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Largescale density functional theory (DFT) calculations provide a powerful tool to investigate the atomic and electronic structure of materials with complex structures. This article reviews a largescale DFT calculation method, the multisite support function (MSSF) method, in the CONQUEST code. MSSFs are linear combinations of the basis functions which belong to a group of atoms in a local region. The method can reduce the computational time while preserving accuracy. The accuracy of MSSFs has been assessed for bulk Si, Al, Fe and NiO and hydrated DNA, which demonstrate the applicability of the MSSFs for varied materials. The applications of MSSFs on large systems with several thousand atoms, which have complex interfaces and nonperiodic structures, indicate that the MSSF method is promising for precise investigations of materials with complex structures.
Atomic structure, electronic structure and the properties of materials correlate with each other strongly. The relationship of materials properties with local structures such as point defects in crystals and interatomic distances in glassy materials have been investigated widely for many years. Recently, wider structures ordered on the nanoscale, such as ring structures in amorphous glass,^{1}^{)} ionic and molecular positions in biomaterials,^{2}^{)} interfaces between metallic nanoparticle catalysts and substrates,^{3}^{)} and composite dopants and defects in crystals,^{4}^{)} have also been focused on as hyperordered structures, which could have significant influence on materials properties. The recent improvements in experimental measurements and computation techniques have enabled us to investigate such nanoscale complex structures.
For computation, density functional theory (DFT) calculations have been a powerful tool to investigate atomic and electronic structures of condensedphase materials and molecules.^{5}^{)} However, because of the high computational cost of conventional DFT calculation methods, the system size treated by DFT has been limited, up to a thousand atoms in most cases. Therefore, efficient DFT calculation methods are desirable to treat nanoscale structures. Several methods have been proposed to overcome the size limitation of DFT calculations, and we have also proposed the largescale DFT code, CONQUEST.^{6}^{–}^{10}^{)} There are two important methods which enable us to perform largescale calculations with CONQUEST, the linearscaling orderN, or
In the next section, we first briefly review several largescale DFT calculation methods, and then provide more details of our largescale calculation techniques in the CONQUEST code, especially the MSSF method. In Sect. 3, by showing several examples, we demonstrate the applicability of the MSSF method to large systems with complex structures. The final section provides the conclusion.
The DFT total energy E is a functional of electron density, which is the diagonal part of the density matrix. The density matrix can be written in terms of the Kohn–Sham (KS) oneelectron orbitals ψ,^{14}^{,}^{15}^{)}
To overcome this limitation, several largescale DFT calculation methods have been proposed. Since there are several review papers already which cover largescale DFT calculation methods,^{16}^{,}^{17}^{)} here we briefly introduce the methods focusing on two key points, “how to solve the KS equation” and “what kind of functions are used to express the KS orbitals”. The parallelization efficiency of DFT codes, which we do not describe here, is also very important.
For linear scaling solution of the KS equation for large systems, there are several methods such as the divideandconquer (DC) method,^{18}^{–}^{20}^{)} the orbital minimization method (OMM),^{21}^{,}^{22}^{)} the density matrix minimization (DMM) method,^{23}^{,}^{24}^{)} and the fragment molecular orbital (FMO) method.^{25}^{)} In the DC method, the target system is divided into small subsystems with some buffer regions whose electronic structures are calculated exactly, and then the subspace density matrices are combined to construct the density matrix of the whole target system. The DC method is used in many codes such as SIESTA^{26}^{,}^{27}^{)} and OpenMX.^{28}^{)} FMO is similar to the DC method, but the division is based on molecular fragments in proteins and biomolecules. The total energy of the whole system is calculated from the energy of fragments and pairs of fragments without solving for the molecular orbitals (MOs) of the whole system. Recently, a method to obtain molecular orbitals of the whole system from subspace MOs (FMOLCMO) has been also proposed.^{29}^{)} FMO is implemented in GAMESS^{30}^{)} and ABINITMP.^{31}^{)} OMM and DMM are methods which minimize the total energy variationally as a functional of orbitals or density matrices, instead of solving KS equations directly. OMM is used in FEMTECK^{32}^{)} and SIESTA, and DMM is used in CONQUEST and ONETEP.^{33}^{)} There are also iterative calculation methods such as the Fermioperator expansion method^{34}^{)} in BigDFT^{35}^{)} and the secondorder tracecorrecting (TC2) method^{36}^{)} in ErgoSCF.^{37}^{)} CP2K also uses an iterative
