4.5 Converging SCF Calculations

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:

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

       The threshold for switching between DIIS extrapolation and direct minimization of the SCF energy is 10-THRESH_DIIS_SWITCH when SCF_ALGORITHM is DIIS_GDM or DIIS_DM. See also MAX_DIIS_CYCLES

Example 4.8  Input for a UHF calculation using geometric direct minimization (GDM) on the phenyl radical, after initial iterations with DIIS.

   0   2
   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

   BASIS           = 6-31G*
   METHOD          = hf
   SCF_ALGORITHM   = diis_gdm
   THRESH          = 10