Chemistry Reference
In-Depth Information
TABLE 6.1
Physical Properties and the Estimated Level of Dificulty in Extracting them
from an Exact Solution Density for a Coulomb Interacting Many-Electron
Problem
Easy
Intermediate
Dificult
Finite.systems
Number.of.electrons
KS.orbitals
Exact.wave.function
Homo.eigenvalue
KS.eigenspectrum
Total.energy
Hartree.energy,.nuclear.energy
Noninteracting.kinetic.energy,.
T
s
Exchange-correlation.energy,.
E
XC
Moments.of.the.density
KS.linear.response.functions,.χ
S
Linear.response.functions
Nuclear.positions.and.charges
general.have.no.physical.signiicance..They.are.not.electron.orbitals,.but.rather.they.
are.simply.mathematical.objects.that.reproduce.the.exact.density.and.noninteracting.
kinetic.energy..Knowledge.of.an.exact.density.permits.calculation.of.useful.physical.
quantities.like.the.total.electronic.energy..In.practice,.we.use.the.minimization.of.
an.energy.functional.to.ind.the.density,.but.here,.we.stress.the.discreteness.of.the.
two.steps.in.the.KS.scheme.as.a.way.of.illustrating.the.strengths.and.shortcomings.
of.the.theory.
As. a. thought. exercise,. imagine. that. you. are. given. an. exact. electronic. density.
distribution.either.as.a.highly.resolved.data.set.on.a.grid.or.a.complicated.analytic.
function..In.principle,.you.could.extract.all.physical.quantities.and.even.nonmea-
surable. objects. such. as. the. Schrödinger. many-particle. wave. function.. Table. 6.1.
lists. the. relative. work. in. inding. various. physical. and. auxiliary. properties. from.
the.density..
Easy
.means.an.exact,.explicit.formula.is.known.as.a.functional.of.the.
density..
Intermediate
.means.that.an.explicit.numeric.scheme.can.be.written.down.
and.if.cutoff.errors.and.other.practicalities.do.not.get.in.the.way,.the.quantity.could.
be.extracted..
Dificult
.means.that.no.exact.scheme.is.known.to.obtain.the.quan-
tity.as.a.function.of.the.density.and.perhaps.even.as.a.functional.of.the.potential!.
Most. of. the. KS. quantities. are. of. the. intermediate. type.. Thus,. KS. gives. a. bridge.
between.dificult.to.extract.properties.like.the.wave.function.and.total.energy,.and.
more. directly. extractable. quantities. like. density. moments. and. total. numbers. of.
electrons.
There. are. many. similarities. between. the. machinery. of. DFT. and. an. effective.
potential.theory,.so.KS.DFT.is.often.misidentiied.as.a.MFT.(See.Figure.6.2)..We.
identify.the.following.two.main.sources.of.this.confusion:
.
1..
Limitations on the accuracy of approximate functionals
.. DFT. may. be.
an. exact. theory,. but. all. practical. KS. calculations. rely. on. approximate.
exchange-correlation. density. functionals. and. give. approximate. densities..
More.accurate.and.sophisticated.functionals.must.be.designed..As.the.com-
plexity.of.functionals.increase,.so.does.the.expense.of.the.calculation,.even-
tually.establishing.a.practical.limit.
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