arXiv:1711.10993v1 [cond-mat.mtrl-sci] 29 Nov 2017 in ihsutrn hehl nry o rtu re- activation tritium neutron low low energy, and threshold expan- tention sputtering thermal high high of point, coefficient sion, melting low high conductivity, their thermal to years, ap- promis- thanks of past very candidates, design as the ing the emerged During have is materials energy Tungsten-based safe components. and reactor clean propriate of source a as igesl-nesiil n vacancies defects, and smallest self-interstitials the of single characteristics elucidate and to properties crucial the been have calculations quest, of this principles In first models irradiation. predictive under build evolution magnetic microstructure to and fundamental as electronic such is as well defects, properties, as these proper- structure of atomic transport nature their and the de- mechanical Understanding produce their ties. will alter which that levels fects radiation high sustain will ftenme fKh-hmobtl,ie tis it i.e. power orbitals, third Kohn-Sham the of as number pre- scales the More system of cost the approaches. computational to ab-initio the respect other cisely, DFT with to scaling compared particular, size In better much and approach. a precision this offers between by balance offered good efficiency the calcula- to mechanical thanks quantum tions principles first for method Theory tional with studied be can defects few methods. a these just with clusters only otn F acltosaeuulyol oei this cubic the in reduce to done possible only still is and usually it Fortunately, atoms, are regime. hundred calculations some DFT to routine sizes system accessible the iersaigDTcluain o ag ugtnsystems Tungsten large for calculations DFT scaling Linear n ftebgcalne oad h s fFusion of use the towards challenges big the of One vrteps eae onSa K)DniyFunc- Density (KS) Kohn-Sham decades past the Over eetees hsihrn ui cln tl limits still scaling cubic inherent this Nevertheless, tpa Mohr, Stephan 2 est arx nti ae eso ht nefiieelectr finite once that, show we the paper of temperat version this finite scaling a In forlinear as developed large matrix. metals, were are to density methods that applied problems scaling straightforwardly tackle linear be recent to these the allowed as atoms, has hundred However, scali methods some cubic DFT to intrinsic size ing the system While accessible applications. the of range wide hnlresprel r required. appro are electronic an supercells the such large investigating that when for believe approaches We key novel a towards Reactors. plays Fusion which Tungsten, for of bulk number materials on the based to examples respect prototype with linearly scales that treatment aoaoyo tmcadSldSaePyis onl Unive Cornell , State Solid and Atomic of Laboratory 4,5 est ucinlTer DT a eoetequasi-stand the become has (DFT) Theory Functional Density .INTRODUCTION I. 1 acln uecmuigCne BC,C od ioa29, Girona Jordi C/ (BSC), Center Supercomputing Barcelona DT a eoetems popular most the become has (DFT) 1 acEixarch, Marc 4 nv rnbeAps NCMM L INAC-MEM, Alpes, Grenoble Univ. 3 CE,P.Luı opns2,000Breoa Spain Barcelona, 08010 23, Llu´ıs Companys Pg. 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OPENMX, there is an implementation of a divide-and- namely the density matrix, is represented in a separable conquer approach within a Krylov subspace that allows way via a set of localized and adaptive basis functions to perform linear scaling calculations for metals; how- {φα(r)}, from now on also called “support functions”, as ever the method seems to require a careful tuning before 41 ′ αβ ′ it can be applied to large systems . Furthermore it has F (r, r )= φα(r)K φα(r ) , (1) been shown that the approach by Suryanarayana et. al.48, Xα,β which calculates the electronic charge density and the to- K tal energy directly by performing Gauss quadratures over where the matrix denotes the “density kernel”. From this expression we can easily get the electronic charge the spectrum of the Hamiltonian and in this way is capa- ρ r F r, r ble to reach