Density Functional Calculations of Metal Clusters: Structure, Dynamics, and Reactivity - Metallic Systems: A Quantum Chemist's Perspective - page 292

this.igure..As.Figure.9.5.shows,.the.phase.space.sampling.is,.however,.very.similar.
to.that.obtained.with.the.previous.thermostat..Thus,.this.thermostat.is.not.suited.for.
the.calculation.of.thermodynamic.averages.either.
The.so-called.constant.kinetic.temperature.dynamics.is.generated.by.the.follow-
ing.equations.of.motion.[89]:

p

r

A
A

(9.71)

= ,

A

. .

p (9.72)

= F − ξ p . .

.

A A A

The.quantity.ξ.is.a.kind.of.“friction.coeficient,”.which.varies.such.that. T ( t ).is.con-
strained.to.a.constant.value,.i.e.,.to.guarantee

⎛ ⎞

d
dt

∑ p 2

T t

( ) ∝

⎟ =

0 .

(9.73)

⎜

A

⎝ ⎠

A

. .

The.constraint.is.chosen.in.order.to.perturb.the.classical.equations.of.motion.as.little.
as.possible..This.is.achieved.by.making.ξ.a.Lagrange.multiplier,.which.minimizes.the.
difference.between.constrained.and.Newtonian.trajectories..The.resulting.expression.is

∑
∑

p F ⋅

A A

ξ =

A

.

(9.74)

2

p

A

. .

A

This.method.generates.conigurational.properties.of.the.canonical.ensemble;.how-
ever,.the.momentum.distribution.is.not.canonical.as.can.be.seen.from.the.BOMD.
simulation.of.the.copper.dimer.shown.in.Figure.9.6.
The.last.scheme.for.controlling.the.temperature.discussed.here.is.the.Nosé-Hoover.
(chain).thermostat.[90-92]..In.a.ictitious.system.that.contains.the.real.system.and.
the.thermostat,.an.extra.degree.of.freedom.is.introduced.that.represents.the.thermal.
reservoir..In.this.way,.the.constant.temperature.simulation.can.be.performed.deter-
ministically.by.a.reformulation.of.the.Lagrange.equations.of.motion.for.the.extended.
system..The.additional.degree.of.freedom.is.represented.by.the.coordinate. s .and.its.
conjugated.momentum. p s .
The. Hamiltonian. describing. the. extended. system. consisting. of. the. real. system.
and.a.heat.bath.is.given.by

N

2 2

N

1
2

∑

p p

Q

f

A s

H = + V({ }) r + + ln . s

(9.75)

A

2

m s 2 β

A

.

A

.

The.irst.two.terms.are.the.kinetic.and.potential.energy.of.the.real.system.we.are.
interested.in..The.last.two.terms.correspond.to.the.extra.degree.of.freedom.for.the.
heat.bath..Here,.β.denotes.1/( k B T ).with. T .being.the.temperature.of.the.heat.bath.