Compact Orbitals Enable Low-Cost Linear-Scaling Ab Initio Molecular Dynamics for Weakly-Interacting Systems Hayden Scheiber,1, A) Yifei Shi,1 and Rustam Z

Compact orbitals enable low-cost linear-scaling ab initio molecular dynamics for weakly-interacting systems Hayden Scheiber,1, a) Yifei Shi,1 and Rustam Z. Khaliullin1, b) Department of Chemistry, McGill University, 801 Sherbrooke St. West, Montreal, QC H3A 0B8, Canada Today, ab initio molecular dynamics (AIMD) relies on the locality of one-electron density matrices to achieve linear growth of computation time with systems size, crucial in large-scale simulations. While Kohn-Sham orbitals strictly localized within predefined radii can offer substantial computational advantages over density matrices, such compact orbitals are not used in AIMD because a compact representation of the electronic ground state is difficult to find. Here, a robust method for maintaining compact orbitals close to the ground state is coupled with a modified Langevin integrator to produce stable nuclear dynamics for molecular and ionic systems. This eliminates a density matrix optimization and enables first orbital-only linear-scaling AIMD. An application to liquid water demonstrates that low computational overhead of the new method makes it ideal for routine medium-scale simulations while its linear-scaling complexity allows to extend first- principle studies of molecular systems to completely new physical phenomena on previously inaccessible length scales.

Since the unification of molecular dynamics and den- LS methods restrict their use in dynamical simulations sity functional theory (DFT)1, ab initio molecular dy- to very short time scales, systems of low dimensions, namics (AIMD) has become an important tool to study and low-quality minimal basis sets6,18–20. On typical processes in molecules and materials. Unfortunately, the length and time scales required in practical and accurate computational cost of the conventional Kohn-Sham (KS) AIMD simulations, LS DFT still cannot compete with DFT grows cubically with the number of atoms, which the straightforward low-cost cubically-scaling KS DFT. severely limits the length scales accessible by AIMD. To In this work, we present an AIMD method that over- address this issue, substantial efforts have been directed comes difficulties of DFT based on compact orbitals to to the development of linear-scaling (LS) DFT. achieve LS with extremely low computational overhead. In all LS DFT methods, the delocalized eigenstates To demonstrate advantages of the new method we ap- of the effective KS Hamiltonian must be replaced with plied it here to systems of weakly-interacting molecules. an alternative set of local electronic descriptors. Most However, the same approach is readily applicable to LS methods2–5 explore the natural locality of the one- systems of strongly-interacting fragments that do not electron density matrix (DM). However, the DM DFT be- form strong covalent bonds such as ionic materials— comes advantageous only for impractically large systems salts, liquids, and semiconductors. A generalization of when accurate multifunction basis sets are used3,6–8. the method to all finite-gap systems, including covalently This issue is rectified in optimal-basis DM methods9–11 bonded atoms, will be reported later. that contract large basis sets into a small number of new localized functions and then optimize the DM in The new AIMD method utilizes a recently developed the contracted basis. Despite becoming the most popu- LS DFT8 based on absolutely localized molecular orbitals lar approach to LS DFT, the efficiency of these methods (ALMOs) – compact orbitals first described in Ref. 21. is hampered by the costly optimization of both the con- Unlike delocalized KS orbitals, each ALMO has its own tracted orbitals and the DM12. From this point of view, a localization center and a predefined localization radius 8,21 direct variation of compact molecular orbitals—orbitals Rc that typically includes nearby atoms or molecules . that are strictly localized within predefined regions—is In the current implementation, a localization center is preferable because LS can be