Molecular dynamics implementation cheat sheet

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A cheat-sheet for implementing a molecular dynamics simulation
Source: "Nanosystems: Molecular Machinery, Manufacturing, and Computation"

The MM2 model (3.3.2.)

Vs=12ks(rr0)2[1kcubic(rr0)]

… stretching (3.4)
Vθ=12kθ(θθ0)2[1+ksextic(θθ0)4]
… bending (3.5)
ks=kθ/r20
… per length angular stiffness ksks/20
at (r0,θ0)
(3.6)
Vω=12[V1(1cos(1ω))+V2(1cos(2ω))+V3(1+cos(3ω))]
… torsion (3.7)
Vvdw=ϵvdw[2.48×105exp(12.5rrvdw0)1.924(rrvdw0)6]
… vdW attraction & non-bonded Pauli repulsion (3.8)
Vsθ=ksθ(θθ0)[(rArA0)+(rBrB0)]
… stretch-bend interaction (3.9)


ks,CC=440N/m

r0,CC=111.3pm
… (Table 3.2.)
kθ,CCC=450zJ/rad2
θ0,CCC=1.911rad=109.47
… (Table 3.3.)
ϵvdw,Csp3=357yJ
rvdw,Csp3=190pm
… well depth ϵvdw
varies widely for other atoms (~10x) (Table 3.1.)
V1,CCCC=1.39zJ, V2,CCCC=1.88zJ, V3,CCCC=0.65zJ
… (Table 3.5.)
ksθ,CCC=1.2nN/rad
… (Table 3.6.)
kcubic=?m1
ksextic=?rad4

nnano=109

ppico=1012

zzepto=1021
yyocto=1024

Bonds under large loads (beyond MM2) (3.3.3.)

Vmorse=De(1exp[β(rr0)]21)

… tensile (3.10)
Vlippincott=De[1exp(ksr0(rr0)22Der)],  rr0
… tensile – more accurate for larger distances (3.15)
For compressive loads Vvdw
see (3.8) above Fvdw=rVvdw
… (3.16)
Note: Popular Lennard-Jones 6-12 potential is too steep in repulsive regime!
MM2 potential within 10% of experiment for 0.5rvdw
(>100yJ
repulsion) 🙂

β=ks/(2De)

… (3.13)
DeD0+2ks/μ=D0+ω2;  μ=m1m2m1+m2
… (3.14)
D0
… potential well depth
De,CC=556yJ
ks,CC=440N/m
r0,CC=152.3pm
… (Table 3.8)

Verlet Integration (not in Nanosystems)

x(t+Δt)=2x(t)x(tΔt)+a(t)Δt2+O(Δt4)


a(t)=mxV(t)

MM2 vs MM3 (Chapter 3.3.2.g.)

★ MM2 prioritizes accuracy in energy and geometry
★ MM3 prioritizes accuracy of vibration frequencies
★ MM3 predicts greater angle-bending stiffness by ~1.5x or more
★ MM3 has a more complex functional form – e.g. it adds:
– stretch-torsion interaction, cubic bending, quartic stretching
★ MM3 has lower energies forces and stiffness in deep repulsive regime by ~10%
★ MM2 overall seems like a more conservative (safe side wrong) choice for diamondoid nanomachinery

Bond cleavage & radical coupling (3.4.2)

The Morse potential Vmorse

can approximate homolytic reactions
Vantimorse=12De(1+exp[β(rr0)]21)
… unpaired spins (3.21)
But unpaired spins being the limiting factor in reaction rate should
typically be avoidable in piezomechanosynthesis.
See Nanosystems Chapter 8.4.3.b. Radical coupling and intersystem crossing.

Nonbonded & large compression limits (3.3.3.b.)

ks,vdw12.5rvdw0Fvdw3.5×1010m1Fvdw

… (3.18)
Vvdw0.08rvdw0Fvdw2.9×1011mFvdw
…(3.19)

Abstraction reactions (Chapter 3.4.3.)

VLEPS

extended London-Eyring-Polyano-Sato potential – details omitted here

Continuum models of van der Waals attraction (Chapter 3.5.1.)

Hamaker constant – details omitted here

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