Many-body approaches to studies of electronic systems: Hartree-Fock theory and Density ...

24 Pages·2013·0.35 MB·English

by  Dick Furnstahl

Chapter 15
Many-body approaches to studies of electronic
systems: Hartree-Fock theory and Density
Functional Theory
AbstractThischapterpresentstheHartree-Fockmethodwithanemphasisoncomputingthe
energiesofselectedclosed-shell atoms.
15.1 Introduction
A theoretical understanding of the behavior of quantum mechanical systems with many in-
teracting particles, normally called many-body systems, is a great challenge and provides
fundamental insights into systems governed by quantum mechanics, as well as offering po-
tential areas of industrial applications, from semi-conductor physics to the construction of
quantum gates. The capabilitytosimulate quantum mechanicalsystems withmanyinteract-
ingparticlesiscrucialforadvancesinsuchrapidlydevelopingfieldslikematerialsscience.
However,mostquantum mechanicalsystems ofinterestinphysics consist ofa largenum-
berofinteractingparticles.ThetotalnumberofparticlesNisusuallysufficientlylargethatan
exactsolution(viz., inclosed form)cannotbefound.One needs thereforereliablenumerical
methodsforstudyingquantummechanicalsystemswithmanyparticles.
Studiesofmany-bodysystemsspanfromourunderstandingofthestrongforcewithquarks
and gluons as degrees of freedom, the spectacular macroscopic manifestations of quantal
phenomena such as Bose-Einstein condensation with millions of atoms forming a coherent
state, to properties of new materials, with electrons as effective degrees of freedom. The
length scales range from few micrometers and nanometers, typical scales met in materials
science, to 10−15 10−18 m, a relevant length scale for the strong interaction. Energies can
−
span from few meV to GeV or even TeV. In some cases the basic interaction between the in-
teractingparticlesiswell-known.AgoodexampleistheCoulombforce,familiarfromstudies
ofatoms,moleculesand condensed matterphysics. Inother cases,such asforthestrong in-
teraction between neutrons and protons (commonly dubbed as nucleons) or dense quantum
liquidsonehastoresorttoparameterizationsoftheunderlyinginterparticleinteractions.But
the system can also span over much larger dimensions as well, with neutron stars as one of
theclassicalobjects.Thisstaristheendpointofmassivestarswhichhaveuseduptheirfuel.
A neutron star, as its name suggests, is composed mainly of neutrons, with a small fraction
of protons and probablyquarks in its inner parts. The star is extremely dense and compact,
with a radiusof approximately10 km and a mass which is roughly 1.5 times thatof our sun.
The quantum mechanical pressure which is set up by the interacting particles counteracts
the gravitational forces, hindering thus a gravitational collapse. To describe a neutron star
oneneedstosolveSchrödinger’sequationforapproximately1054 interactingparticles!
With a given interparticle potential and the kinetic energy of the system, one can in turn
define the so-called many-particleHamiltonianHˆ whichenters the solution of Schrödinger’s
485
41856Many-bodyapproachestostudiesofelectronicsystems:Hartree-FocktheoryandDensityFunctionalTheory
equation or Dirac’sequation in case relativistic effects need to be included. For many parti-
cles, Schrödinger’s equation is an integro-differential equation whose complexity increases
exponentially with increasing numbers of particles and states that the system can access.
Unfortunately, apartfrom some few analyticallysolvable problems and one and two-particle
systems that can be treated numerically exactly via the solution of sets of partial differen-
tialequations,thetypicalabsenceofanexactlysolvable(onclosedform)contributionto the
many-particleHamiltonianmeansthatweneedreliablenumericalmany-bodymethods.These
methods should allow for controlled approximations and provide a computational scheme
whichaccountsforsuccessive many-bodycorrectionsinasystematicway.
Typicalexamplesofpopularmany-bodymethodsarecoupled-clustermethods[93–97],var-
ious types of Monte Carlo methods [98–100], perturbative many-body methods [101–103],
Green’s function methods [104,105], the density-matrix renormalization group [106,107],
density functional theory [108] and ab initio density functional theory [109–111], and large-
scale diagonalization methods [112–114], just to mention a few. The physics of the system
hints at which many-body methods to use. For systems with strong correlations among the
constituents, methods based on mean-field theory such as Hartree-Fock theory and density
functional theory are normally ruled out. This applies also to perturbative methods, unless
onecanrenormalizethepartsoftheinteractionwhichcauseproblems.
