P

T

A typical experiment in a virtual space1. Some material is put in a container at fixed T &

P.2. 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.

3. An apparatus is plugged to measure an observable (a macroscopic quantity) as an average over all the microstates produced from thermal fluctuation.

P

T

How would you build a model system representing a microstate of a water boiler (L~10 cm)? N = ?

P

T

microscopic states (microstates)or microscopic configurationsunder external

constraints (N or , V or P, T or E,

etc.) Ensemble (micro-canonical,

canonical, grand canonical, etc.)

Themodynamic limit (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 cubic crystal state

- N = 10 x 10 x 10 = 103 : ~60% surface atoms- N = 104: ~30% surface atoms- N = 105: ~13% aurface atoms- N = 106: ~6% surface atoms (but big computational system!)

Ns/N ~ 6 x N2/3 / N ~ 6 / N1/3

(exact calculation: Ns = 6 x(N1/3-2)2 + 12 x (N1/3-2) + 8. For N = 103, 49% surface atoms)

Periodic boundary conditions (PBC) – Born & von Karman (1912)

(from Allen & Tildesley)

- 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.

A … H: images of the cell

Cell does not have to be cubic.

Periodic boundary conditions (PBC) – Born & von Karman (1912)

of Schrödinger cat

Periodic boundary condition and nonbonded interactions

usually non-bonded pair interaction

Lrc

L

x x x x

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.)

Cutoff for Long-Range Non-bonded Interactions

Only interactions consideredOnly interactions considered

5000-atom system

Minimizing discontinuity. Spline, a possible choice

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.

Effective potential = actual potential smoothing function S(r)

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

Cutoff for Long-Range Non-bonded Interactions

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)

Periodic boundary condition: Implementation (2d case)

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;

y

0 Lx/2-Lx/2

Ly/2

-Ly/2

i

xi

yi

xxi-Lx

2. Scaled (between [-0.5,0.5]) coordinates (better to handle any cell shape): orthorombic cell 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;