Mathematical Stereochemistry

457 Pages·2015·3.343 MB·English

by  Fujita S.

Fujita
MathematicalStereochemistry
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ISSN0932-0776,e-ISSN1865-7117
Shinsaku Fujita
Mathematical
Stereochemistry
|
Author
ShinsakuFujita
ShonanInstituteofChemoinformaticsandMathematicalChemistry
Kaneko479-7
258-0019Kanagawa-ken
Japan
[email protected]
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Preface
Modernstereochemistryisbasedonanaccumulatedsetoftermswhichdonotalwaysstem
frommathematicalfoundations.Theaccumulatedsetoftermshasseeminglyworkedwellto
discussvariousaspectsofstereochemistryinaqualitativefashion.However,theinsufficient
mathematicalfoundationshaveprovideduswithseriousconfusioneveninqualitativedis-
cussions,whetherornotweareawareofsuchinsufficiency.Theconfusionhasitssources
inthemostfundamentalconceptsandterminologyofmodernstereochemistryasfollows:
1. Misleading Dichotomy Between Enantiomers and Diastereomers: The term ‘di-
astereomers’isdefinedbythewell-known(butmisleading)dichotomyinmodernstere-
ochemistry: “Diastereomers are stereoisomers other than enantiomers.” However, di-
astereomers are incapable of constructing a definite set of molecular entities. On the
otherhand,apairofenantiomersconstructsadefinitesetofmolecularentities,which
iscategorizedtobeanequivalenceclass,mathematicallyspeaking.Asaresult,theex-
pressionofthedichotomyconsistsofsuchtermsasbelongingtodifferentcategories.
Moreover,itdoesnotrefertoachiralmolecules.Hence,thedichotomyisconcludedto
bemisleadingeventhoughitworksasakeyconceptsupportingthetheoreticalfounda-
tionsofmodernstereochemistry.Unfortunately,allofthetextbooksonorganicchem-
istry and on stereochemistry adopt the misleading dichotomy so as to reproduce and
spreadunconsciousconfusion.
2. Confusion of Reflections with Permutations: In modern stereochemistry, there ap-
pearsfrequentlysuchanexpressionas“TheregulartetrahedronofthepointgroupT
d
functionsasapermutationcenterorskeletonwithfourequivalentsites,andmodelsof
stereoisomersaregeneratedbypermutationoftheligandsamongthesesites.”Thisex-
pressionunconsciouslycausestheconfusionofreflections(containedinthepointgroup
T )withpermutations(ofligands).Strictlyspeaking,suchpermutationsarecontrolled
d
bythesymmetric group ofdegree 4(S[4]),whichisakind ofpermutation group iso-
morphictothepointgroupT .Theregulartetrahedronbelongstothesymmetricgroup
d
S[4] as well asto the point groupT . Thereby, the two groupsS[4] andT have been
d d
frequentlymixedupwitheachotherinmodernstereochemistry.
3. MisleadingConceptsof‘StereogenicUnits’,‘ChiralityUnits’,and‘Pseudoasym-
metric Units’: To specify absolute configurations, the Cahn-Ingold-Prelog (CIP)
system of stereochemical notation claims that ‘stereogenic units’ for assigning R/S-
stereodescriptors consist of ‘chirality units’ and ‘pseudoasymmetric units’. Thus,
R/S-stereodescriptors are assigned both to enantiomers (‘chirality units’ specified by
uppercaselabels‘R’and‘S’)anddiastereomers(‘pseudoasymmetricunits’specifiedby
lowercaselabels‘r’and‘s’).ThismeansthattheassignmentofR/S-stereodescriptors
does not obey a single criterion so long as we obey the-state-of-the-art terminology
oftheCIPsystem.Theterm‘stereogenicunits’(=‘chiralityunits’plus‘pseudoasym-
metric units’), which is considered to generate stereoisomerism, has been coined to
rationalize as if the assignment of R/S-stereodescriptors would obey a single crite-
vi Preface
rion. Thereby, stereogenicity is misleadingly presumed to connote chirality, whereas
stereogenicity and chirality should be independent of each other. As a result, the re-
lationship between stereogenicity and chirality has not been fully demonstrated in
modernstereochemistry.