a linear complexity, also works for metallic density as ( ) = ( ) and use it for the construction of the Kohn-Sham Hamiltonian, systems49. These pioneering examples demonstrate that, even if 1 H[ρ]= − ∇2 + V [ρ]+ V , (2) their application is possible, reduced scaling approaches 2 KS P SP for large metallic systems have to be considered under a ′ ρ(r ) r′ different perspective. As for such systems the number of where the Kohn-Sham potential VKS[ρ] = |r−r′| d + degrees of freedom is very large and the electronic struc- VXC[ρ] contains the Hartree and exchange-correlationR 50 ture is complicated, first-principles calculation become a potential, and VP SP denotes the that useful tool only when complementing other approaches, is used to describe the union of the nuclei and core elec- like for instance force fields, which are unable to provide trons. This Hamiltonian operator gives rise to the Hamil- quantum-mechanical information. When such kind of in- tonian matrix H, defined as formation is required, like for example when studying the arrangement of close to a defective region, in- Hαβ = φα(r)H(r)φβ (r)dr , (3) vestigation techniques like the ones presented above are Z or utmost importance. which can then be used to determine a new density ker- BigDFT In this paper we report on the capabilities of nel; methods to do so will be discussed later. to perform reduced and eventually also linear scaling In order to obtain a linear scaling behavior, it is calculations for a metallic system at finite temperature. necessary to employ a set of localized support func- Whereas previous publications have highlighted in de- tions that eventually lead to sparse matrices. What tail the accuracy, efficiency and linear scaling capabili- distinguishes BigDFT from other similar approaches ties of this code for systems with a finite HOMO-LUMO 43,44 is the special set of localized support functions that gap , metals have so far not been considered. In it uses. These are expanded in an underlying basis this paper we demonstrate that the basic algorithm of set of Daubechies wavelets51 and are optimized in-situ. BigDFT remains stable also for systems with vanishing Daubechies offer the outstanding property of be- gap and thus allows to routinely perform accurate linear ing at the same time orthogonal, systematic and exhibit- scaling calculation for large metallic systems without the ing compact support. The in-situ optimization, together need of additional adjustments. As specific example we with an imposed approximative orthonormality, results have chosen Tungsten due to its relevance for finding safe in a set of quasi-systematic support functions offering a and long-lasting materials for fusion reactors. very high precision. This allows to work with a mini- The outline of this paper is as follows: In Sec. II we mal set of support functions, meaning that only very few focus on the theoretical background, with Sec. II A sum- functions per atom are necessary to obtain a very high marizing the principles of the linear scaling version of accuracy. Obviously, this in-situ optimization comes at BigDFT BigDFT , and Sec. IIB discussing how can some cost compared to an approach working with a fixed mitigate the challenges arising for metallic systems. In set of non-optimized support functions. However, in the Sec. III we then present numerical results, with Sec. IIIA latter case we would require a much larger set to obtain demonstrating the precision that we obtain with the lin- the same precision52, and all matrix operations in the BigDFT ear scaling version of and Sec. IIIB showing subspace of the support functions, whose scaling is cu- various performance figures. bic in the worst case, would become considerably more costly. Apart from that, the use of a minimal basis set also has additional advantages, as for instance an easy II. THEORY and accurate fragment identification and associated pop- ulation analysis for large systems52, which can be used A. Overview of the algorithm for a reliable effective electrostatic embedding53.

