Density Functional Calculations of Metal Clusters: Structure, Dynamics, and Reactivity - Metallic Systems: A Quantum Chemist's Perspective - page 274

= ∑ x k

ρ( ) (9.17)

r ( ). r

k

.

k

.

Several.suggestions.for.the.choice.of.the.auxiliary.functions. k ( r ).exist.in.the.literature.
[14-18]..In.earlier.deMon.versions,.atom-centered.primitive.Cartesian.Gaussian.aux-
iliary.functions.were.used.for.the.expansion.of.the.approximated.density..They.were.
grouped.together.to.auxiliary.function.sets.that.share.the.same.exponent.[15,16].
With.the.above.LCGTO.expansion.for.the.density.and.the.approximated.density,.
the.minimization.of.the.ε 2 .error.with.respect.to.the.Coulomb.itting.coeficients. x k .
yields

∂
∂

ε

∑ , ∑

2

(9.18)

= − P μν k + x l k ≡ ∀ 0 k .

μν l

x

k

. .

μ ν l

For.convenience,.we.cast.this.minimization.in.an.inhomogeneous.equation.system,.
G x .=. J ,.with.the.Coulomb.matrix. G .and.the.Coulomb.vector. J .containing.elements.
of.the.form

∑

G = k l ; J = P μν k . (9.19)

kl k μν

μ ν ,

. .

It.can.be.shown.that. G .is.symmetric.and.positive.deinite.[19]..Thus,.the.above.equa-
tion.systems.can.always.be.solved,.and.the.corresponding.itting.coeficients.can.be.
determined..This.procedure.is.nowadays.also.common.in.wave-function.approaches.
where.it.is.often.called.the.Resolution.of.the.Identity.approximation.[20]..With.the.
variational. itting. of. the. Coulomb. potential,. the. computational. bottleneck. in. Xα.
methods.shifted.to.the.calculation.of.the.Xα.potential..For.this.reason,.a.second.aux-
iliary.function.set.was.introduced.to.it.this.potential,.too..However,.the.itting.of.the.
Xα.potential.(and.later.of.the.exchange-correlation.potential).as.suggested.by.Sambe.
and.Felton.[21].represents.a.least-squares.it.on.a.grid..As.a.result,.the.corresponding.
energy.is.not.variational..Nevertheless,.approximate.gradients.were.now.available,.
and,.therefore,.structure.optimizations.became.possible.

9.2.4 h iStory of de m on
The. deMon. program. system. was. developed. for. DFT. calculations. of. atoms,. mol-
ecules,.and.solids..Its.irst.widely.available.version.appeared.in.1992..It.was.based.on.
the.PhD.work.of.A..St-Amant.performed.under.the.supervision.of.D.R..Salahub.at.
the.Université.de.Montréal.[22]..The.name. deMon .stands.for.“ de nsity.of. Mon treal.”.
Shortly.after.its.irst.appearance,.the.original.deMon.code.was.substantially.modi-
ied. for. commercialization. by. BIOSYM. Technologies.. The. beta-release. of. this.
version. appeared. in. 1993.. It. also. served. as. the. basis. for. the. deMon-KS. series. of.
programs.developed.in.Montreal.until.1997.[23]..The.focus.of.this.program.devel-
opment. was. mainly. toward. molecular. property. calculations. with. LCGTO. DFT.