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High resolution spectroscopy of the Vanadium oxide B⁴ [pi] -X⁴ [sigma] ⁻ (0,0) Band Berno, Bob 1992

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HIGH RESOLUTION SPECTROSCOPY OF THEVANADIUM OXIDE B 4Π—X4 E- (0,0) BANDByBOB BERNOB. Sc. (Chemistry) University of Waterloo, 1989a thesis submitted in partial fulfillment ofthe requirements for the degree ofMaster of Scienceinthe Faculty of Graduate StudiesDepartment of ChemistryWe accept this thesis as conformingto the required standardThe University of British ColumbiaDecember Π992© BOB BERNO, Π992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(SignatureDepartment of ^C-:\i•e-kAAA,‘:,-'v-i The University of British ColumbiaVancouver, CanadaDate ^ \14"DE-6 (2/88)AbstractThe B4 11—X4 E- (0,0) band of VO has been recorded at sub-Doppler resolution byintermodulated fluorescence spectroscopy. Spectral linewidths of 60 MHz were typical;which enabled the hyperfine structure due to the 51 V nucleus (I = I) to be resolvedfor most of the observed branches. The hyperfine structure of the B 4Il state is narrowexcept where it is heavily perturbed by the v = 2 level of the a 2 E+ state. The electronconfiguration of the a 2 E+ state was determined to be (4s0) 1 (3d6) 2 because of the largeFermi contact interaction which arose from an unpaired electron having primarily metal4s atomic orbital character.The transition frequencies were fit to a model which included the rotational, fine andhyperfine structure of the BSI , X4 E - and a 2 E+ states. The B 4 11 /a 2 E+ interactionrequired the inclusion of an effective higher order spin-orbit parameter as well as a newhyperfine parameter, denoted by parameter e. The new hyperfine parameter is requiredto describe the hyperfine interactions between 411 and 2 E+ states.The fit included 3211 data points and gave an r.m.s. error of 0.00038 cm -1 .iiTable of ContentsAbstract^ iiList of Tables viList of Figures^ viiAcknowledgements^ ixDedication^ x1 Introduction 12 Experimental 62.1 Introduction ^ 62.2 The Calibration System ^ 102.3 Saturation Spectroscopy 122.4 Lamb Dips and Intermodulated Fluorescence ^ 162.5 Wavelength Resolved Fluorescence Spectroscopy 203 Energy Expressions and the Hamiltonian 243.1 Introduction ^ 243.2 Perturbations 253.3 Hund's Coupling Cases ^ 283.3.1^Case (ap) 283.3.2^Case (bpi) ^ 30iii3.4 Hamiltonian Matrix Elements for the X 4 E- state ^ 313.4.1 Rotational Structure  ^333.4.2 Fine Structure ^333.4.3 Magnetic Hyperfine Hamiltonian ^  373.4.4 The Electric Quadrupole Interaction  383.5 The Hamiltonian for the 4H  upper state ^  393.5.1 Rotational and Fine Structure  393.5.2 A-type Doubling ^  403.5.3 Magnetic Hyperfine Interactions ^  413.5.4 Electric Quadrupole Interaction  433.6 The Hamiltonian for the a2E+ State ^  433.7 The 2 E+ - 411 Matrix Elements  444 Analysis of the Spectra^ 474.1 Introduction  474.2 The Ground State of VO ^  504.2.1 The Spin-spin and Spin-rotation Interactions ^ 504.2.2 The Hyperfine Splitting in the Ground State  524.3 The B4II State ^  584.3.1 The Spin-orbit Splitting of the B 4I1 State ^584.3.2 The A-type Doubling in the B4Il State ^  604.4 The Interaction Between the B 4II and a2E+ States  605 Results^ 676 Discussion^ 706.1 Rotational Structure  ^70iv6.2 Electron Configurations  ^716.3 The Molecular Spin-Orbit Parameters ^  736.4 The B411 /a 2E+ Perturbation  777 Conclusions^ 80Bibliography 82A The Line Assignments of the VO B 4II—X4 E- (0,0) Band.^85List of Tables5.1 The constants for the X 4 E- (v = 0) state of VO. ^ 675.2 The constants for the a 2 E+ (v = 2) state of VO 685.3 The constants for the B4 1-1 (v = 0) state of VO. ^ 696.1 The rotational constants and average bond lengths of the states of the VOB4H—X 4 E- transition. ^ 716.2 Table of the equilibrium rotational constants from the B 411—X4 E-transition^ 726.3 Table of the four sub-band origins of the B4 1-1 state^ 746.4 The band origin and higher spin-orbit parameters of the B4 1-1 state. 756.5 Calculations of the anharmonic oscillator overlap integrals^ 79viList of Figures1.1 The molecular orbital diagram for VO.  ^21.2 Selected electronic states of VO ^42.1 Schematic diagram of the experimental apparatus for IMF spectroscopy ^ 72.2 A schematic diagram of the calibration system. ^  102.3 Plots of the velocity distributions of the E2 (a); and Ei. (b) levels when;an intense laser beam with vector , ^frequency w passes through thesample cell  ^142.4 Plots of the saturated (ces (w)) and unsaturated (ao (w)) absorption coeffi-cients as a function of excitation frequency (w)  ^152.5 a) Illustration of the effects of a saturating standing wave on the velocitydistribution of molecules in the sample cell; b) the corresponding absorp-tion coefficient for the sample  172.6 Plot of the absorption coefficient for two closely spaced transitions. Theunresolved line is shown in a), and the two Lamb dips representing thetwo line positions are shown in b).   192.7 An example of possible relaxations in a simple system after excitation ofa Q-branch line a), and a P-branch line b)^  223.1 Illustration of an avoided crossing between two states.  ^263.2 Hund's case (ap) coupling scheme.  ^293.3 Hund's case (b0j ) coupling scheme^  323.4 The hyperfine Hamiltonian Matrix for 411 states interacting with 2 E+ states. 46vii4.1 The head of the SQ31 branch; illustrating the density of the B 411—X4 E- (0,0)band structure. ^  484.2 Fortrat diagrams showing transitions involving (a) the e-parity compo-nents, and (b) the f-parity components of the F 1 upper spin state  494.3 The energies of the four electron spin components of the X 4 E- state of VO. 514.4 The hyperfine energy level splittings for the F2 and F3 spin states of theX4 E- state of VO. ^  534.5 (a) The sQ31 (9) and (b) the R4 (7) lines showing how the hyperfine struc-tures are mirrored. ^  554.6 Plot of the sR32 (14) line including induced lines from the internal hyperfineperturbation. ^  574.7 The upper state electronic term energies as a function of (J W^594.8 Plot of the A-type splittings of the four spin states of /3 411  ^614.9 The hyperfine energy levels of the B 411_ i f and a2E+ states.  ^634.10 The hyperfine energy levels of the B 411_i e and a 2E,+ states.  ^644.11 The hyperfine widths of (a) the Q i branch, and (b) the °P12 branch. .^65viiiAcknowledgementsAs I put the finishing touches on my masters thesis, I am reminded of the many peoplewho were involved with this work. These people are too numerous to be mentioned here,however I wish to express genuine gratitude to those who made appreciable contribu-tions. Most notable of these was Dr Anthony Merer, whose tremendous enthusiasm wasinspirational and his quantum mechanical expertise and superior spectroscopic insightwere invaluable.I would also like to thank: Allan Adam, Bob Bower, and Photos G. Hajigeorgiou,who all at one time or another helped me collect the spectra which appear in this thesis;Mark Barnes, who helped program the matrix elements, despite suffering from 8 amdyslexia; Chris Chan, whose electronics wizardry and fantastic efficiency keeps the labrunning; and the members of the Gourmet Club, for smoothing over those occasionalrough periods when things in the lab were not working as planned.To the rest of the people who helped make my stay at UBC highly rewarding andenjoyable, I have not forgotten you. I thank you all and look forward to seeing you andworking with you all again.ixI would like to dedicate this to my parents. They continue to love me andsupport me, despite not understanding what I have been doing these pastseveral years.xChapter 1IntroductionThe nuclei in the neighbourhood of iron in the periodic table are the most stable ofall; therefore the elements of the 3d transition series, which surround iron, have veryhigh nuclear stabilities, and are among the most abundant elements in the Universe,not counting hydrogen and helium. The processes that synthesize these elements takeplace when a comparatively heavy star runs out of its hydrogen "fuel", and explodes asa supernova. Heavy stars of this type live short but brilliant lives, and in their violentdeath throes generate immense quantities of the transition elements, which are blownout into space. Later generations of stars can condense from interstellar gas clouds thatcontain such "recycled" material, and are described as "metal-rich" in the jargon of theastrophysicists. Our Sun is a star in this class, as are many of the stars in its immediatevicinity.The optical spectra of the cooler stars containing recycled supernova material aredominated by band systems of the 3d transition metal monoxides, for various reasons.First, oxygen is also one of the more abundant elements and, second, the 3d oxides haveparticularly high dissociation energies, so that they can survive in the relatively hightemperature environments of the stellar atmospheres [1]; most importantly, the oxideshave prominent electronic band systems throughout the visible and near infra-red regionsof the spectrum.Much of the astrophysics of cool stars is merely high-temperature laboratory chem-istry applied to astronomical objects [1], so that the spectroscopy of the 3d monoxides is190Cr►►►►►►►►Chapter 1. Introduction^ 2100*Oxygen►►► 2p37rFigure 1.1: The molecular orbital diagram for VO.an important topic in this field. Of very considerable importance is VO which, after TiO,is the second most abundant molecule found in the spectra of the cool M-type stars [2].Bands in the near infra-red region, near 1.06 p, were in fact attributed to VO by Kuiperet al [3], some time before laboratory work by Lagerqvist et al [4] was able to confirmthe assignment. These observations have generated a strong interest in the VO moleculewhich continues to the present day.The first reasonably detailed theoretical calculations on VO [5] predicted that theground state is 4 E - , from the valence electron configuration (9(7)(1(5) 2 , with the 2 .6, statefrom the configuration (9(7) 2 (18) lying only a very small amount above (see Figure 1.1).The 4 E - nature of the ground state was later confirmed experimentally from electronspin resonance spectra of VO isolated in an argon matrix [6].Chapter 1. Introduction^ 3In their early gas-phase studies of the electronic spectrum of VO, Richards and Bar-row [7] observed large hyperfine splittings in the ground state. Further investigationsof the hyperfine structure led to their discovery of "internal hyperfine perturbations" inthe ground state [8]. Internal hyperfine perturbations also occur in the C 4E- excitedstate [9]: Cheung and coworkers [10] obtained sub-Doppler resolution spectra of theC-X transition using the saturation technique known as intermodulated fluorescencespectroscopy, and were able to characterize the perturbations in the C 4 E- and X4E -states in detail. More recently the C 4 E- state has been used as the intermediate in apulsed field ionization study of VO that led to an accurate determination of its ionizationpotential [11].Another strong system of absorption bands is observed in the spectra of cool M-type stars in the 0.74 - 0.83 y region. Although this system had been tentatively as-signed to VO [2], it was not conclusively identified as belonging to VO until Keenan andSchroeder [12] were able to obtain emission spectra (from an electric arc containing V 2 05powder) which matched the astronomical data. This band system is now recognized asbeing the B 4 1-1—X4 E- electronic transition of VO.The present thesis is concerned with the rotational and hyperfine structure of the (0,0)band of the B4 11—X 4 E- transition of VO. The system had been recorded previously inemission at Doppler-limited resolution by Cheung and coworkers [13], and shown tocontain many intense branches with varying hyperfine line widths. This variation inthe line widths results mostly from the huge hyperfine effects in the ground state, sincethe upper state turns out to have comparatively narrow hyperfine structure. Rotationalassignments could be made for the branches where the hyperfine widths in the upperand lower states happen to cancel and produce comparatively sharp rotational lines, butmany of the branches are hyperfine-broadened to the extent that they are not identifiablein the emission spectra. A further complication is that the B 411 upper state suffers large7E010000 -0)a)cw0Chapter 1. Introduction^ 420000 -c4r 2 TrMIN ,I■ 01•■■ ■B41T052 620* 82 71.Electron ConfigurationFigure 1.2: Selected electronic states of VO.Chapter 1. Introduction^ 5rotational perturbations caused by spin-orbit interaction with the 2 E+ state that comesfrom the same electron configuration as the ground state. Although some of the detailsof the perturbations could be worked out from the Doppler-limited spectra [13], a fullaccount has required that spectra of the transition be obtained at sub-Doppler resolution.It has not been possible to obtain such spectra until quite recently, because of the lackof a suitably intense tunable laser source in the 0.8 a region. With the development, inthe past three years, of commercial continuous-wave Ti:sapphire ring lasers, the 0.8 pregion has become more easily accessible for high resolution studies. This thesis reportsa full sub-Doppler analysis of the (0,0) band of the B41I—X 4 E - system of VO near 0.8 p,including a detailed treatment of the hyperfine effects, and a very complete account ofthe rotational perturbations caused by the a 2 E+ state.Chapter 2Experimental2.1 IntroductionAlthough VO has been found to be quite stable in the atmospheres of the cooler M-typestars[2], it is not found as a stable diatomic molecule in normal terrestrial environments.Therefore, to carry out a high resolution spectroscopic study, VO had to be producedunder non-equilibrium conditions from a stable precursor.To that end, VO was produced in an electrodeless microwave discharge operating at2450 MHz through a flowing mixture of VOC13 and an appropriate carrier gas. Argonwas mostly used as the carrier gas, at a total pressure of approximately one torr. Later,when increased sensitivity was needed to record the high J lines at longer wavelengths(800 nm), a higher pressure of carrier gas was necessary to increase the populations of thehigher rotational levels. However, the argon emission lines in the region being detectedbecame much more intense at this higher pressure, thus increasing the background noise.As a result there was no improvement in the signal-to-noise ratio. Consequently, theargon was replaced by helium, since there are almost no emission lines of helium in thisregion. The shorter wavelength helium emission lines from the discharge could be filteredout before reaching the photo-multiplier tube (PMT), so even though the experiment wasrun at higher pressure than the argon experiments, the background noise was reduced.This improvement in sensitivity came at the cost of reduced resolution because of pressurebroadening.6V^•© 6,®aV^VBURLEIGH^STABILIZEDWAVEMETER 750 MHzETALONSHV® VChapter 2. Experimental 70 Ar ION PUMP LASER^® VACUUM CHAMBER0 Ti:SAPPHIRE LASER^® PHOTOMULTIPLIER^TUBE0 He-Ne LASER^® LOCK-IN AMPLIFIER® 50:50 BEAMSPLITTER^0 U HOLLOW CATHODE LAMPC MECHANICAL CHOPPER^e CHART RECORDER© PHOTODIODE^© p-VAX COMPUTERFigure 2.1: Schematic diagram of the experimental apparatus for IMF spectroscopy.Chapter 2. Experimental^ 8The Doppler linewidth for room temperature VO at around 12 500 cm -1 is approx-imately 600 MHz (0.02 cm -1 ). The hyperfine splittings arising from the 51 V nucleus(nuclear spin, I = and the complexities of the rotational structure lead to considerableblending around the bandheads and at the perturbations, which causes much of the detailto be lost because of blending at Doppler-limited resolution. Consequently, sub-Dopplerspectra of VO had to be recorded. These sub-Doppler spectra were recorded using thetechnique of intermodulated fluorescence (IMF) spectroscopy [15].A schematic illustration of the experimental set-up is shown in Figure 2.1. A CoherentInc. Model 1-20 continuous wave argon ion laser was used to pump a tunable Ti:sapphirelaser (Coherent Inc. Model 899-21) in the region from 12 390 cm - ' to 12 740 cm -1 .A portion of the beam from the Ti:sapphire laser was split off by a beamsplitter, andwas sent to the calibration system which determined the absolute frequency of the light.The calibration system will be discussed in greater detail in the next Section.The remaining laser light was passed through a 50/50 beamsplitter, producing twoequal intensity coherent beams. These beams were oriented so that they passed throughthe sample cell exactly antiparallel to each other as shown in Figure 2.1. To stop thelaser beams from feeding back into the Ti:sapphire laser, an optical diode was insertedjust prior to the 50/50 beamsplitter.Before reaching the sample cell, these two portions of the laser beam passed througha mechanical chopper with three rings of holes punched in it. One ring consisted of 28holes, the second ring consisted of 36 holes and the third of 64 holes. When one portion ofthe laser beam (I,) passes through the first ring of holes, it will be chopped twenty-eighttimes for every revolution of the chopper. The frequency that the first laser is choppedat is thus:Fl = 28FehopChapter 2. Experimental^ 9where Fchop is the frequency of the chopper in revolutions per second. Similarly, thechopping frequency for the second arm, when passed through the second ring of holes, isF2 = 36Fchop .A HeNe laser beam is passed through the third ring of holes, and is thus chopped at:Frei = 64Fchop= F1 + F2Fchop was typically between 20 to 25 revolutions per second. The laser power in eacharm was of the order of 100 mW, dropping to 50 mW/arm at longer wavelengths. Ulti-mately, the drop in laser power at these longer wavelengths coupled with the sharp dropin the quantum efficiency of the photo-multiplier tube (Hamamatsu Model R928) didnot permit spectra in the region from 12 390 cm -1 to 12 450 cm -1 to be recorded usingthe IMF technique. This spectral region was covered at Doppler limited resolution only.The linewidths of most of the IMF spectra were found to be better than 60 MHz. How-ever, when the increased pressure of helium was needed to improve sensitivity, pressurebroadening from the helium carrier gas caused the line width to increase to 100 MHz.The perturbation of the B4 II state by the a 2 E+ state produced extremely complicatedspectra at the avoided crossings. Not only are the positions of the lines shifted, but theintensities of the already weak lines are further reduced since some of the intensity istransfered to extra