# 4.5.4 Geometric Direct Minimization (GDM)

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 $10^{-\mbox{{\small THRESH\_DIIS\_SWITCH}}}$ when SCF_ALGORITHM is DIIS_GDM or DIIS_DM. See also MAX_DIIS_CYCLES TYPE: INTEGER DEFAULT: 2 OPTIONS: User-defined. RECOMMENDATION: None Example 4.8 Input for a UHF calculation using geometric direct minimization (GDM) on the phenyl radical, after initial iterations with DIIS. $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