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98 lines
4.1 KiB
Markdown
98 lines
4.1 KiB
Markdown
# Local Resolution of Identity
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## Introduction
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Density functional theory (DFT) calculations in CP2K employ the Gaussian and plane waves (GPW)
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method. In GPW, the description of the total density on realspace grids is typically the
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computationally most expensive part. By introducing a local resolution-of-the-identity (LRI)
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approach, the linear scaling of the GPW approach can be retained, while reducing the prefactor for
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the grid operations. The combined approach, LRIGPW, is comprehensively described in [](#Golze2017b).
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In LRIGPW, the atomic pair densities $\rho_{\mathrm{AB}}$ are approximated by an expansion in a set
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of fit functions centered at atom A $\{f_i^{\mathrm{A}}(\mathbf{r})\}$ and atom B
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$\{f_j^{\mathrm{B}}(\mathbf{r})\}$,
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$$
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\rho_{\mathrm{AB}}\approx \sum_i{a_i^{\mathrm{A},(\mathrm{AB})}f_i^{\mathrm{A}}}(\mathbf{r}) +
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\sum_j{a_j^{\mathrm{B},(\mathrm{AB})}f_j^{\mathrm{B}}}(\mathbf{r}).
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$$
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The fit functions are also Gaussian-type functions and provided as auxiliary basis set.
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## How to use it
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LRIGPW is specified in the [QS](#CP2K_INPUT.FORCE_EVAL.DFT.QS) section by setting METHOD LRIGPW.
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```
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&QS
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METHOD LRIGPW
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&LRIGPW
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LRI_OVERLAP_MATRIX INVERSE
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SHG_LRI_INTEGRALS
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&END
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&END QS
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```
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Further specifications can be given in the [LRIGPW](#CP2K_INPUT.FORCE_EVAL.DFT.QS.LRIGPW)
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subsection. LRIGPW requires additionally an auxiliary basis set as input.
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```
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&DFT
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BASIS_SET_FILE_NAME BASIS_LRIGPW_AUXMOLOPT
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BASIS_SET_FILE_NAME BASIS_MOLOPT
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...
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&END DFT
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&SUBSYS
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&KIND O
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BASIS_SET DZVP-MOLOPT-GTH
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POTENTIAL GTH-PBE-q6
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LRI_BASIS_SET LRI-DZVP-MOLOPT-GTH-MEDIUM
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&END KIND
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...
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&END SUBSYS
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```
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Auxiliary basis sets are available for the MOLOPT basis sets. All auxiliary basis sets have been
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generated by simple geometric progression without any need for further optimization. These basis
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sets are available in different sizes: `MEDIUM` and `LARGE`. Using the large auxiliary basis sets,
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the accuracy is improved, but the computational overhead increases.
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The LRI auxiliary basis sets are generally quite large leading to a potentially ill-conditioned
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overlap matrix, Equation (10) in [](#Golze2017b). The inversion of this matrix can thus be numerical
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instable.
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If the SCF is not converging, set
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[LRI_OVERLAP_MATRIX](#CP2K_INPUT.FORCE_EVAL.DFT.QS.LRIGPW.LRI_OVERLAP_MATRIX) to `AUTOSELECT`. In
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this case, the atomic pairs are identified that have extremely large condition numbers. For these
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pairs, the pseudoinverse instead of the regular inverse is calculated. The threshold for the
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condition number can be given by
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[MAX_CONDITION_NUM](#CP2K_INPUT.FORCE_EVAL.DFT.QS.LRIGPW.MAX_CONDITION_NUM).
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The LRI integrals, Equations (31)-(34) in [](#Golze2017b), are calculated prior to the SCF. The
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traditionally used Obara-Saika scheme is computationally too demanding here. Therefore, a more
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efficient integral scheme based on solid harmonic Gaussians (SHG) is employed and invoked by
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[SHG_LRI_INTEGRALS](#CP2K_INPUT.FORCE_EVAL.DFT.QS.LRIGPW.SHG_LRI_INTEGRALS), see [](#Golze2017) for
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details.
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## When to use it
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LRIGPW is only beneficial when the operations on the realspace grids, i.e. collocation of the
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density and integration of the potential, are dominating the timings. This is typically not the case
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for metallic systems, where the diagonalization of the Kohn-Sham matrix contributes strongly to the
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computational cost. LRIGPW is efficient for condensed phase systems such as liquids, molecular
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crystals etc. Particularly large speed-ups can be obtained for:
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- non-orthorhombic cells
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- large grid cutoffs
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- many SCF steps
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Using LRI, the SCF step is accelerated and therefore single point calculations profit most. For
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molecular dynamics, where the wave function can be extrapolated from the previous step, the SCF
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converges quickly. Also in this case speed-ups can be obtained depending on the grid cutoff and
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system. Note that LRIGPW comes with higher memory requirements than the standard GPW scheme.
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However, this is typically not a problem on HPC platforms, but might limit the usage on smaller
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clusters.
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## Example input files
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- Ice XV: <path:lrigpw_example.inp>
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