Increasing the Applicability of Density Functional Theory

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Title:
Increasing the Applicability of Density Functional Theory
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1 online resource (132 p.)
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english
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Verma,Prakash
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University of Florida
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Gainesville, Fla.
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Degree:
Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Chemistry
Committee Chair:
Bartlett, Rodney J
Committee Members:
Reynolds, John R
Deumens, Erik
Ohrn, Nils Y
Biswas, Amlan

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Subjects / Keywords:
bartlett -- dft -- non -- oep -- rpa
Chemistry -- Dissertations, Academic -- UF
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Chemistry thesis, Ph.D.
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theses   ( marcgt )
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Abstract:
According to density functional theory(DFT), the density is a sufficient variable for the description of degenerate or non-degenerate ground state of molecules. The density is generally obtained by solving the self-consistent Kohn-Sham (KS) equations rather than by direct minimization of the energy. An attractive feature that the KS procedure offers is it provides a pictorial molecular orbital view of the molecules and the electronic spectra in terms of the KS orbitals. Besides their role in frontier molecular orbital theory, certain quantitative features of these orbitals should be pertinent. According to Bartlett?s theorem, the negative of KS eigenvalue associated with such an occupied orbitals should be a good approximation for its principal vertical ionization energy; however, in practice it differs considerably. However delta SCF calculations (doing two different KS-SCF calculations, one for the neutral and another one for ionic species) give reasonably correct ionization energies attesting to the general accuracy of the KS functional. The failure of the method to relate eigenvectors to vertical ionization energies are due to the lack of the consistency among DFT three key ingredients (functional, potential and density). Our research has been aimed at understanding these consistency conditions. We know that the spatial behavior of the explicit density dependent potential is incorrect (it decays too fast in the asymptotic regions and does not show shell oscillations) and it does not provide correct eigenvalues, thus density obtained from the eigenvectors could be questionable as well. To prove this finding we devise a Non variational DFT, meaning it is non variational in the sense that the density is obtained from wave-function theory rather than by minimizing the DFT functional. The Becke exchange and the LYP correlation energy evaluated using the Hartree-Fock (HF) density augmented the HF kinetic energy and classical coulomb interaction is used to obtain the DFT energy. We also developed an analytical gradient scheme for such an approach and showed that our non-variational approach provides much better transition states and activation energies than variational DFT. In order to assess the quality of the potential, we used Bartlett?s theorem as our criteria and compared the negative of the KS eigenvalues obtained using the 52 different explicit density dependent exchange-correlation potential in the KS operator with the principle vertical ionization energies. None of the explicit density dependent potential provide a good approximation of the vertical ionization energies in terms of the negative of the KS eigenvalue spectra. Even, the HOMO energies are far from the desired accuracy. However, when the KS operator has the orbital dependent exchange-correlation potential such as OEP2-sc, we obtain a good approximation of the vertical ionization energies of the valence orbitals in terms of the negative of the KS eigenvalue spectra. Having an equal amount of local exact exchange and non-local (the Hartree-Fock like exchange operator) exchange potential coupled with MBPT-2sc correlation potential in the KS operators provide a good approximation of the vertical ionization energies for all the orbitals in terms of the negative of the KS eigenvalues and show the importance of the non-local exchange potential in the quest for the consistent DFT approach. We have gathered more concrete numerical results that KSs orbitals are not merely an auxiliary construct to obtain the density and the density gradient, but do have meaning and the proper amount of non-local exchange potential is imperative to get core ionization energies as the negative of the KS eigenvalues. We also question, the recent prominently used approach in DFT community to design a exchange and correlation functional ( e.g. M05, M06, B3LYP, B2-PLYP, DF2-D etc.) by combining the stand alone well known functional such as SLATER exchange ,BECKE exchange, Hartree-Fock exchange PBE exchange, LYP correlation, MP2 correlation etc with the parameters obtained by doing mindless data mining. We call this an a, b ,c approach of the DFT and show first by constructing a potential corresponding to one of the most widely successful functional B2-PLYP or the double hybrid and asking the question if it provides a consistent potential, an absolute average deviation of 2 eV for ionization energies proves that the potential obtained does not have same degree of exactness as the functional has, second by constructing our own potential instead of a functional that will provide vertical ionization energies in terms of the negative of the KS eigenvalues spectra within a desired accuracy of 1 eV, that these approaches are nothing but an elegant ways of suppressing the inherit deficiencies of the explicit density dependent exchange-correlation functional or the potential. The exact exchange-correlation functional can be obtained using the adiabatic connection and the fluctuation dissipation framework via an integration over the coupling constant and the dynamic density-density response function. The various approximations of the dynamic density-density response function can be obtained using the non-interacting density-density response and the exchange-correlation kernel. The unperturbed density-density response function after doing the coupling constant integration gives the exact exchange energy (non-local exchange energy) in terms of reference orbitals. And the simplest approximation of the coupling constant can be obtained by ignoring the kernel, which is often called the random-phase approximation. Using the plasmon model or the equivalence between the RPA and the ring-CCD, frequency integration and coupling strength integration can be avoided. Using our OEP procedure (by insisting that the density correction the KS determinant should vanish) we obtain ring-CCD correlation potential which is equivalent to the RPA correlation potential. Shell oscillations demonstrated by the exact potential, can be reproduced by the RPA potential as well as the MBPT-2 potential. The RPA correlation potential has a correct long range behavior. We also improved the RPA potential by exploiting the relationship between RPA and coupled cluster theory, considering various ways to include singles excitation effects, and additional double excitation diagrams. Potentials obtained showed similar spatial behavior as the RPA potential except they are correct in the near nucleus region as well. The potential are computationally demanding but in the right direction towards a consistent DFT as not only potential show correct spatial behavior and HOMO energies are within desired accuracy but also the total energies. The self-consistent KS calculation with the RPA correlation potential in the KS operator is always avoided as it is too computationally demanding and the RPA energies are obtained using some sets of reference orbitals. It is argued that the KS orbitals obtained from the explicit density dependent potential ( e.g. PBE) in the KS operator are better than the Hartree-Fock orbitals for the RPA. We have investigated that argument and find that for the RPA correlation energy evaluation, choice of the PBE orbitals vs the Hartree-Fock orbitals are irrelevant unless one do the problem in correctly in other words, how to define RPA is time-dependent Hartree-Fock with exchange being neglected or the time dependent DFT with no kernel in linear response function? The RPA with the time-dependent DFT framework overestimate the RPA correlation energy four fold compared to the RPA with the time-dependent HF framework using either the PBE or the HF orbitals. The binding energy curve for argon dimer reveal surprising results, one with incorrect RPA correlation energy provide the best answer. We have also obtained results by including the singles contributions and obtained the similar behavior.
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Statement of Responsibility:
by Prakash Verma.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Bartlett, Rodney J.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-08-31

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INCREASINGTHEAPPLICABILITYOFDENSITYFUNCTIONALTHEORYByPRAKASHVERMAADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011PrakashVerma 2

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Tomyparents 3

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ACKNOWLEDGMENTS Iwouldliketoshowmygratitudetomysupervisor,RodneyJ.Bartlett,whosementorshipenabledmetodevelopanunderstandingofthesubject.Ithankthemembersofmysupervisorycommittee,Dr.Ohrn,Dr.Deumens,Dr.ReynoldsandDr.Biswasforbeingpartofmyintellectualdevelopmentprocessandputtingtheirtimeandeffortaspartofmycommittee.IoffermanythankstoAjithPerera,TomWatson,Ann,VictorandBartlettgroupforsharingandteachingdifferentaspectsoftheoreticalchemistry.Finally,IwouldliketoacknowledgemyfamilyfortheirsupportandJohn,Dmitry,Seni,ManojandofcourseJessicaforsharingfuntimewithme. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 15 1.1ComputingElectronicStructureisaDifcultProblem ............ 15 1.2DensityFunctionalApproachtoMany-BodyElectronicStructureProblem 17 1.3DesiredPropertiesofvks ............................ 28 1.4HowtoObtainthexc-PotentialsUsingtheOEPProcedure ......... 31 1.4.1DirectFunctionalDerivative ...................... 32 1.4.2TotalEnergyMinimizationUsingSharpandHartonCondition ... 33 1.4.3ExactDensityorDensityInvarianceCondition ............ 34 2NON-VARIATIONALDENSITYFUNCTIONALTHEORYFORPREDICTINGTRANSITIONSTATEANDBARRIERHEIGHTS. ................. 41 2.1Motivation .................................... 41 2.2AnalyticalGradientSchemeforNon-VariationalDFT ............ 46 2.3NumericalAnalysis ............................... 49 3DENSITYFUNCTIONALORBITALENERGIESASIONIZATIONPOTENTIAL:THEROLEOFNON-LOCALEXCHANGE. .................... 55 3.1Introduction ................................... 56 3.2PerformanceoftheExplicitDensityDependentFunctional ......... 57 3.3ImportanceoftheNon-LocalExchangePotential .............. 60 3.4Conclusion ................................... 62 4DOESADOUBLEHYBRIDDENSITYFUNCTIONALPROVIDEACONSISTENTPOTENTIAL? ..................................... 64 4.1WhyDoWeNeedB2-PLYPPotential? .................... 64 4.2Grimme'sDoubleHybridDensityFunctional(B2-PLYP) .......... 66 4.3PotentialCorrespondingtotheB2-PLYPFunctional ............. 68 4.4NumericalResults ............................... 72 4.5Analysis ..................................... 74 5

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5CORRELATIONPOTENTIALCORRESPONDINGTOTHERANDOMPHASEAPPROXIMATIONANDBEYONDINDENSITYFUNCTIONALTHEORY .... 80 5.1Introduction ................................... 80 5.2Theory ...................................... 83 5.3ResultsandDiscussion ............................ 90 6REFERENCEPROBLEMINTHERPAANDHOWTOGOBEYONDTHERPA? 97 6.1BestReferencefortheRPA .......................... 98 6.2HowtoGoBeyondRPA? ........................... 100 7ISITPOSSIBLETOCONSTRUCTAPOTENTIALUSINGANA,B,CAPPROACHTODFTTHATWILLGIVEEIGENVALUESASAGOODAPPROXIMATIONTOTHEVERTICALIONIZATIONENERGIES ................... 105 7.1PerformanceofBHLYPvwnPotential ..................... 107 7.2QualityoftheBHLYPvwnDensityandOrbitals ............... 112 8CONCLUSION .................................... 122 REFERENCES ....................................... 126 BIOGRAPHICALSKETCH ................................ 132 6

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LISTOFTABLES Table page 1-1DensitymomentsofNeatom. ............................ 29 2-1BarrierHeightusingtheQCISDoptimizedstructures ............... 51 2-2BarrierHeight .................................... 52 3-1ComparisonofthenegativeoftheOEPeigenvalueswiththeexperimentalverticalionizationenergies ............................. 61 3-2ComparisonofthenegativeoftheOEPeigenvalueswiththeIP-EOM-CCSDionizationenergies ................................. 62 3-3ComparisonofthenegativeoftheHybrid-OEPeigenvalueswiththeexperimentalverticalionizationenergies ............................. 62 3-4ComparisonofthenegativeoftheHybrid-OEPeigenvalueswiththeIP-EOM-CCSDverticalionizationenergies ............................. 62 4-1Totalenergycomparison ............................... 74 4-2DF2OrbitalenergiesvsverticalIPs ........................ 75 4-3Core,ValenceandHOMOVIPsfromtheDF2eigenvalues ............ 75 4-4DF2OrbitalenergiesvsIP-EOM-CCSD ...................... 76 4-5DF2Orbitalenergiescomparison ......................... 76 4-6ComparisonoftheDF2Core,ValenceandHOMOeigenvalues ......... 77 5-1ComparisonofthenegativeoftheHOMOeigenvalues .............. 94 5-2ComparisonofthetotalenergiesinHartree .................... 95 6-1ComparisonoftheRPAcorrelationenergiesevaluatedusingdifferentreferenceorbitals. ........................................ 100 7-1ComparisonofthenegativeoftheBHLYPvwneigenvalueswiththeexperimentalverticalionizationenergies ............................. 108 7-2ComparisonoftheionizationenergiesobtainedbyESCFBHLYPvwn ......... 109 7-3ComparisonoftheionizationenergiesobtainedbyESCFBHLYP ........... 110 7-4ComparisonoftheionizationenergiesobtainedbyESCFgBHLYPvwn ......... 111 7-5DeviationsoftheverticalionizationenergiesfromthenegativeoftheHOMOcorrectedeigenvalue ................................. 111 7

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7-6BarrierHeightusingBHLYPvwndensityandBHLYPfunctional ......... 114 7-7BarrierHeightusingBHLYPvwndensityandHF+RPAfunctional ........ 115 7-8BarrierHeightusingBHLYPvwndensityandHF+RPA+T1functional ...... 116 7-9BarrierHeightsc-BHLYPvwnorbitalsandHF+RPAfunctional .......... 117 7-10BarrierHeightusingusingsc-BHLYPvwnorbitalsandHF+RPA+T1functional 118 7-11Comparisonoftheerrorsassociatedwithbarrierheightestimation ....... 119 8

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LISTOFFIGURES Figure page 1-1Percentagedeviationofthedensitymoments ................... 30 1-2CoupledClusterdensitydiagrams ......................... 35 1-3Singleamplitudediagrams ............................. 37 1-4Doubleamplitudediagrams ............................. 37 1-5Singleamplitudeorder-by-orderdiagrams ..................... 38 1-6Singleamplitudediagramswithimplicitdoublecontribution ........... 38 1-7CCDdoubleamplitude ................................ 38 1-8linCCDdoubleamplitude .............................. 39 1-9MBPT-2doubleamplitude .............................. 39 1-10rCCDdoubleamplitude ............................... 39 1-11exactexchangepotential .............................. 39 1-12correlationpotentialpart1 .............................. 39 1-13correlationpotentialpart2 .............................. 39 3-1PerformanceofDFTpotentialforalloccupiedorbitals .............. 59 3-2PerformanceofDFTpotentialforcoreoccupiedorbitals ............. 59 3-3PerformanceofDFTpotentialforvalenceoccupiedorbitals ........... 59 3-4PerformanceofDFTpotentialfortheHOMOorbitals ............... 60 4-1DF2orbitalenergiesvsverticalIPs ......................... 74 4-2DF2OrbitalenergiesvsEOM-CCSD ........................ 75 4-3DF2OrbitalenergiesDeviations .......................... 76 5-1CoupledClusterdensitydiagrams ......................... 86 5-2OEPexchangepotential ............................... 89 5-3OEPcorrelationpotentialparta. .......................... 89 5-4OEPcorrelationpotentialpartb. .......................... 89 5-5Heliumcorrelationpotentialcomparison ...................... 91 9

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5-6Beryliumcorrelationpotentialcomparison ..................... 92 5-7Neoncorrelationpotentialcomparison ....................... 92 5-8Argoncorrelationpotentialcomparison ....................... 93 5-9Magnesiumcorrelationpotentialcomparison ................... 93 6-1RPAcorrelationenergies .............................. 99 6-2BindingenergycurveforAr2 ............................ 103 6-3BindingenergycurveforAr2 ............................ 103 7-1ComparisonoftheerrorsassociatedwithbarrierheightestimationinKcal/mol 119 7-2RPABindingenergycurveforHe2usingdifferentorbitals ............. 120 7-3RPABindingenergycurveforNe2usingdifferentorbitals ............. 120 7-4RPABindingenergycurveforAr2usingdifferentorbitals ............. 121 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyINCREASINGTHEAPPLICABILITYOFDENSITYFUNCTIONALTHEORYByPrakashVermaAugust2011Chair:RodneyJ.BartlettMajor:Chemistry Accordingtodensityfunctionaltheory(DFT),thedensityisasufcientvariableforthedescriptionofdegenerateornon-degenerategroundstateofmolecules.Thedensityisgenerallyobtainedbysolvingtheself-consistentKohn-Sham(KS)equationsratherthanbydirectminimizationoftheenergy.AnattractivefeaturethattheKSprocedureoffersisitprovidesapictorialmolecularorbitalviewofthemoleculesandtheelectronicspectraintermsoftheKSorbitals.Besidestheirroleinfrontiermolecularorbitaltheory,certainquantitativefeaturesoftheseorbitalsshouldbepertinent.AccordingtoBartlett0stheorem,thenegativeofKSeigenvalueassociatedwithsuchanoccupiedorbitalsshouldbeagoodapproximationforitsprincipalverticalionizationenergy;however,inpracticeitdiffersconsiderably.HoweverSCFcalculations(doingtwodifferentKS-SCFcalculations,onefortheneutralandanotheroneforionicspecies)givereasonablycorrectionizationenergiesattestingtothegeneralaccuracyoftheKSfunctional.ThefailureofthemethodtorelateeigenvectorstoverticalionizationenergiesareduetothelackoftheconsistencyamongDFTthreekeyingredients(functional,potentialanddensity). Ourresearchhasbeenaimedatunderstandingtheseconsistencyconditions.Weknowthatthespatialbehavioroftheexplicitdensitydependentpotentialisincorrect(itdecaystoofastintheasymptoticregionsanddoesnotshowshelloscillations)anditdoesnotprovidecorrecteigenvalues,thusdensityobtainedfromtheeigenvectors 11

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couldbequestionableaswell.ToprovethisndingwedeviseaNonvariationalDFT,meaningitisnonvariationalinthesensethatthedensityisobtainedfromwave-functiontheoryratherthanbyminimizingtheDFTfunctional.TheBeckeexchangeandtheLYPcorrelationenergyevaluatedusingtheHartree-Fock(HF)densityaugmentedtheHFkineticenergyandclassicalcoulombinteractionisusedtoobtaintheDFTenergy.Wealsodevelopedananalyticalgradientschemeforsuchanapproachandshowedthatournon-variationalapproachprovidesmuchbettertransitionstatesandactivationenergiesthanvariationalDFT. Inordertoassessthequalityofthepotential,weusedBartlett'stheoremasourcriteriaandcomparedthenegativeoftheKSeigenvaluesobtainedusingthe52differentexplicitdensitydependentexchange-correlationpotentialintheKSoperatorwiththeprincipleverticalionizationenergies.NoneoftheexplicitdensitydependentpotentialprovideagoodapproximationoftheverticalionizationenergiesintermsofthenegativeoftheKSeigenvaluespectra.Even,theHOMOenergiesarefarfromthedesiredaccuracy.However,whentheKSoperatorhastheorbitaldependentexchange-correlationpotentialsuchasOEP2-sc,weobtainagoodapproximationoftheverticalionizationenergiesofthevalenceorbitalsintermsofthenegativeoftheKSeigenvaluespectra.Havinganequalamountoflocalexactexchangeandnon-local(theHartree-Focklikeexchangeoperator)exchangepotentialcoupledwithMBPT-2sccorrelationpotentialintheKSoperatorsprovideagoodapproximationoftheverticalionizationenergiesforalltheorbitalsintermsofthenegativeoftheKSeigenvaluesandshowtheimportanceofthenon-localexchangepotentialinthequestfortheconsistentDFTapproach.WehavegatheredmoreconcretenumericalresultsthatKSsorbitalsarenotmerelyanauxiliaryconstructtoobtainthedensityandthedensitygradient,butdohavemeaningandtheproperamountofnon-localexchangepotentialisimperativetogetcoreionizationenergiesasthenegativeoftheKSeigenvalues. 12

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Wealsoquestion,therecentprominentlyusedapproachinDFTcommunitytodesignaexchangeandcorrelationfunctional(e.g.M05,M06,B3LYP,B2-PLYP,DF2-Detc.)bycombiningthestandalonewellknownfunctionalsuchasSLATERexchange,BECKEexchange,Hartree-FockexchangePBEexchange,LYPcorrelation,MP2correlationetcwiththeparametersobtainedbydoingmindlessdatamining.Wecallthisana,b,capproachoftheDFTandshowrstbyconstructingapotentialcorrespondingtooneofthemostwidelysuccessfulfunctionalB2-PLYPorthedoublehybridandaskingthequestionifitprovidesaconsistentpotential,anabsoluteaveragedeviationof2eVforionizationenergiesprovesthatthepotentialobtaineddoesnothavesamedegreeofexactnessasthefunctionalhas,secondbyconstructingourownpotentialinsteadofafunctionalthatwillprovideverticalionizationenergiesintermsofthenegativeoftheKSeigenvaluesspectrawithinadesiredaccuracyof1eV,thattheseapproachesarenothingbutanelegantwaysofsuppressingtheinheritdecienciesoftheexplicitdensitydependentexchange-correlationfunctionalorthepotential. Theexactexchange-correlationfunctionalcanbeobtainedusingtheadiabaticconnectionandtheuctuationdissipationframeworkviaanintegrationoverthecouplingconstantandthedynamicdensity-densityresponsefunction.Thevariousapproximationsofthedynamicdensity-densityresponsefunctioncanbeobtainedusingthenon-interactingdensity-densityresponseandtheexchange-correlationkernel.Theunperturbeddensity-densityresponsefunctionafterdoingthecouplingconstantintegrationgivestheexactexchangeenergy(non-localexchangeenergy)intermsofreferenceorbitals.Andthesimplestapproximationofthecouplingconstantcanbeobtainedbyignoringthekernel,whichisoftencalledtherandom-phaseapproximation.UsingtheplasmonmodelortheequivalencebetweentheRPAandthering-CCD,frequencyintegrationandcouplingstrengthintegrationcanbeavoided.UsingourOEPprocedure(byinsistingthatthedensitycorrectiontheKSdeterminantshouldvanish)weobtainring-CCDcorrelationpotentialwhichisequivalenttotheRPAcorrelationpotential. 13

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Shelloscillationsdemonstratedbytheexactpotential,canbereproducedbytheRPApotentialaswellastheMBPT-2potential.TheRPAcorrelationpotentialhasacorrectlongrangebehavior.WealsoimprovedtheRPApotentialbyexploitingtherelationshipbetweenRPAandcoupledclustertheory,consideringvariouswaystoincludesinglesexcitationeffects,andadditionaldoubleexcitationdiagrams.PotentialsobtainedshowedsimilarspatialbehaviorastheRPApotentialexcepttheyarecorrectinthenearnucleusregionaswell.ThepotentialarecomputationallydemandingbutintherightdirectiontowardsaconsistentDFTasnotonlypotentialshowcorrectspatialbehaviorandHOMOenergiesarewithindesiredaccuracybutalsothetotalenergies. Theself-consistentKScalculationwiththeRPAcorrelationpotentialintheKSoperatorisalwaysavoidedasitistoocomputationallydemandingandtheRPAenergiesareobtainedusingsomesetsofreferenceorbitals.ItisarguedthattheKSorbitalsobtainedfromtheexplicitdensitydependentpotential(e.g.PBE)intheKSoperatorarebetterthantheHartree-FockorbitalsfortheRPA.WehaveinvestigatedthatargumentandndthatfortheRPAcorrelationenergyevaluation,choiceofthePBEorbitalsvstheHartree-Fockorbitalsareirrelevantunlessonedotheproblemincorrectlyinotherwords,howtodeneRPAistime-dependentHartree-FockwithexchangebeingneglectedorthetimedependentDFTwithnokernelinlinearresponsefunction?TheRPAwiththetime-dependentDFTframeworkoverestimatetheRPAcorrelationenergyfourfoldcomparedtotheRPAwiththetime-dependentHFframeworkusingeitherthePBEortheHForbitals.Thebindingenergycurveforargondimerrevealsurprisingresults,onewithincorrectRPAcorrelationenergyprovidethebestanswer.Wehavealsoobtainedresultsbyincludingthesinglescontributionsandobtainedthesimilarbehavior. 14

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CHAPTER1INTRODUCTION 1.1ComputingElectronicStructureisaDifcultProblem SincetheinteractionsthatdescribeatomsandelectroninmoleculesareknownthroughtheSchrodingerequation,computingmolecularpropertiesanddesigningmaterialbecomesidealresearchtoolsduetocostandversatility.)]TJ /F3 11.955 Tf 9.29 0 Td[(i@ @t(~x,t)=^H(t)(~x,t), (1) ^H=)]TJ /F4 7.97 Tf 12.71 14.95 Td[(elecXi=11 2r2i)]TJ /F4 7.97 Tf 11.95 14.95 Td[(nucl.XA=11 2r2A)]TJ /F4 7.97 Tf 12.14 14.95 Td[(elec.Xinucl.XAZA j~ri)]TJ /F11 11.955 Tf 12.6 3.16 Td[(~RAj+elecXi
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withrespecttothechangeofthecoordinatesofanytwoelectrons(Pauliexclusionprinciple)[ 2 ]. j(~x)i=Xkck(^AjNeYi=1 ki(~ri,i))i=Xkck(^AjNeYi=1j ki(~ri)(i))i, (1) j ii=No.ofbasisXciji, (1) ji=Xdji=Xdxmynzle)]TJ /F13 7.97 Tf 6.59 0 Td[(j~r)]TJ /F13 7.97 Tf 7.02 2.1 Td[(~RAj2, (1) ^A=N!Xn=1()]TJ /F5 11.955 Tf 9.29 0 Td[(1)0n^Pn. (1) Toretainthecorrectfermionsymmetry,theantisymmetrizeroperator^A( 1 )isintroducedwhichkeepstheproductofthemolecularspinorbitalsantisymmetricwithrespecttoapermutationofthelabelsofanypairofelectrons(operatorPnintheequation( 1 )generatethenthpermutationoftheelectronlabels1,2,3,..N)withthepropersignchange(pnintheequation( 1 )isthenumberoftranspositionsorthesimpleinterchangesrequiredtoobtainthepermutation).Since,theHamiltonianisspin-independentandcommuteswiththeprojectionofspinontothez-direction,theSzoperator,theeigenfunctionofSzwillalsobeeigenfunctionoftheHamiltonianandspin-orbitalscanbewrittenastheproductofspatialandspinfunctions(Equation( 1 )).Themolecularorbitalsareexpandedintoabasisset.AtomicorbitalsorthesolutionsofthehydrogenicSchrodingerequationareconsideredthebestchoiceofsuchabasis.However,theseatomicfunctionsinvolveexponentialfunctionsandevaluatingtheresultingmulti-centeredintegralsformoleculesbecomescomputationally 16

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expensive.Primitivegaussianscanbecontractedinasuitablemannertoreplacetheexponentialfunction,andintegralevaluationbecomeeasier,butduetothenatureofthegaussianfunction,itfallsoffrapidlyattheasymptoticregionandhasacuspproblemonthenucleus[ 3 ].Thusasuitablechoiceofthebasissetbecomecrucial.UsingtheseexpansionstheexactelectronicwavefunctioncanbewrittenasalinearcombinationofallpossibleN-electronSlaterdeterminantsfromthecompletesetofthespin-orbitals( 1 ). Approximatingtheexactelectronicwavefunctionbythesingledeterminantandthetroublinginter-electronicpartoftheHamiltonianbyasumofthetermscorrespondingtoeachelectronmovinginsomeaverageeldofalltheotherelectrons,onecanconvertthemany-bodyproblemintoasingleparticleproblem.Theresultingsingleparticleequationoffersanattractiveadvantageasthepictorialviewofthemoleculecanbeobtainedintermsofthesingleparticlewavefunctionconsistingofmolecularorbitals.Hartree-Focktheoryisanexampleofsuchasingle-particletheory.However,thereisnocorrelationamongtheelectrons[ 4 ]. TheKohn-Sham(KS)densityfunctionaltheoryisanotherexampleofasingleparticletheoryandallthemany-bodyeffectsareformallyembeddedintotheexchangecorrelationoperator[ 5 6 ].Densityfunctionaltheory(DFT)isapotentialwaytoavoidsolvingtheSchrodingerequationandtoprovideinformationaboutthemoleculebyjustknowingtheexactelectronicdensity[ 7 ].Since,theexactdensityofthefullyinteractingsystemcanbeobtainedbysolvingthesingle-particleKSequations[ 8 ],usingDFTtocalculatetheelectronicpropertiesofrealmaterialshasnowadaysturnedintoanomnipresentendeavor. 1.2DensityFunctionalApproachtoMany-BodyElectronicStructureProblem ThedensityFunctionalmethodisanapproachwherethedensityisthemainvariable,asoneonlyneedspartialinformationofthefullwavefunctioninordertoobtainthetotalgroundstateenergyandmostofitsproperties.Sincetheelectronic 17

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Hamiltonian( 1 )containsone-particleandtwo-particleoperators,intheprocessofevaluatingtheexpectationvalueoftheHamiltonianmostofthedegreesofthefreedomareintegratedfromtheequation.Electronareindistinguishable,thusthenuclear-electronattractionoperatorcanbewrittenasasumofsingleparticle(Equation( 1 ),usingthedenitionofthelocaloperator(Equation( 1 ))andthedensityoperator(Equation( 1 )),nuclear-electronattractionenergy(Equation( 1 )and( 1 ))canbeobtainedintermsoftheelectronicdensity(Equation( 1 )). E=hj^Hji, (1) ^O(r,r0)=^O(r)(r)]TJ /F10 11.955 Tf 11.96 0 Td[(r0), (1) ^(r)=NXi=1(r)]TJ /F10 11.955 Tf 11.95 0 Td[(ri), (1) (r)=hj^(r)ji=Zdr2....drNd2.....dNj(r,;r2,2;....,rN,N)j2, (1) ENuclear)]TJ /F4 7.97 Tf 6.59 0 Td[(electron=hj^Vextji=hjNXi=1^vext(ri)ji, (1) ^vext(ri)=)]TJ /F4 7.97 Tf 11.29 14.95 Td[(No.ofAtomsXA=1ZA jri)]TJ /F10 11.955 Tf 11.95 0 Td[(RAj, (1) 18

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ENuclear)]TJ /F4 7.97 Tf 6.59 0 Td[(electron=hjNXi=1^vext(ri)ji=Zdrvext(r)hj(r)]TJ /F10 11.955 Tf 11.96 0 Td[(ri)ji=Zdrvext(r)hj^(r)ji=Zdrvext(r)(r), (1) Similarly,thekineticenergyoftheelectrons( 1 )canbewrittenintermsoftheone-particledensitymatrix( 1 ),thediagonalofwhichisequaltotheelectronicdensity(Equation( 1 )and( 1 )).EKinetic)]TJ /F4 7.97 Tf 6.58 0 Td[(Energy=)]TJ /F5 11.955 Tf 10.49 8.09 Td[(1 2NXi=1hjr2iji=)]TJ /F5 11.955 Tf 10.49 8.09 Td[(1 2Z(r1,1;...rN,N)r21(r1,1;...rN,N)dr1,d1,...drN,dN=)]TJ /F5 11.955 Tf 10.49 8.09 Td[(1 2X,0Z[r21,0(r,r0)](r=r0)dr, (1) (r)=1,(r,r), (1) (r)=X(r), (1) Theelectron-electroninteractionenergy(Equation( 1 ))canbeobtainedintermsofthediagonaloftwo-particledensitymatrix(Equation( 1 )). 19

