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Evaluation of molecular electronic wavefunctions by electron momentum spectroscopy Hollebone, Bruce Paul 1994

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EVALUATION OF MOLECULAR ELECTRONIC WAVEFUNCTIONS BYELECTRON MOMENTUM SPECTROSCOPYByBruce Paul HolleboneB. Sc. (Chemistry and Physics) McMaster University, 1989A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF CHEMISTRYWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITIS11 COLUMBIASeptember 1994® Bruce Paul Hollebone, 1994In presenting this thesis in partial fulfillment of therequirements for an advanced degree at the University of BritishColumbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission forextensive copying of this thesis for scholarly purposes may begranted by the head of my department or by his or herrepresentatives. It is understood that copying or publication ofthis thesis for financial gain shall not be allowed without mywritten permission.(Signature)____________________Department of__________________The University of Britf’h ColumbiaVancouver, CanadaDate tiL ltL(AbstractThe valence binding energy spectra and orbital electron momentum profiles of HF, HC1and C2H4 have been measured by symmetric non-coplanar electron momentum spectroscopy (EMS) at an impact energy of 1200 eV. The measured momentum profiles arecompared with calculated cross-sections using a range of wavefunctions from minimal basis to essentially Hartree-Fock limit in quality. The effects of correlation and relaxationon the calculated momentum profiles are investigated using multi-reference singles anddoubles configuration interaction (MRSD-CI) calculations of the full ion-neutral overlapdistributions. The effects of using several different configuration spaces is investigated inthe case of the lowest binding energy momentum profile of C2H4. The experimental momentum resolution is accounted for in all of the calculated momentum profiles using theGaussian-weighted planar grid procedure. The effects of vibrational averaging are alsoinvestigated in the case of the highest occupied molecular orbital (HOMO) of ethylene.For all three molecules, the effects of final state correlation on the valence binding energyspectrum are investigated using MRSD-CI calculations and the results are compared withother calculations and with experiment.The measurements for HF and HC1 are significantly different from previously reportedEMS profiles for several ionization processes. There is also a significant discrepancybetween SCF limit calculations and the measured HF hr momentum profile. However,with the inclusion of electronic correlation and relaxation effects using MRSD-CI methodsreasonable agreement between theory and experiment is achieved. For the other HFprofiles and all the valence profiles of HC1 the inclusion of many-body effects is found toprovide little further improvement in the already quite good fit of theory to experiment11at the near Hartree Fock limit level.The present measurements of the valence orbital momentum profiles for C2H4 arein generally good quantitative agreement with Hartree-Fock results using moderate tolarge basis set as well as with much more sophisticated many-body treatments. Theobserved momentum profiles of ethylene, like those of acetylene, are well described at thenear energy-converged Hartree-Fock level. As was observed previously in acetylene, whilethe consideration of many-electron interactions greatly improves both total energies andother electron properties, it has little effect on the shapes of the theoretical momentumprofiles. Vibrational motion is also found not to have a large effect on the calculatedshape of the 1b3 ir-system momentum distribution.New EMS measurements are also reported in this work for the outermost valenceir orbitals of the two series C2H4,C2H3F, CH2F,C2HF3 and C2F4 and (CH3)2C0,CH3HO and H2CO. These sample studies of substitution effects in the C=C and C=Oir-systems have revealed trends in both series of compounds which are consistent with abinitio calculations. For the fluoro-ethylenes, near Hartree-Fock limit theory is found toprovide a very good description of the EMS momentum profiles. The theoretical models of these molecules imply that increasing fluorination both removes electron densityfrom between the carbon centres, weakening the ir-bond and causes the formation ofC-F anti-bonds in the HOMO. Significant differences are found between the measuredand theoretical momentum profiles for the methyl substituted series of formaldehyde,acetaldehyde and acetone even at the MRSD-CI level. For these three molecules, methylsubstitution is found to have a net electron-withdrawing effect (compared with hydrogen)on the C=O ir-system.‘UTable of ContentsAbstract iiList of Tables viiList of Figures viiiList of Abbreviations xUnits xiiAcknowledgements Xiii1 Introduction 11.1 Electron Momentum Spectroscopy 31.1.1 Binding Energy Spectra 41.1.2 Momentum Profiles 81.2 Context of this work 101.2.1 Hydrides 121.2.2 Ethylene 141.2.3 Functional group substitution 141.3 Overview of the thesis 151.3.1 Outline of the thesis 162 Theory 192.1 The Scattering Kinematics 19iv2.2 Electronic Structure Calculations2.3 The Scattering Cross-section3 Experimental Method3.1 The EMS Spectrometer3.2 Coincidence Processing3.3 Computer Processing3.4 Operating Conditions3.5 Treatment of Results4 Hydrogen Fluoride 484.1 Calculations4.2 Results and Discussion4.2.1 Binding Energy Spectra .4.2.2 Comparison Experimental and5 Hydrogen Chloride5.1 Calculations5.2 Results and Discussion5.2.1 Binding Energy Spectrum5.2.2 Comparison of Experimental and Calculated6 Ethylene6.1 Calculations6.2 Results and Discussion6.2.1 Binding Energy Spectra6.2.2 Comparison of Experimental and Theoretical Momentum Profiles2227323237394345505153Theoretical Momentum Profiles . . 56Momentum Profiles.69717477829396100100108V6.2.3 Basis Set Saturation and Correlation and Relaxation Effects in the1b3 Momentum Profile of Ethylene 1216.2.4 Electron Density Maps for C2H4 in Position and Momentum Space 1256.2.5 Vibrational Effects in the 1b3 Momentum Distribution 1276.2.6 Comparison with Multi-channel EMS Results 1307 The Effects of Fluoro-substitution on the Ethylene ir-System 1347.1 Calculations 1357.2 Comparison of Experimental and Theoretical Momentum Profiles . . . . 1377.3 Trends in the Electronic Densities with Increasing Fluorination 1438 The Effects of Methyl-substitution on the Formaldehyde ir-system 1478.1 Calculations 1488.2 Comparison of Experimental and Theoretical Momentum Profiles . . . . 1528.3 Trends in the Electronic Densities of H2CO, CH3HO and (CH3)2C0.. . 1629 Concluding Remarks 168A Signal Conditioning Electronics 174Bibliography 177viList of Tables4.1 Calculated and experimental properties for hydrogen fluoride . . . 525.1 Calculated and experimental properties of hydrogen chloride . . 755.2 Ionization energies and intensities for hydrogen chloride 766.1 Calculated and experimental properties for ethylene 1016.2 Ionization energies and intensities for ethylene 1037.1 Electronic properties of selected fluoro-ethylenes 1388.1 Electronic properties of H2CO, CH3HO and (CH3)2C0 153vi’List of Figures2.1 Symmetric Non-Coplanar Scattering Kinematics 20Diagram of the EMS spectrometerSchematic of Coincidence Processing ElectronicsCoincidence Time Correlation Spectrum4.1 554.2 574.3 614.4 644.5 675.1 795.2 815.3 855.4 885.5 906.1. . . . 1046.2 1066.3. . . . 1096.4 1116.5 1126.6 1133.13.23.3343841Experimental HF Valence Binding Energy SpectraExperimental and Synthetic HF binding energy spectra . .The hr momentum profile of HFThe 3cr momentum profile of HFThe 2cr momentum profile of HFExperimental HC1 Valence Binding Energy Spectrum .Experimental and Synthetic HC1 binding energy spectra .The 2ir momentum profile of HC1The 5cr momentum profile of HC1The 4cr momentum profile of HC1Experimental C2H4 valence binding energy spectraExperimental and synthetic C2H4 binding energy spectra . . .Simultaneous experimental C2H4 Valence Binding Energy SpectraThe 1b3 momentum profile of C2H4The lb39 momentum profile of C2H4The 3a9 momentum profile of C2H4viii6.7 The 1b2 momentum profile of C2H4 1146.8 The 2b1 momentum profile of C2H4 1156.9 The 2a9 momentum profile of C2114 1166.10 Correlation and relaxation effects in the 1b3 momentum profile of C2H4 1246.11 Vibrational averaging effects in the 1b3 momentum profile of C2H4 . . . 1316.12 Comparison of single- and multi-channel EMS results for C2H4 1337.1 Momentum profiles of selected fluorinated ethylenes 1407.2 Electron densities and density differences for C2F4 and C2H4 1458.1 The 5b2 momentum profile of (CR3)2CO 1558.2 The lOa’ momentum profile of CH3RO 1578.3 The 2b momentum profile of H2CO 160A.1 Signal Conditioning Electronics Board Schematic 175A.2 Signal Conditioning Electronics Connector Diagram 176ixList of Abbreviations2ph-TDA two-particle-hole Tamm-Dancoff ApproximationADC analog to digital converterBE binding energyBES binding energy spectrumCEM channel electron multiplier or channeltronCFD constant fraction discriminatorCGTO contracted Gaussian type orbitalCI configuration interaction (calculation)CMA cylindrical mirror analyzerDAC digital to analog converterDWIA distorted wave impulse approximationEMS electron momentum spectroscopyfwhm full width at half-maximumG(CI) a configuration interaction wavefunction based on a Gaussian typeorbital basis set.GF Greens function (calculation)GTO Gaussian type orbital, often used to mean the molecular wavefunction that is a Gaussian Type Orbital Hartree-Fock result.GW-PG Gaussian weighted planar grid, a method to account for instrumental resolutionHF Hartree-Fock (calculation)HOMO highest occupied molecular orbitalxIP ionization potentialMD momentum distribution calculated using the target Hartree-Fockapproximation.MRSD-CI multireference configuration interaction calculation withsingle and double excitations.NHOMO next highest occupied molecular orbitalNO natural orbitalODD orthogonalized direct diagonalization, a Greens function methodOVD overlap distributionPES photoelectron spectroscopyPWIA plane wave impulse approximation—see equation (2.20)RHF restricted Hartree-FockSAC symmetry adapted configuration.SCA single channel analyzerSCF self consistent field—in this thesis, always referring to a restrictedHartree-Fock calculation (RHF)TAC time to amplitude converterTHFA target Hartree-Fock approximation—see equation (2.21)TMP theoretical momentum profile—an MD or OVD modified to accountfor instrumental resolution. Directly comparable with an XMP.XMP experimental momentum profile—the angular distribution from anEMS experiment plotted as a function of momentum. Directly comparable only with a TMP, not an MD or OVD.XPS X-ray photoelectron spectroscopyxiUnitsA system of reduced units convenient for discussing quantities on the atomic scale hasbeen used in this thesis. In the system of atomic units, the Bohr radius a0, Planck’sconstant h, the charge of an electron e and the permitivity of free space 4ire0, are allconsidered to be dimensionless quantities of magnitude 1.Quantity 1 au =Distance a0 5.2918 x 10” mMomentum 1.9929 x 10_24 kg•m•s1Energy 4ireoao 4.3598 x 108 J2 Rydberg 27.21161 eVCharge e 1.6022 CDipole Moment a0e 8.4785 x 10_li C’m2.5418 DebyeQuadrupole Moment age 4.4867 x 102i C•m21.3451 x 10_26 esu•cm2Note that “hartree” is the standard IUPAC name for the atomic unit of energy.xiiAcknowledgementsThis thesis would not have been possible without the patience and wisdom of my researchsupervisor, Dr. Chris Brion, to whom I owe a great debt of thanks. It has been a greatprivilege both to have been able to work with him and to have known him personally.Special thanks also belong to my collaborators, Professor E.R. Davidson at IndianaUniversity, and his co-workers: C. Boyle, D. Feller and Y. Wang, who provided manyof the theoretical strengths of this manuscript, including the many large and complexwavefunctions presented in chapters 4—8.I would also like to thank those whom I have had the pleasure to know and workwith here a U.B.C., who gave me advice for the many things I did not know, help withmy fumbling fingers, suffered through my manuscripts and who were always ready to gofor coffee: J. Au, Dr. G.R. Burton, Dr. N. Cann, Dr. M.E. Casida, Dr. W.F. Chan,Professor D.P. Chong, Dr. S.A. Clark, Dr. G. Cooper, P. DufFy, N. Lermer, J.J. Neville,T. Olney, J. Rolke, Dr. B. Todd and Dr. Y. Zheng.Grateful mention must be made of the excellent technical and support staff in theElectrical and Mechanical shops of the Chemistry Department who were able to keep thespectrometer running for “just one more study:” M. Coschizza, B. Greene, E. Gomm,M. Hatton, B. Henderson, D. Jones, B. Snapkauskas and D. Tomkin.I am indebted to Professor D. Fleming for allowing the use of his cylinder of HC1 andalso to Professor L. Weiler for providing the sample of acetaldehyde.Finally, I would like to acknowledge the receipt of an NSERC post-graduate scholarship and the funding of my research by the Natural Sciences and Engineering ResearchCouncil of Canada.xli’I like relativity and quantum theoriesBecause I don’t understand themAnd they make me feel as if space shiftedAbout like a swan that can’t settleRefusing to sit still and be measuredAnd as if the atom were an impulsive thingAlways changing its mind.—D.H. LawrencexivChapter 1IntroductionElectronic wavefunctions and orbitals are fundamental to our understanding of atomsand molecules. The electronic density, which is a function of the square of the quantummechanical wavefunction, plays an important role in determining the chemical behaviourand reactivity of a compound, particularly in the “long-range” or “large-r”, low momentum regions. Good descriptions of the electron density over all regions of position andmomentum space are necessary to understand and predict a wide range of chemical andphysical phenomena. The electronic wavefunction and thus the charge distribution of amolecule can be calculated from first principles by variational ab initio procedures andchecked by comparison with molecular properties; however the use of restricted treatments often makes this evaluation difficult. It is well known, for example, that energyoptimization methods do not necessarily lead to accurate dipole moments for polar molecules [1].The “binary (e,2e)” experiment was proposed about twenty-five years ago as a methodfor measuring electron density distributions of atoms as a function of momentum inorder to provide a quantitative test for quantum mechanical wavefunctions [2,3]. Thistechnique involves the ionization of a target atom or molecule (M) by a energetic electron,(1.1)1Chapter 1. Introduction 2with the two outgoing electrons being detected in coincidence. If the momenta of theincoming and outgoing electrons are determined, then the angular distribution of thescattered electrons can be related to the momentum of the electron in the target moleculeprior to being struck (i.e., ionized).The first orbital momentum distributions were reported in 1972 by Camilloni et al.for C is graphite films [4]. Momentum distributions for the individual 3p and 3s orbitalsof free argon atoms, were published the following year by Weigold et al. [5]. The firstbinary (e,2e) experiment on a free molecule was performed by Hood et al. for methane [6].The interpretation of the (e,2e) scattering cross-section was subsequently developed withmuch input from experimental results [7—9] and is now well understood [10—17].The interpretations of the first studies of argon [5, 18] revealed that for a particularscattering geometry, known as the symmetric non-coplanar geometry, the measured cross-sections were essentially independent of kinematic factors, provided the impact energywas sufficiently high. In particular, spectra obtained with a high impact energy usinga symmetric non-coplanar geometry depended primarily on the ionization channel ofthe target species and were not greatly influenced by factors such as outgoing electronenergies and scattering angles.It was also found [5, 18] that for fast, high energy collisions, the initial state of thetarget species dominated the electron scattering cross-section. If the electronic state ofthe molecule is approximated by a simple Hartree-Fock wavefunction then the ionization process can be interpreted as the removal of an electron from a canonical orbital.Hence, by determining the electron momentum distribution at a fixed binding energy,this experiment is said to provide an “image” of the corresponding initial state orbitalin momentum space [14—17]. Because of this momentum space “imaging” capability, theChapter 1. Introduction 3binary (e,2e) scattering experiment using the symmetric non-coplanar kinematics cameto be known as electron momentum spectroscopy (EMS).1.1 Electron Momentum SpectroscopyElectron Momentum Spectroscopy (EMS) can be used to measure ionization potentialsand also distributions of orbital electron momenta for gas-phase atoms and molecules. Inan EMS experiment, a beam of high energy electrons ionizes the target species and thetwo outgoing electrons, scattered and ejected, are detected in coincidence. The energy ofthe incident electron beam, or impact energy, is E0. For a fast collision (E0 much greaterthan the electron binding energy), the energy transfer to the molecule, e, is equal to theionization potential of the struck electron. The outgoing electrons are selected to haveequal, fixed polar angles (6 = 62) and energies (E1 =E2). The out-of-plane or azimuthalangle, 4’, can be varied at a fixed impact energy to obtain a momentum profile. If 4’ iskept fixed, a binding energy spectrum can be collected by scanning the incident beamenergy.The results obtained by EMS can be used in several different ways. Binding energyspectra are useful for determining ionization potentials, particularly for high ionizationenergies and for assigning the symmetry of final ion states. Momentum profiles provide aquantitative test for wavefunctions and corresponding electron densities and can also beused to gain insight into chemical properties and behavior. The properties and uses ofthe two kinds of measurements that EMS provides are discussed in detail in the followingsections.Chapter 1. Introduction 41.1.1 Binding Energy SpectraA binding energy spectrum (BES) is measured in EMS by varying the incident electronenergy at constant (and thus to a close approximation at a certain momentum). Maxima in the cross-section are observed for resonant ionizations, much as in photoelectronspectroscopy (PES). With the single-channel EMS spectrometer used in the present work,the energy resolution is practically limited to 1.7 eV fwhm, primarily because of the lowcoincidence count rates. In contrast, modern PES atomic line source spectrometers havemuch higher energy resolutions ( 0.02 eV fwhm). New generations of multi-channelEMS instruments, now under development, may allow for monochromation of the incident electron beam to improve the instrumental energy resolution; however this is notpractical with the present single-channel spectrometer. Despite the limits imposed bythe energy resolution, EMS has some distinct advantages over PES which are discussedbelow.High energy resolution photoelectron spectroscopy is routinely performed with atomicemission line sources. The most commonly used resonance lines, He I and He II, haveenergies of 21.2 eV and 40.8 eV respectively. While He I sources are excellent for studiesof outer valence ionization processes below 20 eV, higher binding energy studies areobviously not possible with this technique. Studies using He II light sources are frequentlycontaminated above 20 eV by stray light from an often appreciable He I “shadow” andby helium self-ionization at 24.6 eV. Therefore, helium resonance lamps are of limitedusefulness for the investigation of the higher energy, inner valence ionization manifoldswhich often have a rich array of structures well above the usable range of these sources [19,20]. By contrast, EMS can easily access higher binding energies by simply increasing theelectron beam impact energy, albeit with lower energy resolution (see chapter 2). SomeChapter 1. Introduction 5of the first studies of inner valence ionization manifolds were performed by EMS [21—25].Atomic inner-shell X-ray emission sources (such as the Al and Mg Kc. lines at 1486.6 eVand 1253.6 eV respectively) provide higher energy X-ray photoelectron spectroscopy(XPS) lamps. Electron synchrotron radiation, a high energy, high intensity light source,provides the additional advantage of being tunable over a wide energy range with suitable monochromation. However the energy widths of these sources are much wider thanemission line widths of helium I and II resonance lamps, and are often comparable tothe best energy resolution achievable by EMS instruments. Synchrotron sources alsosuffer from higher order light contamination due to the difficulties of vacuum UV andX-ray monochromator design. PES using high energy light sources, e.g. X-ray emissionand synchrotron radiation, also often have a problem with stray light contamination.In addition in photoelectron spectroscopy the spectra are complicated at lower electronkinetic energies by rising, non-linear backgrounds. As will be discussed in more detail inchapter 3, the background is constant in an EMS experiment and easily removed duringthe collection of a spectrum.While ionization potentials can be determined very accurately using PES, assigningthe final ion state and hence the overall symmetry of the ionization process is oftenquite difficult. Because of the dependence of the PES signal on the (hv,e) reactionkinematics, including the incident photon energy and the ejected electron energy andexit angle, PES intensities offer limited insight into the parentage of an ionization process [26, 27]. Indirect methods, such as variation of the incident light energy (possibleonly with monochromated synchrotron sources) [28], determination of anisotropy parameters (/3) [291 and isotope substitution effects [30] may be used in order to acquireinformation leading to an assignment of the process. However, the results of such investigations are often very difficult to interpret and can lead to incorrect assignments [31—33].Chapter 1. Introduction 6In contrast, the EMS cross-section provides direct information about the symmetry ofthe orbital origin of a final ion state [34]. Distinctive shapes of the cross-section as afunction of electron momentum for each symmetry manifold usually allow unambiguousassignment of the orbital parentage of the ionization process using EMS.Compared to PES, electron momentum spectroscopy provides the following advantages for the study of ionization processes [26,27]:• A wide binding energy range, with well-defined incident energy;• A constant, easily removed, background over the entire energy range;• Direct and usually unambiguous assignment of ionization parentage;The most severe limitation of EMS (for the current generation of single-channel spectrometers) is the wide energy resolution which precludes the study of many interestingspecies. New multi-channel instrumentation, currently under development, should provide significant improvements in the coincidence count rates and thus improvements inboth energy resolution and statistical precision.As mentioned above, EMS is very useful for the investigation of the symmetry of anionization process. One of the foremost applications of this has been the identificationof the parentage of “satellite” structures which arise from many-body interactions andwhich thus cannot be rationalized by a simple orbital model of ionization. Such phenomena are particularly common in the higher energy, inner valence region of bindingenergy spectra [19]. In the Hartree-Fock (independent particle) picture, ionization ofeach orbital results in one peak in the binding energy spectrum (neglecting spin-orbitand nuclear motion effects), with an energy corresponding to the energy of the orbitalChapter 1. Introduction 7(Koopmans theorem [35]). With the inclusion of many-body effects, including electroniccorrelation and relaxation this simple picture becomes much more complex. Each ionization process can lead to multiple final ion states, i.e., a manifold of final states [19,20].For configuration interaction methods, this corresponds to different roots of the finalion state [36]. The sum of the intensities of all of the ionization peaks having the sameparentage must be equal to the number of electrons removed (The spectroscopic sumrule—see chapter 2).Ionization spectra are interpreted and peaks assigned by comparison of the experimental separation energy spectra with theoretical results. For the outer valence, lowerbinding energy states, this is usually straightforward and can be done with Hartree-Focklevel calculations. For the inner valence satellite structures much more detailed theoretical treatments are needed to provide a model of the experimental spectra such asconfiguration interaction [36] or Greens function methods [19,20]. Accurate predictionof inner valence ionization potentials is very difficult and usually only approximate evenfor the largest, most complex calculations. Because of the simple relationship betweenmeasured intensity and electronic density, binding energy spectra obtained by EMS are auseful test of calculated separation energies and intensities or pole strengths. It should beemphasized that while PES can give very good values for ionization potentials, the peakintensities are very difficult to relate to theoretical pole strengths because of the strong dependence of the PES cross-section on the light energy and outgoing photoelectron energyand angle. Since the EMS cross-section corresponds to the pole strength as a function ofmomentum, it is more suited for comparison with theoretical results than PES. All theinner valence spectra obtained in the course of the present work (chapters 4, 5 and 6), areused to test the results of a variety of many-body theoretical techniques including multireference singles and doubles configuration interaction (MRSD-CI) and Greens’ functionChapter 1. Introduction 8methods.1.1.2 Momentum ProfilesBy stepping through the azimuthal angle 4 at a constant incident electron beam energy(i.e. at a constant binding energy), a profile of intensities as a function of momentum, p,is collected. The values of can be related to the momentum of the bound electron justprior to impact, p. The distribution of the number of coincidences as a function of p isreferred to as an experimental momentum profile (XMP).As mentioned above, one of the simplest models of ionization is the removal of anelectron from a canonical Hartree-Fock orbital. For EMS this corresponds to the TargetHartree-Fock approximation (THFA). Using the THFA, the momentum profile measuredby EMS can be interpreted as the orbital momentum density. The EMS cross-sectionis more realistically approximated using the square of the ion and neutral wavefunctionoverlap as a function of momentum. The use of the full ion-neutral overlap allows forthe evaluation by EMS of the effects of electron correlation in the ground state and bothcorrelation and relaxation in the final ion state.Experimental momentum profiles are useful for a number of purposes:• By comparison with the calculated results of both Hartree-Fock charge densitiesand many-body ion-neutral overlaps, XMP’s offer a quantitative check of the lowmomentum region of a wavefunction. Most ab initio methods involve energy minimization as the only criterion for obtaining the final wavefunction. While thevariational theorem is in principle a sufficient condition for obtaining a perfectwavefunction, in practice the use of restricted basis sets limits its effectiveness.Chapter 1. Introduction gThis is the source of the well-known problem of varying convergence rates of calculated electronic properties, such as dipole moment [1], with basis set size. Sinceenergy minimization emphasizes the “small-r” or large momentum parts of thewavefunction, the low momentum regions of the electronic density can be significantly underestimated by a variational procedure [37]. Because EMS momentumprofiles are obtained at low momentum (0—3 au), accurately reproducing themprovides a complementary constraint on a calculated wavefunction. To reproducean XMP in the large-r, low momentum regions, additional functions, particularlydiffuse and polarization primitives, can be added to basis sets [38,39].• As mentioned in section 1.1.1, the ability of EMS to select a particular ionizationenergy allows the identification of satellite structure. If initial state correlation isnot very important, then the shapes of the momentum profiles for different ionization processes sharing the same orbital parentage are almost identical. Thus,by comparing the shapes of the observed profiles the initial orbital origins of thesatellites can (usually) be determined.• By measuring the momentum profiles of corresponding orbitals for a series of compounds insight can be gained into the nature of the bonding and chemical behaviorof the compounds. For example, when used in combination with calculated densitymaps in position and momentum space, EMS results have proven useful in studiesof the methyl inductive effect in amines [40—44]. In a classic study, Leung and Brionused EMS to investigate the chemical bond in molecular hydrogen [45,46].Calculations of momentum distributions from wavefunctions are straightforward frombasis sets if the exponents and orbital coefficients are provided, as they often are in theliterature. The calculation [47] involves a Fourier transform of the basis functions intoChapter 1. Introduction 10momentum space and a spherical average to take account of the random orientationof the target species. The calculations performed in this thesis are either performedusing existing basis sets from the literature or are based on original ones developed inconjunction with co-workers at Indiana University.1.2 Context of this workIn the late 1970’s and early 1980’s EMS studies of many atoms and small molecules,including C2H,C2H4,C2H6,H2CO, C6H,C2H3F, C2H31 and many of the secondand third row1 hydrides were published2.These early studies included those of Cook etal. for H2S [49], Hamnett et al. for PH3 [22], Hood et al. for H20 [50] and NH3 [51],and C2H4 by Dixon et al. [24]. The momentum profiles reported in these studies wereobtained using single channel instruments with poor momentum resolution and withonly partial energy separation of the orbitals. Frequently, the EMS measurements forthe individual orbitals of the same molecule could not be quantitatively compared withtheory, because the various momentum profiles had not been correctly normalized tothe binding energy spectrum intensities. In addition, the EMS measurements were oftencompared with calculations of only very modest size using small to medium size basissets at the Hartree-Fock level. Many-body effects were not investigated in most cases.Furthermore, the effects of instrumental momentum resolution (later found to be a verynecessary consideration) were either ignored or treated inadequately in these early studieswhen comparing the EMS momentum profiles with calculations. Nevertheless, these pilotstudies provided a useful proof of concept for the EMS technique, although they were oflimited value for detailed evaluation of high-quality theoretical wavefunctions.1n the context of this thesis, I use row 2 to mean the elements Li—Ne and row 3, the elements Na—Ar.2A bibliography of these and more recent studies can be found in references [17,48]Chapter 1. Introduction 11With the development of improved EMS spectrometers in the past decade [52], moredetailed investigations have become possible. Careful, precise measurements of the noblegases [52—54] and molecular hydrogen [45,46] have demonstrated the utility of EMS asa sensitive test of very large, near energy converged calculations and also as a probe ofchemical bonding for H2. New quantitative measurements [31,32, 37,55—57] of many ofthe previously studied small molecules have provided valuable insight into the qualityof existing calculations. An important goal of these newer EMS studies has been thedevelopment of high-quality basis sets which reproduce a wide range of electronic properties with very good accuracy (see for example ref. [58]). The basis sets developed inthe course of these studies have been shown to give not only good total energies andfits to the experimental EMS momentum profiles but also excellent results for a widerange of one-electron properties such as the dipole moments. These three properties areof particular interest since:• the total energy observable is dominated by the short-range spatial part of the wavefunction near the nuclei (i.e. the large momentum part).• the dipole moment emphasizes the mid-range part of the radial wavefunction (i.e.medium momentum or position values).