lines arising from the a 2 E+ state. Assignments of these lines nearthe avoided crossings were made using the technique of wavelength resolved fluorescence(WRF) spectroscopy. This will be described in further detail in Section 2.5.Chapter 2. Experimental^ 10Wavemeterstabilized 750 MHz etalonpot lens iris meterpol.HeNe laser^irispol.automaticto computerlock-inopto-isolatorto computer e 1 I \passive 150 MHz etalon+4) O^level control^HV^H 0^ I0amplifierto^ )oscilloscope^ramped 750 MHz etalonring laserto expt.function generatorFigure 2.2: A schematic diagram of the calibration system.2.2 The Calibration SystemIn the past, laser excitation data were calibrated against a reference fluorescence spec-trum of 12 or Te2 [16]. With the advent of high resolution techniques such as laser inducedfluorescence (LIF) molecular beam experiments and IMF spectroscopy, more accuratemethods of calibration became necessary because the characteristic uncertainties in theiodine line positions were larger than the uncertainty in the spectra being calibrated.The spectra of the B4 11—X 4 E - (0,0) band of VO were calibrated using the system illus-trated schematically in Figure 2.2. The key component in this system is the evacuated,temperature and pressure stabilized Fabry-Perot etalon. The cavity length of the etalonis accurately fixed by a piezoelectric driver servolocked to one particular interferencefringe of a polarization-stabilized HeNe laser line, so that the relative frequencies of theinterference fringes (also referred to as markers) are well known.patChapter 2. Experimental^ 11The piezoelectric driver mentioned earlier carries one of the confocal mirrors of the750 MHz etalon. A modulating voltage is applied to the piezo in such a way that whenthe 632.8 nm line from a stabilized HeNe laser enters the interference cavity, the positionof the mirror is locked so that the frequency of the HeNe line is at the maximum of oneparticular fringe. The free spectral range (FSR) of the cavity is thus invariant to changesin room temperature or atmospheric pressure. When the absolute frequency of one fringeis known, and the order number of the other fringes with respect to that one fringe isknown, then the frequency of any marker can be calculated using the expression[16]= nwo (2.1)nowhere n o and wo refer to the order number and frequency of the marker whose frequencyis known, while 77 and w are the known order number and unknown frequency of theother marker.Since the frequency of the HeNe line used to lock the etalon is well known, it ,ispossible to use its frequency as the standard. In practice, this poses a problem in regionsfar from the 632.8 nm HeNe line because the reflectivity of the etalon mirrors, and hencethe fringe spacing, shows a slight wavelength dependence. Therefore, a Burleigh modelWA-20VIS wavemeter was used to identify the markers by giving their frequencies to±0.02 cm -1 . To obtain highly accurate frequency determinations, opto-galvanic spectrafrom a uranium:neon hollow cathode lamp were recorded along with the VO spectra.The uranium line positions were taken from the uranium emission atlas[17], and thefrequencies of the 750 MHz etalon fringes could thus be determined by a least sqares fit.In the 800 nm region of the spectrum, the FSR of the stabilized etalon was found to be0.025046 cm -1 (750.859 MHz).Chapter 2. Experimental^ 122.3 Saturation SpectroscopyFreshman chemistry textbooks teach students that the absorption of radiation by thesample molecules follows the Beer-Lambert Law:If^he -`c1^(2.2)where If and I, are the final and initial light intensities, e is the extinction coefficients,C is the sample concentration in moles per litre and 1 is the path length in centimetersthrough the sample cell for the radiation. e is a constant of the system, and thus theconcentration of the sample molecules in the cell can be calculated from the ratio of theinitial and final radiation intensities I a and If respectively as:In IfII, elThe extinction coeficient (E) has a wavelength dependence defined by the transitionenergy between the states in question. In accurate spectroscopic studies, e is replacedby the absorption coefficient (cr0 ), which for a sample experiencing a weak oscillatingelectric field of frequency w and direction T, such that k is parallel to the z axis, can beexpressed as[19]ao(w) — 120.00N0 °(vz /vP) 2 dv,4 \Fr vp L. (w - wo — k • uz ) 2 — (-y/2) 2 (2.3)In this equation vp V2kBT Im, cro is the absorption cross-section, -y is the sum ofthe radiative and nonradiative decay constants, AN D is the difference in number densitybetween the upper state (N2 ) and the lower state (N1 ) (i. e. AN0 = N2 - N1 ), and k isthe magnitude of k in the z direction.• The Beer-Lambert Law works well when the population density of the upper state(N2 ) is considerably less the the population density in the lower state (N 1 ). However,1 Some textbooks use a for the extinction coeficient instead of E.Chapter 2. Experimental^ 13if the incident electric field intensity is increased to the point where the lower stateis depopulated at an appreciably faster rate than the rate of relaxation from the upperstate, then the Beer-Lambert Law breaks down, and the transition is said to be saturated.Under these conditions, ao becomes dependent on the incident electric field intensity.Saturation is commonly observed when lasers are used as excitation light sourcesbecause of the high light intensities typically generated. To illustrate the effects of satu-ration, consider a sample cell containing molecules with a thermal velocity distribution.'When monochromatic laser light with frequency w and vector k passes through the cell,there will be a depletion of the population density of molecules in the absorbing state atenergy E1 if their velocity components are defined by:w - (6 + Ai)) = w12 (2.4)where w12 = (E2 — El )/hi and 6w is the linewidth of the laser. If k is once again chosento be parallel to the z direction, then Equation 2.4 becomes:w — k • (v, Avz) wi2 6w (2.5)The velocity distributions of molecules in the E1 and E2 levels resulting from this intenselaser light are shown in Figure 2.3. The dip in the n i (vz ) population distribution isknown as a Bennett Hole[18]. The spectral width -y, of the Bennett hole is related to -y(the sum of the radiative and nonradiative decay constants) by= + Sowhere So is the value of the saturation parameter at the transition frequency wo [19]. Atoptical wavelengths, -ys is much narrower than the Doppler profile. However, this sub-Doppler depletion in the population distribution of the El state cannot be observed bysimply passing a single saturating laser through the sample cell. Tuning the frequency ofChapter 2. Experimental^ 14N (v z )V z=0 v2(w)V zFigure 2.3: Plots of the velocity distributions of the E2 (a); and E1 (b) levels when anintense laser beam with vector ic and frequency w passes through the sample cellChapter 2. Experimental^ 15-w041111---Figure 2.4: Plots of the saturated (a s (w)) and unsaturated (ao (w)) absorption coefficientsas a function of excitation frequency (w)the monochromatic laser will simply move the Bennett hole to another part of the velocitydistribution such that the observed spectral line would follow the dotted line shown inFigure 2.4. The expression for the absorption coefficient for molecules experiencing asaturating radiation field is given by[19]:7 2 0.0AN0 too^e-(v./vP)2 dv,a s (w) =  (2.6)40T-v7,^(w — wo — k • v z ) 2 — (-4/2) 2This expression closely resembles the expression in the weak field approximation (Equa-tion 2.3). Evaluation of the integral, with the assumption that -y, is much less than theDoppler width givesas(w) = ao(w)( 1 So)-112 .^ (2.7)Clearly, if the saturation parameter is small, then the absorption coefficient approaches^Chapter 2. Experimental^ 16the value obtained in the weak field approximation.2.4 Lamb Dips and Intermodulated FluorescenceIn order to probe the Bennett hole in the velocity distribution profile, a second radiationsource is required. One way to introduce a second radiation source would be simplyto reflect the laser beam back through the sample cell antiparallel to the incident laserbeam. Under these conditions the total electric field experienced by the molecules in thecell can be expressed as the sum of two oscillating electric fields:E^+Eoe — i(wt-l-kz)^Eo e —i(wt— kz)Eo cos (cot + kz) Eo cos (wt — kz)2E0 cos wt cos kz^ (2.8)The result of having two E fields interacting with the ensemble of molecules in the cell isthe production of two holes at v z = +(coo — co)/k . When w = coo the two holes convergeto one hole in the population distribution of twice the depth. The change in populationdue to the saturating standing wave radiation can be expressed as[19]:^An z (vz ) = Ano (vz ) [1—^ (7/2)2S0 (7/2) 2 S0 (2.9)When w = coo this expression reduces to2An s (vz ) = Ano (vz ) [1 — 2So (11^(2.10)7sThe effect of a saturating standing wave on the change in population of the lower state(An(vz )) is illustrated in Figure 2.5(a), where the dotted line represents the case wherew = coo and the solid line represents w > wo +(wo — w — kvz ) 2 + (75 /2) 2 (wo —^kv,)2 + (75/2) 2as( )vz=0Chapter 2. Experimental^ 17=Figure 2.5: a) Illustration of the effects of a saturating standing wave on the velocitydistribution of molecules in the sample cell; b) the corresponding absorption coefficientfor the sample.Chapter 2. Experimental^ 18The expression for the absorption coefficient for this standing wave experiment,as (u) = ao [1 —^(1 + (u) w(:)321+2)2(78 /2)2 )] '^(2.11)shows that, when the laser is tuned off resonance, a(w) a o (wo ) (1 — ja) , but, whenthe laser is tuned to w = wo, a(w) = ao(wo) (1 — So) . Hence, setting up a standingwave in the sample cell produces a dip in the absorption curve of spectral width -y, atthe transition frequency w o . At optical wavelengths, 7, is usually much less than theDoppler width and thus the line position of the transition, seen as a Lamb dip, can nowbe measured with much higher precision.In addition to improved precision, this technique also affords the ability to resolvelines that were blended at Doppler limited resolution. This capability is illustrated inFigure 2.6, which represents two transitions so close to each other that their Dopplerbroadened line profiles would be completely blended. The two tiny dips on either side ofthe Doppler profile represent the two line positions.Lamb dip spectroscopy has limited usefulness when the spectra become very dense.Under these conditions the Lamb dips can become lost in the mass of Doppler limitedline profiles.One way to avoid this problem is through the use of intermodulated fluorescencespectroscopy (IMF)(see Section 2.1). IMF is a very sensitive saturation technique foreliminating the residual Doppler profile. Like the Lamb dip set-up for absorption experi-ments, an IMF experiment requires two counter-propagating beams of radiation throughthe sample cell. However, IMF differs in that instead of simply reflecting the radiationback through the sample cell (thus setting up a standing wave), the incident wave is firstsplit into two equal intensity components, h and 12. The two beams are chopped at twodifferent frequencies, F 1 and F2 respectively, so the intensities of the laser beams enteringr n N /1\^. ^ N1^1^. \/ \ I^\(.02//co, w20)cot CL) 2Blended line— — Doppler limited w 1line profile— — — Doppler limited w 2line profile/A^1,/ \II/ I*ft ••••• •••■• •=10.^••■••Chapter 2. Experimental^ 19Figure 2.6: Plot of the absorption coefficient for two closely spaced transitions. Theunresolved line is shown in a), and the two Lamb dips representing the two line positionsare shown in b).Chapter 2. Experimental^ 20the sample cell are given by:Il = 2 (1 + cos 27rFi t)and12 = 2 (1 -I- cos 27rF2t)neglecting higher order terms. The intensity of the fluorescence emitted by the moleculesexperiencing these two counter-propagating laser beams is found to be[1 9]:OC^+ 12)which reduces to2OC no (11^1-2)^B1r-yR^+ 12 ) 2when w = wo. B12 is the Einstein coefficient for stimulated emission and R is the sumof all relaxation processes. The linear terms give fluorescence modulated at F 1 and F2,while the quadratic terms are responsible for fluorescence modulated at (F1 + F2 ) and(F1 — F2 ). Sorem and Schawlow[15] demonstrated that by detecting fluorescence at thesum frequency, (F1 + F2 ), it is possible to record sub-Doppler spectra of the transitionswhile the background is greatly suppressed. Thus IMF is a good technique when sub-Doppler resolution of weak fluorescence transitions is desired.2.5 Wavelength Resolved Fluorescence SpectroscopyWhen a molecule is excited to a higher electronic state as a result of absorption of aphoton, there may exist more than one relaxation path back down to the lower state.An example of such a process is illustrated in Figure 2.7. This example shows a casewhere three lines are emitted as a result of excitation. Simply measuring the totalfluorescence emitted by the molecules reveals only that a transition has occurred, andChapter 2. Experimental^ 21gives no information about the assignment. On the other hand, the assignment canbe obtained from wavelength-resolved fluorescence (WRF) spectra. For instance, in thefirst example the WRF spectrum consists of a strong line between two weaker lines. Thegreater intensity of the centre line is due to unavoidable scattered laser light from theexcitation laser, not because the relaxation will preferentially follow that path. Thispattern in the WRF spectrum confirms the assignment to a Q-branch transition.In the second example, however, the intense line is the lowest frequency line of thethree because, in this case, a P-branch line has been excited. Similarly, for an R-branchtransition, the most intense of the three lines would be the one at highest frequency.Hence, the observed patterns of these WRF spectra serve to confirm the branch assign-ment of the particular excitation.Not only can branch assignments be made from WRF data, but if the lower stateis already well known, then the J-assignments follow from ground state combinationdifferences. For example, if the transition shown in Figure 2.7(a) was excited, then thefluorescence line positions can be predicted to be at:P(3) = Ei(2) — E1(3)Q(2) w Ei (2) — Ei(2)R(1) = w Ei(2) — E,(1).WRF spectra can be obtained by two different but related methods. In both casesthe fluorescence signal is focussed onto the entrance slit of a spectrometer. In the firstmethod, the wavelength passing through the exit slit is scanned by rotating the grating;the signal is detected by a PMT, and the dispersed spectrum is recorded sequentially.In a more efficient approach, the whole spectrum is recorded simultaneously. Earlymethods of simultaneous detection involved replacing the exit slit of the spectrometerChapter 2. Experimental^ 22(b)Ek(3)Ek(2)Ek(1)P(4)0 (3 )R(2)Laser'(a)•^•P(3)0 (2 )R(1)Laser •^•0(2)^ P(4)E i(4)V• •E ;(3)E;(2)V•E ;(1)Figure 2.7: An example of possible relaxations in a simple system after excitation of aQ-branch line a), and a P-branch line b).Chapter 2. Experimental^ 23with a photographic plate. The modern version of the photographic plate is the diodearray detector (DAD). The advantages of the DAD over the photographic plate are thatthe spectra can be analysed immediately and the intensity information stored digitally,which means it can be easily transferred to a computer.A SPEX model 1702 spectrometer was used in this study, with the exit slit replacedby an EG&G model 1421-1024-G DAD. The detector was cooled to -20 °C and an EG&Gmodel 1461 detector interface was used so that the experiment could be controlled by acomputer. The width of the entrance slit was varied between 35 am and 60 pm, and theexposure times varied from less than one second for the strongest fluorescence signals toas long as two minutes for very weak signals.Chapter 3Energy Expressions and the Hamiltonian3.1 IntroductionBefore the Hamiltonian matrix elements for the states of the B 4H—X4 E- transition ofVO are described in detail, some basic principles of quantum mechanics will be reviewed.The time-independent SchrOdinger equation is the fundamental expression giving thestationary state energies for a system,7-(0 = EV).^ (3.1)In this equation^is the total Hamiltonian operator,^represents the eigenfunctiondescribing the particular state of interest, and E is the eigenvalue or energy of the state.Despite the simple appearance of the SchrOdinger equation, it usually cannot be solvedanalytically. Instead, a convenient set of basis functions ch i is chosen such that(3.2)When the eigenfunction of the time-independent SchrOdinger equation is replaced by alinear combination of orthogonal basis functions 15i , the problem of solving Equation 3.1becomes that of calculating the roots of the secular determinant^17-123 — Ebij I = 0,^ (3.3)Where the matrix elements 7-t ii are defined as^xtj = I 0t 7-l0j dr,^ (3.4)24Chapter 3. Energy Expressions and the Hamiltonian^ 25and 8,i is the Kronecker delta (i.e. zero if i j and 1 if i = j).Any complete set of wavefunctions Oi would be sufficient, but in practice a basis setis chosen such that the Hamiltonian matrix is most nearly diagonal. Two different baseswere used to model the angular momentum couplings in the B4 11—X4 E- transition ofVO. The ground state was best described by a case (bp j) basis while case (an) couplingapplied in the upper state.3.2 PerturbationsSince both the X 4 E - and the B411 states show rotational perturbations in the observedband structure, the theory will be briefly reviewed.A state is said to be perturbed if the observed branch structure deviates from thatpredicted by simple theory. Rotational perturbations arise from terms neglected in theBorn-Oppenheimer separation of electronic and nuclear motions, or from relativistic ef-fects such as spin-orbit interaction. The Hamiltonian must be written as= n(o) (3.5)where H(o) is the "zero order" rotational Hamiltonian, and If is responsible for theirregularity.Consider a perturbation between two states whose zero order wavefunctions or "basisfunctions", are q5 1 and 02 ; there will be interaction matrix elements of the type (0 1 17-t'102 )which are responsible for the rotational perturbation. The Hamiltonian matrix elementsare given by(011 1-(M 101) = H11 = El, (3.6)(021 7-1(°) 102) = H22 = E21 (3.7)(011 7-002) = H12, (3.8)Chapter 3. Energy Expressions and the Hamiltonian^ 26J(J + 1)Figure 3.1: Illustration of an avoided crossing between two states.andwhich can be written as:(021VI01) = H21,^ (3.9)The 2 x 2 Hamiltonian matrix is assumed to be Hermitian, such that H12 = H21. Theeigenvalues,[1Hil H121121 1122(E1 E2 )^1 ^EA =^+ ^+ 2 V (E i — E2 ) 2 + 41-1,2.22 (3.10)Chapter 3. Energy Expressions and the Hamiltonian^ 27(E1 E2 )^1 ^EB =^2^2v(Ei — E2) 2 + 411?2, (3.11)represent the energies of the two perturbed states. The eigenfunctions, OA and LB, aregiven by C S ^01—s C^02(3.12)where c = = V ir-d^— H22, and k = Vd2 4H 2v 2k^2k d = H11^ 12'This is illustrated in Figure 3.1. The dotted lines represent the unperturbed states,which have energies E 1 and E2; these are functions of J, and are assumed to cross atsome value of J(J + 1). The solid lines represent the observed energies EA and EB, asgiven by Equations (3.10) and (3.11). Where E 1 and E2 cross, the levels EA and EBshow an "avoided crossing", and are separated by twice the perturbation matrix elementH12.The relative intensities of transitions to the perturbed levels can be calculated giventhe appropriate transition moments. Consider the transition moments p i and p 2 , tothe basis states cb i and c2 respectively. From Equation (3.12), the transitions to theperturbed eigenstates are given by(0AI ft Ix) = c ( cbil it l x) + 4021 ft ix)S/1 2(OBI it lx) = —s(g11 it IX) + c(02112 IX)—3 /1 1 + C/1 2The intensity is proportional to the square of the transition moment; thus .1,4 4.