Table Of ContentMathematical Physics
in Theoretical
Chemistry
Developments in Physical &
Theoretical Chemistry
Series Editor
JamesE.House
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2018 PhysicalChemistryofGas–LiquidInterfaces
JenniferA.FaustandJamesE.House,Editors
2019 MathematicalPhysicsinTheoreticalChemistry
S.M.BlinderandJ.E.House,Editors
Developments in Physical &
Theoretical Chemistry
J. E. House, Series Editor
Mathematical Physics
in Theoretical
Chemistry
Edited by
S. M. Blinder
UniversityofMichigan,AnnArbor,MIandWolframResearch,
Champaign,IL,USA
J. E. House
Illinois WesleyanUniversity,Bloomington, IL;andIllinoisState
University,Normal,IL,USA
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Contributors
S.M.Blinder
UniversityofMichigan,AnnArbor,MI,UnitedStates
CailaBruzzese
DepartmentofChemistry,BrockUniversity,St.Catharines,Ontario,Canada
KimberlyJordanBurch
Department of Mathematics, Indiana University of Pennsylvania, Indiana,
PA,UnitedStates
AndrewL.Cooksy
Department of Chemistry and Biochemistry, San Diego State University,
SanDiego,CA,UnitedStates
GuidoFano
UniversityofBologna,Bologna,Italy
JamesW.Furness
DepartmentofPhysicsandEngineeringPhysics,TulaneUniversity,NewOrleans,
LA,UnitedStates
DavidZ.Goodson
Department of Chemistry and Biochemistry, University of Massachusetts
Dartmouth,NorthDartmouth,MA,UnitedStates
JustinK.Kirkland
DepartmentofChemistry,UniversityofTennessee,Knoxville,TN,UnitedStates
ErrolLewars
DepartmentofChemistry,TrentUniversity,Peterborough,ON,Canada
DevinA.Matthews
InstituteforComputationalEngineeringandSciences,TheUniversityofTexasat
Austin,Austin,TX,UnitedStates
EgorOspadov
Department of Physics, Brock University, St. Catharines; Department of Chem-
istry,TheUniversityofWesternOntario,London,Ontario,Canada
StuartM.Rothstein
DepartmentofPhysics;DepartmentofChemistry,BrockUniversity,St.Catharines,
Ontario,Canada
JohnF.Stanton
DepartmentofChemistry,UniversityofFlorida,Gainesville,FL,UnitedStates
JianweiSun
DepartmentofPhysicsandEngineeringPhysics,TulaneUniversity,NewOrleans,
LA,UnitedStates
xi
xii Contributors
JacobTownsend
DepartmentofChemistry,UniversityofTennessee,Knoxville,TN,UnitedStates
IngaS.Ulusoy
Department of Chemistry, Michigan State University, East Lansing, MI,
UnitedStates
KonstantinosD.Vogiatzis
DepartmentofChemistry,UniversityofTennessee,Knoxville,TN,UnitedStates
AngelaK.Wilson
Department of Chemistry, Michigan State University, East Lansing, MI,
UnitedStates
YuboZhang
DepartmentofPhysicsandEngineeringPhysics,TulaneUniversity,NewOrleans,
LA,UnitedStates
Mathematical physics in
theoretical chemistry
CONTENTS
(i) TheHartree-Fockapproximation(S.M.Blinder)
(ii) SlaterandGaussianbasisfunctionsandcomputationofmolecularintegrals
(A.K.Wilson)
(iii) Post-Hartree-Fockmethods:Configurationinteraction,many-body
perturbationtheory,couple-clustertheory(K.D.Vogiatzis)
(iv) Density-functionaltheory(J.Sun)
(v) Vibrationalenergiesandpartitionfunctions(A.L.Cooksy)
(vi) QuantumMonte-Carlo(S.M.Rothstein)
(vii) Computationalchemistryonpersonalcomputers(E.G.Lewars)
(viii) Chemicalapplicationsofgraphtheory(K.J.Burch)
(ix) Singularityanalysisinquantumchemistry(D.Z.Goodson)
(x) Diagrammaticmethodsinquantumchemistry(J.F.Stanton)
(xi) Quantumchemistryonaquantumcomputer(G.FanoandS.M.Blinder)
INTRODUCTION
Theoreticalchemistryprovidesasystematicaccountofthelawsgoverningchemical
phenomenainmatter.Itappliesphysicsandmathematicstodescribethestructureand
interactionofatomsandmolecules,thefundamentalunitsofmatter.Throughtheend
of the 19th century, chemistry remained predominantly a descriptive and empirical
science.1True,therehadbeendevelopedbythenaconsistentquantitativefoundation
based on the notions of atomic and molecular weights, combining proportions,
thermodynamicquantities,andthefundamentalideasofmolecularstereochemistry.
