SCF Convergence Guidelines for ADF
**********************************

The self-consistent field (SCF) method is the standard algorithm for
finding electronic structure configurations within Hartree-Fock and
density functional theory. SCF is an iterative procedure and can,
depending on the situation at hand, sometimes be difficult to
converge. Indeed, convergence problems occur in many different types
of chemical systems. These problems are most frequently encountered
when the electronic structure exhibits a very small HOMO-LUMO gap, in
systems with d- and f-elements with localized open-shell
configurations, and in transition state structures with dissociating
bonds. Finally, many SCF convergence problems are rooted in
nonphysical calculation setups, such as high-energy geometries or an
inappropriate description of the electronic structure.

We advise you to check and test the following list of options and
aspects when encountering SCF convergence issues, starting from the
most common and trivial ones:

1. Ensure that the atomistic system under study is realistic. More
   specifically, check for the proper values of bond lengths, angles,
   and other internal degrees of freedom in the geometry. Unless
   specified otherwise, AMS expects atomic coordinates in Å. Also,
   when copying atomic structures into the graphical user interface,
   check that the imported structure is complete and that no atoms got
   lost during the process.

2. The initial electronic structure is typically initialized as linear
   combinations of atomic configurations. However, a moderately (but
   not fully) converged electronic structure from, say, an SCF
   iteration conducted previously, likely represents a better initial
   guess already. Indeed, in the next step of a geometry optimization,
   this moderately converged electronic structure information is
   reused as initial guess. The SCF iterations of the following
   geometry steps then usually converge even if the first electronic
   structure calculation does not. In the case of single-point
   calculations, the electronic structure needs to be read in via a
   manual restart.

3. Assert that the correct spin multiplicity of the system is used.
   Open-shell configurations should be computed in a spin-unrestricted
   or, if necessary, a spin-orbit coupling formalism. You need to
   manually set the spin component (see the tutorial on spin coupling
   within an iron complex for more details). For non-converging open-
   shell systems, the evolution of the SCF errors during the iteration
   might also provide some insight into the problem. Strongly
   fluctuating errors may indicate an electronic configuration far
   away from any stationary point or an improper description of the
   electronic structure by the approximation used.

4. Change to a different SCF convergence acceleration method, like
MESA, LISTi, or EDIIS. To this end, examine the documentation section
for further details about the available SCF acceleration methods and
the parameters controlling them. In the graphical user interface these
options are available under **Details → SCF** and **Details → SCF
Convergence Details**. The performance of these methods was tested for
a variety of chemical systems that are difficult to converge. The
corresponding results are depicted in Fig. 1 below and show that
significant changes in the convergence behavior can be achieved with
different accelerators.

   [image]Number of SCF iterations needed to converge a series of
   different types of chemical systems with the ADIIS+SDIIS (default),
   MESA, LISTi, and EDIIS accelerator methods. Non-converged SCF
   iterations are depicted by dashed borders.

5. The Augmented Roothaan-Hall (ARH) method is another alternative,
   though computationally more expensive, convergence acceleration
   method. ARH directly minimizes the system's total energy as a
   function of the density matrix using a preconditioned conjugate-
   gradient method with a trust-radius approach. As shown in Fig. 2
   below, ARH can in some situations be a viable alternative for
   difficult systems.

   [image]Number of SCF iterations needed to converge a series of
   different types of chemical systems with the ADIIS+SDIIS (default)
   and ARH accelerators. Non-converged SCF iterations are depicted by
   dashed borders.

6. Some parameters of the DIIS algorithm may also be changed manually.
   Note that sometimes it is necessary to try multiple values and
   combinations of values until the desired outcome is achieved:

* **Mixing** denotes the fraction of the computed Fock matrix that is
  added when constructing the next guess for this matrix. More
  specifically, the Fock matrix resulting from the current guess for
  the electron density is combined with the corresponding matrices
  from multiple previous SCF iteration steps by the DIIS algorithm in
  order to construct the next guess. The mixing parameter controls the
  proportion of the computed Fock matrix in this linear combination,
  whereas a value higher than the default value of 0.2 corresponds to
  a more aggressive acceleration, while a lower value will lead to a
  more stable iteration and should be used for problematic cases.

* **Mixing1** corresponds to the mixing parameter used in the very
  first SCF cycle. Its default of 0.2 should mainly be altered only in
  situations where one attempts to slowly converge the electronic
  structure starting from a restart file (see above).

* **N** is the number of DIIS expansion vectors used for the SCF
  acceleration (default N=10). An input value smaller than 2 disables
  DIIS. A higher number of expansion vectors (e.g. up to N=25) makes
  the SCF iteration more stable, while a smaller number makes it more
  aggressive.

* **Cyc** is the number of initial SCF iteration steps after which
  SDIIS will start (default Cyc=5). An initial equilibration will take
  place in the cycles before; a higher value of Cyc causes a more
  stable computation.

* As an example, the following parameter values can be used as a
  starting point for a slow but steady SCF iteration of a difficult
  system:

   SCF
     DIIS
       N 25
       Cyc 30
     End
     Mixing 0.015
     Mixing1 0.09
   End

Besides the methods discussed above, other algorithms and techniques
can be used to converge a problematic SCF calculation. However, as
opposed to the aforementioned options, the techniques listed below
slightly alter the end result, which needs to be carefully tested:

7. Electron smearing simulates a finite electron temperature by using
   fractional occupation numbers to distribute electrons over multiple
   electronic levels. This is particularly helpful to overcome
   convergence issues in larger systems exhibiting many near-
   degenerate levels. As electron smearing alters the system's total
   energy, the value of this parameter should be kept as low as
   possible, e.g. by using multiple restarts with successively smaller
   smearing values.

8. The level shifting technique artificially raises the energy of
   unoccupied (virtual) electronic levels and can be used to overcome
   SCF convergence problems as well. It will, however, give incorrect
   values for properties involving virtual levels, such as excitation
   energies, response properties, and NMR shifts. Likewise, the
   electronic structure of metallic systems with a vanishing HOMO-LUMO
   gap might be inadequately described by this technique.