achieved with significantly defined as a set of all Gaussian atomic orbitals of one fewer variables. Advantages of the orbitals-only LS DFT molecule. However, the approach can use other local and are especially pronounced in accurate calculations that nonlocal basis sets22,23. The key feature of ALMO DFT require many basis functions per atom. Unfortunately, is that its one-electron wavefunctions are constructed in the development of promising orbital-based LS methods a two-stage self-consistent-field (SCF) procedure8 to cir- has been all but abandoned13,14 because of the inherently cumvent the problem of the sluggish variational optimiza- difficult optimization of localized orbitals2,13–17. tion emphasized above. In the first stage, ALMOs are Thus, despite impressive progress of the LS descrip- constrained to their localization centers24 whereas, in the tion of the electronic and atomic structure of large static second stage, ALMOs are relaxed to allow delocalization systems6,18, the high computational overhead of existing onto the neighbor molecules within their localization radius Rc. To achieve a robust optimization in the prob- lematic second stage, it is important to keep the delocalization component of the trial wavefunction orthogonal a)Electronic mail: [email protected] to the fixed orbitals obtained in the first stage. For math- b)Electronic mail: [email protected] ematical details, see the ALMO SCF method in Ref. 8. 2

ALMO constraints imposed by Rc prohibit electron zero systematically by increasing Rc and decreasing ǫSCF. density transfer between distant molecules, but retain Our approach to compensate for the missing δfiα(t) all other types of interaction such as long-range elec- term is inspired by the methodology introduced into trostatic, exchange, polarization, and—if the exchange- AIMD by Krajewski et al.27, formalized by Kühne et correlation (XC) functional includes them—dispersion al.28 and rationalized by Dai et al.29 before becoming in- interactions25. Since the importance of electron transfer formally known as the second generation Car-Parrinello decays exponentially with distance in finite-gap materi- molecular dynamics30. Adopting the principle of Refs. 27 als2, the ALMO approximation is expected to provide and 28, ALMO AIMD is chosen to be governed by the a natural and accurate representation of the electronic Langevin equation of motion that can be written in terms structure of molecular systems. Because of the greatly of the unknown reference forces reduced number of electronic descriptors and the robust KS γ optimization, the computational complexity of ALMO mirïα = fiα (t) − γmir˙iα + Riα(t), (2) DFT grows linearly with the number of molecules while where mi is the mass of atom i, riα is its position along its computational overhead remains very low. These fea- γ dimension α, γ is the Langevin scaling factor, and Riα(t) tures make ALMO DFT a promising method for accurate is the stochastic force represented by a zero-mean white AIMD simulations of large molecular systems. Gaussian noise The challenge of adopting ALMO DFT for dynamical γ simulations arises from the slightly nonvariational char- hRiα(t)i =0, (3) acter of the localized orbitals. While ALMOs are varia- γ γ ′ ′ hR (t)R (t )i =2kBTγmiδij δαβδ(t − t ). (4) tionally optimized in both SCF stages, the occupied sub- iα jβ space defined in the first stage must remain fixed dur- The last relation means that, for any value of γ, the ing the second stage to ensure convergence. In addition, damping and stochastic terms are in perfect balance and electron transfer effects can suddenly become inactive in trajectories generated with Eq. (2) will sample the canon- the course of a dynamical simulation when a neighboring ical ensemble at a specified temperature T 31. In the limit molecule crosses the localization threshold Rc. Futher- γ → 0, the Newton equation is recovered and the micro- more, the variational optimization in any AIMD method canonical ensemble is sampled. is never complete in practice and interrupted once the The main assumption of ALMO AIMD is that the error maximum norm of the gradient of the energy with respect in the ALMO forces is well