• the momentum value at the the maximum (PMAX) of the spherically averaged momentum profile emphasizes the long range part of the radial wavefunction (i.e. thelow momentum part)3.3This relationship is due to the Fourier relationship of momentum and position spaces. Since the“period” in one space of a Fourier transform is inversely related to the “frequency” in the other, affectingthe long range radial part of a wavefunction changes the short range momentum part and vice-versa.Since the part of the EMS cross-section of interest is at low momentum, it emphasizes the diffuse partof a wavefunction. While the relationship between momentum and position space can be expressedanalytically for atomic electron densities [172]; there is only approximate correspondence in moleculardistributions.Chapter 1. Introduction 12These three properties, considered together, have been found to provide a valuablediagnostic for the evaluation and design of “universal” wavefunctions which give verygood results for a wide range of properties [31,32,37,52,55—58]. The high quality basissets evaluated and developed in this way have led to higher-level calculations in collaborative theoretical and EMS studies of larger molecules such as CH3N2 [42,59] and(CH3)20 [60]. Eventually such studies for larger molecules may provide important experimental input into computer-aided molecular design.1.2.1 HydridesThe hydrides of the main group (IVA—VIJA, IUPAC groups 14 to 17), row 2 and 3elements are isoelectronic with the noble gas atoms Ne and Ar respectively. Their electronic structures are sufficiently simple that very high quality calculations are possibleand they have long been considered as “benchmark” systems in theoretical molecularquantum chemistry. These molecules are particularly amenable to study by EMS because of their ready availability and the wide energy spacing of their valence orbitals.As such, studies of these hydrides are a good test of both the interpretation of the EMSexperiment and of ab initio theory.Because of their fundamental importance for calculations on larger systems, thesehydride molecules have been the focus of a series of collaborative studies over the pastdecade [31,32,37,55—58] involving a comparison of high momentum resolution EMS measurements of momentum profiles made at The University of British Columbia with calculations using near Hartree-Fock limit and also MRSD-CI wavefunctions developed byProfessor E.R. Davidson and his group at Indiana University. Prior to the work in thisthesis the molecules studied in this collaboration include the hydrides involving the heavyChapter 1. Introduction 13atoms of groups VIA, VA and IVA, i.e., H20 [37], NH3 [55] and CH4 [56] from row 2 andH2S [31], PH3 [32] and SiH4 [57] from row 3. In addition the respective isoelectronic noblegases Ne and Ar [52] have also been studied under identical conditions. The completevalence shell binding energy spectra and all valence shell electron momentum profileshave been measured for these atoms and molecules.A particular impetus for the series of these hydride molecule studies was provided bypreliminary EMS measurements of water [50] and ammonia [51]. When compared withthe then existing theory, thought to be essentially converged to the Hartree-Fock energylimit, significant shape mismatches were found at low momentum between the EMSand theoretical results for some of the outer valence momentum profiles. Further andmore detailed investigation by both EMS and theoretical methods of H20, D20 [37] andNH3 [55] revealed that very large, diffuse and highly saturated basis sets together withthe inclusion of correlation and relaxation were necessary for good agreement with theexperimental results. The momentum distributions of the third row hydrides (SiH4 [31],PH3 [32] and H2S [57]) and CR4 [56] on the other hand, showed good agreement with nearHartree-Fock limit electron densities, and the effects of correlation and relaxation on themomentum distributions were found to be negligible. Earlier measurements of HF andHC1 [61] at the Flinders University of South Australia compared similar experimentalresults with modest calculations. However, discrepancies in shape and intensity werefound with theory for some of the orbitals of both HF and HC1. In a later comparison [62]of the HF results [61] with much larger calculations the discrepancies between experimentand theory remained.In order to complete the study of the hydrides, new high momentum resolution EMSmeasurements of HF and HC1 have been performed in the course of this thesis, withthe same EMS instrumentation used for the rest of the Group IVA—VIA hydrides andChapter 1. Introduction 14noble gases [31,32,37,52,55—58]. Corresponding high quality MRSD-CI and very largebasis set Hartree-Fock calculations have been performed for HC1 in collaboration withDr. E.R. Davidson and co-workers at Indiana University.1.2.2 EthyleneAs is the case for the hydrides, ethylene is considered to be one of the fundamentalsystems in computational quantum chemistry. One of the smallest and simplest doublybonded molecules, C2H4 has been widely studied both theoretically and experimentally.While C2H4 has been studied previously by EMS [24,63], the results were obtainedat low momentum and energy resolution compared with current instrumentation. Therelative normalizations of the momentum profiles were not obtained in either of theearlier studies [24,63] which reduces the usefulness the earlier EMS measurements forthe purposes of quantitative evaluation of theory.In view of the unsatisfactory state of the existing EMS results for ethylene, momentum profiles for all the ionization processes have been obtained with high momentumresolution. These are compared with a wide variety of SCF and MRSD-CI configuration calculations, performed in conjunction with co-workers at Indiana University. Thiswork also completes the set of the simplest hydrocarbons, methane [56], ethylene andacetylene [64,65].Chapter 1. Introduction 151.2.3 Functional group substitutionPrior to this thesis, the effects of alkyl- and fluoro-substitution on electronic charge densities have been investigated by electron momentum spectroscopy for several molecules [40—44,59]. It has been found in the course of these studies that increasing methylation ofcompounds containing “lone-pair” orbitals resulted in a net electron withdrawing effectof the methyl groups.Studies of the HOMO’s of the methyl- and fluoro-substituted amines, NH3NH2C3,NH(CH3)2,N(CH3) and NF3 [41,42,55, 59] show an increase in the low momentumintensity with increasing methyl substitution. It was argued [41,42] that the increasein intensity at low momentum corresponded to an increasing contribution from H isand F 2s orbitals. This effect resulted in a delocalization of the nitrogen “lone-pair”charge density with increasing methylation. Fluoro-substitution was found to have asimilar effect [41,42]. A later study of the momentum distributions of the methylatedseries H20 [37], CH3O [66] and (CH3)20[60,67] revealed a similar delocalization of theoxygen “lone pairs” with increasing alkylation.In this thesis the effects of substitution on simple ir systems are investigated. Fluorination of the double bond of ethylene is studied in the series C2H4,C2H3F, CH2F,C2HF3 and C2F4 by comparison of the experimental momentum profiles of the HOMO’swith profiles calculated using Hartree-Fock wavefunctions. The alkylation studies areextended to investigate the effects of methyl substitution of the carbonyl (C=O) functional group. The momentum distributions of the outermost valence orbitals have beenobtained for acetaldehyde (CH3CHO) and acetone ((CH3)2C0). These results, togetherwith the outermost momentum profile of formaldehyde (H2CO) previously measured byBawagan et al. [34] are compared with new Hartree-Fock and configuration interactionChapter 1. Introduction 16calculations of the momentum profiles.1.3 Overview of the thesisThis thesis presents EMS measurements of the binding energy spectra and momentumprofiles of all of the valence orbitals of HF, HC1 and C2H4. Electron momentum spectroscopy has also been used to investigate the effects of fluorination on the ir-system ofC2H4for C2H3F,CH2F,C2HF3and C2F4 and for the methyl substitution series H2CO,CH3HO and (CH3)2C0. The implications to the nature of chemical bonding in each ofthese systems are discussed in the context of the theoretical and experimental results.1.3.1 Outline of the thesisChapter 2 presents a brief discussion of the theoretical framework necessary to interpretthe experimental results. In chapter 3 the EMS spectrometer and coincidence collectionelectronics are detailed. In addition, the data analysis and normalization procedures arediscussed. Chapters 4—8 each comprise a complete study and may be read in any orderafter the first three chapters.Chapter 4 presents the EMS results for HF. The valence orbital momentum profilesand binding energy spectrum of hydrogen fluoride have been obtained using electron momentum spectroscopy at an impact energy of 1200 eV. The momentum profile measurements are compared with calculations ranging from minimum basis set to highly saturatedbasis sets essentially at the Hartree-Fock limit and also with numerical Hartree-Fock calculations. The importance of electron correlation and relaxation effects are investigatedusing MRSD-CI calculations of the full ion-neutral overlap distribution. The bindingChapter 1. Introduction 17energy spectrum is compared with several many-body pole calculations taken from theliterature using a variety of ab initio methods. This work has been published as B.P.Hollebone, Y. Zheng, C. E. Brion, E. R. Davidson and D. Feller, in Chem. Phys. 171(1993) p303 [68].In chapter 5 a related study of the valence orbital electron momentum profiles ofhydrogen chloride is presented. The measured momentum profiles are compared withcalculated cross-sections using a range of wavefunctions from minimal basis to essentiallyHartree-Fock limit in quality. The effects of correlation and relaxation on the calculatedmomentum profiles are investigated by MRSD-CI calculations of the full ion-neutraloverlap distributions. The effects of final state correlation on the valence binding energyspectrum are also investigated using MRSD-CI calculations and the results are comparedwith other calculations and with experiment. This study has appeared in the literatureas B.P. Hollebone, C. E. Brion, E. R. Davidson and C. Boyle, in Chem. Phys. 173(1993) p193 [33].Chapter 6 presents an EMS study of the valence orbitals of ethylene. The momentumprofiles and the complete valence shell binding energy spectrum (8 to 51 eV) of ethylene(C2H4) have been measured. The results are compared with SCF calculations usingbasis sets ranging from minimal to near Hartree-Fock limit quality. Many-body effectsare investigated by MRSD-CI techniques using many different reference and configurationspace choices. The effects of vibrational averaging on the HOMO momentum profile arealso investigated. The observed satellite structure in the binding energy spectrum isinvestigated by comparison with the MRSD-CI results obtained in this work and withother many-body calculations from the literature.In chapter 7 the results of the study of effects of successive fluorine substitution on theChapter 1. Introduction 18electron density of the highest occupied ir-orbital of ethylene are presented. Momentumprofiles have been obtained for the outermost ir-orbitals of the moleculesC2H3F,CH2F,C2HF3 and C2F4. These are compared with calculated distributions using low to highquality basis sets at the Hartree-Fock independent particle model level.In chapter 8 an application of EMS to the study of methyl substitution of H2CO ispresented. The outermost valence electron momentum profiles have been obtained foracetone (5b2) and acetaldehyde (lOa’). These are compared with calculated profiles ofwide ranging quality from minimal basis STO-3G to near Hartree-Fock limit SCF treatments, together with the outermost momentum profile of formaldehyde (2b) measuredpreviously by Bawagan et a!. [34]. The effects of the addition of diffuse functions tothe basis sets are examined for a number of properties. Many-body corrections to theSCF model are evaluated by multi-reference singles and doubles configuration interactioncalculations of all three molecules. This work has been published as B.P. Hollebone, P.Duffy, C.E. Brion, Y. Wang and E.R. Davidson, Chem. Phys. 178 (1993) 25 [69].Chapter 9 contains general conclusions arrived at over the course of this work as wellas suggestions for future work.Chapter 2Theory2.1 The Scattering KinematicsThe purpose of the EMS experiment is to determine the binding energies and individualmomentum distributions of electrons in atoms and molecules. Electron momentum spectroscopy is based on the binary (e,2e) scattering process which involves the ionization ofa target (M) by an energetic electron and detection of the scattered and ejected electrons in coincidence. A schematic of the collision process in the symmetric non-coplanargeometry is shown in figure 2.1. The energies and directions (and thus momenta) of theincoming and outgoing electrons are defined experimentally by the geometric arrangement and electrostatic potentials of the electron optics and energy analyzers. The energyof the incoming electron beam or impact energy, is defined to be E0 with correspondingmomentum Eo is typically 1200 eV +e, where c is equal to the binding (separation)energy of the ionized electron. The outgoing electrons have energies E1 and E2 and momentum vectors j5 and 2• The angles 6 and 92 are the polar angles formed between theoutgoing electrons’ momentum vectors and the direction of the forward scattered beam.The angle 4 is the relative azimuthal angle between the two planes (go, and (jio, p2)(see figure 2.1).If the electron-electron scattering event is considered to be a two-body collision then19Chapter 2. Theory 20Figure 2.1: A schematic of the binary (e,2e) reaction in the symmetric non-coplanar geometryChapter 2. Theory 21the momentum of the electron in the target molecule prior to being stuck (p) can befound by vector subtraction. In this case the ion resulting from the collision is a spectatorto the (e,2e) reaction. This binary encounter approximation [70] can be realized usingthe impulse approximation [71] which requires a large momentum transfer between thecolliding electrons.The momentum transfer (1?—— j3) from the incident electron (eo) to the target (M) is maximized through the choice of the scattering geometry. A convenient arrangement for determining momentum distributions is called the symmetric non-coplanargeometry [10] (Shown in figure 2.1). The impact energy, E0, is chosen to be very largein comparison with the ionization energy of the orbital being studied (typically E0 >1200 eV=44.1 au). The kinematics are referred to as symmetric with the outgoing electron energies and polar angles taken to be equal (E1 = E2 = 600 eV and 0 = 0 = 02 = 45°)and non-coplanar because q is allowed to vary. This choice of geometry maximizes themagnitude of the momentum transfer K [10, 72, 73].For a two-body collision between the electrons conservation of momentum gives,PP1+P2P0 (2.1)where is the momentum of the struck electron just prior to the collision. The energybalance gives,Eo=E1+E2e (2.2)where e is the energy transferred to the target molecule. If the kinetic energy transferto the molecule is neglected (the binary encounter approximation), then e is simply theionization potential of the struck electron.Chapter 2. Theory 22For the symmetric non-coplanar geometry (figure 2.1),2p = [(2pi cosO —p0) + (2pi sinOsm(4)/2)) ] (2.3)where p = I I Pi = I P1 I and Pc = I I. By measuring the scattering intensities for arange of 4’, a probability distribution as a function of momentum can be obtained. Byscanning e at constant 4), a binding energy spectrum at nearly constant momentum canbe collected (since pg = /2ij = ,/(44.1au + ), there is a small dependence of p on2.2 Electronic Structure CalculationsWithin the Born-Oppenheimer approximation, the electronic wavefunction of an atomor molecule is, in principle, completely described by the electronic Schrödinger equation:ufelWel = EeiI’ei. (2.4)However, for a molecule with N electrons, equation (2.4) is an N-body problem andtherefore can only be dealt with by approximate methods. In this thesis, two techniques have been used to obtain molecular wavefunctions; the self-consistent-field (SCF)Hartree-Fock-Roothaan (or simply Hartree-Fock) treatment and configuration interactionmethods.The Hartree-Fock (HF) method was developed in the late 1920’s and early 1930’sfrom intuitive, physical arguments as a model of electronic structure (for a review andproof of the Hartree-Fock procedure see reference [74,75]). In this procedure, the totalN-electron Hartree-Fock wavefunction is expressed as the anti-symmetrized product ofone-electron orbital wavefunctions, 4, which are composed of a spatial part and a spinfunction (a or /3) and are thus referred to as spin-orbitals. The N-electron Hamiltonian,Chapter 2. Theory 23Hei, is reduced to the one-electron Fock operator, 1’. For an electron v,fr(v)(v) = (2.5)Where the Fock operator is given by (in atomic units),N/2P(v) = )CORE(V) + E[2ij(v) — 1(3(v)] (2.6)j=1where,1 2fcoRE(v) = —-Vu — — 2.72 afor all a, nuclei in the molecule. The Coulomb operator, Jj, and the exchange operator,k, for interactions of electron v with electron u are defined as:3g(v) = g(v) J (u) 12 (2.8)k,g(v) = (v) f j(u)9(u)dT (2.9),.vuwhere g is any arbitrary function and the integrals are over all variables of electron u.It should be noted that these last two operators involve the interaction of an electron vwith a smeared-out distribution for electron u, rather than with a dynamic point chargeand is referred to as a field interaction. Note also that E depends on its eigenfunctions, which are not known initially. The N Hartree-Fock equations (2.5) must therefore besolved by an iterative process.In 1951, Roothaan proposed using a complete set of b well behaved functions Xa, torepresent the Hartree-Fock orbitals [75],b= (2.10)Prior to this advance, the differential Fock equations were evaluated numerically, andwere only tractable for atoms and small linear molecules. Roothaan’s procedure can berepresented with matrix algebra and is very amenable to computer-based calculation.Chapter 2. Theory 24Substituting equation (2.10) into the Hartree-Fock equation gives:C3aFX = ec,,x3 (2.11)Multiplication by x gives a set of b equations with unknowns c3,:caj(Fra—ejSrs)=O, r=1...b (2.12)where,Fr3(XrIPIX),Sr8IXs>.For a non-trivial solution of the set of coefficients c33,det I F,.3 — 0 (2.13)This is known as the Hartree-Fock secular equation.The iterative procedure is started by an initial guess of the individual one-electronorbitals ç. The Fock operator is then constructed from equations (2.7) to (2.9). Thematrix elements, F,.3 and S,., are then computed and used to find the set of orbitalenergies ej, from the secular equation (2.13). The values of ej are used in turn to find thecoefficients c33 from equation (2.12) which gives an improved approximation of the orbitalsq5,. The procedure is then repeated using the improved orbitals until the improvementin total energy from one cycle to the next is smaller than a preset value, typically lessthan i0 hartrees.The quality of the final result is very sensitive to the choice of basis functions Xs• Verygood approximations of the Hartree-Fock orbitals can be made using Slater functions,Xs r’’eYr, which are called Slater type orbitals (STO’s). However, for STO’sthe matrix elements F,.3 and S,.3 are computationally expensive, particularly for the twoelectron terms (r s). A less efficient, but much more tractable choice of basis setChapter 2. Theory 25functions are Gaussian-type orbitals (GTO’s), Xs o rn_1e_3r2Yr. Because individualGTO’s are not a very good description of the Hartree-Fock orbitals, 4, the basis functionsXs are often made up of a sum of Gaussian primitive functions, and are typically optimizedon atoms or molecular fragments (see for example, reference [59]).Small basis sets (low b) usually give a poor approximation of the Hartree-Fock orbitalsqf,. These basis sets are said to be far from the true Hartree-Fock limit. Larger basis sets,with carefully chosen exponents, can give wavefunctions and total energies that are veryclose to convergence to the limit of the Hartree-Fock treatment. Large basis sets, whichgive good results for many properties are said to be saturated. Often it is useful to addbasis functions with very small exponents to a basis set, these are referred to as diffusefunctions. Supplemental functions with higher angular momentum (such a p function ina hydrogen basis set) are referred to as polarization functions.It can be seen from equation (2.12) that the number of matrix elements that need tobe evaluated scales as b4 (where b is the size of the basis set). Thus, it is very desirableto choose a basis set that is as small as possible, especially for large, many-electronmolecules. The quality of a basis set can be evaluated by the comparison of the totalenergy with other calculations (and sometimes with an “experimental” value) and withother experimentally derived properties such as the EMS cross-section (using the TargetHartree-Fock Approximation described in the next section). Note that, except whereindicated in the text for the highest level treatments, the Hartree-Fock calculations in thepresent work were performed at the University of British Columbia, using the Gaussianseries of programs [76] with additional post-processing software developed in-house bymany authors [77].It must be remembered that the electron-electron interactions are treated as averagesChapter 2. Theory 26in the Hartree-Fock procedure. For a more accurate model of electronic structure theinstantaneous electron interactions must be included. For example, if one electron at oneinstant is close to the nucleus in helium, then it is energetically favorable that the otherbe far away, due to the repulsive electron-electron force. To fully describe these electroncorrelations methods other than the Hartree-Fock treatment must be used. One such,used extensively in this thesis, is the configuration interaction (CI) approach. The CItechnique is described in reference [36].In a CI calculation the total electronic wavefunction Ji, is written as an expansion ofa complete set of functions.It turns out to be very convenient to use the HartreeFock ground state configuration o, which is called the reference state, and its excitedconfigurations for this purpose. The excitations of the ground state are normally writtenas 4 for an excitation from the orbital to the virtual orbital i. An SD-Cl wavefunctionis written as the sum of all the configurations:‘I’ = 4 + Ea + (2.14)tj4LLZincluding only single and double excitations in equation (2.14). In a very similar mannerto that outlined for the Hartree-Fock method above, the matrix elements Hij and Sjjare constructed from the n-electron Hamiltonian for the configurations I and J. Thecoefficients a are then solved variationally using the secular equation:det I H1., — ES1 1= 0 (2.15)In general, it is not necessary to include all of the possible configurations to obtain“good” total energies or wavefunctions. Selecting the important configurations, thosewhich have the largest coefficients and contributions to the total energy, is a difficultprocedure, which requires considerable experience.Chapter 2. Theory 27The procedure outlined above is referred to as single reference CI, since only theHartree Fock ground state is used to derive the configurations. However, if several configurations contribute strongly to the total wavefunction then the single reference techniqueis not very efficient. A multiple reference configuration interaction treatment (MR-CT)is usually performed in two (or more) stages. First a singles and doubles CI calculationis performed, as described above. The most important configurations (chosen by somecombination of total energy and coefficient contributions) are then each used as referencestates to generate further configurations and the coefficients computed again. It shouldbe noted that all of the CI and corresponding high-level SCF calculations in the presentwork were kindly provided by my collaborators, Professor E.R. Davidson and co-workersat Indiana University.2.3 The Scattering Cross-sectionThe relationship of the binary (e,2e) scattering cross-section to molecular wavefunctionshas been reviewed in detail in references [10,26, 78]. A brief discussion of the scatteringtheory relevant to EMS is presented here.The (e,2e) cross-section, 0EM5, in atomic units, is given by: [10]0EMS = (2ir) P1P2 _Jaci I 12 (2.16)where T1 is a quantity called the scattering amplitude. The matrix element IT1 12 isthe probability of a transition from the initial state prior to collision to the final ionstate. The spherical average fd accounts for the random orientation of the gaseoustarget molecules. Averaging over the vibrational, rotational and translational states ofthe initial and final states of the target should also be included. However, rotational andChapter 2. Theory 28translational effects are too small to be observed experimentally because of the relativelybroad instrumental energy resolution (see chapter 3). Vibrational effects are neglected bycalculating the cross-section at the molecular equilibrium geometry (see references [10, 79]and the discussion in chapter 6).Using the binary encounter [70] and impulse approximations [71], the scattering amplitude T1 can be rewritten in terms of the incoming Xo() and outgoing xi(i), x2(Melectron waves and the initial and final wavefunctions of the target molecule, ‘l and‘I’’ respectively. The EMS cross-section can then be written as,GEMS= 4 3P1P2 fdf I (X1Q)x2Q5)’1!’ I T I ‘I’Xo()) 12 (2.17)where T is the transition operator for the (e,2e) process. Assuming a two-body collisionbetween electrons and neglecting the effects of spin, the transition operator T is thespin-averaged Mott scattering matrix, Tm [11],GEMS = 47rP1P2 J dQ I (1 I Tm I > 12 I (xiQ5t) X2(P2) I ‘I’ Xo()) 12 (2.18)where = (j5 + p) and A = (j5— 2) are the relative initial and final momentarespectively. The first matrix element in this expression, I (1c’ I Tm I ) 12, is the half-off-shell Mott scattering cross-section, fee, which has the form [11],1 2,rv [ 1 1 1 1 ( Io_I21fee i + — cosivln ii (2.19)(21r2)2 e2_1 I—2I Io_iI2 Ipo—v21 \ io—thi2)Jwhere1Ii—MNote that fee is a kinematic term—it does not depend on the target wavefunction andis therefore the same for every EMS measurement. For the symmetric non-coplanarscattering geometry, the Mott cross-section is very nearly constant for a wide range ofbinding energies and for p 2 au [52].Chapter 2. Theory 29Very far away from the scattering centre and at sufficiently high kinetic energies,the incoming and outgoing electrons behave as plane waves, x = e However, nearthe target species the electron waves are distorted by the molecular potential. Thedistortion of the scattering waves can be accounted for using the Distorted Wave ImpactApproximation (DWIA) for atomic targets but this has been found to be intractable formolecules [10]. If the electrons are treated as plane waves for the entire reaction furthersimplifications can be made. The assumption that the electrons are plane waves has beenshown to be valid by experiment for a sufficiently high impact beam energy (E0 1000eV) and low momentum of the struck electron (I j5 I 2 au) [10,80]. The use of planewaves and the binary encounter approximation (and hence the impulse approximation)is referred to as the Plane Wave Impulse Approximation (PWIA). Dropping the constantkinematic terms including fee from equation (2.18) and assuming the electrons are planewaves gives,EMS JdfZ I (N-1 I ‘I’) 12 (2.20)where is a plane-wave spin orbital representing the target electron. The overlap of theion and neutral wavefunctions in equation (2.20), when calculated using configurationinteraction wavefunctions as described in the previous section, or some other many-bodyformulation, can be used to assess the importance of electron correlation and relaxationeffects in the measured cross-section. It should be noted that for many-body calculationsseparate treatments of both the ion and the neutral states must be computed, thenthe overlap between the two calculated. A spherically averaged momentum distributioncalculated using equation (2.20) is referred to as an overlap distribution or OVD. Notethat an OVD should not be compared directly with an experimental momentum profile,since it does not include any accounting for the spectrometer resolution function. Thiswill be discussed in detail below.Chapter 2. Theory 30If initial state (i.e. in the neutral ground state molecule) electron correlation effectsare negligible then the Target Hartree-Fock Approximation (THFA) [10] may be furtherapplied to equation (2.20). The ground state is expressed as a single Slater determinantand the ion state as a linear combination of determinants. The EMS cross-section thenreduces to:OC s? Jdf I &Q3) 12 (2.21)where ‘Mpl is the canonical Hartree-Fock momentum space (orbital) wavefunction for theth electron. The spectroscopic factor (or pole strength) S is simply the probability offinding the (qSj)’ one-hole configuration in the final ion state f. Under these conditionsa manifold of final ion states f is formed when the jth initial state orbital is ionizedand, according to the spectroscopic sum rule the sum over all final states, E1 = 1,although only configurations of the same symmetry will contribute. This implies thatby summing all the observed intensity of each final ion state manifold, all the orbitalsin a molecule can be normalized to a common relative intensity scale, with appropriateallowance for degeneracies. This can be done by integrating areas of the binding energyspectrum and is discussed in detail in section 3.5.When final ion state correlation and relaxation effects are negligible, the ion wave-function can be expressed as a single Slater determinant, using the same basis set as forthe neutral. In this case S =1 which is equivalent to the frozen orbital (Koopmans)approximation. The quantity f dfZ I 12 is the spherically averaged one-electron(independent particle or orbital) momentum distribution (MD). Therefore, within thePWIA and the frozen orbital descriptions, electron momentum spectroscopy can be considered to provide direct “imaging” of the orbital electron density in momentum space.Note that q5,() is the Fourier transform of the more familiar one-electron position spaceorbital wavefunction q(). Again, as for the OVD, because it does not include anyChapter 2. Theory 31consideration of the instrumental resolution a calculated spherically averaged MD shouldnot be compared directly with experimental results.It is necessary to include the finite instrumental angular (momentum) resolution effects for a meaningful quantitative comparison of theoretical OVD’s or MD’s with EMSresults. In order to correctly model the actual experimental conditions of finite acceptanceangles for 0 and , the OVD’s and MD’s calculated using equations (2.20) and (2.21)respectively must be folded with the correct momentum (angular) resolution functionbefore comparison with experiment. This is most accurately done using the GaussianWeighted—Planar Grid or GW-PG method of Duffy et al. [81]. In this procedure, theinstrumental resolution function is first constructed as a Gaussian function on a grid inthe experimental kinematic variables 0 and 4. The size of the grid reflects the instrumental analyser slit widths. The resolution function is then transformed into momentumspace using equation (2.3) and “folded” with the calculated momentum distribution. Theresulting “folded” theory, is referred to as a theoretical momentum profile (TMP) andcan be directly compared to an experimental momentum profile (XMP) recorded by thespectrometer. When the instrumental resolution function has been correctly incorporated into the calculation, the resulting TMP’s (for sufficiently high-level theory) havebeen shown to give good agreement with the experimental profiles, as was demonstratedfor the lb1 HOMO of H20 [81,82].Chapter 3Experimental MethodAll of the experimental work reported in this thesis was obtained on a single-channelsymmetric non-coplanar spectrometer. The design and operation of this instrument hasbeen described in detail previously [52, 77,83—85]. The main features are summarizedbelow together with a description of several new improvements carried out during thepresent work.3.1 The EMS SpectrometerA diagram of the EMS spectrometer used in this work is shown in figure 3.1. While thebasic instrument is the same as that used for all other EMS studies performed in thislaboratory, the vacuum system and electron gun were rebuilt and the computer interfaceand control program were redesigned during the course of this work. These modificationshave provided considerable improvements in instrument performance and data collection.In order to satisfy the requirements of the binary encounter approximation discussedon page 21, a high energy, (E0 = l200eV+€) collimated electron beam is collided with agaseous target. The collision occurs in a well-defined manner, such that the scattered andejected electrons can be energy and angle selected to satisfy the symmetric non-coplanargeometry, i.e., O = 02 = 45° and E1 =E2 = 600 eV. The electrons are then detected in32Chapter 3. Experimental Method 33coincident pairs and the accidental coincidence events removed from the signal. Both theimpact energy E0 and the azimuthal angle are under computer control for scanningbinding energy spectra and momentum profiles respectively.The electron optics and analyzers have been constructed out of brass to reduce theeffects of magnetic interference on the incident and outgoing electron paths. The sprayplates (SP1, GP, SP2) which collimate the electron beam and which are used to monitorits characteristics are made of molybdenum to reduce secondary electron production byhigh energy electron impact. The support structures and vacuum housing are constructedout of aluminum and sealed with rubber 0-rings. The system is pumped by two Edwardsturbomolecular pumps, a 450 £/s pump on the main chamber and a 300 £/s pump differentially evacuating the electron gun chamber. Both the main chamber which containsthe analyzers and turntable and the electron gun section which contains the filament,electron gun body and first deflector set Dl, are pumped from below to ensure rapidpump-down and to afford easy access to the electron gun and main chambers. The differential pumping aperture which divides the spectrometer into separately pumped gunand main chambers is the spray plate SP1. The outlets of the turbomolecular pumps aremaintained at a low vacuum (10-2 torr) by a rotary pump and the exhaust is vented intoa fume hood. The base pressure of both the main and gun vacuum chambers is typicallybelow iO torr.The electron beam is produced by heating a thoriated (0.5%) tungsten wire witha current of approximately 2.2 amps DC. The filament position can be adjusted fromoutside the vacuum housing to optimize performance. This arrangement greatly reducesthe time necessary to replace a filament when compared with the previously fixed filamentmount. The vacuum isolation of the gun chamber also increases the filament lifetimecompared with the previous instrument design. The electrons are collected and focusedChapter 3. Experimental Method 34Fixed - MovableØCEMCMA\\•Turntable -— - ii ii.’’ — —-_ISP1: D2ii Ii Servo[1 Motor—JElectron GunFigure 3.1: A Diagram of the EMS Spectrometer.SP1, SP2, GP Spray Plates CMA Cylindrical Mirror AnalyzerDl, D2 Electrostatic Deflectors SectionsFC Faraday Cup CEM Channel Electron MultiplierZL Zoom Lenses GC Gas CellChapter 3. Experimental Method 35by a commercial electron gun body, a Cliftronics CE5AH, normally used in oscilloscopes.Due to the high temperature of the filament, the electron gun elements are constructedof (non-magnetic) stainless steel. The gun body is mounted such that the film causedby evaporation of the filament does not cause voltage breakdowns on the supportinginsulators during extended periods of operation. The differential pumping of the gunchamber relative to the main chamber also reduces film build-up and extends the filamentlifetime. The filament, grid, anode and centre element of the Einzel focusing lens arefloated at (1200 V + &) relative to the grounded collision region by a modified Fluke412B high voltage power supply. This potential causes the electrons to be acceleratedout of the gun toward the collision region. The value of e can be varied under computercontrol over a range of 100 V. The grid is normally slightly negative with respect tothe filament potential, typically 0 to —10 V. The anode potential depends greatly on theparticular characteristics of a filament and is +50—+190 V with respect to the filament.The first and last elements of the focusing lens are at ground potential, the middle elementis tunable in a range 0 to +200 V relative to the filament potential.The electron beam is aligned and steered into the collision region by the electrostaticquadrupole deflectors Dl and D2. Beam alignment and collimation is monitored bythe currents on the spray plates SP1, GP and SP2 and the Faraday cup, FC. The gascell (GC) is open to the spectrometer only through the apertures GP and SP2 for theelectron beam and exit slits to the analyzers. The opening to the fixed analyzer is asmall rectangular slit. To provide the movable analyzer with a clear view of the collisionregion over the angles 4’ = ±35°, a slot subtending 80° about the electron beam axishas been cut in the gas cell, opposite to the fixed slit.The exit angles from the collision region of the scattered and ejected electrons aredefined by apertures at the front of the three element “zoom” lenses. The effective angleChapter 3. Experimental Method 36resolution of the instrument as defined by these apertures is J.SO = 1.00 and 4 = 0.6°fwhm [81]. This gives an effective instrumental momentum resolution of .