Theaimofthisandthenextthreechaptersistopresenttoyoumany-bodymethodswhich
can be used to study properties of atoms, molecules, systems in the solid state and nuclear
physics.Welimittheattentiontonon-relativisticquantummechanics.
In this chapter we limit ourselves to studies of electronic systems such atoms, molecules
and quantum dots, as discussed partly in chapter 14 as well. Using the Born-Oppenheimer
approximationwerewroteSchrödinger’sequationforN electronsas
N 1 N Z N 1
∑ ∇2 ∑ +∑ Ψ(R)=EΨ(R),
!−i=12 i −i=1ri i<jrij"
whereweletRrepresentthepositionswhichtheNelectronscantake,thatisR= r1,r2,...,rN .
{ }
With more than one electron present we cannot find an solution on a closed form and must
resort to numerical efforts. In this chapter we will examine Hartree-Fock theory applied to
theatomicproblem.However, the machineryweexpose caneasilybeextended to studiesof
moleculesortwo-dimensionalsystemslikequantumdots.
For atoms and molecules, the electron-electron interaction is rather weak compared with
theattractionfromthenucleus.Anindependentparticlepictureisthereforeaviablefirststep
towardsthesolutionofSchrödinger’sequation.Weassumethereforethateachelectronssees
aneffective field set up by the other electrons. This leadsto an integro-differentialequation
andmethodslikeHartree-Focktheorydiscussedinthischapter.
Inpracticalterms,fortheHartree-Fockmethodweendupsolvingaone-particleequation,
as is the case for the hydrogen atom but modified due to the screening from the other elec-
trons. This modified single-particle equation reads (see Eq. (14.15for the hydrogen case) in
atomicunits
1 d2 l(l+1) Z
u (r)+ +Φ(r)+F u (r)=e u (r).
−2dr2 nl 2r2 − r nl nl nl nl
# $
Thefunctionu isthesolutionoftheradialpartoftheSchrödingerequationandthefunctions
nl
Φ(r)andF arethecorrectionsduetothescreeningfromtheotherelectrons.Wewillderive
nl
theseequationsinthenextsection.
Thetotalone-particlewavefunction,seechapter14is
ψnlmlsms =φnlml(r)ξms(s)
15.2 Hartree-Focktheory 487
withsisthe spin(1/2forelectrons), ms isthe spinprojectionms= 1/2,andthe spatialpart
±
is
φ (r)=R (r)Y (rˆ)
nlml nl lml
with Y the spherical harmonics discussed in chapter 14 and u =rR . The other quantum
nl nl
numbers are the orbital momentum l and its projection m = l, l+1,...,l 1,l and the
l
− − −
principalquantumnumbern=nr+l+1,withnr thenumberofnodesofagivensingle-particle
wave function. All results are in atomic units, meaning that the energy is given by e =
nl
Z2/2n2 andtheradiusisdimensionless.
−
We obtain then a modified single-particle eigenfunction which in turn can be used as an
input in a variational Monte Carlo calculation of the ground state of a specific atom. This
is the aim of the next chapter. Since Hartree-Fock theory does not treat correctly the role
of many-body correlations, the hope is that performing a Monte Carlo calculation we may
improveourresultsbyobtainingabetteragreementwithexperiment.
In the next chapter we focus on the variational Monte Carlo method as a way to improve
upon the Hartree-Fock results. Although the variational Monte Carlo approach will improve
our agreement with experiment compared with the Hartree-Fock results, there are still fur-
ther possibilities for improvement. This is provided by Green’s function Monte Carlo meth-
ods, which allow for an in principle exact calculation. The diffusion Monte Carlo method is
discussed in chapter 17, with anapplicationto studies of Bose-Einstein condensation. Other
many-bodymethodssuchaslarge-scalediagonalizationandcoupled-clustertheoriesaredis-
cussedinRef.[115].
15.2 Hartree-Fock theory
Hartree-Fock theory [95,116] is one of the simplest approximate theories for solving the
many-body Hamiltonian. It is based on a simple approximationto the true many-body wave-
function; that the wave-function is given by a single Slater determinant of N orthonormal
single-particlewavefunctions1
ψnlmlsms =φnlml(r)ξms(s).