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Eelectron)]TJ /F4 7.97 Tf 6.58 0 Td[(electron=1 2NXi6=jhj1 jri)]TJ /F10 11.955 Tf 11.96 0 Td[(rjjji=1 2Zdx1dx2Z(r1,1;r2,2;..xN)1 jr1)]TJ /F10 11.955 Tf 11.95 0 Td[(r2j(r1,1;r2,2;..xN)dx3..dxN=1 2X1,2Z21,2(r1,r2)1 jr1)]TJ /F10 11.955 Tf 11.95 0 Td[(r2jdr1dr2, (1) 1,0(r,r0)=Z(r,;x2;..xN)(r0,0;x2;..,xN)dx2..dxN, (1) 2,0(r,r0)=Z(r,;x3;..xN)(r0,0;x3;..,xN)dx3..dxN, (1) TheexpectationvalueoftheelectronicHamiltoniancanbeexpressedintermsofone-particledensitymatrix( 1 )andthediagonalofthetwo-particledensitymatrix(Equation( 1 ))as E=)]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2X,0[r21,0(r,r0)]r0=rdr+Zdrvext(r)(r)+1 2X,0Z1 jr)]TJ /F10 11.955 Tf 11.96 0 Td[(r0j2,0(r,r0)drdr0, (1) UsingtheHellman-Feynmantheoremonecantransformthecalculationofthetotalkineticenergyoftheelectronsintoacalculationofthekineticenergyofnon-interactingelectronplusasmallcorrectiontothecorrelationpartoftheinteractionenergy.ThisisdonebyscalingdownthestrengthoftheCoulombinter-electroninteractionoftencalled-integrationoradiabaticconnectionframework,thatwillreducethescreeningbytheotherelectronsoftheattractionofoneelectrontothenucleus.Thatwillhaveadrastic 20

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effectontheelectrondensityasthedensitywillpileupatthenucleus.Thereforeanexternaldependentone-bodypotentialisintroducedinthesysteminsuchawaythatthedensityisnotaffectedbythereductionintheinter-electronCoulombinteraction.ResultingexpectationvalueoftheelectronicHamiltonian( 1 )canbeexpressedintermsoftheeigenfunctionoftheone-electronSchrodingerequation( 1 ))thediagonalofthetwo-particledensitymatrixofthereducedinter-electroninteraction. E=)]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2XNXi+Zdrvext(r)(r)+1 2X,0ZdrZdr0Z10d1 jr)]TJ /F10 11.955 Tf 11.96 0 Td[(r0j2,0(r,r0), (1) WecanndE0,thetotalenergy( 1 )ofasystemofNnon-interactingelectronsmovingintheexternalpotentialv0ext(~r)andhavingthesamedensity( 1 )asthefullyinteracting(=1)system,bysolvingtheone-electronSchrodingerequation( 1 ). f)]TJ /F5 11.955 Tf 16.47 8.08 Td[(1 2r2+v0ext(r)gp="pp, (1) E0=NXk=1"i=NXi=1hijr2jii+Zv0ext(r)(r)dr, (1) (r)=NXk=1jij2, (1) TheenergyEofthesystem,whichischangedduetoreductionoftheinter-electroninteractioncanbeobtainedbyusingtheHellman-FeynmantheoremintermsofE0.ThederivativeofEwithrespecttocanbeobtainedbydifferentiatingtheexplicit 21

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dependentpartoftheHamiltonian.DuetothestationarypropertiesofthegroundstateenergyE,theexplicit-dependanceoftheground-statewavefunctiondoesnotappearintheequation. @E @=hj@H @ji=hj@^ext @ji+hj^e)]TJ /F4 7.97 Tf 6.59 0 Td[(eji, (1) Since,thedependenceinvext(~r)isdenedsuchthatthedensityofthefullyinteractingsystematallisequaltothenon-interactingelectrondensity,integratingwithrespecttoweobtainEintermsofE0(Equation( 1 )) E)]TJ /F3 11.955 Tf 11.95 0 Td[(E0=Z[ext(r))]TJ /F3 11.955 Tf 11.96 0 Td[(v0ext(r)](r)dr+1 2X,0ZdrZdr0Z10d1 jr)]TJ /F10 11.955 Tf 11.95 0 Td[(r0j2,0(r,r0), (1) Wecanfurthersimplifyourenergyequationbyexpressingthediagonalofthetwo-particledensitymatrix(Equation( 1 ))intermsoftheproductsofthedensitiesandthepaircorrelationfunctiong(Equation( 1 )).Paircorrelationfunctiongisthemeasureofthecorrelation.Iftheelectronsarecompletelyindependent,thengisequalto1andthetwoparticledensitymatrixisjusttheproductofone-electrondensities.Theamountofcorrelationismeasuredbyhowmuchsmallergisfromunity. 2,0(r,r0)=(r)0(r0)g,0(r,r0), (1) 2,0(r,r0)=(r)0(r0)~g,0(r,r0), (1) ~g(r,r0)=Z10g(r,r0)d, (1) 22

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ItisevidentfromEquation( 1 )thatthetotalenergyofaninteractingmany-electronsystemcanalwaysbefoundbydoingone-electrontheory.However,onemustndtheaone-electronpotentialv0ext(~r)thatwillgivethecorrespondingnon-interactingdensityequaltothedensityoftheinteractingsystem(Equation( 1 ))andaveragepair-correlationfunction~g(Equation( 1 )).OnecanseparatetheclassicalcoulombinteractiontermandwritetheenergyexpressionasEquation( 1 ),wherethelasttermhasalltheexchange-correlationeffects. E=)]TJ /F5 11.955 Tf 10.49 8.08 Td[(1 2NXihijr2jii+Zdrvext(r)(r)+Zdrdr01 jr)]TJ /F10 11.955 Tf 11.95 0 Td[(r0j(r)(r0)~g(r,r0), (1) E=)]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2NXi+Zdrvext(r)(r)+Zdrdr01 jr)]TJ /F10 11.955 Tf 11.95 0 Td[(r0j(r)(r0)+Zdrdr01 jr)]TJ /F10 11.955 Tf 11.95 0 Td[(r0j(r)(r0)(~g(r,r0))]TJ /F5 11.955 Tf 11.96 0 Td[(1), (1) Abovementioned,one-electrontheoryisnothingbuttheKohn-Shamone-electronformulationofthedensityfunctionaltheory,whichpostulatestheexistenceofanauxiliarysystemofNnoninteractingelectronsmovinginanexternallocalpotentialvks(~r),whosesingledeterminantwavefunctionconsistsofthelowestNone-electronorbitalswhichprovidesadensityequaltotheexactdensityoftheinteractingelectronsystemwithpotentialvext(~r).Thus,theKohn-ShamHamiltonian(Equation( 1 ))isjustasumofone-electronHamiltonians. 23

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^Hks=Xi^hks=Xi)]TJ /F5 11.955 Tf 10.49 8.09 Td[(1 2r2(i)+vks(~ri)^hks(1)i="ii(1)=j1(1)2(2)...N(N)jks(r)=NXiXsigmaji(r,)j2=exact(r), (1) TherstHohenbergKohntheorem[ 7 ],Forisolatedmany-electronsystem,itsground-stateone-electrondensity(r)determinesuniquelytheexternalpotentialvext(r)(towithinatrivialconstant),establishestheuniqueexistenceofvks(~r),thatthereexistanuniquevks(~r)withthepropertythattherstNoccupiedorbitalsdensitiessumuptotheexactdensity.Itisdifculttoproveinanexactwaythatitisalwayspossibletondasuchexternalpotential.Thisisknownasthenon-interacting-representabilityofthedensityintheDFTliterature.However,inpracticeitisoftenassumedthattoadesireddegreeofaccuracytheexistenceofthevks(~r)isgranted. UsingthesecondHohenbergKohntheorem[ 7 ],theground-stateenergyofamany-electronsystemcanbeobtainedastheminimumoftheenergyfunctional.Ignoringtheelectron-electroninteraction,thekineticenergyofthenon-interactingelectronTocanbewrittenas(Equation( 1 )).ThegroundstatedensityisthedensitythatminimizestheT0andsatisfytheEulerequation(Equation( 1 )),wheretheLagrangemultiplier0isassociatedwiththeconstraintR(r)dr=N. T0=hj^Tji=Xihij)]TJ /F5 11.955 Tf 19.13 8.08 Td[(1 2r2ijii, (1) T0 (r)=)]TJ /F3 11.955 Tf 9.3 0 Td[(vks(r)+0, (1) 24

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Theenergyfunctional( 1 )canbewrittenintermsoftheuniversalHohenbergKohnfunctionalFHK( 1 ).ThefunctionalFHKisuniversalinthesensethatitdoesnotdependontheexternalpotential.Minimizingtheenergyfunctionalprovidestheexactground-statedensitywhichalsosatisestheEulerequation( 1 ),whereistheLagrangemultiplierassociatedwiththeconstraintR(r)dr=N.ByinsertingEquation( 1 )and( 1 )intothefunctionalderivativeoftheFHKfunctional,weobtainthevkspotential( 1 ). E[]=FHK[]+Z(r)vext(r)dr, (1) FHK[]=T0[]+Zdrdr01 jr)]TJ /F10 11.955 Tf 11.95 0 Td[(r0j(r)(r0)+Exc[], (1) Exc[]=Zdrdr01 jr)]TJ /F10 11.955 Tf 11.96 0 Td[(r0j(r)(r0)(~g(r,r0))]TJ /F5 11.955 Tf 11.95 0 Td[(1), (1) FHK =)]TJ /F3 11.955 Tf 9.3 0 Td[(vext[r]+, (1) ^Vks(r)=^vext(r)+^VH(r)+^vxc(r)+, (1) TheHartreepotentialVH( 1 )intheequation( 1 )isjustthefunctionalderivativeoftheclassicalcoulombinteractionenergy.While,Vxctheexchange-correlationpotentialisthefunctionalderivativeoftheexchange-correlationenergyExcwithrespecttothedensity( 1 ). 25

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VH[r]=Z1 jr)]TJ /F10 11.955 Tf 11.96 0 Td[(r0j(r0)dr0, (1) vxc[r]=Exc (r), (1) Weobtaintheexactexpressionfortheexchange-correlationpotentialVxc[](Equation( 1 ))usingtheexactexpressionfortheexchange-correlationfunctional(Equation( 1 ))andthefunctionalderivativerelation(Equation( 1 )) vxc[r]=Z(r0)(~g(r,r0))]TJ /F5 11.955 Tf 11.95 0 Td[(1)1 jr)]TJ /F10 11.955 Tf 11.95 0 Td[(r0jdr0+1 2Z(r)(r0)~g(r,r0) (r)1 jr)]TJ /F10 11.955 Tf 11.96 0 Td[(r0jdrdr0, (1) InsteadofobtainingVxc[]explicitlyintermsofaverage-pair-correlationfunction~ganditsfunctionalderivativewithrespecttodensity,asdoneintheequation( 1 ),allwidelyusedDFTapproaches(e.g.LDA,GGA,meta-GGAetc)predominantlyconstructsomereasonableapproximationtotheexchange-correlationfunctionalExc[]intermsoftheexplicitdensitydependentquantities(e.g.density,gradientofdensity,Laplacianofthedensityetc.)andVxc[]isobtainedbyfunctionallydifferentiatingtheapproximateExcwithrespecttothedensity.Thus,Vxchasanimplicitdependenceontheaverage-pair-correlationfunction~ganditsfunctionalderivativewithrespecttodensity.Forexample,manyapproximateexchange-correlationfunctionalsExc[]havealocaloragradientcorrectedform( 1 )[ 9 17 ].(wedescribeanyfunctionalaslocalifitcanbewrittenastheexpectationvalueofaone-electronoperatorasitincludesgradientcorrectedfunctionals,whichinDFTliteratureoftencallednon-local.Inourview,intrinsicallymulti-electronfunctionalsarethenon-local,e.g.Hartree-Fockexchangefunctional) 26

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Exc=Zf(,,,,)dr, (1) =jrj2,=r.r,=jrj2, (1) Thegeneralformoftheexchange-correlationfunctionalisgiveninthe( 1 ),wherefisthefunctionofthe,andtheirgradients,and.Also,andarespinvariables.Thecorrespondingone-electronexchange-correlationpotentialVxcisobtainedbyusingthecalculusofvariation[ 5 ].Thepartoftheone-electronexchange-correlationpotentialisgivenbyEquation( 1 ),andsimilarexpressionforthepartcanbeobtained. Vxc=f )]TJ /F5 11.955 Tf 11.95 0 Td[(2r.(f r))-222(r.(f r), (1) TheBecke[ 18 ]gradient-correctedexchangefunctionalissuchanexample,whichhastheformfBeckex=4=3Bg(x)+4=3Bg(x),whereBg(x)=)]TJ /F7 7.97 Tf 10.49 4.71 Td[(3 2(3 4)1=3)]TJ /F4 7.97 Tf 31.82 4.71 Td[(bx2 1+6bxsinh)]TJ /F16 5.978 Tf 5.76 0 Td[(1xwithx= 4=3,x= 4=3andbisanempiricalparameter.Corresponding,Beckeexchangepotentialcanbeobtainedbyusingtherelevantderivative@fBeckex @=4 34=3[Bg(x))]TJ /F9 11.955 Tf -423.3 -23.91 Td[(x_Bg(x)],@fBeckex @=1 2)]TJ /F7 7.97 Tf 6.58 0 Td[(1=2_Bg(x),and_Bg(x)=@Bg(x) @=6b2x2[x=(x2+1)1=2)]TJ /F4 7.97 Tf 6.59 0 Td[(sinh)]TJ /F16 5.978 Tf 5.75 0 Td[(1x])]TJ /F7 7.97 Tf 6.59 0 Td[(2bx (1+6bxsinh)]TJ /F16 5.978 Tf 5.76 0 Td[(1x)2. Obtainingtheexchange-correlationpotentialfromthecorrespondingexchange-correlationfunctionalbyfollowingtheaboveprocedureasdonefortheBeckeexchangepotential,denitelyhascertaincomputationaladvantages.Buttherearealsomanydisadvantagessuchasfunctionaldifferentiationbecomessingularatcertainpointsanditcoulddrivedensityfarfromcorrectvalues. ApartfromtheexactexpressionfortheVxc[]intermsoftheaverage-paircorrelationfunctionanditsfunctionalderivativewithrespecttothedensity,thereare 27

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differentexactexpressionsavailabletoobtainthepotential(Vxc[])separatelyfromthefunctional(Exc[]).Forinstance,theSham-Schlulterequation[ 19 ]offersawaytoconstructionanaccurateexchange-correlationpotential(Vxc[]).TheOptimizedEffectivePotential(OEP)ortheOptimizedPotentialMethod(OPM)areothers[ 20 ].Therearethreewaysonecanobtainthepotential.Firstthedirectfunctionalderivativeway,secondthetotalenergyminimizationusingtheSharp-Hartoncondition[ 21 ]andthirdthedensityinvariancecondition[ 22 ].(Moredetaileddiscussioninsubsection 1.4 ) 1.3DesiredPropertiesofvks Knowingvks( 1 ),inprinciple,providesanopportunitytoknowtheexactsolutionofthemany-electronproblembyjustdoingaone-electroncalculation( 1 ).Duetotheabsenceoftheexactanalyticalformofthevks,ithastobeapproximatedandapproximatevksmustsatisfyceratinpropertiesinordertohaveagoodapproximationoftheexactvks. f)]TJ /F5 11.955 Tf 16.47 8.09 Td[(1 2r2+^vks(r)gp="pp, (1) ^vks(r)=^vext(r)+^VH(r)+^vxc(r)+, (1) 1. vksisauniquelocal/non-localpotentialconnectingthecorrelatedsingleparticleproblemtointeractingmanybodyproblem.Whichissubjectedtothe-representabilityproblemofDFT.Forallpracticalpurposesitisoftenassumedonecansatisfytheconditiontoadesirednumericalaccuracy. 2. Inprinciplevksshouldyieldstheexactone-electrondensity(~r).Duetotheabsenceofagoodapproximationfortheexchange-correlationpotential,itishardtoattain.Thewidelyusedexchange-correlationpotentialhasanexplicitdensity 28

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dependance,forexampletheBeckeexchangepotentialandLYPcorrelationpotential.ThedensityobtainedusingtheBLYPpotentialintheVkspotentialprovidesadensityinferiortotheuncorrelateddensityoftheHartree-Fock.Forexample,Table 1-1 comparedthedensitymomentmoment=Prni(ri)oftheBLYPandHFwiththeCCSD(T)densitymoments.AsevidentfromTable 1-1 ,theHFdensityislittlebetterthantheBLYPdensity.Chapter 2 elaboratesthisndinginmoredetail.There,wetaketheHartree-FockdensityandobtainedBLYPfunctionalvaluestoobtainbarrierheightsofthereactionandcompareitthebarrierheightobtainedusingtheBLYPself-consistentdensityandthecorrespondingBLYPfunctional.ItisinterestingtonotethatanonlocalpotentialsuchasHFexchangeandlocalpotentiallikeBeckehasanoppositeeffectonthedensity 1-1 .ThisisoftenusedinDFTtosuppressthedeciencyofthefunctional,asshowninthechapter 4 thatalthoughonecanhidethedecienciesofthefunctionalbythesemi-empiricalcombinationofthefunctionalbutitisverypronouncedinthepotential.WehavealsoshowninChapter 7 thatispossibletosuppressthedeciencyofexplicitdensitydependentpotentialaswell,butoneloosestheconsistencyconditionbetweenthefunctionalandthepotential. Table1-1. ComparisonofthedensitymomentsoftheBLYPandHFwithrespecttoCCSD(T) Methodsr)]TJ /F7 7.97 Tf 6.59 0 Td[(2r)]TJ /F7 7.97 Tf 6.59 0 Td[(1rr2r3r4r5BLYP415.0931.108.019.8816.1533.3184.20HF414.5031.117.899.3714.3927.2061.62CCSD(T)414.8831.107.969.6215.1629.6569.86 3. "HOMO=)]TJ /F3 11.955 Tf 9.3 0 Td[(IPexact[ 23 ].Theasymptoticbehavioroftheexactdensityisgovernedbytherstionizationenergy((~r)=e)]TJ /F7 7.97 Tf 6.59 0 Td[(2p 2Ip~rforlarge~r)[ 24 ],thusthenegativeofthehighestoccupiedmolecularenergyshouldbeequaltotherstexactverticalionizationenergy"HOMO=)]TJ /F3 11.955 Tf 9.3 0 Td[(IPexact.InChapter 3 westudiedmorethan50widely 29

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Figure1-1. Percentagedeviationofthedensitymoments usedexplicitdensitydependentpotentialsandtocompareHOMOeigenvalueswiththeverticalionizationenergiesandfoundthatnoneoftheexplicitdensitydependentpotentialprovideHOMOenergiestoadesiredaccuracy.However,goodpotentialslikeOEPpotentialprovideexcellentresults. 4. Fromthetime-dependentdensityfunctionaltheoryarguments,theKSorbitalenergydifferencesaretherstapproximationoftheexcitationenergies.ThusallthevirtualorbitalswhicharethesolutionoftheSCF-KSprocedurecontainsphysicalmeaning[ 25 ]. 5. "i)]TJ /F3 11.955 Tf 22.81 0 Td[(IPexact.AccordingtoBartlett'stheorem[ 26 ]UsingTDDFTintheadiabaticapproximation,andsimplymakingtheassumptionthatanelectronisexcitedintothecontinuum,subjecttoalocalkernelandusingthefactthatallintegralsinvolvingcontinuumandoccupiedorbitalsvanishfromtheAandBmatrices,theeigenvaluesoftheKSequationsusingrealisticOEPbasedpotentialsshouldap-proximatelycorrespondtoionizationpotentials.Inchapter 3 weprovidenumericalproofofthetheorem.ItisinterestingtonotethatonecanwritethetotalenergyasPni"iminuscorrectiontermsfordoublecountingoftheelectron-electroninteraction,E=Pni"i)]TJ /F12 11.955 Tf 12.3 9.63 Td[(R(1)(2)=r12d1d2+(Exc)]TJ /F12 11.955 Tf 12.29 9.63 Td[(Rvxcd).IhaveshowninChapter 2 thatthecorrelatedKSdensityisinferiortotheuncorrelatedHFdensity 30

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andinChapter 3 IhaveshownnoneoftheDFTfunctionaleigenvaluesareclosetotheexpectedvalue,obtainingreasonabletotalenergydeantlyinvolveserrorcancelation.Thus,itishardtotrusttheexplicitdensitydependentfunctionalwhoseorbitalenergiesarenotwithinthedesiredaccuracy. 6. VksandcomponentsofVkshavecharacteristicspatialbehavior,suchasshelloscillationofcorrelationpotentialinatoms. 1.4HowtoObtainthexc-PotentialsUsingtheOEPProcedure Duetomanyshortcomingsoftheprominentlyusedexplicitdensitydependentfunctionals[ 27 ]suchasLDA,GGA,meta-GGA,etc.thereisaneedtoexploitinterconnectionsbetweenthedensityfunctionaltheoryandthewavefunctiontheorytoobtainbestofthebothworldsasinabinitioDFT[ 26 ].Since,theKohn-ShameigenvaluesandeigenfunctionsprovideinformationthatcannotbeeasilyextractedfromtheKS-densityanditsgradient,ithasbeenarguedthatexchange-correlationfunctional(xc-functional)obtainedfromthewavefunctionapproachprovidebetterapproximationstotheunknownexchange-correlationfunctionalthanthatoftheexplicitdensitydependentfunctionalssuchasLDA,GGA,etc.Exchange-correlationfunctionalsobtainedfromcoupled-clusterandmany-bodyperturbationtheoryprovidearoutetodeneaseriesofDFTmethodsthatwillconvergetotherightanswerinthelimitofbasisandcorrelationandnotsufferfromtheshortcomingoftheexplicitdensitydependentfunctionalssuchself-interactionerrorandlackofdispersioninteraction[ 22 ].UsingOEP(OptimizedEffectivepotential)procedurecorrespondingeffectiveone-particleexchange-correlationpotentialscanbeobtained.Assumingthexc-functionalhasthegeneralformExc[p,"p],thepotentialcanbeobtainedusingthefollowingthreeways. 1. DirectFunctionalderivative 2. TotalenergyminimizationusingSharp-Hartoncondition 3. Exactdensityordensityinvariancecondition 31

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1.4.1DirectFunctionalDerivative Thesimplestwaytoderivethexc-potentialcorrespondingtotheorbitaldependentxc-functionalortheOEPequationisthetransformationofthefunctionalderivative(Equation( 1 ))intoaderivativewithrespecttopand"p,usingthechainruleforfunctionaldifferentiation.Theideaistodetermineforwhichderivativetheexpressionsareknown,andexpressthederivativeofinterestintermsoftheknownderivatives.Since,theanalyticalexpressionfortheexchange-correlationfunctionalintermsoftheorbitalsandorbitalenergiesareavailable,thederivativeoftheenergieswiththeorbitalsandorbitalenergiescanbeobtaineddirectly.TheresponseoftheorbitalsandorbitalenergiestoainnitesimallysmallchangeinthepotentialisreadilyavailablethroughthelinearresponseKohn-Shamequationsandsoistheresponseofthedensity. Vxc(r)=Exc[p,"p] (r)=Zdr0vks(r0) (r)Exc vks(r0)=Zdr0vks(r0) (r)XpfZdr"[yp(r") vks(r0)Exc yp(r")+c.c.]+"p vks(r0)Exc "pg, (1) Thefunctionalderivativeyp=vksand"p=vksareevaluatedbyvaryingvksinnitesimallyandlookinghowpand"preact.Usingrstorderperturbationtheoryweobtain.yp(r) vks(r0)=Xq6=pyp(r0)q(r0) "p)]TJ /F11 11.955 Tf 11.96 0 Td[("qq(r), (1) "p vks(r0)=yp(r)p(r), (1) 32

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Similarly,thestaticKSresponsefunctionareobtained.(r) vks(r0)=ks(r,r0)=XpXqyp(r)q(r)yq(r0)yp(r0) "p)]TJ /F11 11.955 Tf 11.95 0 Td[("q, (1) Bymultiplying( 1 )byksandintegratingoverr,oneobtainstheOEPequation,whichisFredholmequationoftherstkind.Zdr0ks(r,r0)Vxc=!(r)=Exc vks(r), (1) !(r)=XpfXq6=pyp(r)q(r) "p)]TJ /F11 11.955 Tf 11.96 0 Td[("qZdr0[yq(r0)Exc yp(r0)+c.c.]+jk(r)j2Exc "pg, (1) Equation( 1 )isthecentralequationoftheOEPprocedure.Itallowsthecalculationofthevxcpotential.Also,theOEPequation( 1 )islinear,soeachcomponentofExccanbetreatedseparately.OEPiscomputationallydemandingcomparedtoexplicitdensitydependentfunctionalsuchasLDA,GGA,etc.asonehastosolveOEPequationateachSCFiteration. 1.4.2TotalEnergyMinimizationUsingSharpandHartonCondition AccordingtoSharpandHartoncondition[ 21 ],thestandardminimizationofEtotwithrespecttocanbesubstitutedbyaminimizationwithrespecttovks.Etot[k,"k] vks(r)=0, (1) Thederivativein( 1 )canheobtainedasin( 1 ), Etot[p,"p] vks(r)=XpfZdr"[yp(r") vks(r0)Etot yp(r")+c.c.]+"p vks(r0)Etot "pg, (1) 33

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Inadditiontotheingredientswhicharealreadyknown,( 1 )containsthefunctionalderivativeofEtotwithrespecttopand"p,whichcanbeeasilyevaluated.Etot yp(r)=[)]TJ 10.5 8.09 Td[(r2 2+vext(r)+vH(r)]p(r)+Exc yp(r), (1) Etot "p=Exc "p, (1) OnecanthenusetheKSequationtorewriteEtot=yp,Etot yp(r)=["p)]TJ /F3 11.955 Tf 11.95 0 Td[(vxc(r)]p(r)+Exc yp(r), (1) FinallyweobtainXpZdr0fyp(r)Xq6=pq(r)q(r0) "q)]TJ /F11 11.955 Tf 11.95 0 Td[("p[p(r0)(vxc(r0))]TJ /F11 11.955 Tf 11.95 0 Td[("p)+Exc yp(r0)]+c.cg+Xpj(r)j2Exc "p=0, (1) 1.4.3ExactDensityorDensityInvarianceCondition AccordingtotheKohn-ShamprescriptiontheKohn-Shamdeterminanthastogivethecorrectdensity.Thus,imposingtheconditionthatanycorrectionstotheconvergedsingledeterminantvanishonecanuniquelydeneVxc.Forinstance,densitymatrixcorrectionfromthecoupledclusterwavefunctioncanbewrittenas[ 22 ], qp=h0j[eTyfpyqgeT]cj0i, (1) Forsimplicity,werestrictourselveswiththecoupledclustersinglesanddoubleswavefunctionandinitiallyconsideronlythelineartermsofeTyandeTinthedensitycorrection, 34

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qp=h0j[(1+Ty1+Ty2)fpyqg(1+T1+T2)]cj0i, (1) Whichcanbeconvenientlyrepresentedbythediagrams1-4inthegure 1-2 Figure1-2. CoupledClusterdensitydiagrams diagram1+c.c.=h0jT1j0i+h0jTy1j0i, (1) diagram2=h0jTy1T1j0i, (1) diagram3=h0jTy2T2j0i, (1) diagram4+c.c.=h0jTy1T2j0i+h0jTy2T1j0i, (1) Theclusteramplitudecanbeformallywrittenastheinnitesummationofclassesoflinkeddiagramswhichcontainallpossibleconnectedwavefunctioncontributions[ 4 ]. ^Tj0i=1Xn=1(^R0^VN)nj0iC, (1) 35

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Where^VN=^f0N+^W(andnotingthat^HN=(^H0)N+^VN).ThenormalorderHamiltoniancanbewritteninsecond-quantizedas^HN=Xp,qhpj^fjqifpyqg+1 4Xp,q,r,shpqjjrsipyqysr, (1) TheusualFockoperator^f=^h+^vHFarisesduetonormalordering.where,^vHF=^J)]TJ /F5 11.955 Tf 14.04 2.66 Td[(^K=PjRj(2)(1)]TJ /F5 11.955 Tf 13.55 2.66 Td[(^P12)=r12j(2)d2.Since,wearesolvinganeffectiveeigenvalueequation^hejpi=(^h+^u)jpi="pjpiwhere^u=^J+^Vxc,wecanwritetheFockmatrixelementsintermsoftheeffectivehamiltonianeigenvectorashpj^fjqi=hpj^hejqipq)]TJ -442.43 -23.91 Td[(hpj^K+^Vxcjqi.ThereexistdifferentchoicesoftheseparationoftheHamiltonian.However,werestrictourselveswiththeseparationoftheHamiltonianthatisinvarianttoanytransformationamongtheoccupiedorexcitedorbitalspace.(^H0)N=Xpfppfpypg+Xi6=jfijfiyjg+Xa6=bfabfaybg, (1)^VN=Xi,afai[fayig+fiyag]+W, (1) Windicatesthetwo-particleterm.WearefreetorotatetheoccupiedandunoccupiedorbitalstomakeasemicanonicaltransformationoftheFockmatrixtoeliminatetheoff-diagonalterms,f~i~j=f~a~b=0Thuswehave(^H0)N=X~pf~p~pf~py~pg, (1)^VN=X~i,~af~a~i[f~ay~ig+f~iy~ag]+W, (1) Ourresolventoperatorbecomes^R0=Xn6=0j~a1,..ani1,..inih~a1,..ani1,..inj f~i1~i1+...f~in~in)]TJ /F3 11.955 Tf 11.95 0 Td[(f~a1~a1)]TJ /F5 11.955 Tf 11.96 0 Td[(...)]TJ /F3 11.955 Tf 11.95 0 Td[(f~an~an, (1) 36