- 0.1 au. Thelenses retard the electrons by 500 eV to the pass energy of the analyzers, 100 eV. The“launch angle” or angle of entry of the electron into each energy analyser is controlledby a set of electrostatic deflectors in both x and y. The so-called “magic voltages” onthese deflectors are necessary to correct an error in the analyzer manufacture [84]. Theenergy analyzers are 135° sector cylindrical mirror analyzers (CMA) with logarithmicallyspaced end correctors which maintain a uniform radial electric field in the analyzer [86].The CMA’s are tilted away from the electron beam at an angle of 2.7° from the verticalto allow for second order focusing (at 42.3°) while maintaining a polar entrance angle ofO = 450 [86]. Since the energy resolution of this CMA is 1% of the pass energy, the choiceof a 100 eV pass energy gives an analyzer resolution of 1 eV fwhm for an incident beamenergy of greater than 1200 eV. Including the intrinsic energy spread in the incidentelectron beam and the effects of both analyzers, the coincidence energy resolution of thespectrometer is approximately 1.7 eV fwhm. One of the analyzers sits on a turntablewhich can rotate about the axis of the incident beam; the other is fixed. This allows thecomputer control to set any angle 4 from —35° to +35°.The electrons are collected by Phillips B318 AL/Ol channel electron multipliers (CEM)running in saturated count rate mode (constant pulse height distribution). The entrancecone of each CEM floats at the potential of the inner cylinder section of the analyzer.The back ends of the channeltrons are biased at —3 to —4 kV relative to ground. Theoutput signal is capacitively decoupled from this high voltage and passed on to the signalprocessing electronics at ground potential.Chapter 3. Experimental Method 373.2 Coincidence ProcessingA schematic of the signal processing electronics is shown in figure 3.2. Each CEM signal,after being decoupled from the channeltron high voltage, is passed through a feedthroughin the vacuum chamber wall to an Ortec 9301 preamplifier.The output of the preamplifier is sent into a NIM standard timing filtering amplifier (TFA Ortec 454) which cleansup and amplifies the signal, producing a standard NIM output pulse impedance matchedto the rest of the NIM electronics. The signal is then discriminated by an Ortec 463constant fraction discriminator to further reduce noise. The discriminators have twooutputs, a fast iOns NIM signal and a slow 5is TTL output. The TTL outputs are sentto a rate meter so the operator can monitor the “singles” count rates on both the movableand fixed channels. The slow TTL output from the movable channel CFD is collectedby the computer using counter/timer #3 (see below) for use as channel normalization.The fast NIM signal from the movable discriminator is used to start the Ortec 467time to amplitude converter (TAC). The fixed NIM signal is delayed by 30 ns and is usedas the TAC stop pulse. The particular TAC used in this instrument has been found tobe unresponsive to signal delays of less than 15 ns. The TAC generates a voltage levelbetween 0 and 10 V which is linearly proportional to the time between stop and startpulses. The maximum voltage represents a delay time of approximately 200 ns. TheTAC is run in “anti-coincidence” mode, which means that the gate input on the TAC isignored and start signals are accepted as long as the TAC is not busy. If a start signalis received without a matching stop signal within the maximum time delay (200 ns) theTAC resets itself, taking 5 s to do so. During this reset period any input to the TAC,start or stop, is ignored. The maximum singles rate on the start channel is thus 200 kHz.‘The original signal transistors of the 9301’s have been replaced by two 2N5179 signal transistors andone PN491x series transistor. These provide an amplification of lOx.Chapter 3. Experimental Method 38Figure 3.2: A Schematic of the coincidence processing electronics and computer control.TFA Timing Filtering Amplifier CFD Constant Fraction DiscriminatorTAC Time to Amplitude Converter C/T Counter/TimerADC Analog to Digital Converter DAC Digital to Analog ConverterChapter 3. Experimental Method 39Under normal operating conditions, singles rates are 1—2 kHz so the TAC processing timeis not a limiting factor for coincidence detection.The output voltage level from the TAC is sent to an analog to digital converter (AD C)on a Labmaster DMA board (manufactured by Scientific Solutions). Some care is takento ensure that the same ground potential is used by the ADC and the TAC. Poor groundcontact can destroy the time correlation information by causing apparent fluctuations inthe TAC output voltage level.The time taken by the ADC to perform a conversion is approximately 30 /15. Themaximum coincidence processing rate is therefore 33 kllz. For this single channel instrument, we have found that 1—4 coincidences per second is a very high rate, thus the 30us “dead time” does not impose a significant limit on the collection efficiency. Since theTAC output pulse is only 5/1s long the voltage level is held in a sample and hold daughterboard (built by the UBC Chemistry Department Electronics Shop—see diagram in appendix A) during the analog voltage to digital conversion. The conversion is initiated bythe TRUE STOP signal from the TAC. This signal is captured by a monostable and heldtrue (+5V) during the 3OILs conversion process. Counter #4 on the Labmaster DMA isincremented each time a conversion event is completed.3.3 Computer ProcessingIn the present work a new data acquisition system based on an IBM AT (80286) typecomputer was designed and implemented. The collection algorithm is as follows. Atthe beginning of each scan the computer sets the constant parameter, (binding energy efor an experimental momentum profile (XMP) or angle 4 for a binding energy spectrumChapter 3. Experimental Method 40(BES) ) by setting a voltage on either DAC 0 or 1. Each point in the spectrum is thenvisited, by rotating the turn-table with DAC 1 to set 4’ for an XMP and by setting e usingDAC 0 for a BES. For each point the computer clears the array used to contain the ADCresults, initiates the ADC collection, then zeros and starts counter #3 on the LabmasterDMA board. This counter (#3) monitors the slow TTL singles output from the movableCFD. It is used to ensure that the collection interval for each point is independent ofvariations in the movable analyser count rate. During the collection period the analog-to-digital conversion results are automatically stored in an array without intervention by thecomputer (using direct memory addressing). Counter #4 is incremented automatically atthe end of each conversion. When the movable analyzer singles counter (#3) has reacheda preset value (usually 15,000 counts) the computer stops the collection process thenreads the number of conversions from counter #4. The collected array of ADC values(-4095 to +4096) is converted into signal delay times. The time spectrum is then binnedinto “coincidence” and “background” windows (see figure 3.3). Note that the backgroundin the time spectrum is non-zero. This pillar of accidental or background coincidencesis caused by processes other than an (e,2e) reaction and can easily be removed from thetrue signal, in the manner described below.The coincidence window is centred on the peak which is positioned at the delay time(‘ 30 ns) between the movable and fixed CEM signals. The window width is chosen suchthat all of the coincidence peak falls inside the window with a minimum of background(typically 10 ns wide). The width of the coincidence peak depends on many factors. Forexample, variations in the flight times of the electrons through the CMA’s and improperdiscrimination of the channeltron signals both cause broadening in the time spectrumpeak. The background window is positioned at longer delay times and is chosen tobe exactly eight times as wide as the coincidence window (normally 80 ns wide). TheChapter 3. Experimental Method 41...: Coincidence BackgroundWindow Window (80 ns)H(lOns)..00 100 200Time Delay (ns)Figure 3.3: A Time Correlation spectrum of coincidences measured while collecting data on the 3pXMP of argon for 60 hours, 7 minutes. The number of coincidences is plotted as a function of the timedelay between the start and stop pulses arriving at the TAC.Chapter 3. Experimental Method 42background or random events form a Poisson distribution, the probability of observinga correlation between two unrelated events, given that either event channel occurs atrandom. Since the width of the Poisson distribution depends on the single electron countrates of the movable and fixed channels, typically 200—1000 s_i, the difference in probabilities between a random coincidence occurring with a 30 ns correlation and anotherwith a 200 ns time correlation is less than 1 part in 10,000. Thus the background countrate is considered effectively constant over the time correlation space of the experiment.The true count rate, after removing the background coincidences is given by,y=8c— b (3.1)where y is the number of true counts, c is the sum of the coincidence counts in thecoincidence window, b is the number of counts in the background window and 8 is therelative width of the background to coincidence windows. If the errors are assumed to bethe square root of the counts (i.e., ic = Lb = /) then the error in the true countsyis,Ly = V’64c + b. (3.2)Note that through this procedure the number of “true” coincidences is essentially multiplied by the factor 8. This is done for convenience as y remains an integer. Since onlyrelative intensities are compared with theory, this arbitrary factor has no effect on thedata analysis.After the values of the true coincidences and the associated error estimate have beencalculated for the point in the angle distribution or binding energy spectrum, the computer screen is updated and the keyboard buffer is checked to see if the operator hasrequested program termination. At the end of a complete scan, the data is saved todisk and printed out. If desired, several spectra may be collected sequentially. This isChapter 3. Experimental Method 43often useful when collecting binding energy spectra and ensures that the “simultaneous”spectra are on the same relative intensity scale.3.4 Operating ConditionsThe spectrometer must be carefully calibrated for several parameters before it can beconsidered stable and sufficiently characterized for the long running times necessary formeasurements with reasonable statistics.• The energy analyzers and retarding lenses must be calibrated to the correct passenergy (100 eV) and checked for their energy resolution (E = 1 eV). The analyzers and lenses are calibrated by elastic scattering of 600 eV electrons by argonatoms and tuning the voltages on the outer elements of the CMA’s so that a maximum count rate is observed on the channeltrons. By varying the impact energyslightly, the energy resolution of the CMA’s can be determined. As the analyzerpower supplies are very stable this procedure is done infrequently, however, thevoltages of the analyzers, lenses and the retarding voltage are checked at the startof every spectrum.• The servo-motor control must be calibrated to set faithfully the correct turntableposition to ensure proper control of qf. This is usually done after complete spectrometer shutdown, or if the reading on the turntable (observable through a window in the vacuum chamber) is inconsistent with the computer value. This mustbe monitored carefully during normal operation of the instrument.• The movable analyzer singles rate must remain constant through all angles 4’ inorder to ensure a uniform collection efficiency for all values of momentum. AlthoughChapter 3. Experimental Method 44this is minimized by normalizing the coincidence collection time to a preset numberof movable analyzer counts, a large variation in the singles rate is symptomatic of apoorly tuned electron beam. The variation is normally checked when the q anglesare calibrated, and can be adjusted by tuning the electron beam and, in extremecases, by adjusting the “magic voltages” which control the launch angles into thecylindrical mirror analyzers.• The channeltron signals must be amplified and discriminated such that the maximum amount of true signal will be collected while line noise and impedance mismatch “ringing” is minimized. These effects are reflected as spurious peaks in thetime spectrum and often indicates preamp failure.• Before each study, the instrumental performance including the momentum resolution are checked by collecting an argon 3p momentum profile (15.8 eV BE) andcomparing with an existing experimental and theoretical standard [52]. This provides a good “benchmark” for measuring overall system performance and collectionefficiency. The energy resolution is monitored using the Ar 3p binding energy spectrum.• When a sample gas is first let into the spectrometer, the binding energy scaleinitially fluctuates by as much as 2—3 eV. This is caused by a “contact potential”interaction of the gas with the hot filament and the spectrometer surfaces. Anequilibrium is quickly reached after which the contact potential effectively adds orsubtracts a small constant amount to the incident beam energy. Absolute bindingenergy measurements are possible using the EMS spectrometer if a calibration gasis admitted simultaneously (see section 3.5 below).Chapter 3. Experimental Method 45When commencing a study, the system is pumped down to the base pressure of about1 x i0 torr. The sample gas is then let in slowly until an ion gauge ambient pressureof 1.5— 2.5 x 10—6 torr is reached. Typically this corresponds to singles count rates of500-1000 Hz on the movable channel. The system is allowed to equilibrate, so that astable contact potential is reached. The electron beam is tuned to maximize the currenton the Faraday cup and to minimize the current on the spray plates SP1, GP and SP2.In normal operation the current on the Faraday cup is 50 A. The analyzer and retardingvoltages are checked, then a spectrum can be started.Occasionally a very large noise event, i.e., a huge number of counts in either thecoincidence (positive) or background (negative) windows, will occur in a single run. Thesespurious signals are usually caused by either a failing preamplifier or by outside FM noisepickup by the signal leads. The time spectrum is a valuable diagnostic for detecting theserandom occurrences. The printouts of each run can be scrutinized for spurious signalsand individual runs containing noise deleted from the data set. Such events are veryinfrequent and do not pose a major problem to data collection.3.5 Treatment of ResultsAs mentioned in the previous section the absolute voltage scale for each molecule studiedis different due to contact potentials. However the relative energy peak positions in anEMS binding energy spectrum are correct. The binding energy scale is made absoluteby calibrating the EMS peak positions using vertical ionization potentials from highresolution PES. Typically He(I) spectra are used as such studies afford high energyresolution measurements of the ionization potentials. The specific PES studies used tocalibrate the energy scale for each molecule in the present work will be discussed in theChapter 3. Experimental Method 46appropriate chapters.Momentum profiles are derived from the 4 angle spectra by calculating the momentumvalues using equation (2.3). For a given azimuthal angle q, the momentum increases fromlow binding energy to high (variations of a few percent in 10 eV) with greater effects forsmall values of çf. Therefore, binding energy spectra are taken at constant angle, but notconstant momentum.For the same molecule, momentum spectra at different binding energies are oftenmeasured independently due to the long collection times required and the necessities ofrecalibrating and checking the spectrometer during a study. The intensities of momentumprofiles determined in this manner for different values of have no direct relationship witheach other. To place them on the same relative intensity scale a binding energy spectrumscanning an energy range wide enough to cover all of the orbitals of interest is collected.The peaks in the BES are then fitted by Gaussian functions with positions fixed at thevertical ionization potentials obtained by high resolution PES. It should be noted thatthe use of Gaussian functions is an approximation of the true shape of a Franck-Condonionization envelope. However, the use of Gaussian functions have been found in practiceto provide a sufficiently accurate and computationaly simple model of the peaks in the lowresolution EMS ionization spectra. The widths of the Gaussians used in the fit are alsofixed using a combination of the widths of the ro-vibrational manifold for each ionizationpeak (again estimated from high resolution PES measurements) and the known EMSinstrumental energy resolution (1.7 eV fwhm). The the areas of the fitted Gaussians foreach ionization symmetry are then used to normalize the momentum distributions at theappropriate qS values (i.e., at a particular momentum). For outer valence ionizations,this is normally the area of one Gaussian per ionization process (S = 1). For innervalence transitions with strong satellite spectra, often many Gaussian functions or theChapter 3. Experimental Method 47summed area under a region of the energy spectrum is needed to include all the ionizationintensity, or manifold of the same symmetry s’ = 1). Often, two wide energy rangeBES are collected simultaneously at two different angles q5 (see the end of section 3.3)which provides a double check on the normalization procedure as well as being useful forthe assignment of ion states.Note that the fitted peak areas in the binding energy spectra include any degeneracies in the ionization process. Theoretical results are usually reported for single orbitaloccupancy and their intensities must therefore be multiplied by the corresponding orbitaldegeneracy for comparison with EMS results.When all the momentum profiles for a molecule are put on the same relative intensityscale in this manner, multiplication of all of the profiles by a single factor is all thatis needed for comparison with theory. This “single point normalization” at a givenmomentum as the procedure is called, is normally done for one profile, for which thebest experimental statistics are available, with the best calculation for that profile. Theintensities of the other experimental distributions are then multiplied by this same factorfor comparison with all theoretical results. This allows a quantitative comparison oftheory and experiment for all the profiles of a molecule based on the normalizationof theory to experiment for one profile at one particular value of momentum. In thisway, EMS results can be used to test both shapes and relative intensities of calculatedprofiles.Chapter 4Hydrogen FluorideThe first complete EMS measurements of the valence orbital momentum profiles of HFwere reported in 1980, by Brion et al. [61] using a spectrometer at The Flinders Universityof South Australia. These were obtained using reasonably similar instrumental conditionsto those of the spectrometer subsequently used for the studies of the other hydrides [31,32,37,55—57] at The University of British Columbia. However, in the original studyof HF [61], the measurements were only compared with SCF and ion-neutral overlapmomentum profile calculations carried out using extremely limited basis sets and withoutany momentum resolution folding of theory. In addition significant shape differences withtheory were observed for the hr and 3o measured electron momentum profiles while the2o profile showed a large intensity mismatch, being only .—‘ 60% of the calculated intensity.A subsequent re-assessment [62] of the original valence orbital momentum profile data forHF [61] and a comparison with new high-level 116-GTO self-consistent field (SCF) andnumerical Hartree-Fock, as well as 1 16-.G(CI) ion-neutral overlap calculations recovering85% of the ground state correlation energy, was made in 1990 using the (now proventoo approximate [81,82]) Lp resolution folding procedure. This study [62] indicatedvery large differences between experiment and theory, particularly in the case of the hr(HOMO) momentum profile which showed much greater experimental intensity at lowmomentum than even the CI overlap description. Subsequently, in unpublished work, the48Chapter 4. Hydrogen Fluoride 49newer GW-PG resolution folding procedure [81] was applied to the original HF data [61]using the same acceptance angles as in the University of British Columbia spectrometer.While use of the GW-PG procedure resulted in greatly improved agreement betweentheory and experiment, it was unclear as to whether the remaining small but significantdiscrepancies for the hr momentum profile of HF were due to experimental error orto the different experimental arrangement and possibly different acceptance angles inthe spectrometer used in reference [61] compared with those in the spectrometer at theUniversity of British Columbia.In view of the uncertain and incomplete situation concerning the HF measurements,new EMS measurements for HF have been completed. These results are compared withthe very high level SCF and MRSD-CI calculations used in reference [62] but using therefined GW-PG resolution folding method [81]. It should be noted that HF is not only afundamental “benchmark” hydride molecule but also the simplest heteronuclear diatomicspecies available as a stable gaseous sample. In addition, since it is a diatomic species,hydrogen fluoride can be treated exactly at the Hartree-Fock level by means of numericalHartree-Fock methods [87,88]. The latter perspective permits a test of the saturationof a Gaussian “Hartree-Fock limit” basis set which for practical considerations mustbe finite. It is also of interest and importance to establish whether the low momentumdiscrepancies that apparently exist between theory and experiment for the hr momentumprofile of HF at the Hartree-Fock limit are accounted for by a correctly resolution foldedhigh level MRSD-C1 treatment as was found in the cases of H20 [37] and NH3 [55].It should be noted that, following the completion (and publication [68]) of the workpresented in this chapter, an EMS study of HF was published in late 1993 [89]. This newstudy [89] generally confirms the EMS results presented here, although there exist somedifferences with the results of this chapter with regard to the intensities of the hr andChapter 4. Hydrogen Fluoride 503o momentum profiles.The sample of HF was obtained from a Matheson lecture bottle and was of 99.9%stated purity. No impurities were evident in the recorded spectra. However great care wastaken to repeatedly purge the sample inlet system with HF because of the well known,highly efficient, ability of HF to displace all other adsorbed gases. As demonstratedpreviously [61,90,91], failure to observe these precautions results in severe spectral contamination and spurious results, no matter how high the purity of the gaseous sample inthe cylinder.4.1 CalculationsTheoretical momentum profiles have been calculated for the valence electron densitiesof HF using equations (2.20) or (2.21) in conjunction with the GW-PG momentum resolution folding procedures [81] in order to allow for the effects of the finite angularacceptance in the calculations. All calculations were performed at the “experimental”equilibrium geometry, = 0.9170 A [92,93]. The momentum distributions (MD) havebeen calculated via equation (2.21) using a range of independent particle SCF wavefunctions ranging in quality from minimum basis set to the Hartree-Fock limit. NumericalHartree-Fock calculations [87,88] have also been considered. In addition the effects ofelectron correlation and relaxation have been investigated by calculating the ion-neutraloverlap distribution (OVD) using a multi-reference singles and doubles configuration interaction treatment. The wavefunctions used (1 through 5c, see also figures 4.3—4.5) aredescribed briefly below and relevant properties are shown in table 4.1.Chapter 4. Hydrogen Fluoride 511 STO-3G: A minimal basis set, effectively of single zeta quality, using a singlecontraction of three Gaussian functions for each basis function. It was designed byPople and coworkers [94].2 4-31G: This is a split valence basis which has a minimal, single function descriptionof the fluorine is core and essentially a double zeta description of the valence shell.It was developed by Ditchfield et al. [95].3 Snyder & Basch: The Gaussian basis set of Snyder and Basch [96]. This is aGaussian basis set essentially equivalent in quality to a double zeta basis.4 Numerical HF: The numerical Hartree-Fock calculation reported earlier [87,88]and used in reference [62].5 116-GTO: The extensive Gaussian basis reported earlier by Davidson et al. [62] as117-GTO. It should be noted that there are 116 and not the 117 contracted basisfunctions as originally reported in reference [62].Sc 116-G(CI): This uses multi-reference singles and doubles configuration interaction (MRSD-CI) wavefunctions [62] for neutral hydrogen fluoride and its cationicstates. This permits calculation of the full ion-neutral overlap distribution (OVD)according to equation (2.20). The details of this frozen core MRSD-CI treatmentare given in reference [62].4.2 Results and DiscussionThe binding energy spectra and momentum profiles are conveniently discussed with reference to the ground state, independent particle, electron configuration of HF which mayChapter 4. Hydrogen Fluoride 52Table 4.1: Calculated and experimental properties for hydrogen fluorideCalculation° Wavefunction Ref. Contracted Energy PD pMAxbBasis Set [F]/[H] (hartree) (au) (au)1 STO-3G [94] [2s, lp]/[ls] -98.5708 -0.5069 1.152 4-31G [95] [3s, lp]/[2s1 -99.8873 -0.8976 0.943 Snyder & Basch [96] [4s, lp]/[2s -100.0150 -0.9348 0.844 Numerical HF [97] -100.0708 -0.7561 0.725 116 GTO [62] [14s, lOp, 6d, 2f]/ -100.0706 -0.7562 0.72[6s, 4p, 2d]5c 116-G(CI) [62] frozen core MRSD-CI -100.3661 -0.7106 0.70Experimental-100.460c 07068d 0.68asee figures 4.3—4.5 and text, section 4.1bFor the HF 17r momentum profile, see figure 4.3.cExperimentally derived, non-relativistic, non-vibrating, infinite nuclear mass energy, from [58]dDerived by extrapolation for a non-vibrating molecule. From [98,99]Chapter 4. Hydrogen Fluoride 53be written as:(lo.)2 (2) (3u)2 (hr)4core valenceThe valence shell vertical ionization potentials of HF have been reported to be 16.12eV (17r), 19.89 eV (3a) and 39.63 eV (2a) from photoelectron spectroscopy measurements [100—102] and these values were supported by subsequent EMS measurements [61].4.2.1 Binding Energy SpectraFigure 4.1 shows the binding energy spectra of HF observed in the present work from11—58 eV at = 8° and from 28—58 eV at = 0° on a common intensity scale and at animpact energy of 1200 eV + binding energy. The energy resolution is 1.7 eV fwhm. Thespectral relative intensities in figure 4.1 are consistent with those observed up to 54 eVin an earlier EMS measurement [61] at 2.1 eV fwhm under essentially identical kinematicconditions. The fitted Gaussians in figure 4.1 reflect the experimental energy resolution,folded with the estimated natural width as observed for the (lir)’ and (3)’ bandsusing high resolution photoelectron spectroscopy [100, 101]. The width of the main peakfor the (2)1 process reflects a Gaussian fit to the main concentration of intensity at39.7 eV. A comparison of the spectra at = 0° and 8° above 28 eV suggests additionalstructure in the 32—38 eV region. A similar long shoulder on the low energy side of themain (2r)1 peak has been observed in PES studies [101] and previous EMS studies [61].In agreement with earlier EMS spectra [61] additional weak satellite intensity is observedat higher energies to the limit of the data at 58 eV. The earlier EMS studies [61] haveconfirmed that the (lir)’ and (3a)’ processes correspond to the removal of a “p-type”electron with greater intensity observed at 8° than at 0°. In contrast the relative intensitydistributions observed in the = 0° and 8° spectra of figure 4.1 suggest that the intensityChapter 4. Hydrogen Fluoride 54in the 32—58 eV region belongs dominantly to the “s-type” (2o’ manifold. Howeverit is clear that most of the (2)’ intensity is located in the 39.7 eV peak as suggestedby a number of different many-body calculations [61,62, 103] which each predict several(2o’ poles concentrated in the 37—44 eV region. The relative intensities in the g = 8°binding energy spectrum as given by the fitted peaks for the (l7r)1 and (3o’ processesand by the total integrated counts from 32—58 eV at 00 for the (2u)’ process havebeen used, in section 4.2.2, to provide the correct relative normalization of the measuredexperimental momentum profiles for the three valence orbitals. Normalization on the= 0 binding energy spectrum in the case of the 2a XMP is preferable in order toeliminate any contributions due to “p-type” (lir)1 and/or (3u)’ poles in this regionsince many-body calculations [62, 103] have indicated the possibility of such processes.In figure 4.2 the experimental EMS binding energy spectrum at 4 = 8° is compared with synthetic spectra using the results from two previously published [62, 103]many-body calculations which involve large basis sets. The MRSD-CI calculation in figure 4.2(b) employs the pole strengths obtained with the 116-G(CI) wavefunction usedin the present work and also in reference [62]. The spectrum in figure 4.2(c) is basedon the orthogonalized direct diagonalization (ODD) Greens’ function result reported byBaker (the “extended manifold” calculation in reference [103]). The pole strengths fromeach of the MRSD-CI (see table 2 of reference [62]) and ODD many-body calculationsare multiplied by the heights of the theoretical momentum profiles of the appropriatesymmetry (i.e. lir, 3o or 2u) as given by the 116-G(CI) calculations of the OVD’s onfigures 4.3—4.5 respectively. It can be seen from figure 4.2 that both calculations give avery good quantitative reproduction of the outer valence peaks and a reasonable semiquantitative description of the inner valence region. Both calculations attribute most ofthe inner valence intensity to the (2)_1 process. The MRSD-CI calculation reproducesChapter 4. Hydrogen Fluoride 55>U)Cci)Ccici)105010Binding Energy (eV)601050Figure 4.1: EMS binding energy spectra of hydrogen fluoride from 11 to 58 eV at = 8° and 0°,obtained at an impact energy of (1200 eV +binding energy). The solid lines represent Gaussian fits tothe peaks.20 30 40 50Chapter 4. Hydrogen Fluoride 56the relative shape of the inner valence region quite well but somewhat overestimates theintensity whereas the ODD calculation predicts a 2cr concentration which is broader thanthe measured spectrum.4.2.2 Comparison of Experimental and Theoretical Momentum ProfilesThe experimental (solid circles) and various theoretical momentum profiles (lines) for thehr, 3cr and 2cr valence orbitals of HF are shown in figures 4.3, 4.4 and 4.5 respectively.These three figures are all on the same (relative) intensity scale. The experimental measurements for each of the three valence orbitals have been made at binding energies of16.1, 19.9 and 39.7 eV respectively. The three measured momentum profiles have beenput on a common relative intensity scale by normalizing to the respective areas in thebinding energy spectra (at ç6 = 8° for the hr and 3cr; at = 00 for the 2cr) shown infigure 4.1 and discussed in the previous section. The hr XMP was then height normalizedto the 116-G(CI) calculation (figure 4.3). With this single point normalization betweentheory and experiment (described in detail in section 3.5) all other calculations and themeasurements for all three cross-sections are compared on the same relative intensity scalein figures 4.3—4.5. Appropriate allowance has been made for the orbital degeneracies. Itshould also be noted that all MD and OVD calculations were originally normalized suchthat they integrate to unit probability over all (p) space and we have chosen to showthe 116-G(CI) calculations in figures figures 4.3—4.5 with a “scaled-up” pole strength ofunity (see table 2 of reference [62] for the pole strength as calculated for each ionizationprocess). With this representation any difference between the 116-G(CI) calculationsand the respective measured electron momentum profiles is an indication that the corresponding binding energy spectrum peak area used for normalization (see above) doesnot include all the pole strength. This implies that the “missing pole strength” is located57c:5>00>CO0)C10 40 60Binding Energy (eV)Figure 4.2: Experimental and synthetic binding energy spectra for HF. (a) EMS spectrum at 4 = 8°.(b) Multi-reference singles and doubles CI calculation, 1 16-GCI [62]. (c) Orthogonalized Direct Diagonalization Greens’ function calculation, reference [103]. In figures (b) and (c) the vertical bars representthe predicted energies and intensities as given by the calculated pole strengths (MRSD-CI [62] in (b),ODD [103] in (c)) at = 8°.Chapter 4. Hydrogen Fluoride20 30 50Chapter 4. Hydrogen Fluoride 58somewhere else in the binding energy spectrum as suggested by the calculations [62].Quite different and less stringent normalizations between theory and experiment wereemployed in the earlier work of Brion et a!. [61], in addition to the use of much lesssophisticated wavefunctions. In reference [61] the different calculations were separatelyheight normalized to the respective hr and 3cr experimental data, instead of maintainingcorrect relative normalizations for theory. Also the scaling factors between theory andexperiment, i.e., 3cr (0.9) and 2cr (0.6), were not shown on figure 4 of reference [61] butwere discussed in the text. The earlier EMS experimental measurements (open circles) ofBrion et a!. [61] at 1200 eV are also shown on figures 4.3, 4.4 and 4.5 for comparison. Itshould also be noted that transcription errors have recently been discovered in the earlierreported 1200 eV EMS experimental data as presented in figure 4 of reference [61] andfigure 1 of reference [62] for the 2cr orbital. The corrected 2cr data points for the data inreference [61] are shown by the open circles on figure 4.5 of the present work.In the lower panels of figures 4.3—4.5 are shown the respective momentum and position space density maps calculated for an oriented HF molecule using the numericalHartree-Fock SCF wavefunction. These serve to illustrate the bonding and non-bondingcharacteristics of the respective valence orbitals in the complementary momentum andposition space representations. The momentum density maps are directly related to thepresently reported spherically-averaged experimental and theoretical momentum profilesshown in the upper panels of figures 4.3—4.5 while the position density maps show themore familiar spatial charge distributions with reference to the nuclei. The density mapsshown in figures 4.3—4.5 together with those reported earlier for CH4 [56], NH3 [55],H20 [80], SiH4 [57], PH3 [32], H2S [31] and HC1 [33] provide useful position and momenturn space comparisons of the bonding electron density characteristics across the row 2and 3 main group element hydrides of Groups IVA through VIIA.Chapter 4. Hydrogen Fluoride 59We first compare the present EMS momentum profile measurements with those reported earlier for HF [61]. From figures 4.3 and 4.5 it can be seen that the two setsof measurements are in excellent quantitative agreement in the case of the hr and 2crelectrons. However in the case of the 3cr electron (figure 4.4) the earlier 1200 eV measurements [61] are clearly significantly lower than the present data and the best calculations(see discussion below) particularly in the region below 0.6 au. It should be pointed outthat the present measurements and the earlier 1200 eV EMS data [61] were obtainedunder very similar experimental conditions, involving the same design of three-element(6:1) retarding lenses, apertures and acceptance angles in both 0 and 4. In the earlierEMS study of HF [61] momentum profile measurements were also reported at an impactenergy of 400 eV plus the binding energy, with the retarding lenses operated with a 2:1ratio. These earlier 400 eV results [61] have not been shown on figures 4.3—4.5 since thedifferent retarding lens ratio may result in somewhat different acceptance angles whichwould change the GW-PG folding procedure. Nevertheless as can be seen from figure 4of reference [61], the original 400 and 1200 eV momentum profile data were in very closeagreement for the hr and 2cr distributions although in the case of the 3cr measurementsbelow 0.6 au, the 1200 eV data were shifted to slightly higher momentum comparedwith those at 400 eV. This is consistent with the above noted anomaly between the twodata sets shown in figure 4.4 and supports the view that the original 1200 eV data [61] forthe 3cr electron of HF are significantly in error below .