We use hereafter the shorthand ψ (r)=ψ (r), where α now contains all the quantum
nlmlsms α
numbersneededtospecifyaparticularsingle-particleorbital.
TheSlaterdeterminantcanthenbewrittenas
ψα(r1)ψα(r2) ... ψα(rN)
1 %ψβ(r1)ψβ(r2) ... ψβ(rN)%
Φ(r1,r2,...,rN,α,β,...,ν)= √N!%%% ... ... ... ... %%%. (15.1)
% %
% %
%%ψν(r1) ψβ(r2) ... ψβ(rN)%%
% %
% %
Here the variables ri include the coordinates of%spin and space of partic%le i. The quantum
numbersα,β,...,ν encompass all possible quantum numbers needed to specify a particular
system.Asanexample,considertheNeonatom,withtenelectronswhichcanfillthe1s,2sand
2p single-particle orbitals. Due to the spin projections ms and orbital momentum projections
m,the1sand2sstateshaveadegeneracyof2(2l+1)=2whilethe2porbitalhasadegeneracy
l
1 We limit ourselves to a restricted Hartree-Fock approach and assume that all the lowest-lying orbits are
filled.Thisconstitutesanapproachsuitableforsystemswithfilledshells.Thetheoryweoutlineistherefore
applicabletosystemswhichexhibitso-calledmagicnumberslikethenoblegases,closed-shellnucleilike16O
and40Caandquantumdotswithmagicnumberfillings.
41858Many-bodyapproachestostudiesofelectronicsystems:Hartree-FocktheoryandDensityFunctionalTheory
of 2(2l+1)2(2 1+1)=6. This leads to ten possible values forα,β,...,ν. Fig. 15.1 shows the
·
possiblequantumnumberswhichthetenfirstelementscanhave.
s p
L
K H
s p s p
L
K He Li
s p s p s p s p
n=2
n=1 Be B C N
s p s p s p
n=2
n=1 O F Ne
Fig. 15.1 Theelectronicconfigurations for thetenfirst elements.We letan arrow which pointsupward to
representastatewithms=1/2whileanarrowwhichpointsdownwardshasms= 1/2.
−
If we consider the helium atom with two electrons in the 1s state, we can write the total
Slaterdeterminantas
1 ψ (r )ψ (r )
Φ(r1,r2,α,β)= √2%ψαβ(r11)ψαβ(r22)%, (15.2)
% %
with α=nlmlsms =(1001/21/2) and β=nlmlsm%%s =(1001/2−1%%/2) or using ms =1/2=↑ and
ms = 1/2= as α=nlmlsms =(1001/2 ) and β=nlmlsms =(1001/2 ). It is normal to skip
− ↓ ↑ ↓
thequantumnumberoftheone-electronspin.Weintroducethereforetheshorthandnlm or
l
↑
nlm )foraparticularstate.WritingouttheSlaterdeterminant
l
↓
1
Φ(r1,r2,α,β)= √2 ψα(r1)ψβ(r2)−ψβ(r1)ψγ(r2) , (15.3)
& ’
we see that the Slater determinant is antisymmetric with respect to the permutation of two
particles,thatis
Φ(r ,r ,α,β)= Φ(r ,r ,α,β),
1 2 2 1
−
Forthreeelectronswehavethegeneralexpression
ψ (r )ψ (r )ψ (r )
1 α 1 α 2 α 3
Φ(r1,r2,r3,α,β,γ)= √3!%ψβ(r1)ψβ(r2)ψβ(r3)%. (15.4)
%ψ(r ) ψ(r ) ψ(r )%
% γ 1 γ 2 γ 3 %
% %
% %
Computingthedeterminantgives % %
Φ(r ,r ,r ,α,β,γ)= 1 ψ (r )ψ (r )ψ(r )+ψ (r )ψ(r )ψ (r )+ψ(r )ψ (r )ψ (r )
1 2 3 √3! α 1 β 2 γ 3 β 1 γ 2 α 3 γ 1 α 2 β 3 −
ψγ&(r1)ψβ(r2)ψα(r3)−ψβ(r1)ψα(r2)ψγ(r3)−ψα(r1)ψγ(r2)ψβ(r3) .(15.5)
’
Wenoteagainthatthewave-functionisantisymmetricwithrespectto aninterchangeofany
two electrons, as required by the Pauli principle. For an N-body Slater determinant we have
thus(omittingthequantumnumbersα,β,...,ν)
15.2 Hartree-Focktheory 489
Φ(r1,r2,...,ri,...,rj,...rN)= Φ(r1,r2,...,rj,...,ri,...rN).