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Wecanfurtherclassifytheconnecteddiagramsbelongingtotheoperator^Tbythenumberofmofpairsofexternallinestheycontain,uptoN(thenumberofelectrons), ^T=NXm=1^Tm, (1) Where,^Tmisanoperatorthatcreatesmpairsofhole-particleslines,singleclusteramplitudewillcreateonepairofhole-particlelines 1-3 whiledoubleswillcreatetwohole-particleslines 1-4 ,andsoon. Figure1-3. Singleamplitudediagrams Figure1-4. Doubleamplitudediagrams Eachofthe^Tmdiagramsrepresentsasumofperturbationtheoryconnectedwavefunctiondiagramsbutextendingtoallorders.Theorder-by-orderexpansionofthe^Tmcanbewrittenas^Tm=P1n=1^T(n)m,where,^T(n)mj0i=f(^R0^VN)nj0igC,m.Thesubscriptsindicatetherestrictiontonth-orderconnecteddiagramswithmexternalhole-particlelinepairs.Expanding^T1intermsoftheMBPTdiagramsarehelpfulinextractingtheexchangepotentialpartfromtheexchange-correlationpotential.Asshownlatertherstordercorrectioninthedensitygivetheexactexchangepotentialwhilehigherordersprovidethecorrelationpotential.ForsimplicityweignorethecouplingofT1withT2,howevertherewillbeanimplicitcontribution 1-6 ofT2intheT1asT1isexpandedindifferentordersoftheperturbation. T2canbetakenasCoupledclusterdoubles 1-7 orbyignoringthequadratictermslinearCCD 1-8 orjustthedoubleintegralscaledbythedenominatorinother 37

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Figure1-5. Singleamplitudeorder-by-orderdiagrams Figure1-6. Singleamplitudediagramswithimplicitdoublecontribution wordsMBPT-2doubles 1-9 .InfactwecantaketheT2obtainedaftersolvingthedirectring-CCDdoubleequationaswell 1-10 Figure1-7. CCDdoubleamplitude Expandingthedensitycorrectiondiagrams 1-2 byordersofperturbationusingT1 1-6 andT2( 1-7 1-8 1-9 1-10 ),wewillgetthedensitydiagrams 1-11 1-12 1-13 38

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Figure1-8. linCCDdoubleamplitude Figure1-9. MBPT-2doubleamplitude Figure1-10. rCCDdoubleamplitude Figure1-11. exactexchangepotential Figure1-12. correlationpotentialpart1 Figure1-13. correlationpotentialpart2 39

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where,inthediagramsmeansfpyqg=p(x1)q(x1)fpyqg.Andj)-99()]TJ /F12 11.955 Tf 26.27 8.96 Td[(N=j)-99()]TJ /F3 11.955 Tf 24.28 0 Td[(O+j)-99()]TJ /F3 11.955 Tf 24.28 0 Td[(x.Thej)-99()]TJ /F3 11.955 Tf 24.28 0 Td[(Orepresents^vHF=^J+^K,whilethej)-98()]TJ /F3 11.955 Tf 24.28 0 Td[(x=^u.Oneelectronpotential^ucanbeexpandedas^u=^u(1)+^u(2)+...=^J+^V1xc(=^Vx)+^V2xc(=^Vc). Exactexchangepotentialcanbeobtainedfromthedensitydiagram5asequation( 1 ),whilecorrelationpotential( 1 )areobtainedfromthesummingdiagram6to15.ObtainedOEPEquationappearslikeaFredholmequationofrstkind.Equation( 1 )usedinthecalculationofthepotentialcorrespondingtolinCCD,CCD,direct-ring-CCD(RPA)andtheMBPT-2functional. ai^Vexx dia+c.c.=ai dia+c.c., (1) ai^Vc=dia+c.c.=ai[1 2Xjcbtcbij=dia+c.c.)]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2Xkjbtabkj=dia)]TJ /F3 11.955 Tf 11.95 0 Td[(c.c+Xjbtabij=dia+c.c.+Xjb=djbdia+c.c+Xjbtabij=djb+c.c.]+ab[=dibdia+Xijc1 2tcaijtcbij])]TJ /F11 11.955 Tf 11.95 0 Td[(ij[=diadja+1 2Xkabtkjabtabki], (1) 40

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CHAPTER2NON-VARIATIONALDENSITYFUNCTIONALTHEORYFORPREDICTINGTRANSITIONSTATEANDBARRIERHEIGHTS. Ananalyticalgradientschemeforthenon-variationaldensityfunctionaltheoryhavebeendevelopedandtestedforthecalculationoftransitionstatesgeometriesandbarrierheightsforthehydrogentransferreactions.Itisnon-variationalinthesensethattheorbitalsandthedensitiesusedtoevaluatetheDFTenergiesarenotobtainedbysolvingthecorrespondingKohn-Shamself-consistentequations.RathertheHartree-fockdeterminantdensitiesandorbitalsareused,thusavoidingthenumericalintegrationinvolvedintheevaluationofexchange-correlationpotentialinKS-SCFprocedure.ThentheirdensitiesareinsertedintothefunctionalBLYPtoapproximatethecorrelationcorrection.Obtainedbarrierheightsarecomparedwiththeself-consistentHartree-Fock,withtheself-consistentorvariationalKohn-Sham.Non-variationalDFTbarrierheightsaremuchbetterthanvariationalDFTandHF. 2.1Motivation Inordertoobtainthegroundstateenergyofamolecule,oneneedstondthelowesteigenvalueofthe^H=Eeigenvalueproblem,whichiscomputationallyverydemandingduetothepresenceofthePNi,j1 ~rijtermintheHamiltonian^H( 2 ).^H=)]TJ /F5 11.955 Tf 10.5 8.09 Td[(1 2NXir2i+NXiMXAZA ~ri)]TJ /F11 11.955 Tf 12.61 3.15 Td[(~RA+1 2NXi,j1 ~rij, (2) However,onecanndanapproximatesolutionofthismany-bodyproblembydoingasingleparticletheory.Onepossiblewaytosolvethemany-bodyproblemistoapproximatethetroublingPNi,j1 ~rijasasumoftermscorrespondingtoeachelectronexperiencinganaveragepotentialduetoallotherelectrons.TheHartree-Fockissuchanexamplebutitexplicitlylackstheelectroncorrelation[ 4 ].Densityfunctionaltheory(DFT)isanalternativewaytoprovideapossiblesolutionoftheabovemany-bodyproblem[ 5 6 ],wherethedensity,aoneparticlequantityisassumedthefundamentalvariable( 2 ).TheexistencetheoremsofDFTtellusitisexactinprinciple[ 7 ],but 41

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practicalapproachesarehighlyapproximate.TherearetwofundamentalbottlenecksinDFT.First,thesocalleduniversalfunctionalFHK( 2 )containskineticenergyoftheelectronsandtheelectron-electroninteraction.Ithasanunknownexchange-correlationfunctionalpart.Hence,approximationsoftheunknownexchange-correlationfunctional( 2 )becomecrucialforthesuccessofDFT[ 27 ]. E[]=FKS[]+Z(r)vext(r)dr, (2) FHK[]=T0[]+Zdrdr01 jr)]TJ /F10 11.955 Tf 11.95 0 Td[(r0j(r)(r0)+Exc[], (2) Exc[]=Zdrdr01 jr)]TJ /F10 11.955 Tf 11.96 0 Td[(r0j(r)(r0)(~g(r,r0))]TJ /F5 11.955 Tf 11.95 0 Td[(1), (2) Howtondtheexactdensityofthemany-electronsystemisthesecondbottleneckofDFT.Theself-consistentKohn-Shamapproach[ 8 ]isthewidelyacceptedsolutionforthesecondbottleneck.TheKSapproachreliesonconstructingactitioushksoperatorfornon-interactingelectronsmovinginanexternalpotentialvks( 2 )thatwillprovidethegroundstatedensityasasumoftherstNoccupiedorbitaldensitiesequaltotheexactgroundstatedensityoftheinteractingmany-electronsystemks=exact. ^hks(1)i="ii(1)^hks=)]TJ /F5 11.955 Tf 10.49 8.09 Td[(1 2r2(i)+vks(~ri)ks(r)=NXiji(r)j2=exact(r), (2) Thepossibilityofalwaysndingsuchanexternalpotentialthatwillprovideanon-interactingdensity[ 28 ]equaltotheinteractinggroundstatedensityisdifcultto 42

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proveinanexactway.But,inpracticeitisoftenassumedthattoadesireddegreeofaccuracytheexistenceofthe^vks(~r)( 2 )isgranted. vks(r)=vext(r)+VH(r)+vxc(r)+, (2) The^vks(~r)( 2 )containsnuclear-electroninteractionterm^vext( 2 ),theHartreepotential^VH( 2 ),whichisjustthefunctionalderivativeoftheclassicalcoulombinteractionenergywithrespecttothedensityandanexchange-correlationpotential^vxc( 2 ),whichisthefunctionalderivativeoftheunknownexchange-correlationfunctionalwithrespecttothedensity.Andisjustaadditiveconstant. ^vext(ri)=)]TJ /F4 7.97 Tf 11.29 14.95 Td[(No.ofAtomsXA=1ZA jri)]TJ /F10 11.955 Tf 11.95 0 Td[(RAj, (2) VH[r]=Z1 jr)]TJ /F10 11.955 Tf 11.96 0 Td[(r0j(r0)dr0, (2) vxc[r]=Exc (r), (2) AsevidentfromEquation( 2 ),theformof^vxcbecomesvitalforndingtheexactdensityofthesystemasexceptforthevxcexactanalyticalformofallothercomponentsofthevxcsuchasvextandVHareknown.WidelyusedDFTunknownexchange-correlationfunctional(forinstanceLDA[ 8 29 ],GGA[ 30 33 ],meta-GGAandhyperGGA[ 33 34 ])haveanexplicitdensitydependence,andtheyaremodeledusingthehomogenouselectrongasdensity.InhomogeneityisintroducedbyincludingthegradientandtheLaplacianofthedensityinthefunctional.Thesefunctionalshavemanyshortcomingssuchasaself-interactionerror[ 35 ],alackofdispersioninteraction 43

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[ 36 ]etc.Insteadofanindependentestimationoftheexchange-correlationpotentialvxc,itisalwaysconstructeddirectlybyfunctionallydifferentiatingtheexchange-correlationfunctionalExcwithrespecttothedensity[ 5 6 ],thusthepotentialvxcinheritsthedecienciesofthefunctionalExc.Thesedecienciesdonotoftenmanifestthemselvesinpredictingstablemoleculegeometries,vibrationalfrequencies,bondenergiesandheatofreactionasareasonablycorrectdensitiescanprovidegoodDFTfunctionalvalues[ 27 ].ForinstanceOliphantandBartlett[ 37 ]showedthatwhentheHartree-Fockself-consistentdensitiesandorbitalsareusedtoevaluatetheDFTenergiesinvolvingtheBeckeexchangefunctional[ 38 ]andtheLYP[ 39 ]correlationfunctionalonthesubsetofG2dataset[ 40 ],todetermineequilibriumstructures,harmonicfrequencies,dipolemomentsandtotalatomizationenergies,resultsobtainedareactuallyquitecomparablewiththecorrespondingaverageaccuraciesdeterminedusingtheCCSD(T)method.However,whenbondsarepartiallyformedsuchasoccurwithtransitionstategeometriesandbarrierheightdetermination,thesedefectsaremorepronounced[ 41 ].Therehavebeensubstantialeffortstoincludedifferentamountsofnon-localexchangeinadynamic(Longrange-correctedDFT)[ 42 ]ornon-dynamic(B3LYPetc.)[ 43 ]fashion,inself-consistenthybriddensityfunctionaltheorytominimizetheself-interactionerrorsandthusimprovebarrierheightestimations.Thepresentstudyaimsatextendingtheabovenon-variationalapproachofusingtheHartree-FockdensitiesandorbitalsasinputtoobtainaccurateDFTenergiesofmoleculartrnaitionstateandanyarbitrarypointsontheBorn-Oppenhiemerpotentialenergysurface.Thestudyaimsatinvestigating,howHartree-Fockdensities,whichareself-interactionerrorfreebuthavenocorrelationeffects,betterthantheKS-SCFdensitiesandenergiesatanyarbitrarypointsonaBorn-Oppenhiemerpotentialenergysurface.ItisthecommonperceptionthattheKS-SCFprocedurearejustameremathematicalconstructtoobtainthedensityandKS-eigenvaluesandeigenfunctionhavenophysicalcontentbesides,sumoflowestNorbitalsdensityshouldgiveanexactgroundstatedensityof 44

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theinteractingparticleandHOMOenergiesshouldbeequaltorstverticalionizationenergy.However,webelievethatforaconsistentDFT,agoodfunctionalshouldprovideanequallygoodpotential,andagoodpotentialshouldnotonlyprovideagooddensityandthuscorrectorbitalsbutalsoshouldsatisfytheone-particleenergeticcriteriathatthenegativeoftheeigenvalues,notjusttheHOMObutallorbitalsshouldbeagoodapproximationoftheverticalionizationenergies.InquestofsuchaconsistentDFT,weseekaconsistentDFTfunctionalthatshouldprovidethepotentialthatwillgiveaconsistentdensity.Forinstance,adensityobtainedfromtheSCF-BLYPprocedureshouldhaveexchange-correlationeffectsoftheBLYPfunctionalandfunctionalvaluesobtainedfromthecorrelateddensitymustbesuperiortothatofuncorrelateddensityliketheHartree-Fockdensityatallpointsinthepotentialenergysurface. DoingthequantitativeanalysisofthebarrierheightestimationswiththeBLYPDFTfunctionalevaluatedusingtheSCF-BLYPcorrelateddensity(thevariationaldensity)andtheHartree-Fockdensity(non-variationaldensity),weareabletoshowthatcorrelationeffectsofBLYParenotproperlyincludedinthedensity.Yettheshortcomingsofthefunctionalismorepronouncedinthepotential.UsingtheHartree-FockdensitytoobtainbetterresultsthanfromtheSCF-KSdensityhascomputationaladvantages.As,theanalyticalformoftheexplicitdensitydependentfunctional(suchLDA,GGA,etc.)aresuchthatitisimpossibletoevaluatethemanalytically,onehastoreplyona3Dnumericalintegrationscheme[ 38 ].Furhtermore,since,mostoftheDFTapplicationsinvolvesusingthehybridfunctionalsthatcontainacertainamountoftheHartree-Focknon-localexchange( 2 ).ThePinKS-operator( 2 )istheone-particledensitymatrix(P=P+P,P=PiCiCi,spincouldbeor),whereCexpansioncoefcient('p(~r)=PCp(~r))ofthemolecularorbitals'pintheatomicbasis.Thus,thecostofthedoingSCFDFTcalculationsisgreaterthanthatfordoingtheHFSCFcalculationduetotheextracomputationaleffortsrequiredtodothenumericalintegrationtoobtainthematrixelementoftheVxcoperator.Although,for 45

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largermoleculescomputationalcostofthenumericalquadraturewillbesignicantlylessthanthatbuildingtheKS/HFoperatoranddiagonalizingtheKS/HFmatrixifwecanavoidthe3Dnumericalintegrationandobtainadensitythatprovidesbetterresults,itisworthinvestigatingsuchanon-variationalapproach. ^hKS,=h+XfP(j))]TJ /F3 11.955 Tf 11.96 0 Td[(aP(j)g+(Vxc), (2) ^fHF,=h+XfP(j))]TJ /F3 11.955 Tf 11.96 0 Td[(P(j)g, (2) AfterestablishingthattheBLYPfunctionalevaluatedwithanon-variationaldensityobtainedfromtheHartree-FockSCFprocedureissuperiortothatofKS-SCF-BLYPvariationaldensity,wedevelopananalyticalgradientschemeforsuchanon-variationalapproachtoexpeditethetransitionstatesearchesandgeometriesoptimization. 2.2AnalyticalGradientSchemeforNon-VariationalDFT WedenethevariationalDFTastheapproachwhereDFTfunctional( 2 )valuesareevaluatedusingthedensityandorbitals,whichareobtainedbysolvingtheKS-eigenvaluesproblemandtheKSoperatorcontains( 2 )aexchange-correlationpotentialthatisconstructedbyfunctionallydifferentiatingthecorrespondingexchange-correlationfunctionalwithrespecttothedensity. EVariational)]TJ /F4 7.97 Tf 6.59 0 Td[(DFT=ENuclear)]TJ /F4 7.97 Tf 6.59 0 Td[(Nuclear)]TJ /F4 7.97 Tf 6.59 0 Td[(repulsion)]TJ /F4 7.97 Tf 6.59 0 Td[(energy+Eelectron)]TJ /F4 7.97 Tf 6.58 0 Td[(kinetic)]TJ /F4 7.97 Tf 6.58 0 Td[(energy('KS)]TJ /F4 7.97 Tf 6.59 0 Td[(DFT)+EClassical)]TJ /F4 7.97 Tf 6.59 0 Td[(coulomb)]TJ /F4 7.97 Tf 6.59 0 Td[(electron)]TJ /F4 7.97 Tf 6.58 0 Td[(electron)]TJ /F4 7.97 Tf 6.59 0 Td[(enegry('KS)]TJ /F4 7.97 Tf 6.59 0 Td[(DFT)+EBecke)]TJ /F4 7.97 Tf 6.59 0 Td[(LYPExchange)]TJ /F4 7.97 Tf 6.59 0 Td[(Correlation)]TJ /F4 7.97 Tf 6.58 0 Td[(energy(KS)]TJ /F4 7.97 Tf 6.59 0 Td[(DFT,rKS)]TJ /F4 7.97 Tf 6.58 0 Td[(DFT), (2) 46

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^hKS'p="p'p^hKS=(j)]TJ /F5 11.955 Tf 19.13 8.08 Td[(1 2r2)]TJ /F4 7.97 Tf 16.56 14.94 Td[(MXAZA ~rA1j)+XP(j)+(VBecke)]TJ /F4 7.97 Tf 6.59 0 Td[(LYPxc), (2) While,wedenethenon-variationalDFTastheapproachwhereDFTfunctional( 2 )valuesareevaluatedusingthedensityandorbitals,whichareobtainedbysolvingtheHartree-Fockeigenvaluesproblem( 2 )andthusthedensityisun-correlatedbuthasnoself-interactionerror. ENon)]TJ /F4 7.97 Tf 6.59 0 Td[(Variational)]TJ /F4 7.97 Tf 6.58 0 Td[(DFT=ENuclear)]TJ /F4 7.97 Tf 6.58 0 Td[(Nuclear)]TJ /F4 7.97 Tf 6.59 0 Td[(repulsion)]TJ /F4 7.97 Tf 6.58 0 Td[(energy+Eelectron)]TJ /F4 7.97 Tf 6.58 0 Td[(kinetic)]TJ /F4 7.97 Tf 6.58 0 Td[(energy('HF)+EClassical)]TJ /F4 7.97 Tf 6.59 0 Td[(coulomb)]TJ /F4 7.97 Tf 6.58 0 Td[(electron)]TJ /F4 7.97 Tf 6.59 0 Td[(electron)]TJ /F4 7.97 Tf 6.58 0 Td[(enegry('HF)+EBecke)]TJ /F4 7.97 Tf 6.58 0 Td[(LYPExchange)]TJ /F4 7.97 Tf 6.59 0 Td[(Correlation)]TJ /F4 7.97 Tf 6.58 0 Td[(energy(HF,rHF), (2) ^fHF'p="p'p^fHF,=(j)]TJ /F5 11.955 Tf 19.13 8.09 Td[(1 2r2)]TJ /F4 7.97 Tf 16.55 14.95 Td[(MXAZA ~rA1j)+XfP(j))]TJ /F3 11.955 Tf 11.96 0 Td[(P(j)g, (2) ThederivativeoftheNon-variationalDFTenergywithrespecttothenucleardisplacementisgivenas( 2 ). dENon)]TJ /F4 7.97 Tf 6.58 0 Td[(Variational)]TJ /F4 7.97 Tf 6.59 0 Td[(DFT dRx=@ENon)]TJ /F4 7.97 Tf 6.58 0 Td[(Variational)]TJ /F4 7.97 Tf 6.59 0 Td[(DFT @Rx+@ENon)]TJ /F4 7.97 Tf 6.58 0 Td[(Variational)]TJ /F4 7.97 Tf 6.59 0 Td[(DFT @P@P @Rx, (2) dENon)]TJ /F4 7.97 Tf 6.59 0 Td[(Variational)]TJ /F4 7.97 Tf 6.58 0 Td[(DFT dRx=@ENon)]TJ /F4 7.97 Tf 6.59 0 Td[(Variational)]TJ /F4 7.97 Tf 6.58 0 Td[(DFT @RxP+XX,hKS)]TJ /F13 7.97 Tf 6.59 0 Td[(@P @Rx, (2) 47

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Thesecondtermintheequation( 2 )isduetofactthattheorbitalsarenotvariationallyoptimized.Formofthegradientscheme( 2 )tobeeasilyimplementedintheDFTgradientcodediffersfromthevariationalDFTgradientschemebythelasttwoterms.FirstFKSijmatrixelementisnotdiagonalastheorbitalsarenottheself-consistentsolutionoftheKSeigenvalueproblem( 2 )butareconstructedfromtheHartree-Fockorbitals.Second,itrequiresthelinearresponseoftheHartree-FockorbitalsUxia. dENon)]TJ /F4 7.97 Tf 6.59 0 Td[(Variational)]TJ /F4 7.97 Tf 6.58 0 Td[(DFT dx=X,Phx+1 2XPP(j)x+Exxc(P,P))]TJ /F12 11.955 Tf 11.96 11.35 Td[(XXiXjSxij,hKSij,+2XXiXahKSai,Uxai,, (2) Thersttermintheequation( 2 )istheone-electronderivativetermandsecondtermoftheequationisthetwo-electronderivativeterm,whichcanbeevaluatedanalyticallyandarereadilyavailablefortheHF/DFTgradientscheme.Thethirdtermoftheequationisthegradientoftheexchange-correlationfunctional.Since,theformofthewidelyusedexchange-correlationfunctionalissuchthatitcannotbeevaluatedanalytically,Becke's[ 38 ]atomcenterednumericalquadratureisoftenused.Since,thequadratureweightvariesasatom-centered-gridmoves,quadratureweightderivativeneedtobeincludedaswell[ 44 ]. Exc=Zg(,,....)d~r=atomsXAgridpointsXiwAig(i,i,...;~rAi), (2) rBExc=XAXiwAirBg(i,i,...;~rAi)+XAXig(i,i,...;~rAi)rBwAi, (2) ThederivativeoftheoverlapmatrixelementsSxij,canbeevaluatedanalyticallyandtheyareavailablereadilyinanyHF/DFTgradientscheme.TheKSoperatorhKSisconstructedusingtheorbitalsobtainedfromtheSCFHartree-Fockcalculation.TheU 48

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inthelasttermoftheequation( 2 )aretheweightsoftheMOcoefcientsCintheexpansionofthepartialderivativeoftheC'swithrespecttoaperturbation. Ci=@Ci @=MOXpUpiCp,C=C0U(), (2) Theexplicitcalculationoftheparticle-holeblockoftheUwouldrequirearepeatedsolutionofcoupledperturbedHartree-Fockequation( 2 )[ 45 ]fordifferentperturbations.The[]intheequation( 2 )meansonlytheexplicitderivativesofatomicbasiswithrespectofareincluded. (A+B)U=)]TJ /F3 11.955 Tf 9.3 0 Td[(X,Aai,bj=("a)]TJ /F11 11.955 Tf 11.96 0 Td[("i)ijab)-221(hajjjbii,Bai,bj=habjjiji,Xai=haj^hjii)-223(hajii[]"i)]TJ /F12 11.955 Tf 11.95 11.35 Td[(Xj,khajjjikihkjji[]+Xjhajjjiji[], (2) However,usingtheinterchangetheoremorthez-vectormethod[ 46 ],wecanrewritethe2PaihKSaiUai=2PaiZaiXaiwhereZisdenedas(A+B)TZ=hKSandavoidthedirectcalculationoftheUai. 2.3NumericalAnalysis RelevantequationsareimplementedintheACESII[ 47 ]programssuiteandMG3basissetareusedforallthecalculationsreportedinthisstudy.Table 2-1 containsBarrierHeightsdeterminedusingthereactants,productsandtransitionstategeometries[ 41 ]optimizedattheQCISD[ 48 ]leveloftheoryandtheMG3basissetasreportedinthereference[ 41 ].FirstcolumnofthetableIrepresentsthehydrogentransferreaction,secondcolumnrepresentsthereferenceactivationenergiesfortheforwardreactions(Vf)andthebackwardreactions(Vb)takenfromthereference[ 41 ].Thirdcolumn 49

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represents,barrierheights(transitionstateenergy-reactantsenergies)obtainedfromtheHartree-Fockcalculation,thesubsequentcolumnrepresentingbarrierheightsobtainedfromusingtheHartree-FockorbitalstoevaluatetheDFTenergyusingtheBeckeexchangeandLYPcorrelationfunctional.Inotherwordsournon-variationalDFTenergyisasdenedinEquation( 2 ).ThelastcolumnofTable 2-1 hasactivationenergyvaluesobtainedfromself-consistentKScalculationswithBLYPintheKSoperatorandevaluatingthecorrespondingDFTenergyfunctionalasdenedintheequation( 2 )withtheBeckeexchange. TherstcolumnofTable 7-6 representsthehydrogentransferreaction,thesecondcolumnrepresentsthereferenceactivationenergiesfortheforwardreactions(Vf)andthebackwardreactions(Vb)takenfromthereference(truhlar).Thirdcolumnrepresent,barrierheights(transitionstateenergy-reactantsenergies)obtainedfromtheHartree-Fockcalculationsdoneonthegeometriesofthereactantandtransitionstatesoptimizedbythehartree-Fockmethod,thesubsequentcolumnrepresentsbarrierheightsobtainedfromusingtheHartree-FockorbitalstoevaluatetheDFTenergyhavingtheBeckeexchangeandLYPcorrelationfunctional,inotherwordsournon-variationalDFTenergyasdenedintheequation( 2 )usingthegeometriesofthereactantandtransitionstatesoptimizedbythenon-variationalDFT.ThelastcolumnofTable 7-6 hasactivationenergiesvaluesobtainedfromtheself-consistentKScalculationwiththeBeckeexchangeandLYPcorrelationintheKSoperatorandevaluatingthecorrespondingDFTenergyfunctionalasdenedintheequation( 2 )withtheBeckeexchangeusingthegeometriesofthereactantandtransitionstatesoptimizedbythevariationalDFT(orBLYPDFT). AsevidentfromTable 2-1 theHartree-Fockoverestimatesthebarrierheights(barrierheightRef)]TJ /F3 11.955 Tf 12.65 0 Td[(barrierheightHF)bymorethan11kcal/mol.WhilevariationalDFT,underestimatesitbymorethan(barrierheightRef)]TJ /F3 11.955 Tf 12.46 0 Td[(barrierheightBLYP)8kcal/mol.Itisinterestingtonotethatnon-variationalDFTprovidebarrierheights(barrierheightRef)]TJ ET BT /F1 11.955 Tf 227.35 -687.85 Td[(50

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Table2-1. HydrogentransferBarrierheights(inKcal/mol)ofHTBH44set[ 41 ] ReactionsRefHFDFTnon)]TJ /F4 7.97 Tf 6.59 0 Td[(varBLYP Cl+H2)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>HCl+HVf8.720.56.12.5Vb5.613.71.5-2.6OH+H2)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H+H2OVf5.721.25.6-3.2Vb22.025.019.210.2CH3+H2)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H+CH4Vf12.122.010.77.2Vb15.023.611.07.8H+CH3OH)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>CH2OH+H2Vf7.320.25.70.9Vb13.824.314.811.8H+H2)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>H2+HVf9.617.65.32.9Vb9.617.65.32.9OH+NH3)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>H2O+NH2Vf3.227.94.4-8.8Vb13.232.114.81.6F+H2)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>H+HFVf1.813.12.1-11.1Vb33.226.633.420.1H+PH3)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>PH2+H2Vf3.210.11.4-2.5Vb25.230.225.921.9H+ClH0)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>HCl+H0Vf18.031.414.510.1Vb18.031.414.510.1OH+H)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>H2+OVf10.117.27.91.4Vb13.132.78.61.4H+t)]TJ /F3 11.955 Tf 11.96 0 Td[(N2H2)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H2+N2HVf5.99.88.7-6.6Vb41.047.745.235.8H+H2S)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>H2+HSVf3.612.01.4-2.0Vb17.427.218.114.4O+HCl)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>OH+ClVf9.833.110.4-8.3Vb9.924.414.4-3.2CH4+NH)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>NH2+CH3Vf22.739.519.613.6Vb8.420.49.83.5NH2+CH4)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>CH3+NH3Vf14.532.014.08.1Vb17.930.016.810.5AverageError-11.20.98.3AverageAbsoluteError11.62.28.3AbsoluteMaximumError24.74.518.1AbsoluteMinimumError3.00.92.0 51

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Table2-2. Comparisonofthebarrierheights(inKcal/mol)ofthehydrogentransferreactionsusingthenon-variationalDFT,variationalDFTandtheHartree-Fockoptimizedstructures ReactionsRefHFDFTnon)]TJ /F4 7.97 Tf 6.59 0 Td[(varBLYP Cl+H2)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>HCl+HVf8.721.46.94.4Vb5.614.62.0-0.5OH+H2)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H+H2OVf5.724.15.9-0.1Vb22.028.019.213.4CH3+H2)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H+CH4Vf12.122.110.87.2Vb15.023.711.17.8H+CH3OH)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>CH2OH+H2Vf7.320.35.81.2Vb13.824.314.911.9H+H2)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>H2+HVf9.617.65.32.9Vb9.617.65.32.9OH+NH3)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>H2O+NH2Vf3.228.46.9-9.4Vb13.232.617.01.0F+H2)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>H+HFVf1.820.6-0.00.0Vb33.234.032.731.4H+PH3)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>PH2+H2Vf3.211.71.70.0Vb25.231.626.424.4H+ClH0)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>HCl+H0Vf18.031.714.810.0Vb18.031.714.810.0OH+H)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>H2+OVf10.118.19.31.5Vb13.133.310.31.4H+t)]TJ /F3 11.955 Tf 11.96 0 Td[(N2H2)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H2+N2HVf5.912.311.70.1Vb41.049.948.342.5H+H2S)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>H2+HSVf3.612.92.1-0.1Vb17.428.018.616.3O+HCl)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>OH+ClVf9.834.912.0-3.3Vb9.926.515.91.8CH4+NH)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>NH2+CH3Vf22.739.519.513.6Vb8.420.49.43.5NH2+CH4)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>CH3+NH3Vf14.532.113.88.0Vb17.930.116.510.3AverageError-12.50.36.2AverageAbsoluteError12.52.56.3AbsoluteMaximumError25.27.313.1AbsoluteMinimumError0.80.21.0 52

