Long-range interactions: P3M, MMMxD, ELC, MEMD ... - ESPResSo

http://www.icp.uni-stuttgart.deLong-range interactions:P 3 M, MMMxD, ELC, MEMD and ICC ∗Axel ArnoldInstitute for Computational PhysicsUniversität StuttgartOctober 10, 2012

Long-range interactionshttp://www.icp.uni-stuttgart.de• gravity• Coulomb interaction• dipolar interaction• hydrodynamic interactionA. Arnold Long-range interactions 2/40

Electrostatics in periodic boundary conditionshttp://www.icp.uni-stuttgart.deAlternatively from potentialU = 1 2• Poisson’s equation:N∑q i φ(r i )i=1N∑∇ 2 φ(r) = 4π δ(r j − r)q j ,j=1imposing periodic boundary conditions• intrinsic solution• difference to spherical sum known• intrinsic U is periodic in coordinates r iA. Arnold Long-range interactions 4/40

Electrostatics in ESPResSohttp://www.icp.uni-stuttgart.de• requires myconfig.h-switch ELECTROSTATICS• switching on:inter coulomb • methods and their parameters: next 2 hours• switching off:inter coulomb 0• getting l B , method and parameters:intercoulombreturns e. g.{ coulomb 1.0 p3m 7 .75 8 5 0 .1138 0.0}{ coulomb epsilon 80 .0 n_interpol 32768 mesh_off 0.5 0.5 0.5}A. Arnold Long-range interactions 5/40

Assigning chargeshttp://www.icp.uni-stuttgart.de• assigning charges to particlespart 0 pos 0 0 0 q 1part 1 pos 0.5 0 0 q -1.5• adding a charged plateconstraint plate height sigma • plate parallel to xy-plane at z = h, charge density σ• requires 2D periodicity• adding a charged rodconstraint rod center lambda • rod parallel to z-axis at (x, y) = (c x , c y ), line charge density λ• requires 1D periodicityA. Arnold Long-range interactions 6/40

The Ewald methodhttp://www.icp.uni-stuttgart.deP. P. Ewald, 1888 — 1985Coulomb potential has 2 problems1. singular at each particle position2. very slowly decayingIdea: separate the two problems!• one smooth potential — Fourier space• one short-ranged potential — real spaceA. Arnold Long-range interactions 7/40

Ewald: splitting the potentialhttp://www.icp.uni-stuttgart.decharge distributionρ = ∑N ∑n∈LZ 3 i=1q i δ(r − r i − n)= +replace δ by Gaussians of width α −1 :ρ Gauss (r) = ( α/ √ π ) 3 e−α 2 r 2δ(r) = ρ Gauss (r) + [δ(r) − ρ Gauss (r)]A. Arnold Long-range interactions 8/40

The Ewald formulahttp://www.icp.uni-stuttgart.dewithU (r) = 1 2∑∑ ′m∈Z 3 i,jU (k) = 12L 3 ∑k≠0U (s) = − α √ π∑U = U (r) + U (k) + U (s)q i q jerfc(α|r ij + mL|)|r ij + mL|4πk 2 e−k 2 /4α 2 |̂ρ(k)| 2iq 2 iforces from differentiationF i = − ∂∂r iUreal space correctionGaussians in k-spaceGaussian self interactionA. Arnold Long-range interactions 9/40

The dipole termhttp://www.icp.uni-stuttgart.deɛ = ɛ ′ 32 1 2( ɛ = 13 1 0 1 3U (d) 2π ∑=(1 + 2ɛ ′ )L 3 i2 1 2ɛ = ɛ ′3q i r i) 2• Ewald sum derived from Poisson’s formula ⇒ intrinsic potential• spherical summation leads to additional term U (d)• polarizable material of dielectric constant ɛ ′ outside sphere• ɛ ′ = 1 corresponds to mathematical sum• ɛ ′ = ∞ (metallic boundary conditions) ⇔ intrinsic potentialA. Arnold Long-range interactions 10/40

