Regularity and approximability of electronic wave functions

197 Pages·2010·1.37 MB·English

by  Harry Yserentant (auth.)

Lecture Notes in Mathematics (cid:50)(cid:48)(cid:48)(cid:48)
Editors:
J.-M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris

Harry Yserentant
Regularity
and Approximability
of Electronic
Wave Functions
ABC
HarryYserentant
TechnischeUniversitätBerlin
InstitutfürMathematik
Straßedes17.Juni136
10623Berlin
Germany
[email protected]
ISBN:978-3-642-12247-7 e-ISBN:978-3-642-12248-4
DOI:10.1007/978-3-642-12248-4
SpringerHeidelbergDordrechtLondonNewYork
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2010927755
MathematicsSubjectClassification(2000):35J10,35B65,41A25,41A63,68Q17
(cid:176)c Springer-VerlagBerlinHeidelberg2010
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Preface
The electronic Schro¨dinger equation describes the motion of N electrons under
Coulombinteractionforcesinafieldofclampednuclei.Solutionsofthisequation
depend on 3N variables, three spatial dimensions for each electron. Approximat-
ingthesolutionsisthusinordinatelychallenging,anditisconventionallybelieved
that a reduction to simplified models, such as those of the Hartree-Fock method
ordensityfunctionaltheory,istheonlytenableapproach.Thisbookseekstocon-
vincethereaderthatthisconventionalwisdomneednotbeironclad:theregularity
ofthesolutions,whichincreaseswiththenumberofelectrons,thedecaybehavior
of their mixed derivatives, and the antisymmetry enforced by the Pauli principle
contribute properties that allow these functions to be approximatedwith an order
of complexity which comes arbitrarily close to that for a system of one or two
electrons. The present notes arose from lectures that I gave in Berlin during the
academic year 2008/09 to introduce beginning graduate students of mathematics
intothissubject.Theyarekeptonanintermediatelevelthatshouldbeaccessibleto
anaudienceofthiskindaswellastophysicistsandtheoreticalchemistswithacor-
respondingmathematicaltraining.Thetextrequiresa goodknowledgeofanalysis
totheextenttaughtatGermanuniversitiesinthefirsttwoyearsofstudy,including
LebesgueintegrationandsomebasicfactsonBanachandHilbertspaces(comple-
tion,orthogonality,projectiontheorem,Lax-Milgramtheorem,weakconvergence),
butnodeeperknowledgeofthetheoryofpartialdifferentialequations,offunctional
analysis, or quantum theory. I thank everybody with whom I had the opportunity
to discuss the topic during the past years, my coworkersboth from Tu¨bingenand
Berlin, above all Jerry Gagelman, who read this text very carefully, found many
inconsistencies, and to whom I owe many hints to improve my English, and par-
ticularly my colleagues Hanns Ruder, who raised my awareness of the physical
background,andReinholdSchneider,whogenerouslysharedallhisknowledgeand
insightintoquantum-chemicalapproximationmethods.TheDeutscheForschungs-
gemeinschaft supported my work through several projects, inside and outside the
DFG-ResearchCenterMATHEON.IdedicatethisbooktomysonsKlausandMax.
Berlin,September2009 HarryYserentant
v

