Quantum Chemical Models of Electronic Transitions of ... · atoms, and one p electron for every...
Transcript of Quantum Chemical Models of Electronic Transitions of ... · atoms, and one p electron for every...
QuantumChemicalModelsofElectronicTransitionsofConjugatedDyes2
BrittanySchaum 1March2014
(Partner:YuantaoChen)
Abstract
UV-Visspectroscopywasusedtodeterminetheenergyoftransition,∆𝐸,fromthehighest
occupiedmolecularorbital(HOMO)tothelowestunoccupiedmolecularorbital(LUMO)ofseveral
cationicdyes.Eachdyecontainsaconjugated,hydrocarbonπsystemwithnitrogensateachend,which
actaspotentialbarriersfortheπelectronsinthesesystems.Inthisexperiment,wehavetreatedeach
dyeasaone-dimensional“particleinabox”inordertosimplifythecalculationof∆𝐸.Wealsoderiveeq.
12and13fromthismodelinordertopredictl#$%,whichwecomparedwithexperimentaldata.Our
initialapproximationofl#$%(eq.12)waspoor,with~20%erroronaverage,butwhenavariable
parameter,α,wasadded,calculationswerequiteaccurate,with<5%error.Thecalculatedvalueforα
was0.682543.Gaussianwasalsousedtopredictl#$%,alsowithlargeerror,~20%.However,
experimentaldataforl#$%wasquiteaccurate,with<1%errorcomparedtoliteraturevalues.
I. Introduction
Ourunderstandingofthebehaviourofelectronsowesitselftoquantummechanics.Quantum
mechanicsrevealsthatparticlesexhibitwave-likepropertiesontheatomicscale,andthus,existin
discretestateswithcharacteristicenergylevels.Sincetheamountofenergyanelectroncanhavein
non-continuous,itfollowsthatanelectroncanonlygainorloseanamountofenergyexactlyequalto
thedifferenceinenergybetweenitscurrentstateandanotherdiscretestate.Itisknownthatelectrons
gainorloseenergyintheformofelectromagneticradiation,whichofcoursealwayshasacharacteristic
wavelength.Theelectronictransitionswhichoccurbetweenatomicandalsocomplexmolecularorbitals
canbeobservedbyexploitingthisfact.Thelowestenergytransitionofelectronsinamoleculeis
typicallythetransitionbetweentheHOMO(highestoccupiedmolecularorbital)andtheLUMO(lowest
unoccupiedmolecularorbital).Therelationshipbetweenthedifferenceinenergybetweentwo
electronicstatesandthecorrespondingwavelengthofelectromagneticradiationis4
DE=()l (1)
whereDE,h,c,andlrepresentthechangeinenergy,Planck’sconstant,andthespeedandwavelength
oftheelectromagneticradiation,respectively.Itisclearfromeq.1thatlowerenergytransitionsina
moleculecorrespondtolongerwavelengths.Whentheseenergeticdifferencesarelessthanabout70
kcal/mol,visiblelight(electromagneticradiationintherangeof~ 380-760nm)isemitted/absorbed.
Moleculeswithconspicuouscolortendtobeoneswithhighlyconjugatedp-systems.Fourexamplesof
suchmolecules,cationicdyeswithextendedp-systems,werestudiedinthisexperiment.
