Exploring the Borderland between Physics and Chemistry: Theoretical Methods in the Study of Atomic Clusters - Metallic Systems: A Quantum Chemist's Perspective - page 388

plus.polarization.with.the.Stuttgart.[77].effective.core.potential.(SVP/STUTT).basis.
(27. basis. functions. comprised. of. 55. primitive. gaussians). and. the. LANL2DZ. [76].
basis.set.developed.by.the.Los.Alamos.group.(24.basis.functions.comprised.of.44.
primitive.gaussians).with.its.respective.effective.core.potential..The.calculated.elec-
tronic.structures.of.gold.nanoclusters.of.different.sizes.(Au n ,.with. n .=.2,.14,.28,.38,.
56,. and. 68). reveal. that. they. exhibit. (with. the. exception. of. Au 2 ). a. core-shell. geo-
metric.arrangement.of.Au.atoms.and.that.permanent.size-dependent.spin.polariza-
tion.appears.without.geometry.relaxation.for.bare.clusters.even.though.bulk.gold.is.
diamagnetic.[73]..The.spin-polarized.ground.states.for.clusters.are.favorable.due.to.
the.hybridization.of.the. s .and. d .orbitals,.and.bare.octahedral.clusters.are.expected.
to.be.magnetic.for.cluster.sizes.of.approximately.38.atoms.and.larger..Much.larger.
clusters.will.be.diamagnetic.when.the.surface-to-volume.ratio.is.small.and.the.core.
diamagnetism.prevails.
Making.use.of.these.indings,.we.developed.a.spin-spin.Ising.interaction.model.
[79].that.explains.the.origin.of.the.size.dependency.of.magnetization.in.Au.clusters..
This.model.combines.the.bulk.diamagnetic.response.of.the.core.with.surface.(shell).
ferromagnetism. behavior. (as. suggested. by. the. results. in. Ref.. [73]).. In. this. model,.
the. maximum. entropy. formalism. is. used. in. order. to. obtain. an. average. tempera-
ture-dependent.magnetization.of.bare.Au.nanoparticles.within.a.mean-ield.theory..
Accordingly,.the.total.Hamiltonian. H T .can.be.partitioned.into.a.core. H c .and.a.surface.
H s .contribution:

H = H + H . (11.6)

.

T c s

with

N c N N N N

c s c c

1
2

∑ j ∑ ∑ ∑ ∑

c sc 2

H = − J ⋅ S S ⋅ − J ⋅ S S ⋅ − k S ⋅ . (11.7)

c ij i j ij i j c z

i

i < j i j i

.

and

N N N N N s

∑

s s s c

2

1
2

∑ ∑ ∑ ∑

s sc

ˆ S i

H = − J ⋅ S S ⋅ − J ⋅ S S ⋅ − k n ⋅ ⋅ . (11.8)

s ij i j ij i j s

i < j j i j i

.

In.Equations.11.7.and.11.8,. N c .and. N s .indicate.the.number.of.core.and.surface.Au.
atoms,. respectively,. such. that. the. total. number. of. atoms. N = N c .+. N s .. The. interac-
tion.terms.are.partitioned.into. N s ( N s .−.1).surface.atom.interactions,. N s N c .core.atom.
interactions.with.the.surface.atoms.and. N c ( N c .−.1).core.atoms.interactions.with.cou-
pling.functions. J s ,. J cs ,. J sc ,.and. J c ,.respectively.(with. J i x > 0 .for.ferromagnetic.interac-
tions)..In.addition,.the.last.terms.in.Equations.11.7.and.11.8.describe.the.anisotropy.
term.spin.with.the.Hamiltonian.assumed.to.be.aligned.along.the.radial.direction. ˆ r ,.
with. k c .and. k s .being.the.anisotropy.constants.for.the.core.and.surface.spins,.respec-
tively..After.some.algebraic.manipulations.making.use.of.the.maximum.spin.entropy.