1549380323-Statistical Mechanics Theory and Molecular Simulation

234 Isobaric ensembles

into eqn. (5.7.5) to yield

L(s, ̇s) =

1

2

∑

i

mi

∑

α,β,γ

hαβs ̇i,βhαγs ̇i,γ−U(hs 1 ,...,hsN)

=

1

2

∑

α,β,γ

hαβhαγ

∑

i

mis ̇i,βs ̇i,γ−U(hs 1 ,...,hsN). (5.7.7)

A component of the momentumπjconjugate tosjis computed according to

πj,λ=

∂L

∂s ̇j,λ

. (5.7.8)

The trickiest part of this derivative is keeping track of the indices. Since all of the
indices in eqn. (5.7.7) are summed over orcontracted, eqn. (5.7.7) contains many
terms. The only terms that contribute to the momentum in eqn. (5.7.8) are those for
whichi=jandβ=λorγ=λ. The easiest way to keep track of the bookkeeping is
to replace factors of ̇si,βor ̇si,γwithδijδβλandδijδγλ, respectively, when computing
the derivative, and then perform the sums with the aid of the Kroenecker deltas:

πj,λ=

1

2

∑

α,β,γ

hαβhαγ

∑

i

mi[δijδβλs ̇i,γ+ ̇si,βδijδγλ]

=

1

2

mj





∑

α,γ

hαλhαγs ̇j,γ+

∑

α,β

hαβhαλs ̇j,β



. (5.7.9)

Since the two sums appearing in the last line of eqn. (5.7.9) are the same, the factor
of 1/2 can be cancelled, yielding

πj,λ=mj

∑

α,γ

hαλhαγs ̇j,γ=mj

∑

α

r ̇j,αhαλ. (5.7.10)

Writing this in vector notation, we find

πj=mj ̇rjh=pjh (5.7.11)

or
pj=πjh−^1. (5.7.12)

Thus, we see thatπjmust be multipliedon the rightbyh−^1.
Having obtained the Lagrangian in scaled coordinates and the momentum trans-
formation from the Lagrangian, we must now derive the Hamiltonian inorder to
determine the canonical partition function. The Hamiltonian is given by the Legendre
transform rule:
H=

∑

i

πi·s ̇i−L=

∑

i

∑

α

πi,αs ̇i,α−L. (5.7.13)

Using the fact thats ̇i=h−^1 r ̇i=h−^1 pi/mitogether with eqn. (5.7.12) to substitute
piin terms ofπi, the Hamiltonian becomes