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Optical and infrared spectra of some unstable molecules 1987
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Title | Optical and infrared spectra of some unstable molecules |
Creator |
Barry, Judith Anne |
Publisher | University of British Columbia |
Date Created | 2010-09-21 |
Date Issued | 2010-09-21 |
Date | 1987 |
Description | Some unstable gaseous molecules, cobalt oxide (CoO), niobium nitride (NbN) and aminoborane (NH₂BH₂), were studied by high resolution optical spectroscopy. A portion of the "red" system of CoO, from 7000 Å to 5800 Å, was measured using laser induced fluorescence techniques. Three bands of the system, with origins at 6338 Å, 6411 Å and 6436 Å, were rotationally analyzed. The lower levels of these parallel bands are the Ω = 7/2 and 5/2 spin-orbit components of a ⁴∆i electronic state. Available evidence indicates that this is the ground state of the molecule; its bond length is 1.631 Å. This work completes the determination of the ground state symmetries for the entire series of first row diatomic transition metal oxides. The hyperfine structure in the ground state is very small, supporting a σ²δ³π² electron configuration. The upper state, assigned as σδ³π²σ*, has large positive hyperfine splittings that follow a case (aβ) pattern; it is heavily perturbed, both rotationally and vibrationally. The sub-Doppler spectrum of the ³Φ₋³∆ system of NbN was measured by intermodulated fluorescence techniques, and the hyperfine structure analyzed. Second order spin-orbit interactions have shifted the ³Φ₃₋³∆₂ subband 40 cm⁻¹ to the blue of its central first order position. The perturbations to the spin-orbit components were so extensive that five hyperfine constants, rather than three, were required to fit the data to the case (a) Hamiltonian. The ³∆₋³Φ system of NbN is the first instance where this has been observed. The magnetic hyperfine constants indicate that all components of the ³∆ and ³Φ spin orbit manifolds may be affected, though the ³∆ state interacts most strongly, presumably by the coupling of the ³∆₂ component with the ¹∆ state having the same configuration. The Fermi contact interactions in the ³∆ substates are large and positive, consistent with a σ¹δ¹ configuration. In the ³Φ state the (b + c) hyperfine constants are negative, as expected from a π¹δ¹ configuration. The ³∆ and ³Φ bond lengths are 1.6618 Å and 1.6712 Å, respectively, which are intermediate between those of ZrN and MoN. The Fourier transform infrared spectrum of the V7 BH₂ wagging fundamental of NH₂BH₂ was rotationally analyzed. A set of effective rotational and centrifugal distortion constants was determined, but the band shows extensive perturbations by Coriolis interactions with the nearby V5 and V11 fundamentals. A complete analysis could not be made without an analysis of the V5-V7-V11 Coriolis interactions, which is currently not possible because the very small dipole derivative of the V5 vibration has prevented its analysis. |
Subject |
Molecular Spectra Infrared Spectroscopy |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2010-09-21 |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0046956 |
Degree |
Doctor of Philosophy - PhD |
Program |
Chemistry |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/28619 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0046956/source |
