3. Accuracy
In this section, we introduce common parameters influencing the accuracy of a calculation. The optimal value of certain parameters can be determined using physical reasoning whereas the optimal value of other parameters must be determined by trial and error.
3.1. Grid resolution
RESCU uses uniform Cartesian grids to discretize many quantities: the local pseudopotential, the pseudopotential projectors, the Hartree potential, the semilocal part of the exchangecorrelation potential, the numerical atomic orbitals, the wavefunctions, etc. The accuracy of the numerical description of all these objects depends intrinsically on the underlying grid resolution. There are a few keywords to set the grid fineness in RESCU.
domain.lowres
A positive real number which corresponds to the distance between grid points along each dimension. This value is only a target since the actual resolution must yield an integer number of grid points along each direction. A typical value is 0.4 Bohr. This is the most transparent and preferred keyword to set the real space resolution.domain.cgridn
A 1\(\times\)3 array of integers. It tells RESCU the number of grid points along each dimension. The user must be careful to choose a value which suits the lattice vectors and pseudopotentials. This keyword has precedence overdomain.lowres
.
Prec. 
Real grids 
Planewave grids 

Low 


Medium 


High 


In gridbased calculations (LCAO.status = false
), several other
keywords control precision: domain.highres
(doublegrid technique),
interpolation.vloc
(how to evaluate the local effective potential),
interpolation.method
, domain.vnlReal
(nonlocal pseudopotential
representation), etc. Nonexperts need not worry about understanding and
mastering all aspects of the algorithms used by RESCU, and hence it
generally suffices to set domain.lowres
along with
option.precision
which can take the values listed in Table 3.1.1.
The values are quite selfexplanatory.
First, one may choose between real space and planewave grids. The former
are more computationally efficient, but less precise at a fixed grid
resolution. The latter may be competitive in smaller systems (say up to
100 atoms), but is simply too memory hungry in large systems (say more
than 1000 atoms). Next, the precision of the grids (and the underlying
computational methods) is controlled by means of low
, med
or
high
. Basically, the lower the precision, the lower the
computational and memory requirements (again, this is true at a fixed
grid resolution). To illustrate the above points, the total energy of
the \(Au_4\) system as a function of resolution for various
option.precision
values is shown in
Fig. 3.1.1.
3.2. Brillouin zone sampling
Periodic materials are usually simulated invoking Bloch’s theorem to divide the large (infinite) problem into independent subproblems (dependent on the crystal momentum) coupled via the density. For most purposes (e.g. band energy, density), averaging over the socalled first Brillouin zone is required, and hence the accuracy of the calculation depends directly on its sampling. RESCU implements an algorithm to generate MonkhorstPack grids. Such grids are \(\Gamma\)centered for odd grid counts and shifted by half for even grid counts. The main control parameters are the following
kpoint.gridn
is a 1\(\times\)3 integer array defining the number of grid points along each dimension.kpoint.shift
is also a 1\(\times\)3 array allowing the user to apply a global shift on the MonkhorstPack grid. Note that \(\Gamma\)centered even numbered grids can be obtained settingkpoint.shift
equal to [0.5 0.5 0.5].kpoint.sampling
determines the scheme used to calculate the density of states, occupancy, etc. The allowed values arefermidirac
,gauss
,methfesselpaxton
,tetrahedron
andtetrahedron+blochl
which are selfexplanatory. Whilefermidirac
andgauss
are generally appropriate for \(\Gamma\)point calculations,methfesselpaxton
is suitable for metals and other systems with a nontrivial density of states at the Fermi energy. The tetrahedron method is applicable as long as there are more than four kpoints and it is implemented following Blöchl’s method [BJA94]. Ifkpoint.sampling
is equal totetrahedron+blochl
, second order corrections to the Fermi energy are included in the calculation of the occupancies.kpoint.sigma
determines the width of the density of states’ shape functions (the shape of a single energy level). This is only pertinent ifkpoint.sampling
is equal tofermidirac
,gauss
ormethfesselpaxton
. Convergence may be unattainable if this value is too small (the default value is 0.001 Hartree). The units can be controlled usingunits.kpoint.sigma
.kpoint.mpord
sets the order of the MethfesselPaxton shape functions.
3.3. Pseudopotentials
The fidelity of the pseudopotential depends directly on the
completeness of the projectors of the nonlocal component. By default,
RESCU will use all projectors found in a pseudopotential file.
Nevertheless, the principal quantum number and orbital angular momenta
can be controlled using the keywords element.vnlNcutoff
and
element.vnlLcutoff
respectively.
3.4. Atomic orbital basis
The Nanoacademic pseudopotential database files generally include a set
of polarized doublezeta orbitals generated following the SIESTA
method [SAGG02]. When doing numerical atomic
orbital (NAO) calculations, it is crucial to have a basis which
approximately spans the KohnSham occupied subspace and the groundstate
density. By default, RESCU will use all orbitals found in a
pseudopotential file. However, it is possible to constrain the atomic
basis of an element (e.g. to check convergence in the basis size) using
the keywords element.aoLcutoff
and element.aoZcutoff
. Those
limit the orbital angular momentum and \(\zeta\)number of the basis
functions respectively.
Peter E. Blöchl, O. Jepsen, and O. K. Andersen. Improved tetrahedron method for Brillouinzone integrations. Phys. Rev. B 49 (23 June 1994), pp. 16223–16233.
José M Soler, Emilio Artacho, Julian D Gale, Alberto Garcı́a, Javier Junquera, Pablo Ordejón, and Daniel SánchezPortal. The SIESTA method for ab initio order N materials simulation. Journal of Physics: Condensed Matter 14.11 (2002), p. 2745.