Figure1.Structuresandnamesofdyesstudied
Thestatesofelectronsandtheircorrespondingenergiescanalsobedescribedbythenon-
relativistic(time-independent)Schrödingerequation4
− (+
,-+#∇/ + 𝑉 𝑟 𝛹(𝑟) = 𝐸𝛹(𝑟) (2)
wheremisthemassoftheelectron,∇/istheLaplacianoperator,V(r)istheposition-dependent
potentialoperator,Eistheenergyoftheelectron,and𝛹(𝑟)isthewavefunction(whererrepresents
thepositioncoordinates).Whennopotentialisactingupontheelectron(orifitisnegligible),eq.2
reducesto
− (+
,-+#∇/ 𝛹(𝑟) = 𝐸𝛹(𝑟) (3)
Suchisthecasefora“particleinabox,”oraparticlewhichisconfinedtoanareaofzero(orconstant)
potentialbyinfinitepotential“walls.”Infact,thewavefunctionforaone-dimensionalparticleinabox
is
𝛹 𝑥 = /8sin <-%
8 (4)
whereListhe“length”ofthebox,orthelengthofthespacetowhichtheparticleisconfined.Inhighly
conjugatedp-systems,suchasthoseinthemoleculesshowninfigure1,moleculescanbethoughtofas
essentiallyplanar,withallp orbitalsparalleltooneanotherandwithp electronsmovingfreelywithin
this p system,accordingtoErichHückel.This“free-electron”model3,althoughanapproximation,is
quitesimpleaswellasaccuratewhencomparedtoexperimentalresults.Analysisofthemolecular
orbitalsofthemoleculesshowninfigure1wouldbeotherwisequitecomplex,andindeed,computer
softwarecanalsobeusedforthisanalysis.
Ifthe“freeelectron”modelistobeusedtoanalyzethedyesshowninfigure1,thebenzene
ringsareignoredandthep systemisthoughttoconsistofonep-orbitalforeachofthetwonitrogen
atoms,plusoneforeachcarbonatominthechainconnectingthem.Itisalsoassumedthatpotential
energyremainsconstanteverywherealongthischain,andthatthepotentialessentiallyreachesinfinity
justbeyondthenitrogenatoms.The“particles”inthisparticleinaboxmodelare,ofcourse,the
p electronsofthisconjugatedsystem,andsolvingeq.3(aneigenvalueequation)bysubstitutingeq.4
(anappropriateeigenfunctionforeq.3)wecanobtaintheirenergylevels(theeigenvaluesofthis
eigenfunction):
𝐸< = (+<+
,#8+ (5)
wherenisanyinteger>0.AccordingtothePauliexclusionprincipal,thenumberofelectronswhich
occupyanygivenenergylevelcannotexceedtwo,whichthereforemeansthatthegroundstateofany
moleculewithNp electronswillhave=/filledenergylevelsifNiseven,and=>?
/filledlevelsifNisodd.
Then-valuewhichcorrespondstotheHOMOwouldbe
𝑛ABCB =
𝑁2𝑖𝑓𝑁𝑖𝑠𝑒𝑣𝑒𝑛
𝑁 + 12
𝑖𝑓𝑁𝑖𝑠𝑜𝑑𝑑
andthen-valuewhichcorrespondstotheLUMOwouldbe
𝑛8NCB =
𝑁 + 22
𝑖𝑓𝑁𝑖𝑠𝑒𝑣𝑒𝑛
𝑁 + 32
𝑖𝑓𝑁𝑖𝑠𝑜𝑑𝑑
DEforthelowestenergyelectronictransitionisthatofatransitionbetweentheHOMOandLUMO,andisequalto
∆𝐸 = (+
,#8+𝑛8NCB/ −𝑛ABCB/ =
(+
,#8+=+
P+ 𝑁 + 1 − =+
P= (+
,#8+𝑁 + 1 𝑖𝑓𝑁𝑖𝑠𝑒𝑣𝑒𝑛
(+
,#8+=+
P+Q=
/+ R
P− =+
P+ 𝑁 + 1 = (+
,#8+𝑁 + S
PifNisodd
(8)
Forthecationicdyesstudiedinthisexperiment,therearethreep electronsfromthetwonitrogen
atoms,andonep electronforeverycarbonatominthechainconnectingthem.Ifweletp=thenumber
ofcarbonatomsinthep systemofeachdye,itfollowsthatN=p+3.Sincethenumberofcarbonatoms
inthechainofeachdyeisodd,Nwillalwaysbeodd,sinceN=p+3andpisalwaysanoddnumber.