For the functions to represent the KS orbitals, planewave basis functions have been often used with the periodic boundary condition for solids and surfaces, while Gaussian or Slatertype atomicorbital (AO) basis functions have been popular for isolated systems such as molecules and clusters. For largescale calculations, using local orbital functions to express the KS orbitals is crucial. Pseudoatomic orbitals (PAOs) are atomic orbital functions constrained to go to zero at a cutoff, which avoids having longrange tails. PAOs are often used in largescale DFT codes such as SIESTA,^{39}^{,}^{40}^{)} OpenMX,^{28}^{,}^{41}^{)} and CONQUEST.^{42}^{,}^{43}^{)} There are also basis functions defined on a regular grid similarly to plane waves, which can be systematically converged, such as Bspline functions in CONQUEST,^{44}^{)} periodic cardinal sine functions in ONETEP,^{45}^{)} and wavelet functions in BigDFT.^{46}^{)} CP2K uses Gaussian functions to describe KS orbitals and represents the electron density on a grid.^{47}^{)} There are also codes using the finite difference method, for example RSDFT^{48}^{)} and PARSEC.^{49}^{)}
In this subsection, we explain more details about our largescale DFT code CONQUEST. In CONQUEST, the density matrix in Eq. (1) is expressed by using local orbital functions ϕ, called “support functions”, so that the density matrix
Two kinds of basis functions, Bspline (blip) functions^{44}^{)} and PAO functions,^{42}^{,}^{43}^{)} are available in CONQUEST. Blip functions are finiteelement functions akin to planewave basis functions, i.e., the accuracy of the support functions can be improved systematically by making the blip functions finer. However, the optimization of the linearcombination coefficients for fine blip functions can be computationally expensive. On the other hand, the computational cost of PAOs is much cheaper than blip functions. PAOs are atomic orbital functions consisting of numerical radial functions R and spherical harmonic functions Y,
When we use PAOs as the support functions without any contraction, which are called “primitive” PAOs, the coefficients c in Eq. (4) are 1 or 0. When we contract multipleζ PAOs to smaller number of support functions, the c values are optimized numerically by, for example, the conjugate gradient method. The number of support functions can be reduced to the singleζ (SZ) size for each angular momentum functions by contraction, keeping the spatial symmetry of the primitive PAOs. For example, the primitive tripleζ (TZ) PAOs and tripleζ plus triple polarization (TZTP) PAOs are contracted to SZ and SZPsized support functions, respectively. Since the computational cost scales cubically to the number of support functions in both the exact diagonalization and the
For the KS equation solver, CONQUEST supports both exact diagonalization and the DMM to optimize electron density. With the diagonalization method, the computational cost scales cubically with the size of the electronic Hamiltonian, i.e., the number of atoms N. On the other hand, the computational cost of DMM is linear with the system size. In DMM, the electronic structure is optimized by minimizing the total energy E with respect to the auxiliary density matrix
Since the computational cost of both the exact diagonalization and the
Ideally, the linearcombination coefficients
The calculation accuracy with ϕ will depend on the choice of
By optimizing
In this subsection, the accuracy of MSSF is investigated by checking the
First, we investigate the accuracy of the calculated energies for bulk Si and Al systems with the local density approximation (LDA)^{57}^{)} exchange–correlation functional. Figure 1 shows the deviations of the DFT total energies by the MSSFs from those by the primitive PAOs.^{11}^{)} The TZP PAOs (3s, 3p, d) consisting of 17 functions are contracted to four MSSFs for both Si and Al atoms. The energy deviations of the MSSFs decrease exponentially as
Figure 1. (Color online) Deviation of total DFT energy per atom of bulk Si and Al of MSSFs with respect to the multisite range