— 0.6 au. Thus it can be concludedthat, with the exception of the p < 0.6 au region of the 3cr data, the earlier [61] andpresent EMS measurements for the valence electron experimental momentum profiles ofHF are in good quantitative agreement.Also shown on figures 4.3—4.5 are the theoretical momentum profiles calculated usingequations (2.21) and (2.20) with the respective SCF and MRSD-CI wavefunctions asChapter 4. Hydrogen Fluoride 60presented in section 4.1 and also in table 4.1. All MD and OVD calculations have beenresolution folded with the GW-PG procedure recently developed by Duffy et al. [81]rather than the less accurate I.p Gaussian function used by Davidson et al. for theearlier studies of HF in reference [62]. In comparing EMS data, obtained for a widerange of atoms and molecules [15, 16,81] using the present instrument, with near SCFlimit and MRSD-CI calculations it has been shown that use of the GW-PG resolutionprocedures lead to excellent quantitative agreement between sufficiently high level theoryand experiment in all cases. In contrast use of the Lp Gaussian method often resultedin relatively poor agreement, especially in the cases of the outermost valence electronsof the highly polar second row hydrides H20 and NH3 (in particular compare the tpfolded MRSD-CI calculations in the original reports of references [37] and [55] with thecorresponding GW-PG folded results shown in references [15] and [104]). Similarly asignificant overall improvement in agreement between experiment and the 116-G(CI)calculations is now obtained for HF, using the GW-PG folding method, particularly inthe case of the hr outermost electron (compare figure 4.3 of the present work with figure1 of reference [62]).In comparing the THFA (i.e. SCF) calculations of the momentum profile with experiment for the hr electron (figure 4.3) it can be seen that a very poor description resultsfrom the limited basis set STO-3G wavefunction [94]. A continuous improvement occursas the basis set is successively enlarged to 4-31G [95], Snyder and Basch [96] and then tothe highly saturated and diffuse 1 16-GTO basis [62]. It can be seen from table 4.1 andfigure 4.3 that not only the predicted total energy but also the dipole moment and position of the maximum (PMAX) of the hr momentum profile are all successively improvedin going through the range of wavefunctions from STO-3G to 116-GTO. Furthermorethe excellent agreement between the (exact) numerical Hartree-Fock and the 116-GTO>U)-4-,>-I-,QMomentum (au)MOMENTUM DENSITY POSITION DENSITY(au) (cu)Figure 4.3: The hr experimental and theoretical momentum profiles of hydrogen fluoride (upper panel).The solid circles are the present data and the open circles are the earlier EMS data of reference [61].The lower panels show the momentum and position space density maps for an oriented HF moleculecalculated using the numerical Hartree-Fock wavefunction (see table 4.1). The contours shown are 0.01,0.02, 0.05, 0.1, 0.2, 0.5, 1.0, 2.0, 5.0, 10.0, 20.0, 50.0 and 90.0% of the maximum density.Chapter 4. Hydrogen Fluoride 615cHF lrv16.1 eV10 -501Theory5c 11 6—G(Cl)5 116—GTO4 Numerical HF3 Snyder & Bosch2 4—31G1 STO—3GExperiment• This Worko Ref. [61]0I I I0 1 2 384—8—8—4—4 0 4 8HF—8 ——8 —4 0 4 8Chapter 4. Hydrogen Fluoride 62predictions (calculations 4 and 5 respectively) for all three properties, indicates that thetrue Hartree-Fock limit has effectively been reached at the 1 16-OTO level. Nevertheless,from figure 4.3 it can be seen that an appreciable discrepancy still exists between thetheoretical and experimental momentum profiles even at the Hartree-Fock limit (calculations 4 and 5). Clearly the independent particle (orbital) model, even in its most accuraterepresentation (i.e. the numerical calculation which gives the true Hartree-Fock limit),fails to provide an accurate quantitative description of the hr XMP of HF, particularlyin the low momentum region, below p = 1 au, where experiment shows slightly higherintensity. This situation is similar to that observed earlier for the outermost electrons(orbitals) of 1120 [15,37] and NH3 [15, 55].Incorporation of electron correlation and relaxation effects using the 1 16-C(CI) frozencore description (see reference [62] and also table 4.1) of the ion and neutral totalelectronic wavefunctions to calculate the OVD according to the prescription of equation (2.20), with GW-PG folding, results in greatly improved agreement with experimentcompared with the SCF limit calculations for the (17r)’ process (compare calculation 5cwith calculations 4 and 5 on figure 4.3). However, small but significant discrepancies stillremain in the case of HF, even with the large CI description of Davidson et. al. [62]. Thisis in contrast to the situation for H20 [15,81,82, 104] and NH3 [15, 104] where essentiallycomplete agreement has recently been demonstrated at an equivalent MRSD-CI level,when using the GW-PG folding [81]. In particular, it can be seen for the hr momentumprofile of HF (figure 4.3 and table 4.1) that the experimental data (PMAX 0.68 au)is slightly displaced towards low momentum with respect to the 116-G(CI) calculation(PMAx = 0.70 au), not only at low momentum below the maximum, but also on the highmomentum side of the profile. The good agreement between the present and earlier [61]EMS measurements for the hr momentum profile would seem to eliminate the possibilityChapter 4. Hydrogen Fluoride 63that the discrepancy between theory and experiment is due to incorrect data or to inadequate evaluation of the absolute 4 scale (which would lead in turn to inaccuracies in theexperimental momentum scale—i.e. a “sideways shift” of the data in figures 4.3—4.5). Thepossibility therefore remains that the 1 16-G(CI) calculation used for the (lir)1 processin HF is still not sufficiently converged. This particular frozen-core MRSD-CI calculation(see reference [62] and table 4.1) recovers 76% of the total ground state correlationenergy and greater than 98% of the estimated MRSD correlation energy available fromthis basis set.The fact that saturated basis sets with diffuse functions and also electron correlation effects are needed to describe the hr experimental momentum profile of HF mayhave important implications for the quantum mechanical modeling of phenomena suchas intra-molecular hydrogen bonding, moment properties and chemical reactivity whichare all strongly influenced by details of the long range (spatial) charge distribution.Clearly in such applications close attention must be given to an adequate descriptionof the long range “tail” of the wavefunction since even very good SCF wavefunctionsresult in somewhat limited descriptions of the EMS cross-section in regions of the XMPcorresponding to the outer spatial (low momentum) regions of phase space for such ahighly polar species as HF.The lower panels of figure 4.3 show the momentum and position space electron densitydistributions for the hr orbital of an oriented HF molecule, calculated with the numericalHartree-Fock wavefunction. While the experimental measurements indicate the need totake into account electron correlation effects as discussed above, these density mapscan be considered to give a quite good quantitative indication of the valence electrondensity distributions. Note that the location of atomic centres are a position spacephenomenon and as such do not exist on the momentum space plots. The electronChapter 4. Hydrogen Fluoride>Cl)Ca)4JCa)>-4-,0a).I____ 30•—4 0 4 864HF3o-19.1 eV1Theory (94%)5c 11 6—G(CI)5 116—GTO4 Numerical HF3 Snyder & Bosch2 4—31G1 STO—3GExperiment• This Worko Ref. [61)6-4-2-0-840—4—8—80‘ I——I1 2Momentum (au)MOMENTUM DENSITY POSITION DENSITY834H0 F—4—8—8Figure 4.4: The 30’ experimental and theoretical momentum profiles of hydrogen fluoride (upper panel).The solid circles are the present data and the open circles are the earlier EMS data of reference [61].All calculations have been scaled by a factor of 0.94. The lower panels show the momentum andposition space density maps for an oriented HF molecule calculated using the numerical Hartree-Fockwavefunction (see table 4.1). The contours shown are 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1.0, 2.0, 5.0, 10.0,20.0, 50.0 and 90.0% of the maximum density.—4 0 4 8(cu) (cu)Chapter 4. Hydrogen Fluoride 65momentum information in the nuclear region would be located at the virtual p-spaceboundary at infinite momentum [45]. The momentum space maps exhibit inversionsymmetry as a consequence of momentum conservation while all other symmetry elementsare preserved in both spaces. The side panels on each map show the correspondingtransverse and longitudinal slices of the density distributions along the respective dottedlines. Both maps convey essentially the same information but in very different ways,offering complementary perspectives on molecular bonding and electronic structure.It is apparent from the position map that the highly localized hr molecular orbital isessentially equivalent to a non-bonding lone pair orbital (valence bond theory description)on the fluorine atom. This is confirmed by the smooth contours of the momentum densitymap which shows no evidence at large p of the “crinkles” due to interference effects (oftencalled “bond oscillations”) which are the momentum space manifestation of chemicalbonding contributions [14,80, 173]In the case of the 3o electron (figure 4.4) the shape of the momentum profile corresponding to the presently reported measurements is quite well reproduced by the 116-G(CI) calculation and progressively less well by the various SCF calculations as the basisset becomes simpler with the STO-3G calculation providing an extremely poor description. It should be noted that all calculations of the 3o theoretical momentum profile havebeen reduced in intensity by a factor of 0.94 and this may reflect some loss of (3o1intensity due to other undetected small contributions from many-body effects located athigher binding energy (i.e. S < 1 for the “main” pole at 19.9 eV, see equation (2.21) )or merely uncertainties in the normalization procedures. However the present suggestionof missing (3o’ pole strength (on the basis of hr normalization of 116-G(CI) theory(at unit pole strength) and experiment) is supported by the earlier EMS study of HF inwhich a similar scaling factor of 0.9 was used in comparing the 3o calculation with theChapter 4. Hydrogen Fluoride 66normalized experimental intensity. From figure 4.4 it can be seen that the earlier 1200 eVmeasurements below p 0.6 au are probably significantly in error because they are substantially different in shape from both the present data and the calculations. In thisregard it can be deduced, from a consideration of figure 4 of reference [61], that theshape of the earlier reported momentum profile measurements at 400 eV is much more inaccord with the present measurement and with theory. It appears that correlation andrelaxation must be taken into account in predicting the 3o momentum profile. The momentum and position density maps illustrate the strong a-bonding nature of this orbitalwhich is dominantly responsible for the single covalent bond in HF.The experimental and theoretical momentum profiles for the 2o electron are shownin figure 4.5 (upper panel). As discussed above plotting errors have recently been foundin the 2 experimental data of the earlier study [61]. The corrected values appropriate toreference [61] are shown on figure 4.5 of the present work and are in excellent agreementwith the present data. All calculations reproduce well the shape of the experimentalmomentum profile but the measured intensity is only 60% of the predicted value. Thisapparent “intensity loss” may be due to the presence of additional (2)_’ poles at energiesabove the limit (58 eV) of the presently reported binding energy spectrum (figure 4.1)or more likely due to significant distorted electron wave effects at the relatively highbinding energy involved (39.7 eV). EMS measurements at higher impact energies mightbe informative in this regard. The momentum and position density maps both indicatethe small but significant bonding contribution from the 2o orbital according to molecularorbital theory. This is in contrast to the non-bonding character attributed to the orbitalin simple valence bond descriptions.In summary, the present measurements of the valence shell binding energy spectra>CI)Ca)>I.0a)Figure 4.5: The 2o experimental and theoretical momentum profiles of hydrogen fluoride (upper panel).The solid circles are the present data and the open circles are the earlier EMS data of reference [61].The lower panels show the momentum and position space density maps for an oriented HF moleculecalculated using the numerical Hartree-Fock wavefunction (see table 4.1). The contours shown are 0.01,0.02, 0.05, 0.1, 0.2, 0.5, 1.0, 2.0, 5.0, 10.0, 20.0, 50.0 and 90.0% of the maximum density.67HF2u39.7 eVTheory5c 11 6—G(Cl)5 116—GTO4 Numerical HF3 Snyder & Basch2 4—31G1 STO—3GExperiment• This Worko Ref. (61]Chapter 4. Hydrogen Fluoride20__100840—40 1 2 3(au)POSITION DENSITYMomentumMOMENTUM DENSITY840—4. 2uH-I I I I—8—8 —4 0 4 8(au)—8—8 —4 0 4 8(au)Chapter 4. Hydrogen Fluoride 68and momentum profiles of hydrogen fluoride, generally confirm the accuracy of earlier reported measurements [61] for the hr and 2o results although some errors are indicated forthe earlier 3 measurements. The previously reported discrepancies [62] between experiment and near SCF limit or MRSD-CI calculations of the momentum profiles are shownto be largely eliminated when the recently developed and much more accurate GW-PGmomentum resolution folding procedures [81] are used. The results confirm the fact that,as in the case of the outermost electrons of H20 and NH3, the measured electron momentum profile corresponding to the removal of a hr electron of HF is significantly morecompact (in momentum space) than is predicted by the independent particle model, evenwhen using exact (i.e. numerical Hartree-Fock) SCF wavefunctions. Although incorporation of electron correlation and relaxation effects using large MRSD-CI wavefunctionsin full ion-neutral overlap calculations results in greatly improved agreement with experiment, some small discrepancies remain in the case of the shape of the hr momentumprofile. The present studies therefore suggest that the possibility of even further improvement in the MRSD-CI wavefunctions for the hydrogen fluoride molecule and its cationicstates should be investigated.Chapter 5Hydrogen ChlorideThe same EMS study from the Flinders University of South Australia in 1980 [61] discussed in chapter 4 for HF also reported measurements of the valence shell of hydrogenchloride. The HOMO 2ir experimental momentum profile was in reasonably good agreement with the modest SCF calculations performed at the time. However, the calculated50 profile showed significant mismatch for both shape and intensity between 0.5 and1.0 au when compared with experiment. For the 4o inner valence electron momentumprofile a considerable intensity mismatch was observed, with experiment being only 72%of theory even when account was taken of all the observed inner valence many-bodystates. In this earlier study [61] only relatively small SCF calculations were used (34-GTO being the largest). The effects of correlation and relaxation were investigated byion-neutral overlap amplitude calculations using a limited Greens’ function many-bodytreatment. It is also noteworthy that the instrumental momentum resolution was notincluded in the calculated momentum profiles.The earlier EMS binding energy spectrum of HC1 [61] exhibited extensive manybody structures in the (4)1 inner valence region. In this respect the inner valencespectrum of HC1 is qualitatively similar to those observed for the other third row hydridesPH3 [32] and H2S [31]. However, the spectra of HC1 were obtained at a lower energyresolution than those measured subsequently for the other hydride molecules in this69Chapter 5. Hydrogen Chloride 70series of studies (approximately 2.1 eV fwhm compared with 1.7 eV fwhm). Thus onlythe broadest features in the many-body inner valence binding energy spectrum of HC1were resolved [61]. The binding energy spectra of HC1 at two azimuthal angles and alsothe momentum profile measurements [61] clearly indicated that the structure above 25 eVwas dominantly due to the (4o1 process as suggested by the associated Greens functioncalculations [611.Following the earlier EMS study by Brion et al. [61], two higher resolution X-rayphotoelectron spectroscopy (PES) studies [29,30] of the inner valence binding energymanifold of HC1 have been published suggesting other possible assignments for some ofthe inner valence final ion states. In addition, improved many-body calculations havealso been published [105].In view of the uncertainties in the earlier EMS measurements for HC1 and the otherdevelopments in both theory and experiment that have occurred over the past decade,new EMS measurements of the valence electron profiles and binding energy spectrum ofHC1 have been performed in the present work. The measured binding energy spectrum isalso compared with new MRSD-CI calculations. The measured momentum profiles arecompared with new high-level 96-GTO and earlier-reported 107-GTO [58], near HartreeFock limit calculations. In addition, the effects of both initial ground state and finalion state correlation and relaxation on the momentum profiles are investigated by meansof MRSD-CI calculations of the full ion-neutral overlap amplitude using the 96-GTObasis. AU calculated momentum profiles also include the more accurate accounting ofthe instrumental momentum resolution of Duffy et al. [81].The HC1 sample used for the present work was taken from a cylinder (from MathesonGas) in which the stated Cl2 impurity was less than 1%. No impurities were observed inChapter 5. Hydrogen Chloride 71the binding energy spectra.5.1 CalculationsTheoretical momentum profiles have been calculated for the valence shell electrons ofhydrogen chloride using a range of SCF wave functions (via equation (2.21)) varying inquality from a minimal basis set to near the Hartree-Fock limit. In addition ion-neutraloverlap distributions have been calculated (equation (2.20)) using multireference singles and doubles configuration interaction (MRSD-CI) calculations for both the molecular ground state and the excited ion state(s) to investigate the effects of correlationand relaxation on the theoretical momentum profiles. The MRSD-CI calculations usedthe near Hartree-Fock limit basis set to construct the ground state and molecular ionmany-body wavefunctions in order to compute the ion-neutral overlap distributions (seeequation (2.20)). Finite experimental momentum resolution was incorporated into all calculated momentum profiles using the Gaussian-Weighted Planar Grid (GW-PG) methodof Duffy et a!. [81]. All calculations were performed at the experimental bond length of1.2746 A (2.4086 a.u.) [106]. Selected one-electron properties and calculation details arepresented in table 5.1 and further details of the wavefunctions are summarized below.1 STO-3G: This is the single zeta, 10-GTO minimal basis set proposed by Popleand co-workers [94].2 4-31G: This is a split-valence basis developed by Ditchfield, Hehre and Pople [95].For HC1 it is a 15-GTO basis set.3 Cade and Huo: This basis set was taken from work originally reported by Cadeand Huo [107] and extended by McLean and Yoshimine [108]. It is composed of aChapter 5. Hydrogen Chloride 72set of Slater functions comprised of (7s,7p,2d,lf/3s,2p,1 d) on chlorine and hydrogenrespectively. This was shown to be essentially converged in total energy to theHartree-Fock limit by McLean and Yoshimine, who also extended the basis set byadding further polarization and diffuse functions, finding a change in total energyof only 1.6 mhartree.4 107-GTO: This [14s, lOp, 4d, 2f/6s, 4p, ld] contracted Gaussian basis was reportedearlier by Feller, Boyle and Davidson [58]. The improved description of the innercore region in this wavefunction achieved with tight s and p functions results in thelowest SCF total energy of any basis set used in the present work (see table 5.1).5 96-GTO: The basis set for this Hartree-Fock calculation was developed for thepresent work by C. Boyle and E.R. Davidson and was chosen to saturate the diffusepart of the wavefunction, i.e., with special emphasis on the large-r, low-p tail. Itconsists of an even-tempered (21s, l4p, 4d, 2f/lOs, 3p, 2d) primitive set contractedto the final [12s, lOp, 4d, 2f/6s, 3p, ld] 96-GTO basis. To avoid linear dependencein the basis set the s components of the cartesian d functions and the p componentsof the cartesian f functions have been removed. The basis set was based on oneused for previous work on PH3 [32]. The chlorine d and f exponents were takenfrom Table III of Feller et al. [58] which were calculated using an even-temperedrestriction applied to previously energy-optimized exponents [109, 110] for Cl andH atoms.5c 96-G(CI): This CI calculation was performed by my collaborators C. Boyle andE.R. Davidson using the 96-GTO basis set. A Hartree-Fock singly and doublyexcited configuration interaction (HF SD-Cl) was first performed followed by amulti-reference singly- and doubly-excited configuration interaction (MRSD-CI)Chapier 5. Hydrogen Chloride 73calculation. These calculations used a frozen chlorine 1s2, 2s 2p6 core. Since valence electron convergence has been shown to improve (i.e. fewer configurations areneeded to recover more correlation energy) if the virtual orbitals are transformedinto K-orbitals [111—113), all of the unoccupied orbitals from the RHF ground-statecalculation were transformed to K-orbitals to form the molecular orbital basis forthe CI treatment. The SD-CT calculation for the neutral molecule used 4,657 (out ofa possible 16522) spin-adapted Hartree-Fock singly- and doubly-excited configurations by keeping all the singles and choosing the double excitations (by second-orderperturbation theory) such that all the remainder had a total contributions to thetotal energy of less than 1 mhartree.The configuration space for the multireference singles and doubles calculation wasselected using second-order Rayleigh-Schrödinger perturbation theory with a reference space chosen from the HF SD-Cl treatment. For the neutral molecule and the(27r)’ and (5)’ cationic states, the reference space included all configurationswith coefficients greater than 0.030. For the neutral molecule, out of a possible787,887 doubly or singly excited configurations 13,969 were kept. Selected properties for the neutral molecule are given in table 5.1.The calculations for the ion states were done with the same atomic orbital basisas the neutral using a similar process, but no separate SCF calculation was donefor the ions. The matrix element in equation (2.20) thus uses the same molecularorbital basis for both the final ion and initial neutral states which greatly simplifiesthe calculation. Valence shell ionization potentials of HC1 were calculated for boththe (2ir)_1 and (5o’ states, using MRSD-CI wavefunctions with 23,858 and 22,869configurations respectively. The calculated ionization potentials listed in table 5.2agree well with experiment and indicate that the neutral and ion wavefunctions areChapter 5. Hydrogen Chloride 74of comparable accuracy.In order to get an estimate of the many satellite peaks associated with the (4)’hole, an ion calculation on a large number of 2E+ states was performed. Thereference space for this was chosen from a calculation on the first 15 states using all2 hole-i particle configurations (relative to the neutral molecule). All configurationswith coefficients greater than 0.1 in any state were selected for the final referencespace. The reference space was then completed by adding all configurations requiredto obtain a set of configurations closed under C symmetry. A perturbationselected MRSD-CI calculation was then performed with configurations selected thatcontributed most to any of the first 11 states. The final CI matrix dimension was22,572. The lowest 2E energy found in this calculation was 0.3 eV higher than theenergy found for the (5)’ hole by the method described in the previous paragraph.Because of limitations in the basis set, the decreased number of configurations perstate and the higher density of states at higher energy, the calculations for thehigher 2E+ states are less accurate than the calculation for the (5)_1 hole.5.2 Results and DiscussionHydrogen chloride has symmetry C and an electronic ground state of’ E. The independent particle electron configuration of HC1 may be written as:o2 2u 32 1ir4o2 2ir”,core valenceThe outer valence shell ionization potentials have been reported to be 12.80 eV((2i-)’) and 16.60 eV ((5o’) by photoelectron spectroscopy [117, 118]. Earlier EMS [61]Table5.1:CalculatedandExperimentalPropertiesof HydrogenChiorideaProperty6STO-3G4-31GCade&Huo107-GTO96-GTOThisWork96G(CI)cExperimental(a.u.)[94][95][107,108][58]ThisWorkHFSD-CIMRSD-CIValueCalculation12345ScTotalEnergy(hartree)-455.1348-459.5631-460.1119-460.1123-460.1119-460.3294-460.3415-460.886’sBasisSet:Chlorine3s2p4s3p7s7p2dlfl4slOp4d2fl2slOp4d2fHydrogenis2s3s2pid6s4pld6s3pldDipolemoment,11D0.68220.73270.47800.47670.47670.45620.44790.43611ePMAX(2ir)_10.740.570.540.5350.53—0.530.52PMAx(5o)_10.660.610.590.590.59—0.580.57Quadrupolemoment,O2.78062.80812.80782.71682.70192.79’<t2>e34.259734.138734. 138934.000034.0938<P4>e889918891222891313891264<6H>e0.38660.38660.39380.3941<öci>e3186.173184.593184.793184.68<1/rH>8.00078.00108.00108.00978.0088<l/rc,>64.821864.820864.820764.841164.8347<qz/41>-0.00920.00540.00540.0009-0.00330.0000<qz/r1>-0.06820.00190.0013-0.0006-0.00030.0000qH0.28020.28010.28030.28430.2857qci3.56233.59993.60023.49963.4515acalculationsperformedattheexperimentalequilibriumbondlengthof1.2746A(2.4086a.u.)Ref.[106,114].bProperties(ina.u.)definedas:i’=<£qjzj >;e22=<IDq(3z—r)>,summingoverallnucleiandelectrons;<r2>e=<IDr>and<P4>e=<£p>,summingovertheelectronsonly; <6,>e=<£6(r,)>,summingoverallelectrons onatomiccentren;q=thefieldgradientatcentren.Seeref.[58]eThepresentCIcalculationswereperformedwithafrozenchlorinecore,i.e.,thecalculatedenergydoesnotincludecorecorrelationcontributions.dExperimentally..derived, non-relativistic,non-vibrating,infinitemassnucleienergy(includingcorrelatedcorecontributionstothetotalenergy).ep[115,116]forgroundstateJ=0,v=0,H35C1‘Ref.[115]Chapter 5. Hydrogen Chloride 76Table 5.2: Experimental and calculated ionization potentials for hydrogen chlorideIon pa PESb EMSC ADC(4)d MRSD-CIState [30] [29] [61] This Work [105] This Worke(2ir)1 12.80 12.80 12.8 12.8 12.53 (0.91)’ 2ir 12.55 (0.85)’ 2,r(5o.)_1 16.60 16.66 16.5 16.6 16.43 (0.90) 5o 16.54 (0.86) 5o(4u)’ 1g 23.65 23.90 23.26 (0.15) 4o 24.36 (0.22) 4o2 25.85 26.00 25.8 25.8 25.84 (0.39) 4o 26.85 (0.34) 4u3 28.50 28.67 27.92 (0.04) 44 29.80 29.96 29.27 (0.01) 45 32.00 32.24 30.47 (0.04) 431.73 (0.01) 4o33.07 (0.01) 4o 32.48 (0.02) 4o6 33.2 33.69 32.8 33.26 (0.09) 4o7 34.65 34.67 33.71 (0.01) 4u8 > 35.5 37.00 37.8 36.14 (0.02) 4o 36.44 (0.01) 5u(0.01) 4u36.90 (0.24) 4oaAlKa, 1487eVbusing Synchrotron Radiation at a photon energy of 65 eVcAt an impact energy of 1200 eV + binding energy.dusing basis set I of [105] [14s,11p,4d/5s,2p]—(10s,8p,4d/3s,2p) i.e. 63-GTO.eFrozen core on chlorine for this 96-G(CI) calculation.1Pole Strengths ( 0.01) given in brackets.9Line numbers in the inner valence region are as in references [29,30] and as shown onfigures 5.1 and 5.2.Chapter 5. Hydrogen Chloride 77and PES studies [29,30] have shown that the (4o’ inner valence region of the bindingenergy spectrum of HC1 is split by many-body effects into many states in the regionabove 23 eV up to at least 40 eV. The major (4u)1 pole is at 25.8 eV [29, 30, 61].5.2.1 Binding Energy SpectrumThe EMS binding energy spectrum of HC1 from 5—48 eV is shown in figure 5.1. Thisspectrum was recorded at an impact energy of 1200 eV (plus the binding energy) andat a relative azimuthal angle, = 4°. The energy resolution was 1.7 eV fwhm. Theenergy scale was calibrated using the vertical ionization potential for the (2ir)1 peak asmeasured by high resolution He(I) photoelectron spectroscopy [117,118]. Gaussian peakshapes have been fitted to the main peaks throughout the spectrum in figure 5.1 using thecorresponding Franck-Condon widths (folded with the EMS instrumental energy width of1.7 eV fwhm) as estimated from photoelectron spectroscopy measurements [29,30, 117].In the inner valence region the relative peak positions in the fit were determined usingthe first eight peaks reported in this region by Svensson et al. [30] (see table 5.2). TheGaussians fitted to the inner valence region (above 23 eV) are labeled 1 through 8 inaccordance with the nomenclature used in the earlier photoelectron work of Adam [29]and Svensson et al. [30]. The measured and calculated ionization potentials determinedin the present work are compared with previously reported values in table 5.2. Theenergy spacing of the (2ir)1 and (5tr)’ outer valence peaks in the EMS spectrum infigure 5.1 corresponds well with the PES measurements [117, 118]. The present spectrumis also consistent with the previous EMS study [61]. The (2ir)’ peak has very littlevibrational broadening [117,118] and all line width is therefore mainly attributable to theinstrumental resolution of s1 .7 eV fwhm. This is in keeping with ionization from a “lonepair”, non-bonding orbital. In contrast, the (5o1 line shows significant broadening,Chapter 5. Hydrogen Chloride 78beyond the instrumental contribution, due to the strongly bonding nature of this orbital.In the previous lower resolution EMS study [61], all of the intensity above 23 eV wasassigned to the (4o.)_1 inner valence process on the basis of the relative intensities inthe binding energy spectra measured at = 0° and 8°. Consideration of these spectraindicated that the two outer valence peaks corresponded to “p-type” processes (i.e. thosehaving a maximum in the momentum profile for 4> 0°) and that the inner valence regionwas dominantly due to ionizations of “s-type” symmetry, (i.e. those having a maximumin the momentum profile at 4 = 0°). These conclusions were further confirmed bymeasurements of momentum profiles in both the outer and inner valence regions. Inparticular the measured momentum profiles corresponding to the three most prominentinner valence peaks at 25.8, 32.8 and 37.8 eV (i.e. lines 2, 6 and 8 respectively in figure 5.1of the present work) all fitted well for shape with the (4o’ calculated momentumdistributions.Since the earlier EMS measurement [61], two higher resolution photoelectron studiesof the inner valence binding energy spectrum have been reported by Adam [29] and bySvensson et al. [30]. The energy positions resulting from these PES studies are reported intable 5.2. From a /3 angular distribution parameter analysis [29] and isotopic substitutioneffects [30] it was suggested from the photoelectron studies that peaks 3 (at 28.6eV)and 4 (at 29.9eV) did not belong to the (4,.)_1 manifold of initial states. Although theintensity in the region of peaks 3 and 4 is very low compared with most of the rest of theinner valence manifold in both the somewhat lower energy resolution previous [61] andpresent EMS measurements (see figure 5.1) of the binding energy spectrum, the relativeintensities at 4 = 0° and 8° [61] in this region are not consistent with these interpretations [29,30]. The new generation of higher resolution EMS spectrometers presentlyunder development may provide further insight into these assignments. Meanwhile newChapter 5. Hydrogen Chloride 7910>.—U)CCa)>0ci)010 40 50Binding Energy (eV)Figure 5.1: Valence shell binding energy spectrum of hydrogen chloride measured by electron momenturn spectroscopy at an impact energy of 1200 eV (plus the binding energy) and at a relative azimuthalangle of 4°. The solid line through the points is the sum of fitted Gaussian functions (see text for details).20 30Chapter 5. Hydrogen Chloride 80many body calculations, such as the presently reported MRSD-CI calculation and therecent Greens’ function study by von Niessen et al. [105]), predict that the vast majorityof poles above 23 eV belong to the 4o manifold (> 99%). On the basis of the evidencefrom the earlier EMS study [61] and the theoretical work all inner valence ionizationshave been assigned to be of (4o1 character in the present work.In figure 5.2, the present EMS binding energy spectrum (figure 5.2a) is comparedto the synthetic ionization spectra generated from the present 96-GTO based MRSDCI calculation (figure 5.2b) and also from a 63-GTO based ADC(4) Greens’ functioncalculation recently reported by von Niessen et el. [105] (figure 5.2c). The syntheticspectra were calculated using the respective pole positions and strengths from table 5.2(for poles strengths 0.01) with the latter being multiplied by the relative intensitiesof the respective 96-G(CI) theoretical momentum profiles (see figures 5.3—5.5 below) at= 4° to allow for the momentum dependence. The theoretical spectra were then foldedwith Gaussians of appropriate width using Franck-Condon widths from photoelectronspectroscopy [30, 117, 118] and the EMS instrumental resolution of 1.7 eV fwhm. Thecalculated intensities in figures 5.2b and c reflect the appropriate orbital degeneracies.The calculated and measured spectra are normalized to each other at the maxima of the(2ir)_1 peaks.Both calculations show excellent agreement with experiment for the outer valence((2ir)_1 and (5u)’) peaks for both energy and relative intensity. Although the innervalence ionization lines are less well modeled by the calculations there is still reasonablesemi-quantitative agreement with experiment. Although the energies of lines 1 and 2in the (4o1 manifold are reproduced well by both calculations, the intensities do notappear to be entirely in accord with experiment, especially for the line at ‘— 23 eV whichis higher in the MRSD-CI calculation. Even though von Niessen et al. [105] state that81C>0-a0>crCci)C10 40 50Binding Energy (eV)Figure 5.2: Experimental and synthetic binding energy spectra for HC1. (a) EMS spectrum (qS = 4°).