Thus,eq.8reducesto
∆𝐸 = (+
,#8+𝑛8NCB/ −𝑛ABCB/ = (+
,#8+𝑁 + 1 (9)
Inthisexperiment,thelowest-energyelectronictransitionofeachdyepicturedinfigure1was
analyzedusingUV-Visspectroscopy.Sincethistransitioninthesedyesoccursuponabsorptionof
electromagneticradiationinthevisiblelightregion,themostprominentabsorbancebandforeach
moleculeshouldbebetween380and760nm.Ifwesubstituteeq.1intoeq.9andsolveforl,wehave
l#$% = ,#)8+
((=>?) (10)
wherel#$%isthewavelengthofmaximumabsorbance,sinceaBoltzmann5distributionpredictsthat
thewavelengthofmaximumabsorbancewillbethatofthelowestenergyelectronictransition(that
whichoccursbetweentheHOMOandLUMO).Lforeachdyeisestimatedtobethesumofthelengthof
thecarbonchainbetweenthenitrogenatomsandonebondlengthoneachside.Sincethenumberof
bondsbetweeneachnitrogenatomisp+1,ifeachbondlengthisestimatedtobeapproximatelythe
same(duetoconjugation),say,alengthl,thenL=(p+3)l.Ifwesubstitutethisintoeq.10andusethe
factthatN=p+3,wehave
l#$% = ,#)X+ Y>Q +
( Y>P (11)
Ifwepredictthatlisapproximatelyequaltothebondlengthobservedinbenzene(1.39Å),and
substitutethemassofanelectron,thespeedoflight,andPlanck’sconstant,wecansaythat
l#$% = 63.7 Y>Q +
Y>P (12)
approximately,wherel#$%isinnanometers.Sinceitismorelikelythatthepotentialateachendofthe
chainrisesgraduallyratherthanjumpingtoinfinity,wecanaddavariableparameter,α,toLtoaccount
forthisadditionallength(thiswillalsohelpmaketheapproximationofthebondlengthsinthechain
moreappropriate,sincetheymaynotbeexactly1.39Å)bysayinginsteadthatL=(p + 3 + α)l.Thus,eq.
12becomes
l#$% = 63.7 Y>Q>] +
Y>P (13)
whereαshouldremainconstantforthisseriesofmolecules(theonlydifferencebetweeneachmolecule
isthelengthoftheinternitrogenouscarbonchain).Inthisexperiment,calculationsofl#$%usingeq.13
werecomparedtoexperimentalobservationsofl#$%foreachdyeinordertooptimizeα.
Additionally,thecomputerprogram,Gaussian,wasusedtopredictthelowest-energy
conformationofeachstructureshowninfigure1,andthentocalculatetheenergyoftheHOMOand
LUMOofeachdye(whichcanbeusedtoestimatel#$%).Theseresultswerealsocomparedto
experimentaldata.
II. Experimentalmethod
Part1.AnOceanOpticsSpectrometerwasusedtoobtainUV-Visspectraofeachcationicdye
showninfigure1.Methanolicsolutionsofeachdyewerecreated,andconcentrationswereadjusted
untilabsorbancebandshadmaximumintensitiesjustbelow1(allconcentrationswereontheorderof
micromolar).TheabsorbancedataforeachdyewasrecordedandplottedinMicrosoftExcel.Thisdata
wasusedtodeterminel#$%foreachdye.
Part2.Eq.13wasused,withastartingvalueof1forα,inordertoobtaintheoreticalvaluesfor
l#$%foreachdye.InMicrosoftExcel,theSolveradd-inwasusedtooptimizeα;αwasallowedto
varyuntiltheminimumvalueforthesumofthedifferenceofsquaresofeachtheoreticaland
experimentall#$%wasobtained.
Part3.Thecomputerprogram,Gaussian,wasusedtocalculatel#$%foreachdyeshown
infigure1.First,semi-empiricalAM1geometryoptimizationsforeachdyewereperformed
(hydrogenatomsweresubstitutedfortheethylgroupsattachedtonitrogenineachdyewhen
constructingeachmolecule).Next,morerigorousgeometricaloptimizationwasperformedatthe
HF/3-21Glevel.Theoutputofthesegeometricoptimizationswereusedinordertoperformsingle-
pointcalculationsattheDFT/3-21GandDFT/6-31+level.TheMOeditorinGaussianwasusedin
ordertoobtaintheenergyoftheHOMOsandLUMOsofeachdyeforeachcalculationmethod.