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OPTICAL AND INFRARED SPECTRA OF SOME UNSTABLE MOLECULES by JUDITH ANNE BARRY B.S. (Hon.), San Francisco State University, 1981 M.S., San Francisco State University, 1983 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Chemistry We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA 9 November I987 ©Judith Anne Barry, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date ABSTRACT ii Some unstable gaseous molecules, cobalt oxide (CoO), niobium nitride (NbN) and aminoborane (NH2BH2), were studied by high resolution optical spectroscopy. A portion of the "red" system of CoO, from 7000 A to 5800 A, was measured using laser induced fluorescence techniques. Three bands of the system, with origins at 6338 A, 6411 A and 6436 A, were rotationally analyzed. The lower levels of these parallel bands are the ft = 7/2 and 5/2 spin-orbit components of a 4 Aj electronic state. Available evidence indicates that this is the ground state of the molecule; its bond length is 1.631 A. This work completes the determination of the ground state symmetries for the entire series of first row diatomic transition metal oxides. The hyperfine structure in the ground state is very small, supporting a CT283TI2 electron configuration. The upper state, assigned as ob3n2o*, has large positive hyperfine splittings that follow a case (ap) pattern; it is heavily perturbed, both rotationally and vibrationally. The sub-Doppler spectrum of the 3<x>-3A system of NbN was measured by intermodulated fluorescence techniques, and the hyperfine structure analyzed. Second order spin-orbit interactions have shifted the 3o>3-3A2 subband 40 cm- 1 to the blue of its central first order position. The perturbations to the spin-orbit components were so extensive that five hyperfine constants, rather than three, were required to fit the data to the case (a) Hamiltonian. The 3 A - s O system of NbN is the first instance where this has been observed. The magnetic hyperfine constants indicate that all components of iii the 3 A and 3 0 spin orbit manifolds may be affected, though the 3 A state interacts most strongly, presumably by the coupling of the 3 A 2 component with the 1 A state having the same configuration. The Fermi contact interactions in the 3 A substates are large and positive, consistent with a a 1 8 1 configuration. In the 3 0 state the (b + c) hyperfine constants are negative, as expected from a 7 t 1 6 1 configuration. The 3 A and 3 0 bond lengths are 1.6618 A and 1.6712 A, respectively, which are intermediate between those of ZrN and MoN. The Fourier transform infrared spectrum of the V 7 BH2 wagging fundamental of N H 2 B H 2 was rotationally analyzed. A set of effective rotational and centrifugal distortion constants was determined, but the band shows extensive perturbations by Coriolis interactions with the nearby V5 and v n fundamentals. A complete analysis could not be made without an analysis of the V 5 - V 7 - V H Coriolis interactions, which is currently not possible because the very small dipole derivative of the V 5 vibration has prevented its analysis. iv TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iv LIST OF TABLES vii LIST OF FIGURES viii CHAPTER I. ELECTRONIC TRANSITIONS IN HETERONUCLEAR DIATOMICS 1 I.A. Some Properties of Angular Momenta 1 I.B. Spherical Harmonics and Spherical Tensor Operators 4 I.C. Selection Rules and Hund's Coupling Cases 11 I. D. The Hamiltonian 21 I.D.1. Nuclear rotational Hamiltonian 21 I.D.2. Spin Hamiltonian 22 I.D.3. Magnetic hyperfine interactions 25 I.D.3.a. The sign of nuclear coupling constants in transition metal complexes 27 I.D.3.a.i. The sign of the Fermi contact interaction 27 I.D.3.a.ii. The sign of the dipolar nuclear hyperfine interaction 30 I.D.4. The nuclear electric quadrupole interaction 31 I.D.5. A-Doubling 37 CHAPTER II. THE COMPUTERIZED LASER-INDUCED FLUORESCENCE EXPERIMENTS 40 II. A. Experimental Details 40 II.B. Intermodulated Fluorescence 43 II.C. Computerization 49 V CHAPTER III. ROTATIONAL ANALYSIS OF THE RED SYSTEM OF COBALT OXIDE 52 III.A. Introduction 52 III.B. Experimental 56 III.B.1. Synthesis of gaseous cobalt oxide 56 III.B.2. The spectrum 57 III.C. Analysis 60 111.0.1. Rotational analysis of the 6338 A subband 60 Ill.C.l.a. Rotational constants and hyperfine structure....60 