−
As another example, consider the Slater determinant for the ground state of beryllium.
Thissystemismadeupoffourelectronsandweassumethattheseelectronsfillthe1sand2s
hydrogen-likeorbits.Theradialpartofthesingle-particlecouldalsoberepresentedbyother
single-particlewavefunctionssuchasthosegivenbytheharmonicoscillator.
TheansatzfortheSlaterdeterminantcanthenbewrittenas
ψ (r )ψ (r )ψ (r )ψ (r )
100 1 100 2 100 3 100 4
↑ ↑ ↑ ↑
1 ψ (r )ψ (r )ψ (r )ψ (r )
Φ(r1,r2,,r3,r4,α,β,γ,δ)= √4!%%%ψ120000↓↑(r11)ψ120000↓↑(r22)ψ120000↓↑(r33)ψ120000↓↑(r44)%%%.
%ψ (r )ψ (r )ψ (r )ψ (r )%
% 200 1 200 2 200 3 200 4 %
% ↓ ↓ ↓ ↓ %
% %
We choose an ordering where columns re%present the spatial positions of vario%us electrons
whilerowsrefertospecificquantumnumbers.
NotethattheSlaterdeterminantaswritteniszerosincethespatialwavefunctionsforthe
spin up and spin down states are equal. However, we can rewrite it as the product of two
Slaterdeterminants,oneforspinupandoneforspindown.Ingeneralwecanrewriteitas
Φ(r ,r ,,r ,r ,α,β,γ,δ)=Det (1,2)Det (3,4) Det (1,3)Det (2,4)
1 2 3 4
↑ ↓ − ↑ ↓
Det (1,4)Det (3,2)+Det (2,3)Det (1,4) Det (2,4)Det (1,3)
− ↑ ↓ ↑ ↓ − ↑ ↓
+Det (3,4)Det (1,2),
↑ ↓
wherewehavedefined
1 ψ (r )ψ (r )
Det (1,2)= 100↑ 1 100↑ 2 ,
↑ %√2ψ200↑(r1)ψ200↑(r2)%
% %
and % %
% 1 ψ (r )ψ (r )%
Det (3,4)= 100↓ 3 100↓ 4 .
↓ %√2ψ200↓(r3)ψ200↓(r4)%
% %
The total determinant is still zero! Inour%variationalMonte Ca%rlo calculationsthis will obvi-
% %
ouslycauseproblems.
We want to avoid to sum over spin variables, in particular when the interaction does not
depend on spin. It can be shown, see for example Moskowitz etal[117,118], that for the
variational energy we can approximate the Slater determinant as the product of a spin up
andaspindownSlaterdeterminant
Φ(r ,r ,,r ,r ,α,β,γ,δ)∝Det (1,2)Det (3,4),
1 2 3 4
↑ ↓
ormoregenerallyas
Φ(r1,r2,...rN)∝Det Det ,
↑ ↓
wherewehavetheSlaterdeterminantastheproductofaspinuppartinvolvingthenumberof
electronswithspinuponly(twoinberylliumandfiveinneon)andaspindownpartinvolving
theelectronswithspindown.
This ansatz is not antisymmetric under the exchange of electrons with opposite spins but
it can be shown that it gives the same expectation value for the energy as the full Slater
determinant as long as the Hamiltonian is spin independent. It is left as an exercise to the
reader to show this. However, beforewe canprove thisneed toset up the expectationvalue
ofagiventwo-particleHamiltonianusingaSlaterdeterminant.
