Converging Parameters

There are various important parameters in CONQUEST that affect the convergence of the total energy, and need to be tested. Integrals are calculated on a grid; the density matrix is found approximately; a self-consistent charge density is calculated; and support functions are, in some modes of operations, optimised. These parameters are described here.

Integration Grid

While many integrals are calculated analytically or on fine grids that move with the atoms, there are still some integrals that must be found numerically, and CONQUEST uses an orthorhombic, uniform grid to evaluate these integrals (this grid is also used for the Fourier transforms involved in finding the Hartree potential). The spacing of the grid will affect the accuracy of the calculation, and it is important to test the convergence of the total energy with the grid spacing.

The grid spacing can be set intuitively using an energy (which corresponds to the kinetic energy of the shortest wavelength wave that can be represented on the grid). In atomic units, \(E = k^2/2\) with \(k = \pi/\delta\) for grid spacing \(\delta\). The cutoff is set with the parameter:

Grid.GridCutoff E

where E is an energy in Hartrees. The grid spacing can also be set manually, by specifying the number of grid points in each direction:

Grid.PointsAlongX N
Grid.PointsAlongY N
Grid.PointsAlongZ N

If setting the grid in this manner, it is important to understand a little more about the internal workings of CONQUEST. The grid is divided up into blocks (the default size is 4 by 4 by 4), and the number of grid points in any direction must correspond to an integer multiple of the block size in that direction. The block size can be set by the user:

Grid.InBlockX N
Grid.InBlockY N
Grid.InBlockZ N

Note that the blocks play a role in parallelisation and memory use, so that large blocks may require larger memory per process; we recommend block sizes no larger than 8 grid points in each direction. There is also, at present, a restriction on the total number of grid points in anuy direction, that it must have prime factors of only 2, 3 and 5. This will be removed in a future release.

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Finding the density matrix

As discussed in the section on finding the ground state, the density matrix is found either with exact diagonaliation, or the linear scaling approach. These two methods require different convergence tests, and are described separately.

Diagonalisation: Brillouin Zone Sampling

The sampling of the Brillouin zone must be tested for convergence, and the parameters are described here. The convergence of charge density will be faster than detailed electronic structure such as density of states (DOS), and it will be more accurate for these types of calculations to generate a converged charge density, and then run non self-consistently (see the section on self consistency) with appropriate k-point sampling.

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Linear Scaling

The range applied to the density matrix (DM.L_range) determines the accuracy of the calculation, as well as the computational time required (as the number of non-zero elements will increase based on a sphere with the radius of the range, the time will increase roughly proportional to the cube of the range). In almost all circumstances, it is best to operate with a range which converges energy differences and forces, rather than the absolute energy. Testing for this convergence is an essential part of the preparation for production calculations.

The tolerance applied to the density matrix optimisation (minE.Ltolerance) must be chosen to give adequate convergence of the energy and forces. The tolerance is applied to the residual in the calculation, defined as:

\[R = \sqrt{\sum_{i\alpha j\beta} \partial E/\partial L_{i\alpha j\beta} \cdot \partial E/\partial L_{i\alpha j\beta} }\]

The dot product uses the inverse of the overlap matrix as the metric.

The approximate, sparse inversion of the overlap matrix is performed before the optimisation of the density matrix. The method used, Hotelling’s method (a version of a Newton-Raphson approach) is iterative and terminates when the characteristic quantity \(\Omega\) increases. On termination, if \(\Omega\) is below the tolerance DM.InvSTolerance then the inverse is accepted; otherwise it is set to the identity (the density matrix optimisation will proceed in this case, but is likely to be inefficient). We define:

\[\Omega = (Tr[I - TS])^2\]

where \(T\) is the approximate inverse. The range for the inverse must be chosen (Atom.InvSRange in the species block); by default it is same as the support function range (which is then doubled to give the matrix range) but can be increased. The behaviour of the inversion with range is not simple, and must be carefully characterised if necessary.

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The standard self-consistency approach uses the Pulay RMM method, and should be robust in most cases. It can be monitored via the residual, which is currently defined as the standard RMS difference in charge density:

\[R = \sqrt{\int \mathrm{d}\mathbf{r}\mid \rho^{out}(\mathbf{r}) - \rho^{in}(\mathbf{r})\mid^2}\]

where \(\rho^{in}\) is the input charge density for an iteration, and \(\rho^{out}\) is the resulting output charge density. The SCF cycle is terminated when this residual is less than the parameter minE.SCTolerance. The maximum number of iterations is set with SC.MaxIters (defaults to 50).

There are various further approaches and parameters which can be used if the SCF cycle is proving hard to converge. As is standard, the input for a given iteration is made by combining the charge density from a certain number of previous steps (SC.MaxPulay, default 5). The balance between input and output charge densities from these previous steps is set with SC.LinearMixingFactor (default 0.5; N.B. for spin polarised calculations, SC.LinearMixingFactor_SpinDown can be set separately). Reducing this quantity may well improve stability, but slow down the rate of convergence.

Kerker-style preconditioning (damping long wavelength charge variations) can be selected using SC.KerkerPreCondition T (this is most useful in metallic and small gap systems). The preconditioning is a weighting applied in reciprocal space:

\[K = \frac{1}{1+q^2_0/q^2}\]

where \(q_0\) is set with SC.KerkerFactor (default 0.1). This is often very helpful with slow convergence or instability.

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Support Functions

The parameters relevant to support functions depend on the basis set that is used. In the case of pseudo-atomic orbitals (PAOs), when support functions are primitive PAOs, the only relevant parameter is the basis set size, which is set when the ion files are generated. It is important to test the accuracy of a given basis set carefully for the problem that is to be modelled.

When using multi-site support functions (MSSF), the key parameter is the radius of the MSSF (Atom.MultisiteRange in the atomic specification block). As this is increased, the accuracy of the calculation will also increase, but with increased computational effort. Full details of the MSSF (and related OSSF) approach are given in the section on multi-site support functions.

For the blip basis functions, the spacing of the grid where the blips are defined is key (Atom.SupportGridSpacing in the atomic specification block), and is directly related to an equivalent plane wave cutoff (via \(k_{bg} = \pi/\delta\) and \(E_{PW} = k_{bg}^2/2\), where \(\delta\) is the grid spacing in Bohr radii and \(E_{PW}\) is in Hartrees). For a particular grid spacing, the energy will converge monotonically with support function radius (Atom.SupportFunctionRange in the atomic specification block). A small support function radius will introduce some approximation to the result, but improve computational performance. It is vital to characterise both blip grid spacing and support function radius in any calculation. A full discussion of the blip function basis is found here.

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