Symmetry of Intramolecular Quantum Dynamics

445 Pages·2012·12.445 MB·English

by  Alexander V. Burenin

De Gruyter Studies in Mathematical Physics 11
Editors
MichaelEfroimsky,Bethesda,USA
LeonardGamberg,Reading,USA
DmitryGitman,SãoPaulo,Brasil
AlexanderLazarian, Madison,USA
BorisSmirnov,Moscow,Russia

Alexander V. Burenin
Symmetry of Intramolecular
Quantum Dynamics
Translated by Alexey V. Krayev
De Gruyter
PhysicsandAstronomyClassification2010:87.15.hp,31.30.Gs,33.30.-i,33.57.+c,33.15.kr,
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Preface
Themolecule isacomplexmultiparticle system,andinisolatedstate itsinternaldy-
namicscanbedescribed,toagoodapproximation,neglectingthenuclearandelectron
spin-related contributions to the Hamiltonian. The symmetry properties of a purely
spatialHamiltonianaredeterminedbythesymmetrypropertiesofspaceandtime(ex-
ternal symmetry) andbyrequirementsimposedonpermutationofidentical particles
(internal symmetry). However, when we try to solve equationsof motionwith such
aHamiltonianbythemethodsofperturbationtheory,weunexpectedlyface theneed
to introduce an additional internal geometric symmetry group that characterizes the
molecule.Thisisamatterofprincipleinthisapproach,sinceotherwiseitisimpossi-
bletowriteapproximateequationsofmotion.Thebasicworkingapproximationisthe
Born–Oppenheimer(BO)approximation[10,14,35],whichintroducestheconceptof
theeffectivepotentialofnuclearinteractioninagivenelectronicstateand,asaconse-
quence,theconceptofasetofequilibriumconfigurationscorrespondingtotheminima
ofthispotential.Qualitatively,moleculescanbedividedintorigidandnonrigidones.
Forrigidmoleculesinnondegenerateelectronicstates,thechoiceoftheeffectivepo-
tential with oneminimumis quiteadequate, whereas for nonrigidmoleculesseveral
such minima should be taken into account because the internal motion includes the
transitionsbetweenthem.Ithaslongbeenunderstoodthatforrigidmoleculesanad-
ditionalgeometricgroupshouldbeselectedintheformofapointgroupoftheirunique
equilibriumconfiguration,whichbydefinition[60,64]includesallgeometricsymme-
tryelementsofthisstructureasawhole.Itiscommonlyassumedthatthisgroupand
the corresponding inferences are corollaries of the BO approximation, i.e., only in
this approximation can we speak of a certain geometric structuring of internal mo-
tions.Buteveninthissimplestcase there isnoclear ideaofanapplicabilitydomain
of the pointgroup. Two essentially different opinionsare available in the literature.
According to one of them [60,64], the point groupcharacterizes the total (electron-
vibration-rotational) internal motion when deviations from the equilibrium position
are sufficientlysmall. However, whata “sufficientlysmalldeviation”meansis quite
uncertain. The alternative pointof view [15,16,39]isthat thepointgroupdescribes
the symmetry of vibrational and electronic motions only and is inapplicable to ro-
tational motion and, hence, to the total internal motion. As a result, analysis of the
totalmotionisbasedontheso-calledcompletenuclearpermutation-inversion(CNPI)
group [15,16,39]. Such contradictions in the status of empirically introduced point
groups are connected with the absence of a definite point of view on their nature.
Therefore, averyimportantfeature ofthebookisthestatementthatthesegroupsare
vi Preface
implicitordynamicallyinvariantgroupsofsymmetryofarigorousproblemofinternal
coordinatemotion.Despitethefactthatnomethodisknowntodatethatwouldenable
onetoobtainsuchagroupfromstudiesoftheequationsofrigorousspatialdynamics,
thisstatementcanbelogicallyjustifiedonthebasisofanalysisoftheobservedprop-
erties of a molecular system. It is interesting that such a point of view may change
dramaticallysomegeneralconceptsofamolecularsystem:
1. A characteristic propertyof amolecular systemisthe existenceof rotationalmo-
tion of the system as a whole. This means that a molecular system is a certain
structure (“microcrystal”), inwhichinternal motionsoftheparticles are basically
collective.Thesymmetryofthisstructure ischaracterized byanimplicitgeomet-
ric group.It appears thatthe conceptitself ofthe structure of a molecular system
can be introduced into the description only by using the BO approximation. The
correctconfigurationspaceofcollectivemotionsisconstructedseparatelyineach
electronicstate.Inotherwords,weproceedtothedomainofdescriptionbounded
byoneelectronicstate.Forsuchaboundeddomain,implicitsymmetryisreplaced
byitsexplicitcounterpart.
