1549380323-Statistical Mechanics Theory and Molecular Simulation

244 Isobaric ensembles 0 2 4 6 8 10 q 0 0.5 1 1.5 P (q ) NumericalAnalytical 0 2 4 6 8 10 L 0 0.1 0.2 0.3 0.4 0.5 P (L ) Fig. 5. ...

Molecular dynamics 245 η ̇j= pηj Qj , ξ ̇j= pξj Q′j p ̇ηj=Gj− pηj+1 Qj+1 pηj p ̇ηM=GM p ̇ξj=G′j− pξj+1 Q′j+1 pξj p ̇ξM=G′M, (5.1 ...

246 Isobaric ensembles p ̇i,α=Fi,α− ∑ β pg,αβ Wg pi,β− 1 dN Tr [pg] Wg pi,α− pη 1 Q 1 pi,α h ̇αβ= ∑ γ pg,αγhγβ Wg p ̇g,αβ= det(h ...

Virials 247 Z= ∫ dNpdNrdhdpgdη 1 dηcdξ 1 dξcdMpηdMpξ[det(h)]^1 −dedNη^1 +ηced (^2) ξ 1 +ξc ×δ  H(r,p) + ∑M j=1 [ p^2 ηj 2 Qj + ...

248 Isobaric ensembles Ri= ∑n ∑α=1mi,αri,α n α=1mi,α . (5.11.2) We saw in Section 1.11 that the center-of-mass motion of each mo ...

Integrating the MTK equations 249 in Problem 5.8. These equations can easily be generalized for the isothermal-isobaric ensemble ...

250 Isobaric ensembles ×exp (iLǫ, 1 ∆t) exp (iL 1 ∆t) ×exp ( iL 2 ∆t 2 ) exp ( iLǫ, 2 ∆t 2 ) ×exp ( iLNHC−part ∆t 2 ) exp ( iLNH ...

Integrating the MTK equations 251 Integrating eqns. (5.10.2) for the fully flexible case employs the same basic factor- ization ...

252 Isobaric ensembles v(d)g =OvgOT, (5.12.17) wherev (d) g is a diagonal matrix with the eigenvalues ofvgon the diagonal. The c ...

Integrating the MTK equations 253 explicit symmetrization of the pressure tensorPαβ(int). That is, we can simply replace occurre ...

254 Isobaric ensembles P(fast)= 〈 1 dV {N ∑ i=1 [ p^2 i mi +ri·F(fast)i ] −dV ∂U(fast) ∂V }〉 P(slow)= 〈 1 dV {N ∑ i=1 ri·F (slow ...

Constraints: The ROLL algorithm 255 0 10 20 30 40 50 60 70 t (ps) 36 38 40 42 44 46 48 L (Å) 0 2 4 6 8 10 r (Å) 0 0.5 1 1.5 2 g( ...

256 Isobaric ensembles 0.6 0.65 0.7 0.75 0.8 r* 0 5 10 15 20 25 30 P( r*) Fig. 5.5Density distribution for the argon system atP= ...

Constraints: The ROLL algorithm 257 In order to tackle this problem, we need to modify the SHAKE and RATTLE algo- rithms of Sect ...

258 Isobaric ensembles “ROLL scalars.” (In the fully flexible cell case, these scalars are replaced by 3×3 matri- ces.) Note the ...

Constraints: The ROLL algorithm 259 (5.13.8) if the dimensionality is not too large, or as a time-saving measure, we neglect the ...

260 Isobaric ensembles vi(∆t) = { vi(∆t/2)e−αvǫ∆t/^2 + ∆t 2 mi ( Fi(∆t) + ∑ k μkF(c,ik)(∆t) ) ×e−αvǫ∆t/^4 [ sinh(αvǫ∆t/4) αvǫ∆t/ ...

Problems 261 Substituting eqns. (5.13.15) into eqn. (5.13.17) yields ∑ i F(c,ik)(∆t)· [ v(1)i + 1 mi RFv(μ,∆t)Si(∆t) ∑ l δμ ̃(1) ...

262 Isobaric ensembles 5.1. Show that the distribution function for the isothermal-isobaricensemble can be derived starting from ...

Problems 263 p ̇i=− ∂U ∂ri − pǫ W pi− pη Q pi V ̇=dV pǫ W p ̇ǫ=dV(P(int)−P)− pη Q pǫ η ̇= pη Q p ̇η= ∑N i=1 p^2 i mi + p^2 ǫ W − ...