Mesh-based Ewald methodshttp://www.icp.uni-stuttgart.deR. W. HockneyJ. W. Eastwood• replace k-space Fourier sum by discrete FFT• discrete FT is exact — constant real space cutoff• computational order O(N log N)• most frequently used methods:• P 3 M: optimal method• PME• SPMER. W. Hockney and J. W. Eastwood,Computer Simulation Using Particles, 1988M. Deserno and C. Holm, JCP 109:7678, 1998A. Arnold Long-range interactions 12/40

Steps of P 3 Mhttp://www.icp.uni-stuttgart.de1. {r i , q i } → ρ(r): interpolate charges onto a grid(window functions: cardinal B-splines)2. ρ(r) → ̂ρ(k): Fourier transform charge distribution3. ˆφ(k) = Ĝ(k)ˆρ(k): solve Poisson’s equation by multiplicationwith optimal influence function Ĝ(k)(in continuum: product of Green’s function 4π andk 2Fourier transform of Gaussians e −k 2 /4α 2 )4. ik ˆφ(k) → Ê(k): obtain field by Fourier space differentiation4. Ê(k) → E(r): Fourier transform field back5. E(r) → {r i , F i }: interpolate field at position of chargesto obtain forces F i = q i E iA. Arnold Long-range interactions 13/40

Charge assignmenthttp://www.icp.uni-stuttgart.deq5q1aqq6q 3 q4q 2q7 8• interpolate charges onto h-spaced gridM (P ) (x)10.80.60.40.2001123425637x4567ρ M (r p ) = 1 h 3N ∑i=1q i W (p) (r p − r i )• W (p) (r) cardinal B-splines in P 3 M / SPMEA. Arnold Long-range interactions 14/40

Optimal influence functionhttp://www.icp.uni-stuttgart.deĜ opt (k) = h 6 ik · ∑m∈Z 3 Ŵ (p)2 (k + 2π h m)̂R(k + 2π h m)|k| 2 [ ∑m∈Z 3 Ŵ (p)2 (k + 2π h m) ] 2• aliasing of continuum forcêR(k) = −ik 4πk 2 e−k 2 /4α 2with differentiation, Green’s function and transform of Gaussian• minimizes the rms force error functionalQ[F] := 1 ∫h 3 d 3 r [ F(r; r 1 ) − R(r) ] 2h 3 d 3 r 1∫VA. Arnold Long-range interactions 15/40

http://www.icp.uni-stuttgart.deWhy to control errorsrms force error ∆F =√ 〈(F exact − F Ewald ) 2〉 =∆F1010.10.010.0010.0001r max=1, k max=10r max=2, k max=10r max=1, k max=201e-050 1 2 3 4 5α√1NN∑i=1∆F 2i• optimal α brings orders of magnitude of accuracy• at given required accuracy, find fastest cutoffs• compare algorithms at the same accuracyA. Arnold Long-range interactions 16/40

How to: error estimateshttp://www.icp.uni-stuttgart.deKolafa and Perram:∆F∆F real ≈1010.10.010.001total errorreal space estimatek-space estimate0.00010 1 2 3 4 5α∑ q2√Ni2)√(−αrmax L exp 2 r 23 maxHockney and Eastwood:∆F Fourier ≈√∑ q2√NiQ[Ĝopt(k)]L 3A. Arnold Long-range interactions 17/40

P 3 M in ESPResSo (H. Limbach, AA)http://www.icp.uni-stuttgart.de• tune P 3 M for rms force error τinter coulomb p3m tune accuracy \[ r_cut ] [ mesh ] [ cao ] [ alpha ]inter coulomb [ epsilon ɛ ∞] [ mesh_off o x o y o z ]• ɛ ∞ is dielectric constant at infinity (defaults to “metallic”)• o x , o y , o z shifts the mesh origin (defaults to (0, 0, 0))• tunable parameters are (can be fixed)α Ewald splitting parameter (don’t fix!)r max real space cutoff (0 to retune)n M = L/h mesh size (0 to retune)p charge assignment spline order p (0 to retune)• manually set parameters (dangerous!)inter coulomb p3m A. Arnold Long-range interactions 18/40