Contents
1 IntroductionandOutline...................................................... 1
2 FourierAnalysis................................................................ 13
2.1 RapidlyDecreasingFunctions............................................ 13
2.2 IntegrableandSquareIntegrableFunctions ............................. 17
2.3 SpacesofWeaklyDifferentiableFunctions.............................. 20
2.4 FourierandLaplaceTransformation..................................... 25
3 TheBasicsofQuantumMechanics .......................................... 27
3.1 Waves,WavePackets,andWaveEquations ............................. 28
3.2 TheSchro¨dingerEquationforaFreeParticle ........................... 29
3.3 TheMathematicalFrameworkofQuantumMechanics................. 32
3.4 TheHarmonicOscillatorandItsEigenfunctions........................ 37
3.5 TheWeakFormoftheSchro¨dingerEquation ........................... 44
3.6 TheQuantumMechanicsofMulti-ParticleSystems.................... 46
4 TheElectronicSchro¨dingerEquation....................................... 51
4.1 TheHardyInequalityandtheInteractionEnergy....................... 52
4.2 SpinandthePauliPrinciple .............................................. 55
5 SpectrumandExponentialDecay............................................ 59
5.1 TheMinimumEnergyandtheIonizationThreshold.................... 60
5.2 DiscreteandEssentialSpectrum......................................... 61
5.3 TheRayleigh-RitzMethod ............................................... 71
5.4 TheLowerBoundoftheEssentialSpectrum............................ 78
5.5 TheExponentialDecayoftheEigenfunctions .......................... 82
6 ExistenceandDecayofMixedDerivatives.................................. 87
6.1 AModifiedEigenvalueProblem ......................................... 88
6.2 SpacesofFunctionswithHigh-OrderMixedDerivatives .............. 91
6.3 EstimatesfortheLow-OrderTerms,Part1.............................. 93
6.4 EstimatesfortheLow-OrderTerms,Part2.............................. 97
vii
viii Contents
6.5 TheRegularityoftheWeightedEigenfunctions.........................103
6.6 AtomsasModelSystems.................................................111
6.7 TheExponentialDecayoftheMixedDerivatives.......................115
7 EigenfunctionExpansions ....................................................117
7.1 DiscreteRegularity........................................................118
7.2 Antisymmetry.............................................................123
7.3 HyperbolicCrossSpaces .................................................124
8 ConvergenceRatesandComplexityBounds................................127
8.1 TheGrowthoftheEigenvaluesinthe3d-Case..........................128
8.2 ADimensionEstimateforHyperbolicCrossSpaces ...................131
8.3 AnAsymptoticBound....................................................134
8.4 AProofoftheEstimateforthePartitionNumbers......................135
8.5 TheComplexityoftheQuantumN-BodyProblem .....................139
9 TheRadial-AngularDecomposition.........................................141
9.1 Three-DimensionalSphericalHarmonics................................142
9.2 TheDecompositionofN-ParticleWaveFunctions......................151
9.3 TheRadialSchro¨dingerEquation........................................154
9.4 AnExcursustotheCoulombProblem...................................159
9.5 TheHarmonicOscillator .................................................162
9.6 EigenfunctionExpansionsRevisited.....................................164
9.7 ApproximationbyGaussFunctions......................................165
9.8 TheEffectofScaling .....................................................167
Appendix:TheStandardBasisoftheSphericalHarmonics ..................170
References...........................................................................177
Index.................................................................................181
Chapter 1
Introduction and Outline
Theapproximationofhigh-dimensionalfunctions,whethertheybegivenexplicitly
orimplicitlyassolutionsofdifferentialequations,representsoneofthegrandchal-
lengesofappliedmathematics.High-dimensionalproblemsariseinmanyfieldsof
applicationsuch asdata analysisand statistics, butfirst ofallin the sciences. One
of the most notorious and complicated problems of this type is the Schro¨dinger
equation.The Schro¨dingerequationformsthe basis of quantummechanicsand is
offundamentalimportanceforourunderstandingofatomsandmolecules.Itlinks
chemistrytophysicsanddescribesasystemofelectronsandnucleithatinteractby
Coulombattractionandrepulsionforces.AsproposedbyBornandOppenheimerin
thenascencyofquantummechanics,theslowermotionofthenucleiismostlysepa-
ratedfromthatoftheelectrons.ThisresultsintheelectronicSchro¨dingerequation,
theproblemtofindtheeigenvaluesandeigenfunctionsoftheHamiltonoperator
H = −1 ∑N Δ − ∑N ∑K Zν + 1 ∑N 1 (1.1)
2 i=1 i i=1ν=1|xi−aν| 2i,j=1|xi−xj|
i(cid:2)=j
writtendownhereindimensionlessformoratomicunits.Itactsonfunctionswith
argumentsx ,...,x ∈R3,whichareassociatedwiththepositionsoftheconsidered
1 N
electrons.TheaνarethefixedpositionsofthenucleiandthepositivevaluesZνthe
chargesofthenucleiinmultiplesoftheabsoluteelectroncharge.
The mathematical theory of the Schro¨dinger equation for a system of charged
particles is today a central, highly developed part of mathematical physics. Start-
ingpointwasKato’swork[48]inwhichheshowedthatHamiltonoperatorsofthe
given form fit into the abstract framework that was laid by von Neumann [64] a
short time after Schro¨dinger [73] set up his equation and Born and Oppenheimer
[11]simplifiedit.AnimportantbreakthroughwastheHunziker-vanWinter-Zhislin
theorem [46,90,98], which states that the spectrum of an atom or molecule con-
sists ofisolatedeigenvaluesλ ≤λ<Σoffinitemultiplicitybetweena minimum
0
eigenvalueλ andaionizationboundΣandanessentialspectrumλ≥Σ.Themath-
0
ematicaltheoryoftheSchro¨dingerequationtraditionallycentersonspectraltheory.
Of at least equal importance in the given context are the regularity properties of
the eigenfunctions,whose study began with [49]. For newer developmentsin this
H.Yserentant,RegularityandApproximabilityofElectronicWaveFunctions, 1
LectureNotesinMathematics2000,DOI10.1007/978-3-642-12248-4 1,
(cid:2)c Springer-VerlagBerlinHeidelberg2010