Computer simulation methods (2) Dr. Vania Calandrini
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1 Computer simulation methods (2) Dr. Vania Calandrini
2 in the previous lecture: time average versus ensemble average MC versus MD simulations equipartition theorem (=> computing T) virial theorem (=> computing P) position distribution function sampling: mechanical properties versus entropic properties
3 Setting up and running a simulation - energy model - initial configuration possibly close to the state we wish to simulate but avoiding hot spots ( for example by performing energy minimization), or using a crystal structure, or by theoretical modeling,... - equilibration phase the system evolves from initial configuration to achieve stable structural and thermodynamical properties. -production phase to compute the properties of the system
4 Boundaries size of a MD system ~ particles << strong boundary effects Periodic boundary conditions (PBC): it enable a simulation to be performed using a relatively small number of particles in such a way that the particles experience forces as if they were in the bulk fluid The original box is replicated in all directions to give a periodic array. Should a particle leave the box during the simulation then it is replaced by an image particle that enters from the opposite side. The number of particles in the central box remains constant. replicas of a 2D box of particles The coordinates of the particles in the images boxes can be computed by adding or subtracting integral multiples of the box side
5 Fundamental cells Any cell shape can be used provided it fills all of the space by translation operations of the central box. The idea is to choose a cell that reflects the underlying geometry of the system (to minimize the number of simulated particles), and that allows at the same time an efficient calculation of the images.
6 Remarks: Limitations of periodic boundary conditions (PBC) : It is not possible to achieve system fluctuations that have a wavelength greater than the length of the cell (example near liquid-gas critical point). The cell size should be large compared to the range over which the interactions act, in order to minimize possible unwanted long-range order (example the interactions of a particle with its replicas). For relatively short range L-J potential the cell should have a side L ~ 6σ. For longer-range electrostatic interactions the situation is more critical and some long-range order may be imposed upon the system.
7 Non periodic boundary methods PBC are not always used in computer simulations: Some system, such as liquid droplets or van der Waals clusters, inherently contain a boundary. Periodic boundaries may cause difficulties when simulating inhomogeneous systems and/or systems that are not at equilibrium. In other cases the use of pbc would require a prohibitive number of atoms to be included in the simulation (example: the water box in which we simulate a protein have to be large enough to guarantee that the protein does not interact with its own replicas): # simulating isolated molecules in vacuo, although vacuum boundaries tend to minimize the surface area and so may distort the shape if the system is not spherical inducing more compact conformations. # simulating a molecules surrounded by a skin of solvent (equivalent to a molecule in a drop). Boundaries problem transferred to the solvent-vacuum interface. # reaction zone: all atoms within a given radius of interest are fully simulated, while outside they are fixed or restrained to the initial positions by harmonic potentials. This may introduce artificial behavior preventing natural changes to occur.
8 Monitoring the equilibration equilibration: the system evolves from the initial configuration to reach equilibrium. Monitored properties: - energy - temperature - pressure - order parameter (measure of the degree of order or disorder): Verlet order parameter: to measure the translational order in a liquid: Viellard-Baron rotational order parameter: to measure the rotational order of linear molecules: system initially in a facecentered cubic lattice: ri(0)=(l,m,n) a/2 λ(0)=1 for t, ri are randomly distributed and λ 0 γ is the angle between the current and original position of molecule i. P1 1 when the molecules are aligned, P1 0 when the molecules are randomly oriented. - mean squared displacement: measure of the space explored by a random walker for an unconfined motion the MSD grows with time, while for a confined motion MSD saturates to a finite value
9 Short-range an long-range interactions If the potential drops down to zero faster than r d, where r is the separation between two particles and d the dimensionality of the problem, it is called short ranged, otherwise it is long ranged. Short-range interactions: truncation and minimum image convention (ex. LJ): The potential energy of a particle gets the main contributions from the particles below a threshold distance. A cut-off can thus be introduced with the minimum image convention. 1. cutoff: correction here g(r) = ρ(r)/ρ0 is the radial distribution function: it describes how density varies as a function of distance from a reference particle. In eq. ( ), the cutoff RC may be chosen such that g(r) = 1 for r > RC. The longrange correction Ulrc can thus be computed analytically at the end of the simulation ; its value converges to a finite value for a LJ potentials, ( ) hint: E (LJ) lrc ( ) σ 12 =8πρN 9r 6 σ6 9 3r Rc 3 Rc
