Charge equilibration Taylor expansion of energy of atom A @E E A (Q) =E A0 + Q A + 1 @Q A 0 2 Q2 A @ 2 E @Q 2 A 0 +... The corresponding energy of cation/anion and neutral atom E A (+1) = E A0 + @E @Q E A ( 1) = E A0 @E @Q + 1 A 0 2 A 0 + 1 2 @ 2 E @Q 2 @ 2 E @Q 2 A 0 A 0 E A (0) = E A0
Charge equilibration Ionization potential IP = E A (+1) E A (0) = Electron affinity @E + 1 @Q A 0 2 @E 1 EA = E A (0) E A ( 1) = @Q A 0 2 1 @E (IP + EA)= 2 @Q @ 2 E IP EA = @Q 2 =? A 0 A 0 = A @ 2 E @Q 2 @ 2 E @Q 2 A 0 A 0 electronegativity: ability of atom to gain or loose an electron
Charge equilibration How about the 2nd derivative? IP EA = @ 2 E @Q 2 A 0 =? Consider simple scheme of a single molecular orbital -1 Neutral +1 E HF =2h A + J AA E HF = h A E HF =0
Charge equilibration HF energies -1 Neutral +1 E HF =2h A + J AA E HF = h A E HF =0 IP & EA IP = h A EA = h A J AA @ 2 E Coulomb repulsion between IP EA = @Q 2 = J AA 2 electrons in orbital A 0
Charge equilibration Back to the energy of atom A E A (Q) =E A0 + 0 AQ A + 1 2 J 0 AAQ 2 A Total electrostatic energy E(Q 1,Q 2,...Q N )= X A E A + X A<B Q A Q B J AB Chemical potential A = @E @Q A = 0 A + Q A J 0 AA + X A<B Q B J AB
Charge equilibration Chemical potential Equilibrium A = @E @Q A = 0 A + Q A J 0 AA + X A<B Q B J AB A = B =... N-1 equations Total charge Q tot = NX A Q A
Charge equilibration Chemical potential A = @E @Q A = Three parameters: 0 A + Q A J 0 AA + X A<B Mulliken-Pauling electronegativity JAA 0 1 RA 0 Atom size 0 A = 1 2 J 0 AA Parr and Pearson hardness 0, 0 and 0 Q B J AB Inverse separation between atom A-B shielding constant for JAB
Charge equilibration Calculation of partial charge from atomic structure and chemical parameters SCF: expensive in computer time Not long range because need a cutoff most of the time ~ 10A ACKS2: Atom-Condensed Kohn-Sham DFT approximated to second order From Adri van Duin CH121
Long-range Coulomb Computation is very time consuming (N 2 ) Even more complicated by PBC (infinite replica) (NboxN 2 ) E 1/r 6 1/r Poor convergence due to its long-range nature 1. Ewald summation (N 3/2 ): E Coul = C distance q i q j hr 3 ij + 1 3i 1/3 split long-range/short-range real-space/reciprocal-space 2. Particle-particle particle-mesh (PPPM) method (NlogN) de Leeuw et al. Proc. Roy. Soc. Lond. A 373, 27 (1980) de Leeuw et al. Proc. Roy. Soc. Lond. A 373, 57 (1980)
BO 0 ij = exp +exp +exp Example reaxff: parametrization apple p bo,1 rij r 0 apple p bo,3 rij r 0 apple p bo,5 rij r 0 pbo,2 pbo,4 pbo,6 E bond = D e BO ij f(bo ij ) De BOij De BOij E over = f(bo ij ) E angle = 1 exp 3BO 3 a i 1 1+exp( i) 1 exp 3BO 3 b ka k a exp k b ( 0) 2 etc.
Example ReaxFF: parametrization Bond dissociation curve Angle bending
Example ReaxFF: parametrization Angle bending Reaction path Dihedral angles
Example ReaxFF: parametrization Dihedral angles ΔE = +1.98 kcal/mol Echair=-1094597.73 kcal/mol Eboat=-1094595.76 kcal/mol ΔE = +7.92 kcal/mol Eplanar= -1094589.81 kcal/mol
Example ReaxFF: parametrization Over/Under-coordination +
Example ReaxFF: parametrization Monte-Carlo Initial guess of parameters Loop over parameters Pnew(i) = P(i) + rand Error: If err < olderr: P(i) = Pnew(i) err = X i (E MD E DFT ) 2 E 0
Example ReaxFF: Development Wide range of element parametrized + some interaction between different elements
Example ReaxFF: chemistry and more
Reactive force field ReaxFF: van Duin, Adri C. T.; Dasgupta, Siddharth; Lorant, Francois; Goddard, William A. (2001). "ReaxFF: A Reactive Force Field for Hydrocarbons". The Journal of Physical Chemistry A 105 (41): 9396 9409 Adri C. T. van Duin, Alejandro Strachan, Shannon Stewman, Qingsong Zhang, Xin Xu and William A. Goddard, III ReaxFFSiO Reactive Force Field for Silicon and Silicon Oxide Systems Anthony K. Rap and William A. Goddard III Charge equilibration for molecular dynamics Charge-Optimized Many-Body (COMB) potential: T.-R. Shan, B. D. Devine, T. W. Kemper, S. B. Sinnott, and S. R. Phillpot, Phys. Rev. B 81, 125328 (2010) The reactive empirical bond-order (REBO): Stuart, Tutein, Harrison, J Chem Phys, 112, 6472-6486 (2000)
Conclusion: interatomic potentials Empirical potential type Usage Feature Names Pair-potential Noble gaz E Z LJ, Morse, Buckingham Pair-functional Metals E Z EAM Many-body potentials Semiconductors Angles Tersoff, SW Pair-potential /Reactive Organic materials Many-body/ BO DREIDING/ ReaxFF