Ill.C.l.b. Perturbations 67 III.C.2. Rotational analysis of the 6436 A subband 70 III. C.3. Rotational analysis of the 6411 A subband 72 III. D. Discussion 78 CHAPTER IV. HYPERFINE ANALYSIS OF NIOBIUM NITRIDE 84 IV. A. Introduction 84 IV.B. Experimental 86 IV. B.1. Synthesis 86 IV.B.2. Description of the 3<D - 3 A spectrum 86 IV. C. Non-Linear Least Squares Fitting of Spectroscopic Data....97 IV. D. Results and Discussion 102 CHAPTER V. ROTATIONAL ANALYSIS OF THE v 7 BAND OF AMINOBORANE 113 V. A. Background 113 V. B. The Michelson Interferometer and Fourier Transform Spectroscopy 117 V.C. Experimental 126 V.D. The Asymmetric Rotor 127 vi V.E. The Rotational Hamiltonian 134 V.E.1. The Hamiltonian without vibration interaction 134 V.E.2. Coriolis interaction 137 V.F. Band Analysis and Discussion 138 Appendix I. NbN 3 0- 3A Correlation Matrix 145 Appendix II. Transitions of the 3<D-3A System of NbN 147 Appendix 11. A. 3<I>2-3Ai 147 Appendix II.B. 3<D3-3A2 152 Appendix II.C. 30>4-3A3 155 Appendix III. Transitions of the V7 band of NH2 1 1BH2 160 REFERENCES 170 vii LIST OF TABLES 3.1. The most prominent bandheads in the 7000 to 5800 A broadband emission spectrum of gaseous CoO 58 3.11. Assigned lines for the 6338 A band of CoO (4A7/2-4A7/2) with the lower state combination differences, A2F", in cm- 1 63 3.III. Rotational constants for the analyzed bands of the red system of CoO 66 3.IV. Assigned lines from the 6436 A (4A7/2-4A7/2) band of CoO....73 3.V. Assigned lines from the 6411 A (4A5/2-4As/2) band of CoO....74 3.VI. Ground states and configurations of the first row diatomic transition metal oxides, with the fundamental frequency AG 1/2, B and r for the v"=0 state, and the spin-orbit interval AA for the orbitally degenerate electronic states 83 4.1. Molecular constants for the 3 o- 3 A system of NbN 105 4.II. Rotational constants obtained for the 3 o - 3 A system of NbN with the AD and y parameters fixed to zero 110 5.1. Vibrational fundamentals of gaseous NH2 1 1BH2 116 5.II. Character table for the C2v point group 128 5.III. Character sets for an asymmetric top rotational wavefunction in the C2v point group 132 5.IV. Molecular constants of the V 7 band of NH2 1 1BH2 143 viii LIST OF FIGURES 1.1. Polar and Cartesian coordinates 5 1.2. Vector diagram for Hund's coupling case (a) 13 1.3. Vector diagram for Hund's coupling case (b) 15 1.4. Vector diagram for Hund's coupling case (c) 18 2.1. Gaussian inhomogeneously Doppler-broadened velocity population profile 44 2.2. Schematic diagram of the intermodulated fluorescence experiment 46 2.3. a. The formation of crossover resonances, b. Stick diagram of a spectrum with four of the forbidden transitions that can accompany a AF = AJ = 0 Q transition 48 2.4. Schematic diagram of the laser-induced fluorescence experiment interfaced to the PDP-11/23 microcomputer 51 3.1. Energy level diagram of a diatomic 3d transition metal oxide..54 3.2. Broadband laser excitation spectrum of the three bands of gaseous CoO analyzed in this work 59 3.3. Bandhead of the Q' = Q" = 7/2 transition at 6338 A 62 3.4. Upper state energy levels of the 4A 7/2 - 4&7/2 6338 A band....68 3.5. A section of the spectrum of the 6338 A band containing A- doubling, two avoided crossings, and extra lines 69 3.6. Upper state energy levels of the 4A7/2 - 4 A 7/2 6436 A band....71 3.7. Bandhead of the Q' = Q" = 5/2 transition at 6411 A 75 3.8. Upper state energy levels of the 4As/2 - 4As/2 6411 A band....76 4.1. Broadband spectrum of the 3 0 - 3 A system of NbN 87 ix LIST OF FIGURES (cont.) 4.2. The Q-head regions of the a) 3 0 2 - 3 A - i , b) 3 0 3 - 3 A 2 , and c) 30>4- 3 A 3 subbands of NbN 88 4.3. The beginning of the Q head of the 3<&2-3Ai subband 90 4.4. The higher J portion of the 3 3>2- 3 Ai Q head, and the first resolved Q lines 91 4.5. a) R(1), b) R(2), and c) R(3) of the 3 <D 2 - 3 A 1 subband, illustrating