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barrierheightnon)]TJ /F4 7.97 Tf 6.59 0 Td[(var)]TJ /F4 7.97 Tf 6.58 0 Td[(DFT)within2.2Kcal/moleaccuracy.Weseeasimilarbehavior 7-6 ,whenbarrierheightsareobtainedusingthecorrespondingmethodtooptimizestructuresofthereactantsandtransitionstate.Non-variationalDFTnotonlyprovidesbettersinglepointenergiesbutthestructureofthereactantsandtransitionstatesobtainedarequitecompetitive.Thisaddtothedebate,howcanyoutrustaDFTfunctionalwhosecorrespondingpotentialisnotprovidingaconsistentdensity?Oradeciencyofthefunctionalsuchasitsself-interactionerrormaynotbevisibleatthefunctionallevelbutitseffectonthedensityisquitepronounced. Usingouranalyticalgradientschemeforthenon-variationalDFT,wecanexpeditetransitionstatesearchesandstructureoptimizationandshowthatthissimplenon-variationalapproachisquitecompetitiveinndingbarrierheights,oneoftheknownproblemsoftraditionalvariationalDFT.ItisoftenarguedthatKSself-consistentprocedureisjustamathematicalconstructtoobtainanon-interactingelectrondensitywhichisassumedtobeagoodapproximationoftheexactgroundstatedensity.WebelievethattheKSself-consistentprocedureisnotmerelyanauxiliaryconstructbutbeingasingleparticletheory,itprovidesthepossibilityofobtainingausefulviewofmoleculesintermsoftheKSeigenvaluesandionizationenergies,providedaconsistentDFTfunctionalisused.Byconsistentwemean,theKSpotentialshouldbeasgoodastheDFTfunctionalis.Thus,theassociatedeigenvectorswillprovideagoodapproximationoftheexactdensityandthenegativeoftheeigenvalueswillbeagoodestimateoftheverticalionizationenergies.Since,computationallyfeasibleformsofsuchaconsistentfunctionalisstilladream,eigenvaluesassociatedwiththeKSoperatorhavingawidelyusedexplicitdensitydependentexchange-correlationpotentialarecompletelymeaningless(nextchapter),eventheHOMOisnotevenclosetotherstprincipleionizationenergy(nextchapter)coupledwithourndingthatthedensityobtainedarenowayneartotheexactdensityasanuncorrelateddensitysuchasthatfromHartree-fockgivesbetteranswer,itseemsredundanttodoKS-SCFcalculationstoobtainthedensities.As 53

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thenon-variationalDFTapproach,thatavoidstheKSSCFapproachandprovidesanalternativeandcomputationallycheaperapproachthatgivesbetterdensitiesthantheSCF-KSeigenvectors(variationalDFT)provide.WedoanextrastepofsolvingaCPHFequationintheanalyticalgradientschemefornon-variationalDFTcomparedtothevariationalDFT,butavoidnumericalquadratureateachSCFiterationtoconstructexchange-correlationpotentials.SolvingCPHFequationshaslesscomputationalcost. 54

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CHAPTER3DENSITYFUNCTIONALORBITALENERGIESASIONIZATIONPOTENTIAL:THEROLEOFNON-LOCALEXCHANGE. ThisstudydemonstratesthatitisdifculttoobtainreasonableagreementbetweenthenegativeoftheKohn-ShamEigenvaluesandtheverticalionizationenergies,whentheKohn-Shamoperatorhasanexplicitdensitydependentexchange-correlationpotentialsuchasthelocaldensityapproximations,thegeneralizedgradientapproximation,thehypergeneralizedgradientapproximationandthemetageneralizedgradientapproximation.However,theorbitaldependent(OEP)potentialssuchasMBPT-2sc,providereasonableagreementbetweenthevalenceKohn-Shameigenvaluesandthenegativeoftheverticalionizationenergies.However,theamountofnon-localexchangeintheKohn-ShamoperatorbecomesvitaltogetreasonableagreementbetweenthecoreKohn-Shameigenvaluesandthenegativeoftheverticalionizationenergies.Acertainamountofnon-localexchangecoupledwithanexplicitdensitydependentfunctionalhavebeenadvocatedinthefunctionalaswellinthepotentialleveltodiminishtheinherentshortcomingsoftheexplicitdensitydependentfunctional/potentailsuchasself-interactionerrors.TheseapproachesareknownasthehybridDFT,asmostofthemapproachusetheHartree-Focknon-localexchangepotentialinsteadofconvertingthenon-localexchangefunctionalintoalocalOptimizedeffectivepotential(OEP)exchangepotential,intheKSoperator.DFTfunctionalsevaluatedusingtheKSoperatorthatcontainsnon-localexchange(HF-focklikeexchangeoperator)orlocal(OEP)exchangepotential,remainunchanged.However,thechoiceofnon-localvslocalexchangepotentialintheKSoperatorhasahugeimpactontheKSeigenvalues,especiallythecoreeigenvalues.Oneoverestimatesandtheotheroneunderestimatestheverticalionizationenergiesasthenegativeofitseigenvalues.InthequestforaconsistentDFT,consistentinthesensethatcorrespondingpotentialshouldprovideagoodapproximationoftheverticalionizationenergiesasthenegativeoftheeigenvaluesandaccuratetotalenergysimultaneously,weadvocatetheuseof50%non-local 55

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exchangeand50%localOEPexchangepotentialcoupledwithMBPT-2sccorrelationpotentialintheKSoperator. 3.1Introduction Theideaofhavingasingleparticlecorrelatedtheoryisdesirableasitcansimultaneouslyprovidesanorbitalviewofmoleculesandcan,inprinciple,providecorrelatedone-particleresults[ 49 ].TheDyson'sequation[ 50 ]isanexamplewhichprovidesthenegativeoftheverticalionizationsenergiesassociatedwithDyson'sorbitals.ThechargedensitycanalsobeobtainedasthesumoftheoccupiedDyson'sorbitals.However,theDyson'sorbitalsareneitherorthogonalnornormalized,makingallthemany-bodyeffectssuchascorrelationandrelaxation[ 51 ],dynamicpolarization[ 52 ]andcorrectionstotheself-interactiontobeembeddedintoanenergydependentself-energyoperator.Therstapproximationtotheself-energyoperatoristhenon-localexchangeoperatorwhichgivestheHartree-FockmethodwhileKoopmans'theoremprovidestheconnectionbetweentheorbitalenergyandtheionizationenergies,butitlacksthemany-bodyeffectssuchascorrelation.Italsolackstherelaxationeffectsintheorbitals. UsingtheKohn-Shamframework,onecaninprincipleapproximatetheself-energyoperatorwithanenergyindependentlocalKSexchange-correlationpotentialthatcontainsallmany-bodyeffects.Suchapotentialisconceivableinprinciple,butitisvirtuallyimpossibletoobtaininpractice.Mostoftenitisobtainedasafunctionalderivativeofanapproximateexchange-correlationfunctional[ 5 ],[ 6 ].Aninitialexchangeandcorrelationfunctionalisthatobtainedbyapproximatingthepair-correlationfunctionbythehomogenouselectrongasdensity[ 8 29 ].Inhomogeneitycanbeintroducedbyusingthegradientofthedensity[ 30 33 ]anditsLaplacian[ 33 34 ]andbyusinganoptimizedamountofthenon-localHFexchangeoperator[ 43 ],[ 42 ],[ 53 ].Theseapproximateexchange-correlationfunctionalshavemanyshortcomings,suchasalackofdispersioneffects[ 36 ],anincorrectasymptoticbehavior[ 54 ],[ 55 ] 56

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andaself-interactionerror[ 35 ].Thesedecienciesarebetterdemonstratedbytheeigenvaluesratherthanthefunctional,asthefunctionalsarerelativelyinsensitivetothechangeindensity.Areasonablycorrectdensitycanprovideagoodenergy,whichisbothanassetandadetrimenttothedevelopmentofmorerigorousandaccuratetheories.Assuming-representabilityisattainabletothedesiredaccuracyinKS-DFT[ 28 ],thesumofthelowestNeigenvectorsshouldgivetheexactdensityprovidedthattheexactpotentialbeingusedintheKSself-consistenteigenvaluecalculations.OurpremiseisthatinaconsistentDFT,thenegativeoftheKSeigenvaluesshouldbereasonablyclosetotheverticalionizationenergies[ 56 ],[ 49 ],andthisone-electronpropertyisacrucialconsequenceoftheaccuracyoftheexchange-correlationpotential. Inthisstudy,wewillobtainverticalionizationenergiesasthenegativeoftheKSeigenvaluespectrausingexplicitdensitydependentfunctionaltoaneVaccuracy.First,weestablishtheerrorassociatedwiththecommonlyusedexplicitdensitydependentpotentials[ 56 ]toobtainverticalionizationenergiesasthenegativeoftheKS-eigenvaluespectra.Second,wereplacetheexplicitdensitydependentpotentialwithanorbitaldependentpotentialwhichiscorrectthroughsecondordercorrelationinthewave-functionapproach[ 57 ]andshowtheimportanceofthenon-localexchangepotentialintheKSoperatorinordertogetthecoreverticalionizationenergiesintermsoftheKSorbitals. 3.2PerformanceoftheExplicitDensityDependentFunctional ToseehowgoodavailablefunctionsareinprovidingtheverticalionizationenergiesasthenegativeoftheKohn-shameigenvaluespectra,weusemorethan50availablefunctionalintheself-consistentKScalculationsforthemolecules:HF,HCN,CO,N2,C2H2,F2,NH3,H2O,H2CO,CH4andC2H4(testset1)usingDunning'striple-zetabasis[ 58 ].TheexperimentalgeometriesaretakenfromtheNISTdatabase[ 59 ]andexperimentalverticalionizationenergiesaretakenfromthe[ 60 ]and[ 61 ].Figures 3-1 57

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3-2 3-3 and 3-4 showtheperformanceofthevariousfunctionalsinascendingorderoftheabsoluteaverageerrors(AAE).AAEiscalculatedastheabsoluteaveragedifferencebetweentheexperimentalverticalionizationenergiesandthenegativeoftheKSoccupiedeigenvalues.ResultsfromFigure 3-1 suggeststhatthebestfunctionalsthatprovideionizationenergiesastheeigenvaluespectraaretheonesthathavethehighestamountofthenon-localexchangepotentiale.g.BHHLYPhas50%non-localexchangeandgivesanabsoluteaveragedeviationof2.35eV.ExchangeonlypotentialssuchasSLATER,PBEX,OPTX,etc.havethelargestAAEofmorethan12eV.However,correlationonlyfunctionalssuchasLYP,OP,PBEC,PW91C,etc.withnon-localexchangepotentialhaveAAEaround6.5eV.HybridfunctionalssuchastheB3LYP,X3LYP,P3PW91,etc.haveAAEaround7.5.Figure 3-2 tellsasimilarstory.TheonlyDFTpotentialsthatprovidemeaningfulcoreorbitalenergiesaretheonesthatcontainamaximumamountofnon-localexchangee.g.BHHLYP,M06-2X,M06-HF.OtherpotentialshavelargerAAE.AsimilarconclusioncanbedrawnfromthevalenceorbitalAAEresults(Figure 3-3 ).Fromtheexactdensityconditionthehighestoccupiedorbitalsshouldbeequaltotheverticalionizationenergy,however,datafromFigure 3-4 tellsusthatnoneoftheDFTfunctionalsprovideHOMOenergiestothedesiredaccuracy.InfactcorrelationonlyDFTfunctionalssuchasLYP,PBEC,PW91C,etc.withanon-localexchangeoperatorperformbetterthanpureDFTfunctionalssuchasBOPetc.OnlythehighlyparameterizedfunctionalM06-HFprovidesHOMOAAElessthan1eV.butitfailsbadlyforothertheorbitals. Fromourdata,weconcludethat,oftheconventionalfunctionalsBHHLYPfunctionalisthebestatgivingverticalionizationenergiesastheeigenvaluespectra.However,itisstillfarfromourdesiredaccuracy.Also,ourdatasuggestthattheamountofthenon-localexchangeplaysavitalroleinobtainingreasonablyaccurateverticalionizationenergies.Whetherthatisduetothephysicsoftheproblemortothefactthatthenon-localexchangesimplyaddanotherparametertoexploitispertinent. 58

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Figure3-1. PerformanceofDFTpotentialforthecharacterizationoftheverticalionizationenergiesineVasthenegativeofeigenvaluesoftheKSoperator(()]TJ /F11 11.955 Tf 9.3 0 Td[(i)]TJ /F3 11.955 Tf 11.95 0 Td[(Ii),i2alloccupiedorbitals) Figure3-2. PerformanceofDFTpotentialforthecharacterizationoftheverticalionizationenergiesineVasthenegativeofeigenvaluesoftheKSoperator(()]TJ /F11 11.955 Tf 9.3 0 Td[(i)]TJ /F3 11.955 Tf 11.95 0 Td[(Ii),i2coreoccupiedorbitals) Figure3-3. PerformanceofDFTpotentialforthecharacterizationoftheverticalionizationenergiesineVasthenegativeofeigenvaluesoftheKSoperator(()]TJ /F11 11.955 Tf 9.3 0 Td[(i)]TJ /F3 11.955 Tf 11.95 0 Td[(Ii),i2valenceoccupiedorbitals) 59

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Figure3-4. PerformanceofDFTpotentialforthecharacterizationoftheverticalionizationenergiesineVasthenegativeofeigenvaluesoftheKSoperator(()]TJ /F11 11.955 Tf 9.3 0 Td[(i)]TJ /F3 11.955 Tf 11.95 0 Td[(Ii),i2HOMOorbitals) 3.3ImportanceoftheNon-LocalExchangePotential WeperformedSCForbitaldependentDFT(OEP)calculationswiththeexactlocalexchangepotentialandMBPT-2sccorrelationpotential[ 22 ]onthetestset1.SemicanonicalorbitalsandBartlett'sgeneralizedMBPTpartitioningoftheHamiltonianavoidsmuchoftheconvergenceproblemintheKS-DFTperturbationtheory[ 62 ].ThisGMBPTchoiceisalsotheonlyonethatsuitablymaintainsoccupied-occupiedandvirtual-virtualorbitalinvariance.WeusethesamebasissetfortheauxiliaryexpansionofOEPpotentialandSCFcalculations,anuncontractedtriple-zetasetofanatomicnaturalorbital[ 63 ].ThisallowsustomakecomparisontoIP-EOM-CCinthesamebasis,orpotentiallyafullCIresultsiftheywerepossible.Weperformedtwosetofcalculations,rstsetofcalculationsinvolveslocalOEPpotentialintheKS-OEP-operatorEquation( 3 )andsecondsetinvolvesadding50%ofthenon-localHF-likeexchangepotentialto50%ofthelocalOEPexchangepotential(Equation( 3 ))intheKSoperator[ 64 ].Fexx+oep2)]TJ /F4 7.97 Tf 6.59 0 Td[(sc=h+P(j)+(vexxx)+(vope2c) (3) 60

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Fhf+exx+oep2)]TJ /F4 7.97 Tf 6.58 0 Td[(sc=h+P(j)+1 2(vexxx)+1 2(Vnlxx)+(vope2c) (3) Since,thenon-localexchangeandthelocalexact-exchangearederivedfromthesameexchange-functionalwearefreetochooseourpotentialhoweverwewant,withoutsacricingthetotalenergydetermination[ 26 ].Weknowthatthenon-localandlocalexchangeoperatorshaveoppositeeffectsonthecoreorbitals,sohavingequalamountsofbothoperatorsbalancesucheffects.FromTables 3-1 3-2 3-4 and 3-4 itisevidentthatcontrarytoanyconventionalDFTcalculationstheOEPpotentialprovidesHOMOandvalenceeigenvaluesveryclosetotheexperimentalverticalionizationenergies.However,thecoreeigenvaluesarewrongexceptwhentheKSoperatorscontainsequalamountsofthenon-localexchangeandthelocalexactexchangepotentialandtheMBPT-2functionalisusedforcorrelationintheOEPself-consistentcalculationstogetherwiththetreatmentofexchange.Thus,theyarecorrectthroughsecondorderinMBPTandwillconvergetotheexactanswerinthelimit[ 57 ].Theyhavenofundamentalshortcominglikeexplicitdensitydependentfunctionalhaveliketheintegerdiscontinuityproblem.Thus,theyprovideanexactmodelforcorrelatedone-particletheorythattheyhavetoconvergetotheexactsolutioninthelimitofbasissetandcorrelation[ 56 ].OEPresultsconrmourearlierndingthatoneneedsanoptimumamountofnon-localexchangeoperatorintheSCFcalculation,toobtaingoodcoreeigenvaluesthatgiveionizationenergiesclosetoexperimentalvalues.Also,tosatisfytheexactdensitycondition(HOMO=)]TJ /F3 11.955 Tf 9.3 0 Td[(Ip). Table3-1. ComparisonofthenegativeoftheFexx+oep2)]TJ /F4 7.97 Tf 6.59 0 Td[(sceigenvalueswiththeexperimentalverticalionizationenergiesisgiven,second,third,fourthandfthcolumnhasall,core,valenceandHOMOorbitalsenergydeviationsrespectively. )]TJ /F11 11.955 Tf 9.3 0 Td[(exx+oep2i)]TJ /F3 11.955 Tf 11.95 0 Td[(Ii)]TJ /F11 11.955 Tf 9.3 0 Td[(exx+oep2core)]TJ /F3 11.955 Tf 11.96 0 Td[(Ii)]TJ /F11 11.955 Tf 9.3 0 Td[(exx+oep2valence)]TJ /F3 11.955 Tf 11.96 0 Td[(Ii)]TJ /F11 11.955 Tf 9.29 0 Td[(exx+oep2HOMO)]TJ /F3 11.955 Tf 11.96 0 Td[(IiAverageError-5.67-20.2-0.12-0.04AverageAbsoluteError5.6720.20.850.31MaximumDeviation26.6026.63.680.69MinimumDeviation0.016.010.000.00 61

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Table3-2. ComparisonofthenegativeoftheFexx+oep2)]TJ /F4 7.97 Tf 6.59 0 Td[(sceigenvalueswiththeIP-EOM-CCSDverticalionizationenergiesisgiven,second,third,fourthandfthcolumnhasall,core,valenceandHOMOorbitalsenergydeviationsrespectively. )]TJ /F11 11.955 Tf 9.3 0 Td[(exx+oep2i)]TJ /F3 11.955 Tf 11.95 0 Td[(IIP)]TJ /F4 7.97 Tf 6.58 0 Td[(EOM)]TJ /F4 7.97 Tf 6.58 0 Td[(CCSDi)]TJ /F11 11.955 Tf 9.3 0 Td[(exx+oep2core)]TJ /F3 11.955 Tf 11.96 0 Td[(IIP)]TJ /F4 7.97 Tf 6.59 0 Td[(EOM)]TJ /F4 7.97 Tf 6.59 0 Td[(CCSDi)]TJ /F11 11.955 Tf 9.3 0 Td[(exx+oep2valence)]TJ /F3 11.955 Tf 11.95 0 Td[(IIP)]TJ /F4 7.97 Tf 6.59 0 Td[(EOM)]TJ /F4 7.97 Tf 6.59 0 Td[(CCSDi)]TJ /F11 11.955 Tf 9.3 0 Td[(exx+oep2HOMO)]TJ /F3 11.955 Tf 11.95 0 Td[(IiAverageError-5.88-21.45-0.68-0.20AverageAbsoluteError5.8821.450.680.24MaximumDeviation28.1028.103.850.47MinimumDeviation0.0017.200.000.00 Table3-3. ComparisonofthenegativeoftheFhf+exx+oep2)]TJ /F4 7.97 Tf 6.59 0 Td[(sceigenvalueswiththeexperimentalverticalionizationenergiesisgiven,second,third,fourthandfthcolumnhasall,core,valenceandHOMOorbitalsenergydeviationsrespectively. )]TJ /F11 11.955 Tf 9.29 0 Td[(hf+exx+oep2i)]TJ /F3 11.955 Tf 11.96 0 Td[(Ii)]TJ /F11 11.955 Tf 9.29 0 Td[(hf+exx+oep2core)]TJ /F3 11.955 Tf 11.96 0 Td[(Ii)]TJ /F11 11.955 Tf 9.3 0 Td[(hf+exx+oep2valence)]TJ /F3 11.955 Tf 11.95 0 Td[(Ii)]TJ /F11 11.955 Tf 9.3 0 Td[(hf+exx+oep2HOMO)]TJ /F3 11.955 Tf 11.95 0 Td[(IiAverageError-0.24-0.89-0.00-0.04AverageAbsoluteError0.591.000.450.26MaximumDeviation2.252.251.780.66MinimumDeviation0.010.030.010.01 Table3-4. ComparisonofthenegativeoftheFhf+exx+oep2)]TJ /F4 7.97 Tf 6.59 0 Td[(sceigenvalueswiththeIP-EOM-CCSDverticalionizationenergiesisgiven,second,third,fourthandfthcolumnhasall,core,valenceandHOMOorbitalsenergydeviationsrespectively. )]TJ /F11 11.955 Tf 9.3 0 Td[(hf+exx+oep2i)]TJ /F3 11.955 Tf 11.96 0 Td[(IIPEOM)]TJ /F4 7.97 Tf 6.59 0 Td[(CCSDi)]TJ /F11 11.955 Tf 9.3 0 Td[(hf+exx+oep2core)]TJ /F3 11.955 Tf 11.95 0 Td[(Ii)]TJ /F11 11.955 Tf 9.3 0 Td[(hf+exx+oep2valence)]TJ /F3 11.955 Tf 11.95 0 Td[(IIPEOM)]TJ /F4 7.97 Tf 6.59 0 Td[(CCSDi)]TJ /F11 11.955 Tf 9.29 0 Td[(hf+exx+oep2HOMO)]TJ /F3 11.955 Tf 11.96 0 Td[(IIPEOM)]TJ /F4 7.97 Tf 6.59 0 Td[(CCSDiAverageError-0.36-1.04-0.10-0.22AverageAbsoluteError0.641.120.470.38MaximumDeviation2.192.191.620.74MinimumDeviation0.050.210.050.08 3.4Conclusion Ournumericaldatashowthattheamountofthenon-localexchangeinKSoperatorisimportantinorderforthenegativeoftheeigenvaluespectratogivetheverticalionizationenergieswithinadesirablenumericalaccuracyof1eV.Duetothepoordescriptionofthederivativediscontinuitybythecommonlyusedpotentials,thenegativeoftheeigenvaluesarefarfromtheverticalionizationenergies.Thedescriptionofthederivativediscontinuitycanbecorrectedbyremovingtheself-interactionerrorfromthepotential.Oneoftheeasyxesforself-interactionerrorisincludeanoptimizedamountofthenon-localHFlikeexchangeoperator,asitisevidentintheabilityoftheexplicitdensitydependentpotentialtogivebetterapproximationoftheverticalionizationenergies,e.g.BHHLYP,M06-EXandM06-HF.However,theinherentdeciencyofthe 62

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functionalisstillmanifestedintheapproximationsoftheverticalionizationenergies,astheoptimumamountofnon-localexchangeintheKSoperatorimprovestheionizationenergy-eigenvaluescharacterizationbutthedecienciesaresolarge,thattheyremainfarfromthedesiredaccuracy.TheorbitaldependentpotentialsuchasMBPT-2scdoesnothaveanyinherentdeciencyexceptitislimitedtosecondorderincorrelationandisbasissetdependent.ItservesasthemodelforDFTfunctionaltoascertaintheroleofnon-localexchangepotentialintheKSoperator.Wendthatit'seffectsareverywellmanifestedinthecoreionizationenergies.However,ithaslittleimpactontheotherorbitalenergies. 63

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CHAPTER4DOESADOUBLEHYBRIDDENSITYFUNCTIONALPROVIDEACONSISTENTPOTENTIAL? InKohn-Sham(KS)densityfunctionaltheorytherearetwoapproximationsmade:thefunctionalandthepotentialusedtogeneratetheKSorbitals.Thelatteristypicallymuchmoresensitivetoacorrectdescriptionofexchangeandcorrelation.Asameasureofareliablepotential,weinsistuponthenegativeoftheorbitalenergiesintheKohn-Shameigenvalueequationstobereasonableapproximationstoalltheprincipalionizationpotentialsinamolecule.Thisraisesthequestionofwhetherknown;excellentfunctionalsprovideequallygoodpotentials.Oneofthebetterfunctionalsisthedoublehybridform,whichincludessecond-orderperturbationtheoryaspartofitscorrelationenergyevaluation.Usingoptimizedeffectivepotentials(OEP)proceduresthatderivefromthemethodsdevelopedforabinitioDFTcorrelationpotentials,consistentchoicesforVxcforthefunctionalcanbeobtained.However,thepotentialsarestillfoundtobedecient. 4.1WhyDoWeNeedB2-PLYPPotential? Densityfunctionaltheory[ 7 ](DFT)ispotentiallyawaytoavoidsolvingtheSchrodingerequation,toprovidetheelectronicstructureofatoms,moleculesandsolidsbyjustknowingtheexactdensity.However,therearetwomainobstacles,theunknownexchange-correlationfunctionalandaroutetogettheexactdensity.Thereisnoexactprocedureavailabletoobtainthefunctionalinanexplicitdensitydependentform[ 5 ],[ 6 ].Duetotheunavailabilityoftheexactexplicitdensitydependentformoftheexchange-correlationfunctionalthereareamenagerieofapproximateexplicitdesnitydependentformsintheliterature[ 65 ].Therefunctionalshavemanyshortcomingssuchasself-interactionerrors[ 35 ],noproperinclusionofweakinteraction[ 36 ].Numericallyobservedthatacertainamountofnon-localexchangeimprovestheexplicitdensitydependentfunctionalslikeLDA,GGA.B3LYPthemostwidelyusedfunctionalhas20%non-localexchangefunctional[ 43 ],[ 42 ],[ 53 ].Grimme'sdoublehybridfunctional, 64

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B2-PLYP[ 66 ]isanotherprominentlyusedfunctional,itisdoublehybridinthesensethatitcontainsnotonlyacertainamountofnon-localexchangebutalsoacertainamountofnon-localcorrelationfunctional.Usingaleast-square-tproceduretotheG2/97setofheatofformationdataset,itwasobservedthatinordertocorrectlydescribeelectroncorrelationoneneedstohaveanorbitaldependentnon-localcomponentinexchangeandinthecorrelationfunctional,augmentedbyalocalBeckeexchangeandLYPcorrelationfunctional.Despitemanyinherentshortcomings,differentapproximatefunctionalsareabletoperformreasonablywellwhentestedoncertaindataset[ 67 ].Grimme'sapproachisastepinrightdirectionastheorbitalsprovideinformationthatcannotbeeasilyextractedfromthedensity.However,insteadofdirectminimizationoftheDFTenergyfunctionaltoobtaintheorbitalsanddensity,anauxiliaryindependentparticlemodel(Kohn-Sham)isconstructedthatcontainstheexchange-correlationpotentialwhichisnotobtainedfromtheB2-PLYPfunctional,assumingthatKohn-Shamself-consistentprocedureisjustamathematicalconstructandKSeigenfunctionsandeigenvaluesaremereauxiliaryfunctionsandtheyarephysicallymeaningfulonlyincalculatingtotalenergiesandelectrondensities[ 5 ].However,ourdatafromtheearlierchapter 3 i.e.,thehybridMBPT-2scprovidegoodapproximationofverticalionizationenergieswhilenoneoftheexplicitdensitydependentorhybridfunctionalcomecloseshowedthatKS-SCFprocedureisnotjustamathematicalconstructbutprovideacorrelatedsingleparticlepictureifexchange-correlationpotentialischosencarefully. Inthisstudy,weinvestigatethepotentialcorrespondingtotheB2-PLYPfunctionalinordertoseeifthisaccuratefunctionalprovidesaconsistentpotential.Byconsistencywemeanthepotentialshouldbeasgoodasthefunctional.B2-PLYPisasemi-empiricalcombinationofKS-DFTandsecond-orderMoller-Plessetperturbationtheory(MP2),singlescontributiontoMP2,whichwouldnormallyarisewhenKSorbitalsareusedinmanybodyperturbationtheory[ 4 ]areignoredintheB2-PLYPfunctional.areexplicitlyignoredfromtheB2-PLYPfunctional,whichwouldnormallyarisewhenKSorbitals 65