41950Many-bodyapproachestostudiesofelectronicsystems:Hartree-FocktheoryandDensityFunctionalTheory
15.3 Expectation value of the Hamiltonian with a given Slater
determinant
WerewriteourHamiltonian
N 1 N Z N 1
Hˆ = ∑ ∇2 ∑ +∑ ,
− 2 i − r r
i=1 i=1 i i<j ij
as
N N 1
Hˆ =Hˆ0+HˆI=∑hˆi+ ∑ , (15.6)
r
i=1 i<j=1 ij
where
1 Z
hˆi=−2∇2i −r . (15.7)
i
Thefirstterm of Eq.(15.6), H1, isthesum of theN identicalone-bodyHamiltonianshˆi. Each
individualHamiltonianhˆicontainsthekineticenergyoperatorofanelectronanditspotential
energyduetotheattractionofthenucleus.Thesecondterm,H2,isthesumoftheN(N 1)/2
−
two-bodyinteractionsbetweeneachpairofelectrons.Letusdenotethegroundstateenergy
byE0.Accordingtothevariationalprinciplewehave
E0 E[Φ]= Φ∗HˆΦdτ (15.8)
≤
(
whereΦisatrialfunctionwhichweassumetobenormalized
Φ∗Φdτ=1, (15.9)
(
where we have used the shorthand dτ=dr1dr2...drN. In the Hartree-Fock method the trial
functionistheSlaterdeterminantofEq.(15.1)whichcanberewrittenas
1
Ψ(r1,r2,...,rN,α,β,...,ν)= √N!∑(−)PPψα(r1)ψβ(r2)...ψν(rN)=√N!AΦH, (15.10)
P
where we have introduced the anti-symmetrization operator A defined by the summation
overallpossiblepermutationsoftwoeletrons.Itisdefinedas
1
A = ∑( )PP, (15.11)
N! −
P
withthetheHartree-functiongivenbythesimpleproductofallpossiblesingle-particlefunc-
tion(twoforhelium,fourforberylliumandtenforneon)
ΦH(r1,r2,...,rN,α,β,...,ν)=ψα(r1)ψβ(r2)...ψν(rN). (15.12)
BothHˆ0 andHˆI areinvariantunderelectronpermutations,andhencecommutewithA
[H0,A]=[HI,A]=0. (15.13)
Furthermore,A satisfies
A2=A, (15.14)
sinceeverypermutationoftheSlaterdeterminantreproducesit.TheexpectationvalueofHˆ0
Φ∗Hˆ0Φdτ=N! ΦH∗AHˆ0AΦHdτ
( (
15.3 ExpectationvalueoftheHamiltonianwithagivenSlaterdeterminant 491
isreadilyreducedto
Φ∗Hˆ0Φdτ=N! ΦH∗Hˆ0AΦHdτ,
( (
where we have used eqs. (15.13) and (15.14). The next step is to replace the anti-symmetry
operatorbyitsdefinitioneq.(15.10)andtoreplaceHˆ0 withthesumofone-bodyoperators
N
Φ∗Hˆ0Φdτ=∑∑(−)P ΦH∗hˆiPΦHdτ. (15.15)
( i=1 P (
The integral vanishes if two or more electrons are permuted in only one of the Hartree-
functionsΦH becausetheindividualorbitalsareorthogonal.Weobtainthen
N
Φ∗Hˆ0Φdτ=∑ ΦH∗hˆiΦHdτ. (15.16)
( i=1(
Orthogonalityallowsustofurthersimplifytheintegral,andwearriveatthefollowingexpres-
sionfortheexpectationvaluesofthesumofone-bodyHamiltonians
N
Φ∗Hˆ0Φdτ= ∑ ψµ∗(ri)hˆiψµ(ri)dri. (15.17)
( µ=1(
The expectation value of the two-body Hamiltonian is obtained in a similar manner. We
have
Φ∗HˆIΦdτ=N! ΦH∗AHˆIAΦHdτ, (15.18)
( (
whichreducesto
N 1
Φ∗HˆIΦdτ= ∑ ∑(−)P ΦH∗ r PΦHdτ, (15.19)
( i j=1 P ( ij
≤
byfollowingthesameargumentsasfortheone-bodyHamiltonian.Becauseofthedependence
ontheinter-electronicdistance1/rij,permutationsoftwoelectronsnolongervanish,andwe
get
N 1
Φ∗HˆIΦdτ= ∑ ΦH∗ r (1−Pij)ΦHdτ. (15.20)
( i<j=1( ij
where Pij is the permutation operator that interchanges electrons i and j. Again we use the
assumptionthattheorbitalsareorthogonal,andobtain
1 N N 1 1
Φ∗HˆIΦdτ=2 ∑ ∑ ψµ∗(ri)ψν∗(rj)r ψµ(ri)ψν(rj)dxidxj − ψµ∗(ri)ψν∗(rj)r ψν(ri)ψµ(ri)dxidxj .