approximated by Gaussian ∆ to the electronic descriptors drops below small but nev- noise Riα(t): ertheless finite convergence threshold ǫSCF. These errors ∆ do not affect the accuracy of static ALMO DFT calcula- δfiα(t)= Riα(t) (5) tions, geometry optimization, and Monte-Carlo simulations. Unfortunately they tend to accumulate in molec- that obeys ular dynamics trajectories leading to non-physical sam- ∆ hRiα(t)i =0, (6) pling and eventual failure. Traditional strategies to cope ∆ ∆ ′ ′ with these problems are computationally expensive and hRiα(t)Rjβ (t )i =2kBT ∆miδij δαβδ(t − t ). (7) include computing the nonvariational contribution to the 4,26 This assumption, shown to be well justified, allows us to forces via a variational coupled-perturbed procedure , rewrite the Langevin equation using the ALMO forces increasing Rc, and decreasing ǫSCF. In this work, we propose another approach that ob- ALMO γ+∆ mirïα = fiα (t) − γmir˙iα + Riα (t), (8) viates the need in a coupled-perturbed solver, relaxes tight constraints on Rc and ǫSCF, and thus enables us where the two stochastic terms are combined into one γ+∆ γ ∆ to maintain stable dynamics and to keep the algorithmic Riα = Riα + Riα. The only missing piece in the complexity and cost of simulations low. In our approach, modified Langevin equation is the value of ∆, which de- the forces on atoms are calculated approximately after scribes the strength of the newly introduced stochastic the two-stage ALMO SCF using a straightforward pro- term. This term compensates for imperfections in ALMO cedure that computes only the Hellmann-Feynman and forces and must be adjusted to re-balance the damping Pulay components and neglects the computationally in- and stochastic components in ALMO AIMD. tense nonvariational component of the forces. The differ- In principle, ∆ can be calculated using the integral of ence between these approximate ALMO forces and the Eq. (7) averaged over atoms with different mi reference forces that could be obtained from perfectly ∞ −1 1 converged fully-delocalized KS orbitals is δfiα(t): ∆=(2kBTmi) hδf~i(0) · δf~i(τ)idτ (9) Z−∞ 3 KS ALMO f (t)= f (t)+ δfiα(t), (1) iα iα KS if one can afford computing the reference forces fiα (t) where α is a Cartesian component of the force acting on (i.e. Rc → ∞ and ǫSCF → 0) for a short representa- atom i at time t. δfiα(t) comprises all neglected terms tive AIMD trajectory. In practice, we found (see re- that originate from a finite localization radius Rc and sults below) that this approach is not particularly ac- incomplete SCF optimization. δfiα(t) can be reduced to curate because the δij δαβ assumption in Eq. (7) does not 3

0 -100 -50 0 50 100

FIG. 1. Calculated properties of water using ALMO AIMD FIG. 2. The red line is the Euclidean norm of the instanta- −2 δf~i t i with ǫSCF = 10 a.u. and Rc = 1.6 vdWR (red line) and neous error hk ( )ki , the black line is the magnitude of the fully converged OT reference (black line). (a) RDF, (b) ki- time average of the instantaneous error vector, and the green netic energy distribution (the gray curve shows the theoretical line is the time average of the red line. (a) Rc = 1.6 vdWR −2 Maxwell-Boltzmann distribution), (c) IR spectrum. and ǫSCF = 10 a.u., (b) Rc = 1.6 vdWR and fully converged 1 ~i ~i it ALMO SCF, (c) Normalized ACF 3 hδf (t)·δf (t+τ)i of the instantaneous error in panel (a). strictly hold. Nevertheless, the ACF integral can provide a reasonable starting value of ∆. This value can be fur- 10−6 a.u., orbital transformation (OT) method39. In ther fine-tuned in a series of short trial-and-error ALMO ALMO AIMD, the element-specific cutoff radius for elec- AIMD runs until the average kinetic energy corresponds tron delocalization R was set to 1.6 in units of the el- to the requested temperature h 1 m r˙2i = 3 k T . c 2 i i 2 B ements’ van der Waals radii (vdWR). This localization The inherently stochastic approach presented here radius includes approximately two coordination shells of does not aim to produce fully time-reversible dynamics an average water molecule and was shown to reproduce for atomic nuclei. Nevertheless, it is capable to repro- the reference radial distribution