10 ULJ(2.4Rmin) = 1% ULJ,min ULJ,min ULJ(R = σ) = 0 Rmin = 1.12σ
11 2. minimum image convention: The interaction is calculated with the closest atom or image inside the cutoff. β Example the interaction between α and β atoms: α β in this example the interaction is computed between α and the image βʼ When periodic boundary conditions and cutoff are being used, the cutoff should not be so large that a particle sees its own image => the cutoff have to be no more than half the length of the cell. Minimum image convention allow one to avoid potential energy jumps when one atom involved in an interaction leaves the central box
12 Notes on cutoff: Non-bonded Neighbour List If we would have to calculate the distance between atoms in the system simply to decide whether they are close enough to calculate their interaction energy, there is no gain in computing time. Instead, we should know in advance which atoms to include in the non-bonded calculations. Idea: non-bonded (Verlet) neighbor list = it stores all atoms within the cutoff distance, together with all atoms that are slightly further away than the cutoff distance (remark: in simulation of fluids, atomʼs neighbors do not change significantly over 10 or 20 molecular dynamics time steps or Monte Carlo iterations). A pointer array P indicates where in the neighbor list array L the first neighbor for an atom is located. The last neighbor of atom i is stored in element P[i+1]-1 of the neighbor list (see figure) thus the neighbors of atom i are stored in elements L[P[i]]... L[P[i+1]-1] of the array L. The distance used to compute each atomʼs neighbors should be larger than the cutoff Rc. The neighbor list is updated regularly, with efficient frequency (usually every steps), and when the sum of the maximum displacements of any two atoms exceeds the difference Rc - {neighbor list distance}. Electrostatic + VdW interactions: use two cutoff (twin range method)
13 Notes on cutoff: Group-based Cutoffs When simulating large molecular systems, it is often more convenient to divide the large molecules into ʻgroupsʼ, each of which contains a relatively small number of connected atoms and introduce a group based cutoff. This allows one to avoid spurious numerical effects while fine cancellations occur in the computation of the interaction energy among components around or beyond the cutoff scale (see for example the electrostatic interactions between two water molecules using TIP3P = 8Å: EO-O = +29 kcal/mol, EO-H = kcal/mol, EH-H = 29.2 kcal/mol Eint R = 8Å energy evolution with an atom based cutoff: Rc = 8Å......huge fluctuations! because only some of the pairwise interactions are included here energy evolution without a cutoff Rc = 8Å? How should a molecule be divided into groups? by distinct chemical residues (e.g. amino-acid in proteins and peptides), possibly by groups of zero charge it is better to calculate the interactions on a group-group basis! *** remark: if the groups are electrically neutral, then the leading term in the electrostatic interactions between a pair of groups is the dipole-dipole interaction (~1/r 3 ).
14 Notes on cutoff: Group-based Cutoffs How do we decide weather a particular group-group interaction need to be considered? strategy: if any pair of atoms in the two groups, or alternatively any ʻmarker atomsʼ in the two groups are closer than the cutoff distance. beware of using chemically obvious grouping! long side chains!
15 Arginine Asparaginine
16 Notes on cutoff: Problems with Cutoff and How to Avoid Them A cutoff introduces a discontinuity in both the potential energy and the force near the cutoff value. This creates problems, especially when energy conservation is required. Several solutions exists. 1) Shifted potential: constant term is subtracted from the potential at all values Example: vc = v(rc) Remarks a constant term does not affect the dynamics, it disappears when the potential is differentiated; a shifted potential does change the total energy; it does introduce a discontinuity in the force field at R = Rc 2) Shifted potential: a constant and a linear term are added to the potential...but again, thermodynamic properties will be changed
17 Notes on cutoff: Problems with Cutoff and How to Avoid Them A cutoff introduces a discontinuity in both the potential energy and the force near the cutoff value. This creates problems, especially when energy conservation is required. Several solutions exists. 3) Switching function: polynomial function of the distance multiplying the potential. E.g.:. Drawbacks equilibrium structures are affected... Solution introduce a switching function between two relatively close cutoff values, with first and second derivatives vanishing at the cutoffs. =>forces approaches zero smoothly! NOTE: when a group-based cutoff is used, a better energy conservation can be achieved using the same switching function for all the atoms within a group.
