Table Of ContentIntroduction to
Relativistic Quantum
Chemistry
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Introduction to
RELATIVISTIC QUANTUM
CHEMISTRY
Kenneth G. Dyall
Knut Fægri, Jr.
2007
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Introductiontorelativisticquantumchemistry/KennethG.Dyall,
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Preface
Theemergenceofrelativisticquantumchemistryhasbeenoneofthemoreremarkable
developments within computational chemistry over the past decades. Since the early
workofDirac,relativityhasalwaysbeenapartoftheoverallquantumchemicalpicture,
butithasmostlybeenneglectedonthegroundsthattheeffectswereconsideredsmall
andthemethodstotreatthemwerepoorlydevelopedandexpensivetouse.However,
as nonrelativistic quantum chemistry became more powerful and accurate, the lower
rowsoftheperiodicsystemcamewithinreachofcomputationalstudies,anditbecame
clear that relativistic effects had a significant influence on a number of physical and
chemical properties. The start of the “modern” era of relativistic quantum chemistry
may be traced back to a review article by Pyykkö (1978) and to articles by Pitzer
(1979)andbyPyykköandDesclaux(1979).
The developments in the field have been well documented through articles in sci-
entific journals, conferences and symposia, and review volumes. Unfortunately, this
specialist literature is not easily accessible to newcomers to the field. For many years
the book by Moss, Advanced Molecular Quantum Mechanics (1973), served as an
introduction to the field, but today this book suffers from two major drawbacks—it
is out of print and it does not cover the developments of the past three decades. We
thereforedecidedthattherewasaneedforabooktofillthegapbetweenthestandard
texts on quantum mechanics, which have little if anything on relativity, the advanced
texts,whichtreatrelativityindetailbuthavelittleconnectionwithquantumchemistry,
andtheliterature,wherethereisalargeamountofboththeoryandapplications.
Ourambitionistoprovideamodernintroductiontothefieldofrelativisticquantum
chemistry, aimed at the advanced student and the practicing nonspecialist researcher.
The material has been divided into five parts. Parts I and II provide the necessary
background from classical physics, relativistic quantum mechanics, and group theory.
Part III covers the application of these principles to fully relativistic methods for
quantum chemistry within a four-component framework. Part IV deals with the main
vi PREFACE
approximatemethodsthathavebeendeveloped,andPartVtreatstheconsequencesof
relativityforchemicalbonding.
Thebookisintroductoryinthesensethatitintroducesmanyoftheconceptsneeded
for a firm background in relativistic theory and quantum chemistry but without going
intoallthedetails.Referencesaregiventofullertreatmentsoftheintroductorymaterial.
Ourintentionisnottogiveallthedetailsofproofsthatcanbefoundinotherstandard
works on relativistic quantum mechanics, but rather to present the relevant parts for
the purpose of constructing a relativistic quantum chemistry. It is also introductory in
the sense that the details of many of the methods that are found in the literature are
reproduced or elaborated in this book, so that the educated quantum chemist does not
have to search through the literature for them. Finally, it is introductory in the sense
thatitcontainsdescriptivematerialtodowithrelativisticeffectsonbonding,structure,
and energetics of molecules. We have no ambitions of providing an extensive review
ofwhathasbecomealargeandquiteheterogeneousfield,norofprovidingahistorical
overviewofthedevelopmentofrelativisticmethods.Thespecialistwillprobablyhave
nodifficultyidentifyingoneormore“petsubjects”or“keyreferences”thataremissing.
Wedoprovideaguidetothereviewliteratureinthefield,andalsointhisrespectthe
workisintroductory.
The book started out as a set of lecture notes by KGD for a 1995 graduate course
givenat(then)OdenseUniversity.Thesenotesdevelopedthroughfurtherpresentations,
and the process of turning them into a book was begun in 1996 during a research
visit by KGD to the University of Oslo. Since then the writing has been a gradual
process,hamperedbytheacademicworkuncertaintiesofoneofusandbytheextensive
administrative load carried by the other. The work would not have been possible had
it not been for the support of our employers through this period, for KGD: Eloret
and Schrödinger; for KF: the University of Oslo. Support by the Research Council of
Norway,NASA,andDOEisgratefullyacknowledged,asisthehospitalitybothofus
enjoyedduringresearchvisitstotheIRSAMCofUniversitéPaulSabatierinToulouse.
Mostofall,thisbookcouldneverhavebeenwrittenwithoutsupportandcriticism
fromandfrequentdiscussionswithourfriendsinthefield.Theseincludeinparticular
Harry M. Quiney, Luuk Visscher, Trond Saue, Hans Jørgen Aa. Jensen, and Trygve
Helgaker, as well as most of the people involved in the DIRAC and MOLFDIR col-
laborations. Peter Schwerdfeger did a wonderful job of commenting on a late version
of the manuscript. Trond Saue and Luuk Visscher also provided useful comments on
substantialpartsofafinaldraft.Otherswhohaveverykindlyreadandcommentedon
parts of the manuscript in various stages of completion are Joost v. Stralen, Raimo v.
Handel, LeifVeseth, Werner Kutzelnigg, Timo Fleig, and Hans JørgenAa. Jensen. In
addition to the direct contributions to the book, there are many who have in one way
or another influenced our thinking or contributed to the work on which this book was
based.AmongthosenotalreadymentionedwewouldliketoacknowledgePeterTaylor,
Jeppe Olsen,Wim Nieuwpoort, and GustavoAucar. Finally, we would like to express
our gratitude to Pekka Pyykkö and Ian Grant, both pioneers and leaders in this field,
andtoBerndA.Hess,JaapSnijders,JanAlmlöf,andOddGropen,whoarenolonger
withus,butwhoallfourinvariouswayshavehelpedusinoureffortsinthisfield.