The detailed implementation of the linear scaling algo- rithm of BigDFT has been presented in detail in Refs. 43 B. Advantages of BigDFT for metallic systems and 44. Here we will give a brief overview over the most important concepts. DFT calculations for metallic systems are a challeng- The central quantity on which the algorithm is based, ing task. Due to the non-zero density of states at the 3

1000 1 Fermi energy, the occupation of the eigenstates around density of states that energy value can easily jump between occupied and 900 Fermi function empty during the self-consistency cycle, leading to a phe- 800 0.8 nomenon called “charge sloshing”. A solution to this problem is to introduce a finite electronic temperature, 700 leading to the grand-canonical extension of DFT as de- 600 0.6 54 rived by Mermin . In such a setup, the occupations 500 are smoothed out around the Fermi level and the self- consistency cycle becomes more stable. 400 0.4 In the context of linear scaling approaches, the intro- 300 duction of a finite temperature has the additional advan- 200 0.2 tage that it intensifies the decay properties of the density density of states (arbitrary units) matrix, as mentioned in Sec. I, and thus justifies the ex- 100 ploitation of the nearsightedness principle. Nevertheless, 0 0 linear scaling calculations for metals remain very chal- -1 -0.5 0 0.5 1 lenging. First of all, the used electronic temperatures energy (Hartree) must not be too large — otherwise one would change the physics in a too drastic way — and thus the density FIG. 1. Density of states for the 11x11x11 supercell of bcc Tungsten, containing 2662 atoms. The spectral width is very matrix decays much slower compared to finite gap sys- small, which is a direct consequence of the special properties tems. Moreover, the vanishing gap complicates the calcu- of the support functions used by BigDFT. lation of the density kernel. In BigDFT, we use for this task the CheSS library55 — one of the building-blocks BigDFT 10000 of the program suite — that offers several dif- AO with confinement ferent solvers. In the Fermi Operator Expansion (FOE) AO without confinement method14,15, which is the linear scaling solver available within CheSS, one has to approximate the function that assigns the occupation numbers — typically the Fermi function — with Chebyshev polynomials. Obviously this 1000 method is most efficient if the degree of the polynomial expansion is small. This is the case if first the spectral width of the involved matrices is very small, and second if polynomial degree the Fermi function that must be approximated with the polynomials is smooth. Unfortunately the latter condi- tion is violated for metals, since — even when using a 100 small finite temperature — the Fermi function that must be approximated exhibits a sharp drop at the Fermi en- 0.001 0.01 ergy and we thus have to approximate a rather step-like kT (atomic units) function. As a consequence, it is questionable whether the FOE method can be used in practice for calculations FIG. 2. Polynomial degree used by the FOE method within with metals. CheSS as a function of the electronic temperature, for two set of atomic orbitals (AO). The set that was obtained by solving Fortunately it turns out that the special properties of the atomic Schr¨odinger equation with a confining potential the support functions used by BigDFT lead to such a leads to considerably lower values due to its smaller spectral small spectral width that FOE can still be used for metal- width. lic systems. In Fig. 1 we show the density of states for the 11x11x11 supercell of body centered cubic (bcc) Tung- sten, containing 2662 atoms. As can be seen, the spec- in Ref. 43, a confining potential is used in order to prop- tral width is even smaller than the default [−1, 1] interval erly localize the support functions during the optimiza- for the Chebyshev polynomials. In this way the polyno- tion, and this confinement also seems to help in reducing mial degree required to accurately represent the Fermi the spectral width. In Fig. 2 we show the polynomial function can be kept reasonably small even for metallic degree used by CheSS as function of the temperature systems. for two sets of atomic orbitals. Both were obtained by This small spectral width is a direct consequence of solving the Schr¨odinger equation for the isolated atom two concurrent elements. First of all, the usage of PSP within the pseudopotential approach, but in one case we helps by eliminating the need for the treatments of core additionally added the confining potential. In this latter electrons whilst smoothening the behavior of the valence setup, the resulting set of atomic orbitals exhibits a much KS orbitals close to the nuclei. The second important smaller spectral width (36.7eV) compared to the case point is the special way in which BigDFT optimizes without confinement (186.4 eV), leading to much smaller the support functions. As is explained in more detail polynomial degrees. 4

-208.30 Nevertheless it is important to note that FOE, BigDFT linear DIAG BigDFT linear FOE even though it can be used very efficiently within BigDFT cubic 40 Abinit cubic BigDFT, is an O(N) method designed for very large -208.35 systems. For intermediate system size, alternative 20 solvers within CheSS, such as diagonalization using LA- -208.40 PACK56/ScaLAPACK57 or PEXSI58, might thus be more efficient. The first method does not exploit the -208.45 0 sparsity of the matrices and thus exhibits a cubic scal- 3/2 pressure (GPa)

