A typical experiment in a virtual space

1. A typical experiment in a virtual space • Some material is put in a container at fixed T & P. • The material is in a thermal fluctuation, producing lots of different configurations (a set of microscopic states) for a given amount of time. It is the Mother Nature who generates all the microstates. • An apparatus is plugged to measure an observable (a macroscopic quantity) as an average over all the microstates produced from thermal fluctuation. P P T T microscopic states (microstates) or microscopic configurations P under external constraints (N or , V or P, T or E, etc.)  Ensemble (micro-canonical, canonical, grand canonical, etc.) T How would you build a model system representing a microstate of a water boiler (L~10 cm)? N = ?

2. Themodynamiclimit (V →∞) and simulation • Particles (atoms, molecules, macromolecules, spins, etc.) are confined in a finite-size cell. • Particles are in interaction: Time taken to evaluate the interaction energy or force ~ O(N2). • - bonded interactions (bonds, angles, torsions) to connect atoms to make molecules • - nonbonded interactions (between distant atoms) • Particles on the surface of the cell will experience different interactions from those in the bulk! • The total number of particles is always « small » (with respect to NA): the fraction of • surface particles will significantly alter the average of any observables with respect to • the expected value in the thermodynamic limit (V →∞). Example: simple atomic system with N particles in a simple cubiccrystal state Ns/N ~ 6 x N2/3 / N ~ 6 / N1/3 • N = 10 x 10 x 10 = 103 : ~60% surface atoms • - N = 104: ~30% surface atoms • N = 105: ~13% aurfaceatoms • N = 106: ~6% surface atoms (but bigcomputational system!) (exact calculation: Ns = 6 x(N1/3-2)2 + 12 x (N1/3-2) + 8. For N = 103, 49% surface atoms)

3. Periodicboundary conditions (PBC) – Born & von Karman (1912) (from Allen & Tildesley) A … H: images of the cell Celldoes not have to becubic. • - When a particle leaves the cell, one of its images comes in. • Images are not kept in memory: Particle position after a move is checked and « folded » • back in the cell if necessary. • Surface effects are removed, but the largest fluctuations are ~L (cell size). • If the system exhibits large fluctuations (for example, near a 2nd order phase transition), • PBC will still lead to artefacts (finite-size effects). • - Finite-size effects can be studied by considering cells of different sizes.

4. Periodicboundary conditions (PBC) – Born & von Karman (1912) of Schrödinger cat

5. Periodicboundary condition and nonbonded interactions L rc L usuallynon-bonded pair interaction • 2 possibilities: • minimum image convention: consider only nearest image of a given particle when looking • for interacting partners. Still expensive (~N2 pairs) if the cell is large! • - Example: cell L centered on 1, interactingwith 2 and nearest images of 3, 4 and 5 • cutoff: truncate the interaction potential at a cutoff distance rc (No interaction if the distance • between a pair isgreaterthanrc). Sphere of radius rciscenteredeachparticle. • - Remark: usuallyrc <= L/2 in order to satisfied the minimum image convention.

6. 5000-atom system x x x x Cutoff for Long-Range Non-bonded Interactions • Direct method (simplest) • Interactions are calculated to a cutoff distance. • Interactions beyond this distance are ignored. • Leads to discontinuities in energy and derivatives. • As a pair distance moves in and out of the cutoff • range between calculation steps, the energy jumps. • (since the non-bond energy for that pair is included • in one step and excluded from the next.)

7. Minimizing discontinuity. Spline, a possible choice Effective potential = actual potential  smoothing function S(r) • Switching function S(r) • = 1 for small r • = 1  0 smoothly at intermediate r • = 0 for large r • Should be continuously differentiable • (so that forces can be calculated). • Smoothly turns off non-bond interactions over a range of distances. • Switching range is important. • Upper limit = the cut-off distance. • Too large lower limit (small spline width)  Unrealistic forces may result. • Too small lower limit  The feature of the equilibrium region may be lost.

8. Cutoff for Long-Range Non-bonded Interactions Number of non-bond interactions for a 5000-atom system as a function of cutoff distance vdW energy of a hexapeptide crystal as a function of cutoff distance, which does not converge until 20 Å

9. Estimating Non-bonded (esp. Electrostatic) Energy for Periodic Systems: Ewald Summation For details, read Leach (pp.324-343), Allen & Tildelsley (Ch.5), and reading materials (Kofke)

10. Periodicboundary condition: Implementation (2d case) y Ly/2 1. Real coordinates /* (xi, yi) particle i coordinates */ if (xi > Lx/2) xi = xi – Lx; else if (xi < -Lx/2) xi = xi + Lx; if (yi > Ly/2) yi = yi – Ly; else if (yi < -Ly/2) yi = yi + Ly; i yi xi -Lx/2 0 Lx/2 xi-Lx x -Ly/2 2. Scaled (between [-0.5,0.5]) coordinates (better to handleanycellshape): orthorombiccell case #define NINT(x) ((x) < 0.0 ? (int) ((x) - 0.5) : (int) ((x) + 0.5)) sxi = xi / Lx; /* (sxi, syi) particle i scaled coordinates */ syi = yi / Ly; sxi = NINT(sxi); /* Apply PBC */ syi = NINT(syi); xi = sxi * Lx; /* (xi, yi) particle i folded real coordinates */ yi = syi * Ly;