Chemistry Reference
In-Depth Information
ε
F
Energy
FIGURE 5.3
Smearing.of.proile.of.occupation.number.versus.energy.in.a.metal..The.solid.
blue.curve.shows.the.occupied.states.for.a.metal.at.zero.electronic.temperature;.the.shaded.
green.area.shows.the.occupied.states.smeared.near.the.Fermi.energy.at.inite.electronic.tem-
perature..(Adapted.from.Walker,.B.G..et.al.,.
J. Phys. Condens. Matt
.,.16,.S2575,.2004..With.
permission.)
the. occupation. number. of. electronic. orbitals. as. the. Fermi. energy. is. crossed.. This.
discontinuity. makes. calculations. in. metals. more. challenging. (small. errors. in. the.
charge.density.give.rise.to.large.changes.in.the.electrostatic.potential).and.also.more.
expensive.to.converge.with.respect.to.the.sum.over.all.occupied.states.(since.a.dis-
continuity.needs.to.be.integrated.carefully)..This.latter.problem.can.be.dealt.with.
by.smearing.the.occupation.number.proile,.adding.extra.electronic.states.above.the.
Fermi.energy,.and.allowing.noninteger.occupancies.for.those.states.just.above.and.
below.the.Fermi.energy.(such.an.idea.is.illustrated.in.Figure.5.3)..This.is.equivalent.
to.the.introduction.of.a.inite.electronic.temperature,.and.will.be.discussed.further.
below.
5.3.1.1 Ensemble Density Functional Theory
Introduction.of.a.inite.temperature.into.the.description.of.the.electronic.system.(for.
instance.to.deal.appropriately.with.metallic.systems).can.be.achieved.through.gener-
alization.of.the.energy.functional.to.a.free.energy.functional.
Using.an.early.canonical.approach.introduced.by.Mermin.[60],.the.free.energy.
F
.
can.be.written.as.a.density.functional.including.an.entropy.term:
∫
3
r
F T
[
;{ },{ }]
ϕ
f
=
T
[ ]
ρ
+
E
[ ]
ρ
+
d V
( ) ( )
r
ρ
r
+
E
[ ]
ρ
−
T
S
elec
j
j
e
H
ext
XC
,
β
e
lec
∫
3
=
T
[ ]
ρ
+
E
[ ]
ρ
+
d V
r
( ) ( )
r
ρ
r
+
E
[ ]
ρ
e
H
ext
XC
,
β
1
∑
+
[
f
ln
f
+ −
(
1
f
) ln(
1−
f
j
)]
.
(5.3)
j
j
j
β
.
j
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