ing, whereas the second one scales as O(N), O(N ) and energy per atom (eV) -208.50 O(N 2) for one-, two- and three-dimensional systems, re- -20 spectively. Still, FOE is the method of choice in the limit -208.55 of very large systems since it is the only solver which -40 scales strictly linearly with system size. -208.60 30 31 32 33 34 35 36 37 38 volume per minimal cell (Angstroem3) III. TESTS AND CONSIDERATIONS FIG. 3. Plots of the energy (left axis) and the pressure (right axis) as a function of the cell volume, for the four se- In order to demonstrate the accuracy and performance tups described in Sec. III A 1. The linear scaling version of of the linear scaling version of BigDFT for metallic sys- BigDFT yields results that are consistent with those of the tems, we focus on one specific system, namely bcc Tung- two traditional cubic scaling approaches. sten. All runs were performed using a grid spacing of at most 0.38 atomic units, the exchange-correlation part was described by the PBE functional59, and the Krack correctly described. In Fig. 3 we compare the DoS of the HGH pseudopotential60 was used. As we are interested reference calculation with the cubic version of BigDFT in systems requiring the usage of very large supercells, and the one obtained with the linear version with diago- we did not considered k-points in our calculations. Nev- nalization. As can be seen, both setups yield an identical ertheless, we still choose as test-bed for our approach a DoS for the occupied states. For the unoccupied ones, the bulk-like system that can be easily simulated via k-points linear version of BigDFT shows some deviations. How- and small supercells, in order to verify the accuracy of ever, this is not surprising, since the optimization of the our linear scaling approach. support functions only takes into account the occupied states, and a good accuracy can thus only be expected for the latter. However, as shown in Ref. 65 it is possible A. Accuracy of the linear scaling version to include extra states in the optimization of the support functions in case that the user is interested in low-energy 1. Energy versus volume conduction states. Overall, we see that, thanks to the in-situ optimization of the support functions, the linear As a first test we demonstrate that the linear scaling scaling version of BigDFT is able to correctly reproduce version can accurately calculate the equation of state re- the electronic structure of a metallic system. lating energy and volume. To this end, we scaled the lattice vectors of the Tungsten system by ±4% around its equilibrium value. We compare four different setups: B. Performance (1)/(2) the linear scaling version of BigDFT using a 9x9x9 supercell (containing 1458 atoms) and no k-points, 1. Scaling with system size using as solver both diagonalization (DIAG) and FOE; (3) the cubic scaling version of BigDFT using the 2- As anticipated, we expect that DFT calculations of atoms unit cell and a 9x9x9 k-point mesh; (4) the same metallic systems at large scales will be time-consuming setup as (3), but run with the Abinit code61–64. In Fig. 3 compared to similar simulations for insulators. In Fig. 5 we compare the energy-vs-volume curves for all four se- we show the total runtime as a function of the number of tups. In the same figure we also show, for all four setups, atoms in the system, going from the 4x4x4 supercell (128 the variation of the pressure as a function of the volume. atoms) up to the 12x12x12 supercell (3456 atoms). As As can bee seen from this test, the linear scaling approach can be seen, each of the three approaches that we com- correctly determines the optimal lattice parameter. pare — cubic, linear with diagonalization and linear with FOE — is characterized by a typical system size at which the method outperforms the other ones. For the small 2. Density of states systems, the cubic approach is clearly the fastest one. Above about 500 atoms, the linear approach using diag- As a second test we compare the density of states onalization becomes the method of choice, since the cu- (DoS) in order to verify that the electronic structure is bic scaling of the diagonalization exhibits a rather small 5

BigDFT cubic 100 100 BigDFT linear DIAG

EF 80 80

60 60 Percent

40 40 number of states (arbitrary units)

20 20 Unknown Flib LowLevel Communications Density Kernel Hamiltonian 0 0 -10 -5 0 5 432 686 energy (eV) 1024 1458 2000 2662 3456

FIG. 4. Density of states for the cubic scaling version of FIG. 6. Portions of the overall walltime spent in the different BigDFT and the linear scaling version with diagonalization. sections of the code for some of the FOE runs of Fig. 5. The The energies were shifted such that the Fermi energy lies at overall behavior of the code is similar over a wide range of zero. Up to the Fermi energy, the linear scaling version of system sizes, indicated by the number of atoms on the x-axis. BigDFT yields results that are consistent with those of the The FOE operations to construct the density kernel take a traditional cubic scaling approach. little less than 40% of the overall walltime.

cubic 14 cubic fit linear DIAG cubic fit ditional cubic scaling approach. However, the inspection linear FOE 12 linear fit of the overall walltime clearly shows that the application to metals is much heavier compared to insulators. Con- 10 sidering the CPU-minutes per atom, which can be used

8 as a metric to quantify the computational workload for O(N) codes44, and comparing with values obtained for 6

walltime (hours) systems such as organic of light atoms, reveals that the latter run up to two order of magnitude faster 4 (!) on the same platform.

