Coulomb interactions: P3M, MMMxD, ELC and ICC
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(2) Electrostatics Coulomb energy Pair energy summation. Bjerrum length. http://www.icp.uni-stuttgart.de. N. l X0 qi qj U= B 2 |rij |. lB =. e2 4π0 r kB T. i,j=1. summing up 1/r Coulomb pair potential. electrostatic prefactor ∝ inverse temperature. Bjerrum length lB. for two unit charges:. 1kBT 1lB. A. Arnold. Coulomb interactions. 2/26.
(3) Electrostatics Coulomb energy Pair energy summation. http://www.icp.uni-stuttgart.de. U=. N lB X0 qi qj 2 |rij |. Potential summation N. U=. i=1. i,j=1. summing up 1/r Coulomb pair potential. 1X qi φ(ri ) 2. potential from solving Poisson’s equation. Bjerrum length lB ∇2 φ(r) = −4πlB. N X. qj δ(rj − r). j=1. equivalent approaches A. Arnold. Coulomb interactions. 2/26.
(4) Electrostatics in periodic boundary conditions Coulomb energy. http://www.icp.uni-stuttgart.de. Pair energy summation. Potential summation. N ∞ qi qj lB X X X 0 U= 2 |r ij + mL| 2. N. U=. i=1. S=0 m =S i,j=1. conditionally convergent — summation order important. 1X qi φper (ri ) 2. solve Poisson’s equation imposing periodic boundaries. numerically difficult U not periodic in coordinates ri. U is periodic in coordinates ri. these two calculate something different!. A. Arnold. Coulomb interactions. 3/26.
(5) Where the difference comes from: the dipole term assume summation in periodic shells surrounded by polarizable material of dielectric constant ∞ = ∞. http://www.icp.uni-stuttgart.de. 3. 3 2. 1. 1. 0. 1. 2. 1. 2. =1. 3. Pair energy summation vacuum around: ∞ = 1. 2 3. = ∞. Potential summation periodic: ∞ = ∞. difference to periodic solution is nonperiodic dipole term X 2 2π (d) U = qi ri (1 + 2∞ )L3 i. metallic boundary conditions ∞ = ∞ always safe never use ∞ < ∞ for conducting systems A. Arnold. Coulomb interactions. 4/26.
(6) Electrostatics in ESPResSo requires myconfig.h-switch ELECTROSTATICS switching on: inter coulomb <lB > < method > < parameters >. methods and their parameters: next 2 hours http://www.icp.uni-stuttgart.de. switching off: inter coulomb 0. getting lB , method and parameters: inter coulomb. returns 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. Coulomb interactions. 5/26.
(7) Assigning charges part 0 pos 0 0 0 q 1 part 1 pos 0 .5 0 0 q -1.5. Adding a charged plate. http://www.icp.uni-stuttgart.de. constraint plate height <h> sigma <σ >. plate parallel to xy-plane at z = h, charge density σ requires 2D periodicity (nonperiodic in z). Adding a charged rod constraint rod center <cx > <cy > lambda <λ>. rod parallel to z-axis at (x, y) = (cx , cy ), line charge density λ requires 1D periodicity (periodic in z) A. Arnold. Coulomb interactions. 6/26.
(8) P. P. Ewald, 1888 — 1985 Coulomb potential has 2 problems 1. singular at each particle position 2. very slowly decaying Idea: separate the two problems! one smooth potential — Fourier space one short-ranged potential — real space A. Arnold. Coulomb interactions. P. P. Ewald, Die Berechnung optischer und elektrostatischer Gitterpotentiale, Ann. Phys. 369(3):253, 1921. http://www.icp.uni-stuttgart.de. The Ewald method. 7/26.
(9) Ewald: splitting the potential charge distribution ρ=. N X X. qi δ(r − ri − n). http://www.icp.uni-stuttgart.de. n∈LZ3 i=1. =. +. replace δ by Gaussians of width α−1 : √ 3 2 2 ρGauss (r) = α/ π e−α r δ(r) = ρGauss (r) + [δ(r) − ρGauss (r)] A. Arnold. Coulomb interactions. 8/26.
(10) The Ewald formula U = U (r ) + U (k) + U (s) with U (r ) =. erfc(α|rij + mL|) lB X X0 qi qj 2 |rij + mL| 3 i,j. m∈Z. http://www.icp.uni-stuttgart.de. real space correction. U (k). l X 4π −k 2 /4α2 = B3 e |b ρ(k)|2 2L k2. U (s). αl X 2 = − √B qi π. Gaussians in k -space. k6=0. Gaussian self interaction. i. forces from differentiation ∂ U ∂ri ... coming soon to ESPResSo (on GPU) Fi = −. A. Arnold. Coulomb interactions. 9/26.