Partially periodic systemshttp://www.icp.uni-stuttgart.de• partially p. b. c. for slablike systems (surfaces, thin films)• ... or for cylindrical systems (rods, nanopores)• dielectric contrasts at interfaces• P 3 M cannot be employed straightforwardlyA. Arnold Long-range interactions 19/40

Alternative approaches: screeninghttp://www.icp.uni-stuttgart.deU(r)1010.10.011/rexp(-r)/rexp(-r/4)/r0.0010.1 1 10screened Coulomb potential, screening length β −1 :rU β = ∑ m∈Z 3N ∑i,j=1q i q je −β|r ij +mL||r ij + mL|• U = lim β→0 U β is periodic ⇒ intrinsic solutionA. Arnold Long-range interactions 20/40

Taking the limit: MMM2D far formulahttp://www.icp.uni-stuttgart.deφ β (r) = ∑k,l∈LZ(= 2 ∑ ∑Lp∈ 2π L Z l∈LZ= 2πL 2⎛= 2π ⎝L 2e −β√ (x+k) 2 +(y+l) 2 +z 2√(x + k)2 + (y + l) 2 + z 2∑p,q∈ 2π L Z∑p 2 +q 2 >0K 0( √β2+ p 2 √(y + l) 2 + z 2 ) ) e ipxe − √β 2 +p 2 +q 2 |z|√β2 + p 2 + q 2 eipx e iqye fpq|z|f pq⎞e ipx e iqy + |z| ⎠ + π L 2 β−1 + O β→0 (β)• singularity cancels due to charge neutrality• bad or no convergence for z ≈ 0A. Arnold Long-range interactions 21/40

http://www.icp.uni-stuttgart.deMMM2D near formulaφ β (r) = 4 ∑ ∑(K 0Ll∈LZ p∈ 2π L N− 2 L− 1 L+∑n≥1∑n≥0)p√(y + l) 2 + z 2[ (zb 2n (2π) 2n) ] 2n + iy2n(2n)! R Lcos(px)( −1) 2 ψ (2n) (2 + x L ) + ψ(2n) (2 − x L )n(2n)!( y 2 + z 21√(x + L)2 + y 2 + z + 1√ 2 (x − L)2 + y 2 + z 2L 2) n− 2 Llog (4π) −2πL 3 β−1 + ′ 1|r| + O β→0(β)• uses most special functions math offers• self energy: special case r = 0A. Arnold Long-range interactions 22/40

MMM2D procedurehttp://www.icp.uni-stuttgart.defar formula= +far formula• split system into slices• use near formula for adjacent slices, far for the rest• computation time near formula O(N 2 /B)• far formula O(NR max ) = O(NB 2 )• optimal B ∼ N −1/3 , computation time O(N 5/3 )near formulaA. Arnold Long-range interactions 23/40

Dielectric contrastshttp://www.icp.uni-stuttgart.dezq∆ t ∆ bε tq∆ txl zqε mq∆ bε bq∆ b ∆ t• thin film of water in air, water at metallic plate• take into dielectric contrast by image charges• image charges take form of geometric series• handle using far formulaA. Arnold Long-range interactions 24/40

MMM2D and MMM1D in ESPResSo (AA)http://www.icp.uni-stuttgart.de• using MMM2D tuned for maximal pairwise error τcellsystem layered inter coulomb mmm2d [] \[ dielectric | dielectric-contrasts ]• allows to fix k max (p, q)-space cutoffɛ t , ɛ m , ɛ b dielectric constants or∆ t , ∆ b dielectric contrasts• requires layered cell system• number of layers per CPU n layers = B/N p is tuning parameter• using MMM1D tuned for maximal pairwise error τcellsystem nsquareinter coulomb mmm1d tune • requires all-with-all cell systemA. Arnold Long-range interactions 25/40

