Density Functional Calculations of Metal Clusters: Structure, Dynamics, and Reactivity - Metallic Systems: A Quantum Chemist's Perspective - page 272

Hartree-Fock”.theory.that.introduced.an.orbital-averaged.local.exchange..potential.
that. used. what. we. now. call. the. local. density. approximation. (LDA),. based. on. the.
homogenous. electron. gas. model.. He. introduced. a. parameter,. α,. which. could. be.
adjusted,. for. example,. to. reproduce. Hartree-Fock. exchange. energies. of. atoms..
The resulting.Xα.equations.are

⎡ ⎤

1
2

ρ ( ) r

ʹ
ʹ

∫

2

− ∇ + v ( ) r + d r ʹ + v ( ) r ϕ ( ) r = ε ϕ ( ). r (9.10)

⎢ ⎥ α i i i

r − r

⎣ ⎦

. .

The.local.Xα.potential. v a ( r ).is.deined.as

1 3 /

3
2

⎛
⎜

3 ⎞

⎟

(9.11)

v α ( ) r = − α

π ρ

( ) r .

. .

In.order.to.solve.these.equations.for.inite.molecules,.Slater's.student,.Keith.Johnson,.
adopted. the. so-called. mufin-tin. approximation. from. solid-state. band. structure.
methodology.. The. mufin-tin. potential. involves. taking. a. spherical. average. of. the.
potential. in. spheres. centered. at. each. of. the. nuclei. and. at. an. outer. sphere. that. sur-
rounds. the. whole. molecule.. In. the. interstitial. volumes,. the. potential. was. taken. to.
be.constant..With.these.approximations,.partial.wave.expansions.could.be.used.to.
solve.the.Xα.equations.eficiently..This.is.quite.an.ugly.creature,.but.it.turned.out.
to.be.of.suficient.accuracy.to.provide.reasonably.accurate.orbitals,.orbital.energies,.
and.other.properties.to.allow.insightful.studies.of.quite.complex.systems,.molecules.
and.clusters.that.were.beyond.the.reach.of.traditional.quantum.chemical.methodol-
ogy..Importantly,.in.terms.of.the.eventual.growth.of.DFT.as.a.mainstream.quantum.
chemical. methodology,. the. Xα. period. allowed. a. irst. glimmer. of. encouragement.
(and.much.debate.and.push-back…).that.DFT.might.be.a.part.of.the.quantum.chem-
istry.tool.box..A.core.group.of.researchers,.and.their.students,.became.aware.of.the.
potential.for.applications.of.DFT,.and.some.of.them.also.had.a.background.in.tradi-
tional.quantum.chemistry..Eventually,.the.two.worlds.would.come.together.
Although. the. Xα. scattered-wave. method. yielded. remarkable. results. for. many.
systems,. which. were. beyond. the. reach. of. traditional. quantum. chemistry,. it. had.
important. limitations,. most. importantly,. indeed,. fatally,. the. inability. to. optimize.
geometries. and. determine. structures,. which. is. certainly. at. the. heart. of. chemical.
thinking.. Overcoming. this. shortcoming,. by. bringing. in. concepts. and. techniques.
from. traditional. quantum. chemistry,. was. the. goal. of. the. next. leg. of. the. voyage.
toward.modern.DFT.methodologies.and.computer.codes.

9.2.3 lcgto-x α : B ringing (g auSSian ) q uantum c hemiStry
into dft—n oW g eometrieS c an B e o Ptimized
The. problems. with. Xα. and. geometry. optimizations. were. well. recognized. in. the.
1970s,. including. within. the. “Slater. school.”. Two. key. contributions. allowed. enor-
mous.steps.to.be.taken.toward.bringing.Xα.into.a.better.competitive.position.with.
respect. to. traditional. quantum. chemistry,. while. keeping. the. eventual. advantages.