Simula'ons of atomic scale systems using HPC Hannes Jnsson, Faculty - PowerPoint PPT Presentation

Simula'ons of atomic scale systems using HPC Hannes Jónsson, Faculty of Physical Sciences, University of Iceland A few historical remarks … Computa(onal and theore(cal chemistry (electronic structure and rearrangements of atoms/spins) - Development of algorithms and computer implementaFons - ApplicaFons to a few systems

Computer clusters available to researchers at the University of Iceland 2003 Bjólfur (Beowoulf cluster, 130 computers) located in the basement of VR-III Funded in part by the Icelandic Research Fund (or rather its precursor). 2008 Sól (Sun system, 320 cores, each node 2x4 cores) located in the basement of VR-III Funded in part by the Icelandic Research Fund. 2009 Jötunn (IBM, 168 cores) located at Reiknistofnun in Taeknigardur Purchased by funds from Reiknistofnun. 2011 Nordic HPC in Iceland (HP system, 12 core nodes) located at Thor Data Center Purchased by funds from Denmark, Norway, and Sweden. 2015 Garpur (IBM, 12 and 16 core nodes) located at Reiknistofnun in Neshagi Funded in part by the Icelandic Research Fund. 2017 Garpur2 (??, 100 teraflop) located at Reiknistofnun in Neshagi Funded in part by the Icelandic Research Fund.

Examples of collabora'ons with computer scien'sts at University of Washington in SeaAle: - `A Parallel ImplementaFon of the Car-Parrinello Method by Orbital DecomposiFon', J. Wiggs and H. Jónsson, Computer Physics CommunicaFons, vol. 81, p. 1 (1994). - `A Hybrid DecomposiFon Parallel ImplementaFon of the Car-Parrinello Method', J. Wiggs and H. Jónsson, Computer Physics CommunicaFons, vol. 87, p. 319 (1995). - `Dynamic-Domain-DecomposiFon Parallel Molecular Dynamics', S. Srivilliputhur, I. Ashok, H. Jónsson, G. Kalonji and J. Zarhojan, Comp. Phys. Commun., 102, 44 (1997). - `Parallel Short-Range Molecular Dynamics Using Adhara RunFme System', - S. Srivilliputhur, I. Ashok, H. Jónsson, G. Kalonji and J. Zarhojan, Comp. Phys. Commun., 102, 28 (1997). at University of California San Diego: - 'Parallel implementaFon of gamma-point plane-wave DFT with exact exchange', E. J. Bylaska, K. Tsemekhman, S. B. Baden, J. H. Weare and H. Jónsson, J. Comp. Chem. 32, 5469 (2011). Ongoing collabora'on with computer scien'sts at Aalto University in Finland: - 'Minimum energy path calculaFons with Gaussian process regression', O-P. KoisFnen, E. Maras, A. Vehtari, H. Jónsson. Nanosystems: Physics, Chemistry, MathemaFcs, 7, 925 (2016). - 'Nudged ElasFc Band CalculaFons Accelerated with Gaussian Process Regression’ O-P. KoisFnen, F. Dagbjartsdójr, V. Ásgeirsson, A. Vehtari and H. Jónsson. J. Chem. Phys. (in press).

ComputaFonal chemistry (reikniefnafræði) Use basic laws of physics to calculate properFes of chemicals and materials Predict properFes of new systems and/or help interpret experiments on exisFng systems Tremendous progress in recent years - improved approximaFons to the basic equaFons - improved algorithms - improved implementaFons - increased computer power

Basic tasks in computaFonal chemistry Typically work within the Born-Oppenheimer (adiabaFc) approximaFon. Two step process: 1. For fixed locaFon of the nuclei of the atoms, solve for the electronic structure, i.e. find the distribuFon and energy of the electrons. Energy surface for the locaFon of the atoms and orientaFon of magneFc moment, E(R, ω ) . Need to solve many electron Schrödinger equaFon or equivalent formulaFon (most commonly now density funcFonal theory, DFT ). 2. Move the atoms and rotate spins given the energy from step 1, i.e. on the energy surface. Typically, use classical descripFon of the atoms, but someFmes need to include quantum mechanical descripFon (for example tunneling of atoms or spins). In principle, can solve Newton equaFon to calculate Fme evoluFon (trajectories), but the huge difference in Fme scale between atomic vibraFons and chemical reacFons or other atomic or spin rearrangements between stable states make it necessary to use rate theory (such as transiFon state theory).

