1549380323-Statistical Mechanics Theory and Molecular Simulation

Integrating the MTK equations 249

in Problem 5.8. These equations can easily be generalized for the isothermal-isobaric
ensemble with a molecular virial by coupling Nos ́e-Hoover chain thermostats as in eqns.
(5.9.5). Moreover, starting from the transformation for anisotropic cell fluctuations

Si=h−^1 Ri, (5.11.8)

the algorithm in eqn. (5.11.6) can be turned into an algorithm capable of handling
anisotropic volume fluctuations with a molecular virial.

5.12 Integrating the MTK equations of motion

Integrating the MTK equations is only slightly more difficult than integrating the NHC
equations and builds on the methodology we have already developed.We begin with
the isotropic case, and for the present, we consider a system in which no constraints
are imposed so thatNf=dNandF ̃i=Fi=−∂U/∂ri. In Section 5.13, we will see
how to account for forces of constraint. We first write the totalLiouville operator as

iL=iL 1 +iL 2 +iLǫ, 1 +iLǫ, 2 +iLNHC−baro+iLNHC−part, (5.12.1)

where

iL 1 =

∑N

i=1

[

pi

mi

+

pǫ

W

ri

]

·

∂

∂ri

iL 2 =

∑N

i=1

[

Fi−α

pǫ

W

pi

]

·

∂

∂pi

iLǫ, 1 =

pǫ

W

∂

∂ǫ

iLǫ, 2 =Gǫ

∂

∂pǫ

, (5.12.2)

and the operatorsiLNHC−partandiLNHC−baro are the particle and barostat Nos ́e-
Hoover chain Liouville operators, respectively, which are defined in the last two lines
of eqn. (4.11.5). In eqn. (5.12.2),α= 1 +d/Nf= 1 + 1/N, and

Gǫ=α

∑N

i=1

p^2 i

mi

+

∑N

i=1

ri·Fi−dV

∂U

∂V

−dPV. (5.12.3)

The propagator is factorized following the scheme of eqn. (4.11.8) as

exp(iL∆t) = exp

(

iLNHC−baro

∆t

2

)

exp

(

iLNHC−part

∆t

2

)

×exp

(

iLǫ, 2

∆t

2

)

exp

(

iL 2

∆t

2

)