(b) MRSD-CI calculation, 96-G(CI) this work. (c) ADC(4) Greens’ function calculation, reference [105].In figures (b) and (c) the vertical bars indicate the calculated pole strengths (table 5.2) at 4’ = 4°.Chapter 5. Hydrogen Chloride20 30Chapter 5. Hydrogen Chloride 82they do not expect the ADC(4) Greens’ function calculation to be particularly accurateat higher energies it does provide a better description of the spectrum in the region ofpeak 7 than the MRSD-CI calculation. However the relative intensities are in betteragreement with experiment for the higher energy poles for the MRSD-CI calculation. Itshould be noted that the ADC(4) and MRSD-CI calculations predict only very smallcontributions from the “p-type” (5o-)’ process at 32.92 eV (1%) and 36.44 eV (3%)respectively. It should also be noted that in the present MRSD-CI calculation only rootsof E+ symmetry (i.e. (5or)_1 and (4o’ processes) were calculated for the inner valenceregion. However the ADC(4) calculation suggests that (2ir)1 contributions are verysmall (< 0.01) in this region.5.2.2 Comparison of Experimental and Calculated Momentum ProfilesThe experimental momentum profiles and calculated spherically-averaged momentumprofiles corresponding to the removal of the 2ir, 5o and 4u valence electrons of HC1 areshown in figures 5.3—5.5 respectively, on the same relative intensity scale. The experimental momentum profiles were measured at binding energies of 12.8, 16.6 and 28.5 eVfor the 2ir, 5o and 4 profiles respectively. As was discussed in detail in section 3.5, allthree experimental momentum profiles have been placed on a common relative intensityscale by normalizing them according to the integrated areas of the appropriate regions ofthe binding energy spectrum (figure 5.1) as assigned in section 5.2.1. For the 2ir and 5oprofiles, the areas used were those of the Gaussian peaks that were fitted to the bindingenergy spectrum as discussed in section 5.2.1 and shown in figure 5.1. In the case ofthe inner valence region the integrated area under the binding energy spectrum from 23to 45 eV was used so as to include all of the observed states that have been attributedto the (4o1 processes. Following this procedure the 2ir momentum profile was thenChapter 5. Hydrogen Chloride 83normalized to the 96-G(CI) calculation (figure 5.3). This normalization between theoryand experiment places all theoretical and experimental momentum profiles on the samerelative intensity scale for figures 5.3—5.5. Appropriate allowance has been made in thesefigures for hole state degeneracy. It should also be noted that all MD and OVD calculations were originally normalized (before momentum resolution folding) such that theyintegrate to unit probability over all (j3 and ) space. The 96-G(CI) calculations areshown in figures figures 5.3—5.5 with a “scaled-up” unit pole strength (see table 5.2 forthe pole strength as calculated for each ionization). With this representation any difference between the 96-G(CI) calculations and the respective measured electron momentumprofiles is an indication that the corresponding binding energy spectrum peak area usedfor normalization (see above) does not include all the pole strength. This implies thatthe “missing pole strength” is located somewhere else in the binding energy spectrum assuggested by the calculations.It should be noted that the normalizations used in the earlier work of Brion et al. [61]were less stringent than those used in the present work. In that study [61], the calculations were individually height normalized on the figures to each measured momentumprofile. The scaling factors between theory and experiment for the 5o (0.78) and 4cr(0.72) profiles were given in the text but not included on figure 6 of reference [61]. TheSCF overlap calculations reported by Brion et al. [61] also involved much smaller basissets than those used for the near Hartree-Fock limit 107-GTO, 96-GTO and MRSD-CI96-G(CI) calculations of the present work. In addition no allowance for the instrumentalmomentum resolution was made when comparing the previous calculated and measuredmomentum profiles [61]. As mentioned in chapter 2, adequate inclusion of momentumresolution effects has been found to be essential for a meaningful comparison of theory and experiment for the momentum profiles of the other hydrides of rows two andChapter 5. Hydrogen Chloride 84three [15,81,82]. The experimental momentum profiles from the earlier EMS study ofBrion et al. [61] are also shown in figures 5.3—5.5 (open circles) with the correct relativesingle-point normalization factors included.First, compare the present measurements of the momentum profiles with those reported in the earlier study of Brion et al. [61]. For the 2ir profile in figure 5.3, the twomeasurements are in excellent agreement within the limits of the estimated experimentaluncertainties. This is also the case for the shape of the 4u cross-section in figure 5.5although the intensity, as obtained from normalization on the integrated area of thebinding energy spectrum, is approximately 25% higher in the the present measurement.This is probably due to the improved normalization procedures used in the present measurement of the binding energy spectrum in figure 5.1 and also the fact that the presentintegration extends over a slightly wider energy range (i.e. up to 48 eV in the presentwork and only to ‘—‘ 40 eV in reference [61]). However, in the case of the 5o momentumprofiles (figure 5.4) the two measurements differ significantly in shape. In particular,although the two measurements are in good agreement on the leading edge of the 5c profile, there is a. clear mismatch in the region from p = 0.5 to p = 1.5 au. It is noteworthythat the earlier study [61] also included measurements of the valence orbital momentumprofiles obtained at an impact energy of 400 eV (+ the electron binding energy) andthese results, including those for the 5cr orbital, are consistent with the current 1200 eVmeasurements. It is apparent that the earlier reported 1200 eV 5cr momentum profile isincorrect. The 400 eV results [61] are not shown on figures 5.3—5.5 because the differentretarding lens voltage ratios used at 400 eV result in small differences in instrumentalangular resolution from the situation at 1200 eV.Also shown in figures 5.3—5.5 are the theoretical momentum profiles calculated usingequations (2.21) and (2.20) for the SCF and MRSD-CI wavefunctions described in sectionChapter 5. Hydrogen Chloride 85>C’,CcvCa)>0cv10500 1 2 3Momentum (au)MOMENTUM DENSITY POSITION8DENSITY40—4—8—8 —8—8—4 0 4 8 —4 0 4(au) (au)Figure 5.3: The 2,r measured and calculated spherically averaged momentum profiles of HC1 (upperpanel). The solid dots are the present data and the data of reference [61] are plotted as open circles.The lower panels show the momentum and position space density maps for an oriented HC1 moleculecalculated using the 96-GTO basis set. The contours represent 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1.0, 2.0,5.0, 10.0, 20.0, 50.0 and 90.0% of the maximum density.8Chapter 5. Hydrogen Chloride 863 and table 5.1. All the calculated profiles have been treated with the GW-PG methodof Duffy et al. [81] to account for finite instrumental angular (momentum) resolution.An examination of the calculated 2ir momentum profiles in figure 5.3 reveals successively improving agreement with experiment for the theoretical momentum proffles at theSCF level when proceeding from the smallest, minimal basis STO-3G calculation, to thelargest, 107-GTO and 96-GTO near Hartree-Fock limit wavefunctions. As the basis setbecomes more flexible, as shown in figure 5.3 and table 5.1, the improvement in predictedmomentum profile (see also PMAx in table 5.1) is also accompanied by an improvement inthe calculated dipole moment and total energy. The Slater-type wavefunctions reportedby Cade and Huo [108] and the 107-GTO [58] and 96-GTO calculations (present work)give similar results for the 27r momentum profile although the later has a slightly smallerPMAX value. The minor differences between the 107-GTO and the 96-GTO momentumdistributions arise from the slightly better inner-shell core description in the larger basisset. The other calculated properties (see table 5.1) are very similar and thus indicatethat these calculations are very close to the Hartree-Fock limit. The good agreement between the measured 2ir momentum profile and the near Hartree-Fock limit calculationsis similar to the situation found earlier for the outermost orbitals of the other row threehydrides, SiR4 [57], PH3 [32] and H2S [31]. This indicates that the frozen-orbital approximation is a very good model for the (2ir)’ ionization process from the ground state ofHC1. The inclusion of correlation and relaxation in both the ground and ionic wavefunctions, as demonstrated by the multireference singles and doubles 96-G(CI) calculationfor the (2ir)’ process, shows only a small change in the theoretical momentum profilecompared with the 96-GTO near Hartree-Fock limit prediction, even though significantimprovements in dipole moment and total energy occur (see table 5.1). The relativelysmall change from the SCF result for the calculated momentum profile indicates thatChapter 5. Hydrogen Chloride 87the ion-neutral overlap is very similar to that for a simple Koopmans hole (frozen orbital) calculation. This is again very similar to the observed behavior for the other rowthree hydrides, PH3 [32] and H2S [31] in particular and is in marked contrast to thebehavior observed for the outermost electrons of the corresponding second row hydridesNH3,H20 and HF all of which show large correlation and relaxation effects requiring afully correlated ion-neutral overlap treatment to provide an adequate description of theexperimental momentum profiles [15,37,55,68,81,82].In the case of the 5o electron momentum profiles (figure 5.4) all the calculations havebeen scaled by a factor of 0.94. With this scaling both shape and intensity of the presentmeasurements match quite well with the 96-G(CI) calculation and with the 107-GTO,96-GTO and the Cade and Huo calculations. However, the STO-3G minimal basis setcalculation both overestimates PMAx and underestimates the intensity of the momentumprofile. The 4-31 G calculation overestimates both the position of PMAx and also theintensity. Similar to the situation found for the 2b orbital of H2S [31], this emphasizesthe need for some further accounting of polarization and diffuseness in the basis set.The intensity match for the 96-G(CI) calculation (with the scaling of 0.94 noted above)would mean that a small fraction of the 5o pole strength may be located at higher bindingenergy, as suggested by the MRSD-CI calculation (See table 5.2 and section 5.2.1 above.However, these small intensity differences could be due, at least in part, to uncertaintiesin the normalization factors derived from the binding energy spectrum (figure 5.4). Thegood level of agreement between the better calculations and the present measurementsfurther confirms the view that the earlier reported 5o experimental momentum profile [61]is in error. As in the case of the 2ir profile discussed above, correlation and relaxation donot appear to be particularly important for the description of the 5o’ momentum profilesince there is little difference in shape and only a small difference in intensity betweenChapter 5. Hydrogen Chloride>4-,fjCCa)>-4-,Qa)50Momentum (cu)88(au) (cu)Figure 5.4: The 5c measured and calculated spherically averaged momentum profiles of BCl (upperpanel). The solid dots are the present data and the data of reference [61] are plotted as open circles.The lower panels show the momentum and position space density maps for an oriented HC1 moleculecalculated using the 96-GTO basis set. The contours represent 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1.0, 2.0,5.0, 10.0, 20.0, 50.0 and 90.0% of the maximum density.0 1 2 3MOMENTUM DENSITY POSITION DENSITY840—4—8—850-H-cI—8—4 0 4 8 —8 —4 0 4 8Chapter 5. Hydrogen Chloride 89the 96-GTO and 96-G(CI) calculations.The 4o experimental and theoretical momentum profiles are shown in the upper panelof figure 5.5. All calculations agree quite well for both intensity and shape, indicatingagain that the inclusion of correlation and relaxation effects is not particularly importantfor predicting the momentum profiles of HC1. As discussed earlier, the large difference inintensity between the present measurement and that reported earlier by Brion et al [61]for the 2o XMP is probably due to improved normalization procedures in the presentwork. The good quantitative agreement between the present measured and calculatedmomentum profiles further supports the view that the ionization processes above 23 eVare predominantly of (4u)’ character as is also predicted in the Greens’ function andpresently reported MRSD-CI calculations of the binding energy spectrum (see table 5.2and discussion in section 5.2.1). This is also in accord with the relative intensities ofthe binding energy spectra at = 00 and 8° reported in the previous study [61] whichsuggested that there are no significant “p-type” (i.e. (2ir)’ or (5u)’) contributions tothe inner valence manifold of ion states.The respective momentum and position density maps corresponding to an orientedHC1 molecule are presented in the lower two panels of figures 5.3—5.5. The momentumdensity map shows momenta perpendicular and parallel to the bond axis indicated on theposition density map. These maps were calculated using the 96-GTO SCF descriptionwhich gives a good description of the observed momentum profiles (see above). In positionspace the origin corresponds to the centre of mass. The chlorine atom lies just below theorigin (-0.067 au on the vertical z axis) and the hydrogen atom is at +2.341 au. The sidepanels on each map show the transverse and longitudinal one-dimensional slices of thedensity distributions along the respective dotted lines.Chapter 5. Hydrogen Chloride 90>-JC,,c0Ca)>0a)20 -MOMENTUM DENSITY POSITION DENSITY40-÷—4 0 4 8(cu) (au)Figure 5.5: The 4o measured and calculated spherically averaged momentum profiles of HC1 (upperpanel). The solid dots are the present data and the data of reference [61] are plotted as open circles.The lower panels show the momentum and position space density maps for an oriented HC1 moleculecalculated using the 96-GTO basis set. The contours represent 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1.0, 2.0,5.0, 10.0, 20.0, 50.0 and 90.0% of the maximum density.HCI 40-25.8 eVTheory5c 96—G(CI)5 96—GTO4 107—GTO3 Code & Huo2 4—31G1 STO—3GExperiment• This Worko Ref. [61)0ifo010 -0-840—4—8—81Momentum (au)02 3840—4—8 ——8Hci—4 0 4 8Chapter 5. Hydrogen Chloride 91The maps for the 27r orbital are shown in figure 5.3. Simple models predict thisorbital to be formed from the chlorine 3p atomic orbitals to form a non-bonding doublydegenerate (lone pair) ir orbital. The atomic p orbital-like behavior in both momentumand position space can be seen from the respective density maps which are quite similarto those for argon [52]. As shown in the maps (compare figure 5.4 with figures 5.3 and 5.5)the 5o orbital is the main bonding orbital of HC1, in keeping with a simple valence bondpicture arising from the constructive overlap of the hydrogen is and the chlorine 3Pzorbitals. In position space (lower right panel), the 5cT orbital looks somewhat like a porbital with the lobe on the positive z axis stretched over the hydrogen nucleus. Thebonding nature of the 5 orbital can be seen from the electron density slice along the axis(on the right hand side of the plot), which indicates appreciable intensity between thehydrogen and chlorine nuclei. The momentum space picture likewise reflects the bondingnature of the orbital through the so-called “bond oscillations” [45,52] which cause the“wrinkled” appearance of the momentum map. The increase in density perpendicular tothe bond axis in this diatomic molecule is also a characteristic of bonding in momentumspace [45]. In a primitive model, the 4cr orbital would be described as non-bondingand formed from only the 3s atomic chlorine orbital. However the oval shape of theposition space density distribution is indicative of some slight bonding character and thisis confirmed by weak “wrinkles” (or “bond oscillations”) in the momentum density mapand the increase in density perpendicular to the bond axis at low momentum.In conclusion, the calculated momentum profiles for the valence electrons of HC1 showgood quantitative agreement with the presently obtained experimental data when wavefunctions of near Hartree-Fock limit quality or better are used. Electron correlation andrelaxation effects have very little influence on the predicted electron momentum profileseven though they provide significant improvements in the calculated dipole moment andChapter 5. Hydrogen Chloride 92total energy. In this respect HC1 exhibits similar behavior to the other third row hydrides H2S and PH3 and the isoelectronic noble gas argon. This is in marked contrastto the observed behavior of the second row species HF, H20 and NH3 for which electroncorrelation and relaxation effects must be included for the correct prediction of the outervalence momentum profiles.Chapter 6EthyleneThe electronic structure of unsaturated carbon-carbon bonds has presented a challengingproblem for ab initio theoretical methods. As the simplest doubly bonded hydrocarbonethylene (IUPAC name ethene, C2H4)serves as an excellent prototype of this functionalgroup, amenable to both ab initio theoretical and gas-phase experimental study. Inaddition to its fundamental interest as a quantum chemical benchmark system, ethyleneis also an important compound as a feedstock for applications as diverse as drug synthesisand polymer production. A detailed knowledge of the valence orbital electron densitydistributions of C2H4 is important for predicting its chemical and physical behavior,which in turn requires highly accurate molecular wavefunctions. Only by comparisonwith a carefully selected set of measured electronic properties, including the total energy,quadrupole moment and EMS momentum profile, can theoretical results be properlyevaluated.The first reported studies of ethylene by EMS were the early work of Dixon et al. [24]and of Coplan et al. [63]. Subsequent work has involved studies of the lowest bindingenergy momentum profiles of substituted carbon-carbon double bonds, including measurements of the lb3 of ethylene, but no further studies of the complete valence shellhave been reported. Both alkyl-substitution [119] and vinyl halide [120] studies have beenreported. A similar investigation of the effects of fluorination of ethylene is reported in93Chapter 6. Ethylene 94chapter 7 of the present work.In both of the early full valence shell measurements of ethylene [24,63], the observedmomentum profiles agreed qualitatively with the predictions of modest levels of theory,however no attempts were made to account for the effects of instrumental momentumresolution in the calculations. A further important limitation in both of these studies wasthat the individual orbital momentum profiles were each separately height normalized totheory. More recent work on other systems has clearly demonstrated that collecting allof the experimental momentum profiles on a common intensity scale, with a single-pointnormalization to one calculation (see chapter 3) provides a much more stringent andinformative quantitative comparison of theory and experiment. In addition, the XMP’sreported by Dixon et al. [24] were obtained only for momenta above —‘0.3 au, whichlimited the usefulness of this study for the evaluation of the chemically important, lowmomentum regions (below 0.5 au) of theoretical distributions (see discussion in chapter 4and reference [37]). Independently height normalized momentum profiles [24,63], thecomparatively poor statistics obtained in both studies [24,63] and the limited momentumrange used by Dixon et al. [24], makes these early measurements of limited value for theevaluation and development of theoretical methods.Early X-ray photoelectron studies [121, 122] showed the existence of strong satellitestructure in the high energy inner valence region of the binding energy spectrum of C2H4.The main satellite final ion states were assigned to be (2a9)1 at 23.7 eV and (2b1)’at 27.2 eV. However, the 2ph Tamm-Dancoff Approximation (TDA) Greens’ functioncalculation of Cederbaum et al. [123] and the CI results of Martin and Davidson [124]predicted that the entire inner valence region was dominated by (2a9)1poles, with verylittle (2b1)’ intensity. The EMS measurements [24,63], assigned the two main satelliteChapter 6. Ethylene 95features as (2a9)’ processes, by measurement of the momentum profiles at binding energies of 23.7 and 27.2 eV, thus supporting the general ab initio predictions. However nodefinitive assignment of the other wide ranging satellite intensity was made [24,63]. Thecalculations [123, 124] also predicted an unexpected “twinning” of the (2a9)1poles at theenergy of the most intense experimentally observed peak at 23.7 eV in the inner valenceregion. Measurements of this region of the binding energy spectra by both EMS [24,63]and PES [122,125] as well as a recent high resolution synchrotron radiation PES measurement [28] have not revealed any splitting of this satellite peak. More recently, othermany-body calculations [126, 127] have also reported a single pole for this feature, buthave not quantitatively reproduced the intensities or positions of the spectral features.As the above history clearly demonstrates more detailed experimental and theoreticalinvestigations of C2H4 are required. This chapter presents measurements of the momentum profiles and binding energy spectra for ethylene over a greater binding energy rangeand with better energy resolution than in the previous studies. Careful attention hasbeen taken to obtain measurements of the energetically close 1b39, 3a9 and 1b2 momentum profiles. The EMS binding energy spectra are compared with the results of anew MRSD-CI calculation as well as with previously published Greens’ function [123]and symmetry-adapted cluster CI calculations [126]. The measured momentum profilesare compared with a range of SCF treatments ranging from a minimal STO-3G basis toa near Hartree-Fock limit 196-GTO basis set developed in collaboration with ProfessorE.R. Davidson and Y. Wang in the course of the present work. The effects of correlation and relaxation on all of the valence momentum profiles are also investigated witha new multi-reference singles and doubles configuration interaction overlap calculation(196-G(CI)) using the 196-GTO basis. For the lowest binding energy (1b3)momentumprofile, the effects of increasing size of both the basis set and the CI configuration spaceChapter 6. Ethylene 96are studied. Vibrational averaging is also investigated in the lb3 HOMO momentumdensity at the 196-GTO SCF level.The ethylene sample was taken from a cylinder (Matheson Gas Company) of spectroscopic grade, with stated impurities of less than 1%. The gaseous sample was useddirectly, without further purification. No impurities were observed in any of the spectra.6.1 CalculationsSpherically averaged theoretical momentum profiles (TMP’s) have been calculated forC2H4 from several basis sets of varying quality using the plane wave impulse approximation of the cross-section and the target Hartree-Fock approximation described inchapter 2 (equation (2.21)). In addition, full ion-neutral overlap distributions have beencalculated in the plane wave impulse approximation of the EMS cross-section (equation (2.20)) using several multi-reference singles and doubles excitations configurationinteraction (MRSD-CI) calculations for both the ground state and molecular ion statesto investigate the effects of electron correlation and relaxation on the wavefunctions. TheMRSD-CI calculations were based on the respective near Hartree-Fock limit basis sets.The instrumental angular (momentum) resolution was included in the calculations usingthe Gaussian-Weighted Planar Grid method of Duffy et al [81].Due to the many possible choices of symmetry axes in D2h C2H4 the orbital notationfound in the literature is often inconsistent and can be confusing. For clarity, the socalled Mulliken co-ordinate system has been used in the present study as described byHerzberg [128]. In this set of co-ordinates, the z axis is oriented along the C=C doublebond. The y axis lies in the molecular plane, bisecting the double bond. Using theChapter 6. Ethylene 97right-hand rule, the x axis is then perpendicular to the molecular plane. The moleculargeometry used for all of the calculations in this work is that described by McMurchieand Davidson [129].Details of the calculation methods are described below. Selected properties are givenin table 6.1.1 STO-3G: In this minimal basis set the carbon basis is [2s, lp] and [is] is used forhydrogen. This basis was designed by Pople and co-workers [94].2 4-31G: This split valence basis developed by Ditchfield et al. [95] has a minimaldescription of the C is core and a double zeta description of the valence shell.3 Snyder and Basch: This is a Gaussian basis set developed by Snyder andBasch [96]. The atomic bases consist of contracted Gaussians, two for each occupied orbital. Snyder and Basch showed that this basis set gave very similarresults to using two Slater functions per filled atomic orbital [96] and is thus referred to as a double-zeta basis. This basis was also used in the earlier C2H4 studyof Coplan and co-workers [63].4 6_311++G**: The 6-311G basis set was developed by Krishnan et al. [130]) It hasbeen further augmented by additional diffuse [131] (sp on C and F, s on H) and dand p polarization functions [132] on the heavy atoms and hydrogen respectively, toincrease the basis set flexibility. The final contracted basis set is [5s, 4p, id/4s, lp],for [C / H].5 196-GTO: This near Hartree-Fock limit [6s,7p,3d,lf/5s,2p,ld] highly diffuse 196-GTO basis set was developed at Indiana University by Professor E.R. Davidsonand Y. Wang in the course of the present collaborative work. For the carbonChapter 6. Ethylene 98centres, the primitive basis was chosen to be the (18s,13p) Partridge basis set [133],Partridge’s first fourteen s functions were contracted into two s functions usingthe is and 2s atomic orbital coefficients from an SCF calculation of the (atomiccarbon) 3P state. Similarly, the first seven p functions were contracted into one pfunction using the 2p atomic orbital coefficients. Partridge’s (lOs) basis was alsoused for hydrogen [134]. The first six s functions were contracted into a single sbasis function using the is atomic orbital coefficients for the free atom. The restof the functions were left uncontracted. This contraction scheme lost less than0.1 kcal mol’ in a trial SCF calculation on CR4 [59]. For better reproductionof the “large-r” properties, Dunning’s polarization functions [135} were used tosupplement the Partridge contracted basis sets. For carbon, four additional (3d,lf)polarization functions (od = 1.848, 0.649, 0.228; cr = 0.761) were added. Forhydrogen three additional polarization functions were added: (2p,ld) a = 1.257,0.355; od = 0.916. This basis was further augmented by two p and two d diffuseRydberg functions at the center of the C=C bond with exponents of 0.052 and0.104 for both p and d functions respectively. All the cartesian components of thed and f functions were kept.Sc 196-G(CI): Frozen-core MRSD configuration interaction calculations for both theparent neutral molecule and the radical cation were performed with the full virtualspace. For the neutral molecule, the important configurations of a Hartree-Focksingles and doubles configuration interaction (HFSD-CI) calculation were selectedusing the second-order Rayleigh-Schrödinger perturbation theory (RSPT) with theEpstein-Nesbet partitioning. For the cation, a Koopmans’ Theorem singles anddoubles configuration interaction (KTSD-CI) was performed, in which the leadingChapter 6. Ethylene 99configuration was constructed from the neutral molecular orbitals through Koopmans’ approximation. Only configurations having a RSPT contribution to theenergy exceeding 0.25 x 10_6 hartree were kept for the subsequent variational calculation. Then a full multi-reference singles and doubles configuration interaction(MRSD-CI) calculation was performed with a reference space cut-off of 0.030 forthe coefficient contribution and a cut-off of 0.020 hartrees for the energy contribution. This was followed by a second-order RSPT calculation of the correlationenergy for all configurations outside the reference space. For the neutral molecule, 21 reference configurations were included in the final MRSD-CI calculationwhich generated 23706 spin-adapted configurations and recovered about 80% of theempirical valence correlation energy. For the cation, the 23 selected reference configurations generated 41373 spin-adapted configurations and also recovered about80% of the empirical valence correlation energy.Average natural orbitals were used for the MRSD-CI calculations since it was foundthat the use of the ground state canonical molecular orbitals did not result in goodreproduction of the experimental ionization potentials. In particular, for polesof Ag symmetry the second root of the CI matrix did not necessarily correspondto ionization of the 2ag molecular orbital when using the canonical Hartree-Fockorbitals. Renormalization to average natural orbitals ensures that each root receivesabout the same correlation contribution.The average natural orbitals were obtained from the average density matrix forthe first fifteen roots of each symmetry of the cation. The Dyson orbitals, used tocalculate the final theoretical momentum profiles, were derived from the overlapbetween the correlated ground state of the neutral molecule and each root of thecation for each symmetry. The ground state of the neutral molecule was calculatedChapter 6. Ethylene 100separately with ANO’s for each symmetry. The reference space cut-off for thecation states ranged from 0.0023 to 0.0223 hartree.The calculated pole energies and intensities (Si) are reported in table 6.2 and areshown graphically in figure 6.2 (see section 6.2.1).Further CI results for the 1b3 momentum profile are described in section 6.2.3 below.6.2 Results and DiscussionEthylene belongs to the D2h point group and has an electronic ground state of ‘A1. Inthe Mulliken co-ordinate system the single determinant ground state is written as:lag2 1b2 2a9 2b, 1b2 3a92 1b3g2 1b32core valenceThe valence shell ionization potentials reported in the He (II) PES study of Bieri etal. [125] and the XPS results of Banna and Shirley [122] are presented in table 6.2. Theearlier EMS [24, 63] and XPS studies [121, 122] have shown that the inner valence regionof the binding energy spectrum of ethylene is split into many “satellite” states in theregion above 23 eV up to at least 40 eV. The major (2ag) peak is reported have abinding energy of 23.7 eV [24,63, 122,125].6.2.1 Binding Energy SpectraFigure 6.1 shows the binding energy spectra of C2H4 from 5—51 eV for measurementsat relative azimuthal angles = 10 and 4’ = 90 (impact energy of 1200 eV + bindingenergy) on a common intensity scale and at an energy resolution of 1.7 eV. The energyChapter 6. Ethylene 101Table 6.1: Calculated and experimental properties for ethyleneWavefunction Reference Contracted Basis set Total Energy a PMAX6[C/H] (hartree) (au) (an)1 STO-3G [94] [2s, lp/ls] -77.072233 0.484 0.702 4-31G [95] [3s,2p/2s] -77.921028 1.242 0.623 Snyder & Basch [96] [4s,2p/2s] -78.005504 1.495 0.534 6_311++G** [130—132] [5s,4p, ld/4s, lpj -78.055448 1.506 0.515 196-GTO [6s, 7p, 3d, lf/5s, 2p, ld] -78.069343 1.532 0.475c 196-G(CI) C Natural Orbitals (NO) -78.408495 1.327 0.476 228-GTO C [14s, l2p, 3d, lf/7s, 3p, ld] -78.069664 0.476c 228-G(CI) C NO -78.402716 1.312 0.477 234-GTO C [15s, 12p, 3d, lf/8s, 3p, ld] -78.069578 0.477a 234..G(CI)a C using neutral MO’s -78.402701 1.312 0.477b 234-G(CI)b NO -78.410454 0.477c 234-G(CI)c C NO, low threshold cutoff -78.426308 1.258 0.477d 234-G(CI)d NO, from cation MO’s -78.424229 1.320 0.47Experimental l.l±O.2’ 0.47°Quadrupole moment, O = (3z2 — r2). These values have been calculated at the static equilibrium geometry for a non-rotating, non-vibrating molecule.bMomentum value for maximum in the 1b3 momentum profile. From figure 6.4.cThese SCF and CI calculations were carried out by E.R. Davidson and Y. Wang in the course ofthe present collaborative study.dpcommended value, uncorrected for zero-point motion, obtained by induced birefringence [136].Chapter 6. Ethylene 102scale was calibrated with respect to the (1b3)vertical ionization potential measured byhigh resolution photoelectron spectroscopy [117, 125j. Gaussian peak shapes have beenfitted to the main peaks throughout the spectrum in figure 6.1 using vertical ionizationpotentials and Franck-Condon widths (folded with the EMS instrumental energy widthof 1.7 eV fwhm) estimated from photoelectron spectroscopy measurements [117, 122].Measured and calculated ionization potentials from the literature and those determinedin the present work are shown in table 6.2.In the outer valence region of the = 10 and 9° experimental binding energy spectra,several features can be seen below 21 eV in each spectrum. Peaks due to the (1b3)’ and(2b1) ionization processes are found at binding energies of 10.5 and 19.1 eV respectively, consistent with the results of photoelectron spectroscopy [117, 122, 125]. These twopeaks display characteristic “p-type” behavior, having a greater intensity at = 9° thanat 1°. The energies of the (1b39)’, (3a9)’ and (1b2) ionizations are too closelyspaced to permit complete resolution of the individual peaks in the present EMS bindingenergy spectra, however these processes can be identified using the Gaussian functionsfitted at the energy positions reported by high resolution PES measurements [117, 125].In the binding energy spectrum recorded at = 1° the intense feature at 14.7 eV is dueto the (3a9)1 ionization. The peak in the = 90 binding energy spectrum at 16.6 eV isattributable the (1b2)ionization process. The feature at 12.6 eV between the (1b3)’(10.5 eV) and (1b2)’ (16.6 eV) peaks is mainly due to the (1b39)’ process. Care mustbe taken when performing the Gaussian fits to the data to avoid cross-contaminationbetween the s-type (3a9)’ and p-type (1b32) and (1b2)processes.The XPS spectrum of ethylene reported by Gelius [121] was one of the first observations of satellite structure in molecular binding energy spectra. In previous studies ofC2H4 by EMS [24,63], the structure above 21 eV has been assigned exclusively to theChapter 6. Ethylene 103Table 6.2: Ionization energiesa and intensities [pole strengths] for ethyleneOrbital Experimental MRSD-CI SAC CI 2phTDAbOrigin [1251 [122] This Work [126] [123]A1l ionization energies are in eV.bFom table 3 of ref. [123], non-diagonal Green’s function resultscOnly poles with intensities > 0.01 are shownlb30 10.68 10.51 10.50 [0.84]c B30 10.83 [0.95]c B30 9.76 [0.91]’ B30lb39 12.8 12.85 12.90 [0.81] B39 13.26 [0.93] B39 12.16 [0.90] B393a9 14.8 14.66 14.70 [0.83] Ag 15.07 [0.92] A9 13.79 [0.88] A9lb20 16.0 15.87 15.90 [0.73] B20 16.48 [0.87] B20 15.06 [0.81] B2018.21 [0.02] B20 17.83 [0.02] B202b10 19.1 19.23 19.20 [0.62] B10 19.93 [0.81] B10 18.34 [0.76] B1020.23 [0.07] B10 23.81 [0.01] B10 20.32 [0.03] B1022.19 [0.06] B2023.42 [0.19] Ag 23.12 [0.36] A924.42 [0.08] B10 23.23 [0.02] B202a9 23.6 23.65 24.64 [0.30] A9 24.71 [0.64] A9 23.83 [0.27] A925.14 [0.02] A9 25.63 [0.08] B1027.21 [0.03] A9 26.41 [0.06] B20 26.56 [0.02] B3927.63 [0.01] A9 26.96 [0.07] A927.85 [0.03] A9 27.97 [0.03] A,2a9 27.2 28.74 [0.09] A9 28.84 [0.05] B10 28.05 [0.08] A929.00 [0.05] A9 29.88 [0.02] B39 30.61 [0.04] A929.19 [0.03] A9 30.10 [0.03] A9 30.88 [0.01] A930.03 [0.01] Ag 30.54 [0.01] B102a9 31.3 30.93 [0.05] A9 33.14 [0.01] A933.76 [0.02] A933.84 [0.02] A934.44 [0.01] A934.93 [0.01] B1036.30 [0.01] A939.13 [0.01] A939.34 [0.02] B30dpoles above 35 eV were not reported for the 2ph-TDA calculation [123].Chapter 6. Ethylene>(I)CC)I,CC)>CC)Figure 6.1: EMS binding energy spectra of C2H4 from 6 to 51 eV at 4) = 10 and 90, obtained at animpact energy of (1200 eV +binding energy). The dashed lines represent Gaussian fits to the peaks andthe solid curve is the summed fit.10420151050105010 20 30 40 50Binding Energy (eV)Chapter 6. Ethylene 105(2a9)1 inner valence process on the basis of experimental momentum profiles of the twomain features, at 24 and 27 eV. These profiles have “s-type” behavior, with a maximumintensity at 4 = 00 (near p 0), decreasing monotonically at higher momenta, consistentwith the shapes of calculated 2a9 momentum profiles. Even with the modest statisticsof these earlier studies [24,63], it was clear from the shapes of the observed momentumprofiles that these two large satellite structures were due to (2a9)’ and not (the alsos-type) (3a91 ionization processes, which has a much narrower momentum distribution. This assignment is supported by many-body calculations of pole energy positions,including the Greens’ function treatment of Cederbaum et al. [123], the CI results ofWasada et al. [126] and the MRSD-CI calculated ionization energies of the present work.All three theoretical results predict several (2ag)’ poles concentrated in the 21—28 eVregion. The relative intensity ratio in this region observed between the present 1 to= 9° binding energy spectra (figure 6.1) suggests that the majority of the inner valencestructure out to 51 eV is due to (2a9)1 ionization channels. This assignment is alsosupported by the quality of the fit of the relative integrated areas from 21—51 eV of the= 1° and 9° spectra to the theoretical 2a9 momentum profiles (figure 6.9), discussed inthe next section. For the purpose of normalizing the momentum profiles in section 