Chemistrywascertainlyfarmorerationalthanitsancientrootsinalchemybutwas
still largely a collection of empirical facts about the behavior of matter. Immanuel
Kant,inhisCritiqueofPureReason,claimedthat“inanyspecialdoctrineofnature
there can be only as much proper science as there is mathematics therein.”2 This
canserveasourphilosophicalrationalizationforemphasizingmathematicalmethods
(specificallythefielddesignatedmathematicalphysics)intheoreticalchemistry.
1AveryintriguingaccountofthehistoricaldevelopmentofmodernchemistryisgivenbyMaryJo
Nye[1].
2QuotedintheonlineStanfordEncyclopediaofPhilosophy.
xiii
xiv Mathematical physics in theoretical chemistry
Thedevelopmentsofphysicsinthe20thcenturymadeallofchemistryexplicable,
in principle, by quantum mechanics. As summarized by Dirac: “The underlying
physical laws necessary for the mathematical theory of a large part of physics and
the whole of chemistry are thus completely known, and the difficulty is only that
the exact application of these laws leads to equations much too complicated to be
soluble” [2]. By its very nature, quantum mechanics is mathematical physics and
therebyweestablishtheconnectionwhichisthethemeofthisvolume.However,the
loophole noted by Dirac, the existence of chemical problems too mathematically
complex to be solved exactly, justifies the survival of parts of chemistry as an
empiricalscience.Inthiscategoryaresemiempiricalconceptsofchemicalbonding
and reactivity. This has also led to computational models promoting rational drug
design. These have also stimulated applications of other branches of mathematics,
forexample,informationtheoryandgraphtheoryappliedtothedefinitionofvarious
chemicalindices.
The primary objective of theoretical chemistry is to provide a coherent account
for the structure and properties of atomic and molecular systems. Techniques
adaptedfrommathematicsandtheoreticalphysicsareappliedinattemptstoexplain
and correlate the structures and dynamics of chemical systems. In view of the
immense complexity of chemical systems, theoretical chemistry, in contrast to
theoreticalphysics,generallyusesmoreapproximatemathematicaltechniques,often
supplementedbyempiricalorsemiempiricalmethods.
This volume begins with an introduction to the quantum theory for atoms and
smallmolecules, expanding upon the originalapplications of mathematical physics
in chemistry. This field is now largely subsumed within a subdiscipline known as
computationalchemistry.Chapter1beginswithanintroductiontotheHartree-Fock
method,whichistheconceptualfoundationforcomputationalchemistry.Chapter2
discussesthebasisfunctionsemployedinthesecomputations,nowlargelydominated
byGaussianfunctions.Chapter3describessomepost-Hartree-Fockmethods,which
seektoattain“chemicalaccuracy”inatomicandmolecularcomputations,inpartic-
ular, configuration interaction, many-body perturbation theory, and coupled-cluster
theory. Chapter 10 discusses diagrammatic techniques borrowed from theoretical
physics,whichcanenhancetheefficiencyofcomputations.Chapter7isanaccount
of the development of personal computers and their applications to computational
chemistry.
For larger molecules and condensed matter, alternative approaches, including
density functional theory (Chapter 4) and quantum Monte-Carlo (Chapter 6), are
becoming popular computational methods. Some additional topics covered in this
volumearevibrationalpartitionfunctions(Chapter5),singularityanalysisofpertur-
bationtheories(Chapter9),andchemicalapplicationsofgraphtheory(Chapter8).
Finally,Chapter11introducestheprinciplesofthequantumcomputer,whichhas
the speculative possibility of exponential enhancement of computational power for
theoreticalchemistry,aswellasmanyotherapplications.
Mathematical physics in theoretical chemistry xv
REFERENCES
[1] Nye MJ. From chemical philosophy to theoretical chemistry. Berkeley: University of
CaliforniaPress;1993.
[2] Dirac PAM. Quantum mechanics of many-electron systems. Proc R Soc A (Lond)
1929;123:714–33.
