Eigensolver

Name CGAdditionalTerms
Section SCF::Eigensolver
Type logical
Default no

Used by the cg solver only. Add additional terms during the line minimization, see PTA92, eq. 5.31ff. These terms can improve convergence for some systems, but they are quite costly. If you experience convergence problems, you might try out this option. This feature is still experimental.

Name CGDirection
Section SCF::Eigensolver
Type integer

Used by the cg solver only. The conjugate direction is updated using a certain coefficient to the previous direction. This coeffiction can be computed in different ways. The default is to use Fletcher-Reeves (FR), an alternative is Polak-Ribiere (PR).

Options:

• fletcher:
  The coefficient for Fletcher-Reeves consists of the current norm of the steepest descent vector divided by that of the previous iteration.
  
• polak:
  For the Polak-Ribiere scheme, a product of the current with the previous steepest descent vector is subtracted in the nominator.
  

Name CGEnergyChangeThreshold
Section SCF::Eigensolver
Type float
Default 0.1

Used by the cg solver only. For each band, the CG iterations are stopped when the change in energy is smaller than the change in the first iteration multiplied by this factor. This limits the number of CG iterations for each band, while still showing good convergence for the SCF cycle. The criterion is discussed in Sec. V.B.6 of Payne et al. (1992), Rev. Mod. Phys. 64, 4. The default value is 0.1, which is usually a good choice for LDA and GGA potentials. If you are solving the OEP equation, you might want to set this value to 1e-3 or smaller. In general, smaller values might help if you experience convergence problems. For very small convergence tolerances, choose 0 to disable this criterion.

Name CGOrthogonalizeAll
Section SCF::Eigensolver
Type logical
Default no

Used by the cg solver only. During the cg iterations, the current band can be orthogonalized against all other bands or only against the lower bands. Orthogonalizing against all other bands can improve convergence properties, whereas orthogonalizing against lower bands needs less operations. Moreover, orthogonalizing against all bands can make converging the highest band or unoccupied bands more difficult.

Name ChebyshevFilterBoundMixing
Section SCF::Eigensolver
Type float
Default 0.5

In the first application of the filter, for the first SCF step, the initial lower bound estimate is defined as a linear combination of the smallest and largest eigenvalues of the Hamiltonian. The bound mixing determines the proportion of linear mixing, beta: $b_{lower} = beta min{\lambda} + (\beta - 1) max{\lambda}$ of the smallest and largest eigenvalues.

Name ChebyshevFilterDegree
Section SCF::Eigensolver
Type integer
Default 25

Used by the Chebyshev filter only. The degree of the Chebyshev polynomial used to dampen the interval of eigenstates one is not interested in. If used in conjunction with "OptimizeChebyshevFilterDegree", which is the default, "ChebyshevFilterDegree" defines the the maximum Chebyshev polynomial degree Octopus will consider when determining an approximate, optimal degree.

If not used in conjunction with "OptimizeChebyshevFilterDegree", this defines the polynomial degree used for all SCF steps. The larger the degree, the larger the separation between the subspaces, which will reduce the number of SCF iterations required to reach convergence. However, ChebyshevFilterDegree also directly correlates with the number of matrix-vector products performed on the Hamiltonian per SCF step, and so larger values result in slower SCF cycles. A value in the range 8 <= ChebyshevFilterDegree <= 20 is reasonable.

Name ChebyshevFilterLanczosOrder
Section SCF::Eigensolver
Type integer
Default 5

Used by the Chebyshev filter only. The number of Lanczos iterations used to construct the tridiagonal matrix, from which the upper bound of H is estimated. A value in the range 4 <= ChebyshevFilterLanczosOrder <= 10 is reasonable. Values greater than 10 will raise an assertion.

Name ChebyshevFilterNIter
Section SCF::Eigensolver
Type integer
Default 5

The max number of iterations in the first SCF step of the Chebyshev solver. In practice this does not need to exceed 10, as the eigenstates will vary alot between the first and second SCF steps.

Name Eigensolver
Section SCF::Eigensolver
Type integer

Which eigensolver to use to obtain the lowest eigenvalues and eigenfunctions of the Kohn-Sham Hamiltonian. The default is conjugate gradients (cg), except that when parallelization in states is enabled, the default is rmmdiis.

Options:

• cg:
  Conjugate-gradients algorithm.
  
• plan:
  Preconditioned Lanczos scheme. Ref: Y. Saad, A. Stathopoulos, J. Chelikowsky, K. Wu and S. Ogut, "Solution of Large Eigenvalue Problems in Electronic Structure Calculations", BIT 36, 1 (1996).
  
• evolution:
  (Experimental) Propagation in imaginary time.
  
• rmmdiis:
  Residual minimization scheme, direct inversion in the iterative subspace eigensolver, based on the implementation of Kresse and Furthmüller [Phys. Rev. B 54, 11169 (1996)]. This eigensolver requires almost no orthogonalization so it can be considerably faster than the other options for large systems. To improve its performance a large number of ExtraStates are required (around 10-20% of the number of occupied states). Note: with unocc, you will need to stop the calculation by hand, since the highest states will probably never converge. Usage with more than one block of states per node is experimental, unfortunately.
  
