Troy Van Voorhis, working at Berkeley with Martin Head-Gordon, has developed a novel direct minimization method that is extremely robust, and at the same time is only slightly less efficient than DIIS. This method is called geometric direct minimization (GDM) because it takes steps in an orbital rotation space that correspond properly to the hyper-spherical geometry of that space. In other words, rotations are variables that describe a space which is curved like a many-dimensional sphere. Just like the optimum flight paths for airplanes are not straight lines but great circles, so too are the optimum steps in orbital rotation space. GDM takes this correctly into account, which is the origin of its efficiency and its robustness. For full details, we refer the reader to Ref. 936. GDM is a good alternative to DIIS for SCF jobs that exhibit convergence difficulties with DIIS.

Recently, Barry Dunietz, also working at Berkeley with Martin Head-Gordon, has extended the GDM approach to restricted open-shell SCF calculations. Their results indicate that GDM is much more efficient than the older direct minimization method (DM).

In section 4.5.3, we discussed the fact that DIIS can efficiently
head towards the global SCF minimum in the early iterations. This can be true
even if DIIS fails to converge in later iterations. For this reason, a hybrid
scheme has been implemented which uses the DIIS minimization procedure to
achieve convergence to an intermediate cutoff threshold. Thereafter, the
geometric direct minimization algorithm is used. This scheme combines the
strengths of the two methods quite nicely: the ability of DIIS to recover from
initial guesses that may not be close to the global minimum, and the ability of
GDM to robustly converge to a local minimum, even when the local surface
topology is challenging for DIIS. This is the recommended procedure with which
to invoke GDM (*i.e.*, setting SCF_ALGORITHM = DIIS_GDM). This hybrid
procedure is also compatible with the SAD guess, while GDM itself is not,
because it requires an initial guess set of orbitals. If one wishes to disturb
the initial guess as little as possible before switching on GDM, one should
additionally specify MAX_DIIS_CYCLES = 1 to obtain only a single
Roothaan step (which also serves up a properly orthogonalized set of orbitals).

*$rem* options relevant to GDM are SCF_ALGORITHM which should be
set to either GDM or DIIS_GDM and the following:

MAX_DIIS_CYCLES

The maximum number of DIIS iterations before switching to (geometric) direct
minimization when SCF_ALGORITHM is DIIS_GDM or
DIIS_DM. See also THRESH_DIIS_SWITCH.

TYPE:

INTEGER

DEFAULT:

50

OPTIONS:

1
Only a single Roothaan step before switching to (G)DM
$n$
$n$ DIIS iterations before switching to (G)DM.

RECOMMENDATION:

None

THRESH_DIIS_SWITCH

The threshold for switching between DIIS extrapolation and direct minimization
of the SCF energy is ${10}^{-\text{THRESH\_DIIS\_SWITCH}}$ when
SCF_ALGORITHM is DIIS_GDM or DIIS_DM. See
also MAX_DIIS_CYCLES

TYPE:

INTEGER

DEFAULT:

2

OPTIONS:

User-defined.

RECOMMENDATION:

None

$molecule 0 2 c1 x1 c1 1.0 c2 c1 rc2 x1 90.0 x2 c2 1.0 c1 90.0 x1 0.0 c3 c1 rc3 x1 90.0 c2 tc3 c4 c1 rc3 x1 90.0 c2 -tc3 c5 c3 rc5 c1 ac5 x1 -90.0 c6 c4 rc5 c1 ac5 x1 90.0 h1 c2 rh1 x2 90.0 c1 180.0 h2 c3 rh2 c1 ah2 x1 90.0 h3 c4 rh2 c1 ah2 x1 -90.0 h4 c5 rh4 c3 ah4 c1 180.0 h5 c6 rh4 c4 ah4 c1 180.0 rc2 = 2.672986 rc3 = 1.354498 tc3 = 62.851505 rc5 = 1.372904 ac5 = 116.454370 rh1 = 1.085735 rh2 = 1.085342 ah2 = 122.157328 rh4 = 1.087216 ah4 = 119.523496 $end $rem BASIS = 6-31G* METHOD = hf SCF_ALGORITHM = diis_gdm SCF_CONVERGENCE = 7 THRESH = 10 $end