OnceGaussianwasusedtodeterminetheenergyoftheHOMOandLUMOofeachdye,∆𝐸was
calculatedinordertoobtainatheoreticalvalueforl#$%,usingeq.1(whereeq.1wasrearrangedto
solveforl#$%).ThiswasdonefortheresultsofboththeDFT/3-21GandDFT/6-31G+methodsof
singlepointcalculation.
III. Results
First,anOceanOpticsSpectrometerwasusedtoobtainUV-Visspectraofeachcationicdye
showninfigure1.Methanolicsolutionsofeachdyewerecreated,andconcentrationswereadjusted
untilabsorbancebandshadmaximumintensitiesjustbelow1.Thesespectracanbeviewedinfigure2.
Figure2.AbsorbancespectraofmethanolicsolutionsofDTC,DTCC,DTDC,andDTTCwithconcentrationsof4.3µM,
3.2µM,3.9µM,and5.1µM,respectively,obtainedusinganOceanOpticsspectrometer.
Todeterminel#$%foreachdye,absorptiondatawascopiedintoMicrosoftExcel.Absorbance
valuesforeachdyeweresortedgreatest-to-leastbyExcel,andthecorrespondingwavelengthswere
recorded.Thevaluesobtainedforl#$%foreachdyecanbeviewedintable1.
-0.2
0
0.2
0.4
0.6
0.8
1
370 420 470 520 570 620 670 720 770
Absorbance
Wavelength(nm)
DTDC
DTTC
DTCC
DTC
Dyemolecule Concentration(µM) 𝐴#$% l#$%(nm)
DTC 4.3 0.619 422.9
DTCC 3.2 0.703 556.6
DTDC 3.9 0.613 651.5
DTTC 5.1 0.949 758.5
Table1.ConcentrationofmethanolicsolutionsofDTC,DTCC,DTDC,andDTTC(inµM),theirmaximumabsorbances
(𝐴#$%),andwavelengthat𝐴#$%(l#$%,innanometers).
Next,eq.13wasused,withastartingvalueof1forα,inordertoobtaintheoretical
valuesforl#$%foreachdye.ValuesofpusedforDTC,DTCC,DTDC,andDTTCwere3,5,7,and9,
respectively.InMicrosoftExcel,theSolveradd-inwasusedtooptimizeα.Thiswasdonebyallowingα
tovaryuntiltheminimumvalueforthesumofthedifferenceofsquaresofeachtheoreticaland
experimentall#$%wasobtained.Theresultsofthisoptimizationcanbeviewedintable2.
Dyemolecule p l_%Y_`a#_<b$c (nm) l)$c)dc$b_e (nm) (lfghil)$c))/
DTC 3 422.9
406.3731
273.1396
DTCC 5 556.6
533.5693
530.4146
DTDC 7 651.5
660.8396
87.22772
DTTC 9 758.5
788.1498
879.1102
Total:
1769.892
α: 0.682543
Table2.TheresultsofoptimizationofαusingSolverinMicrosoftExcel.Tableshowsexperimentalvaluesfor
l#$%foreachdye,valuesforl#$%calculatedusingeq.13whileallowingαtovaryforeachdye,thedifferenceof
squaresofthesevalues,thesumofthesedifferencesofsquares,andtheresultingoptimizedvalueforα.
Predictionsforthevalueofl#$%werealsocalculatedwithoutthevariableparameterα,shownin
table2-a,andanexampleofthiscalculationcanbeviewedinfigure3.
DTC: p=3
(12) l#$% = 63.7 Y>Q +
Y>P=63.7 Q>Q +
Q>P=(63.7𝑛𝑚) Qk
l=327.6nm
Figure3.Samplecalculationofl#$%usingeq.12.
Dyemolecule l#$%(nm)
DTC 327.6
DTCC 453.0
DTDC 579.1
DTTC 705.6
Table2-a.Theoreticalvaluesforl#$%calculatedusingeq.12(withoutα),innanometers.