The comparison of the energy–volume (E–V) curves of bulk Si with several
Figure 2. (Color online) E–V curve of bulk Si calculated using MSSFs with LFD, MSSFs with numerical optimization, and primitive TZDP PAOs. The multisite ranges

To check the accuracy of atomic forces, the energies and forces have been investigated for a distorted benzene molecule in which a C–H pair has been shifted away from the center of the benzene ring by 0.5 Å. Table II shows the differences of the energies and the maximum forces by the MSSFs with several

Next, the accuracy of the electronic structure was investigated by checking the density of states (DOS) of a hydrated DNA system with 3,088 atoms.^{12}^{)} The PBE generalized gradient approximation (GGA) functional was used.^{58}^{)} The difference of the DOS calculated with the primitive PAOs and the MSSFs are presented in Fig. 3. The difference is smaller in the occupied states and lowenergy unoccupied states than in the highenergy unoccupied states. The difference in the highenergy unoccupied states is larger with smaller
Figure 3. (Color online) DOS of the hydrated DNA system using primitive (black, solid) and MSSFs with local filter diagonalization (blue, dotted) and numerical optimization (red, dashed). Deviations from the primitive functions are shown for the occupied states. (Reproduced from Ref. 12 with permission from the PCCP Owner Societies.)
The computational efficiency was also investigated for the hydrated DNA system.^{12}^{)} Table III summarizes the times required for matrix construction, diagonalization and gradient calculation with respect to the coefficients

For the spinpolarized systems, there are two possibilities for how to determine MSSF linearcombination coefficients: determining the coefficients for spinup and spindown electrons individually; and using the same coefficients for spinup and spindown electrons by taking their average. The first method leads to spindependent MSSFs. Spindependent MSSFs are obtained by using spinup and spindown
Therefore, we compare the accuracy of spindependent and spinindependent MSSFs for spinpolarized systems: bulk bcc ferromagnetic Fe and cubic antiferromagnetic NiO.
Figure 4. (Color online) E–V curve of (a) bcc ferromagnetic Fe and (b) cubic antiferromagnetic NiO calculated with primitive DZP PAOs, SIMSSF, and SDMSSF.
Figure 5. (Color online) DOS of cubic antiferromagnetic NiO calculated with (a) primitive DZP PAOs, (b) SIMSSF, and (c) SDMSSF. The absolute differences of the DOS from (a) are also shown for (b) and (c). Spinup (red) and spindown (blue) states are shown as positive and negative values, respectively.

To demonstrate the actual applicability of MSSFs to large systems, in this section we show several examples of the applications of MSSFs to large complex structures: moiré graphene on the Rh(111) surface;^{60}^{)} the interfaces in YGaO_{3};^{61}^{)} and PbTiO_{3} films on SrTiO_{3}.^{62}^{)} The applications to nonperiodic systems, hydrated DNA^{59}^{)} and metallic gold nanoparticles, are also shown. The LDA exchange–correlation functional is used for YGaO_{3} and PbTiO_{3} on SrTiO_{3}, and the PBE functional is used for graphene on Rh(111), hydrated DNA and Au nanoparticles. Normconserving pseudopotentials have been used in the calculations with CONQUEST. The other detailed information about the computational conditions such as the number of kpoints and the spatial range of PAOs are found in the corresponding reference papers. In several examples, planewave calculations for comparison have been performed with the PAW pseudopotential^{63}^{)} using the VASP software.^{64}^{,}^{65}^{)}
There have been many reports showing the exotic properties of 2D materials. The structure of 2D materials and their electronic structures are often affected by the interactions with the substrates or interlayer interactions. The target system in Ref. 60 also shows the interesting property that a highly corrugated graphene layer grown on Rh(111) can be flattened by the intercalation of oxygen atoms. For this system, planewave DFT calculations demonstrated that strong interactions between the graphene layer and the substrate are decoupled when oxygen atoms are intercalated in the lowest moiré sites. Such DFT simulations have been performed to clarify the structural and electronic properties of 2D materials, but they were limited in the size of simulation cells and it was difficult to study the case of low concentration of oxygen atoms, large moiré structures, the effect of the edges and so on. It is expected that these obstacles can be overcome if accurate MSSF method can be applied to such 2D materials.