-}c a(cm + 4/ 2 ) 2 ; and^OC^cp2)2. In the case where p, 2 = 0, the above relationssimplify to 1A +._x OC c2 ii?; and^CC s 214. Consequently, as the avoided crossingis approached, the relative intensity of the observed transition to one of the perturbedChapter 3. Energy Expressions and the Hamiltonian^ 28levels will decrease. This diminished intensity will show up as increased intensity for thetransition to the other perturbed level. The total intensity is unchanged since 0+ s 2 = I.3.3 Hund's Coupling CasesTo model the rotational structure of the B 4H—X4 E - transition properly, it is necessaryto choose functions that describe the electron orbital angular momentum, L, the electronspin angular momentum, S, the angular momentum of the nuclear rotation, R, andthe angular momentum arising from the non-zero nuclear spin of the 51 V nucleus, I.These angular momentum vectors can be coupled together in many different ways. Hundconsidered five possible arrangements in which L, S and R can be coupled in linearmolecules, which have become known as Hund's cases (a), (b), (c), (d), and (e)[20]. Later,nuclear spin angular momentum effects have had to be included, to produce subsets ofHund's five coupling cases. Only the two cases needed for this transition, namely case (ap)and case (130j) (following the naming convention described by Townes and Schawlow [21]),will be discussed here. The others may be found elsewhere [21] [22] [23].3.3.1 Case (an)Hund's case (a) describes a system where both L and S are coupled to the internuclearaxis. In a non-spherical system, such as a molecule rather than an atom, L is not a "good"quantum number, though it does have a well-defined projection, A, on the internuclearaxis. Likewise, E represents the well-defined projection of S on the internuclear axis.The total angular momentum J is obtained by adding the rotational angular momentumR to L and S , such thatRd-L-1-S=J.Chapter 3. Energy Expressions and the Hamiltonian^ 29Figure 3.2: Hund's case (a n ) coupling scheme.Chapter 3. Energy Expressions and the Hamiltonian^ 30Since the projection of R along the internuclear axis is zero, the component of the totalangular momentum along the internuclear axis, CV, isSt= IA-FEI.One result of this coupling scheme is that the quantum number J, associated withthe operator for J, will be integral or half-integral, depending on whether SI (or moreprecisely E) is integral or half-integral. Of course J can never be less than ftIn case (ad) the nuclear spin angular momentum I is coupled to J, the total of therotational and electronic angular momentum, according toJ+I=F.This is illustrated vectorially in Figure 3.2. Naturally, the quantum number F cannot beless than zero. The values for F are given by the rules of vector coupling asF^IJ+II , IJ+I -1 I , ... , IJ — IIFor the 51 V nucleus, with nuclear spin ;, there will be eight hyperfine components foreach rotational level, provided J > 32. (There are two isotopes of vanadium found innature. Both have non-zero nuclear spins, though the radioactive 50V, with nuclear spinI = 6, is only found in 0.2% abundance. The dominant stable isotope is 51 V, which hasnuclear spin I =3.3.2 Case (13,34The electron spin angular momentum S is not coupled directly to the internuclear axis bythe electrostatic field in the molecule. Instead it is only the internal magnetic field induced1 It should be noted here that the post-subscripts used on the term symbols of individual spin compo-nents are in fact A + E, not IA + El. For example, the electron spin components for the 4 11 state (A = 1,S = 3 E = a I — 1 -a) are 411 411 4111 ,  411 while the values for' '^2^11^21^1)^-I)2 2 2 IC21 are A 2 1 and 12 2' 2' 2'^2respectively.Chapter 3. Energy Expressions and the Hamiltonian^ 31along the axis by the orbital motion of the electrons that can couple S to the internuclearaxis. If A = 0, then the magnitude of this internal magnetic field is identically zero andhence the electron spin angular momentum is not coupled to the internuclear axis. Thissituation is described as case (b), and occurs in most E states (A = 0) and all stateswith A > 0 but BJ > A. Even in states with non-zero orbital angular momentum, theelectron spin becomes increasingly uncoupled from the internuclear axis by the magneticfield generated by increasing molecular rotation; in such states there is no well definedprojection of S on the internuclear axis.In case (b) coupling, as with case (a), the projection of L along the internuclear axis(A) is a "good" quantum number. L is coupled with R to give N, the total angularmomentum excluding electron and nuclear spins; N then couples with S to give J.There are several possible coupling schemes for the nuclear spin angular momentum I.The most common situation, called case (bpj), is where J is coupled to I to give F. Thisis illustrated in Figure 3.3.2. The overall coupling scheme for case (bp j ) can therefore bedescribed as:R-I-L=NN-I-S=JJ-FI=FOnce again,F :=1 ,1 + II,IJ + I —11,...,IJ — .11where F cannot be less than zero.3.4 Hamiltonian Matrix Elements for the X 4 E- stateSeveral of the interactions that occur in the X 4 E- state occur also in the B 4 11 stateof VO. However, since the ground state is best described by case (bpj) coupling whileChapter 3. Energy Expressions and the Hamiltonian^ 32Figure 3.3: Hund's case (boj ) coupling scheme.Chapter 3. Energy Expressions and the Hamiltonian^ 33case (ao ) coupling applies to the B 4II state, the forms of the individual matrix elementswill differ. The X 4 E- matrix elements are discussed below, while the matrix elementsfor the upper state are given in Section 3.5.3.4.1 Rotational StructureThe general form of the rotational Hamiltonian is given by:Hroi = BR2 — DR4^(3.13)where R = J — L — S. In a E state, where L can be omitted in first order, R becomesJ — S, which is called N. The rotational Hamiltonian and its matrix elements for theX4 E- state are thus:7-trot = BN 2 — DN4^(3.14)(NSJIHrot INSJ) = BN(N +1)— DN 2 (N + 1) 2^(3.15)3.4.2 Fine StructureFine structure describes the interactions of unpaired electrons, carrying spin and orbitalangular momenta, with the molecular rotation and, through dipole-dipole interactions,with each other. The fine structure Hamiltonian can thus be partitioned into spin-orbit,spin-rotation and spin-spin operators. However, higher order terms, as well as crossterms between these different interactions, make evaluation and interpretation of thematrix elements complicated.For example, the first order spin-orbit Hamiltonian,7-( (81,, ) = AL • S^ (3.16)gives zero in an electronic E state, since (L i ) = 0, and the effects of L+ and L_ are notcontained within the E state. However, in second order, the effects of L+ and L_ areChapter 3. Energy Expressions and the Hamiltonian^ 34equivalent to a tensor operator (S,S) acting within the E state, i.e.(S•L)(L•S)^(S, S)(L, L),which turns out to be identical in form to the spin-spin interaction operator. The Hamil-tonian for the dipolar spin-spin interaction is given in terms of a parameter A by [24]7-1„ = -2A (152 - S 2 ) ;3^z (3.17)however, the contribution to A from the second order spin-orbit coupling is indistinguish-able from the dipolar spin-spin interaction, so thatelf = A„ AV).Therefore, the A parameter determined from the fit of the X 4 E- state of VO is aneffective parameter representing the sum of the two effects.In spherical tensor formalism, Equation 3.17 has the formHs, =^ VjAV(S,S),3(3.18)which gives as the matrix elements in case (b) coupling [24](N'SJIFInspin-spin NSJIF)A( -1)N+s-f-J3x(-1)N'(N'J S2 • N2 NN'S[(2N[S(S + 1)(2S + 1)(2S+ 1)(2N' + 1)} 112 .- 1)(2S + 3)]“2(3.19)0^0^0The spin-rotation Hamiltonian gives the energy of the interaction between the electronspins and the magnetic field due to nuclear motion. The general form of the spin-rotationChapter 3. Energy Expressions and the Hamiltonian^ 35Hamiltonian is given by [25]:11.9, = -yR • S^(3.20)= -y(J - L - S) • S= -yN • S, for E states.^(3.21)In a case (b) basis, the spin-rotation Hamiltonian has only diagonal matrix elements,which are given by:(NSJIli„INSJ) = -2y [N(N + 1) + S(S + 1) - J(J + 1)]^(3.22)The third order contribution to the spin-rotation interaction arises in third order per-turbation theory when the matrix elements of the spin-orbit operator are taken twice andthose of the spin-uncoupling term, -2B(Jx Sx + Jy Sy ), are taken once. The Hamiltonianfor this third order spin-rotation interaction is quite complicated to evaluate, but Brownand Milton [26] successfully simplified the case (a) matrix elements to(SE, JQ11-1(321SE ± 1, Ai ± 1) =- 2 -y3 [S(S + 1) - 5E(E ± 1) - 2] [J(J +1) - S1(SZ ± 1)]21x [S(S + 1) - E(E ± 1)1 1^(3.23)The case (b) forms of the third order spin-rotation matrix elements do not simplifysimilarly 2 . As required for the least squares fit of the VO B4 H—X4E- transition itis [28]:(N'SJIFIli (sVINSJIF)= -2 [(2N + 1)(2N' + 1)J(J + 1)(2J +1)1 1122 Brown et al [26] have recently expressed the third order spin-orbit Hamiltonian in a slightly different,but equivalent manner. However, since the subroutine for the ground state matrix elements had alreadybeen written using the previous convention, it was not changed.Chapter 3. Energy Expressions and the Hamiltonian^ 36x [2(2S — 2)(2S — 1)2S(2S + 1)(2S + 2)(2S)^'x E (2x + 1)^3 x 1( ( —1 )N1 Nx=2,4^—1 0 1 0+ 3)(2Sx N0^0+ 4)/3] 1 / 2 -ysN' N xS S 3J^J^1(3.24)The centrifugal distortion corrections to the fine structure are straight-forward forcase (b) coupling, and the Hamiltonian has the form:H.,cd = YD(N • S)N 2 + AD R3Sz2 — S 2 ) , N 2 1 + .^(3.25)In Equation (3.25), the symbol [x, y] + stands for the anti-commutator xy + yx, which isneeded to preserve Hermitian form for the matrices. The diagonal matrix elements forthe centrifugal distortion to the spin-rotation and spin-spin interactions respectively aregiven as [28]1=2-yDN(N + 1) [N(N 1) S(S 1) — J(J + 1)]and1—2-yDN(N 1)R(JSN) (3.26)(NSJIliss,edINSJ) = — 1ADN(N 1)3R(JSN)[R(JSN) + 1] — 4S(S 1)N(N + 1) (3.27)whereR(abc) = a(a 1) — b(b 1) — c(c 1)The off-diagonal matrix elements are given by1 ^[N(N + 1) — (2N — 1)] (N — 2, S^s,cdIN S^AD2 (2N — 1) [(2N + 1)(2N — 3)] 1x Y(JSN)Y(JS,N — 1),where(3.28)•3^ (2N — 1)(2N + 3)Y(abc) = [(a + b+ c+1)(b+ c — a)(a c — b)(a+ b — c 1)0- .N' N 2— c [30(2N + 1)(2N' + 1)1 1/2^S S 1 (-1)N '1 J' J 1(N' 2 N0 0 0(3.30)Chapter 3. Energy Expressions and the Hamiltonian^ 373.4.3 Magnetic Hyperfine HamiltonianWhen one or more of the nuclei of the molecule being studied has non-zero nuclearspin and an appreciable magnetic moment, then magnetic hyperfine interactions willmake significant contributions to the spectrum. The X 4 E- state of VO required threemagnetic hyperfine parameters to describe the observed features accurately.The Hamiltonian operators for the direct interactions between the nuclear magneticdipole moment and an electron spin moment in a E electronic state are given by[21]nmag hf = bI • S + c/zSz^ (3.29)where c represents the dipole-dipole interaction and b incorporates part of this dipole-dipole interaction as well as the Fermi contact interaction, bF, in the relation b = bF-1-The matrix elements are thus(N'SJ'IFInmag h f 1NSJIF)= ( 1)j+I+F F I J'[(2J 1)(2J' + 1)/(/ + 1)(2/ + 1)] 1 / 21 J I[S^'x [S(S + 1)(2S + 1)1 112 (-1)N+s+J,^J NbFJ S 1The third magnetic hyperfine parameter needed to describe the X 4 E- state of VOarises from the third order isotropic hyperfine Hamiltonian, 7-1L30) . This energy operator isanalogous to the third order spin-rotation Hamiltonian, whose matrix elements are givenas Equations (3.23) and (3.24), except that the isotropic hyperfine operator Et biI • siChapter 3. Energy Expressions and the Hamiltonian^ 38replaces the spin-uncoupling operator. The third order isotropic hyperfine Hamiltonianis [10][27]?-l(s)^(5.■/^3) bs  T.4(AI "AL) IA) T i (i) ' T 1 [T 2 (1-2 ), T3 (S3 )1and its matrix elements in case (bp j ) coupling are [10][27](3.31)(N'SJ'IFI7CINSJIF)1^IF^'= -4(-1)J+/+F^I J1 J I[(2J + 1)(2J' + 1)/(/ 1)(2I + 1)1 1/2N' N 2x(-1)N' (N 2 N0 0 0^1J' J .[(2N + 1)(2N' + 1)] 1 /2 S S 3x [35(2S - 2)(2S - 1)2S(2S + 1)(2S + 2)(2S + 3)(2S + 4)/3[ 1 / 2 bs (3.32)3.4.4 The Electric Quadrupole InteractionIn addition to its nuclear magnetic dipole moment, the electric quadrupole moment ofthe V nucleus also contributes to the hyperfine structure of the X 4 E- state of VO. TheHamiltonian operator for the electric quadrupole interaction is given by[10][21]7..t(Q) d^e2 Qqo^- 12) 4/(2/ - 1)resulting in matrix elements(N'SJ/IFInZd INSJIF)- 11 2 ^2 /_ e2 cho4^-/ 0 /(-1)J+I+F IF I J'2(3.33)1S^'^'x(- 1)Ni+s+j [(2J + 1)(2J' + 1)(2N + 1)(2N' + 1)1,,,^N J-2 J NChapter 3. Energy Expressions and the Hamiltonian^ 39x(-1)(N' 2 NN '0 0 0(3.34)3.5 The Hamiltonian for the 4 H upper stateA particular complication in the 4 H upper state arises from the a 2 E+ state which perturbsit heavily. The Hamiltonian for the B 4H state, excluding the effects of the a 2 E+ state,is described in this Section. Since the matrix elements of the B 4H Hamiltonian wereevaluated using a case (an) basis, it was necessary to do likewise for the a 2 E+ state so thatthe interaction matrix elements could be evaluated most simply. The a 2 E+ Hamiltonianis described in Section 3.6 and the perturbation matrix elements are given in Section 3.7.3.5.1 Rotational and Fine StructureThe rotational Hamiltonian operator has already been given as Equation 3.24, but isrepeated here for convenience:nroi = BR 2 - DR4 .^ (3.35)In contrast to the X 4 E- state, where A = 0, the first and third order spin-orbitinteractions are non-zero. The full spin-orbit Hamiltonian to third order is given by[29]:2^ 2 -  11-1„ = AL,S, - (3S2 - S 2) +3^z^2^3S 5 1 The spin-orbit matrix elements in a case (a) basis are thus(A'; SE'; JC/N„IA; SE; J11) = AAE + 3a {3E2 - S(S + 1)]+OE [E 2 - 5S(S + 1) + -51 ] .The spin-rotation Hamiltonian, as expressed in Equation 3.23, is given by= 7R • S,(3.36)(3.37)(3.38)Chapter 3. Energy Expressions and the Hamiltonian^ 40which in a case (a) basis has diagonal elements of the form(A; SE; JS1N, IA; SE; J1) = 7^— S(S 1)] .^(3.39)Since each of the sub-band origins is well defined in a state where case (a) couplingapplies, it is convenient to fit the data to a number of separate sub-band origins instead ofa single band origin with various spin-orbit and spin rotation contributions. The generalexpression for the sub-band origins is= To + AAE + 2—A [3E 2 — S(S 1)] + -y [E 2 — S(S + 1)] + AE [E23S(S +1) + 1]3^ 5(3.40)The spin-rotation parameter y appears in off-diagonal matrix elements (see Figure 3.4)and can therefore still be fitted independently of the sub-band origins. The off-diagonal-y term has the effect of linking the effective rotational constants of adjacent spin-states.As mentioned in sub-section 3.4.2, the spin-spin Hamiltonian is identical in form tothe second order spin-orbit Hamiltonian,27-1„ = -dA (3,5 — S 2 ) ,^ (3.41)so that its matrix elements are indistinguishable from those of M20) .3.5.2 A -type DoublingThe interaction between E states (A = 0) and II states (A = 1) lifts the degeneracy ofthe ±A levels in the H state [22][30]. The matrix elements for molecules in 1 H and 211states have long been known [30], but different conventions have been used to describe theparameters for states of higher multiplicity. This work uses the effective A-type doublingHamiltonian defined by Brown and Merer [31],RLD = 2 (^p q)(S_ + S!) — 2—(p 2q)(4S+ + J_S_) + 2—1 q(J_ + J_2_).^(3.42)Chapter 3. Energy Expressions and the Hamiltonian^ 41This form of the Hamiltonian was chosen because it refers to a Hund's case (a)coupling scheme; it gives matrix elements of the form [31](T1, E 2, J,S211(LDI ± 1, EJC2)1= —2(o p q)[S(S +1) — E(E^[S(S + 1) — (E ± 1)(E ± 2)] 2 (3.43)(T1, E + 1, J, S2 + 117-h,DI ± 1, EJS1)1=2(p + 2q) [S(S + 1) — E(E f 1)1 112 [J(J + 1) — it(f/ 1)1 1 / 2^(3.44)(T 1 ,E,J, 1l 217iLDI f 1,EJC2)1= —2q [J(J + 1) —^ 1)]1/2 [J(J + 1) — (CZ 1)(S2 2)]" 2 .