Computational Physics

8.5 Molecular dynamics methods for different ensembles 223 whereS 2 is the second order integrator, and he fixedαandβby the requ ...

224 Molecular dynamics simulations with proportionality constantkB/2, and therefore this quantity is used in MD to calculate the ...

8.5 Molecular dynamics methods for different ensembles 225 in velocity influences the equilibration time and the kinetic energy ...

226 Molecular dynamics simulations of keeping the temperature constant. This force assumes the form of a friction proportional t ...

8.5 Molecular dynamics methods for different ensembles 227 In the Andersen method, it is not always clear at which rate the velo ...

228 Molecular dynamics simulations First we derive the equations of motion in the usual way: dri dt = ∂H ∂pi = pi ms^2 (8.83a) d ...

8.5 Molecular dynamics methods for different ensembles 229 Furthermore we use the relationδ[f(s)]=δ(s−s 0 )/f′(s)withf(s 0 )=0 a ...

230 Molecular dynamics simulations the equations of motion in real variables: dri′ dt′ = p′i m (8.93a) dp′i dt′ =−∇iV(R)−sp′sp′i ...

8.5 Molecular dynamics methods for different ensembles 231 The disadvantage of the real-time equations is that they are not Hami ...

232 Molecular dynamics simulations The equations of motion now read: dr dt = ∂H ∂pi = pi mV^2 /^3 s^2 (8.101a) ds dt = ∂H ∂ps = ...

8.6 Molecular systems 233 stretch torsion bend Figure 8.4. Different types of motion of atoms within a molecule. bond lengthl: V ...

234 Molecular dynamics simulations vary as the square root of theα-coefficients. In general, the bond length vibrations are the ...

8.6 Molecular systems 235 ω n^ Figure 8.5. The nitrogen molecule.nˆis a unit vector,ωωωis the rotation vector. each of the two a ...

236 Molecular dynamics simulations equations of motion the norm ofnˆis not rigorously conserved; it can suffer from numerical er ...

8.6 Molecular systems 237 Enforcing rigidity via constraints Another method for treating rigid molecules is by imposing holonomi ...

238 Molecular dynamics simulations and the remaining ones are fixed by linear constraints of the form(8.119). In the case of a p ...

8.6 Molecular systems 239 Verlet scheme, predictions for the positions att+hare given by rS( 1 )(t+h)= 2 rS( 1 )(t)−rS( 1 )(t−h) ...

240 Molecular dynamics simulations by chemical bonds can be described in terms of stretching, bending and torsion, as described ...

8.7 Long-range interactions 241 Each step will therefore shift the positions more closely to the point where they all satisfy th ...

242 Molecular dynamics simulations or subtracting a constant to the potential does not alter the forces and hence the dynamics o ...