( µ=1ν=1)( ij ( ij *
(15.21)
The first term is the so-called direct term or Hartree term, while the second is due to the
PauliprincipleandiscalledtheexchangetermortheFockterm.Thefactor1/2isintroduced
becausewenowrunoverallpairstwice.
CombiningEqs.(15.17)and(15.21)weobtainthefunctional
N 1 N N 1
E[Φ]= ∑ ψµ∗(ri)hˆiψµ(ri)dri+2 ∑ ∑ ψµ∗(ri)ψν∗(rj)r ψµ(ri)ψν(rj)dridrj−(15.22)
µ=1( µ=1ν=1)( ij
1
− ψµ∗(ri)ψν∗(rj)r ψν(ri)ψµ(rj)dridrj .
( ij *
41952Many-bodyapproachestostudiesofelectronicsystems:Hartree-FocktheoryandDensityFunctionalTheory
15.4 Derivation of the Hartree-Fock equations
Having obtained the functional E[Φ], we now proceed to the second step of the calculation.
Withthegivenfunctional,wecanembarkonatleasttwotypesofvariationalstrategies:
• WecanvarytheSlaterdeterminantbychangingthespatialpartofthesingle-particlewave
functionsthemselves.
• Wecanexpandthesingle-particlefunctionsinaknownbasisandvarythecoefficients,that
is, the new single-particle wave function a is written as a linear expansion in terms of a
| ’
fixedchosenorthogonalbasis(fortheexampleharmonicoscillator,orLaguerrepolynomi-
alsetc)
ψa=∑Caλψλ.
λ
InthiscasewevarythecoefficientsC .
aλ
WewillderivethepertinentHartree-Fockequationsanddiscusstheprosandconsofthetwo
methods. Both cases lead to a new Slater determinant which is related to the previous one
viaaunitarytransformation.
Beforewe proceed we need however to repeatsome aspects of the calculus of variations.
FormoredetailsseeforexamplethetextofArfken[51].
Wehavealreadymetthevariationalprincipleinchapter14.Wegivehereabriefreminder
onthecalculusofvariations.
15.4.1 Reminderoncalculusofvariations
Thecalculusofvariationsinvolvesproblemswherethequantitytobeminimizedormaximized
isanintegral.
Inthegeneralcasewehaveanintegralofthetype
b ∂Φ
E[Φ]= f(Φ(x), ,r)dr,
(a ∂r
whereEisthequantitywhichissoughtminimizedormaximized.Theproblemisthatalthough
f isafunctionofthevariablesΦ,∂Φ/∂randr,theexactdependenceofΦonrisnotknown.
Thismeansagainthateventhoughtheintegralhasfixedlimitsaandb,thepathofintegration
is not known. Inour case the unknown quantities arethe single-particlewave functions and
wewishtochooseanintegrationpathwhichmakesthefunctionalE[Φ]stationary.Thismeans
that we want to find minima, or maxima or saddle points. In physics we search normally for
minima.
Our task is therefore to find the minimum of E[Φ] so that its variationδE is zero subject
to specific constraints. In our case the constraints appear as the integral which expresses
the orthogonality of the single-particle wave functions. The constraints can be treated via
the technique of Lagrangianmultipliers. We assume the existence of an optimum path, that
is a path for which E[Φ] is stationary. There are infinitely many such paths. The difference
betweentwopathsδΦiscalledthevariationofΦ.
The condition for a stationary value is given by a partial differential equation, which we
herewriteintermsofonevariablex
∂f d ∂f
=0,
∂Φ−dx∂Φx
15.4 DerivationoftheHartree-Fockequations 493
This equation is better better known as Euler’s equation and it can easily be generalized to
morevariables.
Asanexampleconsiderafunctionofthreeindependentvariables f(x,y,z).Forthefunction
f tobeanextremewehave
df =0.
Anecessaryandsufficientconditionis
∂f ∂f ∂f
= = =0,
∂x ∂y ∂z
dueto
∂f ∂f ∂f
df = dx+ dy+ dz.