2. The solutionof the problem of the discrete spectrum of a molecular system con-
ceptually relies on perturbation theory, both in the analytical and numerical ap-
proaches. The point is that the conditions of selection of physically meaningful
solutionsofadiscretespectrumofcollectiveinternalmotionagainstthevastback-
groundofformalsolutionscannotbeformulatedwithoutusingtheBOapproxima-
tion.Thisisexactlywhywepassfromtheproblemwithimplicitsymmetrytothe
problemwiththe samebutexplicit symmetryinonegivenelectronic state. Since
thecorrectchoiceofexplicitsymmetryshouldbeprovided,theproblemofempir-
icallyseekingageometricgrouparises.Forrigidmolecularsystemsinnondegen-
erate electronic states, such a group is a point group of their unique equilibrium
configuration. However, the symmetry of such a configuration is an elementary
consequenceof thesymmetryofinternal dynamics,andnotviceversa asisoften
stated,andthesetwosymmetriescoincideonlyintheaforementionedsimplestcase.
3. WhentheSchrödingerequationdescribingadiscretespectrumofamolecularsys-
temissolvedbythemethodsofperturbationtheory,achainofnested(increasingly
approximate) models is constructed until the exact solution of the model prob-
lembecomespossible.Simultaneously,achainofsymmetrygroupscharacterizing
these models arises. In the first place, the difficulties of solving the Schrödinger
equation are due to the declarative nature of the obtained series of perturbation
theory describing the transitions between the neighboring models. Not only are
the properties of the series unknown, but often it is also impossible to correctly
calculateeventhelower-order corrections.Moreover,thesymmetryrequirements
shouldalsobetakenintoaccount.However,thesituationchangesradicallyifonly
thesymmetrypropertiesareconsideredandthetransitionsbetweenmodelsarede-
Preface vii
scribedbysymmetrymatchings.Todothis,inthegroupsoftheneighboringmod-
elswesingleouttheequivalentelementswithrespecttowhichthewavefunctions
and the operators of physical values should be transformed in the same way. In
otherwords,transitionsbetweentheneighboringmodelsareaccompaniedbycer-
tainnontrivialconstraintsonthecomplianceofsymmetrytypes.Theadvantagesof
suchanapproachareprimarilyduetothefactthatthematchingsarerigorous(!).
4. Internal dynamics can be described on the basis of onlythe symmetry principles
with accuracy uptosome phenomenologicalconstantswhich can bedetermined,
forexample,fromacomparisonoftheoreticalconclusionsandexperimentaldata.
Inthisapproach,configurationspaceofaquantumsystemisnotintroducedinex-
plicitform atall, and,asaconsequence,thewave functionsof thecoordinatesof
this space are not explicitly considered. However, due to its wide philosophical
andtechnical difference, thisapproach isatpresent theonlyonethatcan beused
for the solution of many topical problems pertaining to the internal dynamics of
molecules.Themodelsobtainedrigorouslydescribeallinteractionsofinteresting
types of motionthat are possiblewithin the framework of a givensymmetry and
lead to a simple, purely algebraic scheme of calculation for both the position of
thelevelsintheenergyspectrumandthetransitionintensitiesbetweenthem.Itis
importantthatthecorrectnessofthemodelsislimitedonlytothecorrectchoiceof
theinternaldynamicssymmetry.
Certainly,achangeinthegeneralconceptconcernsnotonlythemolecularsystem
proper,butalsoawidevarietyofotherphysicalsystemsalsorequiringtheintroduction
of an additional internal geometric groupto describe its basically collective internal
motions.Interestingly, an atom does not belongto such systems, and this is exactly
thereasonwhyithasnorotationalmotionofthesystemasawhole.