^(3.45)There exists a contribution to the A-type doubling from hyperfine interactions, butthis will be discussed in the next sub-section.3.5.3 Magnetic Hyperfine InteractionsThe Hamiltonian operator for the interaction between the nuclear magnetic dipole mo-ment and the electron spin magnetic dipole moment in a E state has been given insubsection 3.4.3 asHis h f bI • S c/zSz•^ (3.46)In orbitally degenerate states there is an additional interaction between the nuclear spinangular momentum and the orbital angular momentum of the electrons. This interactionis described by the Hamiltonian[32]:= aI L z^(3.47)The three hyperfine parameters mentioned so far describe the magnetic hyperfine ef-fects in the two parity components of H states equally. However, Frosch and Foley[32]showed that hyperfine contributions to the A-type doubling are possible because part ofChapter 3. Energy Expressions and the Hamiltonian^ 42the dipole-dipole interaction has matrix elements that connect electronic states differ-ing by two units in the orbital angular momentum A. The A-type doubling-hyperfineHamiltonian is given by[32]HAD hf = —1 d^e-2icb4S+)2(3.48)where 0 is the angle giving the direction of the unpaired electron relative to an arbitraryreference plane. Translating into tensorial form, the dipolar Hamiltonian can be describedby the general expression [28]: 7-(mag hf =^0 gpBgNiiNr -3T 1 (I) • 711 (S, C 2 ) (3.49)whereand,T9(s,c2). - E(-1)q-4( 1 2 1 ) T:i(S)T,22(C)41 ,42^ ql q2 — q(3.50)r-3T5 Y211i(C) = \I-47 ao(0 0)r -3 .42 ^(3.51)This form is the most convenient for the calculation of the matrix elements. Omittingthe complexities of the tensor algebra, the diagonal matrix elements of Equations (3.46)-(3.48) areV^[aA (b c)E] R(FIJ)IZIFInmag hfIJCIIF) =^ (3.52)2J(J + 1)where R(FIJ) = F(F 1) - I(I + 1) - J(J + 1). The matrix elements diagonal in J,but off-diagonal in S2, are:(SE, JCIIFIHma g hfISE ± 1, JC2 ± 1, IF)= b[J(J +1) -12(f2+ 1)] 1 /2 [S(S + 1) - E(^1)] 1 /2R(FIJ) x4J(J +1)'(3.53)Chapter 3. Energy Expressions and the Hamiltonian^ 43while the matrix elements off -diagonal in J are(JCIIFInmag hflel — 1, I1I F)^[aA (b + c)E] (J2 — 11 2 ) 112 V (F, I, J) 2J [(2J + 1)(2J — 1)1 1 /2and(SE, J12/F17-tmag hf ISE ± 1, J — 1,12 ± 1, IF)= +b [(J C2)(J S2 — 1)1 1 / 2 [S(S + 1) — E(E 1)1 1 /2V(FIJ) x 4J [(2J + 1)(2J — 1)]'where(3.54)(3.55)V(FIJ) = [(J + I F +1)(F + J — I)(J + I — F)(F + I — J +1)1 1123.5.4 Electric Quadrupole InteractionThe Hamiltonian for the electric quadrupole interaction given in sub-section 3.4.4 for Estates (3.33) also applies for H states. However, while H states will have the same zeroorder term as in E electronic states, there is also a e 2Qq2 term which will have non-zeroelements only in H states. Like the d parameter for magnetic hyperfine effects, thereexists an electric quadrupole interaction which links states differing in A by ±2. Thecomplete electric quadrupole Hamiltonian for H states is given by[33][34]e2 Qq0 (3/1 — 12 )^e2 Qq2 (L2f. + 11)nquad^ (3.56)4/(2/ — 1)^8/(2/ — 1) •3.6 The Hamiltonian for the a 2 E+ StateAlthough E states are generally best described by a Hund's case (b) coupling scheme,the matrix elements for the perturbing a2 E+ state of VO were evaluated using a case (a)basis in order to be consistent with those for the B 4H state and so that the interactionmatrix could be written most simply.4 (x ± 1)(2 E +1 1..ihi 12 E+ ‘) = Tb R(F, I, J) (3.59)Chapter 3. Energy Expressions and the Hamiltonian^ 44Using the Hamiltonian operators described in Section 3.4, the rotational matrix ele-ments for the 2 E+ state are given by(2E+I H I2E+ !) = B (x) (x ± 1) — D (x)2 (x ± 1) 2 — 27 (1 ± x)^(3.57)where x=J+ 2.The sole hyperfine parameter that can be determined for the a 2E+ state that needsto be considered arises from the isotropic HamiltonianRh f = bI • S,^ (3.58)Written in an fie parity basis, its diagonal matrix elements have the formwhile the J^J — 1 matrix elements are(2E+ , 417.ihf 12E+ , j _ 1 ef) = b V( 8,JI , )j ' (1 T 1). (3.60)Those parameters which pertain to the a 2E+ state are designated by a prime (eg.B', D', -y', and b') in Figure 3.4 to distinguish them from the B4Il parameters.3.7 The 2 E+ — 4 11 Matrix ElementsThere exist no direct spin-orbit interactions between the .5 2a- a 2 E+ and 52 77. B411 states ofVO within the single configuration approximation, but higher order mechanisms must ex-ist. The relative sizes of the matrix elements of these higher order spin-orbit interactionscan be evaluated using the Wigner-Eckart theorem,(SI(S'E'A'S217-18o 1SEAC2) = (-1)s'—E'^1 s(S'AMH.011SA) bop , ,^(3.61)—E' q EChapter 3. Energy Expressions and the Hamiltonian^ 45which gives four non-zero matrix elements in a signed basis. Similarly, the hyperfineinteractions between 2 E+ and 4 H states require inclusion of an interaction parameterdenoted by the parameter e, which has four equivalent non-zero matrix elements. Aftertransformation to an e/ f parity basis the matrix elements are given in the form a :(4nd n 12E+ j) =^1 (411117.1 30 112E+) e R(F, I, J) (3.62)2^ 12^8J(J + 1)and\7-1 1 2E+ 8f)^(4111Insoli2E+) 60- R(F, , J) 2 8J(J + 1) (3.63)3 Special acknowledgement to Dr John Brown (Oxford University) for clarifying the matrix elementsof the hyperfine interaction between the 2 E+ and 4 11 states.Chapter 3. Energy Expressions and the Hamiltonian^ 461411 .1/2 fie >n 3/2TsA +(2-5)(8++AD+24+An D)-D(Z 2 -72+13) + 05-.5a-77-4) [2D! i D4 (J+ i))-.13T-4) (13-i7+ Ao400-2D(Z-2)-rup--1., I±./(2 -1 )(Z -A) 1 i q+ 4 4.4 . i(Z-2)Dg+0(2))577,+^(o+ 0+c)]T3/1+(2+1)(B+i-A0-244170) -2^Z-1 [B- i 7 -DID -31frf--1)(20(.14)±0+2q)-0(22+9Z-15) bR-2D(Z+2)--40+1) ±D•s9+4± i(2+ 1 )Do+24;(2-1)(.1+ i )D ^+ Q3 ; i (.1+.pliq+Dp4.2R ±i(Z-2)Dg+eh+ , [a+ (b+c)]+.6(22 1)+(Z+2)D4 +2±4,j77Twq2+39-4240-2104W -.13- (J+ •1i-Xe- i7 -A D 4% ;-71 3 + it 4T41)-D(Z2+13Z+5) -2D(Z+2)-0472 , J; (J+I )[(p+2q)+314"," ±,5[(o+p+q)+(2+2)D.4.0,1;,.,2,,+(2- ON+^- 1+(2+3)^...:-11, +.1i (22 -1)D,+20+01+47 [a --i(b+c)]T-e(Z+1)(B-10,0+2AD-AnD) ±^fixT) ]i (7, +,./3- e-D(Z2+5Z+ 1 ) ; 3(J+i)D.,4+0..,-47 [o-Vb+c)]TC114.13'[zt(J+2)]—olzt(J+1)) 2+171.41± I)— 4P+111i)Z=(J+%2) 2R=F(F+1)-1(1+1)-J(J+1)C(F , I ,J)- 87::_*=?j(j_ i*xl„ )Cmme2 04,[302 -J(J+1))6(F,I,J)aCu =e2 0q2G(F,I,J)/2Figure 3.4: The hyperfine Hamiltonian Matrix for 4H  states interacting with 2 E+ states.Chapter 4Analysis of the Spectra4.1 IntroductionThis study describes the analysis of the B4 1-1—X4E - transition of VO at sufficiently highresolution that the rotational and hyperfine structure could be well characterized. Evenat low resolution, the spectrum of the B4 1"1—X4 E - (0,0) band of VO is fascinating inits complexity. Basically it consists of four red degraded sub-bands resulting from thefour electron spin components of the case (a) 4H upper state. The branches of the foursub-bands overlap considerably, and the spectrum becomes very confused in those regionswhere the high-J lines of one spin state run into the low-J lines of the sub-band to thered, as shown in Figure 4.1.The effects of the smaller splittings of the spin components of the ground state aremore subtle, but are clearly discernible. This is illustrated in Figure 4.2, which showsthe Fortrat diagram of the branches involving the F 1 upper state. The Figure illustratesthe relatively large separation between the Q 1 and QR12 branches which arises primarilyfrom the spin-spin interaction, represented in the Hamiltonian by the term in A. Bycontrast, the °P12 and °Q 13 branches are nearly degenerate, whilst the °R 14 branch liesapproximately 2A away. The magnitude of the spin-spin interaction, and that of thespin-rotation interaction, will be discussed further in the Discussion (Chapter 6).The X4 E - ground state was fairly straight-forward to analyse because there are no lowlying electronic states close enough to cause perturbations; it could therefore be treated as47Irf3NNNN NNCVNN Nin- Tr-) Tr; C') ;^c7) WI Tr). (7- )0 0 0 0 0 0 0 0 0 0ci) (t) (/)^co (I) (i) (I)0CD 0CDC\I}-0CC0 WLOd 1±1o .CN.1 01fZd CCCDC\J0T-CDC\JChapter 4. Analysis of the Spectra^ 48Figure 4.1: The head of the s Q3 1 branch; illustrating the density of the B 4 11—X4 E - (0,0)band structure.Chapter 4. Analysis of the Spectra^ 49IDFigure 4.2: Fortrat diagrams showing transitions involving (a) the e-parity components,and (b) the f-parity components of the F 1 upper spin state.Chapter 4. Analysis of the Spectra^ 50a single isolated state. An interesting internal hyperfine perturbation occurs near N=15between the F2 and F3 electron spin components of the ground state. This perturbationhas been well characterized previously [1][7][8][10][14][35], so that it did not pose a greatproblem. Details of the analysis of this perturbation are given in Section 4.2.2.The B4II—X4 E- transition is further complicated by sizeable spin-orbit perturba-tions between the B4II state and the otherwise unseen 2 E+ state (denoted a) which liesclose to it. The effect of this perturbation is nicely illustrated in the Fortrat diagramshown in Figure 4.2, where it is seen that the a 2 E+ state causes the rotational spacingof each branch to collapse before emerging from the other side of the avoided crossingregion. Because of this perturbation, the B4 II and the a 2 E+ upper states could notbe treated separately, but had to be considered simultaneously, with the appropriateinteraction terms included in the combined rotational and hyperfine Hamiltonian. Inparticular the hyperfine structure of the upper, state behaves anomalously: the charac-teristic narrow hyperfine splittings of the B 4 1.1 state widen as the rotational levels of theB411 and a 2 E+ states approach with increasing J, and then diminish again after eachlocal avoided crossing.4.2 The Ground State of VO4.2.1 The Spin-spin and Spin-rotation InteractionsThe X 4 E - state is best described by case (b 0j ) coupling because the electron spin-spin interaction (A) is much larger than the hyperfine interactions. As mentioned inthe Introduction to this Chapter, the effect of the A parameter is clearly seen in theseparation of the Pi and F4 lines from the F2 and Fg lines. The four spin components ofa 4 E state in pure case (b) coupling follow the exact expression[10]4A — 2y = F2 (N) F3 (N) — F1 (N) — F4 (N).^(4.1)Chapter 4. Analysis of the Spectra^ 510^10^20^30^40^50^60NFigure 4.3: The energies of the four electron spin components of the X 4 E - state of VO.Also, because the splitting between adjacent components varies as 7N, the values of Aand -y can be estimated from pairs of ground state combinations differences of the typeF 2 (N)—F 1 (N) and F3 (N)—F 4 (N), or F 2 (N)—F4 (N) and F3 (N)—F 1 (N).The separations of the four spin components of the ground state as a function of Nare shown in Figure 4.3. If lines are drawn along the means of the F2 and F3 componentsand the F 1 and F4 components, the distance between these two lines is almost exactly2A — -y.The effect of y on the ground state spin components is to give a clear N dependence,Chapter 4. Analysis of the Spectra^ 52as shown in Figure 4.3. The separation between the F2 and F3 spin components isapproximately equal to yN at high values of N, while the F 1 and F4 components showa separation of approximately 3yN at high N.The spin-spin and spin-rotation matrix elements were given explicitly in Chapter 3.4.2.2 The Hyperfine Splitting in the Ground StateSince the C 4 E- state also has small hyperfine splittings, Cheung et al. determined thatthe F 1 :F 2 :F3 :F4 hyperfine widths of the C 4 E --X4 E - system are in the ratio —3 :—1 : 1 : 3 which means that Hund's case (bpi) applies to the X 4 E - state. Consequently,the Pi and F4 lines are the easiest to analyse in the B4 11—X4 E- system because theyshow the widest splittings. The Pi lines could be easily distinguished from the F4 linesbecause in the former, the position of the high-F component (seen as the componentwith greatest intensity) is at low frequency. Conversely, in the F4 lines, the position ofthe high-F component is at high frequency.For the B411—X4 E - transition of VO, the contribution from the upper state tothe hyperfine line width is generally much smaller than the ground state contribution.Therefore the observed hyperfine widths are dominated by the ground state hyperfinesplittings. The electron configuration of the X 4 E - state is o- 45 2 , where the o-orbital isderived from the V 4s atomic orbital. Thus, the large hyperfine splitting of the groundstate is primarily due to the Fermi contact interaction of the unpaired 4scr electron, whichis not present in the B4 11 state where the configuration is 62 7r.The spacing between adjacent hyperfine components of a particular ro-vibrationaltransition decreases with decreasing F. Thus, barring any perturbations or any severeblending, the assignments of the individual ro-vibrational lines to the appropriate spincomponents Pi or F',/t , as well as the F-numbering, were straight-forward, as shown inFigure 4.5. The total splitting observed for the P1 and F4 spin components is typically8^10^12^14^16^18^20^22\ I IN-3N-4N+3F =N+4Chapter 4. Analysis of the Spectra^ 53Figure 4.4: The hyperfine energy level splittings for the F2 and F3 spin states of theX4 E - state of VO.Chapter 4. Analysis of the Spectra^ 54of the order of 0.3 cm -1 .The F-assignments of the F' and F' lines are much more difficult to make than thoseof the PI' and Ffl lines. In 1968, Richards and Barrow [7][8] discovered that transitionsinvolving the F2 (N = J + D and F3 (N = J — D electron spin components in theX4 E - state of VO are doubled near N=15. Although they could not resolve the hyperfinestructure, Richards and Barrow concluded that this unusual doubling occurs because theelectron spin contributions to the total energy in those two spin states are accidentallyequal at N=15 (see Figure 4.3).In the case of the F2 and F3 spin components, the matrix elements responsible forthe perturbation have AJ = ±1 and AN = 0. Although the electron spin contributionsto F 1 (N = J + D and F4 (N = J — D are nearly equal at N=9, no internal hyperfineperturbation is observed. This is because there are no matrix elements having AJ = ±3and AN = 0, which would be required for a direct interaction between these spin states.Even at sub-Doppler resolution, the analysis of this internal hyperfine perturbationis quite complicated. The extra lines that are induced by this perturbation contribute tothe complexity of these spectra. In instances where transitions involving both the F2 andF3 spin states of the ground state are allowed by selection rules, the extra lines inducedby the F2 spin state are almost exactly blended with the main lines of the F3 spin stateand vice versa.If only one of the spin states, for example F2, has a transition allowed by the selectionrules, then the problem of blending with the main lines of the F3 spin state will beeliminated since the F3 transitions will not be observed. However, the total intensity ofthis ro-vibronic transition will be shared between the two eigenstates. An example ofthis effect is seen in the SR 32 branch. Figure 4.6 shows how the intensity from seven ofthe eight hyperfine components of the N=14 line appears nominally as the 83 (14) line.Those lines having greater than fifty percent F' character are marked by dashed linesChapter 4. Analysis of the Spectra^ 55(a)F n = 7 8 9 10 11 12 13 14,^i^1 ,^ ,^ ,I^I^,^ , ,I I ,^,I^i^I ,,t,,^ i:^,,1 112653.2^12653.1^12653.0^12652.9TRANSITION ENERGY/ cm -1F"= 9^8^7 6 5 4 3 211 I^I^I^i^I^1I^1^I 1^I^1^I^II 1 1^I I^I^I^II^1 I^I^1^I^11 1^1I I l^II 1^I^1^11 II^ 1^111^1^1^I 1^1I^1 I^I II1^1 1^,1^111^1(b)I^I^I^I^IIIIII^i^i12706.1^12706.0 12705.9TRANSITION ENERGY/ cm -IFigure 4.5: (a) The SQ31(9) and (b) the R4 ( 7 ) lines showing how the hyperfine structuresare mirrored.Chapter 4. Analysis of the Spectra^ 56while the induced lines (those with greater than fifty percent F3 character) are indicatedby dotted lines. The full complement of eight hyperfine components for the SR32 (14)line is shown, but there are only seven induced lines because there is no F=18 hyperfinecomponent in F3 state for N=14. The F=18 hyperfine line of sR32 (14) is thusunperturbed and appears as a sharp line at 12 653.3064 cm -1 .The hyperfine energy levels for the F2 and F3 spin states are shown in Figure 4.4.This illustration clearly shows the avoided crossing of seven of the eight hyperfine com-ponents for both spin states near N=15. The F = N + 4 and the F = N — 4 hyperfinecomponents of the F2 and F3 spin states respectively are free from this internal hyperfineperturbation. Lines resulting from transitions involving these unperturbed levels nearN=15 will generally appear as intense sharp lines between the broader features of theperturbation.The precise assignment of the spin indices for the individual hyperfine componentsinvolving the F2 and F3 states becomes ambiguous near the perturbation. This is evidentin the assignments of the sR 32 (14) features shown in Figure 4.6. The feature which onfirst inspection appears to be the F=17 hyperfine component of the SR32 (14) main line(at 12 653.4043 cm -1 ) turns out to have predominantly F3 character, 1 and is in fact theinduced F=17 line from the S3 (14) branch. This line is normally forbidden according tothe usual spectroscopic selection rules. The main line hyperfine component appears at12 653.2593 cm -1 .1 A spectral feature is assigned to a particular eigenstate provided that it meets the criterion that thelargest contribution to the total wavefunction comes from the eigenvectors of that eigenstate.'I'I441°P43(25 )NJ A,1^I^1^1^I^1^I^I12653.40 12653.30I^I^112653.20\ kiChapter 4. Analysis of the Spectra^ 5718^17^II16Energy /cm-1Figure 4.6: Plot of the sR 32 (14) line including induced lines from the internal hyperfineperturbation.Chapter 4. Analysis of the Spectra^ 584.3 The /3 411 StateThe B4H state of VO is very strongly perturbed, so that it cannot be described withoutat the same time considering the a 2 E+ perturbing state. This is particularly true of theA-type splitting and the hyperfine structure of the F 1 and F2 spin components, whichare the ones most severely affected by the a 2 E+ state.However, the interaction with the a 2 E+ state does not completely mask all the infor-mation which can be obtained about the B4 H state from the B411—X4 E- (0,0) band.The sub-band origins are essentially unperturbed, and the F3 and F4 electron spin com-ponents are completely free from the first order perturbation effects of the a 2 E+ state.4.3.1 The Spin -orbit Splitting of the B 4H StateSince the four electron spin components of the B4 H state of VO lie some distance apart,it is convenient to determine the origins, To, for each sub-state separately. The spacingof the spin components, which mostly reflects the first order spin-orbit parameter A, isevident in Figure 4.7; it is seen that A is approximately 65 cm -1 . Closer inspection ofthe sub-band origins shows that the separations are not exactly equal, withT5 - T3 = 74.7440 cm -12^2TI — Ti = 65.4295 cm -12^2T1 — T 1 = 53.3464 cm-12^2Further insight into the B 4 11 state provided by these spin-orbit parameters is foundin the Discussion (Chapter 6).Chapter 4. Analysis of the Spectra^ 5912850 ^12800 —12750 — 4115/212700 —4Tr3/212650 —-12600 — 47 1/212550 --- 4.11.-1/212500 —- 2x f-4-12450 —- 2E e+12400 1^1.^1^1^J.^1^1.^1^1 ...^1^I^I^1^1^...