4. MisleadingPresumptionofaSinglePairofChirality/Achirality:TheCIPsystem
forspecifyingabsoluteconfigurationspresumesasinglepairofchirality/achiralityand
disregards the presence of pairwise properties of other types. As a result, a pair of
R/S-stereodescriptors ismisleadinglyclaimedtobeassignedtoapairofenantiomers
with opposite chirality senses, where ‘pseudoasymmetric units’ are regarded as ex-
ceptionalcaseswithoutconsideringsuchpairwisepropertiesofothertypes.Although
‘pseudoasymmetricunits’asdiastereomers(⊂‘stereogenicunits’)areimplicitlypaired
duringthenamingprocessoftheCIPsystem,theuseoftheterm‘reflection-invariant’
foralowercaselabel(‘r’or‘s’)forcesustofocusourattentiononthepairingofenan-
tiomers.Notethatdiastereomersarenotalwayspairedingeneral.
5. Misleading Terms of ‘Stereoheterotopic’, ‘Enantiotopic’, and ‘Diastereotopic’:
Inordertospecifyintramolecularenvironments,pro-R/pro-S-descriptorsareassigned
to ‘enantiotopic’ and ‘diastereotopic’ positions, just as R/S-stereodescriptors are as-
signedtoenantiomers(‘chiralityunits’)anddiastereomers(‘pseudoasymmetricunits’).
Hence, this assignment does not obey a single criterion. To claim that this assign-
ment is seemingly done in a single criterion, the term ‘stereoheterotopic’ has been
coinedbymeansoftheterm‘enantiotopic’plustheterm‘diastereotopic’.Asaresult,
prostereoisomerism and prochirality are confused in the assignment of pro-R/pro-S-
descriptors, just as stereoisomerism (stereogenicity) and chirality are confused in the
assignmentofR/S-stereodescriptors.
These matters should be replaced by a more reasonable set of terms for the purpose
ofunderstandingstereochemistrycomprehensively,asintroduceddetailedlyinChapter1.
Keepinmindthatthesemattersareconcernedwiththemostfundamentalterminologyof
modern stereochemistry. This fundamental terminology stems from the theoretical foun-
dations of modern stereochemistry, which have long been developed rather qualitatively
without mathematical basis since the beginning of stereochemistry by van’t Hoff and Le
Bel. As a result, an ad-hoc remedy without mathematical formulations would repeat the
history,soastoprovidenosubstantialsolutions.Hence,thetaskofaimingatatruerem-
edyinevitablyrequirestherestructuringofthetheoreticalfoundations,whichresultsinthe
renewal of the fundamental terminology of modern stereochemistry. In particular, the re-
structuringshouldbebasedonmathematicalformulations,notonqualitativesubstitutions
ofwords,noronnon-mathematicalcoinageofterms.Suchatrueremedyisthetargetofthe
remainingchaptersofthisbookentitledMathematicalStereochemistry.
Chapters 2 is devoted to the solution of the first matter described above, where the
importance of equivalence relationships and equivalence classes is emphasized. In partic-
ular,anewflowchartforclassifyingvariouskindsofisomersisdevelopedonthebasisof
equivalence classes, i.e., isomers as an equivalence class under an isomeric relationship,
Preface vii
isoskeletomersasanequivalenceclassunderanisoskeletomericrelationship,stereoisomers
as an equivalence class under a stereoisomeric relationship, and (self-)enantiomers as an
equivalenceclassunderanenantiomericrelationship.
Chapter3aimsatthesolutionofthefirstandsecondmattersdescribedabove,where
thepoint-grouptheoryisadoptedasamathematicaldeviceafterdevelopingtheproligand-
promoleculemodelbasedonaskeleton.Thereby,weareabletoreplacethemisleadingdi-
chotomy(thefirstmatterdescribedabove)withanewviewpoint:“Thesetofstereoisomers
consistsofdefinitepairsof(self)-enantiomers,wheretherelationshipbetweentwopairsof
(self)-enantiomersisregardedasbeingdiastereomeric.”Notethatapairofself-enantiomers
indicatesanachiralmolecule.Mathematicallyspeaking,eachpairof(self)-enantiomersis
anequivalenceclassundertheactionofthepointgroupofaskeleton.Eachsetofstereoiso-
mersisanequivalenceclassofanotherkindundertheactionofastereoisomericgroup.The
proligand-promoleculemodelisusefultoavoidtheconfusionofreflectionswithpermuta-
tions,whichhasbeenpointedoutasthesecondmatterdescribedabove.
It should be emphasized that even the newly-defined diastereomeric relationship de-
scribedintheprecedingparagraphdoesnotconstructequivalenceclasses.Withoutdefining
suchastereoisomericgroup,however,itispossibletosolvethefirstandsecondmattersto
agreatextentbytakingaccountofthepointgroupoftheskeleton.Inotherwords,geomet-
ricaspectscanbediscussedbyusingpointgroups,apartfromstereoisomericaspects.This
courseispursuedinChapter4toChapter8.