Thecomputerprogram,Gaussian,wasalsousedtocalculatel#$%foreachdyeshownin
figure1.First,semi-empiricalAM1geometryoptimizationsforeachdyewereperformed(hydrogen
atomsweresubstitutedfortheethylgroupsattachedtonitrogenineachdyewhenconstructingeach
molecule).Next,morerigorousgeometricaloptimizationwasperformedattheHF/3-21Glevel.The
outputofthesegeometricoptimizationswereusedinordertoperformsingle-pointcalculationsatthe
DFT/3-21GandDFT/6-31+level.TheMOeditorinGaussianwasusedinordertoobtaintheenergyof
theHOMOsandLUMOsofeachdyeforeachcalculationmethod.Theresultsofthesecalculationscan
beviewedintable3.
DFT/3-21G: DFT/6-31G+:
𝐸ABCB(H) 𝐸8NCB(H) 𝐸ABCB(H) 𝐸8NCB(H)
DTC -0.33134 -0.19399 -0.32881 -0.19311
DTCC -0.30352 -0.20108 -0.30140 -0.20011
DTDC -0.28650 -0.19870 -0.28448 -0.19744
DTTC -0.27325 -0.19631 -0.27125 -0.19487
Table3.ValuesobtainedfortheenergyoftheHOMOsandLUMOsofDTC,DTCC,DTDC,andDTTCusingDFT/3-21G
andDFT/6-31G+singlepointcalculations,inHartrees(H).
TheMOeditorinGaussianwasalsousedtovisualizetheHOMOsandLUMOsofeachdye.
Samplesofthesevisualizationsareshowninfigures4–7.
Figure4.VisualizationoftheHOMOofDTCusingDFT/6-31G+singlepointcalculationinGaussian.Initialgeometric
optimizationusingAM1canalsobeseen(thestructurewasestimatedtobecompletelyplanar).Oppositephases
arerepresentedbygreenandred,sulfurisrepresentedbyyellowtube,andnitrogenisrepresentedbyabluetube.
Theremainingtubestructure(ingrey)representstheremainingcarbon“frame”ofthemolecule.
Figure5.VisualizationoftheLUMOofDTC,alsousingDFT/6-31G+singlepointcalculationinGaussian
(representationissimilartothatoffigure4).
Figure6.VisualizationoftheHOMOofDTTC,usingDFT/3-21GsinglepointcalculationinGaussian(representation
issimilartothatoffigures4and5).
Figure7.VisualizationoftheLUMOofDTTC,usingDFT/3-21GsinglepointcalculationinGaussian(representation
issimilartothatoffigures4,5,and6).
OnceGaussianwasusedtodeterminetheenergyoftheHOMOandLUMOofeachdye(table3),
thedifferenceinenergybetweentheHOMOandLUMOofeachdyewascalculatedinordertoobtaina
theoreticalvalueforl#$%,usingeq.1(whereeq.1wasrearrangedtosolveforl#$%).Thiswasdone
fortheresultsofboththeDFT/3-21GandDFT/6-31G+methodsofsinglepointcalculation.Since
GaussianyieldsenergyvaluesinunitsofHartrees,thesevalueswereconvertedtoJoulesbeforeusing
eq.1.Anexampleofthiscalculationcanbeviewedinfigure8andtheresultsofthesecalculationscan
beviewedintable4.
DTC,DFT/3-21G: DE=𝐸8NCB-𝐸ABCB=-0.19399H+0.33134H=0.13735H
0.13735HxP.QSR,%?mnopq
?A=5.998x10i?R𝐽
(1) l#$%=()∆t= k.k/k%?m
nuvq∗x /.RR,%?moy<# xS.RR,%?mnozq
=331.7nm
Figure8.Calculationofl#$%ofDTCusingeq.1andDFT/3-21GenergycalculationsfromGaussian(seetable3).
DFT/3-21G: DFT/6-31G+:
∆𝐸(H) l#$%(nm) ∆𝐸(H) l#$%(nm)
DTC 0.13735 331.7 0.13570 335.8
DTCC 0.10244 444.8 0.10129 449.8
DTDC 0.08780 518.9 0.08704 523.5
DTTC 0.07694 592.2 0.07638 596.5
Table4.Calculatedvaluesfor∆𝐸andtheoreticalvaluesforl#$%usingresultsoftable3andthecalculations
showninfigure8.
