Results of the DeltaCodes GGA comparison exercise

# FP Verification

**Results of the DeltaCodes GGA comparison exercise**

The DeltaCodes project is an ongoing effort to test the mutual consistency of different implementations (code projects) of density-functional-theory; the first major success of the project has been the formulation of a standardised test set, consisting of 71 elemental crystals, together with the definition of a simple parameter for assessing the similarity of the results of different codes. These tests compare the equilibrium volume and bulk modulii of the different crystals as calculated by different codes using the popular PBE generalised gradient approximation. A large number of codes have participated in the verification exercise, representing a considerable community engagement, and highlighting the growing importance of accuracy and reliability in the use of DFT. This large community effort is coordinated by Kurt Lejaeghere and Stefaan Cottenier DeltaCodes at Uni Ghent and details of the results may be found in the recent Science article.

The following presents the results of the full-potential **lmf** code for the DeltaCodes exersise. There are two data sets corresponding to the use of an LMTO+LO basis and a plane-wave plus LMTO basis (PMT). The LMTO basis is highly suited for the treatment of dense solids, but is less satisfactory for highly open structures such as molecular crystals where the atom-centred basis functions are not adequate for describing the wavefunction in the large interstitial region. Addition of plane waves to the basis fulfils this requirement very satisfactorily, such that the full-potential **lmf** code can be considered a completely general purpose methodology, suitable for all geometries and structures.

### LMTO results

##### Procedure

####### Basis The basis is the “Large” basis, requested using the HAM_AUTOBAS_LMTO=5, and with LMX given by one larger than the highest occupied $l$-channel. This basis consists of two smoothed Hankel functions per $l,m$ for $l$ upto LMX. Default basis Hankel energies suitable for LDA calculations, EH=-0.1, -0.9Ryd, are used, and the RSMH, the Hankel smoothing radius is automatically generated in the atomic case (code **lmfa**) and used without further optimisation. GMAX, which specifies the smooth mesh part of the potential representation that extends throughout the cell, is chosen such that the total energy is converged to better than 1e-5Rydberg.

Local orbitals play an important role in improving the default LMTO basis both by allowing semi-core states to be included in the valence and, with the inclusion of high-lying local orbitals, they can also be used to improve the quality of the augmentation. Extended local orbitals are automatically assigned to atomic states lying within 2Ryd of the Fermi level. High-lying local orbitals are additionally specified for the d-channel of the transition element block.

####### Augmentation Although the radius for the augmentation plays an important role in the LMTO method, the full potential method is to a high degree insensitive to small variations in the choice of $r_{mt}$. Because the basis is superior within the augmentation region, compared to the interstitial, it is advantagous to make the augmentation spheres as large as possible, and for a given geometry, the maximum radius, which corresponds to touching spheres, should be used. For studying variations in the unit cell volume, a choice exists between having the maximum augmentation radii for all volumes, or to choose a volume with sufficiently small overlap and to use this fixed radius for all volumes tested. This second method has the advantage that the quality of the augmentation is essentially constant for all volumes, although the fraction of volume in the interstitial is increased for dilated volumes. For the full-potential code, both choices satisfactory results.

For increased precision of the radial solver (Schrödinger or Dirac), the radial grid parameter A is decreased from its default value and set at A=0.01; further changes in the radial gridding generally result in total energy changes below the 10microRyd level. ####### Calculation For increased precision, the parameters HAM_TOL and EWALD_TOL are tightened beyond their default tolerances and set to the value 1e-16.

##### Comments

### PMT results

##### Procedure

####### Basis The LMTO basis, with local orbitals, is highly accurate for most crystals but is less suitable for open, molecular systems. The addition of plane-waves is a very effective way of increasing the flexibility of the basis set, and molecular problems can be very satisfactorily addressed with the addition of even a small number of plane-waves in addition to the LMTO Hankel functions.

####### Augmentation As the APW+LMTO basis becomes larger, issues with linear dependency can arrise and numerical stability is improved by a reduction of the augmentation radius. For the tests, the same LMTO basis as above is complemented by an APW basis with PWEMAX=3.0Ryd and the augmentation radius is reduced to 90% of the touching sphere value for each volume.

*Comparison with other codes*

The figure shows the DeltaValues of the PMT dataset compared with reference data from the DeltaCodes project; the level of mutual agreement between **lmf** and the other all-electron codes is excellent, demonstrating essentially identical accuracy between the different all-electron full-potential schemes.

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