In Ref. 60, the accuracy and computationaltime efficiency of the MSSF method with a system, graphene on Rh(111) surface (G/Rh), were investigated. First, the accuracies of the PAOs in the optimized structures of G/Rh and G/O/Rh having small supercells were confirmed by comparing to planewave basis functions. TZDP PAOs of carbon atoms, TZTP PAOs of oxygen atoms and DZP PAOs of rhodium atoms were contracted to form MSSFs with
Figure 6. (Color online) DOS for graphene on a Rh(111) substrate (460 atoms) with no oxygen atom (

For largescale calculations, the computational times by the primitive PAOs and the MSSFs for the G/Rh(111) consisting of 3,088 atoms [shown in Fig. 6(c)] are compared in Table VI. These large systems were too computationally expensive to be treated with planewaves. The calculations were performed with the supercomputer SGI ICE X [Intel Xeon E52680V3 (12 cores, 2.5 GHz)

The next example is an investigation of topological defects in ferroelectric YGaO_{3}.^{61}^{)} Ferroelectric domain walls are attracting broad attention for nextgeneration nanoelectronics. Although the basic properties of simple ferroelectric domain walls can be well described by small DFT calculations, complex domain patterns could not be treated since very large supercells are needed to model the structure. The target in Ref. 61 is a vortex core at which six kinds of structural domains meet. To model the complex structure with a periodic boundary condition, two pairs of vortex/antivortex cores need to be included in the calculation cell, which contains at least about 3,600 atoms. Using the MSSF method with CONQUEST, the atomicscale structure of the vortices and their electronic structures have been investigated. TZDP PAOs were contracted to MSSFs with
Before studying the complex topologically protected vortex cores, calculations of two domain walls in the
Figure 7. (Color) (a) Crystal structure of ferroelectric YGaO_{3} (
Then the topologically protected vortex using a 3,600atom,
Figure 8. (Color) Surface map of (a) Φ and (b) Q in the optimized structure and (c) density of states and (d) electron density around the conduction band minimum [yellow region in (c)] of 3600 atoms
We have investigated a perovskite material, PbTiO_{3} films on SrTiO_{3} substrate.^{62}^{)} Advanced deposition techniques allow the creation of thin film perovskite oxides and layered heterostructures, which demonstrate a wide variety of electrical polarization textures with possible applications in low dimensional functional devices. These textures arise from the interaction of different order parameters, notably antiferrodistortive (AFD) and ferroelectric (FE) distortions. At the surface of PbTiO_{3} (PTO), antiphase rotations of the TiO_{6} octahedra give rise to an AFD c(
We found that seven layers of STO substrate were required to avoid any influence on the PTO; we then built cells with between one and nine layers of PTO for the polar morphologies (described above/shown in Fig. 9). We used MSSFs with
Figure 9. (Color online) The initial supercell configurations for the
We find that the polydomain arrangement is unstable for thicknesses of
Figure 10. (Color online) The local polarization vector fields in the x–z plane for two film thicknesses not including AFD modes. (a) The fluxclosure domains of the
The full exploration of these exotic polarization textures at thin film surfaces is only possible through the use of largescale DFT, as enabled by MSSF in CONQUEST.
