1549380323-Statistical Mechanics Theory and Molecular Simulation

Integrating the MTK equations 251

Integrating eqns. (5.10.2) for the fully flexible case employs the same basic factor-
ization scheme as in eqn. (5.12.4). First, we decompose the total Liouville operator
as
iL=iL 1 +iL 2 +iLg, 1 +iLg, 2 +iLNHC−baro+iLNHC−part, (5.12.10)

where

iL 1 =

∑N

i=1

[

pi

mi

+

pg

Wg

ri

]

·

∂

∂ri

iL 2 =

∑N

i=1

[

Fi−

(

pg

Wg

+

1

Nf

Tr [pg]

Wg

I

)

pi

]

·

∂

∂pi

iLg, 1 =

pgh

Wg

·

∂

∂h

iLg, 2 =Gg

∂

∂pg

, (5.12.11)

with

Gg= det[h](P(int)−IP) +

1

Nf

∑N

i=1

p^2 i

mi

I. (5.12.12)

The propagator is factorized exactly as in eqn. (5.12.4) with the contributions to
iLǫreplaced by the contributions toiLg. In the flexible case, the application of the
operators exp(iL 1 ∆t) and exp(iL 2 ∆t/2) requires solution of the following matrix-
vector equations:
r ̇i=vi+vgri (5.12.13)

v ̇i=

Fi

mi

−vgvi−bTr [vg]vi, (5.12.14)

wherevg=pg/Wg, andb= 1/Nf. In order to solve eqn. (5.12.13), we introduce a
transformation
xi=Ori, (5.12.15)

whereOis a constant orthogonal matrix. We also letui=Ovi. SinceOis orthogonal,
it satisfiesOTO=I. Introducing this transformation into eqn. (5.12.13) yields

Or ̇i=Ovi+Ovgri

x ̇i=ui+OvgOTOri=ui+OvgOTxi, (5.12.16)

where the second line follows from the orthogonality ofO. Now, since the pressure
tensor is symmetric,vgis also symmetric. Therefore, it is possible to chooseOto be
the orthogonal matrix that diagonalizesvgaccording to