the "forbidden" A F * AJ transitions and the crossover resonances.between the rR and qR lines 92 4.6. a) R2, b) R3 and c) R4 lines of the 3 0 3 - 3 A 2 subband of NbN, showing the rR, qR and pR transitions, and the crossover resonances associated with the rR and qR lines 93 4.6. d) R5 and e) R6 lines of the 3<I>3-3A2 subband of NbN., 94 4.7. The reversal of hyperfine structure at high J in the 3 0 4 - 3 A 3 Q branch 95 4.8. Partial energy level diagram for NbN 103 5.1. A polychroomatic sugnal in the frequency domain (above) Fourier transformed into the time domain (below) 122 5.2. A boxcar function D(x). The Fourier transform of a boxcar truncated interferogram is a spectrum with the line shape function F{D(x)} = 2Lsin(2TtvL)/27rvL 124 5.3. The triangular apodization function D(x) (above) produces a spectrum with the line shape function F{D(x)} = 2Lsin(27wL)/(27rvL)2 (below) 125 5.4. Schematic drawing of the OZM NH2BH2 molecule in the x, y, z principal axis system and the a, b, c inertial axis system, showing the C2 ov reflection planes 130 X LIST OF FIGURES (cont.) 5.5. NH2BH2 spectrum of the v 7 band, and the V5 and v n bands with which it undergoes Coriolis interactions 140 5.5. Center of the v 7 band of N H 2 1 1 B H 2 141 1 CHAPTER I ELECTRONIC TRANSITIONS IN HETERONUCLEAR DIATOMIC MOLECULES I.A. Some Properties of Angular Momenta. In a non-rotating molecule, the angular momentum operators J , S and L have the following diagonal matrix elements:1 <JQ| J z |JQ> =tiQ (1.1) <SI| S z |SI> - t i l (1.2) <LA| L z |LA> =tiA (1.3) <JQ| J2|JQ> =fi 2J(J + 1) (1.4) <SI|S2|SX>=fi2S(S + 1) (1.5) <LA| L2||_A> =ti 2 L(L+1) (1.6) J , S and L are the total, spin and orbital angular momenta, respectively; J , S and L are their respective quantum numbers, and Q, L and A are the projection quantum numbers in diatomic molecules (i.e., along the molecular z axis). The ladder operator L+ of a general angular momentum L has the Cartesian form 2 L± = L x ± i L y (1.8) It has the property of transforming state |L,m> into state |L,m±1>, where m is the quantum number of L. For J and S the laddering operations are written:1 <J,Q±1| J T |Jn> =ti[J(J+1) - Q(Q±1)]1/2 (1.9) < S , I ± 1 | S ± | S I > = t i [ S ( S + 1 ) - I ( I ± 1)] 1/2 (1.10) 2 J? in equation (1.9) is not expressed as J± because the commutation relations are different in the space-fixed and molecule-fixed axis systems: 3 JXJY - JYJX = Uz SPACE (1.11) J x J y - JyJx = -Uz MOLECULE (1.12) This leads to a sign reversal upon transformation from the space- fixed to molecule-fixed systems (the anomalous sign of i): J±|JM> = fi[J(J+1) - M(M+1)]1'2 |J,M±1> SPACE (1.13) J T |JK> =fi[J(J+1) - K(K+1)]1/2 |J,K±1> MOLECULE (1.14) Although the motion of the electrons about the axis defines a good quantum number A , L itself is not a good quantum number because a diatomic molecule is not a spherical system. Thus L x and L y do not obey the usual operator equations, and L± is left in the form <L+L. + L.L +>/2, or <L_L>, with the quantity B<Lj_> appearing on the diagonal of the rotational Hamiltonian matrix as a minor, constant electronic isotope shift incorporated into the effective vibrational energy. 1 The dot product of two general angular momentum operators A and B is: A B = A 2 B Z + (A+B. + A.B+)/2 (1.15) The addition of angular momenta j i and J2 to form j results in the coupled eigenfunction |jm>: |jm> = I (-i)ji-j2+m VIJTT / ji j 2 j \ | j imi> | j 2 m 2 > (1.16) mirr)2 \ m i m2 -m/ where |ji m 1 > and |J2m2> are the uncoupled eigenfunctions, the first term is a phase factor, and V2j+1 is a normalization factor. The term in brackets is a coefficient called a Wigner 3-j symbol. Its definition is given by equation (1.16) rearranged as: 4 3 / ii J2 j \ (^p-i2+m I = ^ —<jiJ2mim 2|jm> (1.17) \mi rri2 -my V2j + 1 According to the angular momentum commutation relations for J1J2 and j , 5 the algebraic form for the 3-j symbol is determined by the requirement that mi + m2 = m and |ji - J2I < j < Gi + J2) (the triangle, or vector addition, rule) 4. If these conditions are not satisfied, the vector coupling coefficient <jiJ2m 1 m2|jm> is 0. 