^10^1000^2000^3000^4000^5000^6000( j + 1 / 2 )22Figure 4.7: The upper state electronic term energies as a function of (J + - ) .Chapter 4. Analysis of the Spectra^ 604.3.2 The A-type Doubling in the B 4II StateAll states with A > 0 are doubly degenerate because the projection of L along theinternuclear axis, A, is a signed quantity, and in the absence of other effects, the energiesof the two components, with positive and negative values of A, are the same. Thedegeneracy is lifted by interactions with E states (A = 0) which have no such degeneracy.The resulting separation of the otherwise degenerate levels is known as A-type doubling.The A-type doubling in the B4 II state becomes confused at the avoided crossings withthe a2 E+ state. This fact is evident in Figure 4.8, where the avoided crossings causediscontinuities in the A-type doubling plots. The smallest A-type doubling occurs in theF4 spin component with that in the F3 sub-state being the next smallest. Despite theconfusion caused by the a 2 E+ perturbation, this observed decrease in A-type splitting as52 increases agrees with the predicted trend [22] [23].4.4 The Interaction Between the B 4II and a2 E+ StatesAvoided crossings occur at each place where the B 4 11 and a 2 E+ levels with the sameJ-value happen to lie at approximately equal energy. There are three avoided crossingsin the accessible range of J-values in the B 4II—X4 E - (0,0) band. The best characterisedcrossing affects the 4 Il_2 f levels at J 36.5. It takes the form of a 12 cm -1 gap in thebranches having the 411_ f levels as upper state. The most intense branch of this typeis the Q i . Unfortunately the most severely affected lines of the Q i branch lie under theheads formed by the 4 11_1, branches so that it is not easy to follow the course of the2branch at the most critical places. The pattern of the levels can also be seen in the 0 Q 13and °P12 branches, but these lines are not strong enough to be seen in the sub-Dopplerspectra. In fact, the level structure is quite confused near these avoided crossings, and therotational assignments had to be made with extensive wavelength resolved fluorescence.....................^.............0^1500 3000 4500 6000 7500(J+1/2)2Chapter 4. Analysis of the Spectra^ 61Figure 4.8: Plot of the A-type splittings of the four spin states of B411Chapter 4. Analysis of the Spectra^ 62measurements.Two other avoided crossings between the a 2E+ and the B 4 11 states were similarlycharacterized at Doppler-limited resolution. An avoided crossing pattern in the PQ 12branch with a width of about 22.5 cm -1 gave the magnitude of the interaction betweenthe 4 11_12, levels and the a 2 E+ state, while a similar pattern in the PQ 23 branch, with awidth of 18.8 cm -1 , gives the details of the 4 11 1f/a2 E+ interaction.No direct information describing the interaction between the a 2E+ and the 4 1112 ecomponent could be obtained because the avoided crossing is predicted to occur nearJ=70.5. The VO molecules are produced in the reaction cell at temperatures that aretoo low for these levels to be appreciably populated, and thus no ro-vibrational transitionsinvolving the 4 1112,--2 E e+ interaction were observed.Even though the a 2E+ state cannot interact directly with the F3 and F4 spin com-ponents of the B 4I1 state, these sub-states are nevertheless perturbed in second orderthrough spin-uncoupling interactions. The perturbation manifests itself in the A-typedoubling of the F3 and F4 sub-states (Figure 4.8). The observed effect of the perturba-tion on the A-type splitting of the F3 spin component is significant, and in fact an avoidedcrossing is predicted for J=77.5. The effect on the 4 115 sub-state is much smaller, and it2appears essentially unperturbed.The a 2E+ state has the same electron configuration as the X 4 E- state, namelyo-82. Like the ground state, the a 2 E+ state has wide hyperfine splittings because ofthe large Fermi contact parameter arising from the unpaired 4so electron. Evidence ofthe a 2 E+ perturbation as it affects the hyperfine structure of the upper state is shownin Figure 4.11. Both of the branches shown involve the same upper spin-state, namely. Near J=37.5, this spin state has approximately fifty percent ct 2 E -f character,so the hyperfine structure of this sub-state reflects the a 2 E1: contribution. Consequently,the hyperfine widths of the Q i and °P12 branches become larger near the avoided crossingChapter 4. Analysis of the Spectra^ 63LO^0^LI)^o^Ln1 • ^o o od dddd1l.._10/ ILI —0^I-0^C\J7— 1— 0O^ci^11 1Figure 4.9: The hyperfine energy levels of the B 4 11_1 f and a 2 E -f states.Chapter 4. Analysis of the Spectra^ 64tn-Ln Lc)-cr.)Ln-Ln- o.............iii I iIII I ,III I.,,,,,,III I Iiii i i,,,,^III I IIII I 1^OLn o LO 0T- T 0 0O d 6 61LO 0 LC) 0 LO0 1- 1- C\J (\l616I616I61Figure 4.10: The hyperfine energy levels of the B 411_, and a 2 E - states.Chapter 4. Analysis of the Spectra^ 6510 20 30 40 500.500.40NX — Experimental Measurements(b)E0.30^Ground state internal hyperfine perturbationLLJ0.200. 1010 20 30 40 50Figure 4.11: The hyperfine widths of (a) the Q i branch, and (b) the °P1 2 branch.Chapter 4. Analysis of the Spectra^ 66because of this increased a 2 E+ character. Once past the avoided crossing, the hyperfinewidths return to normal. The predicted hyperfine widths of the nominally-forbiddena2E+ -X 4 E- branches are shown by dashed lines. It should be noted that the threeextra lines of the °P12 branch, whose hyperfine widths were measured, fit the calculatedwidths very well.Chapter 5ResultsThe parameters needed to describe the states of the B 4 1I—X4 E - (0,0) band of VO weredetermined in three stages. First, preliminary assignments of the IMF spectra enabledground state combination differences to be calculated. These combination differences,along with three microwave lines measured by Suenram et al.[36] using FT-microwavespectroscopy, were then used in a fit of the ground state only. This fit served to confirmthat the spectral lines had been correctly assigned, and permitted the assignment ofmany previously unassigned features in the IMF spectra.Parameter Value (cm -1 )To 0.0 —fixed—B ±0.000001710 6 D 0.6491 ±0.00097 0.022426 ±0.0000042.03090 ±0.00004b 0.027435 ±0.000002c —0.00450 ±0.00005e2 Qqo 0.00134 ±0.0004810 5 7D 0.0060 ±0.000510 5 AD 0.038 ±0.00710 5 7i 0.810 ±0.09510 5 13, —1.58 ±0.37Table 5.1: The constants for the X 4 E - (v=0) stateof VOThe second stage involved a rotational fit of the B 4 11—X 4 E - transition. The hyper-fine structure of each rotational line was averaged to estimate the rotational transition67Chapter 5. Results^ 68Parameter Value (cm -1 )To 12432.9406 ±0.0476B 0.54304 ±0.0000310 6 D 0.65 —fixed-7 —0.0396 ±0.0019b 0.0930 ±0.0008(4111p_i joi2E+) 20.430 ±0.076,I I-Lif Insol2E-i- 20.404 ±0.021411_071,012Et 20.417 ±0.020e 0.0 —fixed—Table 5.2: The constants for the a 2 E+ (v=2) stateof VO.energy. Also included in the rotational fit were some high-J emission lines recorded pho-tographically some years ago in this laboratory, and recorded also by Fourier transformmethods at Kitt Peak National Observatory. This fit gave an estimate of the rotationaland electron-spin parameters of the upper state, and a good measure of the spin-orbitinteraction between the B 4 I1 and 0E+ states.The third and final stage involved the full hyperfine fit of the B41I—X4 E- transition.In all, 3211 lines were used in the fit including the three microwave lines. The final fitgave an rms error of 0.00038 cm -1 , and the values of the parameters for the X 4 E- ,a 2 E+ , and B 4 Il states are given in Tables 5.1, 5.2, and 5.3 respectively. The reportederrors represent three standard deviations.The combination difference and rotational fits were run on a DEC-MICROVAX IIminicomputer while the full hyperfine fit was run on an IBM RISC-6000 computer.Chapter 5. Results^ 69Parameter Value (cm -1 )TA 12711.8260 ±0.00012TA 12637.1180 ±0.00012T i 12571.6885 ±0.0019T_ 12518.3421 ±0.0024B 0.5126525 ±0.000001810 6 D 0.6634 ±0.0009104AD -0.69 ±0.0410 6 AD -0.28 ±0.05T/D 0.0 -fixed-7 0.0336 ±0.0002o+p+q 1.131 ±0.001p+2q 0.03609 ±0.00002q 0.0001733 ±0.000001110 5 D o _f_p _Fq 0.16 ±0.0510 6 Dp+ 2,7 -0.0041 ±0.001410 6 D g -0.00037 ±0.00009e2 Qq o 0.00157 ±0.00055a 0.0109 ±0.0002b -0.00898 ±0.00003c -0.00508 ±0.00015d -0.00359 ±0.00003e2Qq2 0.0 -fixed-Table 5.3: The constants for the B 4 11 (v=0)state of VOChapter 6Discussion6.1 Rotational StructureThe spectrum of the VO B 4II—X 4 E- (0,0) band shows red-degraded branch structure,indicating that the effective rotational constant of the X 4 E- state is larger than that ofthe B 4 II state. The rotational constant B (in cm -1 ) is defined byB =87r 2 /c'where h is the Planck constant, c is the speed of light and I is the moment of inertia ofthe molecule, given byI = itr2 .^ (6.2)In this equation it is the reduced mass of the molecule and r is the bond length. Thereduced mass of 51 V160 is 12.1729611 9 amu [38], which enables the bond length in aparticular vibrational level of any electronic state to be calculated from Equations (2.1)and (2.2). The bond lengths in the observed vibrational levels of the a 2 E+ , B4II ,and X4 E - states are shown in Table 6.1. This Table also includes results from thefit of the (1,0) band, obtained by Huang et al.[37], and the (0,1) band of the A4II-X4 E - transition[35].In the rigid rotator approximation it is expected that r, and therefore B, wouldbe invariant to the vibrational level. The results listed in Table 6.1 show that thereis an interaction between rotation and vibration. This interaction is described by theh(6.1)70Chapter 6. Discussion^ 71State By (cm -1 ) r (A)X4 E- (v=0) 0.5463713^±0.0000017 1.5920X4 E- (v=1)a 0.542864^±0.000013 1.5972B4Il (v=0) 0.5126525^±0.0000018 1.6436B4II (v=1) b 0.5094926^±0.0000264 1.6487a2 E+ (v=2) 0.5430436^±0.0000292 1.5969a 2 E+ (v=3)b 0.54044^±0.00013 1.6007Table 6.1: The rotational constants and aver-age bond lengths of the states of the VOB4II—X4 E - transition.(0,1) band, reference 35°Data taken from A4 1I—X4 E- (1,0) band, reference 37bData taken from B4 1I—X 4 E-rotation-vibration coupling constant a, [22],1By = Be - a, (v + -2) + • • • , (6.3)where Be represents the equilibrium rotational constant. Given the value of Be , theequilibrium bond length can be determined. The equilibrium bond lengths are given,along with the values for the rotation-vibration coupling constants and the equilibriumrotational constants, for the B4 1-1 and X 4 E - states in Table 6.2.6.2 Electron ConfigurationsThe ground state valence electron configuration of VO was predicted to be cr62 by Carlsonand Moser[5]. The nearly equal bond lengths of the a 2 E+ and X 4 E- states are evidencethat the two states have the same electron configuration in the single configuration ap-proximation. The very slight difference between the two arises from the interelectronrepulsion in the higher multiplicity state and from configuration mixing of the two stateswith states of other configurations.Chapter 6. Discussion^ 72State Be (cm -1 ) a, (cm -1 ) re (A)X4 E- 0.54812 5 0.00307 1.5837B4 II 0.51423 0.00316 1.6351a2 E+ 0.5495 0.0026 1.582Table 6.2: Table of the equilibrium rotational con-stants from the B 4II—X4 E- transition.In the single configuration approximation, the B 4II state is described by the config-uration 62 7r[14]. The longer equilibrium bond length in the B 4II state compared to thea 2 E+ and X4 E- states suggests that the 47:- molecular orbital of the S 2 ir configurationis slightly more anti-bonding in character than the 9cr molecular orbital occupied in theground state (see Figure 1.1).Strong evidence that one of the electrons in the ground state configuration is in anorbital derived from the vanadium 4s atomic orbital comes from the hyperfine splitting.The Fermi contact interaction is the largest contribution to the hyperfine structure of theX4 E- state. The magnitude of the contact parameter is proportional to the probabilitythat an electron is to be found inside the nucleus. Since only s atomic orbitals havewavefunctions which are non-vanishing at the nucleus, the large Fermi contact interactionindicates the presence of an unpaired electron that has appreciable atomic s character.The Fermi contact parameter, bF , is related to the experimentally determinable magnetichyperfine parameters, b and c, by1bF = b + 3-c (6.4)and is equal to 0.02593 5 cm -1 for the X4 E - state.In a study of the 3d4 4s configuration of atomic 51 V, Childs et al.[43] determinedthe value of the contact parameter of the 4s electron to be 0.1036 cm -1 . Allowing forthe difference in spin multiplicity, which introduces a factor of 3, the Fermi contactChapter 6. Discussion^ 73parameter of the X 4 E- state is 75.1% of the atomic contact parameter, which provesthat the electron configuration of the X 4 E- state indeed has an unpaired electron withprimarily 4s atomic 51V character. In single configuration approximation, this electronmust therefore occupy the (4so) molecular orbital.Similarly, the a 2 E+ state also exhibits large hyperfine splittings; the contact interac-tion is even larger than in the X 4 E- state, and amounts to 89.8% of the value for V(4s).As in the X 4 E - state, there must be an unpaired electron occupying the (4so) MO. Thus,the a2E+ state has the same configuration as the X 4 E- state, namely (4so) 1 (3d8) 2 . Thedifference between the values of the two parameters can be attributed to the degree ofconfiguration interaction in the two states.6.3 The Molecular Spin -Orbit ParametersThe four components of the B4 1-1 state were fitted to a model that used four sub-bandorigins To, rather than spin parameters of high order. The two models are related,through the diagonal elements of the spin-rotation and spin-orbit interactions, byTo = To + AAE + 3 a [3E 2 — S(S + 1)] + 7 [E 2 — S(S + 1)1+i1A {E 3 — (3S 2 + 3S + 1) E/51 . (6.5)Since the spin-rotation parameter y has been determined independently of the sub-band origins, there are four sub-band origins To which can be used to give the fourparameters To , A, A, and ii. The values obtained for the four sub-band origins and7 for the B4 II (v=0) state of VO are given in Table 6.3; the reported error limits arethree standard deviations. The band origin and the three spin-orbit parameters weredetermined with the help of Equation (6.5); they are listed in Table 6.4.Chapter 6. Discussion^ 74Parameter Value (cm-1 )T k 12711.8260 ± 0.00012Ta 12637.1180 ± 0.00012T1 12571.6885 ± 0.00192T_L 12518.3421 ± 0.00242.7 0.0336 ± 0.0002Table 6.3: Table of the four sub-band originsof the B 4 11 state.As expected, the estimate of 65 cm -1 for the first order spin-orbit splitting was closeto the calculated value. The next largest spin-orbit contribution is, not surprisingly, thesecond order parameter A. The A term comes mostly from the interaction between theB411 state and other states from the same configuration. Since the spin-orbit operatorhas matrix elements diagonal in Si, 2 H states, which have ft = 2 and 2 spin componentsonly, will interact with the it = aand 2 sub-states of the B 4I1 state, but not with theIl = 2 2and — sub-states. Hence, the 4113 and the 4 111 spin states will be shifted relative1Ospatial = 1 452+ 452—^(6.6)2^ 2to the others; the specific form of the operator is such that this shift is 4A. The sign ofA suggests that the 2H states lie above the B4Il state, since the two spin states involvedin the interaction are pushed down in energy. However, there are two 2H states given bythe valence electron configuration Pr. Since there is only one parameter A that can bedetermined, there is not enough information to deduce where they both lie.The Slater determinant form for the B 411 wavefunction, omitting the electron spinfactors, is given byThe orbital angular momenta of the 6-electrons cancel, so that the spin-orbit couplingChapter 6. Discussion^ 75Parameter Value (cm -1 )To 12609.8367 ± 0.0017A 64.5989 ± 0.0008A 2.6580 ± 0.0002n —0.4614 ± 0.0005Table 6.4: The band origin and higher spin-orbitparameters of the B 411 state.constant of the B 4H state comes from the 7r-electron. The microscopic spin-orbit param-eter for this 7r-electron is given by[24]a,- = 3AA = 3 x 64.5989 cm -1 = 193.797 cm -1. (6.7)This value for a, is consistent with the values obtained from the A' 4 0:1), A4 I1 and 1 211 stateswhich have configurations cr87r, Q(57- and o- 2 71- respectively[14], thus providing furtherconfirmation that the configuration of the B 4 H state is indeed Pr.The X4 E- state can have no first order spin-orbit effects since A = 0. There are, how-ever, second order spin-orbit interactions between the X 4 E - state and other states fromthe same electron configuration[24]. Perversely, the second order spin-orbit Hamiltonianhas the same operator form as the first order electron spin-spin dipolar Hamiltonian.Consequently, the experimentally determined parameter A for the X 4 E- state is an ef-fective parameter, given byA As, 4- A387where A30 is the second order spin-orbit interaction parameter and A s, is the electronspin-spin interaction parameter. There is no way of estimating A ss except from ab initiocalculations, and in any case it is likely to be small campared to A„; it will not beconsidered further.Aso = 4 E (2Et) E (4E1 )21 (245) 2 4 10412^1.44 cm -1 .121  K4EI Hs° 2Et i2^2 (6.12)Chapter 6. Discussion^ 76The most likely contribution to A„ is from the interaction between the 4 E-1 spin2component of the ground state and the 2 Et spin component of the a 2 E+ state. TheSlater determinants for these sub-states are given by1 4Ei) =^ -d [ b+a 6-13 aa l^lbfi 45- a aal^ I6+ab-a ofl]" (6.8)12 E1) b+oc^cal —15+ 13 b-a acti] . (6.9)Since both states have A = 0, only those terms in the spin-orbit Hamiltonian of the formAA = AE = 0 need be considered. Hence, the microscopic spin-orbit Hamiltonian givenin Chapter 3 can be simplified as= E^(6.10)The spin-orbit matrix element between the 4 E7 and 2 Et spin components is easily shownto be(4E i Rs° 2Et )^4a5 (6.11)where as is the microscopic first order spin-orbit parameter for a 3db electron. Thevalue of as can be obtained from the aba-* 4A state of VO, and is approximately 150cm -1 [14]; this gives the matrix element (6.11) as roughly 245 cm -1 . Since the separation),E(4E7) _E(4E- is 4), A„ is given byThe experimental value of the effective A parameter is 2.03090 cm -1 . Therefore, thespin-orbit contribution from the a 2 E+ state represents 71.0% of the total effective Aparameter of the X 4 E- state of VO.Chapter 6. Discussion^ 776.4 The B 4 11 /a2 E+ PerturbationThe Slater determinant function for the 4111 sub-state is21 4110 = 1 11 6" 8-P ra l + 1 6" 6-ar /3 1 + 16+a 8- #11 •^(6.13)The spin-orbit interaction matrix element between B4I12 and a 2Et can be shown to be2^2zero. The only non-vanishing contributions come from the interactions between:(18+ 13 45- a 01 and - IS+ 3 a a al) , which gives - 2 a3 [1(1 + 1)] 1 ;and between (16*(16- 01 and - 1(5+a 6- o-al) , which gives - 1-a3 [1(1 + 1)]i .These two terms cancel, indicating that the 4 11/2 E+ perturbation cannot occur by a firstorder spin-orbit interaction mechanism.Furthermore, the oW-electron is 4so- and the 7r-electron is 3dr. Consequently, the 1+operator is being required to ladder a 4so electron into a 3dr orbital, that is with A/ 2.Similarly, is required to give A/ = -2 matrix elements. Neither is possible.However, the B4I1 state is indeed perturbed by the a 2 E+ state. Therefore, a higherorder mechanism must exist that links the two states. The relative magnitudes of thevarious matrix elements can be calculated by the Wigner-Eckart theorem:S'(S'E'A'I nao ISEA) = (-1)Si-E^1 (S'n' Moll SA) 45011/•^(6.14)-E' q EThis gives four non-zero elements. In a parity basis, the elements are(4110 I nso I 2E+,P2^2 10 (41I Ilxsoll 2 E+ )^(6.15)and^(411_0 I 12 E+'F2) = ±-2 (411117-(„112 E+) ,^(6.16)Chapter 6. Discussion^ 78where the reduced matrix element ( 411 11 7-18011 2 E+ ) has been defined in Chapter 3 as A.The matrix elements of the 4 11/ 2 E+ spin-orbit perturbation have been given by otherauthors [40][41] as^(411117-(30 12E+,21) = 3 4.