Density Functional Theory Calculations on Cobalt and Platinum Transition Metal Clusters - Metallic Systems: A Quantum Chemist's Perspective - page 356

Both.of.these.calculated.excitation.energies.are.much.better.than.the.ones.obtained.
by. Fortunelli. in. 1999. [95],. where. BPW91. GGA. functional. was. employed. with. a.
double-zeta-plus-polarization.basis.set..The.SO.splitting.of.the.5d.orbital.of.the.Pt.
atom.in.its.singlet.state.was.found.to.be.1.131.eV,.in.reasonable.agreement.with.the.
value.of.1.256.eV.calculated.using.the.all-electron.fully.relativistic.DFT.code.FPLO.
[96].via.employing.Perdew-Wang.[97].local.xc.functional..The.IP.of.Pt.was.calcu-
lated.as.9.319.eV.(SO.included).in.contrast.with.an.experimental.value.of.8.958.eV.
[98]..Huda.et.al..[90].calculated.the.IP.of.Pt.as.9.381.eV.in.their.DFT.study.with.the.
projected.augmented.wave.(PAW).method..The.calculated.electron.afinity.(EA).is.
1.806.eV,. which. is. comparable. to. the. experimental. value. of. 2.123.±.0.001.eV. [99]..
The. discrepancy. between. the. calculated. and. experimental. values. of. the. EA. may.
be.reduced.when.the.experimental.value.is.corrected.for.SO.effects,.which.can.be.
estimated.to.decrease.it.by.about.0.2.eV.[100].
The. ground-state. spin. multiplicity. of. the. Pt 2 . dimer. is. found. to. be. a. triplet. (not.
including.SO.coupling)..The.bond.lengths.of.the.Pt.dimer.were.calculated.as.2.373.
and.2.406.Å.for.non-SO.and.SO.calculations,.respectively,.while.the.experimental.
value.[101].is.2.333.Å..The.corresponding.binding.energies.are.1.354.(non-SO).and.
1.238.eV/atom.(SO).with.respect.to.the.atomic.ground.state..The.experimental.BE.is.
1.57.±.0.01.eV/atom.[102]..Thus,.the.present.method.with.SO.effects.slightly.overes-
timates.the.Pt.dimer.bond.length.and.therefore.underestimates.the.BE..However,.the.
primary.goal.of.the.present.work.is.to.investigate.the.effect.of.the.SO.term.on.the.
relative.stability.of.the.isomers.rather.than.providing.accurate.quantitative.results..
Both. of. the. calculated. vibrational. frequencies. of. 237. (non-SO). and. 219.cm −1 . (SO).
agree.very.well.with.the.measured.value.of.223.cm −1 .[103].
10.3.2.2 Pt 4
We.report.the.relative.stability.of.Pt 4 .isomers.with.and.without.SO.coupling.for.dif-
ferent.spin.multiplicities.in.Table.10.4..The.identical.results.in.Tables.10.4.through.
10.6.with.the.same.Δ E .and.spin.moment.for.SO.cases.correspond.to.identical.local.
minima.structures..Each.SO.calculation.was.performed.after.a.non-SO.one.starting.
from. the. charge. and. spin. densities. obtained. in. the. non-SO. calculation.. Thus,. the.
identical.SO.results.for.a.particular.size.and.isomer.mean.that.the.different.spin.mul-
tiplicities.of.the.non-SO.case.converge.to.the.same.SO.electronic.state.in.the.relax-
ation.process..When.the.SO.coupling.effect.is.included.in.the.calculations,.the.spin.
( S ).and.orbital.angular.momentum.( L ).are.not.conserved.separately,.instead.the.total.
angular.momentum. J = L + S .is.conserved..If.relativistic.effects.are.considered.as.a.
perturbation,.then.one.can.consider. L 2 = L ( L .+.1).and. S 2 = S ( S .+.1).as.being.conserved.
[104]..We.use.the.term.spin.moment.in.Tables.10.4.through.10.7.for.SO.cases.in.this.
sense..The.geometric.structures.and.bond.lengths.of.each.isomer.of.Pt 4 .for.their.most.
stable.spin.multiplicity.are.given.in.Figure.10.6.
Similar. to. the. results. of. our. previous. study. on. small. bare. and. hydrogenated. Pt.
clusters. [39]. where. we. did. not. employ. SO. coupling. effects. and. similar. to. results.
presented.in.Refs..[87,105],.the.calculated.lowest.energy.structure.of.the.Pt.tetramer.
is.a.tetrahedron.regardless.of.whether.SO.coupling.is.included..When.SO.effects.are.
not.considered,.the.ground-state.spin.multiplicity.of.the.tetrahedron.is.3.and.it.has.
the.point.group.symmetry. C 3v ..On.the.other.hand,.when.SO.effects.are.taken.into.