1549380323-Statistical Mechanics Theory and Molecular Simulation

Adiabatic dynamics 351

time-scale integrators of Section 3.11, we write this operator usingthe Trotter theorem
as

exp

(

iL 2

∆t

2

)

= lim

M→∞

[

exp

(

∆t

4 M

∑^3 N

α=1

Fα

∂

∂qα

)

×exp

(

iLref, 2

∆t

2 M

)

exp

(

∆t

4 M

∑^3 N

α=1

Fα

∂

∂qα

)]M

. (8.10.12)

It proves useful to decompose the phase space vector as x = (X,Y,PX,PY,ΓX,ΓY),
whereX denotes the full set of reaction coordinates,PX, their momenta,Y, the
remaining 3N−ncoordinates,PY, their momenta, and ΓXand ΓY, the thermostat
variables associated with the temperaturesTqandT, respectively. Thus, when eqn.
(8.10.12) is substituted into eqn. (8.10.11) and the resulting operator is taken to act
on the initial phase space vector x 0 , the result for heavy, slow reaction coordinates is

Xα(∆t) =Xα,ref[X(0),X ̇(∆t/2),ΓX(0); ∆t]

X ̇α(∆t) =X ̇α,ref[X(0),X ̇(∆t/2),ΓX(0); ∆t]

+

(

∆t

2 m′α

)

2

∆t

∫∆t

∆t/ 2

dt Fα[X(∆t),Yadb(Y(∆t/2),Y ̇(∆t/2),ΓY(∆t/2),X(∆t);t)]

X ̇α(∆t/2) =X ̇α(0)

+

(

∆t

2 m′α

)

2

∆t

∫∆t/ 2

0

dt Fα[X(0),Yadb(Y(0),Y ̇(0),ΓY(0),X(0);t)]

Yγ(∆t/2) =Yγ,adb[Y(0),Y ̇(0),ΓY(0),X(0); ∆t/2]

Y ̇γ(∆t/2) =Y ̇γ,adb[Y(0),Y ̇(0),ΓY(0),X(0); ∆t/2]

Yγ(∆t) =Yγ,adb[Y(∆t/2),Y ̇(∆t/2),ΓY(∆t/2),X(∆t); ∆t]

Y ̇γ(∆t) =Y ̇γ,adb[Y(∆t/2),Y ̇(∆t/2),ΓY(∆t/2),X(∆t); ∆t]. (8.10.13)

In eqn. (8.10.13),Xα,ref[X(0),X ̇(0),ΓX(0); ∆t] represents the evolution ofXα(α=
1 ,...,n) up to time ∆tunder the action of the reference-system operator exp(iLref, 1 ∆t)
starting from the initial conditionsX(0),X ̇(0), ΓX(0), with an analogous meaning
forX ̇α[X(0),X ̇(0),ΓX(0); ∆t].Yγ,adb[Y(0),Y ̇(0),ΓY(0),X(0); ∆t/2] denotes the exact
evolution ofYγ(γ= 1,..., 3 N−n) up to time ∆t/2 under the first action of the operator
exp(iL 2 ∆t/2) given in the form of eqn. (8.10.12) starting from initial conditionsY(0),
Y ̇(0), ΓY(0),X(0) with an analogous meaning forY ̇γ,adb[Y(0),Y ̇(0),ΓY(0),X(0); ∆t/2].
The functions in the last two lines of eqn. (8.10.13) are similarly definedfor the second
action of exp(iL 2 ∆t/2). Although we do not have closed-form expressions for these
functions in general, we do not need them for the present analysis.The important
terms in eqn. (8.10.13) are the time integrals of the forces on the slow reaction coor-
dinates. These time integrals result from the action of the operator exp(iL 2 ∆t/2) on