Inourfamiliesthebookhastakenonalmostmythicalstature.Itiswithsomerelief
thatwearenowabletopresentafinalproduct.Wedothisingratefulacknowledgement
oftheiralmostinfinitetoleranceandsupport.
Notation Conventions
Wehaveadoptedanumberofconventionsinthisbookinordertomaintainaconsistent,
clear,andidentifiablenotation.Asfaraspossiblewehavekepttocommonconventions
for symbols and quantities in quantum chemistry. We have also tried to avoid the
duplication of symbols where possible. These goals conflict to some extent, so some
quantitiesaregivenunconventionalsymbols.Thefollowinglistidentifiessymbolsand
typographyusedthroughoutthebook.
r,ϑ,ϕ forsphericalcoordinates
Ψ forageneralone-particleormultideterminantmany-particlewavefunction
Φ foraone-determinantorCSFmany-particlewavefunction
Ψfortime-dependent4-spinors
θ(t)forthetimepartofaspinororageneraltime-dependentfunction
ψfortime-independent4-spinors,thatisΨ=ψθ
ΨX fortime-dependent2-spinors,withX =L, S
ψX fortime-independent2-spinors,withX =L, S
ξ(ϑ,ϕ,τ)for2-spinorangular-momentumfunctions(includingspin)
φ,φX for2-spinorsinamodifiedrepresentation
χX for2-spinorbasisfunctions
µ
ψ,ψXτ forscalarfunctions
η,α,βforspinfunctions
τ,α,β forspinlabels;α andβ alsodenotethespinfunctions,andβ oneofthe
Diracmatrices;
φ,φXτ forscalarspacefunctions;thesuperscriptsX andτ areforcomponent
andspin.
χX,χX¯ forscalarbasisfunction
µ µ
A,Ω boldfaceformatrices
a,σ boldfaceforvectors
viii NOTATIONCONVENTIONS
a Romantypeforfour-vectors
ˆ ˆ
F,Ω hatsforoperators
h forone-electronHamiltonianmatrixelements
pq
H forN-electronHamiltonianmatrixelements
PQ
In addition to these font conventions, we have adopted some conventions for the
indices of functions. In general, Roman letters are used for Fock space functions (i.e.
orbitalsorspinors),whileGreeklettersareusedforbasisfunctions.Specificrangesof
lettersareusedasfollows:
p,q,r,s,...generalorbitalorspinorindices
i,j,k,l,...occupiedorinactiveorbitalorspinorindices
a,b,c,d,...virtualorbitalorspinorindices
t,u,v,w,...activeorbitalorspinorindices
κ,λ,µ,ν,...basisfunctionindices
Inthisbookwehaveusedtwosystemsofunits.ThefirstistheSIsystem,whichwe
use in the early chapters of the book and in particular for electromagnetic quantities.
Factors of c therefore always represent the speed of light and never a conversion
factor for magnetic units. The second is the Hartree atomic units system, defined by
(cid:1) = e = m = 1. In these units, c ∼ 137. Many physics texts use the system
e
(cid:1)=e =c =1, since they are dealing with particles of different masses. Our concern
isprincipallywiththeelectronandchemistry,andthesizeofrelativisticeffects,which
aremeasuredbyc,soHartreeatomicunitsaremoreappropriate.However,tokeepthe
connection with SI units and to track quantities that involve the charge, the mass, or
spin,thesymbols(cid:1),e,andm≡m areretainedinmuchofthedevelopment,whereas
e
1/4π(cid:17) isusuallyomittedforclarity.
0
Therearesomesituationswhere,intheinterestsofclarity,wehaveallowedsome
inconsistencyorsacrificedsomerigorofexpression.Wedonotusuallymultiplyscalars
by the unit matrix in expressions where the context would demand it—such as where
an operator is a combination of scalar and spin-dependent operators or in a matrix
expression—and we do not always indicate the rank of the unit matrix or the zero
matrixbyasubscript.InsomeplacesthenotationwouldbeoverloadedifI wereused
2
insteadof1,butthematrixnotationistobeinferredfromthecontext.
We have also used both Hˆ and hˆ for the one-electron Hamiltonian operator. The
latter is used for the free-particle Dirac Hamiltonian where a distinction between it
and the full one-electron Hamiltonian is necessary, and is also used in a sum over
one-electron Hamiltonians for a single electron. The former is usually used in formal
developments, and to represent the total Hamiltonian. In many of the formal devel-
opments, the total Hamiltonian is simply the one-electron Hamiltonian, so Hˆ is used.
However,fortheone-electronHamiltonianmatrixelements,lowercaseisalwaysused,
andfortheN-electronHamiltonianmatrixelements,uppercaseisalwaysused.
Contents
Notation Conventions, vii
I: Foundations
1 Introduction, 3
2 Basic Special Relativity, 6
2.1 Inertial Frames and Newtonian Mechanics, 6
2.2 Relativistic Coordinate Transformations, 7
2.3 Transformation of Lengths and Relativistic Invariants, 9
2.4 Transformation of Velocities, 11
2.5 Transformation of Mass, 13
2.6 Relativistic Energy, 14
2.7 Relativistic Momentum, 15
3 Relativistic Electromagnetic Interactions, 17
3.1 The Maxwell Equations, 18
3.2 Potentials and Gauge Transformations, 19
3.3 The Relativistic Potential from a Moving Charge, 22
3.4 The Potential Experienced by a Moving Charge, 24
3.5 The Interaction of Two Charged Particles, 26