18 Long-range interactions (interactions that decay no faster than r -d ; ex. Coulomb): The interactions between all particles in the system must be taken into account. One cannot use simple truncation plus tail corrections. For open boundary conditions (like liquid droplets): the computation is straightforwardly implemented and reduces to a double sum over all pairs of particles. For periodic boundary conditions: the interactions with particles in the central cell as well as all periodic images must be taken into account: U = 1 2 N i,j=1 n q i q j r ij + nl conditionally convergent series, i.e. it depends on the sequence of evaluating the sum where n is a lattice vector and Σn means that for n = 0 it is i j. A variety of methods have been developed to handle long-range forces. Some examples: 1. Ewald summation method: O(N 3/2 ) 2. Particle-Mesh based techniques [by FFT method]: O(N logn ) 3. Reaction field and Image charge methods [for dipole-dipole interactions]: O(N) 4. Cell multipole method (or Fast multiple method): O(N)
19 1. Ewald summation charge density: sum of δ-functions periodic boundary conditions n=0 density charge - n 0 electrostatic potential electrostatic energy n=0: actual cell sum over image cells
20 1. Ewald summation charge density: sum of δ-functions = charged cloud with + opposite sign each point-charge is surrounded by a diffuse charge of opposite sign, such that Σq=0 (screening) + = [long-range potential + self-energy] [short-range potential] idea: φ(r) 1 r f(r) r = + 1 f(r) r computed......in Fourier space...in real space
21 1. Ewald summation - intermezzo: Fourier transform
22 1. Ewald summation - intermezzo: Poisson equation by Fourier transform left part: right part: solution: -
23 1. Ewald summation - computation of long-range term in Fourier space -
24 1. Ewald summation - computation of long-range term in Fourier space inverse Fourier transform: defined for k 0 (consequence of conditional convergence) - the narrower the Gaussian, the faster the series converges (#) (#) converges strongly for large k-vectors and thus contains mainly long range contributions
25 1. Ewald summation - long-range term: correction for self-energy term (#) after two partial integrations we obtain x - to be subtracted from the total Coulomb energy independent of charge position
26 1. Ewald summation - computation of short-range term in real space Using the previous results (point charge & self-energy) one can easily obtain + φ(r) U short-range = 1 2 N i,j=1 n q i q j erfc( α r ij + nl ) r ij + nl convergent sum since erfc(x) decay as exp(-x 2 ) for large x the wider the Gaussian the faster the series converges
27 1. Ewald summation - summary + = [long-range potential + self-energy] N i,j= [short-range potential] n q i q j erfc( α r ij + nl ) r ij + nl (self-energy correction) α governs the relative convergence of the two main series: -- usually α is chosen to reduce the evaluation of the real-space sum to the contributions of particle pairs separated by a distance r<r cut ; -- the reciprocal-space sum is conventionally truncated after a k max value. α, R cut and k max may be chosen in an optimal way to balance the truncation error in each sum.
28 2. Ewald+ method: speed-up by particle-mesh (PM) technique / FFT + = + [Hockney & Eastwood 1998] - [long-range potential + self-energy, Fourier space] [short-range potential, real space] FFT method -- O(N log N): it requires that data are not continuous but are discrete values => discrete charges (o, ) with their continuous coordinates are replaced by a grid-based charge distribution, distributing the atomic point charges over the surrounding grid-points in some fashion so as to reproduce the potential of the charge at the original location. Compromise: the finest the grid, the most accurate the approximation of the original potential, the greater the computational cost per particle.
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