6.2.2,the integrated area from 21—51 eV of the = 10 spectrum has been used for the 2a9XMP to minimize any possible contributions from “p-type” satellite processes.In figure 6.2, the sum of the = 10 and 9° spectra (panel (a)) is compared withthe results of several many body calculations (panels (b), (c) and (d)). The calculations have been folded with the experimental resolution and the same Franck-Condonwidths obtained from the PES spectra [117, 122,125] used to fit figure 6.1, to obtain figures 6.2 (b), (c) and (d). One of the first ab initio many-body calculations for ethylene,a Greens’ function treatment, is shown in figure 6.2 (d). This is a non-diagonal twoChapter 6. Ethylene 106Binding Energy (eV)Figure 6.2: Experimental and synthetic binding energy spectra for C2H4. (a) EMS sum spectrumfor 4’ = 10 + 4, = 9°. The solid curve through the points is the sum of Gaussian functions fitted tothe experimental spectrum. (b) Multi-reference singles and doubles CI calculation, 196-G(CI), Thiswork. (c) Symmetry Adapted cluster CI calculation from reference [126]. (c) 2ph-TDA Greens’ functioncalculation, reference [123].10 20 30 40 50Chapter 6. Ethylene 107particle-hole Tamm-Dancoff calculation taken from the work of Cederbaum et at. [123].The recently published results of the symmetry-adapted cluster calculation (SAC CI) ofWasada et at. [126] are shown in figure 6.2 (c). Also shown (in figure 6.2 (b)) are theenergies and spectroscopic factors of the MRSD-CI roots calculated by E.R. Davidsonand Y. Wang for the present work. It can be seen from figure 6.2 and the pole positionsin table 6.2 that all three calculations give a reasonable quantitative reproduction of theenergy positions of the outer valence peaks, however the Greens’ function results seemto be uniformly shifted by —0.3 eV. The intensities in the outer valence binding energyspectrum are rather different but this is mainly due to the fact that the calculated polestrengths are implicitly integrated over momentum, whereas the experimental spectrareflect the corresponding momentum dependencies. While the three treatments showsome differences in their descriptions of the inner valence region, all attribute most of theinner valence intensity to the (2a9)’ process (see table 6.2). The MRSD-CI calculation(figure 6.2 (b)) reproduces the relative shape of the inner valence region quantitativelybut over-emphasizes the intensity of the low-energy side of the 23.6 eV peak. Whilethe SAC-CT treatment [126] (figure 6.2 (c)) gives a qualitative description of the innervalence shapes with good reproduction of the energies of the major features, it clearlygives incorrect intensities for the poles. In particular, the intensity of the (2a9)1 peakfound at 27.2 eV in the experimental spectra is significantly underestimated. The 2ph-TDA Greens’ function results [123] (figure 6.2 (d)) predict a twinned splitting of the A9ion states in the region of 23 eV instead of one “main line” (see table 6.2). Murray andDavidson [127] concluded that this structure, which is not observed in the present results,or in recent high resolution synchrotron radiation PES measurements [28], is an artifactof the calculation.Chapter 6. Ethylene 1086.2.2 Comparison of Experimental and Theoretical Momentum ProfilesThe experimental and calculated spherically-averaged momentum profiles correspondingto the 1b3, lb3g, 3a9, 1b2,2b1 and 2ag orbitals of ethylene are shown in the upper panelsof figures 6.4—6.9 respectively, on a common relative intensity scale. The experimentalinstrumental angular resolution has been accounted for in all the theoretical momentumprofiles using the GW-PG method [81]. In the lower panels the corresponding momentumand position orbital electron density maps are shown for an oriented ethylene molecule,calculated using the 196-GTO wavefunction.Momentum profiles for the completely energetically resolved (1b3),(2b1)’ and(2a9)’ processes were recorded directly at appropriate binding energies. Note that the1b3 XMP was collected at 10.0 eV on the low energy side of the binding energy peakto avoid contamination from the (1b39)’ at 12.8 eV. These three directly measuredmomentum profiles were placed on a common relative intensity scale according to theappropriate fitted peak areas and integrated regions of the binding energy spectra infigure 6.1 (see section 6.2.1). These XMP’s are shown on figures 6.4, 6.8 and 6.9 as filledcircles. The points plotted as solid squares on each of figures 6.4—6.9 reflect the relativepeak Gaussian areas at q = 1° and 9° of figure 6.1, after undergoing a single pointnormalization with the 2b1 196-G(CI) TMP, as described below. The 1b3 and 2b1momentum profiles were normalized using the Gaussian peak areas fitted to the bindingenergy spectra at q = 10 and 9°. For the 2a9 momentum profile all of the intensity above21 eV to the limit of the data at 51 eV in the = 10 BES (figure 6.1) was used forthe normalization procedure. The integrated area of the 21—51 eV region for the = 90spectrum is also shown on figure 6.9 for comparison, but it was not used for normalizationof the 2ag momentum profile.Chapter 6. Ethylene 109Binding Energy (ev)Figure 6.3: EMS binding energy spectra ofC2114 from 7 to 21 eV at = 0°,2°,4°, 8°, 10°, 12°, 15°,20°and 25° obtained at an electron beam energy of (1200 eV +binding energy). The dashed lines axe theindividual Gaussian deconvolution functions of the data, the solid line is the sum.0=108 12 16 20 8 12 16 20Chapter 6. Ethylene 110The remaining (1b39, 3a9 and 1b2)momentum profiles could not be measured directlysince they are not sufficiently resolved in energy by the present single channel EMSspectrometer. Therefore binding energy spectra were recorded from 8 to 21 eV at tendifferent q5 angles: 0, 2, 4, 6, 8, 10, 12, 15, 20 and 25 degrees. These spectra are shownin figure 6.3. Five Gaussian functions were fitted to each spectrum, using the sameenergy positions and widths (see section 6.2.1) as for the wider energy range spectrafits in figure 6.1. The fitted functions are shown with broken lines on figure 6.3 withtheir sum as a solid curve. Momentum profiles for each of the 1b3, lb3g, 3a9, 1b2and 2b1 orbitals were obtained from the deconvoluted peak areas of the fitted Gaussianfunctions. Since these five momentum profiles were collected “simultaneously” (that is tosay, sequentially—see section 3.3), they share a common relative intensity scale. Theseresults are shown as open circles on figures 6.4—6.8 with error bars estimated from thequality of the fit.The two 2b1 experimental momentum profiles (BE=19.1 eV) were each independently normalized to the 196-G(CI) calculation (figure 6.8). With these normalizationprocedures all theoretical momentum profiles and both sets of experimental measurements share the relative intensity scale common to figures 6.4—6.9.The momentum profiles for the 1b3 and 2b1 orbitals were each obtained both directlyand by deconvolution of the narrow range binding energy spectra 6.3. It can be seen fromfigures 6.4 and 6.8 that the two types of measurements are consistent for each of the 1b3and 2b1 momentum profiles within the limited statistical precision of the data and theprocedures used to fit the binding energy spectra.The momentum value of the intensity maximum of the 1b3 momentum profile infigure 6.4, PMAx, at 0.47 au, is unusually low compared with those of the highest occupiedChapter 6. Ethylene>I.Cl)Cq)Cci)>0c1115,45c 3C2H4 1b31201510500.51 eV215c54321.01 96—G(Cl)1 96—GTO6—311++G**Snyder & Basch4—3WSTO—3GXMPLong Range BES areasValence BES areas0 1 2 3Momentum (au)MOMENTUM DENSITY POSITION DENSITY88—4 0 4 8(au)88—4 0(au)4 8Figure 6.4: The 1b3 experimental and theoretical momentum profiles of ethylene (upper panel).The lower panels show the momentum and position space density maps for an oriented C2H4 moleculecalculated using the 196-GTO wavefunction (see table 6.1). The contours represent 0.01, 0.03, 0.1, 0.3,1.0, 3.0, 10.0, 30.0, and 99.0% of the maximum density.>, 10U)CI.C0>0a)-4 0(cu)Figure 6.5: The 1b3, experimental and theoretical momentum profiles of ethylene (upper panel).The lower panels show the momentum and position space density maps for an oriented C284 moleculecalculated using the 196-GTO wavefunction (see table 6.1). The contours represent 0.01, 0.03, 0.1, 0.3,1.0, 3.0, 10.0, 30.0, and 99.0% of the maximum density.Chapter 6. Ethylene 112234,55cIC2H4 lb3g12.83 eV5c 196—G(Cl)5 196—GTO4 6—311++G**3 Snyder & Bosch2 4—31G1 STO—3G• Long Range BES areaso Valence BES areas5084-0 1 2 3Momentum (au)MOMENTUM DENSITYC2H4lb39POSITION DENSITY8oeC2H41 b394.0-—4 --8-0I I I—4—4 0(cu)4 8I I I4 8>C/)0>C0840—488MOMENTUM DENSITY2 3Figure 6.6: The 3a1 experimental and theoretical momentum profiles of ethylene (upper panel). Thelower panels show the momentum and position space density maps for an oriented C2H4 moleculecalculated using the 196-GTO wavefunction (see table 6.1). The contours represent 0.01, 0.03, 0.1, 0.3,1.0, 3.0, 10.0, 30.0, and 99.0% of the maximum density.Chapter 6. Ethylene301134,523,5c120100CH 3a24 g14.66 eV5c 196—G(Ct)5 196—GTO4 6—311++G**3 Snyder & Bosch2 4—31G1 STO—3G• Long Range BES areaso Valence BES areas0 1Momentum (au)POSITION DENSITY—4 0 4 8(au)—8—8 —4 0(au)4 8>CFigure 6.7: The 1b2 experimental and theoretical momentum profiles of ethylene (upper panel).The lower panels show the momentum and position space density maps for an oriented C2H4 moleculecalculated using the 196.-GTO wavefunction (see table 6.1). The contours represent 0.01, 0.03, 0.1, 0.3,1.0, 3.0, 10.0, 30.0, and 99.0% of the maximum density.Chapter 6. Ethylene20>C/)Cci)I,— 1011455c 4,231C2H4 1b215.87 eV5c5432101 96—G(Cl)1 96—GTO6—311 ++G**Snyder & Bosch4—31GSTO—3GLong Range BES areasValence BES areas0 1 2 3Momentum (au)MOMENTUM DENSITY0840—488POSITION DENSITY840C2H41b2_/\/__I I I—4—4 0(cu)4 8 —4 0 4 8(au)Chapter 6. Ethylene 115>C/)Cci)C>00)c252015105.00—4MOMENTUM DENSITYC2H42b1—8 —4 0(au)POSITION DENSITY3Figure 6.8: The 2b1 experimental and theoretical momentum profiles of ethylene (upper panel).The lower panels show the momentum and position space density maps for an oriented C2H4 moleculecalculated using the 196-GTO wavefunction (see table 6.1). The contours represent 0.01, 0.03, 0.1, 0.3,1.0, 3.0, 10.0, 30.0, and 99.0% of the maximum density.I41 CH 2b2—5c 2 4 lii19.1 eVI f’ 5c 196—G(CI)5 196—GTO4 6—311++G**3 Snyder & Bosch1i 2 4—31Gt \ 1 STO—3G\T •XMP• Long Range BES areas0 Valence BES areasI1Momentum2(au)84-0840—44 8 —4 0(cu)4 8>C,)C-4--)Ca)>0.C2H4—.-20g•% —CI I IFigure 6.9: The 2ag experimental and theoretical momentum profiles of ethylene (upper panel). Thelower panels show the momentum and position space density maps for an oriented C2H4 moleculecalculated using the 196-GTO wavefunction (see table 6.1). The contours represent 0.01, 0.03, 0.1, 0.3,1.0, 3.0, 10.0, 30.0, and 99.0% of the maximum density.Chapter 6. Ethylene 11613,4,55c2C2H423.22a geV5c54321S.1 96—G(CI)1 96—GTO6—311 ++G**Snyder & Bosch4—31GSTO—3GLong Range BES areasXMP806040200840—4—4 0(a u)I0 1 2 3MOMENTUM DENSITY POSITION DENSITYMomentum (cu)840—4884 8 —4 0(au)4 8Chapter 6. Ethylene 117orbital of other small molecules containing row 2 atoms [17]. This indicates that theethylene ir-system is contracted in momentum space, but diffuse in position space. Similarbehaviour was observed for the HOMO XMP of acetylene [64,65] which has a smallervalue of PMAx, and the it2 HOMO XMP of methane [56], which has a more diffusemomentum profile but which is still narrow by comparison with the other hydrides.The good reproduction of the sp behavior of the C2H43a9 momentum profile by all thetheoretical methods (figure 6.6) over the entire momentum range is also an interestingphenomenon. Past experience for other systems has shown that “secondary” highermomentum maxima are very sensitive to the quality and size of the basis sets at theHartree-Fock level of treatment (see, for example references [41,42], and chapter 8). Thedouble maximum in the 3ag momentum profile of C2H4 is a result of the °cc and ac-Hmulti-centre bonding behavior of this orbital (see section 6.2.4).Note also that there is reasonably good agreement for intensity for all 2ag TMP’s withexperiment in figure 6.9. As described in section 6.2.1 earlier, all of the experimentallyobserved intensity (in the BES of figure 6.1) above 20 eV has been assigned to the 2a9ionization manifold. If there are some ionization processes of symmetry other than 2agin this energy region, then the intensity of the 2a9 momentum profile would be highcompared to that of the calculated TMP’s (which are all calculated from normalizedwavefunctions). If some 2a9 poles are outside of the integrated BE spectral region (21—51 eV), then the relative intensity of the 2a9 experimental momentum profile would belower than is expected theoretically. In fact, the 2a9 XMP has a slightly higher intensitythan predicted by most calculations. This difference may be due to experimental uncertainties or small contaminations from poles of other ionization manifolds. Neverthelessagreement between theory and experiment is quite good in figure 6.9 and confirms theoverall assignment of the 20—51 eV BES region to the (2a9)1 ionization process madeChapter 6. Ethylene 118earlier in section 6.2.1. Furthermore, the generally good overall agreement for intensitybetween the integrated areas of the wide-ranging binding energy spectra peak areas (filledsquares) and the higher quality calculations for all of the profiles (figures 6.4—6.8) whichindicates that all the pole strength for these processes is present in the outer valenceregion, further supports assignments the assignment of the majority of the 20—51 eVregion to the (2a9)’ process. However, it should be noted that the integrated area inthe 4 = 90 BE spectrum (figure 6.1), plotted as a filled square at 0.75 au on figure 6.9,does appear to be high when compared with the measured momentum profile. If this isthe case it would indicate small contributions in the ionization energy range above 20 eVfrom other (i.e. p-type) processes.Turning now to an examination of the calculated momentum profiles, a comparisonof the profiles calculated from the 196-G(CI) ion-neutral Dyson overlap using correlatedwavefunctions and the 196-GTO SCF orbital reveals very good agreement for shape foreach orbital (figures 6.4—6.9), although there are small intensity differences for some ofthe theoretical momentum profiles. When compared with the experimental results, thesetwo calculations also give a good reproduction of the experimental momentum profiles forboth intensity and shape, given the limited statistical precision of the experimental dataand the uncertainties in the binding energy peak area determinations (see section 6.2.1)used for the relative normalizations.The SCF calculations (numbers 1—5, see table 6.1 for selected properties) generallyshow progressively improving agreement with the experimental data as basis set size anddiffuseness increases. The minimal basis set STO-3G results (curve 1) have generallythe poorest agreement with the experimental momentum profiles, especially for the 1bXMP (figure 6.4 and also PMAx in table 6.1). However, this limited basis set also gives asurprisingly good description of the rest of the momentum profiles (figures 6.5—6.9). ItChapter 6. Ethylene 119should be noted, however, that the STO-3G quadrupole moment is very low (table 6.1),which is a reflection of the small size of this minimal basis set. In addition, the totalenergy is very poor indicating the low degree of basis set saturation in the STO-3G SCFtreatment.The small (by current standards) 4-31G and DZ basis sets (lines 2 and 3 respectivelyon figures 6.4—6.9) have some additional diffuse flexibility compared with a minimal basisset, particularly for the valence region. The two treatments give better total energiesthan the minimal STO-3G description, although they are still energetically far from theHartree-Fock limit. Compared with the 4-31G basis set, the DZ basis of Snyder andBasch has an additional contracted core function, which results in a slightly better totalenergy (table 6.1). The extra flexibility in the DZ basis is also reflected in figures 6.4—6.9 as a slightly better reproduction of the experimental shapes and relative intensities.This is seen most dramatically for the 1b3 TMP’s (see figure 6.4 and PMAx values intable 6.1) but is also true for all the other profiles. However, the poor descriptions of theHOMO momentum profile for both treatments is characteristic of unsaturated basis sets.Augmentation with polarization [132] and diffuse functions [38,39, 131] has been foundto greatly increase the quality of small basis sets. Note, however the relative insensitivityof the quadrupole moment to the size of these small basis sets. The apparently goodagreement attained by these two treatments is probably accidental, much the same asthe often quite reasonable dipole moments obtained by the STO-3G basis set (see forexample, tables 4.1 and 8.1)It is interesting to note that the 6311++G** basis set (curve 4 on figures 6.4—6.9)produces similar results to the near Hartree-Fock limit 196-GTO calculation for many ofthe theoretical momentum profiles, with much less computational effort. However, thisstandard basis set does much more poorly for the total energy (see table 6.1). The goodChapter 6. Ethylene 120agreement attained by the 6311++G** momentum profiles and quadrupole momentcombined with its over estimation of the total energy (compared with the near HartreeFock limit 196-GTO result) emphasizes the caution that must be taken when evaluatingbasis set quality. While good reproduction of a small set of properties can be attainedwith limited basis sets, it is often at the expense of others. Such is the case here for allof the small to intermediate basis sets, especially for the total energies.The near Hartree-Fock limit 196-OTO momentum profiles (curves 5 on figures 6.4—6.9), as mentioned above, are in good agreement with the experimental data consideringthe uncertainties in the measurements. Note also the great improvement in the totalenergy (table 6.1) using this basis set when compared with the 6311++G** results.As mentioned earlier, there is little change in the agreement with the XMP’s with theinclusion of electron correlation and relaxation effects in the 196-G(CI) overlap calculation. This is similar to behaviour observed for both acetylene [64] and methane [56]for which the inclusion of electron correlation and relaxation was not found to have alarge effect on the shapes of the valence orbital momentum profiles of either molecule.Note however, that there are large improvements in both total energy and quadrupolemoment of ethylene (table 6.1) with the inclusion of correlation and relaxation, althougha small discrepancy remains for the quadrupole moment. However, all of the calculationsin the present work (with the exception of those in section 6.2.5) have been performedat a fixed geometry, while the experimental value reflects a vibrating and rotating molecule. One unusual result for the present 196-G(CI) calculation is the apparent drop inintensity for the 1b3 momentum profile in figure 6.4. In general however, the 196-0(d)TMP’s are very similar in shape and intensity to the results of the 196-OTO Hartree-Fockcalculation.Chapter 6. Ethylene 1216.2.3 Basis Set Saturation and Correlation and Relaxation Effects in the1b3 Momentum Profile of EthyleneAs discussed in the preceding section, the reproduction of some of the electronic propertiesby both the 196-GTO Hartree-Fock and 196-G(CI) MRSD-CI treatments is unsatisfactory. For example, both methods result in small differences from the EMS measurementsin the low momentum region of the 1b3 momentum profile (see figure 6.4) and also givesomewhat unsatisfactory reproductions of the quadrupole moments (table 6.1). Therefore it was decided to investigate (i) basis set saturation and the extent of convergenceof the 196-GTO basis set to the Hartree-Fock limit and (ii) the influence of the improvements in the size of the configuration space available in the MRSD-CI treatment. Allof the calculations in this section were performed in collaboration with E.R. Davidsonand Y. Wang at Indiana University. An outline of the procedures used by Davidson andWang are given below.It has been found in earlier work for NH3 [15, 16, 55j, 1120 [15, 16,37] and HF (chapter 4) that when using MRSD-CI methods, the basis set size and especially the diffusenessand flexibility of the SCF wavefunctions is very important for reproduction of the lowmomentum region of the lowest energy (“HOMO”) XMP’s. It was found for these casesthat achieving a high degree of basis set saturation and convergence not only for totalenergies but also for other properties such as the dipole moment and momentum profileswas an essential precursor to meaningful inclusion of electronic correlation and relaxation.Once effective convergence to the Hartree-Fock limits had been achieved for a range oftest properties (i.e. total energy, dipole moment and the momentum distributions—seethe discussion in chapter 1) the importance of including correlation and relaxation effectsChapter 6. Ethylene 122in the wavefunction for reproduction of the XMP’s for all three second row of these hydrides was clearly demonstrated. In NH3,H20 and HF, the “HOMO” momentum profilewas found to be better reproduced by a Dyson orbital (i.e. the overlap of the ion andneutral correlated wavefunctions) than a canonical Hartree-Fock orbital. The results ofsimilar investigations for C2H4 are shown in figure 6.10. As for all the other calculatedmomentum profiles in figures 6.4—6.9 the effects of finite instrumental momentum resolution have been accounted for in the profiles. Note, however, that in figure 6.10 theexperimental momentum profiles have been independently re-normalized to the 196-GTOTMP.In these further investigations, calculations for ethylene were carried out with thesame even-tempered [14s,lOp,3d,lf/6s,3p,ld] basis set used for C2H [64]. However,trial MRSD-CI calculations with this basis set resulted in a poorer total energy thanfor the 196-G(CI) results. After carefully comparing the 196-GTO and the originalacetylene basis sets, a second, intermediate basis set was formed from the 196-GTObasis set by uncontracting some of the s and p functions and adding one more diffuse s(a8 = 0.033728) and p (a1, = 0.020323) function on the carbon atoms using Partridge’srecommended C— anion supplements [133]. As Partridge did not report a supplementaryfunction for hydrogen, an additional s function with an exponent of 0.018644 was foundby minimizing the SCF energy of H-.All the possible combinations of the whole s, p and d sets of the intermediate basisand the original acetylene basis set were evaluated in atomic and molecular fragmentSCF trial calculations. The fragments used for carbon were C+, C, C— and C2 (3E).Neutral H, H—, H2 and Ht were used for hydrogen. The properties of the final 228-GTObasis, [14s,12p,3d,lf/7s,3p,ld], which corresponds to the lowest-energy combination, canbe found in table 6.1. The cartesian s and p components were removed from d and fChapter 6. Ethylene 123functions respectively. This Gaussian basis set was then used for the 228-G(CI) calculations. For the neutral molecule, 21 reference configurations were included in the finalMRSD-CI calculation which generated 20443 spin-adapted configurations and recoveredabout 80% of the empirical valence correlation energy. Twenty-three reference configurations were included for the cation which generated 35619 spin-adapted configurationsand also recovered about 80% of the expected valence correlation energy. The resultingproperties for the 228-G(CI) calculations are shown in table 6.1. However, very littlechange occurred in the resulting TMP (see figure 6.10). In fact, the 228-GTO and 228-G(CI) TMP’s are almost identical to the 196-GTO and 196-G(CI) results respectively.It is interesting to note (see table 6.1) that while a better SCF energy is achieved usingthe 228-GTO basis set than with the 196-GTO, the 228-G(CI) result is slightly worsethan the total energy obtained by the 196-G(CI) treatment. It is clear that the natureof the SCF basis set as well as the size are important considerations for CI calculations.In view of this situation, the final 228-GTO basis was expanded to a 234-GTO basis,[15s,12p,3d,if/8s,3p,ld], by replacing the last three s functions on carbon with foureven-tempered s functions and uncontracting the last s function from the is hydrogenfunction. These 234-GTO SCF and corresponding (molecular orbital based) CI results(the first CI calculation on this basis set, 234-G(CI)a) are collected in table 6.1. Toensure that the wavefunctions were sufficiently converged, the CI calculation was redonewith natural orbitals obtained from the original MRSD-CI wavefunction of the neutralmolecule. The first natural orbital calculation (the second CI calculation on this basisset, 234-G(CI)b) was performed using the same configuration space cut-offs as the 196-G(CI) treatment. A second natural orbital calculation (the third CI calculation on thisbasis set, 234-G(CI)c) was obtained by lowering the perturbation selection threshold to0.32 x iO hartree, which recovered about 85% of the empirical valence correlation energy,Chapter 6. Ethylene 124>I.(I)Cci)Cci)>Ca)2015100CH lb24 3u10.51 eV7d 234—G(CI)d7c 234—G(CI)c7b 234—G(CI)b7a 234—G(CI)o7 234—GTO6c 228—G(CI)6 228—CTO5c 196—G(CI)5 196—GTO50 1 2 3Momentum (au)Figure 6.10: Correlation and relaxation effects in the 1b3 momentum profile of C2H4Chapter 6. Ethylene 125a 5% improvement. Both of the two natural orbital calculations show a slightly decreasein the intensity of the TMP but the positions of the maxima (PMAx) are changed in bothcases. A final CI calculation (the fourth CI calculation on this basis set, 234-G(CI)d)was performed using natural orbitals which were obtained from the original MRSD-CIwavefunction of the cation. From table 6.1, it can be seen that this calculation obtainedthe second lowest energy for the neutral molecule. However, the corresponding TMP(labeled 7d in figure 6.10) is very similar to the other 234-G(CI) results.Overall, of all the various treatments, the 234-G(CI)c calculation has the best characteristics, giving the lowest value for total energy and the most satisfactory reproductionof both the quadrupole moment and the XMP. However, some small discrepancies remain. For the EMS results these are within the (limited) statistical precision of the singlechannel experimental data. In the case of the quadrupole moment it should be notedthat the calculations involve the equilibrium geometry whereas the experimental resultsare for a vibrating molecule.6.2.4 Electron Density Maps for C2H4 in Position and Momentum SpaceAlso shown on figures 6.4—6.9, below the panels of the momentum profiles, are the orbitalelectron density distribution maps in both position and momentum space for an orientedethylene molecule. They are presented here for pedagogical interest as they comparethe electronic density distributions in position and momentum space for this simplestdoubly-bonded organic molecule. These results complement those presented earlier inthe studies of the other small hydrocarbons CH4 [56] and C2H [64] and also aid in theunderstanding of the shapes of the momentum proffles.Chapter 6. Ethylene 126In the lower panels of figure 6.4, the 1b3 density maps, cut perpendicular to the planeof the molecule along the double bond show the characteristic double lobed shape of thecarbon-carbon 7r-bonding orbital in both position and momentum space. The multiplelobes of the momentum density map are the the result of constructive interference (commonly and misleadingly called “bond oscillations”) of the Gaussian wavefunctions on thecarbon centres [49,52] reflecting the bonding nature of this orbital. The node in the planeof the molecule is reflected in the spherically averaged momentum profile as an intensityminimum at p = 0, giving rise to a “p-type” TMP. The small PMAx values (table 6.1)for the 1b3,, momentum profile are a reflection of the very diffuse nature of this orbitalin position space. In the lower panel of figure 6.5, the lb3g orbital can be seen to be0c—H bonding and at the same time-c anti-bonding. Again, the participation of thisorbital in molecular bonding is reflected as a “interference pattern” in momentum space.Note that the p-type structure of the lb3g momentum profile arises from nodal planesin the plane of and perpendicular to the C-C bond, in contrast to the 1b3 HOMO. The3a9 electron densities (figure 6.6) show that this orbital is both strongly bondingand weakly CC—H bonding. The s-p character of the 3ag momentum profile is a reflectionof this multiple centre bonding. The spatially contracted uc_c density gives rise to astrong contribution at p = 0 in the momentum profile with a broad high momentumtail, while the much more spatially diffuse Oc...H bonding behavior corresponds to to ahighly contracted maximum in momentum space at p = 0. The combination of boththe occ and oc..jj contributions thus give a large maximum at low momentum anda smaller, higher momentum tail in the 3a9 momentum profile. The 1b2 density distributions (figure 6.7) show that this orbital is strongly cTc....H bonding with some lrc_cbonding characteristics. This type of orbital is often labeled a 7rc....H bond. Note that itlies in the molecular plane, perpendicular to the 1b3 7r-system. Like the 1b3 profile, the1b2 TMP is p-type due to a nodal plane along the carbon-carbon axis, perpendicularChapter 6. Ethylene 127to the plane of the molecule. The maps of the 2b1 charge distribution (figure 6.8), inboth position and momentum space, show both the strongly cT..C anti-bonding natureand the weak C-H bonding character of this orbital. The 2b1 orbital primarily involvesthe C .s basis functions, with some hydrogen s contributions. This is a highly contractedorbital in position space and is correspondingly diffuse in momentum space. Finally, the2ag orbital (figure 6.9) is c..c bonding, like the 3a9 orbital, but has less oc_ bondingcharacter. The 2a is composed of mainly carbon s contributions of similar magnitudeto, but in opposite phase of those making up the 2b1 orbital, and has a similarly broadmomentum profile.It is interesting to note that in this full MO picture of the valence electronic density,all orbitals participate to some degree in C-C bonding and, with the exception of the1b3, in the C-H bonding as well. Note also that two orbitals, lb3g and 2b1, are carbon-carbon anti-bonding. This is in contrast to the simple “textbook” valence bond picture ofethylene with a lrc_c and a °cc bond, four directed C-H o bonds and no anti-bondingcharacter.6.2.5 Vibrational Effects in the 1b3 Momentum DistributionIn the previous sections, all of the theoretical calculation have been performed at theequilibrium geometry [129]. This completely ignores vibrational effects on the momentumprofiles which may induce a non-zero value at zero momentum for the “p-type” HOMO(even before resolution folding). The effects of vibrational closure on momentum profileshas not been widely investigated, although studies of water [80, 137] have shown almostnegligible vibrational effects on the lb1 non-bonding HOMO momentum distribution.It is therefore of some interest to investigate the effects of vibrational averaging for aChapter 6. Ethylene 128bonding orbital, such as the HOMO ‘ir 1b3 orbital of ethylene. The calculations in thissection were performed by Wang and Davidson at Indiana University as part of thiscollaboration.Ethylene has 12 normal vibrational modes [128]. Even assuming that in the initialstate only the lowest vibrational levels are occupied, the final state will be vibrationallyhot due to the Franck-Condon overlap for the ionization process. Summing over all thepossible vibrational states, one can rewrite the momentum distribution (() =1 (j5’I’’12 from equation (2.20)) as,p(p) = (W fi c I z7’ H c’3) (II ‘i’ I I1jv fi qf,) (6.1)a V1 LI1 iiwhere ‘i7’ and ‘II are the final and initial state electronic wavefunctions; j5 is a spin-orbital representing the target electron; cb is the vibrational wavefunction (see equation (6.5 below)), where v (z/) denotes a the normal mode in the initial (final) state, andj indicates the th vibration level of the mode v.Ignoring the dependence of ‘1’’ on nuclear coordinate the following identity is truefor the final state vibrational wavefunction,I flL,lj)(H4Vlj 1 I (6.2)j LI1 LI1Thus, from equation (6.1),=1 (N_1 flW> 12. (6.3)One of the consequences of the equation (6.3) is that only those modes which havethe same symmetry as the final electronic state will have non-zero contributions to thecalculated momentum distribution. Since ethylene has only one vibrational mode of B3symmetry [128], the product fl, 4LI reduces to a single vibrational wavefunction for theChapter 6. Ethylene 1291b3 orbital. In this case, the momentum density can then be simplified top(p) = fpo(p,Q) I 12 dQ(6.4)p°(p,Q) = I(13’W’ I ‘I’’) 12where the integral over the normal coordinate Q indicates averaging over the B3 vibrational mode. Note that Po is not the usual electronic structure factor for EMS calculations(in equation (2.20)) because of its dependence on the normal coordinate, Q.Assuming a harmonic oscillator, the vibrational wavefunction , is given by=_2,,I2 (6.5)where a = v3, p is the reduced mass of the molecule and 1/ is the vibration frequencyof the mode.The matrix elements of equation (6.4) can be evaluated exactly (with a coordinatetransform) with the assumption of a harmonic vibrational potential using Gauss-Hermiteintegration. However, the normal coordinate Q for the vibrational motion must be knownin terms of the atomic displacements of the vibrational mode; SXN for centre N. The“B3 [128] mode of ethylene is an out-of-plane “umbrella” motion with the hydrogencentres all moving in opposite phase to the carbon nuclei:2mSx + 4mH6xH = 0(6.6)6x 2mThe normal coordinate Q for this mode is then defined by:SX=QC (6.7)Chapter 6. Ethylene 130where,XT= (xH,xH,xH,xH,xc,x)CT= A(2mc, 2mg, 2mg, 2mg, —4m11 —4mH)and where A is reduced-mass factor which makes matrix C unitless.The spherically averaged p(p) for VB3u = 949.2 cm1 from equation (6.4) and equilibrium geometry momentum density from equation (2.21), both calculated using the196-GTO SCF basis set are shown in figure 6.11. The vibrationally averaged density isshown with a broken line, the equilibrium geometry MD with a solid curve. It is clearfrom figure 6.11 that the effects of averaging over this normal mode are very small anddo not significantly change the shape of the momentum density. While it is not possibleto differentiate differences of this magnitude with the current single-channel instrumentation, these effects may become significant with the development very high count ratespectrometers, such as multi-channel angle dispersive instruments. Note that neithercurve in figure 6.11 includes any consideration of the instrumental resolution function.6.2.6 Comparison with Multi-channel EMS ResultsVery recently, and after all experimental and theoretical work for this thesis was completed, a new high sensitivity multi-channel EMS spectrometer entered operation in thislaboratory [138]. Measurements for several small molecules have been obtained veryrecently with this instrument, including XMP’s of the valence shell region of C2H4.Y. Zheng and J.J. Neville have made available their preliminary results, in advance ofpublication [1391, for comparison with the XMP’s obtained for C2H4on the single-channelEMS spectrometer used in the present work. The new multi-channel instrument collectsa binding energy (e) spectrum over an 8 eV range at a given azimuthal angle () ratherChapter 6. Ethylene 13115- C2H4 1b3/ \ 10.51 eV10 / \ - 196GT0I \ (vibrationally averaged)I \I \(equilibrium geometry)50- •0 1 2 3Momentum (au)Figure 6.11: Vibrational averaging effects in the 1b momentum profile ofC2114Chapter 6. Ethylene 132than the single point (e, q) measured by the single channel instrument. This allows formuch improved energy resolution (1.3 instead of 1.7 eV fwhm) and improved coincidencecounting statistics using the energy dispersive EMS spectrometer. In addition, since itcollects a binding energy spectrum at each of a series of angles , the momentum distributions which are deconvoluted from these BES all share the same relative intensityscale, as was the case for the XMP’s obtained by the lengthy single channel measurements(section 6.2.2) from the ten binding energy spectra in figure 6.3.A summary of the multi-channel and single channel results is shown in figure 6.12.This provides a consistency check for the low precision single channel data of the presentwork. The multi-channel EMS results in figure 6.12 are from a preliminary fit of themulti-channel data using the same energy positions and Franck-Condon widths usedto fit the binding energy spectra in the present single channel work as described insection 6.2.1. The single channel results have each been individually scaled so as toshape match the multichannel results for each XMP. In all cases the scaling factors arein the range 1.0—1.1. The particular factors used for each single channel data set areindicated on figure 6.12. These small intensity differences are likely due to the largerstatistical uncertainties in the peak areas fitted to the single channel binding energyspectra (the procedure is described in section 6.2.1). It can be seen that although thenew multi-channel measurements have understandably much better statistics than thesingle channel results, the two data sets are quite consistent. It appears, however, that theapparent slight mismatch (figures 6.4 and 6.10) between the best theoretical treatmentsand experiment on the leading (low momentum) edge of the 1b3 momentum distributioncan largely be attributed to statistical uncertainties in the single channel data.Chapter 6. Ethylene 133Ethylene Momentum Profiles(. .) SC data o MC data 196—G(CI)21b321b22 lb3g 2 2b112.83 19.1• xl.O • xl.O;,t \4x1.1________________Momentum (au)Figure 6.12: Comparison of energy-dispersive multi-channel (open circles) and single-channel (filledcircles and squares) EMS results. The theoretical curves are the 196-G(CI) MRSD-CI calculation.Chapter ‘7The Effects of Fluoro-substitution on the Ethylene 7r.