Finally,valuesobtainedforl#$%usingeq.12,eq.13,andGaussianwerecomparedto
experimentalvalues(table1).Percenterrorsintheoreticalvalueswereobtainedusingthesample
calculationshowninfigure9andtheresultsofthesecalculationscanbeviewedintable5.
l𝒎𝒂𝒙forDTCusingGaussian,DFT/6-31G+singlepointcalculation:335.8nm
Experimentall𝒎𝒂𝒙forDTC:422.9nm
%error=l𝒕𝒉𝒆𝒐𝒓𝒆𝒕𝒊𝒄𝒂𝒍il𝒆𝒙𝒑𝒆𝒓𝒊𝒎𝒆𝒏𝒕𝒂𝒍
l𝒆𝒙𝒑𝒆𝒓𝒊𝒎𝒆𝒏𝒕𝒂𝒍x100%=-2.06%
Figure9.Samplecalculationofpercenterrorforthetheoreticalcalculationofl#$%forDTCusingGaussianDFT/6-
31G+singlepointcalculation.
%errorincalculationsusing:
Freeelectronapproximation
Gaussiansinglepointcalculation
withoutα(eq.12)
withα(eq.13)
DFT/3-21Gmethod
DFT/6-31+method
DTC -22.5 -3.9 -21.6 -20.6
DTCC -18.6 -4.1 -20.1 -19.2
DTDC -11.1 +1.4 -20.4 -19.6
DTTC -7.0 +3.9 -21.9 -21.4
Table5.Percenterroroftheoreticalvaluesobtainedforl#$%usingeq.12,eq.13,GaussianDFT/3-21G,and
GaussianDFT/6-31G+ascomparedtoexperimentalvalues.
Literaturevaluesforl#$%werealsousedtocalculatepercenterrorinexperimentaland
theoreticaldataobtained.Theseresults,aswellastheliteraturevalues1forl#$%ofeachdye,are
shownintable5-a.
%errorincalculationsusing:
Freeelectronapproximation
Gaussiansinglepointcalculation
withoutα(eq.12)
withα(eq.13)
DFT/3-21Gmethod
DFT/6-31+method
Experimentaldata
Literaturevalues1forl#$%(nm)
DTC -22.7 -4.2 -21.8 -20.8 -0.3 424DTCC -19.1 -4.7 -20.6 -19.7 -0.6 560DTDC -11.6 +0.9 -20.8 -20.1 -0.5 655DTTC -7.8 +3.0 -22.6 -22 -0.8 765
Table5-a.Literaturevaluesforl#$%ofeachdye,andpercentdeviationfromthesevaluesoftheexperimental
data,freeelectronapproximations,andGaussiancalculations.
IV. Discussion
ItwouldseemfromthegivenliteraturevaluesforDTC,DTCC,DTDC,andDTTCthatthe
experimentaldataisquiteaccurate(theabsolutevalueofthepercentdeviationofeachislessthan1%).
Itwouldalsoseemthatapproximationsusingeq.13arealsoquiteaccurate(theabsolutevalueofthe
percentdeviationofeachislessthan5%).However,allotherapproximations,especiallythoseusing
Gaussian,seemtobequiteinaccurate(theabsolutevalueofthepercentdeviationofeachisgreater
than5%,andinfact,oftenexceeds20%,whichisquitehigh).Ingeneral,almostallapproximationsare
underestimationsofl#$%,withonlytwoexceptions(bothofwhichareapproximationsusingeq.13).
