Density Functional Calculations of Metal Clusters: Structure, Dynamics, and Reactivity - Metallic Systems: A Quantum Chemist's Perspective - page 277

this.procedure,.the.exponents.of.the.auxiliary.function.sets.are.generated.according.
to.the.exponent.range.spanned.by.the.given.basis.set..These.auxiliary.function.sets.
are.named.GEN-AX.with.X.=.2,.3,.and.4..A.larger.X.indicates.that.more.auxiliary.
function.sets.are.generated..The.GEN-A2.set.is.for.the.DFT-optimized.double.zeta.
plus.valence.polarization.(DZVP).basis.set.identical.with.the.A2.auxiliary.function.
set..Again,.GEN-A2.auxiliary.function.sets.contain.only. s .and. d .sets..As.an.alterna-
tive,.GEN-AX*.sets.(X.=.2,.3,.4).are.also.available.in.deMon2k..These.sets.possess.
the.same.exponents.as.the.corresponding.GEN-AX.sets.but.also.contain. f .and. g .aux-
iliary.function.sets..For.a.more.detailed.description.of.the.GEN-AX.and.GEN-AX*.
auxiliary.function.sets,.we.refer.the.interested.reader.to.the.literature.[47,48].

9.2.5.1 Auxiliary Density Functional Theory
As.an.alternative.to.the.above-described.least-squares.itting.of.the.exchange-corre-
lation.potential,.the.direct.use.of.the.variationally.itted.density.for.the.calculation.of.
the.exchange-correlation.energy.has.been.investigated.over.the.last.decade.[34,49]..
The.corresponding.ADFT.energy.expression.has.the.form.[34]

1
2

1
2

∑ μν ∑ ∑

E = P H + P μν k x − x x k l + E [ ( )]. ρ r (9.21)

μν μν k k l XC

.

μ ν , μ ν , k l ,

.

Here
μ.and.ν.denote.(contracted).atomic.GTOs
k ‾ .and. l ‾ .denote.primitive.Hermite.Gaussian.auxiliary.functions

The.irst.term.represents,.as.usual,.the.one-electron.core.energy..The.next.two.terms.
result.from.the.variational.itting.of.the.Coulomb.potential.and.substitute.the.four-
center.ERI.term.in.standard.Kohn-Sham.energy.expressions..The.last.term.denotes.
the.exchange-correlation.energy.expression.of.the.selected.functional..It.is.calculated.
using.the.linear.scaling.auxiliary.density. ~ ( r )..The.derivatives.of.this.energy.expres-
sion.with.respect.to.the.density.matrix.elements,.keeping.the.itting.coeficients.con-
stant,.deine.the.corresponding.ADFT.Kohn-Sham.matrix.elements

[ ( )] .

∂
∂

E
P

+ ∂

E ρ r

∑

XC

(9.22)

K = = H + μν k x

μν μν k

∂ P

μν μν

. .

k

The. partial. derivative. of. the. (local). exchange-correlation. energy. functional. with.
respect.to.the.density.matrix.elements.yields.[50]

∂ E [ ( )] ρ r δ E [ ( )]

( )

ρ r ∂

∂

ρ ( ) r

∫

XC XC

= d r .

(9.23)

∂ P δρ r P

.

μν μν

.

As.in.standard.Kohn-Sham.DFT,.the.functional.derivative.of.the.exchange-correla-
tion.energy.with.respect.to.the.approximated.density.deines.the.exchange-correla-
tion.potential.in.ADFT