(11) discrete FT is exact — constant real space cutoff computational order O(N log N) most frequently used methods: P3 M: optimal method PME SPME A. Arnold. Coulomb interactions. M. Deserno and C. Holm, JCP 109:7678, 1998. R. W. Hockney J. W. Eastwood replace k -space Fourier sum by discrete FFT. R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, 1988. http://www.icp.uni-stuttgart.de. Mesh-based Ewald methods. 10/26.
(12) Steps of P3 M 1. {ri , qi } → ρ(r): interpolate charges onto a grid (window functions: cardinal B-splines). http://www.icp.uni-stuttgart.de. 2. ρ(r) → ρb(k):. Fourier transform charge distribution. 3. φ̂(k) = Ĝ(k)ρ̂(k): solve Poisson’s equation by multiplication with optimal influence function Ĝ(k) (in continuum: product of Green’s function 4π and k2 2 /4α2 −k Fourier transform of Gaussians e ) 4. ikφ̂(k) → Ê(k):. obtain field by Fourier space differentiation. b 4. E(k) → E(r):. Fourier transform field back. 5. E(r) → {ri , Fi }: interpolate field at position of charges to obtain forces Fi = qi Ei A. Arnold. Coulomb interactions. 11/26.
(13) Charge assignment q7. q8. 1. 1. 2. 0.8. q5. 3 4. M (P ) (x). q6. 5. 0.6. 6. 7. 0.4. q3. http://www.icp.uni-stuttgart.de. q1. a. q4. 0.2. q2. 0. 0. 1. 2. 3. 4. 5. 6. 7. x. interpolate charges onto h-spaced grid ρM (rp ) =. N 1 X qi W (p) (rp − ri ) h3 i=1. W (p) (r) cardinal B-splines in P3 M / SPME A. Arnold. Coulomb interactions. 12/26.
(14) Optimal influence function. Ĝopt (k) = h. 6 ik. ·. P. 2. |k|2. http://www.icp.uni-stuttgart.de. 2π 2π b h m)R(k + h m) 2 2 P [ (p) (k + 2π m) W m∈Z3 h. m∈Z3. [ (p) (k + W. aliasing of continuum force 4π 2 2 b R(k) = −ik 2 e−k /4α k with differentiation, Green’s function and transform of Gaussian minimizes the rms force error functional Z Z 2 1 3 Q[F ] := 3 d r1 d3 r F(r; r1 ) − R(r) h h3 V A. Arnold. Coulomb interactions. 13/26.
(15) Why to control errors rms force error ∆F =. q. s (Fexact − FEwald )2 =. 10. 1 N. N P i=1. ∆Fi2. rmax=1, kmax=10 rmax=2, kmax=10 rmax=1, kmax=20. 1. http://www.icp.uni-stuttgart.de. ∆F. 0.1 0.01 0.001 0.0001 1e-05 0. 1. 2. 3. 4. 5. α. optimal α brings orders of magnitude of accuracy at given required accuracy, find fastest cutoffs compare algorithms at the same accuracy. A. Arnold. Coulomb interactions. 14/26.
(16) How to: error estimates 10. total error real space estimate k-space estimate. 1. ∆F. 0.1 0.01 0.001 0.0001. http://www.icp.uni-stuttgart.de. 0. 1. 2. 3. 4. 5. α. Kolafa and Perram: ∆Freal. P 2 q 2 2 ≈ √ i p exp −α2 rmax N rmax L3. Hockney and Eastwood: ∆FFourier A. Arnold. P 2 q ≈ √ i N. s Q[Ĝopt (k)] L3. Coulomb interactions. 15/26.
(17) P3 M in ESPResSo (F. Weik, H. Limbach, AA) tune P3 M for rms force error τ. http://www.icp.uni-stuttgart.de. inter coulomb <lB > p3m tune accuracy <τ > \ [ r_cut <rmax >] [ mesh <nM >] [ cao <p >] inter coulomb epsilon ∞. computes optimal α tunes for optimal speed rmax real space cutoff (0 to retune) nM = L/h mesh size (0 to retune) p charge assignment spline order p (0 to retune) fixing parameters speeds up tuning, if you know the optimal value second command to set ∞ (defaults to ∞ (“metallic”)) manually set parameters (dangerous!) inter coulomb <lB > p3m <rmax > <nM > <p> <α>. A. Arnold. Coulomb interactions. 16/26.
(18) http://www.icp.uni-stuttgart.de. Partially periodic systems. partially p. b. c. for slablike systems (surfaces, thin films) ... or for cylindrical systems (rods, nanopores) dielectric contrasts at interfaces P3 M cannot be employed straightforwardly. A. Arnold. Coulomb interactions. 17/26.