-SystemThe quantum chemistry of the derivatives of ethylene (C2H4) has been the subject ofmuch experimental investigation by EMS [119,140—142] . Many of these compounds, inparticular the halogen derivatives, are of topical interest because of the environmentaland health hazards they pose [143].Previous EMS studies of molecules containing a carbon-carbon double bond includemeasurements of the complete valence shells of fluoro-, chloro-, and bromo-ethylene [140,141] and the HOMO’s of the vinyl halides (C2H3X, X=F, Cl, Br and I) [120]. Ying andLeung [144} have also recently published a PES study of of the chioro-substituted isomersof C2H1,including the 1,1-, 1,2-cis- and 1,2-trans-dichioroethylene. In other work,alkyl-substitution of C2H4 has been studied by Tossell et al. [119] and by Ying et al. [142]for CH2=C(CH3)and cis-, and trans-CH3CH=CHC.Fluorination effects have alsobeen studied in the substituted acetylene system,H3CCCCH andF3CCCCF [44].There has been considerable interest in the valence momentum profiles of ethylene [24,63, 119, 120] (and also the results described in chapter 6) and monofluoroethylene [120, 140], but prior to the present work, there have been no EMS studies of the134Chapter 7. The Effects of Fluoro-substitution on the Ethylene ir-System 135the other molecules in the series C2H4,C2H3F, CH2F,C2HF3 and C2F4. Photoelectron spectroscopy studies of this substituted series [100,145] and theoretical calculations [146,147] have revealed little about the quality of wavefunctions necessary for theaccurate reproduction of electronic properties.With the existing lack of experimental data, and because of the chemical interestin the effects of fluorination of the ethylene double bond, the highest occupied valenceorbitals of the series C2H4,C2H3F, CH2F,C2HF3 and C2F4 have been investigatedby EMS. The experimental momentum profiles and other measured electronic propertiesare compared with the results of a range of ab initio SCF calculations, using treatmentsranging from from minimal basis quality to the much larger, very recently developedbasis set of Dunning and co-workers [135,148,149]. In addition to the evaluation of thetheoretical wavefunctions, the effects of fluorination on the carbon-carbon double bondare also of interest. C2F4has one of the weakest carbon-carbon double bonds known [150],and this phenomenon is not well understood theoretically.The experimental samples were commercially obtained at CP grade or better (i.e.99.0 % purity) and used without further purification. No impurities were observed inany of the spectra obtained in the course of this work. It should be noted that the cisand trans-i ,2-difluoroethylene isomers were not commercially available at the time of thisstudy.7.1 CalculationsSpherically averaged momentum profiles have been calculated using a variety of basis setsfor the highest occupied SCF orbitals of the molecules C2H4,C2H3F,CH2F,C2HF3andChapter 7. The Effects of Fluoro-substitution on the Ethylene ir-System 136C2F4 using the Target Hartree-Fock and Plane Wave Impulse approximations of the EMScross-section (see chapter 2). In each case, the instrumental angular resolution functionhas been accounted for using the GW-PG method of Duffy et al. [81]. The calculationsfor ethylene were performed at the equilibrium geometry as given by McMurchie andDavidson [129]. The geometries for the fluorinated compounds were taken from theexperimental and theoretical compilation of Heaton and El-Talbi [146]. The details ofthe various basis sets used for this study are given below. Calculated and measuredproperties are summarized in table 7.1.1 STO-3G: This is a standard minimal basis set designed by Pople and co-workers [94].For C2H4,this basis set provides 14 contracted Gaussian orbitals.2 4-31G: This is a split valence basis, developed by Ditchfield et al. [95]. It has aminimal description of the C and F is cores and a double zeta description of theC and F 2s, 2p and H is valence orbitals. This is a 26-GTO basis set for C2H4.3 6_311++G**: The 6-311G set is the valence triple zeta basis set developed byKrishnan et a!. [130] which supplies a single contraction for the carbon and fluorineis core orbitals and three basis functions for the valence shell. It has been furtheraugmented by additional diffuse (++) [131] (sp on C and F, s on H) and d and ppolarization functions (**) [132] on the heavy atoms and hydrogen respectively, toincrease the basis set flexibility. The final contracted basis set is [5s, 4p, ld/4s, ip],for [C and F/ H], a 72-GTO basis set for ethylene.4 aug-cc-pVTZ: This basis set, like the 6-3iiG above, is also a valence triple zetabasis set and has been recently developed by T.M. Dunning and co-workers [135,148]. The basis has been supplemented with a single set of Dunning’s recommendedaugmentation basis functions, which are designed to ensure near saturation of theChapter 7. The Effects of Fluoro-substitution on the Ethylene ir-System 137basis set [149]. This basis is (us, 6p, 3d, 2f)—*[5s, 4p, 3d, 2f] for the carbon andfluorine centres and (6s, 3p, 2d)—[4s, 3p, 2dj for hydrogen. For C2H4, this is a184-GTO basis.Also included on table 7.1 are the 196-GTO and 196-G(CI) results obtained for ethylene (see chapter 6). Since the 196-GTO basis set is very close to the Hartree-Focklimit, the extent of convergence of the present calculations can be investigated for ethylene by comparison with these large basis set results. It should be noted that the dipoleand quadrupole moments reported for the calculations in table 7.1 have been obtainedfor the equilibrium geometries, as given by Heaton and El-Talbi [146]. In contrast, theexperimental values reflect the vibrational motion of the molecule, for both quantities.7.2 Comparison of Experimental and Theoretical Momentum ProfilesHigh momentum resolution EMS measurements have been made of the outermost valenceelectron momentum profiles for each of C2H3F (2a”), CH2F (2b1), C2HF3 (4a”) andC2F4 (2b3,j. These measurements are shown in figure 7.1, together with the 1b3 experimental momentum profiles obtained for C2H4 (see chapter 6). On this figure the bindingenergy of each XMP is indicated (in electron volts), ranging from 10.5 eV for C2H4to 10.7 eV for CH2F. These values correspond to the vertical ionization potentials ofthe lowest lying peaks observed in the respective photoelectron spectra [100, 145]. Forall four of the fluorinated ethylenes studied in the present work, the peaks correspondingto these lowest lying XMP’s were well resolved in the respective binding energy spectra,with no cross-contamination from other higher energy ionization process. The smallestenergy separation, for C2H4, is 2.3 eV; the largest, for C2F4, is 5.4 eV. The instrumentalChapter 7. The Effects of Fluoro-substitution on the Ethylene zr-System 138Table 7.1: Electronic properties of selected fluoro-ethylenesMolecule Wavefunction Total Energy p Oaaa PMAx(hartree) (Debye) (au) (au)C2H4 STO-3G —77.0722 0.484 0.704-31G —77.9210 1.242 0.626311++G**—78.0554 1.495 0.51aug-cc-pVTZ —78.0624 1.530 0.47196-GTO —78.0693 1.532 0.47196-G(CI) —78.4085 1.327 0.47Experiment 1.1 ± 02b 0.47C2H3F STO-3G —174.5302 0.899 —0.597 0.804-31G —176.6493 2.063 —0.801 0.646311t+G**—176.9361 1.765 —0.331 0.51aug-cc-pVTZ —176.9554 1.597 —0.223 0.50Experiment 1.427c _O.lS±O.lSd 0.50CH2F STO-3G —271.9988 0.660 0.270 0.804-31G —275.3801 2.117 1.601 0.686311++G**—275.8218 1.718 1.680 0.60aug-cc-pVTZ —275.8529 1.552 1.655 0.58Experiment 1.384e 1.8 ± 0.6 0.56C2HF3 STO-3G —369.4477 0.941 —1.234 0.924-31G —374.0882 2.162 —2.845 0.766_311++G**—374.6793 1.854 —2.802 0.75aug-cc-pVTZ —374.7208 1.684 —2.648 0.71Experiment 1•32g —2.6 ± 0.2’s 0.72C2F’4 STO-3G —466.9000 —1.041 1.044-31G —472.8023 —2.264 0.896311++G**—473.5561 —1.448 0.89aug-cc-pVTZ —473.6098 —1.241 0.87Experiment 0.85a Quadrupole Moment component, eaa = (3a2—r2), where a is the principal inertialaxis most parallel to the C=C bond.bRecommended value, uncorrected for zero-point motion, obtained by induced birefringence [136].CFor a vibrating molecule, from reference [151].dFrom Zeeman effect splitting measurements, for a rotating, vibrating molecule [152].eFor a vibrating molecule, from reference [153].1From Zeeman effect splitting measurements, for a rotating, vibrating molecule [154].‘For a vibrating molecule, from reference [155].h From Zeeman effect splitting measurements, for a rotating, vibrating molecule [152].Chapter 7. The Effects of Fluoro-substitution on the Ethylene 7r-System 139energy resolution was 1.7 eV fwhm.On figure 7.1, the experimental momentum profile for each molecule is shown togetherwith the spherically averaged theoretical momentum profiles (TMP’s) calculated from theSCF basis sets described in section 7.1. Note that these TMP’s include consideration ofthe finite instrumental angular resolution using the GW-PG method developed by Duffyand co-workers [81]. The experimental results for each molecule shown on figure 7.1 havebeen independently scaled for a best fit to the corresponding aug-cc-pVTZ momentumprofile. The intensities of the calculated momentum profiles are such that all the SCForbitals integrate to unity over momentum. This allows for the evaluation of the individual basis set calculations for both shape and intensity by comparison with the EMSresults.For all five molecules, the total energies, quadrupole moments and positions of theintensity maxima (PMAx) in the momentum profiles for each calculation and the corresponding experimental values (where available) are shown in table 7.1. For the polarmolecules C2H3F, CH2F and C2HF3,calculated and experimental dipole moments (inDebye) are also shown.The minimal basis set STO-3G momentum profiles are in very poor agreement withthe EMS results (the short dashed lines on figure 7.1), for all molecules. This treatmentalso results in very poor values for the total energies and quadrupole moments (table 7.1).For the three polar molecules (C2H3F,CH2F and C2HF3), this minimal basis set alsoresults in a large underestimation of the dipole moment, and thus the charge separation. These results emphasize the serious limitations of using a minimal description formolecular calculations.The slightly larger 4-31G basis set results show some improvement in agreementChapter 7. The Effects of Fluoro-substitution on the Ethylene ir-System 1402010010010030 3Momentum (au)Figure 7.1: Experimental and theoretical momentum proffles for the lowest energy ionizations ofselected fluorinated ethylenes, including C2H4. Note that the relative (theoretical) scale for the threepanels on the left is twice that of the two right-most graphs.C2H4lb 10.513uC2H2a”3F10.57502b1 10.72 500 1 2 1 2Chapter 7. The Effects of Fluoro-substitution on the Ethylene 7r-System141with the XMP’s (the dash-dot curves on figure 7.1) and also for the total energies andquadrupole moments (table 7.1), when compared with the STO-3G results. The levelof agreement with the XMP’s also improves with increasing fluorine substitution. The4-310 TMP is a poor model of the C2H4 1b3,C2H3F 2a”, and CH2Fb1momentumprofiles, but is in reasonable agreement with the XMP’s the 4a” and 2b3 profiles ofC2HF3and C2F4 respectively. However, the very high dipole moments for the polar speciesand the poor total energies and quadrupole moments for all the molecules (table 7.1),indicate that the 4-31G results are still very far from convergence to the Hartree-Focklimit. The very high dipole moments attained with the 4-31G basis set imply alargeseparation of charge. An examination of the 4-310 molecular orbital coefficients revealslarge contributions from fluorine p atomic orbitals at the expense of contributionsfromthe carbon-carbon yr-system, with increasing fluorination. The electron densitiesaremuch more highly localized on the fluorine centres in the 4-310 calculation than forresults with the larger basis sets. This “over-polarization” behavior of the 4-31Gbasisset has also been observed for HF (see chapter 4) and many other compounds, includingthe derivatives of formaldehyde (see chapter 8).The 6311++G** theoretical momentum profiles (dashed lines on figure 7.1) are inreasonable agreement with the observed momentum profiles for all five molecules. However, as was found for ethylene (see chapter 6, table 6.1) the total energies and quadrupolemoments calculated using this basis set are still far from being converged. Although muchimproved over the 4-31G basis, it can be seen from the electronic properties on table7.1that the 6-311 + +G** electron densities are not very good models of the molecularelectronic structures. It should be noted that the use of the standard polarization [132] anddiffuse [131] supplements does give some improvement over the 6-3110 results which areintermediate in all cases to the 4-310 and 6311++G** calculated properties. However,Chapter 7. The Effects of Fluoro-substitution on the Ethylene ir-System142a comparison of the 6311++G** and aug-cc-pVTZ results (see table 7.1 and discussionbelow) suggests that even more supplements must be added to the 6311++G** basis toachieve basis set saturation.Dunning’s augmented triple-zeta valence results (aug-cc-pVTZ, the solid curve on allpanels of figure 7.1) are in very good agreement with the EMS measurements. This basisset also predicts the best dipole and quadrupole moments and the lowest total energiesof any of the four basis sets used for all molecules in the present work (see table 7.1).A comparison of the total energies, quadrupole moments and PMAx valuesobtained forethylene (table 7.1) shows that the (184-GTO) aug-cc-pVTZ basis provides a descriptionof the electronic structure of C2H4 comparable to that of the highly converged196-GTObasis set. While it was found that electronic correlation and relaxation effectswere negligible for the 1b3 TMP of C2H4 (see chapter 6), many-body considerations are necessaryfor a complete description of the hr HOMO XMP of the fluorine compound HF (seechapter 4). In the present work, since there is generally very good agreement betweenthe experimental and aug-cc-pVTZ SCF results, electronic correlation and relaxationwould be expected to have a minimal effect on the HOMO momentum profilesofC2H3F,CH2F,C2HF3 and C2F4. However, correlation and relaxation effects would be expected to have a large effect on other electronic properties such as the total energies andquadrupole moments as can be seen for C2H4 (see tables 6.1 and 7.1).It remains unclear as to why the CH2F 2b1 experimental momentum profiledoesnot agree with any of the theoretical results at low momentum, considering thegenerallyexcellent agreement of the 6311++G** and aug-cc-pVTZ theoretical momentum profileswith the XMP’s for all the other molecules shown on figure 7.1. It is possiblethat aneven larger and more diffuse basis set and/or the inclusion of electronic correlation andrelaxation effects would remove the remaining discrepancy with the 2b1 EMS result.Chapter 7. The Effects of Fluoro-substitution on the Ethylene ir-System 1437.3 Trends in the Electronic Densities with Increasing FluorinationIt can be seen from figure 7.1 that the position of the intensity maximum (PMAx,seetable 7.1) of the momentum profile shifts to higher momentum with increasing fluorosubstitution of ethylene. The trend of the PMAx values towards higher momentum inthe profiles (table 7.1) reflects both a greater diffuseness in p-space and an increasednodal structure in the molecular 7r-system as fluorination of the double bond increases.The greater diffuseness in momentum space roughly corresponds to a contraction ofthe electron charge density in position space (due to the approximate Fourier inverserelationship between position and momentum space). This localization of the r-spaceelectron density does not necessarily occur on one centre—in this case an examination ofthe molecular orbital coefficients reveals large contributions from localized atomicorbitalson all the fluorine nuclei which implies that a considerable fraction of the ir-electrondensity is localized about the fluorines. Thus, as more fluorine atoms are substitutedfor hydrogens, the localization effect of the electron density intensifies in positionspacewhile the density becomes increasingly diffuse in momentum space. The molecular orbitalcoefficients of the fluorine centres for all of the fluoro-substituted species are alsoseen tobe of opposite phase to the carbon contributions, indicating C-F anti-bondingcharacterin the HOMO ir-orbital. This anti-bonding character introduces a nodal surface betweenthe fluorine and the carbon. Since the electron momentum can be expressed as thederivative of position (j3 = —ih), regions of a high density gradient (e.g., near a node)in position space would be expected to cause an increase in the high momentum regionsof the electron density. In particular, the more nodes in position space, the larger thecontributions to the density at higher momentum, as has been previously observed forpara-dichlorobenzene [156] and carbon dioxide [48]. Thus, the trend of PMAX to valuesof higher momentum with increasing fluorination can also be said to reflect thegreaterChapter 7. The Effects of Fluoro-substitution on the Ethylene ir-System144nodal complexity of the ir orbital C-F anti-bonds.The above effects are dramatically illustrated by a comparison of the charge distributions of C2H4 and C2F4 in figure 7.2. The position and momentum density distributionmaps have been calculated with the aug-cc-pVTZ basis set which gives a good descriptionof the XMP’s (see figure 7.1) and other properties (table 7.1). On figure 7.2 the electrondensities for the highest-occupied molecular orbitals of C2H4 and C2F4 are shown in bothposition (upper row) and momentum space (lower row) for a cross-section through one ofthe ir-system lobes, parallel to and 1.0 au above the primary nodal plane (the molecularplane in position space). Also shown on the right panels are the density difference mapsfor Pc2F4 — Pc2H4 in both position and momentum space. The contours represent 0.01,0.03, 0.1, 0.3, 1.0, 3.0, 10.0, 30.0, and 99.0% of the maximum density. Solidcontours onthe density difference maps indicate regions of higher density in C2F4, broken lines areregions of greater density in C2H4. On all maps, closely spaced contour lines indicateregions of high gradient in the electron density. It can be seen from the position densitymaps and their difference on figure 7.2 that there are regions of high density localized onthe fluorine centres of C2F4,when compared with the charge density near thehydrogennuclei in C2H4 (which do not contribute to the ir orbital). The bonding (and anti-bondingfor C2F4)nature of these orbitals is reflected as the complex structure in the momentumdensity maps. While the density maps convey the same information in both r— andp—space, the bonding character is more easily understood in the position charge distribution maps. While there is a small increased density in the centre of the ir lobe forC2F4, the integral of the electron density along the dashed vertical line on the positiondensity maps, along the C=C bond is greater for C2H4 than for C2F4. This can be seenon the one-dimensional density cut along the C-C axis on the vertical sidepanel of theposition density difference map in figure 7.2. While there is a small enhancement of theChapter 7. The Effects of Fluoro-szibstitution on the Ethylene7r-SystemPOSITION DENSITY145Figure 7.2: Electron densities and density differences for C2F4 and C2114 in position and momentumspace, calculated using the aug-cc-pVTZ basis set. The maps show a cross—section of the electrondensity of one lobe of the ir-system, parallel to and 1.0 au above the primary nodal plane (the plane ofthe molecule in r.space) - The density difference maps shows the difference ofthe first two maps. Thecontours represent 0.01, 0.03, 0.1, 03, 1.0, 3.0, 10.0, 30.0, and 99.0%of the maximum density.DENSITY DIFFERENCE—4—8 —4 0 4 8(cu)car4AA2bMOMENTUM DENSITY MOMENTUM DENSITY DENSITYDIFFERENCE840—4C2H41ba4:*—F I-4 40(cu) a______Car4\1 —C2H4a I40,——4I I—a —4 0(au)4 -a —4 0(ci u)4 0Chapter 7. The Effects of Fluoro-substitution on the Ethylene ir-System 146electron density near the centre-of-mass origin, there are negative “dips” in the densitydifference near the nuclei, between the two carbon centres. As expected by “chemicalintuition”, fluorine is thus a net electron withdrawing group with respect to hydrogenfor this carbon ir-system although there is a small enhancement of the position spaceelectron density at the centre of the C-C bond in C2F4. The ir electron withdrawingeffect of the fluorine group has been seen for the HOMO’s of theC2HNF4_ series byother methods, including by analysis of the out-of-plane components of the quadrupolemoments (O) of the mono-, di- and tri-fluoroethylenes [152, 154] and in C is ionizationspectra [157]. This electron withdrawing behaviour, which weakens ir-bond, is also reflected by the relative C-C bond strengths of C2F4 and C2H4 (.-.s60 kcal/mole and i75kcal/mole respectively) [150].The anti-bonding character of the carbon-fluorine interaction is also readily apparenton the C2F4 position density map. For each additional fluorine substituent, there is anadditional nodal surface present in the ir HOMO thus four such structures are seen onthe C2F4 position density map on figure 7.2. Note that additional nodal surfaces arenot present in the corresponding momentum density map, because these surfaces possessneither radial nor planar symmetry. However, as discussed above, the additional nodalsurfaces in the C2F4 position space electron distribution introduce regions of high densitygradient and thus increase the high momentum parts of the momentum profile.Both the electron withdrawing nature of fluorine and its anti-bonding interaction withthe carbon ir system contribute to the trend to larger PMAxfl the observed momentumprofiles (figure 7.1) with increasing fluorination. The enhanced density on the fluorinecenters coupled with the resulting reduced charge between the carbon nuclei is responsiblefor the weakening of the carbon-carbon ir-bond in C2F4.Chapter 8The Effects of Methyl-substitution on the Formaldehyde ir-systemRecently, in this laboratory and elsewhere, the effects of alkyl- (and fluoro-) substitutionon the HOMO’s of amines [40—43,59] and the increasingly methylated species, H20 [37J,CH3O [66] and (CH3)20 [60,67], have been studied using both EMS and ab initiomethods with a view to investigating the electron withdrawing behavior of thealkylgroups on the hetero atom “lone pair” (HOMO) electrons. In all of these studies the EMSand theoretical results, consistent with a growing body of other experimental evidence,have clearly indicated that the methyl groups are intrinsically electron withdrawing (whencompared with hydrogen) in contrast to traditional intuitive arguments [158].In the present work, these studies are extended to investigate the effects of methylsubstitution on the r HOMO of the carbonyl (C=O) functional group. Molecules of thistype, aldehydes and ketones, are of particular interest to quantum chemistry because theelectronic environment about the carbonyl group is uniquely electropositive when compared with other carbon environments. The study of the HOMO electron densitiesofthe increasingly methylated formaldehyde (H2CO), acetaldehyde (CH3CHO) and acetone((CH3)2C0) series may provide an improved understanding of the “methyl inductive effect.” The experimental momentum profiles of the outermost valence electrons of acetone((CH3)2C0) and acetaldehyde (CH3CHO) have been obtained by electron momentumspectroscopy. These experimental results, together with those for the outermost XMPof147Chapter 8. The Effects of Methyl-substitution on the Formaldehyde ir-system 148formaldehyde (H2CO) reported earlier by Bawagan et al. [34], are compared with calculations of the theoretical momentum profiles using a wide range of calculations includingnew near Hartree-Fock quality SCF and multiple reference single and double excitationconfiguration interaction (MRSD-CI) treatments, carried out in collaboration with Y.Wang and E.R. Davidson. Other calculated molecular properties are also compared withexperimental values in order to test all parts of the electronic wavefunctions [68]. In addition the effects of diffuseness are examined for intermediate quality SCF basis sets for allthree molecules. These measurements and calculations provide the basis for discussionof the differences observed in the electronic structure of the HOMO oxygen “lone-pairs”in both position and momentum space with increasing methyl substitution of the C=Ogroup. Particular attention is given to the effects of orbital symmetry and the nature ofchemical bonding in the outermost orbital in the context of the EMS measurements andthe new ab initio calculations.The acetone and acetaldehyde samples used were spectroscopic grade (> 99.9 %pure) from BDH chemicals. Gaseous impurities dissolved in the samples were removedby repeated freeze-thaw-pump cycles using liquid nitrogen. To avoid variations in theliquid partial pressures, the sample tubes were held at a constant temperature of 00 C byimmersion in an ice bath. This procedure maintained target gas pressures at 1.5x10torr regardless of the ambient temperature.8.1 CalculationsSpherically averaged theoretical momentum profiles (TMP’s) have been calculated fromseveral basis sets of varying quality using the target Hartree-Fock and plane wave impulseapproximations of the EMS cross-section (equation (2.21)). In addition, full ion-neutralChapter 8. The Effects of Methyl-substitution on the Formaldehyde ir-system 149overlap distributions have been calculated from the plane wave impulse approximation(equation (2.20)) using multi-reference singles and doubles excitations configuration interaction (MRSD-CI) calculations of the overlap of the ground and molecular ion statesto investigate the effects of correlation and relaxation on the momentum distributions.The MRSD-CI calculations were based on the near Hartree-Fock limit basis sets for therespective molecule and used neutral molecule orbitals for both the neutral and ion. Theinstrumental angular (momentum) resolution was included in the calculations using theGaussian-Weighted Planar Grid method of Duffy et al [81].The smaller, more limited calculations were performed at the experimental geometriesobtained from averaged rotational (microwave) spectroscopic data (formaldehyde) [159]and zero-point average structures measured by both electron diffraction and microwaverotational spectroscopy for acetone [160] and acetaldehyde [161]. Although these studies determined the C—H bond lengths for the methyl groups of both acetone and acetaldehyde, the orientation of the freely rotating methyl groups on both CH3HO and(CH3)2C0 were not determined. The methyl group for acetaldehyde was chosen to bein an eclipsed conformation with respect to the carbonyl group. For acetone, the methylgeometries were chosen to be in an “half-staggered” conformation (one methyl eclipsed,the other anti to the C=O bond). Previous studies on dimethyl ether [60] have shownthat methyl group orientation has only a small effect on the resulting TMP’s at this levelof calculation.A geometry optimization was done to investigate the energy differences of the rotational conformers of acetone and acetaldehyde using 631G** basis sets at the secondorder Moller-Plesset perturbation (MP2) level. Acetaldehyde was found to have twominimum energy C3 structures. The “staggered” form has one C—H bond anti to theC=O. The “eclipsed” conformer has one C—H bond eclipsing the carbonyl group (veryChapter 8. The Effects of Methyl-substitution on the Formaldehyde ir-system 150similar to the geometry used for the smaller calculations, described above). The eclipsedform was found to be 1.6 mhartrees lower in energy than the staggered form at the 6-31G**/MP2 level. Acetone had three energy minima structures, two with C2,, symmetry,one with C3. All three forms were found to have one C—H bond in each methyl groupin the (C2C=O) molecular plane. The “eclipsed” form has both in-plane C—H bondseclipsed with CO bond; the “staggered” form has the in plane C—H bonds anti tothe ketone C=O bond; the “half-staggered” conformer has one C—H bond eclipsed andone anti. The “half-staggered” conformer was very similar to that used for the smallercalculations. The eclipsed form was found to be the lowest energy structure with thehalf-staggered being 1.2 mhartrees higher in energy and the staggered form 3.6 mhartreeless stable according to the 631G**/MP2 calculations. The eclipsed geometries wereused for the near Hartree-Fock and subsequent MRSD-CI calculations for both acetoneand acetaldehyde.Details of the calculation methods are described below and selected properties aregiven in table 8.1.1 STO-3G: This is a minimal basis set designed by Pople and co-workers [94].2 4.-31G: This is a split valence basis developed by Ditchfield et aL [95].3 4-31G+sp+s: This is an extension to the 4-310 basis above, developed by Casidaand Chong [38,39] and has been extensively used in this laboratory in work onhydrogen suffide [31] and dimethyl ether [60]. The addition of the extra diffusefunctions, s and p on each oxygen and carbon centre and an s function on thehydrogens, has been found, particularly in the dimethyl ether study [60], to greatlyimprove the reproduction of “large-r” properties such as the EMS momentum profiles, with little extra effort. The exponents used were 0.0946 for the diffuse 0 sChapter 8. The Effects of Methyl-substitution on the Formaldehyde ir-system 151and p functions, 0.123 for C s, 0.0663 for C p and 0.036 for the additional H sGaussian function exponent.4 Snyder and Basch: This is the Gaussian basis set of Snyder and Basch [96].5 Near Hartree-Fock SCF: These calculations, developed by Y. Wang and E.R.Davidson as part of this collaborative study, (90-GTO for formaldehyde, 143-GTOfor acetaldehyde and 196-GTO for acetone) are based on the highly converged,Gaussian function basis sets of Partridge [133,134]. These bases were energy optimized to within 4 phartree of the (numerical) Hartree-Fock atomic limit. For C ando the original (18s, l3p) bases [133] were contracted to [6s, 7p]. The hydrogen (lOs)basis of Partridge [134] was contracted to [7s] Gaussian functions. The contractionscheme used lost less than 0.16 mhartree (0.1 kcal/mole) on trial SCF calculationsof CO and CR4 [59]. For better reproduction of the “large-r” properties, Dunning’s“double d” polarization functions [135] were used for the 0 and C exponents (0.645,2.314 and 0.318, 1.097 respectively). A single p polarization function was added tothe hydrogens (exponent 1.30) [59]. A trial calculation on CO using this basis waswithin 3 kcal/mol of the numerical Hartree-Fock result.Sc MRSD-CI: The configuration space for the multi-reference singles and doublesconfiguration interaction calculations for both the molecular target and ion wave-functions were chosen from the results of respective single reference singles anddoubles configuration interaction calculations. For both the ion and neutral themolecular orbitals were from the neutral SCF wavefunction for which the virtualorbitals had been converted to K—orbitals [111—113] to improve the energy convergence. All CI calculations used frozen core electrons, to allow for the most detaileddescription of the valence electrons and most accurate reproduction of electronicproperties possible at the current level of computation.Chapter 8. The Effects of Methyl-substitution on the Formaldehyde ir-system 1526 142-GTO This is the very large 142-GTO SCF calculation for formaldehyde usedin reference [341.6c 142-G(CI) The 142-G(CI) MRSD-CI ion-neutral overlap calculations for H2COof Bawagan et al. [34].8.2 Comparison of Experimental and Theoretical Momentum ProfilesHigh momentum resolution measurements have been made of the outermost valence electron momentum profiles for each of acetone (5b2) and acetaldehyde (lOa’). These resultsare shown in figures 8.1 and 8.2 together with the previously measured [34] outermostXMP for formaldehyde (2b) in figure 8.3. On each figure the binding energy at whicheach particular XMP was measured is noted (9.7 eV for acetone, 10.2 eV for acetaldehydeand 10.9 for formaldehyde). These values correspond to the vertical ionization potentialsof the lowest lying peaks observed in the respective photoelectron spectra [117, 167, 168].It should be noted that, for all three molecules studied, the outermost XMP is well separated in energy from the rest of the valence ionization manifold, which ensures that allthe observed intensity of the momentum profiles is solely due to ionization to the lowestlying ion state and contains no mixing with other ionization processes. Even for thesmallest energy separation between the lowest and second lowest peaks in the photoelectron spectrum (in acetone 2.9 eV [117, 167]), the difference is still much greater thanthe instrumental energy resolution (fwhm 1.7 eV).Each experimental momentum profile is compared on figures 8.1—8.3 with the theoretical momentum distributions (TMP’s) calculated from the SCF basis sets and MRSD-CIChapter 8. The Effects of Methyl-substitution on the Formaldehyde ir-system 153Table 8.1: Electronic properties of H2CO, CH3HO and (CH3)2C0.Molecule Basis Set Total Energy 4uD PMAx(hartree) (Debye) (au)H2CO’ 1 STO-3G -112.354 1.53 0.942 4-31G -113.691 3.01 0.893 4-31G+sp+s -113.699 3.16 0.884 Snyder & Basch (DZ) -113.821 3.11 0.885 90-GTO -113.917 2.94 0.846 142GTOc -113.918 2.86 0.84Hartree-Fock Limit’ -113.92 2.86Sc 90-G(CI) -114.277 2.55 0.846c 142G(CI)c -114.276 2.45 0.84Experimental -1 14.515e 2.33 0.751 STO-3G -150.940 1.77 1.202 4-31G -152.690 3.40 1.103 4-31G+sp+s -152.689 3.60 1.064 Snyder & Basch (DZ) -152.730 3.56 1.065 143-GTO -152.982 3.34 1.045c 143-0(d) -153.451 3.06 1.04Experimental 275h 1.04(CH3)2C01 1 STO-3G -189.534 1.91 0.35 1.392 4-31G -191.676 3.53 0.32 1.233 4-31G+sp+s -191.686 3.75 0.34 1.234 Snyder & Basch (DZ) -191.764 3.67 0.35 1.235 196-OTO -192.043 3.51 0.32 1.245c 196-0(d) -192.629 3.29 0.32 1.23Experimental 2.90 0.32 1.10a Positive dipole moment indicates that the oxygen nucleus is negative (ie C÷ Oj.bGeometry for calculations 1—4, 6 and 6c based on microwave spectroscopic data [159], a zero-pointaverage. Calculations 5 and 5c based on 6-3 1G**/MP2 optimized geometry. See section 8.1.cSee reference [34]dEstimated Hartree Fock energy limit [162].eNon_relativistic, infinite mass energy [163].1 reference [164].g Geometry for calculations 1—4 based on a gas phase ERD and MW zero-point average [160].Calculations 5 and 5c based on 631G**/MP2 optimized geometry. See section 8.1.hSee reference [165]Geometry for calculations 1—4 based on a gas phase ERD and MW zero-point average [161].Calculations 5 and 5c based on 631G**/MP2 optimized geometry. See section 8.1.