Accordingtothepoint-grouptheoryandthenewviewpointdescribedinthepreceding
paragraph, I have developed the unit-subduced-cycle-index (USCI) approach for com-
binatorialenumerationofmoleculesas3Dstructures.Ihavealreadypublishedtwomono-
graphsconcerningthistopic(S.Fujita,SymmetryandCombinatorialEnumerationinChem-
istry,Springer-Verlag,BerlinHeidelberg,1991;andS.Fujita,DiagrammaticalApproachto
Molecular Symmetry and Enumeration of Stereoisomers, Mathematical Chemistry Mono-
graphsSeriesVol.4,Univ.Kragujevac,Kragujevac,2007).
The USCI approach is based on the concepts of sphericities and chirality fittingness
(CF),whichareassignedtoequivalenceclasses(orbits)governedbycosetrepresentations.
Chapter4isdevotedtoanintroductiontotheseconceptsandtosuchrelatedtopicsasthe
purely-geometricconceptofprochirality.
AfterfoundationsofenumerationunderpointgroupsareintroducedinChapter5,the
USCIapproachforsymmetry-itemizedenumerationunderpointgroupsisdemonstratedin
Chapter6.TheUSCIapproachsupportsfourmethodsofcombinatorialenumeration,which
are based on unit subduced cycle indices without or with chirality fittingness (USCIs or
USCI-CFs).TheUSCIsorUSCI-CFsarecalculatedalgebraicallybystartingfromthesub-
ductionofcosetrepresentations.Theconceptofchiralityfittingness(CF),whichisderived
fromtheconceptofsphericities(homospheric,enantiospheric,andhemispheric),controls
the modes of accommodating chiral and achiral proligands for producing promolecules.
The USCI approach works well in quantitative and qualitative discussions to understand
stereochemistryfromthenewviewpointcombinedwithpointgroups.
viii Preface
As a promising method for gross enumeration under point groups, I have developed
theproligandmethod,wheretheconceptofsphericitiesofcyclesisdevisedasasubstan-
tialextensionoftheconceptofsphericitiesoforbitsintheUSCIapproach.Ihavealready
published a monograph on the proligand method and related methods developed by my-
self(S.Fujita,CombinatorialEnumerationofGraphs,Three-DimensionalStructures,and
ChemicalCompounds,MathematicalChemistryMonographsSeriesVol.15,Univ.Kragu-
jevac,Kragujevac,2013). This monograph isasuccessortoPólya-Read’smonograph (G.
PólyaandR.C.Read,CombinatorialEnumerationofGroups,Graphs,andChemicalCom-
pounds,Springer-Verlag,NewYork,1987),wheregraphsofPólya-Read’smonographare
substantiallyextendedtothree-dimensionalstructuresofFujita’smonographonthebasisof
theextendedconceptofsphericitiesofcycles.Chapter7isdevotedtoanintroductionofthe
proligand method. The proligand method iscapableofrecursive calculation,asdiscussed
inChapter8forenumerationofalkanesas3Dstructures.
To avoid the confusion of reflections with permutations (the second matter described
above),weshouldexaminepermutationgroupsincomparisonwithpointgroups.Chapter
9aimsatdemonstratinghowpermutationgroupshavebeenmixedupwithpointgroupsin
modernstereochemistry.TheconceptofRS-permutationgroupsisomorphictopointgroups
isintroducedtosettlethemisleadingsituationsofmodernstereochemistry.
To solve the third matter described above, I have developed the stereoisogram
approach, where point groups and RS-permutation groups are integrated to create RS-
stereoisomericgroups.NotethatusualpermutationgroupsarerestrictedtoRS-permutation
groups,whichareselectedtobeisomorphictothecorrespondingpointgroups.Asshown
in Chapter 10, I have devised stereoisograms as diagrammatic expressions of such RS-
stereoisomeric groups. Each stereoisogram consists of a quadruplet of promolecules,
which are correlated by three relationships, i.e., enantiomeric (due to point groups), RS-
diastereomeric (due to RS-permutation groups), and holantimeric ones (due to ligand-
reflectiongroups).Thereby,suchstereoisogramsarecategorizedintofivetypes(typeI–type
V), so that quadruplets of types I, III, and V are concluded to replace such conventional
terms as ‘stereogenic units’, ‘chirality units’, and ‘pseudoasymmetric units’. To show the
usefulnessofthestereoisogramapproach,Chapter11dealswithstereoisogramsfortetra-
hedralderivatives,whileChapter12dealswithstereoisogramsforallenederivatives.
