Density Functional Calculations of Metal Clusters: Structure, Dynamics, and Reactivity - Metallic Systems: A Quantum Chemist's Perspective - page 289

for.this.task.can.be.considerably.reduced.by.employing.ADFT.[34,86]..In.an.early.
molecular. dynamics. implementation. in. a. precursor. of. the. deMon2k. program,. the.
improved.performance.of.the.ADFT-BOMD.approach.has.been.documented.[40].
The.natural.time.evolution.of.a.classical.system.of. N .particles.in.a.volume. V .can.be.
studied.with.the.above-described.BOMD.method..In.such.simulations,.the.total.energy.
E .is.a.constant.of.motion..Although.thermodynamic.results.can.be.transformed.between.
ensembles,.this.is.only.reliable.in.the.limit.of.macroscopic.system.sizes..Because.many.
experiments.are.performed.at.ixed.temperature.instead.of.ixed.energy,.simulations.in.
the.canonical,. NVT ,.ensemble.are.often.desired..Thermostats.are.designed.to.ensure.
the.molecular.dynamics.sampling.within.this.ensemble,.by.modulating.the.tempera-
ture.of.the.system.in.some.fashion..First,.we.need.to.establish.what.we.mean.by.tem-
perature..In.simulations,.the.“instantaneous.(kinetic).temperature”.is.usually.computed.
from.the.kinetic.energy.of.the.system.using.the.equipartition.theorem..In.other.words,.
the.temperature.is.computed.from.the.system's.total.kinetic.energy..A.thermostat.that.is.
coupled.to.the.motion.of.the.nuclei.thus.ensures.that.the.average.temperature.of.the.sys-
tem.stays.around.a.predeined.value..By.the.equipartition.theorem,.the.instantaneous.
temperature.of.the.system.is.related.to.the.kinetic.energy.as

= 2

K
N f B

(9.66)

T t ( ) ,

. .

where
K .is.the.kinetic.energy.of.the.system
k B .is.the.Boltzmann.constant
N f .is.the.number.of.degrees.of.freedom.of.the.system

A.simple.way.to.alter.the.temperature.of.the.system.is.to.scale.the.velocities.of.the.
particles.(velocity.scaling.thermostat.[87])..If.the.velocities.are.multiplied.by.a.factor.
λ,.then.the.associated.temperature.change.can.be.calculated.as.follows:

N N

2 2

1
2

m ( λ r ) 1

2

m
N k

( r )

∑ ∑

A A A A

Δ T = −

N k

f B f B

A A

(9.67)

2

= ( λ − 1 ) ( ). T t

.

Thus,.it.follows .

T
T ( ) .
.

λ = (9.68)

.

The. simplest. way. to. control. the. temperature. is. then. to. multiply. the. velocities. at.
each.time.step.by.the.factor. λ = T T t / ( ) ,.where. T .is.the.desired.temperature..This.

method.is.usually.used.to.equilibrate.the.system..As.can.be.seen.from.Figure.9.4,.the.
convergence.of.the.average.temperature.(dash.line).and.instantaneous.temperature.
(gray.line).to.the.desired.temperature.(300.K).is.practically.immediate;.however,.the.