Largescale calculation methods are also important to investigate nonperiodic materials such as glassy materials, polymers and biomaterials. For example, in this section, MSSF have been used to model a hydrated DNA system,^{59}^{)} shown in Fig. 11. The solvent water molecules have been treated explicitly in the calculations, therefore the system consists of 634 atoms in the DNA, 932 hydrating water molecules and 9 Mg counterions, in total 3,439 atoms. DZP PAOs with 27,883 primitive functions have been contracted to 7,447 MSSFs. Figure 11(a) compares the DOS of the hydrated DNA calculated by the primitive PAOs and the MSSFs. As discussed in Sect. 2.3.2, the MSSFs have reproduced the DOS of the primitive PAOs with high accuracy for the occupied states but not for unoccupied states. To improve the accuracy of the unoccupied states, we have introduced the SSM.^{59}^{,}^{67}^{,}^{68}^{)} SSM is an interior eigenproblem solver for large sparse matrices, providing the eigenvalues and eigenvectors in given energy regions with high parallel efficiency.^{67}^{,}^{68}^{)} We have performed a SCF calculation to optimize the electronic density using MSSFs and reconstructed the electronic Hamiltonian in primitive PAO basis with the optimized electronic density. Subsequently, oneshot SSM calculations have been performed for the energy region of interest. For the hydrated DNA system, we have performed the SSM calculations for the energy range [
Figure 11. (Color online) (a) DOS calculated using MSSFs (
In this section, we show an example of the calculations of metallic nanoparticles with MSSFs briefly. The sizecontrolled metallic nanoparticles show high catalytic reactivity, and the combination of nanoparticles and substrate is one of the important factors to affect the reactivity. The interface between the nanoparticle and the substrate is a kind of hyperordered structure. We have investigated an Au nanoparticle with 923 atoms in octahedral (Oh) symmetry, consisting of six layers [Fig. 12(a)], using DZP PAOs. The diameter of this sixlayered nanoparticle is about 3 nm, which is close to the sizes used in actual experiments.^{69}^{)} The nanosize calculation model enables us to investigate the sitedependence of the atomic and electronic structures of nanoparticles. Figures 13(a) and 13(b) show the intra and interlayer nearest neighbor atomic distances of the nanoparticle, respectively. The atomic distances of the inner layers are close to those in a bulk fcc system, while they are distributed widely in the outer layers. The wide distribution of the intralayer distances corresponds to the site dependence, i.e., the atomic distances around the center of the faces (about 2.9 Å) are longer than those around the vertices and edges (about 2.8 Å). Figures 14(a) and 14(b) show the projected DOS (pDOS) of an Au atom in a bulk system and at the vertex of the nanoparticle. The electronic structure at the vertex of the nanoparticle is quite different from that in the bulk, where the dband center is shifted closer to the Fermi level, which suggests high reactivity at the vertex.
Figure 12. (Color online) Optimized structures of (a) Au nanoparticle in Oh symmetry with 923 atoms and (b) Au particle on Mg(001) surface.
Figure 13. (a) Intralayer and (b) interlayer nearest atom distances in Au nanoparticle with 923 atoms. Black horizontal lines correspond to the Au–Au distance in a bulk fcc gold (2.95 Å). Abscissa corresponds to the indices of the layers increasing from inner to outer of the nanoparticle. The sixth layer is the surface of the nanoparticle.
Figure 14. PDOS of an Au atom (a) in bulk system and (b) at the vertex of an Au nanoparticle. Fermi levels are set to be zero (dashed line).
Not only the vertices of the nanoparticles but also the interface between the nanoparticles and the substrate have been considered as reaction active sites.^{3}^{)} To treat nanoscale metallic nanoparticles (i.e., not clusters) on substrates, large calculation models with several thousand of atoms are required. With MSSFs, we can treat these large models of the catalytic systems. For example, Fig. 12(b) shows the optimized structures of Au nanoparticles on the MgO(001) substrate with 2,844 atoms in total, in which we removed the bottom part of the nanoparticle as found in experimental observations.^{3}^{,}^{70}^{)} The detailed investigation of atomic and electronic structures and reactivity of metallic nanoparticles on the substrate will be provided in future studies.
We have reviewed largescale calculation methods, especially focusing on the methods in our largescale DFT code CONQUEST.^{6}^{–}^{10}^{)} To model nanoscale complex structures, we often need large calculation models with several thousand atoms or more. DFT is a powerful tool to investigate the atomic and electronic structures of materials with high accuracy, but the computational cost of conventional DFT calculation methods is quite high (cubic scaling to system sizes). Therefore, special calculation techniques to treat such large systems are required.
