Divide-conquer method with localized natural orbitals (DC-LNO) method

The DC-LNO method [51] is a variant of the DC method. The computational efficiency is improved by introducing localized natural orbitals (LNOs) to span the subspace of each atom. The dimension of the resultant subspace is smaller than that in the DC method, leading to the reduction of computational cost. The LNOs are non-iteratively calculated by a low-rank approximation via a local eigendecomposition of a projection operator for the occupied space. As shown in Fig. 19(a), introducing LNOs to represent the long range region of a truncated cluster reduces the computational cost of the DC method while keeping computational accuracy. The method can be applied to not only gapped systems, but also metallic systems as long as the size of truncated clusters is large enough, and typically clusters including more than 200 atoms might be chosen. The functionality is compatible with not only the collinear calculation, but also the non-collinear calculations.

Figure 19: Truncation of a system in the DC method with LNOs. The short range (orange) and long range (yellow) regions are represented by PAOs and LNOs, respectively.

As a first step for O($N$) calculations of the DC-LNO method, one can perform an O($N$) calculation for Si crystal using an input file store in the directory 'work' as

     % mpirun -np 112 ./openmx Si8-LNO.dat | tee si8-lno.std
The calculation was performed in 66 seconds using 112 cores on a Xeon cluster machine of 2.6 GHz. In order for you to start an trial calculation by the DC-LNO method, an input file 'Si8-LNO.dat' is available in the directory 'work'. Since as shown in Fig. 19(b) the three level parallelization has been implemented: atom level, spin level, and diagonalization level, it is expected that the method scales up to, e.g., 40000 CPU cores for a 1000 atom system in the parallel calculations, where we assumed 1000 atoms $\times $ 2 (spin-polarized calculation) $\times $ 20 CPU cores per node, resulting in that the product becomes 40000. The benchmark calculation of the multi-level parallelization will be shown later on. When you try to perform the hybrid parallelization, the following keyword has to be switched on:
     scf.dclno.threading       on    # off|on
In the hybrid parallelization the diagonalization at the bottom level will be parallelized by OpenMP. The computational accuracy and efficiency of the method can be controlled by the following keywords:
     orderN.HoppingRanges     7.0    # 7.0 (Ang.)
     orderN.LNO.Buffer        0.2    # default = 0.2
     orderN.LNO.Occ.Cutoff    0.1    # default = 0.1
The role of the keyword 'orderN.HoppingRanges' is exactly the same as that in the DC method. For each atom a truncated cluster is constructed by picking-up atoms within a sphere whose radius is specified by the keyword 'orderN.HoppingRanges'. Though the proper choice of the parameter depends on systems, a serie of benchmark calculations implies that the accuracy is enough for not only gapped systems, but also metallic systems if 'orderN.HoppingRanges' is set so that the resultant truncated cluste can include 300 atoms. The setting might be regarded as a conservative choice to unsure the accuracy rather than efficiency. So, a compromising choice with respect to both accuracy and efficiency may be in between 200 and 300 atoms. The region in the truncated cluster where the PAOs are replaced by LNOs is determined by the keyword 'orderN.LNO.Buffer'. The 'orderN.LNO.Buffer=0.0' means that PAOs allocated on all the SNAN atoms are replaced by LNOs, while the PAOs on all the SNAN atoms remain unchanged in the case with the 'orderN.LNO.Buffer=1.0' which is equivalent to the DC method. As for the SNAN, please refer the subsection 23.4 'User definition of FNAN+SNAN'.

Figure 20: Absolute error in the total energy (Hatree/atom) for (a) diamond, (b) silicon in the diamond structure, (c) rutile TiO2, (d) BCC lithium, (e) FCC aluminum, and (f) BCC iron as a function of the number of atoms in a truncated cluster calculated by the DC and DC-LNO methods. The experimental lattice constants were used for all the cases.

The orderN.LNO.Buffer of 0.1$\sim$0.2 might be a proper choice with respect to accuracy and efficiency, while the default value is 0.2. The selection of LNOs for each atom $i$ is performed by monitoring eigenvalues of $\Lambda_{{\bf0}i}$ defined with [51]
$\displaystyle \Lambda_{{\bf0}i}
\sum_{{\bf R}j}
\rho_{{\bf0}i,{\bf R}j} S_{{\bf R}j,{\bf0}i},$     (2)

where $\rho_{{\bf0}i,{\bf R}j}$ and $S_{{\bf R}j,{\bf0}i}$ are block elements of density matrix and overlap matrix, respectively. Since the eigenvalues can be understood as population of LNOs,. the LNOs having the population more than orderN.LNO.Occ.Cutoff are chosen as basis functions for the targeted atom, which well span the occupied subspace space. Instead of using 'orderN.LNO.Occ.Cutoff', one can directly specify the number of LNOs for each species by the keyword 'LNOs.Num'. When you define the species as
     Si    Si7.0-s2p2d1   Si_PBE19
     H     H6.0-s2p1      H_PBE19
the number of LNOs can be specified by
       Si    4
       H     1
In this case, the numbers of the LNOs are fixed to 4 and 1 for Si and H, respectively. To avoid a sudden change of the number of LNOs during geometry optimization and molecular dynamics simulations, it might be better to use 'LNOs.Num' rather than orderN.LNO.Occ.Cutoff. The comparison between the DC and DC-LNO methods is shown in Fig. 20. Although the PAOs in the long range region are replaced by the LNOs, it is found that the accuracy is comparable to the DC method both in gapped and metallic systems. As an illustration for applications of the DC-LNO method, we show in Fig. 21 radial distribution functions (RDF) of liquids for silicon, aluminum, lithium, and SiO$_2$. It turns out that in all the cases the DC-LNO method reproduces well the results by the conventional O($N^3$) diagonalization method, and that the obtained RDFs are well compared to other computational results [52,53,54,55].

Figure 21: Total radial distribution function (RDF) of (a) silicon at 3500 K, (b) aluminum at 2500 K, (c) lithium at 800 K, and (d) SiO$_2$ at 3000 K, calculated by the conventional O(N3 ) diagonalization and the DC-LNO methods. The MD simulations were performed for cubic supercells containing 64, 108, 128, and 192 atoms with a fixed lattice constant of 10.86, 12.15, 14.04, 14.25 for silicon, aluminum, lithium, and SiO2, respectively, for 10 ps with the time step of 2 fs. The details of the simulations can be in Ref. [51].

In Fig. 22 the speed-up ratio in the MPI parallelization of the DC-LNO method is shown for non-spin polarized calculations of a diamond supercell containing 64 atoms. Since the multiplicity of spin index is 1, we see a nearly ideal behavior up to 64 MPI processes. Beyond 64 MPI processes the parallelization in the diagonalization level is taken into account on top of the parallelization in the atom level.

Figure 22: Speed-up ratio in the MPI parallelization of the DC-LNO method for a diamond supercell containing 64 atoms, where the cutoff radius rL of 8.0 Åwas used, leading to the numbers of atoms of 239 and 142 in the short and long range regions, and the dimension of matrices of 3675 for the truncated cluster problem. The details of the benchmark calculation can be in Ref. [51].

A superlinear speed-up is observed at 128 and 256 MPI processes, which might be due to an effective use of cache by the reduction of memory usage, and a good scaling is achieved up to 1280 MPI processes at which the parallel efficiency is calculated to be 70% using the elapsed time at 1 MPI process as reference. Since each computer node has 20 CPU cores in this case, it would be reasonable to observe the good caling up to 1280 (=64$\times $20) MPI processes. Thus, we see that the multilevel parallelization is very effective to minimize the computational time in accordance with a recent development of massively parallel computers.