2 Nonetheless, this behavior is not related to the absence of a gap, but is due to the the unbiased nature of the 0 0 500 1000 1500 2000 2500 3000 3500 description, which requires support functions optimized number of atoms in-situ. The reasons for this claim are explained in the following. In Fig. 6 we show the percentage of the time FIG. 5. Scaling of the total runtime as a function of the sys- spent in the different sections of the code for the FOE tem size, for the cubic approach, the linear approach with runs of Fig. 5. We see that about 40% of the time is diagonalization, and the linear approach with FOE. The cal- spent in the application of the KS Hamiltonian, 40% in culations were performed at the Γ point, and the runs were the determination of the density matrix and some 20% in performed in parallel, using 9600 cores (800 MPI tasks with 12 OpenMP threads). communications, and this over a wide range of number of atoms. All these calculations converged in about 15 iterations of the combined self-consistent optimization of the support functions and the density kernel. This fact prefactor. For system sizes beyond about 1000 atoms, shows, on the one hand, that FOE calculations of the however, this cubic scaling starts to become a dominant kernel are not necessarily a bottleneck, even when higher part, and thus the truly linear scaling approach with FOE polynomial degrees are needed, i.e. systems without a becomes the fastest option. gap can be calculated efficiently with our method. On the other hand, it will be of much interest to work with pre-optimized basis functions that exhibit a similar accu- 2. Considerations about the computational cost racy, in the same spirit as the fragment-based approach that has been employed using molecular fragments with We have demonstrated that the linear scaling version BigDFT65,66; this would provide results in only one (in- of BigDFT can offer an unbiased and unconstrained de- stead of 15) iterations and therefore lead to calculations scription of metallic systems and is thus capable of yield- running more than one order of magnitude faster. Work ing results that are of the same quality as those of a tra- is ongoing in this direction. 6

IV. CONCLUSIONS AND OUTLOOK tential of quantum-mechanical investigations of metallic systems at such sizes will only be possible if they are con- sidered as complementary investigation techniques along- In this work we have demonstrated, by applying our side other approaches at this scale. code BigDFT to the case of large Tungsten systems, that it is possible to perform accurate and efficient linear scaling DFT calculations for metals with this code. Even V. ACKNOWLEDGMENTS though the linear scaling version of BigDFT was de- signed — as most other codes — for insulating systems, We acknowledge valuable discussions with Mar´ıaJos´e the obtained performance — considering also the very Caturla and Chu-Chun Fu. S.M. acknowledges support high accuracy of the resulting description — for metal- from the MaX project, which has received funding from lic systems is excellent. We have shown that the results the European Unions Horizon 2020 Research and In- obtained with the linear scaling version of BigDFT are novation Programme under Grant Agreements 676598. of equal quality as those obtained with traditional cu- M.A. acknowledges support from the Novartis Univer- bic scaling approaches, but the reduced scaling allows to sit¨at Basel Excellence Scholarship for Life Sciences and tackle much larger systems. The crossover point between the Swiss National Science Foundation (P300P2-158407, the cubic and linear scaling treatment lies at about 500 P300P2-174475). We gratefully acknowledge the com- atoms. puting resources on Marconi-Fusion under the EURO- Thanks to these achievements, the possibility of ad- fusion project BigDFT4F, from the Swiss National Su- dressing the challenge of unbiased first-principles investi- percomputing Center in Lugano (project s700), the Ex- gation for systems with such a large degree of complexity treme Science and Engineering Discovery Environment opens up new interesting opportunities, as now more re- (XSEDE) (which is supported by National Science Foun- alistic conditions, like for instance lower concentrations dation grant number OCI-1053575), the Bridges system of supercell defects, can be considered. Nevertheless it at the Pittsburgh Supercomputing Center (PSC) (which has to be pointed out that, despite the very good per- is supported by NSF award number ACI-1445606), the formance offered by our approach, calculations like the Quest high performance computing facility at Northwest- one presented remain extremely challenging from a first- ern University, and the National Energy Research Scien- principles point of view. Therefore, exploiting the full po- tific Computing Center (DOE: DE-AC02-05CH11231).

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