4 I.B. Spherical Harmonics and Spherical Tensor Operators. Spherical harmonics, Y|m(0,(p), are orbital angular momentum eigenfunctions normalized to unity on a unit sphere. To be exact they are the eigenfunctions of the differential operators L 2 and L z , corresponding to the eigenvalues 1(1+1) and m: 6 - 7 L2Y|m(e,(p) = l(l + 1)Y|m(e,<p) (1.17) L2Y|m(e,<p) = mY|m(e,<p) (1.18) The angles 6 and <p are the usual polar coordinates as illustrated in Figure 1.1. The differential operators L 2 and L Z ) defined in units where fi = 1, are 6 L z = d/id<p (1.19) L 2 = -[(sin 9)-l(3/39)(sin 93/39) + (sin 29)-l3 2 /39 2] (1.20) Expressed in terms of the orbital angular momentum functions of 9 and cp on the unit sphere, a spherical harmonic is: 8 Y|m(e,q>) = C|(-1)'+m [(l-m)!/(l+m)!]1'2 (sin9)m [3/3(cos9)]'+m x (sin9)2 1 e i m<P (1.21) where ci is a normalization factor: |q| = [(21+1 )!]i/2/(47C)i/2 2>l! (1.22) Associated Legendre polynomials, P | m ( cos 9), are commonly exploited in quantum mechanics because of their connection to spherical harmonics: 6 Y|m(9,(p) = (-)m[(2l+1)(l-m)!/4jt(l+m)!]l/2 P|m(cos 9)eimcp (1.23) where 6 P|m(x) = (1-x2)m/2/2l|! [d'+m/dx'+m](x 2-1)1 (1.24) When the component m = 0, the spherical harmonic and Legendre polynomial differ only by a constant9 Y|0(9,<p) = [(2I+1)/4TC]1/2 P|(cos 9) (1.25) 5 Fig. 1,1. Polar and Cartesian coordinates, in which x = rsinBcosq), y = rsinGsincp, z = rcosG. 8 6 The derivation of expressions describing the coupling of angular momenta, particularly those for the magnetic hyperfine and quadrupolar hyperfine interactions, is often best approached using irreducible spherical tensors. A brief explanation of spherical tensor operators, and the expressions required for their manipulation, follows. Spherical tensor methods are then applied where necessary in subsequent sections to derive the forms employed in the Hamiltonian representing the diatomic molecules in the present work. The spherical components of a vector, or first rank tensor, operator acting on an angular momentum A are related to their Cartesian counterparts by: 2 - 4 T 1 o(A) = A z (1.26) T 1 ± 1(A) = T (A x ± iA y)/V2 (1.27) A spherical tensor T of rank k is defined as a set of 2k+1 quantities ("components") which transform into one another upon rotation from one coordinate system to another (for example, between molecule- and space-fixed axis systems): 1 0 - 1 1 T k q = X TkpDpq(k)(ccpY) (1.28) P where q and p are the components of the tensor in the molecule- and space-fixed axis systems, respectively, and Dpqk(a{3y) is the Wigner rotation matrix. The angles a , p and y are the Euler angles corresponding to the three successive axis rotations required to transform between two coordinate systems. In spectroscopy, a beam of photons (in the space-fixed axis system) induces a change in the molecule in the molecule-fixed system. Wigner rotation 7 matrices function to project from one axis system to another in order to put the photon beam and the molecules being altered by the photons into the same frame of reference. In the reverse direction, from space- to molecule-fixed coordinates, the relation is: T pk = I D p q k>py)Tk q (1.29) q where the complex conjugation of a rotation matrix is given by 0 M K k > P Y ) - ( -1 ) M - K 0-M, -K k (apy) (1.30) The complex conjugation is required to account for the anomalous sign of i. A Wigner rotation matrix is a matrix describing how the eigenfunctions of J 2 and J z , i.e., a spherical harmonic |jm>, transform on coordinate rotation into other functions |jm>: 1 2 D (apY)|jm> = I |jm ,>Dn V m(])(apY) (1.31) Premultiplying