# Local Resolution of Identity

## Introduction

Density functional theory (DFT) calculations in CP2K employ the Gaussian and plane waves (GPW) method. In GPW, the description of the total density on realspace grids is typically the computationally most expensive part. By introducing a local resolution-of-the-identity (LRI) approach, the linear scaling of the GPW approach can be retained, while reducing the prefactor for the grid operations. The combined approach, LRIGPW, is comprehensively described in Golze2017b.

In LRIGPW, the atomic pair densities \(\rho_{\mathrm{AB}}\) are approximated by an expansion in a set of fit functions centered at atom A \(\{f_i^{\mathrm{A}}(\mathbf{r})\}\) and atom B \(\{f_j^{\mathrm{B}}(\mathbf{r})\}\),

\[ \rho_{\mathrm{AB}}\approx \sum_i{a_i^{\mathrm{A},(\mathrm{AB})}f_i^{\mathrm{A}}}(\mathbf{r}) + \sum_j{a_j^{\mathrm{B},(\mathrm{AB})}f_j^{\mathrm{B}}}(\mathbf{r}). \]

The fit functions are also Gaussian-type functions and provided as auxiliary basis set.

## How to use it

LRIGPW is specified in the QS section by setting METHOD LRIGPW.

    &QS
      METHOD LRIGPW
      &LRIGPW
         LRI_OVERLAP_MATRIX INVERSE
         SHG_LRI_INTEGRALS
      &END
    &END QS

Further specifications can be given in the LRIGPW subsection. LRIGPW requires additionally an auxiliary basis set as input.

     &DFT
        BASIS_SET_FILE_NAME BASIS_LRIGPW_AUXMOLOPT
        BASIS_SET_FILE_NAME BASIS_MOLOPT
        ...
     &END DFT
     &SUBSYS
        &KIND O
          BASIS_SET DZVP-MOLOPT-GTH
          POTENTIAL GTH-PBE-q6
          LRI_BASIS_SET LRI-DZVP-MOLOPT-GTH-MEDIUM
        &END KIND
        ...
     &END SUBSYS

Auxiliary basis sets are available for the MOLOPT basis sets. All auxiliary basis sets have been generated by simple geometric progression without any need for further optimization. These basis sets are available in different sizes: `MEDIUM` and `LARGE`. Using the large auxiliary basis sets, the accuracy is improved, but the computational overhead increases.

The LRI auxiliary basis sets are generally quite large leading to a potentially ill-conditioned overlap matrix, Equation (10) in Golze2017b. The inversion of this matrix can thus be numerical instable.

If the SCF is not converging, set LRI_OVERLAP_MATRIX to `AUTOSELECT`. In this case, the atomic pairs are identified that have extremely large condition numbers. For these pairs, the pseudoinverse instead of the regular inverse is calculated. The threshold for the condition number can be given by MAX_CONDITION_NUM.

The LRI integrals, Equations (31)-(34) in Golze2017b, are calculated prior to the SCF. The traditionally used Obara-Saika scheme is computationally too demanding here. Therefore, a more efficient integral scheme based on solid harmonic Gaussians (SHG) is employed and invoked by SHG_LRI_INTEGRALS, see Golze2017 for details.

## When to use it

LRIGPW is only beneficial when the operations on the realspace grids, i.e. collocation of the density and integration of the potential, are dominating the timings. This is typically not the case for metallic systems, where the diagonalization of the Kohn-Sham matrix contributes strongly to the computational cost. LRIGPW is efficient for condensed phase systems such as liquids, molecular crystals etc. Particularly large speed-ups can be obtained for:

- non-orthorhombic cells

- large grid cutoffs

- many SCF steps

Using LRI, the SCF step is accelerated and therefore single point calculations profit most. For molecular dynamics, where the wave function can be extrapolated from the previous step, the SCF converges quickly. Also in this case speed-ups can be obtained depending on the grid cutoff and system. Note that LRIGPW comes with higher memory requirements than the standard GPW scheme. However, this is typically not a problem on HPC platforms, but might limit the usage on smaller clusters.

## Example input files

- Ice XV: `lrigpw_example.inp`