Consideringthefactthattheentireelectromagneticspectrumspanstwenty-fourordersof
magnitude,thatthevisiblelightspectrumishardlyoneorderofmagnitudeinsize,andthatthe
approximationsusingGaussianandeq.12estimatel#$%tobesomewhereintherangeofvisiblelight,
thesemethodsactuallyseemquiteimpressive.However,thereareobviousflawsinthese
approximations.Intheapproximationofl#$%usingeq.12,itisassumedthatthebondlengthbetween
allatomsintheinternitrogenouschainofeachdyemoleculeispreciselythatofcarbon-carbonbondsin
benzene(1.39Å),andthatthereispreciselyonebondlengthofthislengthoneithersideofeach
nitrogenbeforethepotentialofthemoleculereachesinfinity.Inreality,despitethefactthatthisregion
ofeachmoleculeisconjugated,itisnotknownifthesebondlengthsareprecisely1.39Å(buteq.12
assumesthatthisisso).Also,itismuchmorelikelythatinsteadoftherebeinganinfinitepotential
“wall”oneithersideofeachnitrogen,thepotentialmostlikelyrisesgradually,whichwouldallowan
electrontomoveabitfurtherthanexpected.Thisexplainsthepositivevalueofα;Lisprobablyabit
longerthan(p+3)l.Additionally,eveniftherewereinfinitepotential“walls”beyondeithersideofeach
nitrogen,quantumtunnelingpredictsthatthereisacertainprobabilityoffindinganelectronbeyond
thosewalls.DespitethehugeerrorintheGaussianapproximationsof∆E,figures4–7showthatthe
lowestenergyconformationofeachofthedyemoleculesstudiedisplanar,andthatthereisa
“boundary”(anode)justbeyondeachnitrogen,inboththeHOMOandLUMOofeachmolecule.We
haveignoredtheethylgroupsattachedtonitrogenineachmolecule,replacingthemwithhydrogens.In
theory,thisshouldnotaffecttheenergyoftheπsystem,butbecauseoftheplanarnatureoftheπ
systemandthefactthatthecarbontwiceremovedfromnitrogenshouldbeabletomovefreely,thereis
perhapsachancethatthereissomeπorσinteractionfromthiscarbon,whichwouldofcourseaffect
theenergyoftheπsystemandmakeitdeviatefromtheconditionsofthe“freeelectron”model.
CalculationswouldhavetoberepeatedwithGaussianwiththeseethylgroupsaddedinordertosee
whetherornottheyaffecttheenergyoftheHOMOandLUMO(atime-dependentDFTmaybeinorder,
sincethecarbonsinquestionshouldbeinanon-inertialframerelativetotherestofthemolecule).Also,
whileeq.13mayseemmostaccurate,itisstillanartifice,sincewearesimplyusingavariable
parameterto“makeup”forthelackofaccuracyintheassumptionswhichallowedustoderiveit,rather
thanaccountingforphysicalfactorswhichwouldexplainthislackofaccuracy.Onesimplecorrection,to
startwith,wouldbetorecognizethatthepotentialwithintheπsystemshouldnotsimplybezero,but
rather,anon-zeroconstant(orapotentialthatisessentiallyconstant).Thiswouldaffecttheapplication
ofeq.2(thenon-relativisticSchrödingerequation),andthus,wouldresultindifferenteigenvalues
(energies).Thiswouldnotaffect∆𝐸.Butifinstead,anon-zeropotential(V(r))werepresentintheπ
system,theneq.5wouldinsteadread
𝐸< = ℎ/𝑛/
8𝑚𝐿/+𝑉<(𝑟)
andeq.9wouldread
∆𝐸 = (+
,#8+𝑁 + 1 + 𝑉</ 𝑟 − 𝑉<? 𝑟
wherethesubscriptsn1andn2representapossibledependenceofV(r)onn.Sinceusingeq.12predicts
ashorterwavelength(higher∆𝐸)fortheHOMO/LUMOtransitionineverydyemolecule,ifV(r)
dependedonn,therewouldbeagreaterdisparityintheenergyoftheHOMOandLUMO.Explorations
ofapossiblefunctionfor𝑉< 𝑟 shouldbeexploredbyexaminingthediscrepancybetweenexperimental
dataandcalculationsfromeq.12.
References:1.http://www.sigmaaldrich.com/catalog/product/sial/390410?lang=en®ion=US2.AdaptedfromGarland,C.W.,Nibler,J.W,andShoemaker,D.P.ExperimentsinPhysicalChemistry,8thEd.,McGraw-Hill,2009,pp.393-3983.Kuhn,H.J.Chem.Phys.1949,17,11984.Atkins,P.W.,dePaula,J.PhysicalChemistry,8thEd.,NewYork,Freeman,2006,pp.2795.Bennett,C.A.,PrincipalsofPhysicalOptics,1stedition;Danvers,MA;2008