∂x ∂y ∂z
In physical problems the variables x,y,z are often subject to constraints (in our case Φ and
theorthogonalityconstraint)sothattheyarenolongerallindependent.Itispossibleatleast
in principle to use each constraint to eliminate one variable and to proceed with a new and
smallerset ofindependentvarables.
The use of so-called Lagrangian multipliers is an alternative technique when the elimi-
nation of of variables is incovenient or undesirable. Assume that we have an equation of
constraintonthevariablesx,y,z
φ(x,y,z)=0,
resultingin
∂φ ∂φ ∂φ
dφ= dx+ dy+ dz=0.
∂x ∂y ∂z
Nowwecannotsetanymore
∂f ∂f ∂f
= = =0,
∂x ∂y ∂z
if df =0 is wanted because there are now only two independent variables. Assume x and y
aretheindependentvariables.Thendzisnolongerarbitrary.However,wecanaddto
∂f ∂f ∂f
df = dx+ dy+ dz,
∂x ∂y ∂z
amultiplumofdφ,viz.λdφ,resultingin
∂f ∂φ ∂f ∂φ ∂f ∂φ
df+λdφ=( +λ )dx+( +λ )dy+( +λ )dz=0,
∂z ∂x ∂y ∂y ∂z ∂z
whereourmultiplierischosensothat
∂f ∂φ
+λ =0.
∂z ∂z
However,sincewetookdxanddytobearbitrarywemusthave
∂f ∂φ
+λ =0,
∂x ∂x
and
∂f ∂φ
+λ =0.
∂y ∂y
When all these equationsaresatisfied, df =0.We have four unknowns, x,y,z andλ. Actually
we want only x,y,z, there is no need to determine λ. It is therefore often called Lagrange’s
undeterminedmultiplier.Ifwehaveasetofconstraintsφ wehavetheequations
k
41954Many-bodyapproachestostudiesofelectronicsystems:Hartree-FocktheoryandDensityFunctionalTheory
∂f ∂φ
+∑λ k =0.
k
∂xi k ∂xi
Letusspecializetotheexpectationvalueoftheenergyforoneparticleinthree-dimensions.
Thisexpectationvaluereads
E = dxdydzψ∗(x,y,z)Hˆψ(x,y,z),
(
withtheconstraint
dxdydzψ∗(x,y,z)ψ(x,y,z)=1,
(
andaHamiltonian
1
Hˆ = ∇2+V(x,y,z).
−2
The integral involving the kinetic energy can be written as, if we assume periodic boundary
conditionsorthatthefunctionψvanishesstronglyforlargevaluesofx,y,z,
1 1
dxdydzψ∗ ∇2 ψdxdydz=ψ∗∇ψ + dxdydz ∇ψ∗∇ψ.
−2 | 2
( # $ (
Insertingthisexpressionintotheexpectationvaluefortheenergyandtakingthevariational
minimum(usingV(x,y,z)=V)weobtain
1
δE =δ dxdydz ∇ψ∗∇ψ+Vψ∗ψ =0.
2
+( # $,
Therequirementthatthewavefunctionsshouldbeorthogonalgives
dxdydzψ∗ψ=constant,
(
and multiplying it with a Lagrangian multiplier λ and taking the variational minimum we
obtainthefinalvariationalequation
1
δ dxdydz ∇ψ∗∇ψ+Vψ∗ψ λψ∗ψ =0.
2 −
+( # $,
Weintroducethefunction f
1 1
f = 2∇ψ∗∇ψ+Vψ∗ψ−λψ∗ψ= 2(ψx∗ψx+ψy∗ψy+ψz∗ψz)+Vψ∗ψ−λψ∗ψ.
Inournotationherewehavedroppedthedependenceonx,y,zandintroducedtheshorthand
ψx,ψy andψz forthevariousfirstderivatives.
Forψ∗ theEulerequationresultsin
∂f ∂ ∂f ∂ ∂f ∂ ∂f
=0,
∂ψ −∂x∂ψ −∂y∂ψ −∂z∂ψ
∗ x∗ y∗ z∗
whichyields
1
(ψxx+ψyy+ψzz)+Vψ=λψ.
−2
WecanthenidentifytheLagrangianmultiplierastheenergyofthesystem.Thelastequation
is nothing but the standard Schrödinger equation and the variational approach discussed
hereprovidesapowerfulmethodforobtainingapproximatesolutionsofthewavefunction.