The method of Yeh+Berkowitzhttp://www.icp.uni-stuttgart.dereplicated slab systemslab systemreplicated slab system• potential of a charge and its periodic images similar to plate• plates cancel due to charge neutrality2πq iN∑j=1hLLzσ j (|z ji + mL z | + |z ji − mL z |) = 4πq i nL zN∑j=1σ j = 0• leave a gap and hope artificial replicas cancel( ) ∑ 2• requires changed dipole term U (d) = 2πL 3 i q iz iA. Arnold Long-range interactions 26/40

Electrostatic layer correction (ELC)http://www.icp.uni-stuttgart.de• error not known a priori — required gap size?• idea: calculate contribution of image layersLayer correction termU lc = π L 2∑k∈ 2π L Z2k 2 >0N∑i,j=1q i q je |k|z ij+ e −|k|z ijf pq (e fpqLz − 1) ei(kx x ij + k y y ij )• cut off k 2 < R max : computational effort O(NR max )• subtract numerically the ELC term ⇒ smaller gaps• 2-4x faster than plain Yeh+BerkowitzAA, J. de Joannis, and C. Holm, JCP 117:2496, 2002A. Arnold Long-range interactions 27/40

ELC in ESPResSo (AA)http://www.icp.uni-stuttgart.de• using ELC for maximal pairwise error τinter coulomb p3m tune accuracy ...inter coulomb elc [k max >] \[ dielectric | dielectric-contrasts ]• gap size g = L z − h has to be specified• user is responsible to keep a gap (by walls or fixed particles)• gap location unimportant• requires P 3 M to be switched on first• allows to fix k max (p, q)-space cutoff (otherwise tuned)ɛ t , ɛ m , ɛ b dielectric constants or∆ t , ∆ b dielectric contrastsA. Arnold Long-range interactions 28/40

Maxwell Equation Molecular Dynamicshttp://www.icp.uni-stuttgart.deA. C. MaggsJ. C. Maxwell,I. Pasichnyk1831 — 1879B. Dünweg• Electrostatics is quasi–static limit of electrodynamics• Maxwell equations are intrinsically local• dielectric constant can vary spatially• naturally include boundary conditions• ⇒ linear scaling, easy parallelization• problem: speed of lightA. C. Maggs and V. Rosetto, PRL 88:196402, 2002I. Pasichnyk and B. Dünweg, JPCM 16:3999, 2004A. Arnold Long-range interactions 29/40

Car–Parrinello Molecular Dynamicshttp://www.icp.uni-stuttgart.de• Electron dynamics in the nuclear potential isquasi–static limit of coupled electron–nuclei dynamics• problem: speed of electrons• Car–Parrinello: heavy electrons to avoid mismatch of time scales• statistics are still correctA. C. Maggs: electrostatics fromelectrodynamics with reduced speed of lightA. Arnold Long-range interactions 30/40

Maxwell equationshttp://www.icp.uni-stuttgart.deElectrostaticsis c → ∞ limit of∇ · E = 1 ɛ 0ρ and ∇ × E = 0, F = qe E(F = qe E + 1 )ɛ 0 c 2 v × H∇ · E = 1 ρɛ 0∇ × E = − 1 ∂Hɛ 0 c 2 ∂t∇ · H = 0∂E∇ × H = j + ɛ 0∂ρ∂t + ∇ · j = 0∂tWhat if c ≪ ∞?A. Arnold Long-range interactions 31/40