Saving CPU time The force calculation is most expensive part in MD Interaction particle i with all its neighbors ~ N 2 Verlet list Cell list Combination Parallelism
Verlet list or neighbor list δ rc rs For each atom keep a list of the neighbor within rs = rc + δ Calculate the force if rij < rc Update the neighbor list every few timestep
Verlet list Construction Later Too late Good choice for δ Temperature, diffusion speed, density, etc. usually ~ 10-20 timestep but depend of system studied
Verlet list Nneigh i = 1 j = 1 j = 2 Nneigh i = 2 j = 1 j = 2 Nneigh i = 3 j = 1 Number of neighbor of atom i = 1 Index of neighbors of i = 1 Index of neighbors Number of of i = 2 neighbor of atom i = 2 Loop over i = 1 to N Loop over j neighbor of i However building the list still N 2
Linked cell method Bin the atoms into 3D cells of size d rc Only check the atoms in the 26 surrounding cells Binning requires N operations
Combined neighbor-list / linked-cell Bin the atoms in 3D cells every few timestep: build the neighbor-list Compute the force using the neighbor-list: Less atoms in a sphere of volume 3/4πrs 3 than cube 27rc 3 Speed up with Newton s 3rd law: half-list, etc. 2 1 3
Parallelism: Spacial decomposition The physical simulation domain is divided in small 3D boxes, one for each processor Each processor compute the force and update positions within the box Atom are reassign to other processor as they move through the physical domain In order to compute the force, the processor need to know the positions of atoms nearby boxes only (26) proc 1 proc 2 etc. Empty space will slow the scaling down
Periodic boundary conditions ghost cells containing ghost atoms
Parallelism: other algorithm Atom decomposition: Each processor receive a subset of atom N/P Each processor keep updating positions all along the simulation no matter the displacement of the atoms (local integration) Each processor need to know the whole vector position Force decomposition Each processor receive a subset of the force matrix Each processor need to know only part of the position vector J. Comput. Phys. 117, 1-19 (1995)
Macroscopic property from MD Macroscopic property from microscopic simulation Diffusion: Fick s law of diffusion Ex. study diffusion of copper in silica Z r 2 dr @c(r, t) @t = Dr 2 c(r, t) Diffusion constant @ @t <r2 (t) >= 2dD Microscopic (average displacement) Concentration of diffusive specie (defect, etc.) Laplacian Macroscopic (diffusion constant)
Mean square displacement <r 2 (t) >= 1 N NX r i (t) 2 = MSD i=1 Solid (crystal) vibrate around equilibrium Liquid diffusion
Copper diffusion in a-sio2 10 5 10 5 10 0 10 0 Log(MSD) (Å 2 ) 10 5 10 5 10 0 10 5 10 5 800K 10 4 10 2 10 0 1200K 10 4 10 2 10 0 10 5 10 5 10 0 10 5 10 5 2000K 10 4 10 2 10 0 2200K 10 4 10 2 10 0 10 0 10 0 10 5 10 4 10 2 10 0 Log(t) (ns) 1600K 10 5 10 4 10 2 10 0 Log(t) (ns) 2400K Temp%(K)% D%(cm 2 /s)% Temp%(K)% D%(cm 2 /s)% Temp%(K)% D%(cm 2 /s)% 800# 4.5493E*07# 1400# 1.8933e*05# 2000# 6.4334e*05# 1000# 2.3960E*06# 1600# 2.6278e*05# 2200# 9.4395e*05# 1200# 7.3527e*06# 1800# 5.6319e*05# 2400# 1.0102e*04#
Copper diffusion in a-sio2 9 10 Ln(D) vs 1/T Y= 6.2939 6.6220E+03X E A =0.566 ev 11 Ln(D) 12 13 14 E A = 13.0523kcal / mol = 0.566eV 15 4 5 6 7 8 9 10 11 12 13 14 1/T (K 1 ) x 10 4
Velocity auto-correlation function @ @t <r2 (t) >= 2dD x = D = Z 1 With MSD be careful at PBC 0 Z t 0 vdt d < v x ( )v x (0) > Ensemble average velocity-velocity auto-correlation function No need with velocity Limits of velocity auto-correlation function: Small tau Large tau
Velocity auto-correlation function @ @t <r2 (t) >= 2dD x = D = Z 1 If period of tau close to the oscillation of your system: peaks in auto-correlation function 0 Z t 0 vdt d < v x ( )v x (0) > Ensemble average velocity-velocity auto-correlation function correlation Fourier transform of auto-correlation function Integral: D no more correlation
Radial distribution function How density varies as a function of distance from a reference particle Probability finding particle at r away from a given reference particle g(r) =4 r 2 dr
Nudged elastic band Calculate the barrier between two configurations: activation energy 1. Extrapolate the path between the two configurations 2. Relax the structures by constraining successive configurations
Nudged elastic band Dehydrogenation of ethene C 2 H 6 C 2 H 4 + H 2
Copper diffusion in silica Nudged elastic band