^(6.17)and(411_ 1_1 'Hs. 1 2 E+^2^ (6.18)Equations (6.15) and (6.16) can be made to resemble the above expressions if the reducedmatrix element is multiplied by a factor If. However, the relative signs of the 4 11_12interactions differ. This is an interesting result, but there is no doubt that the correctrelative signs are as in Equations (6.15) and (6.16) because it is not possible to get aconverged least squares fit if they are reversed. The conclusion is confirmed by a studyof the B4 11—X 4 E- (1,0) band done by Huang and co-workers[42].The values of the perturbation matrix elements for the B 411 v = 0 and v = 1 levelscan be used to determine the vibrational numbering of the a 2 E+ state. This determina-tion requires that the Born-Oppenheimer approximation holds, so that the perturbationmatrix element can be factorized:(411 ' 10-t 12E+ v') "7" (411 1Helectr onic 1 2 E+ ) (VI VI)^(6.19)The overlap integral, (v Iv') was calculated using numerical integration over the eigen-functions of modified Morse potentials given byU(r)^De [1 — eX13 -131r)1r—rel 2^(6.20)where3(r) = /30 + 01 (r — re ) + 02 (r — 7.0 2^(6.21)Overlap integrals were calculated for several likely assignments for the vibrational num-bering of the a 2 E+ state. The results of these calculations are found in Table 6.5.Chapter 6. Discussion^ 79x (11x + 1) (01x) (1AV )0 0.5144 0.8296 0.6201 0.6052 0.5384 1.1242 0.4051 0.2729 1.4843 0.2200 0.1235 1.781Table 6.5: Calculations of the anharmonic oscillator overlap integrals.The values of the perturbation matrix elements of the v = 1 level of the B 4II statedetermined by Huang at al.[42] and of the v = 0 level determined in this work arev = 111-11 2 E+, v' = x^1) = 28.7 cm -1 (6.22)(4 II, v = 011-11 2E+, v' = x) = 20.42 cm -1 (6.23)respectively. In the approximation that the perturbation matrix element is separable, theratio of these two matrix elements represents the ratio of the two corresponding overlapintegrals, i.e.(4II,v = 11711 2 E+, v' = x + 1)^(11x + 1) (4H , v 01 x 12E+, v' = x)^(01x)When this is compared to the values listed in Table 6.5, then the obvious conclusion isthat x = 2. Therefore, if w e x e is assumed to equal the value for the ground state, thenthe vibrational constants for the perturbing a 2 E+ state arewe = 1024.24 cm -1 ; (6.25)Be = 0.5508 cm'; (6.26)To (v = 0) = 10412. cm-1 . (6.27)= 1.41^(6.24)Chapter 7ConclusionsThe B4H—X4 E- (0,0) band of vanadium monoxide has been well characterized andthe constants have been determined by a least squares fit to spectra recorded at sub-Doppler and Doppler limited resolution. The fit of 3211 data points resulted in thedetermination of 39 constants with an r.m.s. error of 0.00038 cm -1 . Not only havethese highly precise data provided an improved determination of the parameters of theX4 E- state as compared to the values that had been determined previously from sub-Doppler spectra of the C 4 E+—X4 E- transition [10], but they have also afforded the firstcomplete analysis of the B 411 state.Earlier studies of the 13 411 state had only been done at Doppler-limited resolution.These studies had focussed mostly on regions where the hyperfine widths of the up-per state fortuitously cancelled the widths of the ground state, thus producing sharprotational lines; branches that were substantially hyperfine-broadened remained largelyunassigned. This was particularly true near the regions where the B4H state was heavilyperturbed by the a 2 E+ state.However, the use of the technique known as intermodulated fluorescence spectroscopyhas enabled the assignments of many of these branches. The information obtained in thiswork from the regions where the a 2 E+ /B411 interaction is greatest provide details of theperturbing state. These details confirm cp5 2 as the correct assignment for the valenceelectron configuration for the a 2 E+ state, the same configuration as for the X 4 E- state.By comparison, the configuration of the B 4H state is 82 7r.80Chapter 7. Conclusions^ 81The vibrational numbering for the a 2 E+ state has been determined using data fromthe (1,0) band of the B 4II—X 4 E- transition [37]. The v=0 and v=1 levels of theB4Il state are perturbed by the v = 2 and v = 3 levels of the a 2 E+ state respectively.This information has resulted in significant change for the value for the band origin ofthe v=0 level of the a 2 E+ state. When it had been believed that the v=0 level of thea2 E+ state was perturbing the v=0 level of the B 4 I1 state, the band origin had beenplaced at 12430 cm -1 [14], whereas the new vibrational assignment has shifted thisvalue to 10412 cm -1 .The eight hyperfine components arising from the 51 V nucleus (I=i) were generallywell resolved. This enabled the determination of magnetic hyperfine parameters forthe a 2 E+ , X4 E- and B 411 states. The Fermi contact interactions in the X 4 E- anda 2 E+ states produced by far the largest effects; this provides evidence that these statesarise from a configuration with appreciable V 4s atomic orbital character, and thus anon-zero probability that the electron can be found inside the nucleus.This work has shed considerable light on the B 4 11—X 4 E- (0,0) band of VO. Furtherwork could be done on the higher vibrational bands of this transition, which would leadto the characterization of the higher levels of the B 4 11 and a 2 E+ states. Studies of theseother bands would provide further proof of the vibrational numbering, and would giveinformation on the contours of the potential wells of the B 411 and a2 E+ states.Bibliography[1] A. J. Merer, Ann. Rev. Phys. Chem., 40, 407-438 (1989).[2] H. Spinrad, R. F. Wing, Ann. Rev. Aston. Astrophys., 7, 249-302 (1969).[3] G. P. Kuiper, W. Wilson, R. J. Cashman, Astrophys. J. 106, 243, (1947).[4] A. Lagerqvist, L.-E. Selin, Ark. Fys. 11, 429-430 (1957); A. Lagerqvist, L.-E. Selin,Ark. Fys. 12, 553-568 (1957).[5] K. D. Carlson, C. Moser, J. Chem. Phys., 44(9), 3259-3265 (1966).[6] P. H. Kasai, J. Chem. Phys., 49(11), 4979-4984 (1968).[7] D. Richards, R. F. Barrow, Nature, 217, 842 (1968).[8] D. Richards, R. F. Barrow, Nature, 219, 1244-1245 (1968).[9] W. H. Hocking, A. J. Merer, D. J. Milton, Can. J. Phys., 59, 266-270 (1981); A.S-C. Cheung, R. C. Hansen, A. M. Lyyra, A. J. Merer, J. Mol. Spec., 86, 526-533(1981).[10] A. S-C. Cheung, R. C. Hansen, A. J. Merer, J. Mod. 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Hirota, "High Resolution Spectroscopy of Transient Molecules," Springer-Verlag,New York (1985).[34] T. D. Varberg, R. W. Field, A. J. Merer, J. Chem. Phys., 95(3), 1563-1576 (1991).[35] A. S-C. Cheung, A. W. Taylor, A. J. Merer, J. Mol. Spec., 92, 391-409 (1982).[36] R. D. Suenram, G. T. Fraser, F. J. Lovas, G. W. Gillies, J. Mol. Spec., 148, 114-122(1991).[37] D. J. Clouthier, G. Huang, A. J. Merer, J. Mol. Spec., 153, 32-40 (1992).Bibliography^ 84[38] K. P. Huber, G. Herzberg, "Constants of Diatomic Molecules," Von Nostrand Rein-hold Company, Toronto, 1979.[39] F. Hund, Z. Physik, 63, 719 (1930).[40] I. Kovics, Can. J. Phys., 36, 329-351 (1958); I. Kovics,"Rotational Structure inthe Spectra of Diatomic Molecules," Adam Hilger Ltd, London (1969).[41] H. Ito, Y Ozaki, K. Suzuki, T. Kondow, K. Kuchitsu, J. Chem. Phys., 96(6),4195-4204 (1992).[42] G. Huang, S. Huang, D. Clouthier, A. J. Merer, unpublished results.[43] W. J. Childs, 0. Poulsen, L. S. Goodman, H. Crosswhite, Phys. Rev. A, 19(1),168-176 (1979).Appendix AThe Line Assignments of the VO B411—X4 E- (0,0) Band.85Appendix A. The Line Assignments of the VO B 411-X4 E - (0,0) Band.^86ASSIGN J= F=J-7/2TABLE OF ASSIGNED LINES OF THE VO 8-X (0,0) BANDF=J-5/2^F=J-3/2^F=J-1/2^F=J+1/2 F=J+3/2 F=J+5/2 F=J+7/2N= 0 SR21 1.5 12581.2817 12581.2225 12581.1414 12581.0383TR31 1.5 12645.1538 12645.0772 12644.9794N= 1 0P21 2.5 12577.1988* 12577.1187*SR21 2.5 12583.6604 12583.6193 12583.5639 12583.4936 12583.4088R3 0.5 12642.8454* 12642.6776*TR42 1.5 12707.7612* 12707.7447 12707.7200 12707.6865*N= 2 oP21 3.5 12577.6867 12577.6562 12577.6151 12577.5630 12577.4999* 12577.4263SR21 3.5 12586.0496* 12586.0279 12585.9966 12585.9542* 12585.9007 12585.8361 12585.7600*02 2.5 12574.3546 12574.3390 12574.3289 12574.31700024 0.5 12575.0443SR43 1.5 12708.0869 12708.0418 12707.9908 12707.9286SR43 1.5 12708.0869 12708.0501 12708.0022TR42 2.5 12713.6184* 12713.6131* 12713.6022 12713.5880 12713.5626 12713.5408TR42 2.5 12713.5880 12713.5711 12713.5512*N= 3 01)21 4.5 12577.9512 12577.9178 12577.8754 12577.8246 12577.7646 12577.6948SR21 4.5 12588.3567 12588.3316 12588.2979* 12588.2542* 12588.2018 12588.1410 12588.07010024 1.5 12574.5205*TR31 4.5 12652.2395* 12652.2235 12652.2003* 12652.1680 12652.1283 12652.0799 12652.0229* 12651.9576*TR42 3.5 12714.7036 12714.6976* 12714.6870* 12714.6685 12714.6487 12714.6248 12714.5957TR42 3.5 12714.6870* 12714.6727* 12714.6537* 12714.6307* 12714.6026*R4 1.5 12712.7205 12712.7123 12712.7123N. 4 0P21 5.5 12578.1296' 12578.0873 12578.0360 12577.9776 12577.9119SR21 5.5 12590.6336 12590.5836* 12590.5472 12590.5035 12590.4520 12590.3944 12590.3273TR31 5.5 12654.4918 12654.4650 12654.4316 12654.3914 12654.3439* 12654.2897 12654.2280TR42 4.5 12715.8731 12715.8678* 12715.8573 12715.8443 12715.8278 12715.8070 12715.7815 12715.7501TR42 4.5 12715.8678* 12715.8599* 12715.8467' 12715.8311* 12715.8103* 12715.7854* 12715.7547'R4 2.5 12707.6237* 12707.6362 12707.6601* 12707.6785*N= 5 P012 5.5 12519.7837*0P21 6.5 12578.3728* 12578.3503' 12578.3199 12578.2827* 12578.2389* 12578.1896* 12578.1324* 12578.0693SR21 6.5 12592.8316' 12592.8068* 12592.7753* 12592.7387* 12592.6948 12592.6445 12592.5874 12592.5235TR31 6.5 12656.7161 12656.6871 12656.6527 12656.6127 12656.5663* 12656.5134 12656.4542*0P12 5.5 12514.7981Ro21 6.5 12583.9408 12583.9198 12583.8929 12583.8611 12583.8236 12583.7804 12583.7311 12583.6757R4 3.5 12707.1893 12707.2005 12707.2154 12707.2366 12707.2629 12707.2939 12707.3295N. 6 P1 7.5 12523.4910* 12523.4581 12523.4193 12523.3734 12523.3222 12523.2650 12523.2005P012 6.5 12519.7056 12519.6943 12519.6800 12519.6623 12519.6390 12519.6109 12519.57530P21 7.5 12578.4697' 12578.4431 12578.4113* 12578.3728' 12578.3292* 12578.2794 12578.2243 12578.1631*TR31 7.5 12658.9177* 12658.8920 12658.8621' 12658.8266 12658.7866* 12658.7406 12658.6895 12658.63240P12 6.5 12513.8208* 12513.8116* 12513.8016 12513.7868 12513.7441 12513.7151 12513.6783R021 7.5 12584.9089 12584.8853 12584.8569 12584.8242 12584.7867 12584.7431 12584.6947 12584.6413R4 4.5 12706.5973 12706.6087 12706.6258 12706.6477 12706.6750 12706.7076 12706.7452 12706.7880*N= 7 P1 8.5 12523.1984 12523.1586 12523.1137 12523.0621 12523.0060 12522.9441p012 7.5 12519.4418 12519.4332 12519.4218 12519.4077 12519.3892 12519.3666 12519.3374 12519.30090P21 8.5 12578.4985 12578.4687* 12578.4358 12578.3967 12578.3531* 12578.3039 12578.2494 12578.1897*TR31 8.5 12661.0451 12661.0184 12660.9876 12660.9521 12660.9106 12660.8661 12660.8164 12660.76040P12 7.5 12512.6348* 12512.6263* 12512.5998 12512.5800 12512.5568 12512.5268 12512.4895Ro21 8.5 12585.8133 12585.7886 12585.7596 12585.7259 12585.6878 12585.6447 12585.5975 12585.5451S031 8.5 12651.1737 12651.1163* 12651.0805* 12651.0403* 12650.9950 12650.9451 12650.8903R4 5.5 12705.9000 12705.9159 12705.9373 12705.9634 12705.9946 12706.0309 12706.0720 12706.1175* indicates blended linesAppendix A. The Line Assignments of the VO B 41I-X4 E- (0,0) Band.^87^ASSIGN J"^F=J-7/2^F=J-5/2^F=J-3/2^F=J-1/2^F=J+1/2^F=J+3/2^F=J+5/2^F=J+7/2N= 8 P1^9.5^12522.9212*^12522.8888^12522.8536^12522.8138*^12522.7674^12522.7174^12522.6617^12522.6011^P012 8.5^12519.0952^12519.0865^12519.0754^12519.0615^12519.0434^12519.0206^12518.9911^12518.9517N014 6.5^12506.5908^12506.60890P21 9.5 12578.4278^12578.3924^12578.3531^12578.3089^12578.2601^12578.2064^12578.1478*01^9.5^12533.2621*^12533.2318^12533.1949^12533.1539*^12533.1075*^12533.0574^12533.0008^12532.9405OP12 8.5^12511.3694^12511.3603^12511.3491^12511.3345^12511.3160^12511.2930^12511.2625^12511.2230R021 9.5^12586.6554*^12586.6306*^12586.6002*^12586.5644* 12586.5257^12586.4831* 12586.4370^12586.3857S031 9.5^12652.2020*^12652.1739^12652.1420^12652.1063^12652.0659^12652.0211*^12651.9720^12651.9184R4 6.5^12705.1182*^12705.1379*^12705.1624*^12705.1913^12705.2253^12705.2633^12705.3061^12705.3530N= 9 P1 10.5^12522.4908^12522.4583^12522.4222^12522.3805^12522.3350^12522.2846^12522.2310^12522.1716P012 9.5^12518.6684^12518.6614^12518.6504^12518.6370^12518.6198^12518.5956 12518.5241N014 7.5^12504.3079^12504.3301^12504.3538^12504.3838^12504.4163^12504.4535^12504.4943^12504.53860P21 10.5^12578.3459^12578.3145^12578.2794*^12578.2389*^12578.1942*^12578.1460*^12578.0935*^12578.0360*01 10.5^12533.9086^12533.8755^12533.8385*^12533.7969^12533.7510^12533.7002^12533.6446*^12533.5857*0P12 9.5 12509.9889^12509.9707^12509.9477^12509.9170R021 10.5^12587.4328*^12587.4056^12587.3743^12587.3396^12587.3007^12587.2583^12587.2120^12587.1616S031 10.5^12653.1741^12653.1450^12653.1130^12653.0764^12653.0360^12652.9918^12652.9437^12652.8912R4 7.5^12704.2647*^12704.2886*^12704.3158^12704.3468^12704.3826^12704.4221^12704.4659^12704.5136N=10^P1 11.5^12521.9737^12521.9406^12521.9035^12521.8616^12521.8161^12521.7663^12521.7134^12521.6547P012 10.5^12518.1610^12518.1534^12518.1440^12518.1306^12518.1141^12518.0916^12518.0609^12518.0141N014 8.5 12502.0710*^12502.1125*^12502.15570P21 11.5^12578.1631*^12578.1296*^12578.0935*^12578.0535^12578.0095^12577.9614^12577.9088^12577.8523R032 10.5 12638.4305^12638.4179^12638.4009^12638.3761^12638.336201 11.5^12534.4628^12534.4294^12534.3912*^12534.3493*^12534.3027^12534.2529^12534.1978^12534.1395OP12 10.5 12508.5808 12508.5187^12508.4868^12508.4395R021 11.5^12588.1453*^12588.1167*^12588.0852^12588.0502^12588.0107^12587.9686*^12587.9228^12587.8733S031 11.5 12654.0601^12654.0269^12653.9903^12653.9500*^12653.9066*^12653.8587^12653.8076R4 8.5^12703.3515^12703.3767^12703.4052^12703.4382^12703.4750^12703.5158^12703.5600^12703.6078N=11^P1 12.5^12521.3705^12521.3357^12521.2979^12521.2563^12521.2104^12521.1607^12521.1071^12521.0496P012 11.5^12517.5698^12517.5647^12517.5561^12517.5442^12517.5280^12517.5062^12517.4746^12517.4211N014 9.5^12499.4310^12499.4558^12499.4832^12499.5144^12499.5490^12499.5871^12499.6275^12499.67090P21 12.5^12577.9088*^12577.8741*^12577.8366^12577.7961^12577.7520* 12577.7041^12577.6522^12577.59640R23 10.5 12573.7889*^12573.8071RP31 12.5^12642.1043 12642.0394^12642.0018^12641.9614*^12641.9176^12641.8705^12641.8199R032 11.5 12638.2908*^12638.2804*^12638.2637*^12638.2378^12638.191001 12.5 12534.8915^12534.8525^12534.8102^12534.7641^12534.7138^12534.6596^12534.6019OP12 11.5^ 12507.0028*^12506.9705*R021 12.5^12588.7927^12588.7631^12588.7308^12588.6950^12588.6561^12588.6137^12588.5687^12588.5193S031 12.5^12654.9472^12654.9171^12654.8835^12654.8467^12654.8068^12654.7635^12654.7162^12654.66595I132 11.5^12651.1473*^12651.1458*^12651.1414*^12651.1351^12651.1245^12651.1087^12651.0834*^12651.0373*1041 12.5 12727.4463^12727.4112^12727.3736^12727.3323 12727.2391^12727.18671041 12.5 12727.3736^12727.3323^12727.2874*N=12 P1 13.5^12520.6793^12520.6438^12520.6046^12520.5632^12520.5174^12520.4680^12520.4151^12520.3582P012 12.5^12516.8954^12516.8907^12516.8841^12516.8738^12516.8594^12516.8386^12516.8074^12516.7425NP13 11.5^12492.7091*N014 10.5^12496.8455^12496.8715^12496.8999^12496.9325*^12496.9670*^12497.0043^12497.0449^12497.08800P21 13.5^12577.5785*^12577.5441^12577.5068^12577.4661^12577.4216^12577.3737^12577.3223*^12577.26760R23 11.5 12573.5231*^12573.5415RP31 13.5 12641.8609*^12641.8269^12641.7899^12641.7487^12641.7053^12641.6583^12641.6081R032 12.5^ 12638.1060*^12638.1005*^12638.0913^12638.0762^12638.0505^12637.9927SR43 11.5 12710.3068*01 13.5^12535.2984^12535.2624^12535.2235^12535.1803*^12535.1340"^12535.0835*^12535.0305^12534.9729R021 13.5^12589.3738^12589.3439^12589.3106^12589.2748^12589.2357^12589.1937^12589.1481^12589.10005031 13.5^12655.7471^12655.7162^12655.6820^12655.6453^12655.6053^12655.5622^12655.5155*^12655.4660SR32 12.5 12651.9478*^12651.9329^12651.9079^12651.85101041 13.5^12728.4271^12728.3949 12728.3224^12728.2817^12728.2375^12728.1904^12728.1390Appendix A. The Line Assignments of the VO B 411-X4 E- (0,0) Band.