Chemistry Reference
In-Depth Information
r
min
∫
.
(4.1)
2
CN
=
4πρ
r g r dr
( )
.
0
where
ρ.is.the.number.density
r
.is.the.distance
g
(
r
).is.the.correlation.function
Analysis.of.the.geometric.structures.obtained.from.our.simulations.enable.an.under-
standing.of.the.hydration.characteristics.of.the.metal.ion.and.the.structural.proper-
ties.of.aqua.ions..Results.obtained.for.each.metal.ion.solution.are.compared.to.each.
other.in.order.to.understand.the.effect.of.the.size.and.charge.of.the.metal.ion.on.the.
hydration.structures.of.the.metal.ions.and.aqua.ions.
We.now.turn.to.the.computation.of.free.energy.in.our.metal.ion.solutions..In.gen-
eral,.the.lack.of.a.corresponding.microscopic.analogue,.that.is,.a.function.of.conigu-
ration.space.variables.that.may.be.calculated.to.obtain.the.free.energy,.creates.one.of.
the.principal.challenges.in.the.determination.of.free.energy.of.dense.systems..Much.
effort. has. gone. into. the. development. of. free. energy. calculation. methods.. Popular.
methods. include. the. particle. insertion,. particle. deletion. scheme,. and. perturbation.
methods.[22,23,65,90]..Using.our.implementation.of.well-known.classical.molecular.
simulation.free.energy.methods.with.CPMD.simulations,.the.preference.for.Zundel.
or.Eigen.complexes.in.metallic.solutions.was.calculated.via.the.perturbation.method..
The.perturbation.method.gives.the.free.energy.change.between.two.states.
i
.and.
i
+1.
with.Hamiltonians.
H
i
.and.
H
i
+1
.as
−
(
H
−
H
)
⎡
⎢
⎤
⎥
(
)
= −
.
(4.2)
λ
λ
Δ
G
λ
+
−
λ
kT
ln exp
i
+
1
i
i
1
i
kT
.
λ
i
where. λ. is. a. continuous. coupling. parameter. such. that. λ.=.0. at. the. initial. state. and.
λ.=.1.at.the.inal.state..Perturbations.are.broken.into.intermediate.steps.in.which.the.
change.in.the.energy.does.not.exceed.the.value.of.
kT
.(
k
.is.the.Boltzmann.constant.
and.
T
.is.the.temperature)..For.more.details.see.Ref..[24].
For.dimerization.studies.of.Al
3+
.ions.in.water,.the.Becke-Lee-Yang-Parr.(BLYP).
gradient.corrected.functional.was.used.with.Troullier-Martins.pseudopotentials..The.
initial.trajectory.was.taken.from.an.equilibrated.classical.molecular.dynamics.simu-
lation.of.a.single.Al
3+
.ion.solvated.by.64.water.molecules.at.300.K.and.at.a.pressure.
of.0.1.MPa.[65,66]..A.modiied.TIP5P.model.for.water.and.the.UFF.(Universal.Force.
Field).potential.function.were.used.for.the.classical.molecular.simulations.of.Al.ion.
in.water.and.Lorentz-Berthelot.mixing.rules.were.applied.for.calculating.the.cross.
parameters.[1,90]..The.system.was.then.equilibrated.for.15.ps.using.CPMD.followed.
by.a.35.ps.production.run.using.a.time.step.of.0.1.fs.and.an.effective.electron.mass.of.
700.au..A.homogeneous.background.charge.was.applied.to.compensate.for.the.ionic.
charge.. The. temperature. was. controlled. using. a. Nose-Hoover. thermostat,. and. the.
Ewald.mesh.method.was.used.for.long-range.interactions.[47]..The.valence.electron.
wave. functions. were. expanded. in. plane. waves. with. an. energy. cutoff. of. 140.Ry.. In.
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