Themaingoalofthisbookistogiveasystematicdescriptionofquantumintramo-
leculardynamicsonthebasisofthesymmetryprinciplesonly.Inthisrespect,thereis
nocomparablebookintheworldliterature.Ascomparedwiththesecondedition[24],
ithasbeenexpandedsubstantially.Anumberofnewproblemsareconsidered,among
whichadiscussionofthebasicnecessityofusingtheBO approximationforthefor-
mulationitselfoftheproblemoffindingadiscretemolecularspectrumbysolvingthe
stationary Schrödinger equation using analytical and/or numerical methods (Chap-
ter 12), the analysis of nonrigid molecular systems with continuousaxial symmetry
groups(Chapter 15),andadescriptionoftheZeemanandStarkeffects (Chapter19)
areworthyofspecialnotice.Wehavealsorevisedandextendedadiscussionoftheis-
suesstudiedinthesecondedition.Importantly,asaresult,therangeofstudiedtypesof
nonrigidmotionshasbeencompletedinanontrivialway.Inparticular,wehaveadded
analysesofveryinterestingdynamicsofsomemolecularcomplexesandthesimplest
ofthecarbocations,which are theintermediate molecularionsofmanychemical re-
actions. Finally, the applicability of developed methodsin such fields as analysis of
viii Preface
moleculeswithmorethantwoidenticaltops,allowanceforhyperfineinteractionsand
parityviolationeffects,etc.isdemonstratedinChapter20.
Thebookisbasicallyintendedforphysicistsworkinginthefieldofmolecularspec-
troscopyandquantumchemistry.Thereaderisnotexpectedtoknowtheapparatusof
grouprepresentationtheoryneededforapplicationofsymmetrymethodsinquantum
intramoleculardynamicssincethefirstpartofthebookisdedicatedtoit.Foramore
detailed study of almost all issues touched upon, one may consult, e.g., the mono-
graphs [43,50,60,92]. The problems of using a semidirect product of groups and
dynamicgroupsaretheonlymajorexclusions.Thesearediscussedin[8,39].Thesec-
ondpartofthebookconcernsthestate-of-the-artdescriptionofquantumintramolecu-
lardynamicsemployingonlysymmetryprinciples.Thispartnowcomprisesfourteen
chapters(insteadofnineinthesecondedition),andtheconsiderationismainlybased
ontheauthor’sworks[17,18,20,23,28].Thereaderissupposedtoknowatleastfoun-
dationsoftheanalyticaldescriptionofintramolecularmotions.Variousissuesinthis
extensive area can be found in [10,14–16,35,39,60,64]. Additional references are
givenasthe statement unfolds.The appendices containtherequired reference mate-
rial.AppendixVabouttheactionofthedirectioncosinesontherotationalunitvectors
isaddedascomparedwiththesecondedition.
The author is grateful to Professors Yu.S. Makushkin, A.M. Sergeev, B.M.
Smirnov,andV.G.Tyuterevforsupportinghiseffortsaimedatdevelopingthemeth-
odsofsymmetrytheory.
NizhnyNovgorod,Russia AlexanderV.Burenin
September17,2011
Contents
Preface v
I Foundations ofthemathematicalapparatus
1 Basicconceptsofgrouptheory 3
1.1 Thegrouppostulates ....................................... 3
1.2 Subgroup,directproductofgroups,isomorphism,and
homomorphism ........................................... 6
1.3 Cosets.Semidirectproductofgroups .......................... 7
1.4 Conjugacyclasses ......................................... 9
2 Basicconceptsofgrouprepresentationtheory 10
2.1 Linearvectorspaces ....................................... 10
2.2 Operatorsinconfigurationandfunctionspaces................... 13
2.3 Representationsofgroups ................................... 15
2.4 Characters.Decompositionofreduciblerepresentations ........... 17
2.5 Directproductofrepresentations.Symmetricpower .............. 20
2.6 TheClebsch–Gordancoefficients ............................. 23
2.7 Basisfunctionsofirreduciblerepresentations .................... 25
2.8 Irreducibletensoroperators.TheWigner–Eckarttheorem .......... 28
3 Thepermutationgroup 31
3.1 Operationsinthepermutationgroup.Classes .................... 31
3.2 Irreduciblerepresentations.TheYoungdiagramsandtableaux ...... 33
3.3 Basisfunctionsofirreduciblerepresentations .................... 35
3.4 Theconjugaterepresentation ................................. 37
4 Continuousgroups 39
4.1 CompactLiegroups ....................................... 39
4.2 Liegroupoflineartransformations ............................ 41
4.3 Liealgebra.Three-dimensionalrotationgroup ................... 42
4.4 Irreduciblerepresentationsofathree-dimensionalrotationgroup .... 46