(19) Another approach: MMM2D far formula φβ (r ) =. e−β. X. √. (x+k )2 +(y +l)2 +z 2. p (x + k)2 + (y + l)2 + z 2 k ,l∈LZ q q 2 X X 2 2 2 2 = K0 β +p (y + l) + z eipx L 2π p∈ L Z. http://www.icp.uni-stuttgart.de. =. 2π L2. l∈LZ. X p,q∈ 2π Z L. √ 2 2 2 e− β +p +q |z| ipx iqy p e e 2 2 β + p + q2. 2π X = 2 L 2 2. p +q >0. π efpq |z| ipx iqy e e + |z| + 2 β −1 + Oβ→0 (β) fpq L. screened Coulomb interaction in limit of screening length ∞ other formula for z ≈ 0 optimal computation time O(N 5/3 ), comparable to Ewald analogously for 1d, but then O(N 2 ) A. Arnold. Coulomb interactions. 18/26.
(20) Dielectric contrasts z. water. electrode. q∆t ∆b εt q ∆t. membrane. lz. q. http://www.icp.uni-stuttgart.de. q ∆b. εm. x. εb. q ∆b∆t. water. water. electrode. typical two dimensional systems: thin films, slit pores material boundaries =⇒ dielectric contrast take into account polarization by image charges can be handled by MMM2D A. Arnold. Coulomb interactions. 19/26.
(21) MMM2D and MMM1D in ESPResSo (AA) using MMM2D tuned for maximal pairwise error τ. http://www.icp.uni-stuttgart.de. cellsystem layered <nlayers > inter coulomb <lB > mmm2d <τ > [ <kmax >] \ [ dielectric <t > <m > <b > | d i e l e c t r i c - c o n t r a s t s <∆t > <∆b >]. allows to fix kmax (p, q)-space cutoff t , m , b dielectric constants or ∆t , ∆b dielectric contrasts requires layered cell system number of layers per CPU nlayers = B/Np is tuning parameter using MMM1D tuned for maximal pairwise error τ cellsystem nsquare inter coulomb <lB > mmm1d tune <τ >. requires all-with-all cell system A. Arnold. Coulomb interactions. 20/26.
(22) The method of Yeh+Berkowitz replicated slab system slab system h Lz. http://www.icp.uni-stuttgart.de. replicated slab system L. potential of a charge and its periodic images similar to plate plates cancel due to charge neutrality 2πqi. N X. σj (|zji + mLz | + |zji − mLz |) = 4πqi nLz. j=1. N X. σj = 0. j=1. leave a gap and hope artificial replicas cancel 2 2π P (d) requires changed dipole term U = L3 i qi zi A. Arnold. Coulomb interactions. 21/26.
(23) Ulc =. N e|k|zij + e−|k|zij i(kx xij + ky yij ) π X X qi qj e 2 L fpq (efpq Lz − 1) 2π 2 k∈ L Z k2 >0. i,j=1. error not known a priori — required gap size? calculate contribution of image layers subtract numerically ⇒ needs smaller gaps 2-4x faster than plain Yeh+Berkowitz A. Arnold. Coulomb interactions. AA, J. de Joannis, and C. Holm, JCP 117:2496, 2002. http://www.icp.uni-stuttgart.de. Electrostatic layer correction (ELC). 22/26.
(24) ELC in ESPResSo (AA) using ELC for maximal pairwise error τ inter coulomb <lB > p3m tune accuracy <τ 0 > ... inter coulomb elc <τ > <g > [kmax >] \ [ dielectric <t > <m > <b > | d i e l e c t r i c - c o n t r a s t s <∆t > <∆b >]. http://www.icp.uni-stuttgart.de. gap size g = Lz − h has to be specified user is responsible to keep a gap (by walls or fixed particles) gap location unimportant requires P3 M to be switched on first allows to fix kmax t , m , b ∆t , ∆b. A. Arnold. (p, q)-space cutoff (otherwise tuned) dielectric constants or dielectric contrasts. Coulomb interactions. 23/26.
(25) http://www.icp.uni-stuttgart.de. Arbitrarily shaped dielectric surfaces. MMM2D/ELC only handle planar parallel dielectric interfaces what about a nanopore? vesicle? cannot be handled by image charges satisfy boundary constraints for electric field. A. Arnold. Coulomb interactions. 24/26.
(26) ICC∗ algorithm boundary condition at interface. http://www.icp.uni-stuttgart.de. εin Ein · n = εout Eout · n. nk. Ak. can be fulfilled by interface charge density σ=. ε − εout 1 εout in E(σ) 2π εin + εout. represent σ as charges qk at fixed positions at interface solve for qk iteratively, E from standard Coulomb solver ε − εout qkl+1 = (1 − ω) qkl + ωAk in nk · E [qjl ] εin + εout A. Arnold. Coulomb interactions. 25/26.
(27) ICC∗ in ESPResSo (S. Kesselheim) set up the meshed interfaces. http://www.icp.uni-stuttgart.de. dielectric sphere center <x y z > \ radius <r > res <a> eps <εin >. a is mesh size of the generated mesh alternatively wall, pore, cylinder creates Tcl variables with properties of the surface 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 by iccp3m $ n _ i n d u c e d _ c h a r g e s epsilons $icc_epsilons \ normals $icc_normals areas $icc_areas [ sigmas $icc_sigmas ]. A. Arnold. Coulomb interactions. 26/26.
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