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

A typical experiment in a virtual space

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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 = ?

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)

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.

Periodicboundary conditions (PBC) – Born & von Karman (1912)

of Schrödinger cat

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.

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.)

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.

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

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)

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;