1549380323-Statistical Mechanics Theory and Molecular Simulation

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/ 4

]}

Si(∆t), (5.13.12)

which we can also write as

vi(∆t) =

{

Rvv(μ,∆t)vi(∆t/2)

+

∆t

2 mi

RFv(μ,∆t)

[

Fi(∆t) +

∑

k

μkF(c,ik)(∆t)

]}

Si(∆t), (5.13.13)

and forvǫ, we obtain

vǫ(∆t) =

[

vǫ(∆t/2) +

∆t

2 W

Gǫ(μ,∆t)

]

Sǫ(∆t). (5.13.14)

In eqns. (5.13.13) and (5.13.14), the use ofμkandμfor the Lagrange multipliers
indicates that these multipliers are used to enforce the first time derivative of the
constraint conditions as described in Section 3.9. Let ̃μk= (∆t/2)μk, and suppose we

have a good initial guess to the multipliers ̃μ(1)k. Then, ̃μk= ̃μ(1)k +δμ ̃(1)k , and we can
write eqns. (5.13.13) and (5.13.14) in shorthand as

vi(∆t) =v(1)i +

1

mi

RFv(λ,∆t)

∑

k

δμ ̃(1)k F(c,ik)(∆t)Si(∆t)

vǫ(∆t) =v(1)ǫ +

1

W

S ̃ǫ(∆t)

∑

i

∑

k

δμ ̃(1)k ri(∆t)·F(c,ik)(∆t), (5.13.15)

whereS ̃ǫ(∆t) = (∆t/2)Sǫ(∆t).
As in the first half of the ROLL algorithm, we assume that the ROLL scalars and
scaling factors are independent of the multipliers and use eqns. (5.13.15) to determine

the correctionsδ ̃μ(1)k such that the first time derivative of each constraint condition
vanishes. This requires

σ ̇k=

∑N

i=1

Fc,i(k)·r ̇i= 0. (5.13.16)

However, a slight subtlety arises because according to eqns. (5.9.5),r ̇i 6 =vibut rather
r ̇i=vi+vǫri. Thus, eqn. (5.13.16) becomes a condition involving bothvi(∆t) and
vǫ(∆t) att= ∆t:

∑

i

F(c,ik)(∆t)·[vi(∆t) +vǫ(∆t)ri(∆t)] = 0. (5.13.17)