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&region=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