Density Functional Calculations of Metal Clusters: Structure, Dynamics, and Reactivity - Metallic Systems: A Quantum Chemist's Perspective - page 279

From.this.expression,.we.note.immediately.that.the.Kohn-Sham.matrix.elements.in.
ADFT.are.independent.from.density.matrix.elements..As.a.result,.only.the.approxi-
mated. density. (and. the. corresponding. density. derivatives. in. the. case. of. gradient.
corrected.functional).is.numerically.calculated.on.a.grid..Because.these.quantities.
are.linear.scaling.by.construction,.the.necessary.grid.work.is.considerably.reduced..
Moreover,.the.sharing.of.auxiliary.function.exponents.by.the.above.discussed.auxil-
iary.function.sets.decreases.dramatically.the.expensive.exponential.function.evalu-
ations.at.each.grid.point..Besides.the.eficient.calculation.of.the.three-center.ERIs,.
the.simpliied.grid.work.is.the.main.reason.for.the.computational.eficiency.of.the.
ADFT.approach..In.fact,.the.calculation.of.the.Kohn-Sham.potential.in.ADFT.is.
identical.to.orbital-free.DFT.approaches.with.the.auxiliary.function.density.as.the.
basic.variable..Of.course,.Kohn-Sham.orbitals.are.still.used.in.ADFT.for.the.cal-
culation.of.the.kinetic.energy.contribution.and.the.orbital.density,.which.is.the.seed.
for.the.auxiliary.density.via.the.variational.Coulomb.itting..Thus,.ADFT.might.be.
seen.as.a.combination.of.the.basic.ideas.from.the.conventional.Kohn-Sham.method-
ology.with.those.from.orbital-free.DFT..As.the.above.derivation.of.the.Kohn-Sham.
matrix. as. a. partial. derivative. of. the. ADFT. energy. expression. demonstrates. ana-
lytic. energy. derivatives. can. be. formulated. straightforwardly. in. ADFT.. This. is. a.
major. difference. to. other. approaches. that. use. least-squares. itting. techniques. for.
the. approximate. calculation. of. the. exchange-correlation. energy. and. potential.. Of.
course,.this.holds.for.analytic.gradients.too..For.higher.energy.derivatives,.the.so-
called.coupled.perturbed.Kohn-Sham.equations.that.are.close.analogs.of.the.cou-
pled.perturbed.Hartree-Fock.equations.are.usually.employed.to.solve.the.necessary.
systems.of.equations.for.the.perturbed.MO.coeficients.or.density.matrix.elements..
In.order.to.keep.the.necessary.memory.manageable,.iterative.procedures.are.usually.
employed..In.the.next.section,.we.describe.an.alternative.procedure.for.the.calcula-
tion.of.the.perturbed.density.matrix.elements.that.is.based.on.the.ADFT.formalism.
discussed.here.

9.2.5.2 Auxiliary Density Perturbation Theory
For.the.derivation.of.ADPT,.we.start.from.McWeeny's.self-consistent.perturbation.
(SCP).theory.[51-56],.which.represents.a.direct.approach.for.the.calculation.of.the.
perturbed.density.matrix..For.clarity.of.derivation,.we.restrict.ourselves.to.perturba-
tion-independent.basis.and.auxiliary.functions.as.commonly.used.in.polarizability.
calculations.. Under. these. restrictions,. the. perturbed. closed-shell. density. matrix. is.
given.by.the.SCP.formalism.as.[56]

occ uno

( λ )

≡ ∂
∂

P K

∑ ∑

μν ia

P

( λ )

= 2 ( c c + c c ). (9.32)

μν μ ν i a μ a ν i

λ ε − ε

i a

. .

i a

In. this. notation,. λ. denotes. a. perturbation. parameter. (e.g.,. an. external. ield. com-
ponent. in. the. case. of. polarizability. calculations),. and. the. indices. i . and. a . denote.
occupied.( occ ).and.unoccupied.( uno ).MOs,.respectively..The.perturbed.Kohn-Sham.
matrix.in.the.MO.basis.is.given.by