function (RDF) perfectly duce correct dynamical properties of a system as long as in Monte-Carlo simulations8. To check the ability of the γ is set to a small value and partially optimized ALMOs R∆(t) term to compensate for imperfections in ALMO γ −6 remain close to the ground state resulting in ∆ ≪ . forces, we varied ǫSCF between tight 10 a.u. and loose ALMO AIMD was implemented in CP2K, an open 10−2 a.u. 32 −2 source materials modeling package . Accuracy and ef- Even with ǫSCF = 10 , the simulation is stable with ficiency of ALMO AIMD was tested using liquid water the correct average temperature and perfect Maxwell- as an example. This system is challenging because in- Boltzmann distribution (Figure 1b). ∆ was initially es- termolecular electron delocalization is a critical compo- timated at 2 × 10−5 fs−1 using Eq. (9) and then refined nent of hydrogen bonding and must be described cor- heuristically to 6 × 10−5 fs−1. We found that it is easier rectly to reproduce static and dynamical properties of to optimize ∆ when γ is set to zero because of reduced liquid water. A periodic cell containing 125 molecules noise in the trial runs. Analysis of δf~i(t) shows that was simulated for 30 ps at T = 298K and a constant − the error indeed resembles Gaussian white noise. The density of 1.01g · cm 3. Ricci-Ciccotti algorithm33 was mean of the error is zero (black line in Figure 2a). Its used to integrate the Langevin equation. We found that − − ACF decays rapidly (Figure 2c) so that the errors can be γ = 10 3 fs 1 is large enough to thermostat the system considered uncorrelated on time scale of 50fs. Thus the efficiently and small enough not to significantly affect main assumption behind our approach to ALMO AIMD dynamical properties of liquid water. In the dual Gaus- is justified for liquid water. We established that the sian and plane-wave scheme implemented in CP2K34, the main source of error in forces for this system is the loose TZV2P basis set was used to represent molecular orbitals, convergence criterion and not the finite Rc: fully con- and a plane-wave cutoff of 320Ry used to represent elec- verged ALMO SCF calculations remove the oscillating tron density. The XC energy was approximated using component of δf (Figure 2b). We also verified that the the dispersion-corrected PBE functional35,36. Separable ALMO forces converge to the reference forces in the limit 37 norm-conserving pseudopotentials were used and the Rc → ∞ (Figure S1 in the Supporting Information). Brillouin zone was sampled at the Γ-point. The pre- To test the accuracy of ALMO AIMD we used the tra- 38 dictor of the Kolafa scheme was adopted to localized jectory analyzer TRAVIS40 to compute the infrared (IR) 28 orbitals to generate a highly accurate initial ALMOs spectrum, RDF, and diffusion coefficient of liquid water −2 in both SCF stages, which can be brought close to the from both the ALMO trajectory (ǫSCF = 10 a.u. and ground state with just a few SCF steps of the robust Rc =1.6 vdWR) and from the reference trajectory. The −10 2 −1 two-stage optimization procedure. diffusion coefficients DOT = 1.7(1) × 10 m · s and −10 2 −1 The reference forces were calculated with fully delo- DALMO =1.8(4) × 10 m · s and RDFs (Figure 1a) calized electrons using the tightly converged, ǫSCF = are in good agreement. The quality of the ALMO IR 4

FIG. 3. Timing benchmarks for PBE/TZV2P simulations FIG. 4. Weak scalability benchmarks for PBE/TZV2P −2 of liquid water on 256 compute cores. For ALMO methods, ALMO AIMD with Rc = 1.6 vdWR and ǫSCF = 10 a.u. Rc = 1.6 vdWR. Cyan lines represent perfect cubic scaling, Dashed gray lines connect systems simulated on the same whereas gray lines represent perfect linear scaling. number of cores to confirm LS behavior. spectrum (Figure 1c) is good despite minor errors in the on modern HPC platforms. intensity of the OH stretching mode, which is sensitive to To summarize, we demonstrated—for the first time— the precise positions of the centers of localized orbitals. that compact localized orbitals can be utilized to perform These stringent tests show that despite noticeable errors accurate and efficient LS AIMD without concomitant op- in the