subspace of singlet rotations, the stationary point is said to have
a singlet instability and if a negative Hessian eigenvalue is found in the subspace of triplet rotations,
it is said to have a triplet instability 54,56 . Triplet instabilities are connected to breaking the symmetry
between α and β orbitals. If a triplet instability is found, a minimum with a lower energy than the
current stationary point can be found, if the α and β parts are allowed to differ, typically leading to
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a solution which is not an eigenfunction of Ŝ . Hence, the lower minimum could be found by an
unrestricted HF (UHF) optimization. A singlet instability found in the total-symmetric subspace
indicates that the current stationary point is a saddle point and a minimum with lower energy exists
within the subspace. If a singlet instability is found outside the total-symmetric subspace, orbitals of
different symmetries should be mixed to decrease the energy further, changing the symmetry of the
resulting wave function.
The aufbau ordering rule assumes that occupying the orbitals of lowest energy also leads to the
lowest Hartree-Fock energy. This cannot be proven to always apply for restricted HF as it can for
UHF 57 . Thus it is a risk when the aufbau ordering is forced upon an optimization, that a lower
energy with the aufbau ordering broken could exist. However in a study by Dardenne et. al. 58 , in
which different ordering schemes were tested, they found in all cases that the minimum was an
aufbau solution. The aufbau ordering was broken only for saddle points. In our schemes we always
apply the aufbau ordering rule, but if the RH step is level shifted to the end of the optimization, it
can force the convergence to a non-aufbau solution.
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