1549380323-Statistical Mechanics Theory and Molecular Simulation

Constraints: The ROLL algorithm 257

In order to tackle this problem, we need to modify the SHAKE and RATTLE algo-
rithms of Section 3.9. We refer to the modified algorithm as the “ROLL” algorithm.^4
It is worth noting that a version of the ROLL algorithm was developedby Martyna,
et al.(1996), however, the version that will be described here based oneqn. (5.12.4)
is somewhat simpler.
Here, we will only consider the problem of isotropic cell fluctuations;the extension
to fully flexible cells is straightforward, though tedious (Yuet al., 2010). Because of the
highly nonlinear dependence of eqn. (5.12.6) on the Lagrange multipliers, the operators
exp(iLǫ, 1 ∆t) exp(iL 1 ∆t) exp(iL 2 ∆t/2) exp(iLǫ, 2 ∆t/2) must be applied in an iterative
fashion until a self-consistent solution that satisfies the constraints is obtained. The
full evolution of the coordinatesriis obtained by combining eqns. (5.12.6) and (5.12.8)
to give

ri(∆t) =ri(0)evǫ∆t+ ∆tvi(∆t/2)evǫ∆t/^2

sinh(vǫ∆t/2)

vǫ∆t/ 2

=ri(0)evǫ∆t+ ∆tevǫ∆t/^2

sinh(vǫ∆t/2)

vǫ∆t/ 2

(5.13.2)

×

[

v

(NHC)

i e

−αvǫ∆t/ (^2) + ∆t
2 mi

(

Fi(0) +

∑

k

λkF

(k)

c,i(0)

)

sinh(αvǫ∆t/4)

αvǫ∆t/ 4

]

,

or

ri(∆t) =ri(0)evǫ∆t+ ∆tv(NHC)i e−vǫ(α−1)∆t/^2

sinh(vǫ∆t/2)

vǫ∆t/ 2

(5.13.3)

+

∆t^2

2 mi

[

Fi(0) +

∑

k

λkF(c,ik)(0)

]

e−vǫ(α−2)∆t/^4

sinh(vǫ∆t/2)

vǫ∆t/ 2

sinh(αvǫ∆t/4)

αvǫ∆t/ 4

,

whereα= 1+d/Nf. Here,vi(NHC)is the “velocity” generated by the thermostat oper-
ator, exp(iLNHC−part∆t/2). Because the evolution ofvǫis determined by the pressure,
many of the factors in eqn. (5.13.4) depend on the Lagrange multipliers. Thus, let us
write eqn. (5.13.4) in the suggestive shorthand form

ri(∆t) =Rxx(λ,0)ri(0) +Rvx(λ,0)∆tv(NHC)i

+

∆t^2

2 mi

RFx(λ,0)

[

Fi(0) +

∑

k

λkF(c,ik)(0)

]

, (5.13.4)

whereλdenotes the full set of Lagrange multipliers. The factorsRxx(λ,0),Rvx(λ,0)
andRFx(λ,0) denote thevǫ-dependent factors in eqn. (5.13.4); we refer to them as the

(^4) Yes, the “ROLL” moniker does fit well with “SHAKE” and “RATTLE;” however, there is an
actual “rolling” procedure in the ROLL algorithm when used in fully flexible cell calculations.