Lennard-Jones Potential play sheet A common INTER-molecular potential is the Lennard-Jones potential. (Lennard-Jones is one man who appended his wife's name to his own name.)This potential is commonly called the "6-12 potential" for obvious reasons. It is suitable for induced dipole - induced dipole interactions.

σ 1:= One "LJ" unit of length is the distance at which the potential goes through zero.

ε 1:= One "LJ" unit of energy is the absolute value of the energy of maximum attraction.

1. POTENTIAL (Solid line)vlj r( ) 4 ε⋅

σr

⎛⎜⎝

⎞⎠

12σr

⎛⎜⎝

⎞⎠

6

−⎡⎢⎣

⎤⎥⎦

⋅:=

flj r( )r

vlj r( )−dd

:=r

vlj r( )−( )dd

48

r13

24

r7−→2. FORCE (dashed)

When the force is > 0 the nuclei are subjected to a force to increase the relative separation, r.When the force is < 0 the nuclei are subjected to a force to decrease the relative separation.The force is zero at the potential minimum [r = 21/6σ] and is maximally attractive at the potential inflection.

0.75 1 1.25 1.5 1.75 23

2

1

0

1

2

3

vlj r( )

flj r( )

r

Plot in "LJ" units. This plot is then valid for all molecules which interact via induced dipoles. Thequestions are then: How does one scale the distance and energy axes for any given molecule?

UNITS

1. The number density.Consider one particle in a box of with linear dimension if lc = the lattice constant

ρ lc( ) 1

lc3:= LC rho( ) 1

rho0.333333:=

LC 0.2( ) 1.71=For example what size lattice do we need to get a density of 0.2?

This little calculator is useful for generating "box size" for simulations at different densities!

2. Energy and Temperature

The average <KE> = (1/2)<v2> = (3/2)T therefore T = (2/3) <KE> = (1/3)<v2>

Therefore a KE = 1 ===> T=2/3 T = 80K = 1.5 T=1 T = 119K

Now if you start calculations with velocity vectors of equal magnitude, <v2> = <v>2.While this will not be true as the sumulation runs, you can estimate T as ~ (1/3)*vinit

2.3. Planck's constant

Logic: calculate thermal wavelength of Ar at 120K in SI Λ (SI) = 2.52x10-11m = 0.0742 σ calculate thermal wavelength of Ar at T=1 in LJ Λ(LJ) = h * 0.399 σTherefore h(LJ) = 0.186

4. Time Use uncertainly relation. Logic: Calculate δt (sec) if δE is 119K, result 4.03 x10-13. but δt (LJ) = h/1 = 0.186 therefore 0.186 unit of LJ time = 4.03 x10-13 and thus

1 t (LJ) = 2.2 ps

Some LJ conversions from 2nd Virial coefs.molecule ε/k σ(Αο) bo (cc/mole)

Ne 35.6 2.75 26.2Ar 120 3.4 49.8Kr 171 3.6 58.9Xe 220 4.1 86.9

Air 99.2 3.52 55.1N2 95.1 3.70 63.8O2 117.5 3.58 57.8CO 100.2 3.76 67.2CO2 189 4.49 113.9NO 131 3.17 40N20 189 4.59 1.22CH4 148.2 3.82 70.2CF4 152.5 4.7 131.0

The second virial Coefficient of a LJ fluid sig 1:=

A. Calculate the Mayer f functionf r T,( ) e

vlj r( )−T

1−:=

0 1 2 3 41.5

10.5

00.5

11.5

2

f x 1,( )

x

B. Calculate the virial coefficient by integration

B T( ) 2− π⋅0

1000

xf x T,( ) x2⋅⌠⎮⌡

d:=

C. Reduce by the high T limit of 4* the volume of a single object.

Vo 43π⋅

sig2

⎛⎜⎝⎞⎠

3

⋅:=

b T( )B T( )4 Vo⋅

:=

0 2 4 6 8 106

4

2

0

b y( )

y