1549380323-Statistical Mechanics Theory and Molecular Simulation

236 Isobaric ensembles

∂H

∂hαγ

=−

∑

i

∑

μ,ν,λ

πi,μπi,ν

2 mi

∑

ρ,σ

(

h−μρ^1

∂hρσ

∂hαγ

h−σλ^1 h−νλ^1 +h−μλ^1 h−νρ^1

∂hρσ

∂hαγ

)

h−σλ^1

+

∂

∂hαγ

U(hs 1 ,...,hsN). (5.7.22)

Using∂hρσ/∂hαγ=δαρδσγand performing the sums overρandσ, we find

∂H

∂hαγ

=−

∑

i

∑

μ,ν,λ

πi,μπi,ν

2 mi

∑

ρ,σ

(

h−μα^1 h−γλ^1 h−νλ^1 +h−μλ^1 h−να^1 h−γλ^1

)

+

∂

∂hαγ

U(hs 1 ,...,hsN). (5.7.23)

Since

∂

∂hαγ

U(hs 1 ,...,hsN) =

∑

i

∑

μ,ν

∂U

∂(hsi)μ

∂hμν

∂hαγ

si,ν

=

∑

i

∑

μ,ν

∂U

∂(hsi)μ

δαμδγνsi,ν

=

∑

i

∂U

∂(hsi)α

si,γ, (5.7.24)

we arrive at the result

∂H

∂hαγ

=−

∑

i

∑

μ,ν,λ

πi,μπi,ν

2 mi

∑

ρ,σ

(

h−μα^1 h−γλ^1 h−νλ^1 +h−μλ^1 h−να^1 h−γλ^1

)

+

∑

i

∂U

∂(hsi)α

si,γ. (5.7.25)

To obtain the pressure tensor estimator, we must multiply byhβγand sum overγ.
When this is done and the sum overγis performed according to

∑

γhβγh

− 1
γλ=δβλ,
then the sum overλcan be performed as well, yielding

∑

γ

hβγ

∂H

∂hαγ

=−

∑

i

∑

μ,ν

πi,μπi,ν

2 mi

∑

ρ,σ

(

h−μα^1 h−νβ^1 +h−μβ^1 h−να^1

)

+

∑

i

∑

γ

∂U

∂(hsi)α

hβγsi,γ. (5.7.26)

We now recognize that

∑

απi,μh

− 1

μα=pi,α,

∑

νπi,νh

− 1
νβ=pi,β,∂U/∂(hsi) =∂U/∂ri
and

∑

γhβγsi,γ=ri,β. Substituting these results into eqn. (5.7.26) and multiplying
by− 1 /det(h) gives