Self-interacFon error in standard implementaFons of DFT In Kohn-Sham DFT, the energy is esFmated form N d 3 r | r φ i ( r ) | 2 d 3 r ρ ( r ) v ext ( r ) + 1 d 3 r d 3 r 0 ρ ( r ) ρ ( r 0 ) Z Z Z X E [ ρ ( r )] = + | r � r 0 | + E xc [ ρ ( r )] 2 2 i E KS E kin E ext C Orbitals, , introduced only to obtain an accurate esFmate of kineFc energy. φ i ( r ) Total electron density is a sum over orbital densiFes (Note: unitary invariance ) X X | φ i ( r ) | 2 ρ ( r ) = ρ i ( r ) = i i In terms of orbital densiFes, the KS esFmate of the electron Coulomb interacFon is = 1 d 3 r 0 ρ i ( r ) ρ j ( r 0 ) Z Z X X E KS d 3 r C 2 | r − r 0 | i j It includes interacFon of the each orbital density with itself ( i=j terms). A more accurate esFmate, which was used by Hartree, excludes the diagonal terms d 3 r 0 ρ i ( r ) ρ j ( r 0 ) d 3 r 0 ρ i ( r ) ρ i ( r 0 ) H = 1 − 1 Z Z Z Z E C X X = E KS X d 3 r d 3 r C | r − r 0 | | r − r 0 | 2 2 i j 6 = i i Coulomb self-interacFon correcFon

Perdew-Zunger self-interacFon correcFon, PZ-SIC should fix all approximaFons in the kineFc and Coulomb energy, E xc but pracFcal funcFonals are approximate and a self-interacFon error remains. E KS can be corrected explicitly by subtracFng the diagonal terms, but E xc C then also needs to be corrected. For a one-electron system, self-interacFon in a KS funcFonal can be removed by N  1 � Z Z d 3 r 0 ρ i ( r ) ρ i ( r 0 ) X E KS � SIC [ ρ 1 , . . . , ρ N ] = E KS [ ρ ] − d 3 r − E KS xc [ ρ i ] 2 | r − r 0 | i Perdew and Zunger ( PRB 1981) PZ-SIC proposed using this correcFon for many electron systems. Note: A specific set of orbitals minimizes the total energy given by E KS − SIC due to orbital density dependence (ODD). Convenient to work with two sets of orbitals eigenstates of the Hamiltonian Canonical orbitals (give energy) diagonalize the KS Hamiltonian OpFmal orbitals (give locaFon) obtained by minimizing the total energy minimize the ODD energy The two sets are related by a unitary transformaFon, W Now need a new inner loop in SCF to find the W that minimizes E KS-SIC .

VariaFonal implementaFon of PZ-SIC The energy is no longer unitary invariant with respect to the orbitals, now need an inner loop to find opFmal linear combinaFon of the orbitals at each SCF iteraFon. Use algorithm developed in signal processing by Abrudan,Eriksson and Koivunen ( Signal Processing 2009, 89, 1704). Same algorithm can be used to find opFmal local orbitals as post processing of Kohn-Sham or Hartree-Fock orbitals ( S. Lehtola, E. Jónsson and HJ., JCTC 2013, 2014, 2017 ) Has been implemented in the GPAW code, a separate branch. Developed originally to use real space grid but can now also make use of atomic basis sets and plane waves. PAW representaFon of inner electrons. Elvar

Example applicaFon: Energy of two Mn atoms as a funcFon of the distance between them ( Tushar, Aleksei, Elvar ). Two magneFc states: FerromagneFc (green) and anFferromagneFc (red) Standard DFT methodology (PW91 funcFonal) fails, while self-interacFon corrected funcFonal gives results in close agreeement with high level quantum chemistry calculaFons and experiments.

Electronic structure calculaFons are computaFonally demanding Can readily deal with 100 to 200 atoms with DFT. ComputaFonal effort scales as N 3 (where N is number of electrons) Much larger systems can be simulated if an energy surface, E(R, ω ), can be developed. Example: DescripFon of interacFon between H 2 O molecules using mulFpole expansion of the electrostaFcs, including dipole and octupole polarizaFon (SCME potenFal funcFon) QM/MM approach: Divide the system up into a core region that is described by DFT, and an outer region that is described by SCME. ElectrostaFc embedding implemented in GPAW code Asmus and Elvar

Second step: How do the atoms move in chemical reac'ons (or other rearrangements such as diffusion)? Time scale problem: Most interesting transitions are rare events (i.e., much slower than vibrations). • A transiFon with an energy barrier of 0.5 eV and 1000/s a typical pre-exponenFal factor occurs 1000 Fmes per second at room temperature – fast on laboratory scale! • A video of a direct classical dynamics simulaFon 0.5 eV where each vibraFon spans a second in the video would go on for more than 100 years in between such reacFve events – slow on atomic scale! Typically there is a clear separation of time scales, and a statistical approach can be used

Transition State Theory ( Wigner, Eyring 1930s ) IdenFfy a 3 N -1 dimensional dividing surface, that represents a boAleneck for going from the iniFal to a final state: Initial The bottleneck state can be due to an energy ‡ barrier and/or entropy barrier 3 N -1 dimensional dividing surface, . Final ‡ Add thickness σ to define state σ TransiFon State (TS)

HTST - Harmonic approximation to TST : Good for solids at not too high T Energy Approximate the energy surface 2 nd order SP ridge with second order Taylor > k B T expansions, (a) For reactant region expand around the 1 st order SP local energy minimum, (b) For the transiFon state expand around the 1 st order saddle point. R Works well when (1) energy of second order saddle points is significantly higher than k B T over the energy of first order saddle points, and (2) when the potenFal is smooth enough that a second order Taylor approximaFon to the PES is good enough in the region with large staFsFcal weight.