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areusedinmanybodyperturbationtheory.Thus,standardHF-likesecond-orderMoller-PlessetperturbationcorrectionareobtainedusingtheKS-SCFeigenfunctionandeigenvaluesastheinputs.TheKS-operatoreigenvaluesandeigenfunctionsusedtoobtaintheB2-PLYPfunctionaldoesnotincludethepotentialcorrespondingtheMP2.WeimprovetheB2-PLYPKSoperatorFKS)]TJ /F4 7.97 Tf 6.58 0 Td[(B2)]TJ /F4 7.97 Tf 6.59 0 Td[(PLYPbyincludingthemissingMP2correlationpotentialFKS)]TJ /F4 7.97 Tf 6.58 0 Td[(B2)]TJ /F20 7.97 Tf 6.58 0 Td[(oep)]TJ /F4 7.97 Tf 6.58 0 Td[(LYPEigenvaluesfrombothoperatorsarecomparedtoseetheeffectsofintroducingthepotentialcorrespondingtotheMP2intheSCFprocedure.WeexpectthatagoodKSpotentialshouldgivetheverticalionizationenergiesasthenegativeofitseigenvalues.ThefunctionalqualityareassessedbycomparingthetotalenergyobtainedbyusingbothmethodsB2-OEP-LYP(FKS)]TJ /F4 7.97 Tf 6.59 .01 Td[(B2)]TJ /F20 7.97 Tf 6.59 -.01 Td[(oep)]TJ /F4 7.97 Tf 6.59 -.01 Td[(LYP)andB2-PLYP(FKS)]TJ /F4 7.97 Tf 6.59 .01 Td[(B2)]TJ /F4 7.97 Tf 6.59 -.01 Td[(PLYP). 4.2Grimme'sDoubleHybridDensityFunctional(B2-PLYP) Grimme'sdoublehybriddensityfunctional(B2-PLYP)isdenedassumoftheKS-SCFenergyandthescaledamountofthesecondorderMoller-PlessetPerturbation(MP2)functional( 4 ).TheMP2energy( 4 )isthenon-localcorrelationcomponentoftheGrimme'sfunctional.TheKS-SCFenergy(Equation( 4 )),containsthenuclear-nuclearrepulsionenergy(rsttermoftheEquation( 4 )),theone-electroninteractionenergywhichconsistofindependentparticlekineticenergyandelectronnucleusattractionenergy(secondtermoftheequation( 4 )),theclassicalcoulombinteractionenergy(thirdtermoftheequation( 4 )),andtheexchange-correlationenergy(fourthtermsoftheequation( 4 )),whichconsistsof53%(aintheequation( 4 ))ofthenon-localexchange-energy,47%(bintheequation( 4 ))oftheBeckeexchangefunctionaland73%(cintheequation( 4 ))LYPcorrelationenergy. EB2)]TJ /F4 7.97 Tf 6.59 0 Td[(PLYP=Escf+cE2, (4) 66

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E2=Enon)]TJ /F4 7.97 Tf 6.58 0 Td[(local)]TJ /F4 7.97 Tf 6.59 0 Td[(orbital)]TJ /F4 7.97 Tf 6.58 0 Td[(dependentc=1 4occXi,jun)]TJ /F4 7.97 Tf 6.59 0 Td[(occXa,bjhijjjabij2 "i+"j)]TJ /F11 11.955 Tf 11.95 0 Td[("a)]TJ /F11 11.955 Tf 11.96 0 Td[("b, (4) Escf=ENN+hhPi+1 2Z(~r1)(~r2) j~r1)]TJ /F11 11.955 Tf 10.86 .49 Td[(~r2j+EB2)]TJ /F4 7.97 Tf 6.59 0 Td[(LYPxc, (4) EB2)]TJ /F4 7.97 Tf 6.58 0 Td[(LYPxc=aEnon)]TJ /F4 7.97 Tf 6.58 0 Td[(localx+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(a)EBeckex+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(c)ELYPc, (4) EB2)]TJ /F4 7.97 Tf 6.59 0 Td[(PLYPxc=aEnon)]TJ /F4 7.97 Tf 6.59 0 Td[(localx+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(a)EBeckex+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(c)ELYPc+cE2, (4) Enon)]TJ /F4 7.97 Tf 6.59 0 Td[(localx=1 2XXPP(j), (4) (j)=ZZ(~r1)(~r1)r)]TJ /F5 11.955 Tf 7.08 -4.94 Td[(112(~r1)(~r1)d~r1d~r2, (4) Generalizedeigenvalueproblem( 4 )withtheKSoperatordenedasEquation( 4 )issolvedandcorrespondingeigenfunctionsandeigenvaluesareusedtoobtainthequantitiessuchastheone-particledensitymatrixP(P=P+P,P=PiCiCi,spincouldbeor),themolecularorbitals'p('p(~r)=PCp(~r),whereisanatomicorbitals),thedensity(~r)((~r)=Pij'i(~r)j2=+=PP(~r)(~r))anddensitygradient.TheKSoperatorinEquation( 4 )intheatomicbasiscontainstheusualone-electrontermsconsistingofthematrixelementsofthekineticenergyandelectron-nuclearattractionenergy.Theclassicalcoulombinteractionmatrixelementsandthelocalexchange-correlationpotential,areconstructedbyfunctionally 67

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differentiatingtheexchange-correlationfunctionwithrespecttothedensity.ThepotentialcorrespondingtotheMP2functional.Toavoidthefunctionaldifferentiationofthenon-localexchangefunctional,theHartree-Focktypeexchangeoperatorisincludedasthenon-localexchangepotential.Parametersaandbremainthesameinpotentialandinthefunctional. FKS)]TJ /F4 7.97 Tf 6.59 0 Td[(B2)]TJ /F4 7.97 Tf 6.58 0 Td[(LYP(~ri)'i(~ri)="i'i(~ri), (4) FKS)]TJ /F4 7.97 Tf 6.58 0 Td[(B2)]TJ /F4 7.97 Tf 6.58 0 Td[(LYP,=h+XfP(j))]TJ /F3 11.955 Tf 11.95 0 Td[(aP(j)g+(VB2)]TJ /F4 7.97 Tf 6.59 0 Td[(LYPxc), (4) h=hj)]TJ /F5 11.955 Tf 19.13 8.09 Td[(1 2r2)]TJ /F4 7.97 Tf 16.56 14.94 Td[(MXAZA ~rA1ji, (4) VB2)]TJ /F4 7.97 Tf 6.59 0 Td[(LYPxc=(EB2)]TJ /F4 7.97 Tf 6.58 0 Td[(PLYPxc)]TJ /F3 11.955 Tf 11.95 0 Td[(cE2) =(EB2)]TJ /F4 7.97 Tf 6.58 0 Td[(LYPxc)]TJ /F3 11.955 Tf 11.95 0 Td[(cEnon)]TJ /F4 7.97 Tf 6.59 0 Td[(localx) =(1)]TJ /F3 11.955 Tf 11.96 0 Td[(a)EBeckex +(1)]TJ /F3 11.955 Tf 11.96 0 Td[(c)ELYPc (4) 4.3PotentialCorrespondingtotheB2-PLYPFunctional TheB2-PLYPfunctionalsuccessareattributedtoitbeingadoublehybrid,itisadoublehybridinthesensethatithasascaledamountofnon-localorbitaldependentexchangejustliketheHartree-Fockexchangeoperatorandscaledamountofanon-localorbitaldependentcorrelationoperatorasMP2correlationfunctional.InordertoconstructtheactualpotentialcorrespondingtotheB2-PLYfunctional,weneedtointroducethemissingMP2correlationpartintheKS-operator(modiedFKS)]TJ /F4 7.97 Tf 6.59 0 Td[(B2)]TJ /F4 7.97 Tf 6.58 0 Td[(LYP 68

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orFKS)]TJ /F4 7.97 Tf 6.59 0 Td[(B2)]TJ /F4 7.97 Tf 6.59 0 Td[(oep)]TJ /F4 7.97 Tf 6.58 0 Td[(LYP)usingourOEPstrategy.UsingtheSharp-Hortoncondition[ 21 ]whichisequivalenttotheHohenberg-Kohnvariationalprinciple[ 7 ],wecanobtainappropriatepotentialcorrespondingtotheB2-PLYPfunctionalbynotonlyincludingthelocalorbitaldependentcorrelationpotentialcorrespondingtotheMP2correlationfunctionalbutalsobyincludingthelocalexactexchangeorbitaldependentexchangepotentialcorrespondingtothenon-localorbitaldependentexchangepartoftheB2-PLYPfunctional.However,wedidnotincludetheexactlocalexchangeOEPpotentialintheFKS)]TJ /F4 7.97 Tf 6.59 0 Td[(B2)]TJ /F4 7.97 Tf 6.59 0 Td[(oep)]TJ /F4 7.97 Tf 6.58 0 Td[(LYP)operator,choosingtoretainthenon-localexchangepotentialintheFKS)]TJ /F4 7.97 Tf 6.59 0 Td[(B2)]TJ /F4 7.97 Tf 6.59 0 Td[(oep)]TJ /F4 7.97 Tf 6.58 0 Td[(LYP)operatorforconsistencywithB2-PLYPfunctional,asourearlierstudiesshowedthattheamountofthenon-localexchangeintheKSoperatorisimportanttogetthecoreionizationenergiesinasthenegativeofitseigenvalues 3 .Alternatively,thepotentialcorrespondingtotheMP2functionalisobtainedbytransformingitsfunctionalderivativewithrespecttothedensityintoderivativewithrespectto'pand"pbyusingthechainruleforthefunctionaldifferentiation[ 68 ].Theideaistodetermineforwhichderivativestheexpressionsareknownandexpressthederivativeofinterestintermsoftheknownderivatives,whichaftermultiplyingbytheKSstaticresponsefunctionksandintegratingoverrresultsintothewidelyknowntheOEPintegralequation[ 69 ]. voep)]TJ /F4 7.97 Tf 6.58 0 Td[(MP2c(~r)=E2 (~r), (4) voep)]TJ /F4 7.97 Tf 6.59 0 Td[(MP2c(~r)=Zdr0vks(r0) (r)E2 vks(r0)=Zdr0)]TJ /F7 7.97 Tf 6.58 0 Td[(1KS(r0,r)E2 vks(r0), (4) ()]TJ /F5 11.955 Tf 10.49 8.08 Td[(1 2r2+vks)'i="i'i, (4) 69

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(r) vks(r0)=ks(r,r0)=XpXqyp(r)q(r)yq(r0)yp(r0) "p)]TJ /F11 11.955 Tf 11.95 0 Td[("q, (4) Zdr0ks(r,r0)Vxc=!(r)=Exc vks(r), (4) !(r)=XpfXq6=pyp(r)q(r) "p)]TJ /F11 11.955 Tf 11.96 0 Td[("qZdr0[yq(r0)Exc yp(r0)+c.c.]+jk(r)j2Exc "pg, (4) TheOEPintegralequation( 4 )canbetransformedintoalinearmatrixproblembyexpandingpotentialvoep)]TJ /F4 7.97 Tf 6.59 0 Td[(MP2candlinearresponsefunctionksinaniteauxiliarybasisset[ 69 ]. voep)]TJ /F4 7.97 Tf 6.58 0 Td[(MP2c(~r)=Xcg(~r), (4) KS(r,r0)=X(KS)g(~r)g(~r), (4) auxXXc=(E2 vks), (4) Although,OEPintegralequationisill-dened,withasuitablechoiceofdomainonecancircumventthisproblemandobtaintheOEPpotentialuniquelyuptoaconstant.Theexpansioncoefcientsareapproximatedbythesingularvaluedecomposition(SVD)withthegivenSVDthreshold[ 69 ]. 70

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~c=(XSVD)(E2 vks), (4) (E2 vks)=Xia(AUX),ia(RHS)ia+Xij(AUX),ij(RHS)ij+Xab(AUX),ab(RHS)ab, (4) (AUX),pq=(pqj)=Zdr'p(r)'q(r)gp(r), (4) (RHS)ia=Xj,b,chajjjcbihcbjjiji ("i)]TJ /F11 11.955 Tf 11.95 0 Td[("a)("i+"j)]TJ /F11 11.955 Tf 11.96 0 Td[("a)]TJ /F11 11.955 Tf 11.95 0 Td[("b))]TJ /F12 11.955 Tf 11.96 11.35 Td[(Xk,j,bhkjjjibihabjjkji ("i)]TJ /F11 11.955 Tf 11.95 0 Td[("a)("k+"j)]TJ /F11 11.955 Tf 11.96 0 Td[("a)]TJ /F11 11.955 Tf 11.95 0 Td[("b), (4) (RHS)ij=Xk,a,b1 2hkjjjabihabjjkii ("k+"j)]TJ /F11 11.955 Tf 11.95 0 Td[("a)]TJ /F11 11.955 Tf 11.96 0 Td[("b)("i+"k)]TJ /F11 11.955 Tf 11.95 0 Td[("a)]TJ /F11 11.955 Tf 11.96 0 Td[("b))]TJ /F12 11.955 Tf 11.96 11.36 Td[(Xk,j,bhkjjjibihabjjkji ("i)]TJ /F11 11.955 Tf 11.96 0 Td[("a)("k+"j)]TJ /F11 11.955 Tf 11.95 0 Td[("a)]TJ /F11 11.955 Tf 11.95 0 Td[("b), (4) (RHS)ij=Xk,a,b1 2hkjjjabihabjjkii ("k+"j)]TJ /F11 11.955 Tf 11.96 0 Td[("a)]TJ /F11 11.955 Tf 11.96 0 Td[("b)("i+"k)]TJ /F11 11.955 Tf 11.96 0 Td[("a)]TJ /F11 11.955 Tf 11.95 0 Td[("b), (4) (RHS)ab=Xi,j,c1 2hijjjcaihcbjjiji ("i+"j)]TJ /F11 11.955 Tf 11.96 0 Td[("a)]TJ /F11 11.955 Tf 11.95 0 Td[("c)("i+"j)]TJ /F11 11.955 Tf 11.95 0 Td[("b)]TJ /F11 11.955 Tf 11.96 0 Td[("a), (4) FKS)]TJ /F4 7.97 Tf 6.59 0 Td[(B2)]TJ /F4 7.97 Tf 6.59 0 Td[(oep)]TJ /F4 7.97 Tf 6.59 0 Td[(LYP,=h+XfP(j))]TJ /F3 11.955 Tf 11.95 0 Td[(aP(j)g+(VB2)]TJ /F4 7.97 Tf 6.58 0 Td[(LYPxc)+c(voep)]TJ /F4 7.97 Tf 6.59 0 Td[(MP2c), (4) 71

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4.4NumericalResults OurtestsystemsincludeclosedshellmoleculesC2H2,CH4,CO,H2CO,H2O,N2,F2,NH3,HCNandHF.ExperimentalgeometriesareobtainedfromNIST[ 59 ]ComputationalChemistryComparisonandBenchmarkdatabase.Uncontractedtriple-zetasetofatomicnaturalorbitalsbasisset[ 63 ]areusedforbothatomicandpotentialbasisset.TheACESII[ 47 ]programisusedtodoallcalculations. TheKS-SCFcalculationsareperformedwiththreedifferentKS-operatorsFKS)]TJ /F4 7.97 Tf 6.59 0 Td[(B2)]TJ /F4 7.97 Tf 6.59 0 Td[(LYP(Equation( 4 )),FKS)]TJ /F4 7.97 Tf 6.59 0 Td[(B2)]TJ /F4 7.97 Tf 6.59 0 Td[(oep)]TJ /F4 7.97 Tf 6.59 0 Td[(LYP(Equation( 4 ))andFKS)]TJ /F4 7.97 Tf 6.58 0 Td[(B2)]TJ /F4 7.97 Tf 6.58 0 Td[(LYP)]TJ /F4 7.97 Tf 6.59 0 Td[(full.(Equation( 4 )). FKS)]TJ /F4 7.97 Tf 6.59 0 Td[(B2)]TJ /F4 7.97 Tf 6.59 0 Td[(LYP)]TJ /F4 7.97 Tf 6.58 0 Td[(full,=h+XfP(j))]TJ /F3 11.955 Tf 11.95 0 Td[(aP(j)g+(VB2)]TJ /F4 7.97 Tf 6.58 0 Td[(LYP)]TJ /F4 7.97 Tf 6.59 0 Td[(fullxc), (4) TheKSoperatorintheequation( 4 )istheoriginalGrimme'sKSoperator,Equation( 4 )isconsistentGrimme'sKSoperatorasitcontainsapotentialthatisfunctionalderivativeofB2-PLYPfunctionalandhencecontainsOEP-MP2potential.Equation( 4 )isthemodiedGrimme'sKSoperatorastheLYPcorrelationfunctionalamountismodiedfrom73%to100%.TheformoftheenergyfunctionalassociatedwithallthreeKS-operatorsisthesame( 4 )excepttheinputssuchasKSorbitals,densityanddensitygradientusedtoevaluatethefunctionaldependsontheassociatedeigenvalueequation.Forconvenience,wegroupaboveKSoperatorintoonegroupcancalleditDF2.Obviously,thecomputationalcosttosolvetheeigenvalueproblemassociatedwiththeFKS)]TJ /F4 7.97 Tf 6.58 0 Td[(B2)]TJ /F4 7.97 Tf 6.59 0 Td[(LYP)]TJ /F4 7.97 Tf 6.59 0 Td[(fulloperatorismorethanthatofFKS)]TJ /F4 7.97 Tf 6.59 0 Td[(B2)]TJ /F4 7.97 Tf 6.59 0 Td[(LYPorFKS)]TJ /F4 7.97 Tf 6.59 0 Td[(B2)]TJ /F4 7.97 Tf 6.59 0 Td[(LYP)]TJ /F4 7.97 Tf 6.59 0 Td[(fullasoneneedstosolvetheOEPintegralequation.However,itprovidesconsistentpotentialassociatedwiththeB2-PLYPfunctional.Density,densitygradient,KSeigenvaluesobtainedfromsolvingtheeigenvalueproblemassociatedwiththeFKS)]TJ /F4 7.97 Tf 6.59 0 Td[(B2)]TJ /F4 7.97 Tf 6.59 0 Td[(LYP)]TJ /F4 7.97 Tf 6.59 0 Td[(fullwilltellustheeffectofLYPvsOEPMP2correlationpotentialintheKS 72

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operator.Itwillalsotellusthatsmallchangeindensityordensitygradientdonotaffectthetotalenergy,aseventhereasonablecorrectdensityprovidegoodenergyvalues. EB2)]TJ /F4 7.97 Tf 6.59 0 Td[(PLYP=ENN+hhPi+1 2Z(~r1)(~r2) j~r1)]TJ /F11 11.955 Tf 10.86 .49 Td[(~r2j+aEnon)]TJ /F4 7.97 Tf 6.59 0 Td[(localx+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(a)EBeckex+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(c)ELYPc+cE2, (4) Forerroranalysiswedene,AverageError(AE)astheequation( 4 ),AbsoluteAverageError(AAE)astheEquation( 4 ),AverageDeviation(AD)( 4 )andAbsoluteAverageDeviationastheEquation( 4 ).AverageError(AE)=Xi("method1i)]TJ /F11 11.955 Tf 11.96 0 Td[("method2i) n, (4) AbsoluteAverageError(AAE)=Xij("method1i)]TJ /F11 11.955 Tf 11.95 0 Td[("method2i)j n, (4) AverageDeviation(AD)=Xij("method1i)]TJ /F11 11.955 Tf 11.95 0 Td[("method2i))]TJ /F3 11.955 Tf 11.96 0 Td[(AE nj, (4) AbsoluteAbsoluteDeviation(AAD)=Xijj("method1i)]TJ /F11 11.955 Tf 11.96 0 Td[("method2i)j)]TJ /F3 11.955 Tf 17.94 0 Td[(AAE nj, (4) InFigure 4-1 4-2 and 4-3 ,x-axisrepresentsorbitalsofthemoleculesC2H2,CH4,CO,H2CO,H2O,N2,F2,NH3,HCNandHF.(Forinstance,number1to7inx-axisrepresent7orbitalsofthemoleculeC2H2,nextnumberrepresentsCH4allorbitalsandsoon).Y-axisinthegure 4-1 representsdeviationofthenegativeoftheDF2eigenvalueswiththeexperimentverticalionizationenergies.Y-axisinthegure 4-2 representsdeviationofthenegativeoftheDF2eigenvalueswiththeIP-EOM-CCSDionizationenergies.Y-axisinthegure 4-3 representsdeviationoftheDF2eigenvalues. 73

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InTable 4-2 and 4-3 ,IPiistheexperimentalverticalionizationenergies.AllvaluesreportedintheguresandinthetablesareineV. Table4-1. ComparisonoftheDF2totalenergies (EB2)]TJ /F4 7.97 Tf 6.58 0 Td[(PLYP)]TJ /F3 11.955 Tf 11.96 0 Td[(EB2)]TJ /F4 7.97 Tf 6.59 0 Td[(oep)]TJ /F4 7.97 Tf 6.58 0 Td[(LYP)(EB2)]TJ /F4 7.97 Tf 6.58 0 Td[(PLYP)]TJ /F4 7.97 Tf 6.59 0 Td[(full)]TJ /F3 11.955 Tf 11.96 0 Td[(EB2)]TJ /F4 7.97 Tf 6.59 0 Td[(oep)]TJ /F4 7.97 Tf 6.59 0 Td[(LYP) AverageError0.030.03AbsoluteAverageError0.030.03AverageDeviation0.010.01AbsoluteAverageDeviation0.010.01MaximumError0.060.06MinimumError0.000.00 Figure4-1. DF2orbitalenergiesvsverticalIPs 4.5Analysis TotalenergyremainspracticallyunaffectedintheDF2calculations(seeTable 4-1 ),regardlessofwhichKS-operator(FKS)]TJ /F4 7.97 Tf 6.59 0 Td[(B2)]TJ /F4 7.97 Tf 6.59 0 Td[(LYPEquation( 4 ),FKS)]TJ /F4 7.97 Tf 6.59 0 Td[(B2)]TJ /F4 7.97 Tf 6.59 0 Td[(oep)]TJ /F4 7.97 Tf 6.58 0 Td[(LYPEquation( 4 )andFKS)]TJ /F4 7.97 Tf 6.58 0 Td[(B2)]TJ /F4 7.97 Tf 6.59 0 Td[(LYP)]TJ /F4 7.97 Tf 6.59 0 Td[(fullEquation( 4 )).eigenvalues,eigenvector,densityanddensitygradientusedintheB2-PLYP(DF2)functionalmaximumerrorof0.06eV.andAAEof0.03eVisobservedforourtestset.Thus,thefunctionalsarerelatively 74

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Table4-2. ComparisonofthenegativeoftheDF2eigenvalueswithexperimentalverticalionizationenergies ()]TJ /F11 11.955 Tf 9.3 0 Td[("B2)]TJ /F4 7.97 Tf 6.59 0 Td[(PLYPi)]TJ /F3 11.955 Tf 11.95 0 Td[(IPi)()]TJ /F11 11.955 Tf 9.3 0 Td[("B2)]TJ /F4 7.97 Tf 6.58 0 Td[(oep)]TJ /F4 7.97 Tf 6.59 0 Td[(LYPi)]TJ /F3 11.955 Tf 11.95 0 Td[(IPi)()]TJ /F11 11.955 Tf 9.29 0 Td[("B2)]TJ /F4 7.97 Tf 6.58 0 Td[(PLYP)]TJ /F4 7.97 Tf 6.59 0 Td[(fulli)]TJ /F3 11.955 Tf 11.96 0 Td[(IPi) AverageError-1.94-2.17-1.70AbsoluteAverageError1.942.171.71AverageDeviation0.630.570.62AbsoluteAverageDeviation0.610.570.60MaximumError4.214.003.97MinimumError0.220.030.44 Table4-3. DistributionofDF2eigenvalueserrorsintothecore,valenceandHOMOorbitalenergies ()]TJ /F11 11.955 Tf 9.3 0 Td[("B2)]TJ /F4 7.97 Tf 6.59 0 Td[(PLYPi)]TJ /F3 11.955 Tf 11.96 0 Td[(Ik)()]TJ /F11 11.955 Tf 9.3 0 Td[("B2)]TJ /F4 7.97 Tf 6.59 0 Td[(oep)]TJ /F4 7.97 Tf 6.59 0 Td[(LYPi)]TJ /F3 11.955 Tf 11.96 0 Td[(Ik)()]TJ /F11 11.955 Tf 9.3 0 Td[("B2)]TJ /F4 7.97 Tf 6.59 0 Td[(PLYP)]TJ /F4 7.97 Tf 6.59 0 Td[(fulli)]TJ /F3 11.955 Tf 11.96 0 Td[(Ik) AAEforcoreorbitals2.822.492.57AAEforvalenceorbitals1.632.051.40AAEforHOMO1.802.231.57 Figure4-2. DF2OrbitalenergiesvsEOM-CCSD 75

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Table4-4. ComparisonofthenegativeoftheDF2eigenvalueswithIP-EOM-CCSDionizationenergies ()]TJ /F11 11.955 Tf 9.29 0 Td[("B2)]TJ /F4 7.97 Tf 6.58 0 Td[(PLYPi)]TJ /F3 11.955 Tf 11.96 0 Td[(IIP)]TJ /F4 7.97 Tf 6.59 0 Td[(EOM)]TJ /F4 7.97 Tf 6.58 0 Td[(CCSDi)()]TJ /F11 11.955 Tf 9.3 0 Td[("B2)]TJ /F4 7.97 Tf 6.59 0 Td[(oep)]TJ /F4 7.97 Tf 6.58 0 Td[(LYPi)]TJ /F3 11.955 Tf 11.96 0 Td[(IIP)]TJ /F4 7.97 Tf 6.59 0 Td[(EOM)]TJ /F4 7.97 Tf 6.58 0 Td[(CCSDi)()]TJ /F11 11.955 Tf 9.3 0 Td[("B2)]TJ /F4 7.97 Tf 6.59 0 Td[(PLYP)]TJ /F4 7.97 Tf 6.59 0 Td[(fulli)]TJ /F3 11.955 Tf 11.96 0 Td[(IIP)]TJ /F4 7.97 Tf 6.59 0 Td[(EOM)]TJ /F4 7.97 Tf 6.58 0 Td[(CCSDi) AverageError-2.49-2.76-2.20AbsoluteAverageError2.492.762.23AverageDeviation0.980.780.98AbsoluteAverageDeviation0.980.780.98MaximumError5.75.55.47MinimumError0.811.00.60 Figure4-3. DF2OrbitalenergiesDeviations Table4-5. TheDF2eigenvaluescomparison ("B2)]TJ /F4 7.97 Tf 6.58 0 Td[(PLYPi)]TJ /F11 11.955 Tf 11.96 0 Td[("B2)]TJ /F4 7.97 Tf 6.59 0 Td[(oep)]TJ /F4 7.97 Tf 6.58 0 Td[(LYPi)("B2)]TJ /F4 7.97 Tf 6.59 0 Td[(PLYP)]TJ /F4 7.97 Tf 6.59 0 Td[(fulli)]TJ /F11 11.955 Tf 11.96 0 Td[("B2)]TJ /F4 7.97 Tf 6.59 0 Td[(oep)]TJ /F4 7.97 Tf 6.58 0 Td[(LYPi)("B2)]TJ /F4 7.97 Tf 6.59 0 Td[(PLYP)]TJ /F4 7.97 Tf 6.59 0 Td[(fulli)]TJ /F11 11.955 Tf 11.96 0 Td[("B2)]TJ /F4 7.97 Tf 6.59 0 Td[(PLYPi) AverageError-0.26-0.50-0.24AbsoluteAverageError0.440.570.24AverageDeviation0.360.370.01AbsoluteAverageDeviation0.230.300.01MaximumError1.171.450.28MinimumError0.020.010.21 76

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Table4-6. ComparisonoftheDF2core,valenceandHOMOeigenvalues ("B2)]TJ /F4 7.97 Tf 6.59 0 Td[(PLYPi)]TJ /F11 11.955 Tf 11.95 0 Td[("B2)]TJ /F4 7.97 Tf 6.58 0 Td[(oep)]TJ /F4 7.97 Tf 6.59 0 Td[(LYPi)("B2)]TJ /F4 7.97 Tf 6.59 0 Td[(PLYP)]TJ /F4 7.97 Tf 6.58 0 Td[(fulli)]TJ /F11 11.955 Tf 11.95 0 Td[("B2)]TJ /F4 7.97 Tf 6.58 0 Td[(oep)]TJ /F4 7.97 Tf 6.59 0 Td[(LYPi)("B2)]TJ /F4 7.97 Tf 6.59 0 Td[(PLYP)]TJ /F4 7.97 Tf 6.58 0 Td[(fulli)]TJ /F11 11.955 Tf 11.95 0 Td[("B2)]TJ /F4 7.97 Tf 6.59 0 Td[(PLYPi) AEforcoreorbitals0.330.09-0.25AEforvalenceorbitals-0.46-0.70-0.23AEforHOMO-0.42-0.66-0.23 insensitivetothechangeindensityandreasonablycorrectinputsuchaseigenvalues,eigenvector,densityanddensitygradientprovidegoodenergyvalues. TheDF2eigenvaluesarefarfromcorrectas(seeTable 4-2 ).WeknownthatanaccurateKSpotentialshouldprovideagoodapproximation(lessthanaeV)oftheverticalionizationpotential(VIPs)asthenegativeoftheKSeigenvaluesnotonlyforhighestoccupiedorbitalsbutwithalloccupiedorbitals,buttheDF2eigenvaluesarearound2eVwayfromtheVIPsforallorbitals.EventheHOMOenergiesarenotcorrecttothedesiredaccuracy(1.57-2.23evAAE,seeTable 4-3 ).TheoriginalGrimme'sKSoperatorFKS)]TJ /F4 7.97 Tf 6.59 0 Td[(B2)]TJ /F4 7.97 Tf 6.59 0 Td[(LYP(Equation( 4 ))doesnothaveapotentialthatisconsistentwiththeB2-PLYPfunctionalasitlackstheMP2correlationpotentialpartandwasusedjustasamereauxiliaryconstructtogeneratethenecessaryinputtoevaluatetheB2-PLYPfunctional.Thus,anabsoluteaverageerror1.94eV(seeTable 4-2 4-3 andFigure 4-1 )forestimatingtheverticalionizationenergiesasthenegativeoftheFKS)]TJ /F4 7.97 Tf 6.58 0 Td[(B2)]TJ /F4 7.97 Tf 6.58 0 Td[(LYPeigenvaluesisnotsurprising.However,anabsoluteaverageerror2.17eV(seeTable 4-2 4-3 andgure 4-1 )forestimatingtheverticalionizationenergiesasthenegativeoftheconsistentGrimme'sKSFKS)]TJ /F4 7.97 Tf 6.58 0 Td[(B2)]TJ /F4 7.97 Tf 6.59 0 Td[(oep)]TJ /F4 7.97 Tf 6.59 0 Td[(LYPeigenvalues,whichhasthemissingMP2correlationpotentialvoep)]TJ /F4 7.97 Tf 6.58 0 Td[(MP2c,showthatanempiricaltgoodfunctionallikeB2-PLYPdoesnotprovideaconsistentpotential. WealsocomparedthenegativeoftheDF2eigenvalueswithIP-EOM-CCSDionizationenergies,whichisconsideredtobeanexcellentmethodtocalculateverticalionizationenergies.TheIP-EOM-CCSDcomparisonshowasimilarpatternastheexperimentverticalionizationcomparison(seeTable 4-4 andFigure 4-2 ),anAAEof 77