The MSSF method^{11}^{,}^{12}^{)} in CONQUEST makes it possible to improve both computational efficiency and accuracy. MSSFs are linear combinations of basis functions which belong not only to a target atom but also to its neighboring atoms, like local MOs. This MOlike picture of MSSFs enables us to reduce the number of the support functions to a minimalbasis size, while we can increase the number of the basis functions to improve computational accuracy, without increasing the number of the support functions. The linearcombination coefficients can be determined by using the local filter diagonalization method^{11}^{,}^{54}^{,}^{55}^{)} and subsequent numerical optimization.^{12}^{)} The investigations of accuracy for bulk Si and Al, hydrated DNA, bulk Fe and NiO demonstrate that MSSFs are applicable to varied materials such as insulating, semiconducting, metallic, and spinpolarized systems.
Examples of applications of MSSF to large systems with several thousand of atoms have been also shown. The geometry and electronic structure around complex interfaces were investigated using MSSF in these examples.^{59}^{–}^{62}^{)} MSSF have been also applied to nonperiodic materials such as biomolecules and metallic nanoparticle catalysts. The combination of MSSF and the Sakurai–Sugiura method,^{67}^{,}^{68}^{)} an efficient interior eigenproblem solver, enable the MSSFs to be used to investigate the excited states of large systems.^{59}^{)} Thus, we suggest that MSSF is now one of the most promising tools for investigation of hyperordered structures such as interfaces between nanoparticle catalysts and substrates.
Acknowledgment
This work is supported by the World Premier International Research Centre Initiative (WPI Initiative) on Materials Nanoarchitectonics (MANA), JSPS GrantinAid for Transformative Research Areas (A) “HyperOrdered Structures Science” (Grant Nos. JP20H05883 and JP20H05878), JSPS GrantinAid for Scientific Research (Grant No. JP18H01143), and JST PRESTO (Grant No. JPMJPR20T4). This work is also partially supported by a project, JPNP16010, commissioned by the New Energy and Industrial Technology Development Organization (NEDO). Calculations were performed on the Numerical Materials Simulator at NIMS and the supercomputer HA8000 system at Kyushu University in Japan.
The authors are grateful for computational support from the U.K. Materials and Molecular Modeling Hub, which is partially funded by EPSRC (Grant No. EP/P020194), for which access was obtained via the UKCP consortium and funded by EPSRC (Grant Ref. No. EP/P022561/1). We also acknowledge computational support from the UK national highperformance computing service, ARCHER, for which access was obtained via the UKCP consortium and funded by EPSRC (Grant Ref. No. EP/P022561/1).
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Author Biographies
Ayako Nakata (ORCiD: 0000000233116283) is a Principal Researcher at NIMS, a researcher in JSTPREST, and an adjunctive associate professor of cooperative graduate school at University of Tsukuba. She obtained her Ph.D. degree from Waseda university in 2007. She has been a member of the CONQUEST developer team and has mainly worked on the development and applications of the multisite method. She has also interested in theoretical investigation of catalysts (especially metallic nanoparticles) and excitedstate calculations.
David R. Bowler (ORCiD: 0000000178531520) is Professor of Physics at UCL, and a PI in both the London Centre for Nanotechnology and the World Premier Research Institute for Materials Nanoarchitectonics (MANA) in the National Institute for Materials Science (NIMS), Japan. He obtained his Ph.D. degree from Oxford University in 1997. He has driven the development of the massivelyparallel linear scaling density functional theory code, CONQUEST, and collaborates extensively with experimental groups on the growth and properties of nanostructures on semiconductor surfaces.
Tsuyoshi Miyazaki (ORCiD: 0000000335344404) obtained a Ph.D. from University of Tokyo in 1995. He is now a MANA Principal Investigator at the International Center for Nanoarchitectonics (MANA), National Research Institute for Materials Science (NIMS), and a group leader of Firstprinciples Simulation Group in NanoTheory Field of NIMSMANA. He has performed theoretical studies of various materials, such as organic conductors, semiconductor surfaces, nanostructured materials, biological system and so on. He is the coleader of the CONQUEST project, a visiting Professor of School of Engineering at Nagoya University, and a Professor of School of Integrative and Global Majors at University of Tsukuba.