1549380323-Statistical Mechanics Theory and Molecular Simulation

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)

i −dV

∂U(slow)

∂V

}〉

, (5.12.26)

that is, using the definitions of the reference system and correction contributions to
the internal pressure. Another simple choice is

P(fast)=

n

n+ 1

P

P(slow)=

1

n+ 1

P. (5.12.27)

The factorized propagator then takes the form

exp(iL∆t) = exp

(

iLNHC−baro

∆t

2

)

exp

(

iLNHC−part

∆t

2

)

×exp

(

iL(slow)ǫ, 2

∆t

2

)

exp

(

iL(slow) 2

∆t

2

)

×

[

exp

(

iL

(fast)

2

δt

2

)

exp

(

iL

(fast)

ǫ, 2

δt

2

)

×exp (iLǫ, 1 δt) exp (iL 1 δt)

×exp

(

iL(fast)ǫ, 2

δt

2

)

exp

(

iL(fast) 2

δt

2

)]n

×exp

(

iL(slow) 2

∆t

2

)

exp

(

iL(slow)ǫ, 2

∆t

2

)

×exp

(

iLNHC−part

∆t

2

)

exp

(

iLNHC−baro

∆t

2

)

+O(∆t^3 ). (5.12.28)

Note that becauseGǫdepends on the forcesFi, it is necessary to update both the
particles and the barostat in the reference system.
The integrators presented in this section can be generalized to handle systems
with constraints under constant pressure. It is not entirely straightforward, however,
because self-consistency conditions arise from the nonlinearity ofsome of the operators.
A detailed discussion of the implementation of constraints under conditions of constant
pressure can be found in Section 5.13.