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2.49eVfortheoriginalGrimme'sKSoperatorFKS)]TJ /F4 7.97 Tf 6.59 0 Td[(B2)]TJ /F4 7.97 Tf 6.58 0 Td[(LYPeigenvaluesand2.76evAAEforheconsistentGrimme'sKSFKS)]TJ /F4 7.97 Tf 6.58 0 Td[(B2)]TJ /F4 7.97 Tf 6.58 0 Td[(oep)]TJ /F4 7.97 Tf 6.59 0 Td[(LYPeigenvalues. Having,voep)]TJ /F4 7.97 Tf 6.58 0 Td[(MP2cinGrimme'sKSoperatorhasanegligibleimpactonenergy(seeTable 4-1 )andeigenvalues(Table 4-6 andFigure 4-3 ).Grimme'sKSeigenvalues(Table 4-2 andFigure 4-1 )donotdescribeionizationenergytoadesiredaccuracy.100%LYPor73%LYPcorrelationpotentialintheGrimme'sKSoperatorhasverylittleeffectontotalenergies 4-1 andeigenvaluesaredeviatedmorefromtheconsistentGrimme'sKSoperatoreigenvalues(Table 4-6 andFigure 4-3 ).Accordingtoournumericalresult,(Figure 4-3 curveB2PLYPfull-B2PLYP)effectsoftheLYPpotentialintheKSoperatorarelinearbynatureasallorbitalsareshiftedbyapproximatelyequalamount(0.25eV,Table 4-6 andFigure 4-3 )astheamountofLYPpotentialhasincreasesfrom73%to100%.TheeffectsoftheOEP-MP2correlationintheKSoperatorarenon-linearasthecoreandvalenceorbitalsareaffecteddifferently(Table 4-6 andFigure 4-3 ).Therationalgivenbythedouble-hybridfunctionaldesignerstohaveacertainamountofMP2functionalisthatitdescribeslong-rangeelectroncorrelationandregularDFTcorrelationfunctionalsaregoodforshort-rangeelectroncorrelation,thusanempiricaltcanhandlebothtypesofcorrelation.Ournumericalndingssuggestsuchanempiricaltdoesnotworkforthepotential.Thus,oneshouldbemorecarefulwhilecombiningtheLYPandtheOEP-MP2potentialastheformerhaslineareffectontheorbitalswhilethelatterhasnon-lineareffects.Thusintroducingthevoep)]TJ /F4 7.97 Tf 6.59 0 Td[(MP2cintoGrimme'sKSoperator,maketheorbitalenergiesslightworsethantheLYPpotential(Table 4-2 andFigure 4-1 ).OurnumericalresultsshowthatGrimme'spotentialisnotconsistentwithitsfunctionalasitseigenvaluesfailtoprovideagoodapproximationoftheverticalionizationenergyasnegativeoftheeigenvalues.EventheHOMOenergies,whichisthetraditionalmeasureofexactnessofanexchange-correlationfunctional,has1.65eVAAE.Grimme'sapproachisapragmaticwayofdoingJacob'sfthrungDFTcalculationwithoutcomputationally 78

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expensiveorbitaldependentnon-localcorrelationfunctionals.ItissimilartodoingaHF-likesecond-orderMoller-PlessetPerturbationcalculationbutwithKSorbitalswhileignoringtheonebodyterms.Although,itisastepinrightdirectionasKS-orbitalsareusedtocapturetheinformationthatcannotbeextractedbythedensityanddensitygradient,itsufferslikeanyotherlowerrungJacob'sladderfunctionalbynothavingaconsistentpotential.Theenergyprovidedisreasonablycorrectbutpotentialfailsofgiveionizationenergyreasonablyclosetoexperimentalnumbersasnegativeofitseigenvalues Inourstudyweaimtoaddress,whetheritispossibletogetadensitythatisconsistentwiththepotentialandonewherethepotentialisconsistentwiththefunctional.OfcoursethetraditionalroutetogetthefunctionalrstandthengeneratethepotentialasafunctionalderivativeanduseitinKS-operatortodeneddensity,doesnotsatisfythebeforementionedconsistencycondition,althoughwegetreasonablygoodenergyusingtheobtaineddensitywiththetakenfunctional(infact,HFdensitycouldbequitecompetitiveinoncecombinedwithDFTfunctionalinapostSCFmanner)potentialfailstoprovideagoodapproximationoftheverticalionizationenergiesasthenegativeofiteigenvalues. 79

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CHAPTER5CORRELATIONPOTENTIALCORRESPONDINGTOTHERANDOMPHASEAPPROXIMATIONANDBEYONDINDENSITYFUNCTIONALTHEORY Densityfunctionaltheory(DFT)resultsaredeservedlymistrustedduetothepresenceofanunknownexchangecorrelationfunctional,withnowaytoguarantyconvergencetotherightanswer.UseofaknownexchangecorrelationfunctionalbasedonthewavefunctionapproachthatisrightanswerinDFThelpstoalleviatesuchmistrust.Theexchangecorrelationfunctionalscanbewrittenexactlyintermsofthedensity-densityresponsefunctionusingtheAdiabatic-ConnectionandFluctuation-Dissipationframework.TheRandomphaseapproximation(RPA)isthesimplestapproximationforthedensity-densityresponsefunction.SincethecorrelationfunctionalobtainedfromRPAisequivalenttotheringcoupledclusterdoubles(ring-CCD)correlationfunctional,onecangobeyondtheRPAbyaddingmoretermsfromcoupledclustertheory.Inadditiontothering-CCD,wecanalsouselinearizedCCDorthefullCCDamplitudesintheequation.Inprinciple,wecanalsouseCCSD,CCSD(T),etc.SimilarlyonecanobtainMBPT-2correlationpotentialsbyapproximatingthetwo-bodyclusteroperatorwiththeanti-symmetrizedtwo-electronintegralscaledbythedenominator.UsingourOEPstrategy,weobtaincorrelationpotentialcorrespondingtoMBPT(2),RPA(ring-CCD),linear-CCDandCCD.UsingthesemicanonicalchoiceoftheunperturbedHamiltonian,Kohn-Shamself-consistentcalculationsareperformedandspatialbehavioroftheresultingpotentials,totalenergiesandtheHOMOeigenvaluesarecomparedwiththeexactvaluesforsphericalatoms. 5.1Introduction Beingasingleparticlecorrelatedtheory,theKohn-Shamdensityfunctionaltheorynotonlyoffersthepossibilityofobtainingtheexactgroundstateenergyofthenon-relativisticsysteminacomputationallyinexpensiveway[ 5 6 ]butalsoprovidesapictorialrepresentationofthesystemintermsoftheKSmolecularorbitals[ 8 ].However,itcontainsanunknownexchangecorrelationfunctionalthatneedstobe 80

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approximated.Differentapproximationsoftheexchangecorrelationfunctionalsareavailableintheliterature.Mostareobtainedsemi-empiricallyandsomearesaidtobenon-semi-empirical[ 70 ].Themostprominentapproximationsaretheexplicitfunctionalsofthedensityanditsgradientssuchasthelocaldensityapproximation[ 8 29 ](LDA)andgeneralizedgradientapproximation[ 30 33 ](GGA)andtheirextensionssuchashybridGGA[ 33 34 ]thatcontainsanoptimizedamountofthenon-localexchangefunctionalandmetaGGAthatusestheLaplacianofthedensity.Thesemodelfunctionalshaveanumberofdecienciessuchasself-interactionerrors[ 35 ]andthelackofdispersioninteractions[ 36 ].However,doubtlessduetoacancelationoferrorsformanyordinarymolecularpropertiesandsolidstateproblems,theyprovideresultsthatarecomparabletotheaccuracyofthecomputationallymoreinvolvedmanybodywavefunctionapproach. Recently,therehasbeenatremendousinterestinusingorbitaldependentorimplicitdensitydependentexchangecorrelationfunctionalsobtainedfromthewave-functiontheory.Besidesexchangewhichisfairlycommon,correlationpotentialshavealsobeengeneratedfrommanybodyperturbationtheories[ 71 72 ],orcoupled-clustertheoryfunctionals[ 22 73 74 ].Suchfunctionalarenon-local,doincludedispersionapproximately,anddonotsufferfromself-interactionerrors.Theyalsoconvergetotheexactanswerinthelimitofbasisandcorrelation.Finally,unlikeanyoftheDFTmenagerietheyhavethecorrectspatialbehavior.Anotheraspectweexpectisthatsuchacorrelatedorbitaltheoryshouldprovideapproximationsforallprincipalionizationenergies.ThisalsomeansthattheywouldhavetodescribeRydbergexcitations.[ 26 57 74 77 ]. AcorrelationfunctionalbasedonsecondorderperturbationtheorywiththeKSreferenceHamiltonian(KS-MBPT)asproposedbyGorlingandLevy[ 71 72 ]overestimatesthecorrelationeffectsinatomsandmoleculessignicantly,beinginferiortotheHartree-Fockbasedsecondorderperturbationtheory.Frequently,the 81

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selfconsistentsolutionfortheKS-MBPTpotentialisdifcultorimpossibletoobtain[ 78 79 ],becausetheKSchoiceofH0retainsadiagonalformintheperturbation,virtuallyguaranteeingpoorconvergenceforanyniteorderMBPTapproximation.OnecanimprovethedescriptionofthecorrelationeffectsandgetbetterdensitiesandenergiesbyusingtheoptimalchoiceofreferenceHamiltonianinGMBPT[ 62 80 ].Onecanfurtherimprovesuchanapproachbysummingcertaintypesofperturbationdiagramstoinniteorderasdoneincoupled-clustertheory[ 56 ].Therandomphaseapproximationwithoutexchangeoffersapotentiallyattractivestartingpoint,asitisthesimplestapproximationoftheexactcorrelationfunctional[ 81 83 ],thatisobtainedbytheadiabaticconnection[ 82 84 85 ]anductuationdissipationframework[ 86 87 ].Beinganinniteordermethod,itshouldbecomparativelystable.Italsooffersanapproximationtoweakinteractions[ 88 ].TheRPAfunctionalisanorbitaldependentfunctional,writtenintermsofthedensity-densityresponsefunction.Thusitprovidesaseamlessconnectionbetweenthewavefunctionmethodandthedensityfunctionalmethod[ 56 ].DuetothecomplicatedformoftheRPAfunctional,mostoftheefforttousetheRPAfunctionalinDFTareinapost-SCFmanner,i.e.simplyasaaddontotheenergyforanormalDFTresult.[ 89 91 ].ThedirectringRPAortheRPAwithoutexchangedoesnothaveawavefunction.Itlacksthefermionanti-symmetryofaproperwavefunction.Butitdoessomethingofinterest,ithelpstosolvethecoulombproblemastherstapproximationforanysubsequentmoreaccuratefunctional.Thismakesperfectsense,sincethatperturbationisbyfarthelargest.Relegatingtheothereffectslikeexchangetobetreatedinaperturbationmanner. TherehasbeenseveralattemptstoimprovetheRPAfunctional,asapost-scfapproach.ThechoiceoforbitalsusedtoobtaintheRPAfunctionaliscrucial[ 92 ].ItiswellknownthattheRPAfunctionallacksshort-rangecorrelation,thusarange-separatedRPAapproachhasbeenintroducedtotrytoaccountforit[ 91 93 95 ].Todealwithself-correlationproblemsinRPAasecond-orderscreenedexchangehasbeen 82

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introducedthatgobythenameSOSEX[ 96 97 ].Therehavealsobeenattemptstoimprovethedensity-densityresponsefunctionbyintroducingtheLDA/GGAorexactexchangeOEPkernelandtogobeyondtheRPAapproach[ 95 98 ].Theseattemptsresultedinamixedbagofsuccessesandfailures. TheRPAcorrelationfunctionalobtainedwithintheadiabaticconnectionanductionationdissipationframeworkhasacomplicatedformandobtainingtheactualpotentialhasproventobedifcult[ 83 99 ].However,theequivalencebetweentheRPAandacoupledclusterdoublescalculationforjusttheringdiagrams,[ 100 101 ]simpliesthefunctionalformasthereisnoomegaintegrationorintegrationalongtheadiabaticpathinvolved,italsoprovidesthesimplestwaytogobeyondRPA,asdoubleamplitudescanbeobtainedbyusingtheringdiagrams(ring-CCD)orthelineardiagrams(linCCD)orallthediagrams(CCD)orjustbythetwo-electronintegral(MBPT-2).Inthepresentstudy,wehaveobtainedthepotentialcorrespondingtotheRPA(thering-CCD)functionalandbeyondsuchaslinCCD,CCDandCCSD.Theselfconsistentcalculationswiththeabovepotentialsareperformedforsphericalatomsandthespatialbehaviorofthepotentialsiscompared.Accordingtotheexactdensitycondition[ 102 ]((~r)/exp()]TJ /F5 11.955 Tf 9.3 0 Td[(2p 2Ipr)forlarger,thenegativeofthehighestoccupiedmolecularorbitalenergyshouldbeequaltotheexactionizationenergy(HOMO=)]TJ /F3 11.955 Tf 9.3 0 Td[(Ip)oftheconcernedsystem.Hence,therstassessmentoftheaccuracyofthepotentialcanbeassessedbycomparingtheobtainedHOMOeigenvaluewiththeexperimentalionizationenergy.Thepotential,densityandfunctionalqualitycanbefurtherascertainedbycomparingthetotalenergywithhigh-levelCCSD(T)energiesorwiththosefromexperiment. 5.2Theory Usingtheadiabaticconnectionframework,thetotalelectronicenergy(Equation( 5 ))canbewrittenintermsofthenon-interactingelectronickineticenergy,thenuclear-electroninteractionenergyandthescaledelectron-electroninteractionenergy. 83

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ThisisanelegantexpressionasitshowsoneonlyneedsapartialknowledgeofthewavefunctiontoobtaintheexpectationvalueoftheHamiltonian. E=NXi=1hj)]TJ /F5 11.955 Tf 19.13 8.08 Td[(1 2r2ji+Zvext(~r)(~r)d~r+1 2Zd~rZd~r0Z10d1 j~r)]TJ /F11 11.955 Tf 10.87 .49 Td[(~r0j2(~r,~r0), (5) Howeveroneneedstoapproximatethetwo-bodyquantityintermsofone-bodyquantitiesifoneistohaveasingleparticletheory.Thiscanbedonebywritingthereducedpairdensityintermsofthedensity(Equation( 5 ))andthedensityuctuationoperator(Equation( 5 )). ^(~r)=Xi(~r)]TJ /F11 11.955 Tf 10.86 .5 Td[(~r0), (5) ^=^(~r))]TJ /F11 11.955 Tf 11.95 0 Td[((~r), (5) Thenthersttermoftheresultingreducedpairdensity(Equation( 5 ))willgiveustheclassicalcoulombelectron-electroninteractionenergyandthelasttwotermswillgiveusanexpressionfortheexactexchange-correlationenergy(Equation( 5 )).Usingthefrequencyintegrated,zero-temperatureformoftheuctuation-dissipationtheoremwecanrewritetheexactexchange-correlationfunctionalintermsoftheinteractingdensity-densityresponsefunction(Equation( 5 )).Thelinearizedinteractingdensity-densityresponsefunctionandthenon-interactingdensity-densityresponsefunctionarerelatedbyaDyson-likescreeningequation(Equation( 5 )).Itistrivialtoshowthattheexactexpressionfortheexchangefunctionalisnon-local(Hartree-Focktype)exchangefunctionalwiththeKohn-Shamorbitals.Theexactexpressionforthecorrelationenergy(Equation( 5 ))canbeexpressedintermsofthedensity-densityresponsefunction(Equation( 5 )).Byignoringthekernelfromthedensity-density 84

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responsefunction,theRPAcorrelationenergyisobtained(Equation( 5 )).Thisisthesimplestapproximationfortheexactcorrelationfunctional.TheresultingformoftheRPAfunctionaliscomplicatedsothatobtainingthepotentialisquitedifcultcomputationally.However,theequivalencebetweenRPAandring-coupledclusterdoubles,simplifythefunctionalformandpotentialiseasytoobtainusingtheOEPcorrelationpotentialapproachpioneeredinabinitiodft[ 22 ]. 2(~r,~r0)=(~r)(~r0)+<()j^(~r)^(~r0)j()>)]TJ /F11 11.955 Tf 9.29 0 Td[(3(~r)]TJ /F11 11.955 Tf 10.86 .5 Td[(~r0)(~r), (5) EExactxc=1 2Zd~rZd~r0Z10d1 j~r)]TJ /F11 11.955 Tf 10.87 .5 Td[(~r0j[h()j^(~r)^(~r0)j()i)]TJ /F11 11.955 Tf 19.26 0 Td[(3(~r)]TJ /F11 11.955 Tf 10.87 .5 Td[(~r0)(~r)], (5) EExactxc=1 2Z10dZd~rZd~r01 j~r)]TJ /F11 11.955 Tf 10.86 .5 Td[(~r0j[)]TJ /F5 11.955 Tf 10.89 8.09 Td[(1 Z10d!(~r,~r0,i!))]TJ /F11 11.955 Tf 11.96 0 Td[(3(~r)]TJ /F11 11.955 Tf 10.86 .5 Td[(~r0)(~r)], (5) (~r,~r0,i!)=0(~r,~r0,i!)+0(~r,~r0,i!)(vee+fxc,)(~r,~r0,i!), (5) EExactc=h()jveej()i)-222(h0jveej0i, (5) KS(~r,~r0,i!)=Xi,ai(~r)a(~r)i(~r0)a(~r0) i)]TJ /F11 11.955 Tf 11.96 0 Td[(a+i!+c.c., (5) EExactc=)]TJ /F5 11.955 Tf 10.5 8.09 Td[(1 2Z10dZd! 2Zd~rZd~r0vee(~r)]TJ /F11 11.955 Tf 10.86 .5 Td[(~r0)[(~r,~r0,i!))]TJ /F11 11.955 Tf 11.96 0 Td[(0(~r,~r0,i!)], (5) 85

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EExactc=1 2Zd! 2Zd~rZd~r0[ln(1)]TJ /F3 11.955 Tf 11.96 0 Td[(veeKS(i!))+veeKS(i!)], (5) AccordingtotheKohn-ShampremiseanycorrectiontotheconvergedKSdensityintroducedbyimprovementstoiandkshastovanish.Thus,imposingtheconditionthatanycorrectionstothedensityobtainedfromtheconvergedsingledeterminantvanishesonecandeneaVxc.Forinstance,thedensitymatrixcorrectionfromthecoupledclusterwavefunctioncanbewrittenas, qp=h0j[eTyfpyqgeT]cj0i, (5) Forsimplicity,werestrictourselveswiththecoupledclustersinglesanddoubleswavefunctionandconsideronlythelineareTyandeTinthedensitycorrection, qp=h0j[(1+Ty1+Ty2)fpyqg(1+T1+T2)]cj0i, (5) Whichcanbeconvenientlyrepresentedbythediagrams1-4inthegure 5-1 The Figure5-1. CoupledClusterdensitydiagrams clusteramplitudecanbeformallywrittenastheinnitesummationofclassesoflinkeddiagramswhichcontainallpossibleconnectedwavefunctioncontributions. 86

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^Tj0i=1Xn=1(^R0^VN)nj0iC, (5) Where^VN=^f0N+^W(andnotingthat^HN=(^H0)N+^VN).ThenormalorderedHamiltoniancanbewritteninsecond-quantizedas^HN=Xp,qhpj^fjqifpyqg+1 4Xp,q,r,shpqjjrsipyqysr, (5) TheusualFockoperator^f=^h+^vHFarisesduetonormalordering,^vHF=^J)]TJ /F5 11.955 Tf 14.26 2.66 Td[(^K=PjRj(2)(1)]TJ /F5 11.955 Tf 13.54 2.65 Td[(^P12)=r12j(2)d2.Since,wearesolvinganeffectiveeigenvalueequation^hejpi=(^h+^u)jpi="pjpiwhere^u=^J+^Vxc,wecanwritetheFockmatrixelementsintermsoftheeffectivehamiltonianeigenvectorashpj^fjqi=hpj^hejqipq)-234(hpj^K+^Vxcjqi.ThereexistdifferentchoicesoftheseparationoftheHamiltonian.However,werestrictourselvestotheseparationoftheHamiltonianthatisinvarianttoanytransformationamongtheoccupiedorexcitedorbitalspace.(^H0)N=Xpfppfpypg+Xi6=jfijfiyjg+Xa6=bfabfaybg, (5)^VN=Xi,afai[fayig+fiyag]+W, (5) WhereWindicatesthetwo-particleterm.WearefreetorotatetheoccupiedandunoccupiedorbitalstomakeasemicanonicaltransformationoftheFockmatrixtoeliminatetheoff-diagonalterms,f~i~j=f~a~b=0Thuswehave(^H0)N=X~pf~p~pf~py~pg, (5)^VN=X~i,~af~a~i[f~ay~ig+f~iy~ag]+W, (5) 87

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Ourresolventoperatorbecomes^R0=Xn6=0j~a1,..ani1,..inih~a1,..ani1,..inj f~i1~i1+...f~in~in)]TJ /F3 11.955 Tf 11.95 0 Td[(f~a1~a1)]TJ /F5 11.955 Tf 11.96 0 Td[(...,f~an~an, (5) Wecanfurtherclassifytheconnecteddiagramsbelongingtotheoperator^Tbythenumberofmofpairsofexternallinestheycontain,uptoN(thenumberofelectrons), ^T=NXm=1^Tm, (5) Where,^Tmisanoperatorthatcreatesmpairsofhole-particleslines,singleclusteramplitudewillcreateonepairofhole-particlelineswhiledoubleswillcreatetwohole-particleslines,andsoon.Theorder-by-orderexpansionofthe^Tmcanbewrittenas^Tm=P1n=1^T(n)m,where,^T(n)mj0i=f(^R0^VN)nj0igC,m.Thesubscriptsindicatetherestrictiontonth-orderconnecteddiagramswithmexternalhole-particlelinepairs.Expanding^T1intermsoftheMBPTdiagramsishelpfulintheextractingexchangepotentialpartfromtheexchange-correlationpotential.Asshownlater,therst-ordercorrectioninthedensitygivetheexactexchangepotentialwhilehigherordersprovidethecorrelationpotential.ForsimplicityweignorethecouplingofT1withtheT2,howevertherewillbeanimplicitcontributionofT2intoT1asT1isexpandedindifferentorderoftheperturbation.T2canbetakenasthecoupledclusterdoublesapproximationorignoringthequadratictermsthelinearCCDorjustthedoubleintegralscaledbythedenominatorinotherwordsMBPT-2doubles.InfactwecantaketheT2obtainedaftersolvingthedirectring-CCDdoubleequationaswell.Expandingthedensitycorrectiondiagrams 5-1 byordersofperturbationusingT1andT2,wewillgetthefollowingdensitydiagrams 5-2 5-3 and 5-4 where,inthediagramsmeansfpyqg=p(x1)q(x1)fpyqg.Andj)-99()]TJ /F12 11.955 Tf 26.27 8.96 Td[(N=j)-99()]TJ /F3 11.955 Tf 24.28 0 Td[(O+j)-99()]TJ /F3 11.955 Tf 24.28 0 Td[(x.Thej)-99()]TJ /F3 11.955 Tf 24.28 0 Td[(Orepresents^vHF=^J+^K,whilethej)-98()]TJ /F3 11.955 Tf 24.28 0 Td[(x=^u.Oneelectronpotential^ucanbeexpandedas^u=^u(1)+^u(2)+...=^J+^V1xc(=^Vx)+^V2xc(=^Vc). 88

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Figure5-2. OEPexchangepotential Figure5-3. OEPcorrelationpotentialparta. Theexactlocalexchangepotentialcanbeobtainedfromthedensitydiagram5asequation( 1 ),whilethecorrelationpotential( 5 )areobtainedfromsummingdiagram5to15.ObtainedOEPEquationappearslikeaFredholmequationoftherstkind.Equation( 5 )usedinthecalculationofthepotentialcorrespondingtolinCCD,CCD,direct-ring-CCD(RPA)andtheMBPT-2functional. ai^Vexx dia+c.c.=ai dia+c.c., (5) Figure5-4. OEPcorrelationpotentialpartb. 89

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ai^Vc=dia+c.c.=ai[1 2Xjcbtcbij=dia+c.c.)]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2Xkjbtabkj=dia)]TJ /F3 11.955 Tf 11.95 0 Td[(c.c+Xjbtabij=dia+c.c.+Xjb=djbdia+c.c+Xjbtabij=djb+c.c.]+ab[=dibdia+Xijc1 2tcaijtcbij])]TJ /F11 11.955 Tf 11.95 0 Td[(ij[=diadja+1 2Xkabtkjabtabki], (5) 5.3ResultsandDiscussion Self-consistentOEPcalculationswiththeMBPT-2,ring-CCD(RPA),lin-CCD,CCDpotentialsintheKSoperatorareperformedwithanuncontracteddouble-zetasetofatomicnaturalorbitals[ 103 ].TheexpansionoftheOEPpotentialusesthesamefunctionsasintheSCFprocedure.InFigure 5-5 5-6 5-7 5-8 and 5-9 thespatialbehavioroftheRPA,CCDandlin-CCDcorrelationpotentialsforthesphericalatoms(He,Be,Ne,Ar,Mg)arepresentedandcomparedwiththeMBPT-2-sccorrelationpotential.Contrarytotheexplicitdensitydependentcorrelationfunctionals(suchasLDA,GGA,etc.),theorbitaldependentcorrelationfunctionals(suchasRPA,MBPT-2sc,CCD,linCCD)showthecharacteristicshelloscillationoneexpectsfromtheexactcorrelationpotential[ 104 105 ].Theabsenceoftheshort-rangecorrelationinRPAis 90

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clearlymanifestintheRPApotentialplot,astheRPApotentialaretooattractiveneartothenuclearregioncomparedtoMBPT-2sc,linCCDandCCDpotential[ 99 ]. Figure5-5. Heliumcorrelationpotentialcomparison AccordingtoBartlett'stheorem[ 26 49 56 ]andnumericalexperimentsofBaerendsetal.,thenegativeoftheKS'seigenvaluesshouldbeagoodapproximationtotheexperimentalverticalionizationenergy.However,explicitdensitydependantfunctionalsortheirdifferentsemi-empiricalcombinationsfailtoprovideverticalionizationenergiesintermsoftheKSeigenvalues.Eventhehighestoccupiedmolecularorbitaleigenvaluesarequitedifferentfromtheexperimentalionizationenergy.Webelievethatthisisduetothepoorqualityofthepotential[ 106 ].Thereisalsoalackofconsistencybetweenthedensitypotential,functional.Inourearlierstudies,wehaveshownthatthebestfunctionalsdonotprovidegoodpotentials.EventheHartree-FockdensityprovidesabetterDFTfunctionalenergythantheKS-SCFobtaineddensity[ 107 ].AbinitioDFTfunctionalssuchasMBPT-2sc,RPA,lin-CCD,CCDtakenfromthewave-functionapproachareknowntobecorrecttocertainordersandthecorrespondingpotentials 91

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Figure5-6. Beryliumcorrelationpotentialcomparison Figure5-7. Neoncorrelationpotentialcomparison 92

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Figure5-8. Argoncorrelationpotentialcomparison Figure5-9. Magnesiumcorrelationpotentialcomparison 93

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areconsistentwiththefunctionals.ArstmeasureofthequalityofthepotentialistocomparetheHOMOeigenvalueswiththeexperimentalionizationenergies. Table5-1. ComparisonofthenegativeoftheHOMOeigenvalueswithexperimentalionizationenergies AtomExptEXXMBPT-2scRPAlin-CCDCCDHe0.9030.9180.9070.9050.9020.903Be0.3430.3090.3270.3370.3450.342Ne0.7920.8500.7620.7950.7900.792Ar0.5790.5900.5660.5680.5640.566Mg0.2810.2520.2680.2730.2750.274AbsoluteAverageError0.0290.0150.0060.0050.004 InTable 5-1 ,wecomparetheexactexchangeonlyOEP,MBPT-2sc,Ring-CCD(RPA),lin-CCD,CCDHOMOenergieswiththeexperimentalionizationenergiesinHartreeunits[ 105 ].ItisevidentfromthetablethathavingacorrelationpotentialintheKSoperatorwiththeexactexchangeOEPpotentialimprovesthedescriptionoftheHOMO'seigenvalues.Asthedescriptionofdoubleexcitationamplitudesimprovesfromtwo-electron(MBPT-2sc)(0.015AAE)toRPA(direct-ring-CCD)(0.006AAE)tolin-CCD(0.005AAE),CCD(0.004AAE)amplitudesodotheHOMOenergies.WeobservefromtheHOMOenergiesdatatableandpotentialplotsthatthespatiallyincorrectbehavioroftheRPApotentialnearthenuclearregionhasanegligibleeffectontheHOMOeigenvalues.AsonewouldexpectthespatialbehavioroftheRPA,lin-CCD,CCD,MBPT-2scpotentialsforHe,ArandMgaresimilarasaretheirHOMOeigenvalues.TheMBPT-2scpotentialoftheNeonatomisalittleshiftedfromtheotherpotentialsasistheHOMOeigenvalue.TheeffectsofthecorrelationpotentialsintheBerylliumatomarescatteredreectingthequasi-degeneracybetweenthe2sand2porbitals[ 74 ]. Table 5-2 ,showthecomparisonofthetotalenergiescalculatedusingtheSCFKSorbitalsobtainedwiththeexactexchangeonlyOEP,theMBPT-2sc,theRPA(ring-CCD), 94