The crux of the stereoisogram approach (Chapter 10 to Chapter 12) is the rational
introduction of three pairs of attributes, i.e., chirality/achirality, RS-stereogenicity/RS-
astereogenicity,andsclerality/asclerality,whichcorrespondtothethreerelationshipscon-
tainedinastereoisogram.Thesethreepairsofattributesareindependentofeachother,but
may coalesce according to respective stereoisograms of five types. This is sharp contrast
to the fact that modern stereochemistry presumes a single pair of attributes, i.e., chiral-
ity/achirality,andlackstheotherpairsofattributes.Thelimitationsofmodernstereochem-
istryareconcludedtocomefrommisleadingrationalizationbymeansofchirality/achirality
asasinglepair,aspointedoutasthefourthmatterdescribeabove.
Chapter13dealswithstereochemicalnomenclature,wheretheoriginalfoundationsof
the CIP system for assigning R/S-stereodescriptors (the third matter described above) are
Preface ix
clarified to be misleading, so that they should be replaced by new foundations originated
from the stereoisogram approach. Among the three pairs of attributes established by the
stereoisogram approach, a pair of RS-stereogenicity/RS-astereogenicity is concerned with
the foundations of the CIP system. In fact,a pair of chirality/achirality (the fourth matter
describedabove)hasnothingtodowiththeassignabilityofR/S-stereodescriptorsintheCIP
system.ItfollowsthatapairofR/S-stereodescriptorsisgiventoapairofRS-diastereomers
(butnottoapairofenantiomers).
Thestereoisogramapproachisalsoeffectivetosolvethefifthmatterdescribedabove.
As discussed in Chapter 14, the terms enantiotopic, RS-diastereotopic, and holantitopic
have been coined by the stereoisogram approach, so that the conventional set of terms
‘stereoheterotopic’,‘enantiotopic’,and‘diastereotopic’shouldbeabandoned.Thisispar-
allel to the remedy that the present set of terms RS-stereoisomeric, enantiomeric, RS-
diastereomeric, and holantimeric should replace the conventional set of terms ‘stereoiso-
meric’,‘enantiomeric’,and‘diastereomeric’.Finally,Chapter15summarizestheperspec-
tives of mathematical stereochemistry, which would be brought about on the basis of the
stereoisogramapproach.
Υ
AnadditionalaimofthisbookistogiveafieldtestoftheXMTEXsystem,whichhas
been developed by myself as a software for drawing structural formulas. I have released
Υ Υ
Version 5.01 of the XMTEX system in 2013. The XMTEX system is attached by an on-
Υ
line manual of about 800 pages: S. Fujita, XMTEX: Reliable Tool for Drawing Chemical
StructuralFormulas,whichisavailablefromthefollowingWebsites:
http://xymtex.com/fujitas3/xymtex/xym501/manual/xymtex-manualPS.pdfand
http://www.ctan.org/tex-archive/macros/latex/contrib/xymtex/doc/XyMTeX-manual.pdf
Becauseallofthestructuralformulasappearinginthisbookhavebeendrawnbyusingthe
Υ Υ
XMTEXsystem,theusefulnessoftheXMTEXsystemisclearlydemonstrated.
Beforeclosingthepreface,itisworthwhiletoemphasizetheconceptualimportanceof
thestereoisogramapproachfromahistoricalpointofview.
– Fujita’sstereoisogramapproachcreatestheintermediateconceptofRS-stereoisomers,
whichmediatesbetweenenantiomersandstereoisomers.Suchanintermediateconcept
brings about a paradigm shift, so that modern stereochemistry has been restructured
substantiallyonthebasisofmathematicalformulations.Thisfactisparalleltothehis-
torical event that Avogadro’s theory has brought about a paradigm shift in chemistry
bycreatingtheintermediateconceptofmolecule(e.g.,H O),whichmediatesbetween
2
atoms(e.g.,hydrogenatomsandoxygenatoms)andsubstances(e.g.,water).
– Asaresult,Fujita’sstereoisogramapproachhasstoppedthelong-standinggapbetween
van’t Hoff’s way (emphasizing asymmetry and stereoisomerism) and Le Bel’s way
(emphasizing dissymmetry or chirality initiated by Pasteur), which has continuously
puzzledorganicchemistsfromthebeginningofstereochemistryover140years.
Kanagawa,Japan ShinsakuFujita
May2015