^88ASSIGN^J" F=J-7/2 P=J-5/2 F=J-3/2 P=J-1/2 P=J+1/2 F=J+3/2 P=J+5/2N.13^P012 13.5 12516.1091 12516.0894 12516.06070P21^14.5 12577.1761 12577.1407 12577.1031 12577.0623 12577.0176 12576.9701 12576.91910R23 12.5 12573.1755RP31^14.5 12641.6256 12641.5941 12641.5594 12641.5216 12641.4811 12641.4377 12641.3913R032 13.5SR43 12.5 12710.2065* 12710.1968*01^14.5 12535.5796* 12535.5409' 12535.5023 12535.4591* 12535.4129 12535.3631 12535.3092OP12 13.5R021^14.5 12589.8888 12589.8581 12589.8248 12589.7885 12589.7493 12589.7072 12589.6621S031^14.5 12656.4877 12656.4560* 12656.4224 12656.3850 12656.3451 12656.3023 12656.2567*SR32 13.5 12652.7221' 12652.7233* 12652.7233 12652.7170* 12652.7043 12652.6813T041^14.5 12729.2625 12729.2249 12729.1839 12729.1405 12729.0941N=14^P1^15.5 12518.8711 12518.8223 12518.7693P012 14.5 12515.2612 12515.0899*N014 12.5 12491.3925 12491.4187* 12491.4496 12491.4835* 12491.59470P21^15.5 12576.6986 12576.6636 12576.6248 12576.5837 12576.5391 12576.4918 12576.44130R23 13.5 12572.7872 12572.7713 12572.7592 12572.7431* 12572.7456RP31^15.5 12641.3027 12641.2707 12641.2359 12641.1977 12641.1573 12641.1144R032 14.5SR43 13.5 12710.0851*01^15.5 12535.7680 12535.7308 12535.6911 12535.6474 12535.6009 12535.5508 12535.4976OP12 14.5R021^15.5 12590.3385 12590.3067 12590.2722* 12590.2363 12590.1971 12590.1549 12590.1098S031^15.5 12657.1692 12657.1374 12657.1030 12657.0267* 12656.9831 12656.9379SR32 14.5 12653.4246* 12653.4320* 12653.4357* 12653.4357* 12653.4320' 12653.4234* 12653.2593*T041^15.5 12730.1857 12730.1530 12730.1177 12730.0799 12730.0398 12729.9509N=15^P1^16.5 12518.0854 12518.0481 12518.0082 12517.9645 12517.9179 12517.8689 12517.8164P012 15.5 12514.3650 12514.3717 12514.3717 12514.1666 12514.1666PR13 14.5 12514.23660P21^16.5 12576.1470 12576.1113 12576.0726 12576.0312 12575.9865 12575.9392 12575.8890RP31^16.5 12640.9249* 12640.8914 12640.8563 12640.8181 12640.7777 12640.7349 12640.6897'R032 15.5SR43 14.5 12709.913601^16.5 12535.8665 12535.8288 12535.7883* 12535.7445 12535.6976 12535.6474* 12535.59500013 14.5 12499.9718R021 16.5 12590.7205 12590.6891 12590.6544 12590.6173 12590.5784 12590.5361 12590.4915S031 16.5 12657.7909 12657.7586 12657.7245 12657.6871 12657.6475 12657.6050 12657.5601SR32 15.5 12654.0713 12654.0829 12654.0898* 12654.0937* 12653.8960 12653.9026 12653.91661041 16.5N=16^P1^17.5 12517.0477 12517.0098 12516.9687 12516.9251 12516.8795 12516.8296 12516.7789*P012 16.5 12513.3695* 12513.1747 12513.1661 12513.1618 12513.1633'PR13 15.5 12513.2602 12513.2009*0P21^17.5 12575.5204 12575.4839 12575.4448 12575.4033 12575.3591 12575.3121 12575.262202^16.5 12571.8398' 12571.8471* 12571.6497'0R23 15.5 12571.7334* 12571.6994' 12571.6774* 12571.6614'RP31^17.5 12640.4893 12640.4561 12640.4203* 12640.3827 12640.3421 12640.2994 12640.2539R032 16.5SO42 16.5 12709.7896 12709.8066SR43 15.5 12709.6932 12709.6623 12709.6442 12709.6334* 12709.8241* 12709.8241* 12709.8241*01^17.5 12535.8743 12535.8359 12535.7944* 12535.7505* 12535.7035 12535.6540 12535.6009*0013 15.5 12498.0426 12498.0076* 12497.9831R021^17.5 12591.0369* 12591.0044* 12590.9695 12590.9327 12590.8929 12590.8510 12590.8067S031^17.5 12658.3528 12658.3205 12658.2854 12658.2482 12658.2088 12658.1667 12658.1220SR32 16.5 12654.6630* 12654.6792 12654.6904' 12654.4981* 12654.4956 12654.4981* 12654.5080*S3^15.5 12654.5204' 12654.5080' 12654.7000'OP34 14.5 12592.7057 12592.7387* 12592.7753* 12592.8137 12592.8550 12592.8989 12592.94511041^17.5 12731.7519 12731.7186 12731.6841 12731.5193F=J+7/212515.97921255g:7 192712641.341612637.741412535.2525*12503.602612589.613912656.207412652.607712729.043912518.713512515.129712491.637312576.387512572.904812641.0190'12637.435512535.441412501.8091*12590.0622*12656.889412653.306412729.901612517.761512514.195312575.835612640.640712637.074912535.5387*12590.444112657.512212653.946712730.7110'12516.722612513.1747*12575.209312640.206112636.658612709.8153*12535.5453■12590.759312658.074612654.527612592.993912731.4718Appendix A. The Line Assignments of the VO B 411-X4 E- (0,0) Band.ASSIGN^J"^F=J-7/2^F=J-5/2^F=J-3/2^F=J-1/2^F=J+1/2^F=J+3/2^F=J+5/289F=J+7/2N=17^P1^18.5 12515.8868 12515.8451 12515.8009 12515.7544 12515.7055 12515.6532 12515.5981P012 17.5 12512.2691 12512.1216 12512.1015* 12512.07000P21^18.5 12574.8179 12574.7812 12574.7417 12574.7002 12574.6558 12574.6090 12574.5593 12574.5071*02^17.5 12571.0148* 12570.9959* 12570.9818* 12570.9729*0R23 16.5 12571.0911RP31 18.5 12639.9973 12639.9637 12639.8499 12639.8072 12639.7620 12639.7139Ro32 17.5R3^16.512636.3427 12636.199612636.222612636.1864*12636.362812636.1747*12636.376212636.1696* 12636.1696* 12636.1747* 12636.1864*SO42 17.5 12709.4960*SR43 16.5 12709.4232 12709.3760 12709.5166 12709.5306 12709.5413* 12709.5449* 12709.5449* 12709.5413*01^18.5 12535.7919* 12535.7524* 12535.7109 12535.6667 12535.6195* 12535.5700 12535.5173 12535.4615OP12 17.5 12496.0916 12495.9445 12495.9239 12495.9071 12495.8959 12495.8877* 12495.8877 12495.89210013 16.5 12496.0211 12495.9685R021^18.5 12591.2865* 12591.2537 12591.2181 12591.1816* 12591.1416 12591.0994 12591.0548 12591.0078*S031 18.5 12658.8546* 12658.8214* 12658.7866 12658.7492* 12658.7099 12658.6680 12658.6238 12658.5770SR32 17.5 12655.2001* 12655.0575* 12655.0428* 12655.0354* 12655.0296* 12655.0296* 12655.0354* 12655.0488*OP34 15.5 12589.1728 12589.2081 12589.2446 12589.2838* 12589.3254* 12589.3692* 12589.4154 12589.46377041^18.5 12732.4292* 12732.3576 12732.3178 12732.2759* 12732.2314 12732.1843P4^15.5 12662.0149 12662.0480 12662.0850 12662.123011=18^P1^19.5 12514.7151 12514.6759 12514.6345 12514.5909 12514.5448 12514.4952 12514.4436Po12 18.5 12510.9771 12510.9476 12510.9248 12510.9078 12510.8945 12510.8850* 12510.8803* 12510.88030P21^19.5 12574.0398 12574.0029 12573.9634 12573.9213 12573.8771 12573.8301 12573.7795* 12573.7286*02^18.5 12570.2987* 12570.2710* 12570.2506*0R23 17.5 12570.3715 12570.4240 12570.4433*RP31^19.5 12639.4483 12639.4145 12639.3796 12639.3412 12639.3008 12639.2581 12639.2134 12639.1659R032 18.5 12635.7099 12635.6857 12635.6692 12635.6576*R3^17.5 12635.8354 12635.8574 12635.8729So42 18.5 12709.0344* 12709.0107 12708.9940 12708.9827* 12708.9768* 12708.9725* 12708.9768 12708.9850*SR43 17.5 12709.1035 12709.1594 12709.1829 12709.1983 12709.2099* 12709.2178* 12709.2197* 12709.2197*01^19.5 12535.6195 12535.5796 12535.5387 12535.4935 12535.4461 12535.3968 12535.3440 12535.2888OP12 18.5 12493.8355* 12493.8071 12493.7843 12493.7668 12493.7436 12493.73830013 17.5 12493.9076R021^19.5 12591.4687* 12591.4359* 12591.3989* 12591.3625 12591.3229 12591.2811 12591.2367 12591.18965031^19.5 12659.2950 12659.2623 12659.2274 12659.1900 12659.1505 12659.1091 12659.0647 12659.0196SR32 18.5 12655.5567 12655.5327 12655.5064* 12655.5012* 12655.4986* 12655.5012* 12655.5103*OP34 16.5 12585.5815 12585.6162 12585.6533 12585.6929 12585.7346 12585.7785 12585.8247 12585.8729T041^19.5 12733.1240 12733.0912 12733.0186'TR42 18.5 12729.3856 12729.3618 12729.3462 12729.3352* 12729.3298* 12729.3395P4^16.5 12658.5960 12658.6324* 12658.7059* 12658.7491* 12658.8356* 12658.8834N=19^P1^20.5 12513.4216 12513.2972 12513.2502 12513.2009 12513.1496 12513.0949p012 19.5 12509.7147 12509.6846 12509.6608 12509.6423 12509.6273 12509.6164* 12509.6091* 12509.6067PR13 18.5 12509.81060P21 20.5 12573.1854* 12573.1485 12573.1089 12573.0670 12573.0226 12572.9757 12572.9268* 12572.8751OP23 18.5 12530.7290 12530.7703 12530.7902 12530.80260024 17.5 12535.0733 12535.1045 12535.1385 12535.1733 12535.2109 12535.2504 12535.3342RP31 20.5 12638.8428 12638.8089 12638.7730 12638.7348 12638.6948 12638.6525 12638.6074* 12638.5609*R032 19.5 12635.1352 12635.1104 12635.0928 12635.0803* 12635.0725*R3^18.5 12635.2760 12635.2988* 12635.3151* 12635.3271*SO42 19.5 12708.6387* 12708.6142 12708.5976 12708.5861* 12708.5769* 12708.5755* 12708.5755* 12708.5798*SR43 18.5 12708.7339 12708.7798 12708.8035 12708.8207 12708.8336 12708.8424 12708.8481* 12708.8495*01^20.5 12535.3585 12535.3180 12535.2761 12535.2314 12535.1832* 12535.1340* 12535.0814* 12535.0266OP12 19.5 12491.6056 12491.5750 12491.5520 12491.5328 12491.5178 12491.5057 12491.4988 12491.4962oR12 19.5 12531.6505* 12531.6189 12531.5955* 12531.5772 12531.5610 12531.5490 12531.5419 12531.53860013 18.5 12491.7028 12491.7417Ro21 20.5 12591.5849* 12591.5511 12591.5152 12591.4774* 12591.4370* 12591.3957* 12591.3518* 12591.3046s031 20.5 12659.6750 12659.6419 12659.6068 12659.5699 12659.5301 12659.4887 12659.4448 12659.3993SR32 19.5 12655.9675 12655.9430* 12655.9267* 12655.9147* 12655.9076* 12655.9049* 12655.9049* 12655.9112*0P34 17.5 12581.9279 12581.9633 12582.0007 12582.0407 12582.0826 12582.1267 12582.1725 12582.22107041 20.5 12733.7366 12733.6325* 12733.5084* 12733.4628*TR42 19.5 12730.0289 12730.0053 12729.9887 12729.9770* 12729.9738*P4^17.5 12655.1281 12655.1627* 12655.2001 12655.2397* 12655.2811 12655.3250 12655.3705 12655.4185Appendix A. The Line Assignments of the VO B 411-X4 E - (0,0) Band.^90ASSIGN^J" F=J-7/2 F=J-5/2 F=J-3/2 F=J-1/2 F=J+1/2 F=J+3/2 F=J+5/2 F=J+7/2N=20 P1^21.5 12512.0449 12511.9627 12511.9178 12511.8718 12511.8225 12511.7696 12511.7161P012 20.5 12508.3644 12508.3345 12508.3106 12508.2916 12508.2751 12508.2626 12508.2535 12508.24850P21 21.5 12572.2561 12572.2183 12572.1782 12572.1361 12572.0922 12572.0455* 12571.9966 12571.9452OP23 19.5 12527.8280 12527.8626 12527.90380024 18.5 12532.1985 12532.2305 12532.2640 12532.3002 12532.3378 12532.3766 12532.4176 12532.4599SO42 20.5 12708.1900 12708.1672* 12708.1504* 12708.1379* 12708.1275* 12708.1249* 12708.1249* 12708.1257*SR43 19.5 12708.3147 12708.3536 12708.3777 12708.3954 12708.4091 12708.4195 12708.4267 12708.430601^21.5 12535.0084 12534.9681 12534.9257 12534.8805 12534.8332 12534.7834 12534.7308 12534.67570R12 20.5 12531.3282 12531.2973 12531.2736 12531.2526 12531.2365 12531.2238 12531.2138 12531.2079R021 21.5 12591.6335 12591.5996 12591.5636 12591.5255* 12591.4854 12591.4432 12591.3989* 12591.3519*S031 21.5 12659.9935 12659.9608 12659.9252 12659.8877 12659.8483 12659.8076 12659.7643 12659.7186SR32 20.5 12656.3134 12656.2750* 12656.2610* 12656.2526* 12656.2477* 12656.2477* 12656.2512*OP34 18.5 12578.2130 12578.2494* 12578.2870* 12578.3273* 12578.3695 12578.4134* 12578.4594* 12578.5072TR42 20.5 12730.6192 12730.5957 12730.5791 12730.5676* 12730.5598 12730.5559* 12730.5559* 12730.5584*P4^18.5 12651.6106 12651.6468 12651.6846 12651.7242 12651.7663 12651.8101 12651.8562 12651.9038N=21 P1^22.5 12510.5840 12510.5438 12510.5015 12510.4562 12510.4098 12510.3598 12510.3079 12510.2535P012 21.5 12506.9272 12506.8982 12506.8740 12506.8539 12506.8368 12506.8234 12506.8123* 12506.80590P21 22.5 12571.2499 12571.2120 12571.1718 12571.1296 12571.0855 12571.0391 12570.9903 12570.9392OP23 20.5 12524.8811 12524.9000 12524.9133 12524.92200024 19.5 12529.2467 12529.2797 12529.3134 12529.3492 12529.3866 12529.4258 12529.4666 12529.5092SO42 21.5 12707.6904 12707.6515* 12707.6397* 12707.6292 12707.6257* 12707.6237* 12707.6257*SR43 20.5 12707.8454 12707.8799 12707.9034 12707.9213* 12707.9362 12707.9475 12707.9561 12707.961501^22.5 12534.5716 12534.5309 12534.4882 12534.4428 12534.3953 12534.3453 12534.2927 12534.2379QR12 21.5 12530.9162 12530.8864* 12530.8614 12530.8416 12530.8243 12530.8098*. 12530.7982* 12530.7905R021 22.5 12591.6151 12591.5812 12591.5447 12591.5061 12591.4658* 12591.4235 12591.3793* 12591.3329NP24 19.5 12506.9436 12506.9842 12507.1159* 12507.1633S031 22.5 12660.2510* 12660.2180* 12660.1823 12660.1451 12660.1060 12660.0653 12660.0218 12659.9767SR32 21.5 12656.5956 12656.5733 12656.5561 12656.5438* 12656.5347* 12656.5295* 12656.5276* 12656.5295*OP34 19.5 12574.4373 12574.4739 12574.5124* 12574.5529 12574.5951* 12574.6390 12574.6849 12574.7329TR42 21.5 12731.1575 12731.1347 12731.1182 12731.1064 12731.0981*N=22 P1^23.5 12509.0390 12508.9985 12508.9559 12508.9105 12508.8633 12508.8144 12508.7622 12508.7082OP21 23.5 12570.1671 12570.1288 12570.0891 12570.0469 12570.0029 12569.9566 12569.9078 12569.8572OP23 21.5 12521.7978 12521.8247 12521.8428 12521.8557 12521.8652 12521.8717 12521.8754*0024 20.5 12526.2185 12526.2511 12526.2855 12526.3218 12526.3593 12526.3981 12526.4392 12526.4815RP31 23.5 12636.6812 12636.6097* 12636.5719 12636.5319 12636.4896 12636.4451P034 20.5 12592.1420 12592.1782 12592.2160 12592.2562 12592.2978 12592.3411 12592.3860* 12592.4332*SO42 22.5 12707.1406 12707.1194 12707.1022 12707.0899 12707.0809* 12707.0751* 12707.0743* 12707.0743*0P43 21.5 12661.4109 12661.4414 12661.4649 12661.4828 12661.4972 12661.5088 12661.5182 12661.5248SR43 21.5 12707.3260* 12707.3572 12707.3806 12707.3989 12707.4141 12707.4265 12707.4361 12707.443001^23.5 12534.0486 12534.0068 12533.9634 12533.9177 12533.8703 12533.8199 12533.7676 12533.71280R12 22.5 12530.4170 12530.3875 12530.3623 12530.3414 12530.3235 12530.3080 12530.2960 12530.2869R021 23.5 12591.5293 12591.4954 12591.4583 12591.4195 12591.3793* 12591.3371 12591.2930 12591.2465NP24 20.5 12502.7749 12502.8108 12502.8497 12502.8899 12502.9318 12502.9758 12503.0222* 12503.0686*S031 23.5 12660.4462 12660.4131 12660.3781* 12660.3411* 12660.3016 12660.2611 12660.2180' 12660.1729SR32 22.5 12656.8156 12656.7937 12656.7770* 12656.7646 12656.7552' 12656.7511* 12656.7469* 12656.7469*OP34 20.5 12570.6006 12570.6375 12570.6763 12570.7168 12570.7595 12570.8039 12570.8499 12570.8975TR42 22.5 12731.6442 12731.6227 12731.6067 12731.5946 12731.5856P4^20.5 12644.4292 12644.4661 12644.5051 12644.5459 12644.5884 12644.6327 12644.6788 12644.72671=23 P1^24.5 12507.4115 12507.3707 12507.3278 12507.2829 12507.2354 12507.1857 12507.0796P012 23.5 12503.8036PR13 22.5 12504.0196 12504.0414 12504.05600024 21.5 12523.1137 12523.1463 12523.1815 12523.2175 12523.2551 12523.2941 12523.3349 12523.3768*RP31 24.5 12635.8442 12635.8107* 12635.7732 12635.7348 12635.6939 12635.6084 12635.5623SO42 23.5 12706.5401 12706.5195 12706.5034 12706.4910 12706.4820 12706.4759* 12706.4733* 12706.4733*0P43 22.5 12658.7626 12658.7911 12658.8143 12658.8322* 12658.8473* 12658.8602 12658.8706 12658.8779SR43 22.5 12706.7558 12706.7855* 12706.8082 12706.8269 12706.8423 12706.8552 12706.8660 12706.874301^24.5 12533.4392 12533.3975 12533.3537 12533.3079 12533.2597 12533.2100 12533.1574 12533.10280R12 23.5 12529.8315* 12529.8025 12529.7779 12529.7563 12529.7374 12529.7218 12529.7082 12529.6979R021 24.5 12591.3421* 12591.3046* 12591.2658 12591.2256 12591.1829' 12591.1393* 12591.0921NP24 21.5 12498.6132 12498.6498 12498.6887 12498.7288 12498.7704 12498.8146* 12498.9068*S031 24.5 12660.5799 12660.5462* 12660.5114* 12660.4742* 12660.4345* 12660.3947' 12660.3524 12660.3074SR32 23.5 12656.9725' 12656.9513* 12656.9354* 12656.9224* 12656.9126' 12656.9064' 12656.9029* 12656.9029*TR42 23.5 12732.0797 12732.0579 12732.0424 12732.0313 12732.0227 12732.0167P4^21.5 12640.7644 12640.8022 12640.8413 12640.8824 12640.9249* 12640.9701 12641.0160' 12641.0646*ASSIGN^JoAppendix A. The Line Assignments of the VO B 411---X 4 E - (0,0) Band.F=J-7/2^F=J-5/2^F=J-3/2^F=J-1/2^F=J+1/2^F=J+3/2^F=J+5/2^F=J+7/2N=20^P1^21.5 12512.0449 12511.9627 12511.9178 12511.8718 12511.8225 12511.7696 12511.7161p012 20.5 12508.3644 12508.3345 12508.3106 12508.2916 12508.2751 12508.2626 12508.2535 12508.24850P21 21.5 12572.2561 12572.2183 12572.1782 12572.1361 12572.0922 12572.0455* 12571.9966 12571.9452OP23 19.5 12527.8280 12527.8626 12527.90380024 18.5 12532.1985 12532.2305 12532.2640 12532.3002 12532.3378 12532.3766 12532.4176 12532.4599SO42 20.5 12708.1900 12708.1672* 12708.1504* 12708.1379* 12708.1275* 12708.1249* 12708.1249* 12708.1257*SR43 19.5 12708.3147 12708.3536 12708.3777 12708.3954 12708.4091 12708.4195 12708.4267 12708.430601^21.5 12535.0084 12534.9681 12534.9257 12534.8805 12534.8332 12534.7834 12534.7308 12534.67570R12 20.5 12531.3282 12531.2973 12531.2736 12531.2526 12531.2365 12531.2238 12531.2138 12531.2079R021 21.5 12591.6335 12591.5996 12591.5636 12591.5255* 12591.4854 12591.4432 12591.3989* 12591.3519*S031^21.5 12659.9935 12659.9608 12659.9252 12659.8877 12659.8483 12659.8076 12659.7643 12659.7186SR32 20.5 12656.3134 12656.2750* 12656.2610* 12656.2526* 12656.2477* 12656.2477* 12656.2512*0P34 18.5 12578.2130 12578.2494* 12578.2870* 12578.3273* 12578.3695 12578.4134* 12578.4594* 12578.5072TR42 20.5 12730.6192 12730.5957 12730.5791 12730.5676* 12730.5598 12730.5559* 12730.5559* 12730.5584*P4^18.5 12651.6106 12651.6468 12651.6846 12651.7242 12651.7663 12651.8101 12651.8562 12651.9038N=21^P1^22.5 12510.5840 12510.5438 12510.5015 12510.4562 12510.4098 12510.3598 12510.3079 12510.2535P012 21.5 12506.9272 12506.8982 12506.8740 12506.8539 12506.8368 12506.8234 12506.8123* 12506.80590P21 22.5 12571.2499 12571.2120 12571.1718 12571.1296 12571.0855 12571.0391 12570.9903 12570.93920P23 20.5 12524.8811 12524.9000 1 2524.9133 12524.92200024 19.5 12529.2467 12529.2797 12529.3134 12529.3492 12529.3866 12529.4258 12529.4666 12529.5092SO42 21.5 12707.6904 12707.6515* 12707.6397* 12707.6292 12707.6257* 12707.6237* 12707.6257*SR43 20.5 12707.8454 12707.8799 12707.9034 12707.9213* 12707.9362 12707.9475 12707.9561 12707.961501^22.5 12534.5716 12534.5309 12534.4882 12534.4428 12534.3953 12534.3453 12534.2927 12534.2379oR12 21.5 12530.9162 12530.8864* 12530.8614 12530.8416 12530.8243 12530.8098* 12530.7982* 12530.7905R021 22.5 12591.6151 12591.5812 12591.5447 12591.5061 12591.4658* 12591.4235 12591.3793* 12591.3329NP24 19.5 12506.9436 12506.9842 12507.1159* 12507.1633s031 22.5 12660.2510• 12660.2180* 12660.1823 12660.1451 12660.1060 12660.0653 12660.0218 12659.9767SR32 21.5 12656.5956 12656.5733 12656.5561 12656.5438* 12656.5347* 12656.5295* 12656.5276* 12656.5295*OP34 19.5 12574.4373 12574.4739 12574.5124* 12574.5529 12574.5951* 12574.6390 12574.6849 12574.7329TR42 21.5 12731.1575 12731.1347 12731.1182 12731.1064 12731.0981*N=22^P1^23.5 12509.0390 12508.9985 12508.9559 12508.9105 12508.8633 12508.8144 12508.7622 12508.70820P21 23.5 12570.1671 12570.1288 12570.0891 12570.0469 12570.0029 12569.9566 12569.9078 12569.8572OP23 21.5 12521.7978 12521.8247 12521.8428 12521.8557 12521.8652 12521.8717 12521.8754*0024 20.5 12526.2185 12526.2511 12526.2855 12526.3218 12526.3593 12526.3981 12526.4392 12526.4815RP31 23.5 12636.6812 12636.6097• 12636.5719 12636.5319 12636.4896 12636.4451P034 20.5 12592.1420 12592.1782 12592.2160 12592.2562 12592.2978 12592.3411 12592.3860* 12592.4332*SO42 22.5 12707.1406 12707.1194 12707.1022 12707.0899 12707.0809* 12707.0751* 12707.0743* 12707.0743*0p43 21.5 12661.4109 12661.4414 12661.4649 12661.4828 12661.4972 12661.5088 12661.5182 12661.5248SR43 21.5 12707.3260* 12707.3572 12707.3806 12707.3989 12707.4141 12707.4265 12707.4361 12707.443001^23.5 12534.0486 12534.0068 12533.9634 12533.9177 12533.8703 12533.8199 12533.7676 12533.71280R12 22.5 12530.4170 12530.3875 12530.3623 12530.3414 12530.3235 12530.3080 12530.2960 12530.2869R021 23.5 12591.5293 12591.4954 12591.4583 12591.4195 12591.3793* 12591.3371 12591.2930 12591.2465NP24 20.5 12502.7749 12502.8108 12502.8497 12502.8899 12502.9318 12502.9758 12503.0222* 12503.0686*S031 23.5 12660.4462 12660.4131 12660.3781* 12660.3411* 12660.3016 12660.2611 12660.2180* 12660.1729SR32 22.5 12656.8156 12656.7937 12656.7770* 12656.7646 12656.7552* 12656.7511* 12656.7469* 12656.7469*OP34 20.5 12570.6006 12570.6375 12570.6763 12570.7168 12570.7595 12570.8039 12570.8499 12570.8975TR42 22.5 12731.6442 12731.6227 12731.6067 12731.5946 12731.5856P4^20.5 12644.4292 12644.4661 12644.5051 12644.5459 12644.5884 12644.6327 12644.6788 12644.7267N=23^P1^24.5 12507.4115 12507.3707 12507.3278 12507.2829 12507.2354 12507.1857 12507.0796P012 23.5 12503.8036PR13 22.5 12504.0196 12504.0414 12504.05600024 21.5 12523.1137 12523.1463 12523.1815 12523.2175 12523.2551 12523.2941 12523.3349 12523.3768*RP31 24.5 12635.8442 12635.8107* 12635.7732 12635.7348 12635.6939 12635.6084 12635.5623s042 23.5 12706.5401 12706.5195 12706.5034 12706.4910 12706.4820 12706.4759* 12706.4733* 12706.4733*0p43 22.5 12658.7626 12658.7911 12658.8143 12658.8322* 12658.8473* 12658.8602 12658.8706 12658.8779SR43 22.5 12706.7558 12706.7855* 12706.8082 12706.8269 12706.8423 12706.8552 12706.8660 12706.874301^24.5 12533.4392 12533.3975 12533.3537 12533.3079 12533.2597 12533.2100 12533.1574 12533.10280R12 23.5 12529.8315* 12529.8025 12529.7779 12529.7563 12529.7374 12529.7218 12529.7082 12529.6979R021 24.5 12591.3421* 12591.265812591.3046* 12591.2256 12591.1829* 12591.1393* 12591.0921NP24 21.5 12498.6132 12498.6498 12498.6887 12498.7288 12498.7704 12498.8146* 12498.9068*S031 24.5 12660.5799 12660.5462* 12660.5114* 12660.4742• 12660.4345* 12660.3947• 12660.3524 12660.3074SR32 23.5 12656.9725* 12656.9513* 12656.9354* 12656.9224* 12656.9126* 12656.9064' 12656.9029* 12656.9029*TR42 23.5 12732.0797 12732.0579 12732.0424 12732.0313 12732.0227 12732.0167P4^21.5 12640.7644 12640.8022 12640.8413 12640.8824 12640.9249* 12640.9701 12641.0160* 12641.0646*91Appendix A. The Line Assignments of the VO B 41I-X 4 E - (0,0) Band.