ALMO forces (Figure 2a), the compensating R∆(t) timization of the DM. High efficiency of the presented term in the modified Langevin equation makes it possi- method is achieved without sacrificing accuracy with a ble to recover atomic dynamics properly. We would like combination of two techniques: (1) on-the-fly calculation to note that ALMO AIMD could not be stabilized with of approximate forces without lengthy self-consistent op- ∆ = 0. Neither were we able to find any values of ∆ timization of localized orbitals and (2) integration of a that stabilize trajectories generated using perturbative modified Langevin equation of motion that is fine-tuned versions of ALMO DFT8. to retain stable dynamics even with imperfect forces. By To demonstrate the computational efficiency of ALMO obviating the optimization of the DM, the method re- AIMD, we compared the average wall-time per MD step mains remarkably efficient even with large localized basis for a variety of methods in Figure 3. It is important to sets. Using liquid water as an example, we showed that emphasize that the comparison is performed for a three- the new approach enables simulations of molecular sys- dimensional condensed phase system described with an tems on previously inaccessible length scales. The devel- accurate triple-ζ basis set with polarized functions—a oped method will have a significant impact on modeling particularly challenging case for DM-based LS methods. of complex molecular systems (e.g. interfaces or nuclei) ALMO AIMD shows clear LS behavior for all values of making completely new phenomena accessible to AIMD. Generalization of the methodology to systems of strongly ǫSCF, even for medium-size systems. While the second generation Car-Parrinello method decreases the compu- interacting atoms (e.g. covalent crystals) is underway. tational overhead of the cubically-scaling AIMD for small Acknowledgements. The authors are grateful to systems28, ALMO AIMD exploits the modified Langevin Thomas Kühne for insightful discussions. The research concept to substantially reduce the simulation cost for was funded by the Natural Sciences and Engineering Re- systems of all sizes. The crossover point between ALMO search Council of Canada through the Discovery Grant AIMD and cubically scaling methods lies in the region (RGPIN-2016-05059). The authors are grateful to Com- of 256 molecules—length scales routinely accessible with pute Canada and McGill HPC Centre for computer time. AIMD today. 1 Weak scaling benchmarks for very large systems show Car, R.; Parrinello, M. Unified Approach for Molecular-Dynamics and Density-Functional Theory. Phys. Rev. Lett. 1985, 55, 2471. (Figure 4) that localized orbitals are naturally suited for 2Goedecker, S. Linear scaling electronic structure methods. Rev. parallel execution: LS is retained for a wide range of sys- Mod. Phys. 1999, 71, 1085. tems and compute cores. We were able to successfully 3Bowler, D. R.; Miyazaki, T. O(N) methods in electronic structure simulate systems as large as ∼ 105 atoms within rea- calculations. Rep. Prog. Phys. 2012, 75, 36503. 4Kussmann, J.; Beer, M.; Ochsenfeld, C. Linear-scaling self- sonable wall-clock time using only moderate number of consistent field methods for large molecules. Wiley Interdiscip. compute cores—an impressive feat for AIMD considering Rev. Comput. Mol. Sci. 2013, 3, 614–636. that accurate molecular orbitals and the idempotent DM 5Aarons, J.; Sarwar, M.; Thompsett, D.; Skylaris, C.-K. Perspec- are computed on each step. The horizontal line in Fig- tive: Methods for large-scale density functional calculations on ure 4 is shown as a rough guide to time and length scales metallic systems. J. Chem. Phys. 2016, 145, 220901. 6VandeVondele, J.; Borstnik, U.; Hutter, J. Linear Scaling Self- accessible in a fixed wall-clock time given various com- Consistent Field Calculations with Millions of Atoms in the Con- putational resources. It indicates that ALMO AIMD can densed Phase. J. Chem. Theory Comput. 2012, 8, 3565. extend the range of routine simulations to ∼ 104 atoms 7Arita, M.; Bowler, D. R.; Miyazaki, T. Stable and efficient linear 5

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