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Table5-2. ComparisonofthetotalenergiesinHartree ATOMExptCCSD(T)EXXMBPT-2scRPAlin-CCDCCDHe2.9042.9002.8622.8942.9182.9002.900Be14.66714.66214.57214.64114.68914.66614.661Ne128.937128.865128.545128.861128.935128.860128.858Ar527.604527.249526.811527.236527.330527.244527.239Mg200.059199.866199.616199.853199.925199.864199.859AAEw.r.texpt0.3540.1370.0880.1290.125AAEw.r.tCCSD(T)0.2280.0110.0530.0030.002 thelin-CCDandtheCCDpotentialintheKSoperator.TheRPA(ring-CCD)energyismuchclosertotheexperimentalenergies[ 105 ]thantheCCD,thelin-CCD,theMBPT-2scortheexactexchangeOEP,whichmostlyfollowfromthefactthattheRPAexcludesanyexchangetermswhiletheotherconventionalquantumchemistrymethoddonot.ThisisreectedinthecomparisonwithCCSD(T).WeseeconsistentbehavioroftheexactexchangeonlyOEP,MBPT-2sc,lin-CCDandCCDenergiesincomparisontoexperimentorCCSD(T)energies,asthedescriptionofdoubleamplitudesimprovessodoesthetotalenergy.Atthecompletebasissetlimit,wewouldexpectCCSD(T)tobeclosetotheexperimentalvalues,butasseenforthelargeratoms,thechosenbasisdoesn'tachievethatlimit. Itishardtochoosewhichisthebestcorrelationpotentialbasedontheirspatialbehaviorastheyallshowsimilarshelloscillationsandthecorrectlongrangebehavior.TheRPApotentialwhichshowsalackofshortrangecorrelationneartonucleus.But,thisincorrectbehavioroftheRPApotentialhasnegligibleimpactontheHOMOeigenvaluesandlikelyarticiallyimprovesthetotalenergies.MBPT-2sc,lin-CCD,CCDandamethodlikeCCSDinparticular,offersanopportunitytogobeyondRPA.Therstthreeimprovethedescriptionofthedoubleamplitudes,whilethelastputsinthesinglesamplitudesthancannotbeconsideredintheRPA.TheRPAcanbeimprovedbytheintroducingoftheexchangetermsindoublesthusapproximatingthe 95

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doubleamplitudejustbytwo-electronintegral.ItcanbefurtherimprovedbysolvingthelinearCCDorCCDequation.TheequivalenceoftheRPAandring-CCD,providesastraight-forwardwaytogobeyondRPAwithoutworryingaboutthekernelinthedensity-densityresponsefunction.OurresultsshowthattheRPAfunctionalisastepintherightdirectioninthequestforanexactDFTfunctionalasitprovidesafunctionalandanequallygoodpotential,whosespatialbehavioriscorrectinouterpartofatomsanditsHOMOeigenvaluesaregoodapproximationstoexperimentalionizationenergies.Itisanon-localfunctionalwhichdescribesweakinteractionsattime-dependentDFTlevel.Itsuffersfromtheabsenceofshort-rangecorrelationasevidentfromthespatialbehaviorofthepotentialandstillhasaself-correlationproblem,whichcanbecircumventedinseveralways.OursuggestmethodsareoneofthewaytogobeyondRPA. 96

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CHAPTER6REFERENCEPROBLEMINTHERPAANDHOWTOGOBEYONDTHERPA? ItisagrowingfeelingamongtheDFTcommunitythatonlywaytoalleviatetheweaknessesofDFTresultsisbyusingtheknownexchange-correlationfunctionalfromwavefunctiontheory[ 20 ].Theadiabatic-connecteductuation-dissipation(AFCD)energyexpressionprovidestheexactexpressionfortheexchange-correlationenergy[ 108 ],[ 109 ].TheexactexchangeenergyintheAFCDenergyexpressionisnothingbuttheHartree-Fockexchangewiththereferenceorbitals[ 83 ]andcorrelationenergyisgivenintermsofthedensity-densityresponse[ 110 ],[ 81 ].TherearetwofundamentalproblemsassociatedwiththeevaluationoftheACFDenergy,rsthowtodealwiththekernel[ 111 ][ 25 ],[ 112 ]andhowtondthereferenceorbitalstoobtainthedensity-densityresponsefunction[ 90 ].However,byignoringthekernel,onecanmadethesimplestapproximationofthecorrelationenergywhichiscalledtheRandomPhaseApproximation(RPA)[ 110 ],[ 113 ],[ 114 ]asshowninthepreviouschapter 5 .Theplasmonformula[ 115 ]ortheequivalencebetweentheRPAandthering-CCD[ 116 ],[ 100 ],[ 101 ],simplifytheRPAformulaasneitherfrequencyintegrationnorcouplingstrengthintegrationisneeded[ 90 ].Also,itprovidesroutetogobeyondRPAwithoutevenworryingaboutthekernel,aslin-CCDorCCDorMBPT-2asshowninthepreviouschapter 5 .Toavoidconfusion,wewilldenetheRPAasthelinearresponseoftheHartree-Fockequation,inotherwordsthetime-dependentHartree-FockequationwiththeexchangeintegralignoredintheAandBmatrix( 6 ).ThismakesitatimedependentorthogonalHartreemethod.But,onestillneedstondthereferenceorbitals[ 90 ].TheHartree-Fockorbitalsseemsthenaturalchoice,aswehaveshownearlier 2 thattheHFdensityissuperiortotheSCF-KSdensity 2 .WebelievewhenoneusesthePBEorbitalstoevaluatetheRPAmeansalthoughPBEfunctionalisincorrect,theassociatedPBEexchange-correlationpotentialisconsideredtobecorrect,makingtheassociatedeigenvectorarecorrectaswell.Our,previousstudy,howevershowthat 97

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inherentdecienciesaresubduedinthefunctionalbutitismorepronouncedinthepotential,asonedoesnotevengettheHOMOconditionrightHOMO=)]TJ /F3 11.955 Tf 9.3 0 Td[(Ip). 6.1BestReferencefortheRPA AsisevidentfromEquation( 6 ),below,theRPAapproximationoftheadiabatic-connecteductuation-dissipation(AFCD)energyexpressionisequaltotheHartree-FockenergyandtheRPAcorrelationenergyevaluatedusingthereferenceorbitals(EACFD)]TJ /F4 7.97 Tf 6.58 0 Td[(RPA=EHF+ERPA).TheRPAamplitude( 6 )canbeobtainedbysolvingthetime-dependentHartreetypeequation( 6 ). EACFD)]TJ /F4 7.97 Tf 6.59 0 Td[(RPA=NXi=1+Zvext(~r)(~r)d~r+1 2Zd~rZd~r01 j~r)]TJ /F11 11.955 Tf 10.87 .5 Td[(~r0j(~r)(~r0))]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2occXi,jhijjjii+ERPA, (6) ERPA=1 2Tr(BT),T=YX)]TJ /F7 7.97 Tf 6.59 0 Td[(1, (6) 264ABBA375264XY375=!264100)]TJ /F5 11.955 Tf 9.3 0 Td[(1375264XY375, (6) Aia,jb=Fabij)]TJ /F3 11.955 Tf 11.95 0 Td[(Fijab+hijjabi, (6) Bia,jb=hibjaji, (6) 98

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TheRPAcorrelationenergiesareobtainedusingthreedifferentreferenceorbitals:HF,PBEandsemi-canonicalPBE.Usingtime-dependentDFTargumentsonecangetthesamesetofEquations( 6 ).However,thedifferencewillbeintheAmatrix( 6 ).DiagonalFockelementsintheAmatrix( 6 )willbecomecanonicaleigenvaluesoftheKSoperator.ThereisnothingwrongwiththatexceptitchangesthedenitionoftheRPA,asthelinearresponseoftheKSoperatorwithnokernelorthelinearresponseoftheFockequationwiththeexchangeoperatorisneglected.Thenalequationswillhavethesameformsexcepttheydependuponthereferenceorbitals.OnecanstilldenetheRPAasthelinearresponseoftheFockequationwiththeexchangeoperatorneglectedevenwhenusingthePBEtypeKSorbitals,however,oneneedstoconstructtheFockmatrixanddiagonalizetheoccupied-occupiedandthevirtual-virtualblockoftheFockmatrixtomaketheevaluationoftheAmatrixeasier.PresentstudyexploreallthreeoptionsandanswerthequestionwhichisthebestorbitalsfortheRPAcalculation? Figure6-1. RPAcorrelationenergies TheRPAcorrelationenergiesforthesphericalatomssuchasHe,Li+,Be++,Li,Be+,B++,Be,B+,C++,N,O+,F++,Ne,Na+,Mg++,Na,Mg+,Al++,Mg,Al+,Si++, 99

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Table6-1. ComparisonoftheRPAcorrelationenergies(inHartree)evaluatedusingdifferentreferenceorbitals.(HF,semi-canonicalPBE(sc-PBE)andPBEorbitals) RPAHFRPAsc)]TJ /F4 7.97 Tf 6.58 0 Td[(PBERPAsc)]TJ /F4 7.97 Tf 6.59 0 Td[(PBE+T1RPAPBERPAPBE+T1AverageError-0.0326-0.0324-0.0386-0.0942-0.1AverageAbsoluteError0.04500.04310.04770.09420.1 P,S+,Cl++andArareobtainedusingtheaug-pcS-4basis[ 117 ].Theexactcorrelationenergiesareobtainedfromthereference[ 118 ]Figure 6-1 andTable 7-11 showthatthereisnotmuchdifferencebetweenthedirect-ringRPAcorrelationenergyobtainedfromthesemicanonicalPBEorbitals(PBE-sc)andtheHartree-Fockorbitals.But,thedirect-ringRPAcorrelationenergiesobtainedusingthePBEorbitalsarequitedifferentfromtheabovetwo.Infactitoverestimatethecorrelationenergybymorethanfactorof2comparedtotheHF/PBE-scorbitals 7-11 .Anaverageabsoluteerrorof0.0942HartreeisobservedofthePBEorbitalscomparedtoanaverageabsoluteerror0.045fortheHForbitalsand0.0431forthePBE-scorbitals. 6.2HowtoGoBeyondRPA? ThetotalenergyofthesystemcanbewrittenasE=h0j^Hj0i+4E,wherethe^HistheHamiltonianofthesystemand0isconsideredtobeanindependentparticlereferencesuchastheHartree-FockorKSdeterminant.ThereferenceenergyEref=h0j^Hj0iisnothingbuttheHartree-FockenergywiththereferenceorbitalsandthecorrelationenergyEcor=4EcanbewrittenintermsofthecoupledclusterwavefunctionCC=exp(T)0as4E=hoj[HN(1+T1+T2+...)]cjoi.WherethenormalorderHamiltonianisthesumoftheFockoperatorandthetwo-electronoperator.Thecorrelationenergycanbewrittenintermsofthecoupledclustersinglesanddoublesas( 6 ). Ecor=4E=Xi,afi,atai+1 4Xi,j,a,bhijjjabi(tabij)]TJ /F3 11.955 Tf 11.96 0 Td[(taitbj)]TJ /F3 11.955 Tf 11.95 0 Td[(tbitaj), (6) 100

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Thedirect-RingRPAcorrelationenergyisdenedbyassumingtaiandtheexchangeintegralarezero( 6 ).AnaturalwaytogobeyondRPAistoincludethesinglescontributions.FortheHartree-FockreferencetaiiszerobutfortheKSorbitalsitisnot[ 4 ].SinglesaretaiandcouldbeobtainedbysolvingtheCCSDequationswithnocontributionfromthedoubles.TheRPAdoublestabijareobtainedbysolvingtheequation( 6 ).Thesolutionoftheequation( 6 )provideanopportunitytodeneRPAintwoquitedifferentwaysbutresultinghasthesameform. ERPA=1 2Xi,j,a,bhijjabitabij, (6) E=NXi=1+Zvext(~r)(~r)d~r+1 2Zd~rZd~r01 j~r)]TJ /F11 11.955 Tf 10.87 .5 Td[(~r0j(~r)(~r0))]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2occXi,jhijjjiiXi,afiatai+1 4Xi,j,a,bhijjjabi(taitbj)]TJ /F3 11.955 Tf 11.95 0 Td[(tbitaj)+ERPA, (6) E=EHF+T1+ERPA, (6) Wendthatfromthetable 7-11 thatthePBEorbitalsoverestimatetheRPAcorrelationenergybytwofoldcomparedtotheHForsemi-canonicalPBEorbitals.InordertoseethereferenceorbitalseffectonthetotalenergyEtotal=EHF(ref)+ERPA(ref),weconsidertheAr2bindingenergycurveasexample.ThebindingenergiesofAr2areevaluatedusingtheHF,PBEandsemi-canonicalPBE(sc-PBE)orbitals.TheRPAbindingenergiesarecomparedwiththeCCSD(T).aug-cc-pVTZbasissetisusedandbasissetsuperpositionerroristakenintoaccount.AsexpectedCCSDenergieshavevirtuallynodependanceonthereferenceorbitals,theCCSDbindingcurveofthe 101

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HForbitalsandthePBEorbitalsbeingontopofeachotheratallpoints 6-2 .TheRPAisequaltoring-CCDandthusacrudestapproximationofCCSDwithnosinglescouplinganddoublesarerestrictedtoring-diagramswithnoexchangeterms.Infactsinglesarecompletelyneglected.Thiscrudestapproximationisheavilydependentonthereferenceorbitals.ThebindingcurveforAr2usingHFandsemicanonicalPBEorbitalsorjustPBEorbitalsarequitedifferent.Table 7-11 showsthattheRPAcorrelationenergyevaluatedusingthePBEorbitalsisfarfromtheexactcorrelationenergiesascomparedtosemicanonicalPBEorbitalsortheHForbitals.Onewouldthenexpectthatsc-PBEshouldgiveabindingcurveforAr2thatismuchbetterthantheHFasthesc-PBEorbitalshascorrelationandrelaxationthusthereferenceenergywillbefarsuperiorthantheHForbitals.However,thegure 6-2 showthatitisworsethantheHForbitals.ItisinterestingtonotethatthereferenceenergythatistheHartree-fockenergyexpressionevaluatedusingthePBEorsc-PBEorbitalsandtheRPAcorrelationenergyofPBEorbitalsareinferiorcomparedtosc-PBEorbitalsRPAcorrelationenergies,butthePBEbindingcurveforAr2ismuchbetterthanthesc-PBEbindingcurveforAr2.Infact,itislittlebetterthantheHFbindingcurve 6-2 .TheRPAinthe 6-2 meansthereferenceplustheRPAcorrelationenergy. ThenaturalwaytogobeyondRPAistoincludesinglesintheenergyexpressionasshowninEquation( 6 ).Withoutcouplingbetweensinglesanddoubles,wewouldexpectnochangeinthebindingenergycurveofAr2withaHartree-fockreference,asfiaiszeroduetotheBrillouin'stheorem.ThisisnotthecasewiththePBEreferenceorbitalsasfiaisnotzeroandtheamplitudetaihavenitevaluesevenintheabsenceofcouplingbetweensinglesanddoubles.Since,thesemicanonicallytransformedPBEorbitalschangetheoccupied-occupiedandvirtual-virtualblock,whilethevirtual-occupiedblockremainsunaffected;singlescontributionwillbethesameinbothcases(PBEandsc-PBEreference).WerepresentsinglecontributionasT1inFigure 6-3 102

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Figure6-2. BindingenergycurveforAr2 Figure6-3. BindingenergycurveforAr2 103

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AsisevidentfromFigure 6-3 T1hasaremarkableeffectontheRPAbindingcurveoftheAr2obtainedusingPBEorbitals.Thesemi-canonicalPBEvaluesimprovebutnotasmuchasthePBEorbitalsRPAvalues.Infact,theRPAplusT1usingthePBEorbitalsisascorrectastheCCSD(T)bindingenergycurve.Thereasonfortheabovendingispotentiallyacancelationoferrorsascorrelationenergybecomesmorenegative 7-11 afteraddingsinglestothePBE-RPAcorrelationenergy.Doingtheproblemincorrectlywoundseemstoprovidethebestanswer.IfweallowT2andT1tocouplethenwewouldobtaintheCCSDresultshownin 6-2 byvirtueosThouless'theorem. 104

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CHAPTER7ISITPOSSIBLETOCONSTRUCTAPOTENTIALUSINGANA,B,CAPPROACHTODFTTHATWILLGIVEEIGENVALUESASAGOODAPPROXIMATIONTOTHEVERTICALIONIZATIONENERGIES Thesuccessofthedensityfunctionaltheoryreliesonagoodapproximationfortheunknownexchange-correlationfunctional,aswellasthecorrespondingexchange-correlationpotentialthatwillprovideagoodapproximationtotheexactgroundstatedensityasthesumofthelowestNorbitaldensities.However,itisnumericallyobservedthatareasonablycorrectdensityprovidesgoodfunctionalvalues.Asourstudyinchapter 2 showed,eventheHartree-fockdensitiesprovidebetterfunctionalvaluesthantheBLYPSCFdensityatallpointsonthepotentialenergysurface.OnewouldarguethattheKS-SCFprocedureisredundantinthatrespect,ifoneisjustlookingawaytogetadecentapproximationoftheexactgroundstatedensity[ 37 ].Thus,thepotentialqualitydoesnotseemstomatter.MostoftherecentadvancementsofDFTinndingbetterapproximationsoftheunknownexchange-correlationfunctionalechotheaboveargument[ 119 ].Onecanclassifysuchdevelopmentsasana,b,cap-proachtoDFT,intherespectthatmostofthesuccessfulfunctionalssuchasB3LYP[ 43 ],B2PLYP[ 66 ],M05[ 120 ],M06[ 121 ],etc.arebasedonthendingafewparametersa,b,andcwhosevaluesareobtainedusingmindlessdataminingtocombinecertainwelldenedfunctionalssuchLDA[ 122 ],LYP[ 39 ],PBE[ 32 ],Becke[ 31 ],MP2[ 123 ]functional,hartree-fockexchangeetc.Forinstance,thewidelysuccessfulfunctionalB2PLYP[ 66 ]has53%(aintheEquation( 4 ))ofthenon-localexchange-energy,b=47%(bintheEquation( 4 ))oftheBeckeexchangefunctionaland73%(cintheEquation( 4 ))LYPcorrelationenergy.However,theauthorsdoarguepresuminglythatrightanswerrequirebothlocalandnon-localexchangeandlocalandnon-localcorrelation,thelattergivenbyMBPT-2. However,ourdenitionofaconsistentDFTrequires(Chapter 3 )exchange-correlationpotentialtoprovideallverticalionizationenergiesintermsofthenegativeofthe 105

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eigenvaluesbysolvingtheSCFKSequation[ 56 ],Chapter 3 .Wehaveshowedinchapter 4 thatasemi-empiricalcombinationofdifferentfunctionalssuchasaMP2correlationfunctional,LYPcorrelationfunctional,Hartree-FockexchangefunctionalandBeckeexchangefunctionaldonotprovideconsistentpotentialsaswouldberequiredinaconsistentDFT. Inthisstudy,wewanttousethea,b,capproachoftheKS-DFTbut,insteadofdesigninganexchange-correlationfunctionalwewilldesignanexchange-correlationpotentialrequiringthatitprovidesareasonableapproximationofallverticalionizationenergiesasthenegativeoftheSCFKSeigenvalues.Inotherwords,wearetryingtondthebestenergyindependentapproximationoftheself-energyoperatoroftheDysonequationintermsofexplicitdensitydependentpotentials.InordertoconceiveofsuchapotentialweneedtosuppresstheinherentdecienciesoftheDFTfunctional.InabilityofDFTpotentialstoprovideagoodapproximationoftheverticalionizationenergiesintermsoftheKSeigenvalueshasbeenattributedtothelackofthe'integerdiscontinuity'[ 23 ]inthecorrespondingexplicitdensitydependentfunctional,whichstemsfromthepresenceoftheself-interactionerrorinthefunctionals[ 35 ].Numericaldatafromchapter 3 showedthatthepresenceofanoptimumamountofthenon-localexchangenotonlysuppressessuchinheritdecienciesbutisvitalforgettingcoreionizationenergiesintermsofthenegativeoftheKSeigenvalues.However,suchdecienciesofthefunctionalaresopronouncedinthecorrespondingpotentialthatitishardtosuppresstheireffectontheeigenvalues.Thisisthemainreasonforthecommonlyusedexplicitdensitydependentpotentialtonotprovideareasonabledescriptionofverticalionizationenergies(seeChapter 3 ). Thepresentstudyaimsatndingsuchamagicnumbersofcoefcienta,b,cthatwillcombinewellknownpureDFTpotentialsinsuchawaythatwecangetionizationenergiesasthenegativeoftheKSeigenvalueswiththemagicpotentialintheKSoperator.Since,theexchangecorrelationpotentialareuniversalinthesensethatthey 106

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donotdependupontheexternalpotentialasitestimateselectron-electroninteraction[ 5 6 ],onemoleculeshouldbethegoodenoughtondourmagicnumbersa,b,cetc.Itmightnotbetheuniquenumbersandothernumbersaredenitelypossible.Wedonotstrivefortheultimatepotentialwiththea,b,cmagicnumbers.Weintendtoshowthatthea,b,capproachinnothingbutawaytosuppresstheinherentdecienciesoftheexplicitdensitydependentpotentialinacomputationallyelegantmanner. 7.1PerformanceofBHLYPvwnPotential Wehavedesignedahybridtypepotential(BHLYPvwn)( 7 )thatcontains0.45Becke88localexchange[ 31 ],0.55non-localexchange(Hartree-Focklikeexchange),1.00LYP[ 39 ]correlationand1.17VWN[ 124 ]correlationpotential.Parametersareobtainedusingwaterasthetestexample. ^VBHLYPvwnxc=0.45^vBECKEx+0.55^Vnlxx+^vLYPc+1.17^vVWNc, (7) FBHLYPvwn=h+P(j)+0.45(vBECKEx)+0.55(Vnlxx)+(vLYPc)+1.17(vVWNc), (7) FBHLYPvwn'i=BHLYPvwni'i, (7) Self-consistentKScalculation( 7 )areperformedonPyridine,Thiophene,Furan,Acrolein,CO,HCCCN,O3,H2CCCl2,CHF=CF2,HCOOH,OCS,FCN,CS,SiO,CH3CN,CH3F,C2H4,H2CS,HCONH2,NH3,CH3COCH3,CH4,CH3CCH,CH3NC,CH2F2,C2H6,H2O,P2,HCN,C2H2,C2N2,CO2,C3O2,N2,HF,s)]TJ /F3 11.955 Tf 12.85 0 Td[(tetrazine,F2,H2CO,HCl,CHF3,N2O,CH2F2,C2H6andCHF3.ExperimentalgeometriesaretakenfromtheNIST[ 59 ]databaseandexperimentalionizationenergiesaretakenfromthereference[ 60 ]and[ 61 ].DataintheTable 7-1 suggeststhatourdesignedpotentialprovidestheKS 107

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eigenvaluespectra(BHLYPvwncore,BHLYPvwnvalenceandBHLYPvwnHOMO)betterthancommonlyavailablepotentials.FortheourexamplesweobtainHOMOionizationenergieswithin0.5eV,coreionizationenergieslessthan1.5evandoverallionizationenergies(BHLYPvwni-Ip)around1eV. Table7-1. ComparisonofthenegativeoftheFBHLYPvwneigenvalueswiththeexperimentalverticalionizationenergiesisgiven,second,third,fourthandfthcolumnhasall,core,valenceandHOMOorbitalsenergydeviationsrespectively. )]TJ /F11 11.955 Tf 9.3 0 Td[(BHLYPvwni)]TJ /F3 11.955 Tf 11.96 0 Td[(Ii)]TJ /F11 11.955 Tf 9.3 0 Td[(BHLYPvwncore)]TJ /F3 11.955 Tf 11.96 0 Td[(Ii)]TJ /F11 11.955 Tf 9.29 0 Td[(BHLYPvwnvalence)]TJ /F3 11.955 Tf 11.96 0 Td[(Ii)]TJ /F11 11.955 Tf 9.3 0 Td[(BHLYPvwnHOMO)]TJ /F3 11.955 Tf 11.95 0 Td[(IiAverageError0.991.330.860.32AverageAbsoluteError1.061.460.900.44MaximumDeviation4.154.153.261.23MinimumDeviation0.000.040.000.00 ^VBHLYPvwnxc=@EBHLYPvwnxc @, (7) EBHLYPvwnxc=0.45EBECKEx+0.55Enlxx+ELYPc+1.17EVWNc, (7) ESCFBHLYPvwn=ENN+hhPi+1 2Z(~r1)(~r2) j~r1)]TJ /F11 11.955 Tf 10.87 .49 Td[(~r2j+EBHLYPvwnxc+1.17EVWNc, (7) ESCF=EN)]TJ /F3 11.955 Tf 11.95 0 Td[(EN)]TJ /F7 7.97 Tf 6.59 0 Td[(1 (7) FsTOMi=FNKS+FN)]TJ /F7 7.97 Tf 6.58 0 Td[(1KS 2=^h+Xj6=i(jjj)+1 2(iji)+Vxc[sTOM] (7) 108

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sTOM=N+N)]TJ /F7 7.97 Tf 6.59 0 Td[(1 2 (7) FsTOMBHLYPvwn=^h+Xj6=i(jjj)+1 2(iji)+^VBHLYPvwnxc[sTOMBHLYPvwn], (7) ConstructingtheparentBHLYPvwnfunctional( 7 )fromtheBHLYPvwnpotential( 7 ),weexpectedthatitwillhavetoomuchcorrelationandisalsonumericallymanifestedinESCF( 7 )andSlatertransitionoperatormethodcalculation( 7 )(sTOM)[ 125 ](Table 7-2 ),withtheionizationenergiesobtainedhavingmorethan2eVerrorsontestset1(HF,HCN,CO,N2,C2H2,F2,NH3,H2O,H2CO,CH4andC2H4).Since,sTOMiscorrectthroughsecondorder[ 126 ],wendtheequivalencebetweentheESCFandsTOMionizationenergiesasexpected.InconsistenciesbetweenthefunctionalandpotentialareevidentastheeigenvaluesarequitedifferentfromtheESCFandsTOMobtainedionizationenergies(FsTOMBHLYPvwn[sTOMBHLYPvwn])ii. Table7-2. ComparisonoftheionizationenergiesobtainedbyESCFBHLYPvwnwiththeexperimentalverticalionizationenergies,orbitalenergiesandwiththeionizationenergiesobtainedfromslatertransitionoperatormethod. ESCFBHLYPvwn)]TJ /F3 11.955 Tf 11.96 0 Td[(Ii2.76ESCFBHLYPvwn)]TJ /F5 11.955 Tf 11.96 0 Td[((BHLYPvwn)ii2.28ESCFBHLYPvwn)]TJ /F5 11.955 Tf 11.96 0 Td[((FsTOMBHLYPvwn)ii0.17 FBHLYP=h+P(j)+0.45(vBECKEx)+0.55(Vnlxx)+(vLYPc) (7) ESCFBHLYP=ENN+hhPi+1 2Z(~r1)(~r2) j~r1)]TJ /F11 11.955 Tf 10.86 .5 Td[(~r2j+0.45EBECKEx+0.55Enlxx+ELYPc, (7) 109

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Self-consistentKScalculationsareperformedwithouttheVWNpotentialintheKSoperator( 7 )onthetestset1.TotalenergiesarecalculatedwithouttheVWNfunctionalinthetotalenergyexpression( 7 ).TheobtainedESCFionizationenergiesare0.63eVAAEdifferentfromtheexperimentalverticalionizationenergies.ESCFandsTOMionizationenergiesarecomparablewitheachotherwith0.19eVAAE(Table 7-3 ).WhichisnotsurprisingasmostoftheDFTfunctionalprovideagoodcharacterizationoftheionizationenergyintermsoftheESCFandsTOMvalues.Howeverorbitalenergiesarenotwithinadesirableaccuracy.TheESCFionizationenergiesandtheeigenvaluesdifferfromeachotherby1.9eVAAE(Table 7-3 ). Table7-3. ComparisonoftheionizationenergiesobtainedbyESCFBHLYPwiththeexperimentalverticalionizationenergies,orbitalenergiesandwiththeionizationenergiesobtainedfromslatertransitionoperatormethod. ESCFBHLYP)]TJ /F3 11.955 Tf 11.95 0 Td[(Ii0.63ESCFBHLYP)]TJ /F5 11.955 Tf 11.95 0 Td[((BHLYP)ii1.90ESCFBHLYP)]TJ /F5 11.955 Tf 11.95 0 Td[((FsTOMBHLYP)ii0.19 E^BHLYPvwn[BHLYPvwn,'BHLYPvwn]=ENN+hhePi+1 2Z~(~r1)~(~r2) j~r1)]TJ /F11 11.955 Tf 10.86 .49 Td[(~r2j+0.45eEBECKEx+0.55eEnlxx+eELYPc, (7) WeremovedtheVWNfromthefunctional( 7 )butkeptitintheKSoperator( 7 )andndthatESCFionizationenergies(ESCF^BHLYPvwn),orbitalenergies(BHLYPvwn)ii,andexperimentalionizationenergiesarequitecomparabletoeachotherbutquitedifferentfromsTOMionizationenergies(FsTOMBHLYPvwn)ii(Table 7-4 ). FromtheexactHOMOdensityconditionweknowthattheHOMOeigenvalueshouldbeequaltotheexactionizationenergies[ 5 6 ],coupledwiththefactthatE isdeneduptoanarbitraryconstant[ 5 6 ].ThuswearefreetoshifteachorbitalbytheHOMOcorrection(differencebetweenHOMOeigenvalueandexactionization 110

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Table7-4. ComparisonoftheionizationenergiesobtainedbyESCF^BHLYPvwnwiththeexperimentalverticalionizationenergies,orbitalenergiesandwiththeionizationenergiesobtainedfromslatertransitionoperatormethod. ESCF^BHLYPvwn)]TJ /F3 11.955 Tf 11.96 0 Td[(Ii0.63ESCF^BHLYPvwn)]TJ /F5 11.955 Tf 11.96 0 Td[((BHLYPvwn)ii0.40ESCF^BHLYPvwn)]TJ /F5 11.955 Tf 11.96 0 Td[((FsTOMBHLYPvwn)ii2.45 energies).ShiftingtheFBHLYPeigenvaluesbytheHOMO-corrections,wendthatverticalionizationenergiesprovideanimproveddescriptionoftheKSeigenvalues(Table 7-5 ).However,itrequirestheknowledgeoftheHOMO-corrections.WecanalsoaccomplishthisbyintroducingtheVWNcorrelationpotentialintheKSoperator.Ourresultsshowthatit'seffectonthedensityisnegligiblesoitsactslikeasystemdependentconstantshiftforeachorbital. Table7-5. DeviationsoftheverticalionizationenergiesfromthenegativeoftheHOMOcorrectedeigenvaluesofFBHLYP,FBHLYPandFBHLYPvwnKSoperator. FBHLYPKSoperatoreigenvalue1.45HOMO-correctedeigenvaluesofFBHLYP0.53FBHLYPvwnKSoperatoreigenvalue0.63 WealsondthatourconcoctedpotentialperformancesremainthesameifwereplacetheBeckeexchangewithPBExorPW91x,givingamaximumorbitaldifferenceofaround0.15eVobservedforthetestset1. Ournumericaldatesuggestthatweabletosuppresstheinherentdeciencyoftheexplicitdensitydependentpotentialtoobtainmagica,b,cnumbersandconcoctedtheBHLYvwnpotentialtogivetheverticalionizationenergiesasthenegativeoftheeigenvalueswithinthedesirableaccuracyofoneeV.However,thecorrespondingfunctionaldoesnotprovideaccuratetotalenergyproperties,thuswendtheionizationenergiesfromtheESCFandsTOMarenotgood,whichisquitecontrarytothebehavior 111