Electrostatics as a constraint problemhttp://www.icp.uni-stuttgart.deElectrostatic Gauss’ law as constraint∇ · E = 1 ɛ 0ρ∇ · ∂∂t E = 1 ɛ 0∂∂t ρ ⇒ ∂ ∂t E + 1 ɛ 0j − ∇ × ˙Θ = 0• solve as constrained dynamics with Lagrange multiplier• ˙Θ transversal component• equivalent to electrodynamics with ˙Θ ≡ c 2 H• but c freely choosable⇒A. Arnold Long-range interactions 32/40

Discretizationhttp://www.icp.uni-stuttgart.deDiscrete curl• discretize on simple cubic latticeof spacing a• charge on lattice sites• j, E on the links• Θ, H on plaquettesyE(2)(∇ × E) z = ∂ x E y − ∂ y E xE(1)E(3)a= 1 a [(E(3) − E(1)) − (E(2) − E(4))] E(4)xA. Arnold Long-range interactions 33/40

MEMD procedurehttp://www.icp.uni-stuttgart.de1. calculate current j from ∫ ∂V j ds = ∫ V2. calculate Ḣ = −∇ × E3. propagate H4. calculate ˙Θ = c 2 H5. calculate Ė = − 1 ɛ 0j + ∇ × ˙Θ˙ρ dV6. propagate E7. backinterpolate E on particles and calculate forceA. Arnold Long-range interactions 34/40

Self energy correctionhttp://www.icp.uni-stuttgart.de000 111000 111000 111000 11100 1100 1100 1100 11000 111000 111000 111000 111000 111000 111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000 111000 111000 111000 111• a charge “sees” its interpolated charge on the lattice• drives charge to center of mesh cell• solution: calculate lattice Green’s function and subtractA. Arnold Long-range interactions 35/40

Initial electric fieldhttp://www.icp.uni-stuttgart.de• initial E-field required for updating• either use P 3 M & Co. for this• ... or a (slow) global update scheme• allows for spatial inhomogeneous dielectric constantA. Arnold Long-range interactions 36/40

MEMD in ESPResSo (F. Fahrenberger)http://www.icp.uni-stuttgart.de• using MEMDcellsystem domain_decomposition -no_verlet_listinter coulomb maggs [ epsilon ɛ ∞]• requires the domain decomposition cellsystem, but with exactsorting (which implies not using Verlet lists)• requires to setf mass = 1/c 2n M = L/hɛ ∞mass of the field’s degree of freedommesh sizedielectric constant at infinity (defaults to “metallic”)• plain MEMD has artificial dipole term that keeps the initial dipolemoment constant• ESPResSo corrects the dipole term similar to P 3 MA. Arnold Long-range interactions 37/40

Arbitrary dielectric contrastshttp://www.icp.uni-stuttgart.de• MMM2D/ELC only handle planar parallel dielectric interfaces• what about a nanopore? vesicle?• cannot be handled by image chargesA. Arnold Long-range interactions 38/40

ICC ∗ algorithmhttp://www.icp.uni-stuttgart.de• Maxwell equation in dielectric medium• boundary condition at interfaceε in E in · n = ε out E out · n• can be fulfilled by interface charge densityσ = 12π ε ε in − ε outout Eε in + ε out• solve for σ iteratively, E from standard Coulomb solverA. Arnold Long-range interactions 39/40

ICC ∗ in ESPResSo (S. Kesselheim)http://www.icp.uni-stuttgart.de• set up the meshed interfacesdielectric sphere center \radius res eps • a is mesh size of the generated mesh• alternatively wall, pore, cylinder• creates Tcl variables with properties of thesurface points:• n induced charges• icc epsilons: list of dielectric constants, can vary per surface point• icc normals: list of normal vectors• icc areas: list of surface areas• sigmas: optional list of additional surface charge densities• surfaces charges are calculated byiccp3m $n_induced_charges epsilons $icc_epsilons \normals $icc_normals areas $icc_areas [ sigmas $icc_sigmas ]A. Arnold Long-range interactions 40/40