• chebyshev_filter:
  A Chebyshev-filtered subspace iteration method, which avoids most of the explicit computation of eigenvectors and results in a significant speedup over iterative diagonalization methods. This method may be viewed as an approach to solve the original nonlinear Kohn-Sham equation by a nonlinear subspace iteration technique, without emphasizing the intermediate linearized Kohn-Sham eigenvalue problems. For further details, see [Zhou et. al.](http://dx.doi.org/10.1016/j.jcp.2014.06.056)
  

Name EigensolverImaginaryTime
Section SCF::Eigensolver
Type float
Default 0.1

The imaginary-time step that is used in the imaginary-time evolution method (Eigensolver = evolution) to obtain the lowest eigenvalues/eigenvectors. It must satisfy EigensolverImaginaryTime > 0. Increasing this value can make the propagation faster, but could lead to unstable propagations.

Name EigensolverMaxIter
Section SCF::Eigensolver
Type integer

Determines the maximum number of iterations that the eigensolver will perform if the desired tolerance is not achieved. The default is 25 iterations for all eigensolvers except for rmdiis, which performs only 5 iterations. Increasing this value for rmdiis increases the convergence speed, at the cost of an increased memory footprint.

In the case of imaginary time propatation, this variable controls the number of iterations for which the Hxc potential is frozen. Default is 1 for the imaginary time evolution.

Name EigensolverMinimizationIter
Section SCF::Eigensolver
Type integer
Default 0

During the first iterations, the RMMDIIS eigensolver requires some steepest-descent minimizations to improve convergence. This variable determines the number of those minimizations.

Name EigensolverTolerance
Section SCF::Eigensolver
Type float

This is the tolerance for the eigenvectors. The default is 1e-7.

Name OptimizeChebyshevFilterDegree
Section SCF::Eigensolver
Type logical
Default yes

Used by the Chebyshev filter only. Octopus generates a best-estimate for the degree of the Chebyshev polynomial used to filter the subspace. A separate estimate is generated for each state block, per SCF iteration. Note that if running parallelism over states, the block/batch containing the largest eigenstates will converge more slowly and will therefore use a larger degree relative to all other batches. One should therefore avoid setting "ChebyshevFilterDegree" to an excessive value (for example > 50). For more details regarding how the degree is estimated, one can refer to Section 5.4 in "Computer Physics Communications 187 (2015) 98–105" (http://dx.doi.org/10.1016/j.cpc.2014.10.015).

Name Preconditioner
Section SCF::Eigensolver
Type integer

Which preconditioner to use in order to solve the Kohn-Sham equations or the linear-response equations. The default is pre_filter, except for curvilinear coordinates, where no preconditioner is applied by default.

Options:

• no:
  Do not apply preconditioner.
  
• pre_filter:
  Filter preconditioner.
  
• pre_jacobi:
  Jacobi preconditioner. Only the local part of the pseudopotential is used. Not very helpful.
  
• pre_poisson:
  Uses the full Laplacian as preconditioner. The inverse is calculated through the solution of the Poisson equation. This is, of course, very slow.
  
• pre_multigrid:
  Multigrid preconditioner.
  

Name PreconditionerFilterFactor
Section SCF::Eigensolver
Type float

This variable controls how much filter preconditioner is applied. A value of 1.0 means no preconditioning, 0.5 is the standard.

The default is 0.5, except for periodic systems where the default is 0.6.

If you observe that the first eigenvectors are not converging properly, especially for periodic systems, you should increment this value.

The allowed range for this parameter is between 0.5 and 1.0. For other values, the SCF may converge to wrong results.

Name PreconditionerIterationsMiddle
Section SCF::Eigensolver
Type integer

This variable is the number of smoothing iterations on the coarsest grid for the multigrid preconditioner. The default is 1.

Name PreconditionerIterationsPost
Section SCF::Eigensolver
Type integer

This variable is the number of post-smoothing iterations for the multigrid preconditioner. The default is 2.

Name PreconditionerIterationsPre
Section SCF::Eigensolver
Type integer

This variable is the number of pre-smoothing iterations for the multigrid preconditioner. The default is 1.

Name StatesOrthogonalization
Section SCF::Eigensolver
Type integer

The full orthogonalization method used by some eigensolvers. The default is cholesky_serial, except with state parallelization, the default is cholesky_parallel.

Options:

• cholesky_serial:
  Cholesky decomposition implemented using BLAS/LAPACK. Can be used with domain parallelization but not state parallelization.
  
• cholesky_parallel:
  Cholesky decomposition implemented using ScaLAPACK. Compatible with states parallelization.
  
• cgs:
  Classical Gram-Schmidt (CGS) orthogonalization. Can be used with domain parallelization but not state parallelization. The algorithm is defined in Giraud et al., Computers and Mathematics with Applications 50, 1069 (2005).
  
• mgs:
  Modified Gram-Schmidt (MGS) orthogonalization. Can be used with domain parallelization but not state parallelization. The algorithm is defined in Giraud et al., Computers and Mathematics with Applications 50, 1069 (2005).
  
• drcgs:
  Classical Gram-Schmidt orthogonalization with double-step reorthogonalization. Can be used with domain parallelization but not state parallelization. The algorithm is taken from Giraud et al., Computers and Mathematics with Applications 50, 1069 (2005). According to this reference, this is much more precise than CGS or MGS algorithms. The MGS version seems not to improve much the stability and would require more communications over the domains.
  

Name SubspaceDiagonalization
Section SCF::Eigensolver
Type integer
Default standard

Selects the method to perform subspace diagonalization. The default is standard, unless states parallelization is used, when the default is scalapack. Note that this variable is not parsed in the case of the evolution eigensolver.

Options:

• none:
  No subspace diagonalization. WARNING: this will generally give incorrect results.
  
• standard:
  The standard routine. Can be used with domain parallelization but not state parallelization.
  
• scalapack:
  State-parallelized version using ScaLAPACK. (Requires that Octopus was compiled with ScaLAPACK support.)