‘See reference [166]Chapter 8. The Effects of Methyl-substitution on the Formaldehyde ir-system 154ion-neutral overlaps as described in section 8.1 above. Selected properties for each calculation and corresponding experimental values are shown in table 8.1. The SCF basissets used range from a very modest, minimal STO-3G basis to the very much larger nearHartree-Fock treatment. The effects of many-body correlations and electronic relaxationare also seen on figures 8.1—8.3 from the multiple reference space singles and doublesconfiguration interaction calculations (MRSD-CI) of the TMP’s.It has been found in the studies of the hydrides of the row two and three mainblock elements and several other small molecules [15, 16,811, particularly for highly polarspecies such as NH3 [55], H20 [37] and HF (see chapter 4) that an accounting for the finiteinstrumental angular (momentum) resolution is essential for the quantitative comparisonof theoretical and experimental momentum profiles [81,82]. The recently developed,Gaussian weighted planar grid method of Duffy et al [81] has been very successful in thisregard. The EMS instrumental momentum resolution has been accounted for in all theTMP’s on figures 8.1—8.3 using this method.The calculated momentum profiles are compared with the respective experimentalmomentum profiles by a visual fit of the MRSD-CI TMP’s (assuming unit pole strength)with the experimental data for each molecule. Since the wavefunctions are normalizedall of the calculations are on a common relative intensity scale for each molecule. Thisallows for quantitative comparison of the wavefunctions used to generate each momentumprofile.From figures 8.1—8.3 it can be seen that the best descriptions of the observed momentum profiles are achieved by the high-level MRSD-CI calculation (curves 5c), noted asChapter 8. The Effects of Methyl-substitution on the Formaldehyde ir-system>‘(1)CcvCC)>CC)c864200Momentum (a.u.)155MOMENTUM DENSITY—8—8POSITION DENSITY5b2—4 0(o.u.)Figure 8.1: The 5b2 experimental and theoretical momentum profiles of acetone (upper panel). Thelower panels show the momentum and position space density maps for an oriented (CH3)2C0 moleculecalculated using the 196-GTO basis set (see section 8.1). The contours represent 0.01, 0.03, 0.1, 0.3, 1.0,3.0, 10.0, 30.0, and 99.0% of the maximum density.1 2 3—8 —4 0 4 8(c.u.)4 8Chapter 8. The Effects of Methyl-substitution on the Formaldehyde lr-sys tern 15690-G(CI), 143-G(CI) and 196-G(CI) for formaldehyde, acetaldehyde and acetone respectively. The SCF calculations of the momentum profiles show a trend of improving agreement with experiment as the basis set quality improves from the STO-3G (curves 1) whichfit the observed XMP’s very poorly, to the intermediate quality results (4-31G+sp+s andDZ) and above, the near Hartree-Fock results (curves 5, noted as 143-GTO and 196-GTOas for the MRSD-CI above), which are in good agreement with the experimental dataat low momentum for acetone (figure 8.1) and acetaldehyde (figure 8.2). Although thestatistics are poorer for acetone, the measurements suggest that the second maximumis .— 0.2 au lower in momentum than predicted by the calculations for both molecules.This behavior is quantified as the value of pMAxof the momentum profile for formaldehyde, of the PMAx of the second, higher momentum, “lobe” of the acetaldehyde profiles,and of the PMAx values of the low and high momentum “lobes” in the TMP’s of acetone.The calculated and estimated experimental values are shown in table 8.1. Note that thehigher momentum features in the acetone (at 1.1—l .3 au) and acetaldehyde (at ‘—‘1 au)spectra (figures 8.1 and 8.2) are not true radial lobes of the electron density since theEMS cross-section does not go completely to zero between them. However, as discussedin the following section, the electronic distributions have structures that are very similarto radial nodes which are reflected in the spherical averaged density as local minimaand maxima. These features will be referred to as “nodes” and “lobes” respectively, forconvenience.The minimal basis set STO-3G results are in very poor agreement with both theobserved EMS data (curves 1 on figures 8.1—8.3) and electronic properties (see table 8.1).For acetone and acetaldehyde the higher momentum “lobes” are of greater intensitythan the lower momentum ones in clear disagreement with the experimental data. Fromtable 8.1 it can be seen that as well as a poor total energy (compared with the otherChapter 8. The Effects of Methyl-substitution on the Formaldehyde 7r-system>C,)C0C0>00840—4—8—8MOMENTUM DENSITYFigure 8.2: The lOa’ experimental and theoretical momentum profiles of acetaldehyde (upper panel).The lower panels show the momentum and position space density maps for an oriented CH3HO moleculecalculated using the 143-GTO basis set (see section 8.1). The contours represent 0.01, 0.03, 0.1, 0.3, 1.0,3.0, 10.0, 30.0, and 99.0% of the maximum density.15710500 1 2 3Momentum (a.u.)POSITION DENSITY0(o.u.)0(a.u.)Chapter 8. The Effects of Methyl-substitution on the Formaldehyde ir-system 158calculations) the calculated polarity of the C=O bond, as represented by the dipolemoment, is clearly less than the observed value. These problems underline the deficienciesof a minimal basis set model.With only a slightly larger basis set, the 4-31G (curves 2) description provides amuch better model of all three molecules than the STO-3G wave function. In each casethe calculated momentum profiles, while shifted to higher momentum, are in qualitativeagreement for shape with the XMP’s. For formaldehyde [34], while not in very goodagreement with the experimental data points, the 4-31G basis is almost as good as thelarger, intermediate size (4-31G+sp+s and DZ) basis sets. For the two larger molecules,the behavior of the calculated momentum profiles is similar, with the 4-31G having verysimilar PMAx values to the larger, higher quality calculations although there is considerableintensity at higher momentum for the 4-31G TMP’s than for the calculations with bettertotal energies, particularly for acetone (figure 8.1). The improvement of the 4-31G basisover the minimal description is also seen in the total energies and dipole moments intable 8.1. While there are obvious improvements in the respective total energies forall three molecules, the dipole moments of the 4-31G calculations are larger than theexperimental values in each case, indicating an overestimation of the polarity of theC=O bond. This behavior is typical of this basis set for highly polarized bonds (seereferences [37, 55] and chapter 4).Of special interest is the behavior of the basis sets of intermediate size, the 4-31G+sp+sbasis (curves 3) and the double zeta calculation of Snyder and Basch [96] (DZ, curves4). The extended 4-31G+sp+s basis [38,39] calculations agree particularly well with therespective XMP’s, approaching the near HF limit results, considering the relatively smallsize of the basis set ([4s, 3p] for 0 and C, [3s] for H). However the total energies anddipole moments given by this calculation (table 8.1), show only small improvements overChapter 8. The Effects of Methyl-substitution on the Formaldehyde ir-system 159the 4-31G results and are poorer than the (smaller basis set) Snyder and Basch doublezeta results.The double zeta basis set of Snyder and Basch [96] was specifically designed to mimictwo Slater functions per atomic orbital, a contracted one and a more diffuse one. As suchit generally gives better agreement with the XMP’s than the 4-31G TMP’s (which use amore limited core orbital description) and a poorer description than the more diffuse 4-31G+sp+s TMP’s. However, due to the greater number of large exponent, “core” basisfunctions that give a better description of the electron density near the nuclei whichcontributes significantly to the total energy, this basis set results in a much improvedtotal energy and dipole moment than the extended 4-31 G+sp+s basis set.These results again emphasize the care that must be taken when choosing intermediatesize basis sets not very near to the Hartree-Fock minimum energy limit, particularly whenadding diffuse (and polarization) functions which can be tuned for specific properties.Casida and Chong [38,39] chose to add very diffuse functions (i.e. gaussian primitiveswith small exponents) to the 4-31G basis which increase the electron density at long-rangerelative to the density near the nuclei. This has the effect of improving the calculationof those properties, such as the EMS cross-section (the TMP’s), which depend relativelymore on the large radial part of the wavefunction than the short range part. Conversely,properties which depend on the electron density nearer the nucleus, such as total energy,do not improve proportionately. The basis set of Snyder and Basch uses larger exponentsfor its diffuse functions and, more importantly, an additional “core” basis function (i.e.large exponent is function) for each heavy nucleus. This additional flexibility in theSnyder and Basch basis set [96] results in a better total energy than that obtained usingthe 4-310+sp+s basis.>‘.4-,(I)0.4-,CC)>.4-,00Momentum (a.u.)Figure 8.3: The 2b experimental and theoretical momentum profiles of formaldehyde (upper panel).The solid dots are the previously reported experimental data of Bawagan et. oL [34]. The lower panelsshow the momentum and position space density maps for an oriented H2CO molecule calculated usingthe 90-GTO basis set (see section 8.1). The contours represent 0.01, 0.03, 0.1, 0.3, 1.0, 3.0, 10.0, 30.0,and 99.0% of the maximum density.Chapter 8. The Effects of Methyl-substitution on the Formaldehyde ir-system6160656c5cH2CO10.92beV6c65c543211 42—G(CI)1 42—GTO90—G(CI)9 0—GTODz4—31 G+sp+s4—31G (80%)STO —3 C42080 1 2 3MOMENTUM DENSITY2b.POSITION DENSITY4-.4S40—4—8—8—8 ——8I I I—4 0(o.u.)4 8 —4 0 4 8(o.u.)Chapter 8. The Effects of Methyl-substitution on the Formaldehyde ir-system 161An examination of the momentum profiles calculated using the near Hartree-FockSCF basis sets (90-GTO and 142-GTO for H2CO, 143-GTO for CH3HO and 196-GTOfor (CH3)2C0) indicates that good agreement is attained for acetaldehyde and acetonewith the experimental profiles for the low momentum regions. At higher momentum,as with all the other SCF calculations discussed above, the intensities and PuAx’ of thesecond, higher momentum “lobes” of both acetone and acetaldehyde are overestimated(see table 8.1). While there is some improvement in the higher momentum parts ofthe profiles for both these molecules when compared with the more limited calculations,there is still large disagreement with the experimental observations in the high momentumregions. Previous experience suggests this is due to an overestimation of the near-coreelectron densities for these molecules. Use of larger, more flexible and diffuse basissets would likely improve agreement with the experimental momentum profiles. Themomentum profile calculated from the 90-GTO basis set for formaldehyde is considerablylower than the experimental intensity at low momentum (see figure 8.3). This is consistentwith the result using the 142-GTO basis set (also shown in figure 8.3) of the earlier studyof formaldehyde by Bawagan et al. [34]. The 142-GTO basis set has a total energy evencloser to the Hartree-Fock limit. While it is possible that even more diffuse functionswould improve agreement, it is more probable, as with other highly polar molecules suchas NH3 [55], H20 [37] and HF [68], that electron correlation and relaxation effects arerequired to describe the low momentum component of the momentum profiles.However, the direct calculation of the ion-neutral overlap including electronic correlation and relaxation by the MRSD-CI method discussed in section 8.1 only slightlychanges the level of agreement between the measured and calculated momentum profilesfor all three molecules (figures 8.1—8.3). In contrast to the situation for the momentumprofiles, there is a significant improvement (table 8.1) in the calculated total energies andChapter 8. The Effects of Methyl-substitution on the Formaldehyde ir-system 162dipole moments. However, when comparing the dipole moment values on table 8.1, itshould be remembered that the experimental values involve vibrational averaging whereasthe calculations were performed at the equilibrium geometry, as discussed in section 8.1.Although with the inclusion of correlation and relaxation, there is a small change in thecalculated profiles particularly in the high momentum regions for acetone and acetaldehyde, there are still significant discrepancies with the XMP’s. For formaldehyde, thereis significant underestimation of the low momentum cross-section with little change fromthe SCF description of the momentum profile for either the 90-G(CI) or 142-G(CI) calculations (shown on figure 8.3). The extent to which the remaining discrepancies in thecase of formaldehyde are due to errors in the experiment or limitations of the calculations is, at present, unclear. Possible sources of these discrepancies are the neglect ofvibrational averaging (due to zero-point motion) and the use of the plane wave impulseapproximation (equation 2.20). However the PWIA has been found to provide a goodquantitative model of the EMS cross-section in a wide variety of EMS studies under thesame kinematic conditions.8.3 Trends in the Electronic Densities ofH2CO, CH3HO and (CH3)2C0.A deeper appreciation for the shape of the experimental and theoretical momentumprofiles may be gained by consideration of the momentum and position space electrondensity distribution contour maps (in the bottom panels of figures 8.1—8.3). These calculations are based on the near Hartree-Fock SCF results for each molecule (90-GTO forformaldehyde, 143-GTO for acetaldehyde and 196-GTO for acetone) which provide thebest currently available SCF description at an equivalent level for all three molecules.These maps are slices of the electron density in position and momentum space throughChapter 8. The Effects of Methyl-substitution on the Formaldehyde ir-system 163the plane of the carbon and oxygen backbones of each molecule, oriented such that theC=O bonds are parallel to the vertical axes. The side panels show slices of the respectivecharge densities along the vertical (right panel) and horizontal (top panel) axes. For theposition space maps the origin is the molecular centre of mass.While simple valence bond descriptions predict that the outermost valence orbitalsof these three molecules, H2CO, CH3HO and (CH3)2C0 are non-bonding oxygen lonepairs, it is clear from the position density images (lower right panels in figures 8.1—8.3) that these orbitals are not simply pure oxygen p structures. For all three orbitalsthere is considerable density located on the hydrogens and methyl groups bonded tothe C=O group. The bonding character of these HOMO’s is revealed by the wrinkled“cabbage section” appearance of the momentum charge density maps. This phenomenonis a reflection of the interference effects caused by multiple atomic centres participatingin bonding (or anti-bonding) in the momentum space representations of the molecularorbitals. In general, a highly wrinkled appearance indicates strong bonding behavior inan orbital.These observations indicate that the oxygen lone-pair HOMO’s (H2CO 2b,CH3HOlOa’, (CH3)2C0 5b2) are slightly delocalized over the hydrogens for formaldehyde andacetaldehyde and strongly diffused by the methyl groups of CH3HO and (CH3)2C0.Clearly, as methyl groups are substituted for hydrogens these HOMO’s are increasinglydelocalized. A similar delocalization effect has been noticed in the previous work onthe alkyl substituted amines [40—43,59] and on H20 [37], CH3O and (CH3)20[60,67].The implications of these results to an understanding of bonding in these aldehydes andketones are discussed below.The structures in the density maps for the oriented molecules are reflected in theChapter 8. The Effects of Methyl-substitution on the Formaldehyde ir-system 164spherically averaged momentum distributions. The formaldehyde 2b position map (figure 8.3) clearly has C2 symmetry with a nodal plane down the vertical axis, which isalso preserved in the momentum space picture. When the momentum density is spherically averaged, this node, with intensity of zero at p = 0.0 au, is reflected as a nodeat zero momentum in the 2b spherically averaged momentum distribution (figure 8.3,upper panel) . It should be noted that the non-zero intensity at zero momentum in thecalculated momentum profiles in figures 8.1 and 8.3 is due to the fact that they havebeen folded with the instrumental (angular) resolution function in order to compare withexperiment (see section 8.1). Note that any intensity at p = 0 in a spherically averagedmomentum distribution (that does not include any instrumental resolution folding) mustbe due entirely to the s orbital components of the molecular orbital [42,59]. The Fourierrelationship between the position and momentum spaces at p = 0 is:ib(p) =o= -- fb(r)dr (8.1)This implies that only those atomic component orbitals which do not have equal intensitylobes of opposite sign (and thus integrate to 0) will contribute [9], such as s orbitals. Thetwo hydrogen s orbitals make equal and opposite phase contributions to the formaldehyde2b orbital. This gives a density of zero at zero momentum.Given the large discrepancy between the 90-GTO near Hartree-Fock limit SCF calculation and the observed 2b XMP at low momentum the question of how valid thisindependent particle model is for this “orbital” must be asked, although as noted in theprevious section, the inclusion of many body effects does not seem to significantly changeagreement with the XMP. The theoretical 90-GTO results give very similar results tothe 142-GTO calculations of the previous EMS study of formaldehyde by Bawagan etal. [34] for the calculated momentum profiles. A large MRSD-CI calculation using this142-GTO basis set did not give a large change from the SCF result, as is also seen forChapter 8. The Effects of Methyl-substitution on the Formaldehyde ir-system 165the present 90-G(CI) calculations. It was suggested by Bawagan et. al. [34] that, likethe strongly polar water [37] and HF (see chapter 4) molecules, very large basis sets andconsideration of a great deal more configurations are necessary to adequately model thepresumably highly correlated electron density in the molecular and ionic states.Acetaldehyde, with substitution of a methyl group at one of the formaldehyde hydrogen sites, has lost the C2, symmetry of formaldehyde in the lOa’ HOMO and thus thenodal plane through its centre of mass. Therefore, in momentum space there is no nodalplane and considerable density at p = 0. Equivalently, as above, this can be describedas unbalanced atomic s orbital contributions from the hydrogen bonded to the C=Ogroup and the methyl hydrogens. This results in a large s orbital net contribution anddensity at zero momentum. This is observed in the corresponding XMP (figure 8.2), butthere is also an additional “lobe” at higher momentum. It has been suggested by Tossell et. al. [40] for CH3N2and by Rosi et. at. for triethylamine [43] that these highermomentum “lobes” in the experimental and theoretical momentum profiles are due tothe introduction of additional nodal surfaces in the space charge density maps, whencompared with simpler systems. The 143-OTO SCF calculation for CH3HO developedin the present work, indicates a large contribution of the methyl group carbon and themethyl hydrogens. These methyl C—H bonds introduce an additional nodal surface(see figure 8.2) which is not present in the formaldehyde 2b map (figure 8.3). Whentransformed into momentum space and spherically averaged, the resulting calculatedmomentum profile reflects this additional nodal structure as a local minimum. Both thecalculations and the EMS experiment reflect this as a double “lobed” momentum profile.The 5b2 orbital of acetone calculated from the 196-GTO basis set, like the formaldehyde 2b map, reflects its C2, symmetry in the charge density maps in both momentumand position space (figure 8.1). Equal and opposite phases on the p orbital contributionsChapter 8. The Effects of Methyl-substitution on the Formaldehyde ir-system 166of oxygen, the aldehydic carbon and the methyl groups result in a nodal plane alongthe C=O bond in r—space as was the case for the formaldehyde 2b orbital discussedon page 164. The presence of a nodal plane through the position space centre of massorigin results in a TMP with a value of zero at p = 0 au, which is reflected in the XMP(figure 8.1, top panel). As before, this can equivalently be considered to be due to cancellation of the methyl group s components which are of opposite phase. Again, as was thecase for the CH3HO lOa’ HOMO, the acetone 5b2 XMP has a second “lobe” at highermomentum. It is evident that compared with H2CO (figure 8.3), there are two additionalnodal regions caused by the methyl group a- C—H bonds. As described above these resultin an additional minimum at higher momentum in the calculated profiles. This is alsoobserved in the measured momentum profile, although the position secondary maximumappears to be at a much lower momentum than the calculated values (see table 8.1).Further examination of the electron density maps indicates that there is considerableoverlap between the methyl group orbitals and the oxygen atomic p orbitals for boththe calculated acetaldehyde lOa’ and acetone 5b2 HOMO’s. This is consistent with thebehavior of the alkylated amines [40—43,59] and dimethyl ether [60,67] all of whichindicate that with increasing alkyliza.tion there is a reduction of charge on the N and 0lone pairs. This is can be interpreted as a mixing interaction of the oxygen lone-pair porbital and unoccupied irCH orbitals [169].It must be noted that no calculation in the present work, including the near HartreeFock results used to calculated the density maps, gives an entirely satisfactory descriptionof the observed experimental cross-section. Nevertheless, even though larger, more diffuse and polarizable bases and an accounting for electron correlation and relaxation arenecessary for more complete convergence to the XMP’s, experimental dipole momentsand the variational energy limit, the SCF calculations derived in the present work provideChapter 8. The Effects of Methyl-substitution on the Formaldehyde ir-system 167a good semi-quantitative model for the HOMO orbitals of all three molecules.Chapter 9Concluding RemarksThis work has presented EMS measurements of the binding energy spectra and all valence momentum profiles of HF, HC1 and C2H4. In addition the momentum profilesfor the highest occupied molecular orbitals have been studied for two series: C2H3F,CH2F,C2HF3 and C2F4 and (CH3)2C0, CH3HO and H2CO. The experimental resuits have been compared with a range of calculations performed both at The Universityof British Columbia and in collaboration with Professor E.R. Davidson and co-workersat the Department of Chemistry at Indiana University.The generally good agreement of the largest SCF and many-body (MRSD-CI) calculations with the experimental results provides additional confirmation of the EMS interpretation using the plane wave impulse approximation for the high-energy, symmetricnon-coplanar scattering conditions used in the present work. However, some discrepancies remain, particularly for the intensity of the inner valence 2u XMP of HF andthe shape of the 2b momentum profile of H2CO, neither of which are are completelydescribed by the best current levels of theory.The studies of HF and HC1 have completed the series of EMS measurements for theGroup IVA through VITA hydrides of rows 2 and 3. These experimental and theoreticalstudies have produced a consistent set of results for these fundamental systems. In the168Chapter 9. Concluding Remarks 169present work, some errors have been revealed in earlier reported measurements [61] forthe 3u and 5o- momentum profiles of HF and HC1 respectively. For HF, the discrepanciesreported earlier [62] between experiment and large MRSD-CI calculations for the lowestlying lTr momentum profile are shown to be largely eliminated for the present data whenthe more accurate GW-PG momentum resolution folding procedures [81] are used. Theresults also indicate that, as in the case of the lowest energy XMP’s of 1120 and NH3, theobserved hr electron momentum profile of HF is significantly more compact in momentumspace than is predicted by a Hartree-Fock orbital, even when using exact (i.e. numericalHartree-Fock) SCF wavefunctions. Although the use of a full ion-neutral overlap Dysonorbital, which includes electron correlation and relaxation effects using large MRSD-CIwavefunctions, results in greatly improved agreement with experiment, small differenceswith the EMS results remain for the shape of the 17r momentum profile. Further many-body studies of this system including even larger CI or density functional methods wouldbe useful to resolve these differences. For HC1, the theoretical momentum profiles for thevalence region show good quantitative agreement with the experimental data for all caseswhen near Hartree-Fock limit quality wavefunctions are used. In contrast to the situationfound for HF, electron correlation and relaxation effects have very little influence on thepredicted electron momentum profiles even though they provide significant improvementsin the calculated dipole moment and total energy. In this respect HC1 exhibits similarbehavior to that found earlier for the momentum profiles of the other third row hydridesPH3 [32] and H2S [31] and the isoelectronic noble gas argon [52].As mentioned above, these studies of HF and HC1 complete a decade-long study bya series of students in this laboratory [77,84,85, 170] of the group IVA—VIIA hydridesof rows 2 and 3 and the corresponding group VIllA noble gas elements. These studieshave served several purposes. First, the overall good agreement obtained for all of theChapter 9. Concluding Remarks 170outer valence orbitals validates the use of the simple and easily understood plane waveimpulse approximation to interpret the EMS results. Secondly, particularly in the caseof the highly polar row 2 hydrides, NH3 1120 and HF, large improvements in previouslyexisting quantum chemical theory have been made, in large part due to the additionalconstraint provided by the EMS momentum profile [58]. In particular this has resultedin the development of new highly accurate “universal” wavefunctions at both the SCFand CI levels as well as correspondingly improved calculated electronic properties for thebenchmark hydride series [58]. In particular the importance of highly saturated basissets including very diffuse functions has been demonstrated [37,58]. Thirdly, the EMSmeasurements have led to a much improved understanding of the ionization manifoldsof these hydrides, particularly in the energy region above the He I source limit at20 eV. Finally, the hydrides work has also led to a greater understanding of the electronicstructure of these systems, including the chemical bonding.As was the case for HC1 and as also seen earlier for acetylene [64], generally goodagreement for ethylene has been found between the EMS momentum profile results andlarge, highly converged SCF calculations. Good agreement for intensity for the innervalence 2a9 momentum profile was attained with the assignment of all of the observedbinding energy intensity above 21 eV as being due to (2a9)’ ionization processes. Inaddition, the pole positions and strengths of the 196-GTO based MRSD-CI calculationdeveloped in the present collaborative work were found to provide a very good modelof the valence ionization spectrum. The effects of correlation and relaxation were foundto be negligible for the momentum profiles, even when using very large configurationspaces, although there were large improvements for other properties. In addition, vibrational averaging was not found to significantly change the shape of the 1b3 HOMOmomentum profile. Comparison with very recently obtained preliminary multi-channelChapter 9. Concluding Remarks 171EMS measurements confirms the present result, and underscores the limitations imposedby the low coincidence count rates attainable with the single-channel instrument used inthis thesis.Good reproduction of lowest binding energy ir-system momentum profiles of the seriesC2H4,C2H3F, CH2F,C2HF3and C2F4 was found when using near Hartree-Fock limitSCF calculations. The shift to higher momentum observed in the profiles with increasingfluoro-substitution is attributable to increasing localization of the electron density onthe fluorine centres. The fluorine group was found to be both net electron withdrawingand C-F ir anti-bonding when substituted for hydrogen in the ethylene ir-system. Theelectron-withdrawing nature of this group is also responsible for the weak bond observedin C2F4.Good agreement is attained at low momentum for the outermost orbitals of acetaldehyde and acetone but considerable differences are found between the theoretical andexperimental momentum profiles above 1.5 au. Large discrepancies also remain between the 2b XMP of formaldehyde and the results of the near Hartree-Fock limit SCFand MRSD-CI calculations developed in the collaboration in this work and in the study ofBawagan et al. [34]. Consideration of larger, particularly more diffuse and polarizable basis sets for SCF models should improve the agreement at larger momentum for (CH3)2C0and CH3HO and also give even better values for the dipole moments and other properties. While inclusion of correlation and relaxation effects using a multi-reference spacesingles and doubles configuration interaction calculation does greatly improve the totalenergy obtained for all three molecules, the calculated dipole moment is only in moderately good agreement with the experimental values although the effects of vibrationalmotion should also be considered. In addition the inclusion of many-body effects doesnot greatly improve the agreement with the observed momentum profiles. The reasons asChapter 9. Concluding Remarks 172The reasons as to why the measured electron momentum profile corresponding to theremoval of the 2b electron of formaldehyde is still in significant disagreement with allcalculations remain unclear.The range of experimental and theoretical studies presented in this thesis allow forsome general conclusions to be made concerning the usefulness of the EMS techniqueand about the current theory of electronic structure. Clearly the momentum profilesobtained experimentally by EMS provide a complementary check of the normal energyminimization procedures. With the consideration of other properties such as dipoleand/or quadrupo].e moments together with the EMS momentum profiles as well as thetotal energy, a quantitative experimental evaluation can be made of any wavefunction.Although differences between the EMS momentum profiles and theoretical results canoften be made up by the inclusion of sufficiently diffuse functions in a basis set, parallelimprovements for other experimental quantities must be achieved also, particularly forsmaller basis sets, to avoid the basis set “tuning” such as that seen for the 4-31G+sp+sbasis sets in chapter 8. The results presented in this thesis also show that in most cases,the independent electron model used by Hartree-Fock methods works generally quite wellfor sufficiently large and diffuse basis sets. The important exceptions to this overall ruleare those highly polar molecules which contain the atoms of nitrogen, oxygen and fluorine(see chapter 4 for HF, chapter 8 for CH3HO and (CH3)2C0).The use of larger basis sets is becoming ever more practical with the rapid increasesin computing resources. For example, at the beginning of the work presented in thisthesis 4-31G basis sets were a very commonly used basis set while at present, severalnew very much larger basis sets such as those recently developed by Dunning and coworders [135, 148, 149] are coming into routine use. Other calculation techniques, such asChapter 9. Concluding Remarks 173density functional theory, also appear to be very promising. Electron momentum spectroscopy, used in conjunction with other experimental quantities, will continue to be a veryuseful tool for the evaluation and development of new higher-quality molecular wavefunctions and theoretical methods. However, it is clear that the higher sensitivity achieved bymulti-channel instruments, will be necessary for the study of larger molecules with manyelectrons. The limitations of single-channel EMS spectrometers can be seen particularlyin the present studies of C2F4 and (CH3)2C0, both of which required extremely longcollection times (two months each) for even the modest data statistics reported here.With the further development of new energy dispersive [138] and angular-dispersive [171]EMS spectrometers, much higher coincidence count rates, and thus better energy resolution and statistics will permit the investigation of electronic structure in large moleculesand other chemically interesting species including free radicals and molecules in excitedstates. These investigations, using the unique capabilities of EMS as a probe of electron motion will provide new insights and further understanding of electronic structure,chemical bonding and reactivity.Appendix ASignal Conditioning ElectronicsThe following diagram (figure A.1) is a schematic of the sample and hold daughter boardthat was constructed to match the pulse lengths of the Ortec 457 time-to-amplitude converter output voltage level (TPHC OUT) and conversion initiation signal (TRUE STOP)to the pulse lengths required by the inputs of the Scientific Solutions Labmaster DMAanalog-to-digital converter computer board. The second diagram (figure A.2) illustratesthe wiring between the Labmaster DMA connectors and the sample and hold daughterboard. For a description of the function of this unit, please refer to section 3.2.174-‘>1 0qwI-.1.916.. Ia.91 n2,IIJNRYIIFOR’OR.IRIdM1ORCRDFILEsRNLGCONI15CTNO.IDATE’ENILTIT.CHECKED.1igI-1DEC93I1KCh..I.,r9O.p.rt..nt®IELECTRONICTITLESANALoGCONVERSIoNUNITENGINEERINGUnI.r.Iti.•E.C.C)’LI •Oq ‘1,. 09 0 , 0COUNTS.SCIENTIFICSOLUTIONS5$POSITIONTERMINALBLOCK930533135SNONOTE;THEFOLLOWINGARCSHORTEDTOGETHERON9305329-lI11—1213—14IS—Il17-lB19—2121—2223—2425-2027-2129—3131—3233.3435—3637—30> o. C.) o. aSCIENTIFICSOLUTIONS4$POSITIONTERMINALBLOCK936532ANALOGGND.4:.S/HOUTSCIENTIFICSOLUTIONSGAUGHTERBOARD5111221P•Ij’_l:_j4LJa’NOTE;THESILESCREENONTHEDAUGHTERBOARDFORP1.P2ANDPSISINCORRECT.®JELECTRONICENGINEERINGICh..I.*ruD.p.rt..ntIUn.ar.ItuofB.C.iiTITLE;ANALOGCONVERSIONUNITIFOR’DR.ORIONORCTDF1IFItTNO.IDATE;IBUILTBY;CHECKED;IPAGENO.I•,.JnrroII2flF2I-4Bibliography[1] J.D. Goddard and 1.0. Csizmadia. J. Chem. Phys. 68 (1978) 2172.[2] A.E. Glassgold and G. lalongo. Phys. Rev. 175 (1968) 151.[3] Yu.F. Smirnov and V.G. Neudachin. Soviet Phys. JETP Letters 3 (1966) 192;V.0. Neudachin, G.A. Novoskol’tseva and Yu. F. 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