^92ASSIGN^J" F=.1-7/2 F=J-5/2 F=J-3/2 F=.1-1/2 FuJ+1/2 F=J+3/2 F=J+5/2 FzJ+7/2N=24^P1^25.5 12505.7014 12505.6175 12505.5724 12505.5248 12505.4752 12505.4236 12505.3696P012 24.5 12502.1172 12502.0896PR13 23.5 12502.3637 12502.3834 12502.3972OP23 23.5 12515.4614 12515.4837 12515.5013 12515.5140 12515.5241 12515.5315* 12515.5362*0024 22.5 12519.9318 12519.9655 12520.0006 12520.0364 12520.0744 12520.1134 12520.1540 12520.1957P034 22.5 12586.4447* 12586.4814* 12586.5199* 12586.5599* 12586.6002* 12586.6454 12586.6903 12586.7368SO42 24.5 12705.8897 12705.8699 12705.8542 12705.8421 12705.8328 12705.8267 12705.8231* 12705.8231*0P43 23.5 12656.0655 12656.0926 12656.1152 12656.1330 12656.1487 12656.1619 12656.1727 12656.1815SR43 23.5 12706.1358 12706.1635 12706.1858 12706.2045 12706.2206 12706.2343 12706.2456 12706.254804^22.5 12660.5351 12660.5740 12660.6134 12660.6548 12660.6982 12660.7428 12660.7899 12660.838301^25.5 12532.7461 12532.7044 12532.6602 12532.6143 12532.5661 12532.5157 12532.4632 12532.40850R12 24.5 12529.1615 12529.1333* 12529.1082 12529.0869* 12529.0670 12529.0506* 12529.0368* 12529.0252*R021 25.5 12591.1561 12591.1206 12591.0838 12591.0450 12591.0045* 12590.9617 12590.9170 12590.8707P023 23.5 12536.8237 12536.8499 12536.8699 12536.8868 12536.9007 12536.9122 12536.9212* 12536.9277S031 25.5 12660.6518* 12660.6181 12660.5829* 12660.5462* 12660.5074 12660.4670* 12660.4239* 12660.3801SR32 24.5 12657.0300* 12657.0183* 12657.0084* 12657.0012* 12656.9975* 12656.9972*TR42 24.5 12732.4642 12732.4449 12732.4292 12732.4172P4^22.5 12637.0505* 12637.0886 12637.1285* 12637.1699* 12637.2127* 12637.2574 12637.3040 12637.3518*N=25^P1^26.5 12503.9113 12503.8697 12503.8265 12503.7343 12503.6841 12503.6328 12503.5786P012 25.5 12500.3496 12500.3218 12500.2770 12500.2580 12500.2416 12500.2282 12500.2168PR13 24.5 12500.6258 12500.6439 12500.6570 12500.6670OP23 24.5 12512.1787 12512.2002 12512.2167 12512.2296 12512.2399 12512.24800024 23.5 12516.6734 12516.7073 12516.7425 12516.7789 12516.8157 12516.8561* 12516.8954* 12516.9382P034 23.5 12583.5088 12583.5461 12583.5852* 12583.6254 12583.6669* 12583.7102 12583.7551 12583.8016SO42 25.5 12705.1891* 12705.1701* 12705.1549 12705.1426 12705.1334* 12705.1271 12705.1230*0P43 24.5 12653.3186 12653.3445 12653.3664 12653.3848 12653.4009* 12653.4146* 12653.4258* 12653.4356*SR43 24.5 12705.4647 12705.4914 12705.5136 12705.5324 12705.5486 12705.5628 12705.5750 12705.585104^23.5 12657.8127* 12657.8510* 12657.8913 12657.9333 12657.9771 12658.0220 12658.0690 12658.117301^26.5 12531.9707* 12531.9293* 12531.8847 12531.8385 12531.7900* 12531.7396 12531.6871 12531.63230R12 25.5 12528.4092 12528.3811 12528.3564 12528.3345 12528.3146 12528.2972 12528.2826 12528.2703R021 26.5 12590.8681* 12590.8326 12590.7955 12590.7565* 12590.7154* 12590.6724 12590.6280 12590.5819*P023 24.5 12534.5477 12534.5716* 12534.5922 12534.6082 12534.6219* 12534.6334 12534.6430 12534.6499NP24 23.5 12490.1268* 12490.1659* 12490.2066* 12490.2458* 12490.2906* 12490.3353 12490.3817S031 26.5 12660.6613 12660.6275 12660.5922 12660.5554 12660.5167* 12660.4765* 12660.4345* 12660.3898*SR32 25.5 12657.0996* 12657.0798 12657.0508* 12657.0412* 12657.0343 12657.0298* 12657.0280*TR42 25.5 12732.7972 12732.7781 12732.7635 12732.7517N=26^P1^27.5 12501.9109* 12501.8617 12501.7065P012 26.5 12498.5012 12498.4743 12498.4503 12498.4288 12498.4097 12498.3927PR13 25.5 12498.8066 12498.8240 12498.8364 12498.8526OP23 25.5 12508.8201 12508.8417 12508.8571 12508.8699 12508.8802 12508.8881 12508.8942 12508.89770024 24.5 12513.4078 12513.4444 12513.4823 12513.5210 12513.5619 12513.6034P034 24.5 12580.5147 12580.5520 12580.5911 12580.6314 12580.6736 12580.7167 12580.7615 12580.8080SO42 26.5 12704.4378 12704.4194* 12704.4045 12704.3925 12704.3827* 12704.3769* 12704.3727* 12704.3727*SR43 25.5 12704.7428 12704.7686 12704.7904 12704.8095 12704.8262* 12704.8405* 12704.8534* 12704.8643*04^24.5 12655.0400 12655.0789 12655.1200 12655.1627* 12655.2061 12655.2513 12655.2983 12655.346601^27.5 12531.1156 12531.0729 12531.0284 12530.9824 12530.9336 12530.8834 12530.8309 12530.77590R12 26.5 12527.5768 12527.5488 12527.5239 12527.5015 12527.4816 12527.4639 12527.4479 12527.4350R021 27.5 12590.5125 12590.4767 12590.4395* 12590.3998 12590.3587 12590.3157 12590.2722* 12590.2251P023 25.5 12532.2039 12532.2268 12532.2460 12532.2640* 12532.2764 12532.2878 12532.2973 12532.3046PR24 24.5 12536.7217 12536.7585 12536.7975 12536.8376 12536.8784 12536.9212* 12536.9655 12537.01105031 27.5 12660.6083 12660.5740* 12660.5394 12660.5021 12660.4640 12660.4239* 12660.3813* 12660.3382*SR32 26.5 12657.0508* 12657.0343* 12657.0220* 12657.0120* 12657.0046* 12657.0012* 12656.9975*TR42 26.5 12733.0782 12733.0603 12733.0459 12733.0338Appendix A. The Line Assignments of the VO B 411-X4 E- (0,0) Band.^93ASSIGN^J" F=J-7/2 F=J-5/2 F=J-3/2 F=J-1/2 F=J+1/2 F=J+3/2 F=J+5/2 FEJ+7/2N=27^P1^28.5 12500.0481 12500.0036 12499.9575 12499.8606 12499.8089 12499.7546P012 27.5 12496.5730 12496.5456 12496.5223 12496.5010 12496.4815 12496.4644 12496.4489 12496.4362PR13 26.5 12496.9068 12496.9224* 12496.9356 12496.9447 12496.95690024 25.5 12509.9273 12509.9619 12509.9968 12510.0334 12510.0722 12510.1109 12510.1512 12510.1930P034 25.5 12577.4619* 12577.4997* 12577.5382' 12577.5795* 12577.6208* 12577.6642 12577.7091* 12577.7549•SR43 26.5 12703.9698 12703.9948 12704.0167 12704.0357 12704.0524 12704.0676 12704.0808 12704.092304^25.5 12652.2176 12652.2572 12652.2982 12652.3406 12652.3846 12652.4301 12652.4774 12652.525701^28.5 12530.1834 12530.1401 12530.0952* 12530.0483* 12529.9995 12529.9487 12529.8960* 12529.8411*OR12 27.5 12526.6672' 12526.6385 12526.6129 12526.5910* 12526.5710 12526.5525 12526.5368 12526.5223R021 28.5 12590.0895 12590.0533* 12590.0151* 12589.9759 12589.9345 12589.8917* 12589.8473 12589.8009P023 26.5 12529.7927 12529.8150 12529.8335* 12529.8495 12529.8632 12529.8751 12529.8853 12529.8928*PR24 25.5 12534.3354 12534.4504 12534.4917 12534.5337 12534.5781 12534.6219*S031 28.5 12660.4925 12660.4592 12660.4239* 12660.3870* 12660.3488 12660.3074* 12660.2667 12660.2232SR32 27.5 12656.9754* 12656.9569 12656.9418' 12656.9195 12656.9126 12656.9064* 12656.9033*TR42 27.5 12733.3078 12733.2900 12733.2757 12733.2645N=28^P1^29.5 12498.0601 12498.0176 12497.9735 12497.8788 12497.8302 12497.7786 12497.7251P012 28.5 12494.5657 12494.5392 12494.5158 12494.4930 12494.4743 12494.4573 12494.4409 12494.4278PR13 27.5 12494.9283 12494.9433 12494.9561 12494.9644 12494.9714 12494.9762 12494.9789* 12494.9796*OP23 27.5 12501.8746* 12501.8935*0024 26.5 12506.4412 12506.4752 12506.5106 12506.5477 12506.5856 12506.6250 12506.6647 12506.7060P034 26.5 12574.3502* 12574.3881* 12574.4266 12574.4679 12574.5098* 12574.5529* 12574.5975* 12574.6435*OP43 27.5 12644.7789 12644.8031 12644.8244 12644.8435 12644.8599 12644.8748 12644.8885 12644.9002SR43 27.5 12703.1450 12703.1701 12703.1914 12703.2105 12703.2278 12703.2428 12703.2572 12703.269301^29.5 12529.1765 12529.1333 12529.0869* 12529.0407 12528.9913 12528.9404 12528.8870 12528.8324OR12 28.5 12525.6821 12525.6543 12525.6291* 12525.6065 12525.5858 12525.5672 12525.5500 12525.5351R021 29.5 12589.5988 12589.5619 12589.5234* 12589.4841 12589.4431 12589.4000* 12589.3555* 12589.3090*P023 27.5 12527.3147 12527.3359 12527.3540 12527.3697 12527.3834 12527.3949 12527.4049 12527.4135PR24 26.5 12532.1678*S031 29.5 12660.3145* 12660.2805 12660.2452 12660.2085 12660.1702* 12660.1304 12660.0884 12660.0450SR32 28.5 12656.8018 12656.7876 12656.7744 12656.7646* 12656.7552* 12656.7512* 12656.7488*TR42 28.5 12733.4850 12733.4684 12733.4544 12733.4427N=29^P1^30.5 12495.9518 12495.8658 12495.8200 12495.7726* 12495.7229* 12495.6705 12495.6167P012 29.5 12492.4797 12492.4538 12492.4304 12492.4087 12492.3891 12492.3713 12492.3554 12492.3411PR13 28.5 12492.8708 12492.8868*OP23 28.5 12498.2875 12498.3059 12498.3209 12498.3336 12498.3440 12498.35310024 27.5 12502.8778 12502.9117 12502.9475 12502.9847 12503.0222 12503.0620 12503.1024 12503.1435P034 27.5 12571.1786* 12571.2171 12571.2568 12571.2973 12571.3393 12571.3825 12571.4272 12571.47330P43 28.5 12641.8325* 12641.8562* 12641.8786* 12641.8961* 12641.9128* 12641.9286* 12641.9424* 12641.9548*01^30.5 12528.1004 12528.0564 12528.0108 12527.9629 12527.9133* 12527.8626 12527.8091 12527.7540OR12 29.5 12524.6283 12524.6003 12524.5755* 12524.5523 12524.5308 12524.5113 12524.4947* 12524.4787R021 30.5 12589.0400 12589.0030 12588.9645 12588.9245 12588.8832 12588.8406 12588.7955' 12588.7489P023 28.5 12524.7694 12524.7897 12524.8074 12524.8231 12524.8365 12524.8484 12524.8582* 12524.8666PR24 27.5 12529.3603* 12529.3958 12529.4745 12529.5148 12529.5570 12529.6007 12529.6453S031 30.5 12660.0732 12660.0392 12660.0040 12659.9676 12659.9291* 12659.8877* 12659.8483* 12659.8048SR32 29.5 12656.6009 12656.5833 12656.5687 12656.5561 12656.5383* 12656.5330* 12656.5295*TR42 29.5 12733.6097 12733.5937 12733.5799 12733.5685N=30^P1^31.5 12493.7668 12493.7246 12493.6796 12493.6335 12493.5854 12493.5358 12493.4840 12493.4307P012 30.5 12490.3167 12490.2906 12490.2672 12490.2458 12490.2253 12490.2066 12490.1914 12490.1768PR13 29.5 12490.7356 12490.7504 12490.7613 12490.7700 12490.7767 12490.7818OP23 29.5 12494.6253 12494.6424 12494.6585 12494.6708 12494.6806 12494.6896 12494.6964 12494.70190024 28.5 12499.2383 12499.2727 12499.3089 12499.3455 12499.3840 12499.4237 12499.4634 12499.50500P43 29.5 12638.8355 12638.8590 12638.8797 12638.8993 12638.9166 12638.9319 12638.9461 12638.958704^28.5 12643.4486 12643.4896 12643.5312 12643.5749 12643.6193 12643.6654 12643.7127 12643.761701^31.5 12526.9613 12526.9168 12526.8709 12526.8226 12526.7724 12526.7210 12526.6672* 12526.6109*0R12 30.5 12523.5110* 12523.4851* 12523.4581* 12523.4351 12523.4131 12523.3929 12523.3745* 12523.3582R021 31.5 12588.4135 12588.3763* 12588.3385 12588.2979* 12588.2563 12588.2124 12588.1679 12588.1211*P023 29.5 12522.1567 12522.1767 12522.1943 12522.2089 12522.2224 12522.2341 12522.2446 12522.2527PR24 28.5 12526.9261 12526.9680 12527.0117 12527.0555S031 31.5 12659.7689* 12659.7346 12659.6995 12659.6632 12659.6252 12659.5850 12659.5437 12659.5011SR32 30.5 12656.3191 12656.3023 12656.2872*TR42 30.5 12733.6814 12733.6659 12733.6526 12733.6414Appendix A. The Line Assignments of the VO B 411-X4 E - (0,0) Band.^94ASSIGN^J" F=J-7/2 F=J-5/2 F=J-3/2 F=J-I/2 F=J+1/2 F=J+3/2 F=J+5/2 F=J+7/2N=31 P1^32.5 12491.5057 12491.4627 12491.4187 12491.3716 12491.3236 12491.2738 12491.2223 12491.1694OP23 30.5 12490.8856* 12490.9027* 12490.9179 12490.9304 12490.9406 12490.9504 12490.9577 12490.96410024 29.5 12495.5232 12495.5585 12495.5940 12495.6310 12495.6705 12495.7085 12495.7490 12495.78980P43 30.5 12635.7876 12635.8107* 12635.8320 12635.8509 12635.8688 12635.8845* 12635.8989* 12635.912504^29.5 12640.4247 12640.4656 12640.5079 12640.5517* 12640.5969 12640.6428 12640.6899* 12640.738801^32.5 12525.7686 12525.7235 12525.6766 12525.6286 12525.5779 12525.5256 12525.4716 12525.4152OR12 31.5 12522.3403 12522.3134 12522.2868* 12522.2635 12522.2408 12522.2199* 12522.2013 12522.1839*R021 32.5 12587.7195 12587.6822 12587.6435 12587.6030* 12587.5604* 12587.5176 12587.4728 12587.4259P023 30.5 12519.4768 12519.4962 12519.5132* 12519.5280 12519.5414 12519.5531 12519.5631* 12519.5718*PR24 29.5 12524.1136* 12524.1522 12524.1891* 12524.2291* 12524.2694 12524.3111 12524.3545 12524.3982S031 32.5 12659.3671 12659.3317 12659.2950* 12659.2573* 12659.2179 12659.1767 12659.1340SR32 31.5 12655.9737 12655.9573 12655.9430* 12655.9112* 12655.9074*OP34 29.5 12533.3270 12533.3679 12533.4084 12533.4508 12533.4950 12533.5395 12533.5857* 12533.6332*TR42 31.5 12733.7005 12733.6850 12733.6720 12733.6611N=32 0024 30.5 12491.7320* 12491.8035 12491.8405 12491.8792 12491.9188 12491.9591 12492.000004^30.5 12637.5681* 12637.6163* 12637.6655*01^33.5 12524.5366* 12524.4899* 12524.4429 12524.3931 12524.3424 12524.2894 12524.2345 12524.1779OR12 32.5 12521.1312 12521.1031 12521.0763 12521.0513* 12521.0281 12521.0068 12520.9869 12520.9685R021 33.5 12586.9576 12586.9202 12586.8812 12586.8411 12586.7989 12586.7554 12586.6629•P023 31.5 12516.7789* 12516.7931* 12516.8051* 12516.8157* 12516.8236PR24 30.5 12521.3910* 12521.4273 12521.4663 12521.5051 12521.5456 12521.5878 12521.6307 12521.6737*S031 33.5 12658.9703 12658.9358 12658.9009 12658.8266* 12658.7866* 12658.7467*0P32 32.5 12587.6103 12587.5939 12587.5798 12587.5675 12587.5571* 12587.5474* 12587.5433* 12587.5389*OP34 30.5 12528.8801 12528.9198 12528.9615 12529.0041 12529.0481* 12529.0927 12529.1395OR34 30.5 12592.7458 12592.7862 12592.8274 12592.8701 12592.9143 12592.9597 12593.0060 12593.0536TR42 32.5 12733.6659 12733.6526* 12733.6374 12733.6270N=33 01^34.5 12523.2869 12523.2403 12523.1906* 12523.1410 12523.0889 12523.0347 12522.9791 12522.9212*OR12 33.5 12519.8764* 12519.8472* 12519.8218* 12519.7974 12519.7751* 12519.7535 12519.7336*R021 34.5 12586.1288 12586.0908 12586.0515* 12586.0106 12585.9690 12585.9253 12585.8800* 12585.8333*P023 32.5 12513.9158 12513.9339 12513.9504 12513.9653 12513.9785*PR24 31.5 12518;8830*S031 34.5 12658.4759 12658.4416 12658.4067 12658.3702 12658.3320 12658.2929 12658.2519* 12658.2088*0P32 33.5 12585.0965 12585.0804 12585.0660 12585.0540 12585.0426 12585.0355 12585.0282* 12585.0252'SR32 33.5 12655.0929 12655.0789* 12655.0628 12655.0400 12655.0340 12655.0269OP34 31.5 12524.3714 12524.4114 12524.4531 12524.4956* 12524.5401* 12524.5850* 12524.6316* 12524.67880R34 31.5 12590.2813* 12590.3221* 12590.3641 12590.4070 12590.4520* 12590.4969' 12590.5430* 12590.5906*TR42 33.5 12733.5774 12733.5629 12733.5510* 12733.5398 12733.5318 12733.5250 12733.5206* 12733.5180*N=34 01^35.5 12522.0567 12522.0078 12521.9578 12521.9061 12521.8523 12521.7978* 12521.7397 12521.68040R12 34.5 12518.6370 12518.6100 12518.5603* 12518.5365* 12518.5150R021 35.5 12585.2334 12585.1949 12585.1557* 12585.1144 12585.0719 12585.0292* 12584.9826* 12584.9363*S031 35.5 12657.9178 12657.8838 12657.8127* 12657.7741 12657.7353 12657.6947 12657.6524*01,32 34.5 12582.5195 12582.5032 12582.4896 12582.4776 12582.4673 12582.4593 12582.4527 12582.4482OP34 32.5 12519.8835* 12519.9264* 12519.9712 12520.0161* 12520.0620 12520.1101*0R34 32.5 12587.7549 12587.7957 12587.8379 12587.8812' 12587.9703* 12588.0177 12588.0650TR42 34.5 12733.4340* 12733.4216 12733.4089 12733.3981 12733.3897 12733.3837* 12733.3793* 12733.3773*N=35 01^36.5 12520.9046 12520.8541 12520.8023 12520.7481 12520.6921 12520.6351 12520.5745 12520.5133OP12 35.5 12444.6630* 12444.6346* 12444.6108* 12444.5634* 12444.5398*OR12 35.5 12517.4454* 12517.3932 12517.36920013 34.5 12445.2668'R021 36.5 12584.2699 12584.2315* 12584.1501 12584.1077 12584.0642' 12584.0183 12583.9718*P023 34.5 12508.0876 12508.1055 12508.1209 12508.1346 12508.1473 12508.1587 12508.1682 12508.17815031 36.5 12657.2960 12657.2622 12657.2271 12657.1908 12657.0298*0P34 33.5 12515.1684* 12515.2091 12515.3381 12515.3835 12515.4301 12515.4777oR34 33.5 12585.1664 12585.2069 12585.2496' 12585.2927 12585.3373 12585.3828 12585.4296 12585.4767TR42 35.5 12733.2376 12733.1934 12733.1873 12733.1834* 12733.1806*P4^33.5 12592.8949 12592.9382* 12592.9802 12593.0238* 12593.1170 12593.1650 12593.2141Appendix A. The Line Assignments of the VO B 411-X 4 E - (0,0) Band.^95ASSIGN^J" F=J-7/2 F=J-5/2 F=J-3/2 F=J-1/2 F=J+1/2 F=J+3/2 F=J+5/2 F=J+7/2N=36 OP12 36.5 12441.3164* 12441.2874* 12441.2589* 12441.2321* 12441.2032* 12441.1823* 12441.1576* 12441.1380*0013 35.5 12441.9117*P023 35.5 12505.0750 12505.0915 12505.1079* 12505.1209 12505.1334 12505.1452 12505.15470P32 36.5 12577.1780* 12577.1627 12577.1490* 12577.1372 12577.1274 12577.1187* 12577.1134*OP34 34.5 12510.4762 12510.5165 12510.5587 12510.6019 12510.6464 12510.6918 12510.7380 12510.78570R34 34.5 12582.5144 12582.5552 12582.5977 12582.6410 12582.6858 12582.7310 12582.7781 12582.8260*TR42 36.5 12732.9436* 12732.9384* 12732.9330* 12732.9306*R4^34.5 12661.3906 12661.4327* 12661.4771 12661.5223* 12661.5679* 12661.6158 12661.7131*N=37 0013 36.5 12438.6366*SR32 37.5 12652.5681 12652.5535 12652.5396 12652.5257*OP34 35.5 12505.7207 12505.7616 12505.8039 12505.8476 12505.8921 12505.9373 12505.9843 12506.03180R34 35.5 12579.8400 12579.8830 12579.9264 12579.9714 12580.0170 12580.0631 12580.1109TR42 37.5 12732.6311* 12732.6273* 12732.6247*R4^35.5 12659.0784 12659.1212 12659.1657 12659.2108 12659.2573* 12659.3043*N=38 NP02 38.5 12421.3408* 12421.2906* 12421.2381* 12421.0997* 12421.0576*N003 37.5 12421.9083*OP12 38.5 12434.9090* 12434.8740* 12434.7783* 12434.7473* 12434.7156* 12434.6879*0013 37.5 12435.5394*P023 37.5 12498.8807 12498.8946 12498.9068* 12498.9175 12498.9274 12498.9358OP34 36.5 12500.9047 12500.9460 12500.9886 12501.0320 12501.0764 12501.1220 12501.1683 12501.2157N=39 P034 37.5 12536.1839 12536.2228 12536.2636 12536.3048 12536.3901 12536.4345 12536.4799NP02 39.5 12432.1228* 12432.0829* 12432.0423* 12432.0023* 12431.9318* 12431.8976* 12431.8648*01^40.5 12506.5042* 12506.4506 12506.3950 12506.3380 12506.2786 12506.2181 12506.1555 12506.0917OP12 39.5 12419.8280* 12419.7827* 12419.7373* 12419.6944* 12419.6529* 12419.6131* 12419.5738*P023 38.5 12495.6402* 12495.6555 12495.6705* 12495.6843* 12495.6954 12495.7067* 12495.7159OP34 37.5 12496.1549 12496.2437 12496.2900 12496.3371TR42 39.5 12731.8474N=40 NP02 40.5 12429.5138*01^41.5 12505.2216 12505.0520 12504.9919OP12 40.5 12417.6282* 12417.5937* 12417.5602* 12417.5262*0013 39.5 12418.4598*OP34 38.5 12491.0864 12491.1289 12491.1694* 12491.2152 12491.2597 12491.3048 12491.3514 12491.3998N=41 01^42.5 12503.6444 12503.5904 12503.4772OP12 41.5 12415.0641* 12415.0277* 12414.9973* 12414.9632* 12414.9332* 12414.9025* 12414.8714* 12414.8459*N=42 01^43.5 12501.8304 12501.7240 12501.6684OP12 42.5 12411.8015* 12411.6109*N=43 01^44.5 12499.9964 12499.8548 12499.8038 12499.7510* 12499.6987 12499.6444OP12 43.5 12408.1331* 12407.9589*N=44 01^45.5 12497.8033* 12497.7564 12497.7113* 12497.6612 12497.6109* 12497.5609 12497.5070 12497.4537N=45 01^46.5 12495.4723 12495.4261 12495.3788 12495.2820 12495.2320 12495.1791 12495.1251N=46 01^47.5 12493.0225* 12492.9316* 12492.8802* 12492.8339*N=47 '01^48.5 12490.4657 12490.4197 12490.2772* 12490.1217.

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