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ofmostotherDFTfunctionals.However,removingtheextracorrelationfunctional(i.e.scaledVWN)fromtheBHLYvwnfunctionalweobtaintheESCFionizationenergieswithinthedesiredaccuracyofoneeV.TheVWNcorrelationfunctionalisadensitydependentfunctionalbutitseffectoneachorbitalisalmostconstant,thusactingasaconstantshift.ThusshiftingtheBHLYPeigenvaluesbytheHOMOcorrectiongivesthesameerrorintheverticalionizationdeterminationasinBHLYvwn.Weknowthattheexchange-correlationpotentialsaredeneduptoanarbitraryconstant.ThusitappearsthattheVWNpotentialintheKSoperatoristhemissingarbitraryconstant.ThatalsoaccountforthelackofderivativediscontinuityintheDFTfunctional. 7.2QualityoftheBHLYPvwnDensityandOrbitals AsshowninChapter 2 thatuncorrelateddensitysuchasHartree-FockdensityprovidebettercharacterizationofbarrierheightsthanthecorrelateddensityobtainedfromtheBLYPpotential.AsshowninTable 7-11 PBEpotentialhasthesimilarbehavior.Barrierheightfor22reactiononQCISDoptimizedgeometriesusingMG3basissetareobtainedwithBHLYPvwndensityandorbitals.ExplicitdensitydependentfunctionallikePBE,BLYPwhenevaluatedusingcorrespondingSCFKSdensityunderestimatethebarrierheightsastheytendtomaketransitionstatemorestableasevidentfromthetables 2-1 and 7-11 .TheBLYPfunctionalevaluatedwithSCF-BLYPdensityunderestimatethebarrierheightby8.3Kcal/mol(Table 2-1 )forourtestcaseswhilethePBEfunctionalevaluatedusingSCF-PBEdensityunderestimatethebarrierheightby9.7Kcal/mol(Table 7-11 ).Ourstudiesaboutthenon-variationalDFT(Chapter 2 )showedthatshortcomingsofthefunctionalareverypronouncedatthefunctionallevelthanatthecorrespondingpotentiallevelasthequantityobtainedfromthepotentiale.g.eigenvectors(density)andeigenvaluesarefromthedesiredaccuracy.Ourconcoctedpotentialprovidesremarkableimprovementsineigenvaluesasthenegativeofeigenvaluesbecomeagoodapproximationoftheverticalionizationenergies.However,inordertoseethatthesuppressionoferrorinthexc-potential 112

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manifestedinthedensityandtheeigenvectorsatallpointsinthepotentiallyenergysurfacespeciallypointswhereexplicitdensitydependentpotentialhavehardtimeprovideareasonablecorrectdensity(Thepointswherebondsarepartiallyformedorbrokeninthepotentialenergysurfaceareoneofthosepoints),barrierheightfor22reactiononQCISDoptimizedgeometriesusingMG3basissetareobtainedwithBHLYPvwndensityandorbitals.ExplicitdensitydependentfunctionalBHLYP(Equation( 7 ))areevaluatedfor22reactionsusingtheBHLYPvwndensityandorbitals.Datafromthetable(Table 7-6 )showthatbarrierheightobtainedareinareasonablygoodagreementwiththebestreportedvaluesfromtheliteratureasAAEof2.7Kcal/mol(Table 7-6 )areobserved.Time-dependentDFTformofRPAfunctional( 6 )areevaluatedusingtheBHLYPvwnorbitalsfor21reactionsandbarrierheightobtainedareinareasonablygoodagreementwiththebestreportedvaluesfromtheliteratureasAAEof2.9Kcal/mol(Table 7-7 )areobserved.IncludingsinglesfurtherimproveourbarrierheightestimationastheAAEfor21reactioncomesdownto1.6Kcal/mol(Table 7-8 ).However,time-dependentHFformofRPAareevaluatedusingtheBHLYPvwnorbitalsfor21reactionsgiveanAAEof4.5Kcal/mol(Table 7-9 )whichimproveswiththeinclusionofT1asAAEgoesdownto2.9Kcal/mol(Table 7-10 ).ItissurprisingtoseethatbarrierheightsobtainedbyevaluatingTD-DFTformoftheRPAwiththePBEorbitalshaveanAAEof2.2Kcal/mol(Figure 7-1 ,Table 7-11 ),butanefforttoimproveRPAenergybytheinclusionofsingleresultedintoanoverestimationofbarrierheightsastheAAEincreasefrom2.2Kcal/molto7.2Kcal/mol(Figure 7-1 ,Table 7-11 ).WeobtaincontrastingbehaviorwhenbarrierheightsareobtainedusingPBEorbitalswithTD-HFformoftheRPAastheAAEisdecreasedafterinclusionofsinglesfrom6.3Kcal/molto2.8Kcal/mol(Figure 7-1 ,Table 7-11 ).SuchunpredictabilityisnotobservedforBHLYPvwnorbitals(Figure 7-1 ,Table 7-11 )inclusionofsinglesalwaysimprovestheresults. 113

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Table7-6. BarrierheightofthehydrogentransferreactionusingBHLYPvwndensityandBHLYPfunctional ReactionsRefBHLYPvwn Cl+H2)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>HCl+HVf8.78.6Vb5.62.2OH+H2)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H+H2OVf5.77.9Vb22.016.8CH3+H2)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>H+CH4Vf12.112.2Vb15.013.2OH+CH4)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>CH3+H2OVf6.710.9Vb20.218.8H+CH3OH)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>CH2OH+H2Vf7.38.3Vb13.815.7H+H2)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>H2+HVf9.67.1Vb9.67.1OH+NH3)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H2O+NH2Vf3.29.4Vb13.216.5HCl+CH3)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>Cl+CH4Vf1.82.5Vb7.89.9OH+C2H6)]TJ /F11 11.955 Tf 12.62 0 Td[(>H2O+C2H5Vf3.48.3Vb20.720.0F+H2)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>H+HFVf1.82.4Vb33.224.7OH+CH3)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>O+CH4Vf7.89.2Vb13.717.2H+PH3)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>PH2+H2Vf3.22.3Vb25.225.4H+ClH0)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>HCl+H0Vf18.017.8Vb18.017.8OH+H)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>H2+OVf10.18.0Vb13.115.0H+t)]TJ /F3 11.955 Tf 11.96 0 Td[(N2H2)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H2+N2HVf5.91.6Vb41.042.1H+H2S)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>H2+HSVf3.63.0Vb17.419.43O+HCl)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>OH+ClVf9.814.3Vb9.913.7CH4+NH)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>NH2+CH3Vf22.724.3Vb8.410.9C2H6+NH)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>NH2+C2H5Vf18.422.0Vb8.012.5C2H6+NH2)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>NH3+C2H5Vf10.415.7Vb17.820.4NH2+CH4)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>CH3+NH3Vf14.518.0Vb17.918.8s)]TJ /F3 11.955 Tf 11.95 0 Td[(transcis)]TJ /F3 11.955 Tf 11.95 0 Td[(C5H8)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>s)]TJ /F3 11.955 Tf 11.96 0 Td[(transcis)]TJ /F3 11.955 Tf 11.96 0 Td[(C5H8Vf38.445.8Vb38.445.8AverageError-1.1AverageAbsoluteError2.7AbsoluteMaximumError8.5AbsoluteMinimumError0.0 114

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Table7-7. BarrierheightofthehydrogentransferreactionusingBHLYPvwndensityandHF+RPAfunctional ReactionsRefBHLYPvwn Cl+H2)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>HCl+HVf8.711.4Vb5.66.2OH+H2)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>H+H2OVf5.79.2Vb22.022.4CH3+H2)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>H+CH4Vf12.114.2Vb15.018.1OH+CH4)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>CH3+H2OVf6.710.1Vb20.219.5H+CH3OH)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>CH2OH+H2Vf7.313.1Vb13.817.5H+H2)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H2+HVf9.612.8Vb9.612.8OH+NH3)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H2O+NH2Vf3.27.0Vb13.215.5HCl+CH3)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>Cl+CH4Vf1.82.7Vb7.811.8OH+C2H6)]TJ /F11 11.955 Tf 12.62 0 Td[(>H2O+C2H5Vf3.47.1Vb20.719.5F+H2)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H+HFVf1.84.3Vb33.231.4OH+CH3)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>O+CH4Vf7.89.4Vb13.718.5H+PH3)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>PH2+H2Vf3.26.3Vb25.228.1H+ClH0)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>HCl+H0Vf18.022.6Vb18.022.6OH+H)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>H2+OVf10.112.3Vb13.117.5H+t)]TJ /F3 11.955 Tf 11.95 0 Td[(N2H2)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>H2+N2HVf5.95.6Vb41.042.4H+H2S)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H2+HSVf3.67.0Vb17.422.1O+HCl)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>OH+ClVf9.814.1Vb9.914.1CH4+NH)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>NH2+CH3Vf22.725.9Vb8.410.5C2H6+NH)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>NH2+C2H5Vf18.423.5Vb8.011.0C2H6+NH2)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>NH3+C2H5Vf10.415.0Vb17.818.9NH2+CH4)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>CH3+NH3Vf14.517.6Vb17.918.6AverageError-2.7AverageAbsoluteError2.9AbsoluteMaximumError5.8AbsoluteMinimumError0.3 115

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Table7-8. BarrierheightofthehydrogentransferreactionusingBHLYPvwndensityandHF+RPA+T1functional ReactionsRefBHLYPvwn Cl+H2)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>HCl+HVf8.710.6Vb5.64.5OH+H2)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>H+H2OVf5.76.8Vb22.019.5CH3+H2)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>H+CH4Vf12.113.1Vb15.016.2OH+CH4)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>CH3+H2OVf6.77.8Vb20.217.5H+CH3OH)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>CH2OH+H2Vf7.310.7Vb13.816.6H+H2)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H2+HVf9.610.9Vb9.610.9OH+NH3)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H2O+NH2Vf3.24.3Vb13.212.9HCl+CH3)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>Cl+CH4Vf1.81.6Vb7.810.9OH+C2H6)]TJ /F11 11.955 Tf 12.62 0 Td[(>H2O+C2H5Vf3.44.5Vb20.717.4F+H2)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H+HFVf1.80.4Vb33.227.1OH+CH3)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>O+CH4Vf7.87.6Vb13.716.1H+PH3)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>PH2+H2Vf3.23.0Vb25.226.0H+ClH0)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>HCl+H0Vf18.019.8Vb18.019.8OH+H)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>H2+OVf10.19.6Vb13.115.0H+t)]TJ /F3 11.955 Tf 11.95 0 Td[(N2H2)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>H2+N2HVf5.91.2Vb41.039.7H+H2S)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H2+HSVf3.63.9Vb17.420.0O+HCl)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>OH+ClVf9.89.6Vb9.910.4CH4+NH)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>NH2+CH3Vf22.724.1Vb8.48.9C2H6+NH)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>NH2+C2H5Vf18.421.5Vb8.09.6C2H6+NH2)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>NH3+C2H5Vf10.413.4Vb17.817.6NH2+CH4)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>CH3+NH3Vf14.516.2Vb17.917.2AverageError-0.4AverageAbsoluteError1.6AbsoluteMaximumError6.1AbsoluteMinimumError0.1 116

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Table7-9. Barrierheightofthehydrogentransferreactionusingsc-BHLYPvwnorbitalsandHF+RPAfunctional ReactionsRefBHLYPvwn Cl+H2)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>HCl+HVf8.712.3Vb5.67.8OH+H2)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>H+H2OVf5.710.9Vb22.024.5CH3+H2)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>H+CH4Vf12.115.4Vb15.019.7OH+CH4)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>CH3+H2OVf6.712.3Vb20.221.6H+CH3OH)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>CH2OH+H2Vf7.314.9Vb13.818.5H+H2)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H2+HVf9.614.1Vb9.614.1OH+NH3)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H2O+NH2Vf3.210.2Vb13.218.3HCl+CH3)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>Cl+CH4Vf1.84.1Vb7.812.9OH+C2H6)]TJ /F11 11.955 Tf 12.62 0 Td[(>H2O+C2H5Vf3.49.4Vb20.721.5F+H2)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H+HFVf1.86.1Vb33.233.3OH+CH3)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>O+CH4Vf7.811.1Vb13.720.9H+PH3)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>PH2+H2Vf3.27.7Vb25.228.8H+ClH0)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>HCl+H0Vf18.024.6Vb18.024.6OH+H)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>H2+OVf10.114.2Vb13.119.7H+t)]TJ /F3 11.955 Tf 11.95 0 Td[(N2H2)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>H2+N2HVf5.97.6Vb41.043.7H+H2S)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H2+HSVf3.68.5Vb17.422.9O+HCl)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>OH+ClVf9.817.4Vb9.916.5CH4+NH)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>NH2+CH3Vf22.727.9Vb8.412.2C2H6+NH)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>NH2+C2H5Vf18.425.6Vb8.012.7C2H6+NH2)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>NH3+C2H5Vf10.416.9Vb17.821.0NH2+CH4)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>CH3+NH3Vf14.519.5Vb17.920.7AverageError-4.5AverageAbsoluteError4.5AbsoluteMaximumError7.6AbsoluteMinimumError0.1 117

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Table7-10. Barrierheightofthehydrogentransferreactionusingsc-BHLYPvwnorbitalsandHF+RPA+T1functional ReactionsRefBHLYPvwn Cl+H2)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>HCl+HVf8.711.6Vb5.66.1OH+H2)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>H+H2OVf5.78.6Vb22.021.6CH3+H2)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>H+CH4Vf12.114.3Vb15.017.8OH+CH4)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>CH3+H2OVf6.710.1Vb20.219.6H+CH3OH)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>CH2OH+H2Vf7.312.4Vb13.817.6H+H2)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H2+HVf9.612.2Vb9.612.2OH+NH3)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H2O+NH2Vf3.27.4Vb13.215.7HCl+CH3)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>Cl+CH4Vf1.83.0Vb7.812.0OH+C2H6)]TJ /F11 11.955 Tf 12.62 0 Td[(>H2O+C2H5Vf3.46.7Vb20.719.4F+H2)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H+HFVf1.82.2Vb33.229.0OH+CH3)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>O+CH4Vf7.89.4Vb13.718.5H+PH3)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>PH2+H2Vf3.24.4Vb25.226.7H+ClH0)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>HCl+H0Vf18.021.8Vb18.021.8OH+H)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>H2+OVf10.111.6Vb13.117.1H+t)]TJ /F3 11.955 Tf 11.95 0 Td[(N2H2)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>H2+N2HVf5.93.3Vb41.041.0H+H2S)-222()]TJ /F11 11.955 Tf 24.58 0 Td[(>H2+HSVf3.65.5Vb17.420.9O+HCl)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>OH+ClVf9.812.9Vb9.912.8CH4+NH)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>NH2+CH3Vf22.726.1Vb8.410.7C2H6+NH)-222()]TJ /F11 11.955 Tf 24.57 0 Td[(>NH2+C2H5Vf18.423.6Vb8.011.4C2H6+NH2)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>NH3+C2H5Vf10.415.3Vb17.819.7NH2+CH4)-221()]TJ /F11 11.955 Tf 24.57 0 Td[(>CH3+NH3Vf14.518.1Vb17.919.3AverageError-2.2AverageAbsoluteError2.7AbsoluteMaximumError5.2AbsoluteMinimumError0.0 118

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Table7-11. ComparisonoftheerrorsassociatedwithbarrierheightestimationinKcal/mol AverageErrorAverageAbsoluteErrorAbsoluteMaximumErrorAbsoluteMinimumErrorHartreeFock-11.211.624.73.0RPAwithHForbitals-4.75.611.11.3PBE9.79.719.84.6RPAwithPBEorbitals-0.42.27.10.0RPA+T1withPBEorbitals6.97.233.00.3RPAwithsc-PBEorbitals-6.36.316.73.0RPA+T1withsc-PBEorbitals1.12.814.50.1BHLYPwithBHLYPvwnorbitals-1.12.78.50.0RPAwithBHLYPvwnorbitals-2.72.95.80.3RPA+T1withBHLYPvwnorbitals-0.41.66.10.1RPAwithsc-BHLYPvwnorbitals-4.54.57.60.1RPA+T1withsc-BHLYPvwnorbitals2.22.75.20.0 Figure7-1. ComparisonoftheerrorsassociatedwithbarrierheightestimationinKcal/mol InordertofurtherquantifythequalityoftheBHLYPvwnorbitals,bindingenergycurvesofHe2,Ne2andAr2areevaluatedwithHF,PBEandBHLYPvwnorbitalsusingRPAfunctionalinTD-DFTorTD-HFframeworkandalsowiththeinclusionofsingles.TheHe2andNe2bindingenergycurverevealsthatPBEorbitalsarenotusefulinboththeframeworkwithouttheinclusionofsingles,howeveraddingsinglesintheRPAwithinTD-DFTframeworkoverestimatebindingenergy,whileinTD-HFframeworkimprovementisconsistent.But,forAr2bindingenergycurveaddingsingletoTD-DFTRPAfunctional,PBEorbitalsaccidentlybecomeequaltoCCSD(T).However,such 119

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unpredictabilityisnotseenwiththeHForbitalsortheBHLYPvwnorbitals.Ourresultsindicatethatsuppressionofdecienciesoffunctionalinpotentialusingthea,b,capproachtoDFTnotonlyimprovetheeigenvaluesbutalsotheeigenvectorsthusdensityaswell. Figure7-2. RPABindingenergycurveforHe2usingdifferentorbitals Figure7-3. RPABindingenergycurveforNe2usingdifferentorbitals 120

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Figure7-4. RPABindingenergycurveforAr2usingdifferentorbitals 121

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CHAPTER8CONCLUSION ThestudycanbesummarizedasanefforttowardsincreasingtheapplicabilityofKohn-Shamdensityfunctionaltheory.DFT,inprinciple,isanexactcorrelatedsingleparticletheoryasitdependsonaone-particlequantityi.e.density.However,inpractice,DFTisfarfromexactduetothepresenceofaexchangeandcorrelationfunctionalandhowtoobtainanexactdensity.Thereareamenagerieofvariousapproximatefunctionalsavailablewithinherentdeciencies,andtheKohn-ShamSCFprocedureisoftenusedasanauxiliaryconstructtoobtaintheexactgroundstatedensity.ThequantitiesassociatedwiththeKS-SCFproceduresuchastheexchange-correlationpotential,eigenvalueswhichareoftenconsideredasquantitieswithnophysicalrelevanceexceptthesumoftheNlowesteigenvectorsshouldgivethenon-interactingelectrondensityequaltotheexactgroundstateinteractingelectrondensity. Beingasingle-particletheory,theKS-SCFprocedureoffersanattractiveadvantageoverhonestmany-bodytheory,asitonlyrequirescalculationslikethoseofHF.AelectronicstructureofamoleculesisofferedbytheKSeigenvectorsandifKS-DFTprovidesagoodapproximationoftheverticalionizationforalloccupiedorbitalsasthenegativeoftheKS-eigenvalues.ThequalityoftheDFTresultsnotonlydependontheDFTfunctionalbutalsoontheKSeigenvaluesandeigenvector.MostoftherecentadvancementsintheDFTareconcentratedonimprovingthequalityoftheexchange-correlationfunctionalbyusingmindlessdatamining,ignoringcompletelytheimportanceofthequalityoftheexchange-correlationpotential.EffortstowardsincreasingtheapplicabilityofDFTdemandsaconsistentDFT.AconsistentDFTrequiresthepotentialandtheassociateddensitytohavethesameaccuracyasthefunctional.Thepresentstudyshowstheneedfordevelopingsuchaconsistencyconditions.Todate,theonlyKS-DFTmethodtoachievethisisourOEP2-scapproach. 122

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Inchapter 2 wequestionthequalityoftheKSdensityobtainedbyusingtheexplicitdensitydependentpotential(BLYP)intheKSoperator.TheBLYPpotentialareobtainedbyfunctionaldifferentiatingtheBLYPfunctionalwithrespecttothedensity.Theinherentdecienciesofthefunctionallyarenotvisibleinthefunctionalevaluationbutaremorepronouncedinthepotential,astheuncorrelatedHartree-FockdensityprovidebetterfunctionalvaluesthanthecorrespondingcorrelatedKSeigenvectorsatallpointsinthepotentialenergysurface.OurstudynotonlyquestionstheuseoftheKSSCFprocedurewithapproximatepotentialintheKSoperatorbutalsoprovideanon-variationalDFTschemetoestimatebarrierheightsandtransitionstatetoofferaccuraterateconstantcalculationfromasingle,HF-DFThybridapproach. Inchapter 3 wenotonlyobtainednumericaldatatosupportBartlett'stheoremthatthenegativeoftheKSeigenvaluesshouldbeagoodapproximationfortheprincipleverticalionizationenergies,buttoalsoshowthatthenon-localexchangepotential(HF-likeexchangeoperator)intheKSoperatorisvitalforthecoreorbitalionization.NoneoftheavailableexplicitdensitydependentfunctionalsatisfyBartletts'stheoremoreventheHOMOcondition(HOMO=)]TJ /F3 11.955 Tf 9.3 0 Td[(Ip). Bycomparingmanyformsoffunctionalstoexperiment,enablessemi-empiricalKS-DFTtogeneratemanyfunctionalssuchasthedoublehybridortheB2-PLYPfunctional.However,ourstudyshowsthatitdoesnotsuppresstheinherentdeciencyoftheDFTfunctionalasthecorrespondingpotentialdonotsatisfytheconsistentDFTconditions(Chapter 4 ).WeknowthatinherentshortcomingsoftheDFTfunctionalaremorepronouncedinthecorrespondingpotentialthanthefunctional,thusitishardtosuppressthosedecienciesinthepotential.However,weareabletoconstructanexplicitdensitydependentpotentialthatsuppresstheinherentdeciencies.Insteadoftakingasetofmoleculesandttingthe(a,b,c)parameteronthose,wejusttookwaterasatestcasedependingupontheuniversalityofthefunctional.Theseparameterarenotuniquebuttheirrangeis.Theconcoctedpotential(BHLYPvwn)showaneVerrorin 123

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thedeterminationofionizationenergiesintermsoftheKSeigenvalues.Theidea,nottoconcoctapotentialthatwillprovideionizationenergiesintermsofKSspectra,buttoshowthatwithlittleartonecansuppresstheinherentdecienciesoftheexplicitdensitydependentfunctionalandobtainanydesirableresults.Thestudyinthechapter 7 isaperfectexample,onceitisrealizedthatthelocalexplicitdensitydependentexchangepotentialslikeBecke,PBExetcandnon-localpotentiallikeHFexchangeoperatorhaveanoppositeeffectonthecoreionizations,coupledwiththefactthattheexplicitdensitydependentcorrelationpotentialhasalinearoralmostaconstanteffectoneachorbital,itiseasytodesignapotentialthatappearstosuppresstheself-interactionerrorandthelackofanintegerdiscontinuityintheDFTpotential.SuchaconcoctedpotentialshowsreasonableperformancecomparedtootheravailableexplicitdensitydependentDFTpotentials(Chapter 7 ),butdoesnotaccountfortotalenergyproperties.However,forthenotionthatKS-SCFapproachisjustameremathematicalconstructtoobtaindensity,ourdesignedpotentialprovidereasonablecorrecteigenvectoranddensitytobeusedwithconjunctionwithanyresolublecorrectfunctionalsuchasBHLYP,HF+RPAorHF+RPA+T1toprovidebettertotalenergypropertiesthanotherexplicitdensitypotentialasevidentfromthebarrierheightsestimationanddissociationcurvedata(Chapter 7 ). Asevidentfromourstudy,DFTresultscannotbetrustedcompletelyasthefunctionalhavemanyshortcomingsandconcoctedfunctionalsdonotcorrecttheinconsistencies.Obtainedresultsmustsatisfythe'ConsistentDFTconditions',suchasBartlett'stheorem,andatleast,theHOMOcondition(HOMO=)]TJ /F3 11.955 Tf 9.3 0 Td[(Ip).Theuctuation-dissipation-adiabatic-connectedDFTenergyfunctionalprovideanexactexpressionfortheexchange-correlationfunctionalintermsofthedensity-densityresponsefunctionandtheRPAisthesimplestapproximationofthedensity-densityresponsefunction.TheequivalencebetweenRPAandring-CCDnotonlyprovidesaroutetoeasilycomputethecorrespondingRPAexchange-correlationpotentialbutalsotoprovidearoutetogobeyondtheRPA,asMBPT-2,lin-CCD,CCDetcwithout 124

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worryingaboutthecomplexkernelinthedensity-densityresponsefunction.WehaveobtainedpotentialcorrespondingtotheRPA,lin-CCDandCCD,comparedtotheirspatialbehaviorandtheHOMOcondition(HOMO=)]TJ /F3 11.955 Tf 9.3 0 Td[(Ip).TheRPA,lin-CCDandCCDpotentialarecomputationallydemandingbut,isastepindirectdirectiontowardsa'ConsistentDFT'(Chapter 5 ). DuetothecomplexRPApotentialform,theselfconsistentRPAcalculationsareoftenavoidedwithRPAcorrelationenergiesbeingobtainedinapost-scfmanner.Therehavebeendebatesaboutwhichorbitalsarebesttouse,theHFortheKSorbitalswithPBEorsomeotherexplicitdensitydependentpotentialintheKSoperatortoevaluatetheRPAenergy.WendthattheHForbitalsaremuchbetterthanthePBEorbitalstoevaluatetheRPAenergy,unlessonesolvestheproblemincorrectly.OnecangobeyondRRAbyincludingthesinglecontributionstothecorrelationenergy,fortheKSreferenceincludingsinglesismorenaturalasthefiaelementsarenon-zerobutduetotheBrillouins'stheoremthoseelementsarezerointheHartree-Fockreference.Thestudyinthechapter 6 exploreswaystoincludesinglesintotheRPAandthusgobeyondtheRPA.Wendthatdoingtheproblemincorrectly,oftenthecasewithDFT,givesthebestanswerforAr2dissociationenergy. 125

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REFERENCES [1] A.SzaboandN.Ostlund, Modernquantumchemistry:introductiontoadvancedelectronicstructuretheory (DoverPublications,1996). [2] M.PeskinandD.Schroeder,Introductiontoquantumeldtheory,AdvancedBookProgram(Addison-WesleyPub.Co.,1995). [3] W.KutzelniggandJ.D.M.III, J.Chem.Phys.96,4484(1992) [4] B.R.J.,Many-BodyMethodsinChemistryandPhysics:MBPTandCoupled-ClusterTheory(CambridgeUniversityPress,Cambridge,2009). [5] R.G.ParrandY.Weitao,Density-FunctionalTheoryofAtomsandMolecules(OxfordUniversityPress,Oxford,1989). [6] R.M.DreizlerandE.K.U.Gross,Densityfunctionaltheory:anapproachtothequantummany-bodyproblem(Springer,Heidelberg,1990). [7] P.HohenbergandW.Kohn, Phys.Rev.136,B864(1964) [8] W.KohnandL.J.Sham, Phys.Rev.140,A1133(1965) [9] F.Herman,J.P.VanDyke,andI.B.Ortenburger, Phys.Rev.Lett.22,807(1969) [10] J.P.PerdewandW.Yue, Phys.Rev.B33,8800(1986) [11] J.P.Perdew, Phys.Rev.B33,8822(1986) [12] S.K.GhoshandR.G.Parr, Phys.Rev.A34,785(1986) [13] A.D.Becke, J.Chem.Phys.85,7184(1986) [14] A.D.Becke, J.Chem.Phys.88,1053(1988) [15] A.D.Becke, J.Chem.Phys.84,4524(1986) [16] A.Cedillo,J.Robles,andJ.L.Gazquez, Phys.Rev.A38,1697(1988) [17] C.Lee,W.Yang,andR.G.Parr, Phys.Rev.B37,785(1988) [18] A.D.Becke, Phys.Rev.A38,3098(1988) [19] L.J.ShamandM.Schluter, Phys.Rev.Lett.51,1888(1983) [20] S.KummelandL.Kronik, Rev.Mod.Phys.80,3(2008) [21] R.T.SharpandG.K.Horton, Phys.Rev.90,317(1953) [22] R.J.Bartlett,I.Grabowski,S.Hirata,andS.Ivanov, J.Chem.Phys.122,034104(2005) 126

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BIOGRAPHICALSKETCH PrakashVermagrewinasmalltowninIndia,withdesiretobeinthecompanyofsmartpeopleandsolveintriguingproblemlandedhiminoneofthemostprestigiousengineeringcollegeofIndia,theIndianInstituteofTechnologyatBombay.Duringhis5-yearsatIIT-Bombay,hetookmanyclasses,oneoftheclasswastondaresearchtopicanddoexhaustiveliteraturesurveyonthattopic,hechoose,Coulombscatteringandresonance,thatexposedhimtothewonderfulworldofmathematicalchemistry,inanotherwordsonecanrepresentmoleculebyamathematicalequation.AftertakingfewmoreclassesinquantumchemistryanddoingMasterofScienceresearchonelectronpropagator,hisfascinationfortheeldgrewmore.Fortunately,hegotanopportunitytodohisgraduatestudieswiththebestbrainsintheoreticalchemistryatQTP,thehomeofquantumchemistry. 132