equation (1.31) by |jm'>* (i.e., <jm'|) and integrating reduces the right hand side to Dm<mQ) due to the orthogonality of spherical harmonic functions: 1 2 Dm'm«)(aPy) = <jm'|D(apY)|jm> (1.32) A D matrix element with one of its projections equal to zero collapses to a spherical harmonic, which depends on only two angles: 1 2 D^oWy) = (-1)P[4TC/(2I+1)]1/2 Y| p (p,a) SPACE (1.32) D 'oq(apY) = [4ic/(2l+1 )]1'2 Y| q (p, Y ) MOLECULE (1.33) If both projections are zero, the Wigner rotation matrix collapses to a Legendre polynomial: 9- 1 2 D 'oo(ap Y) = P|(cos P) = [47c/(2l+1)]1/2 Y, 0(p,0) (1.34) 8 The Legendre polynomial P|(cos8) is also related to the spherical harmonics by the spherical harmonic addition theorem: P|(C0S 9) = (4n/2l+1) I Y * | m ( 9 i , 9 i ) Y|m(e 2 iq>2) (1-35) m where Y*im(9,cp) = ( - ) m Y | ( . m ( 9 , ( p ) . 6 . 9 , i 3 , i 4 The angles 0 i , 6 2 , 91 and cp2 are as defined by Fig. 1.1 for vectors n and r2, and 6 is the angle between directions (61,91) and (9 2 ,cp2)- Using Racah's modified spherical harmonics to eliminate the factor of [47t/(2l+1)J 1 / 2: 1 6 C|m(9,cp) = [4TC/(2I+1)]1/2 Y|m(9 >(p) (1.36) the spherical harmonic addition theorem b e c o m e s 1 3 ' 1 5 P|(C0S 9) = I C*im(9i ,91) C|m(e2,q>2) (1-37) m o r 1 4 P|(cos 9) = C|(9i ,(pi )C| (9 2 ,cp2) (1.38) The coupling of two tensor operators to form a compound tensor is similar to the addition of two angular momenta given in equation (1.16):"«0 [Tki(1) ® T k 2 ( 2 ) ] q k = £ (-1)ki-k2+q V2k+1 / k i k 2 k\ V^ 1 ^2 <\) x [ r k i q 1 ( 1 ) J k 2 q 2 ( 2 ) ] (1.39) Here the tensor T k i of rank k i , operating on system (1), is coupled to tensor T k 2 [which operates on system (2)]. Shorter, alternative ways of denoting a compound tensor are [T k l(1), Tk2(2)] or, for a tensor of rank ki coupled to itself, [Tk(1,1)], where k = 2ki. If two tensors of the same rank k are coupled to give a scalar , i.e., a quantity invariant to a coordinate rotation, the compound tensor of equation (1.39) is also a scalar, or of rank zero. The resulting expression 9 becomes much simpler and lacks the orientation-dependent 3-j symbol : 1 0 [Tk(1) ® T k ( 2 ) ] 0 ° = (-1)k(2k+1)-1/2Tk(1)Tk(2) (1.40) where the conventional scalar product T k ( i )T k ( 2 ) is given a s : 1 0 ' 1 1 Tk(1)Tk(2) = I (-1)q Tkq(1) Tk.q(2) (1.41) q After a compound tensor equation is written which appropriately represents a particular physical interaction and breaks it into its constituent tensors, the Wigner-Eckart theorem is applied to evaluate the matrix elements T k q of the constituent tensors. According to the theorem the matrix elements of a tensor operator are factored into: 1) a 3-j symbol, which contains information on the geometry or orientation of the angular momentum; 2) a reduced matrix element (denoted by double vertical bars), related to the magnitude of the angular momentum but independent of its direction; and 3) a phase factor. Expressed in terms of the eigenfunctions |Yjm>, where j is the quantum number acted upon by T k , m is the projection of j, and Y contains any remaining quantum numbers not of interest in this particular basis, the Wigner-Eckart theorem i s : 1 6 <YTm,| Tk q |Yjm> = ( - 1)1'^'/ j' k j W j ' l l Tk ||Yj> (1.42) \ -m ' q m/ Note that the reduced matrix element is independent of m. A reduced matrix element is usually worked out by evaluating the simplest type of matrix element and then substituting into the Wigner-Eckart theorem. For example to obtain <J|| T 1(J) ||J>, where J refers to a general angular momentum, we calculate the simplest type of matrix element of T 1 (J) , namely its q = 0 (or z) component: 1 7 10 <J'M'| T 1 0 (J) MM> = 8MM'5jj'M (1.43) This element is non-vanishing only if J'M' = JM. From the Wigner- Eckart theorem (equation 1.42), M = (-1)J-M / j 1 j \ < j | | -p(J) ||J> (1.44) \-M 0 UJ Substitution for the 3-j symbol 1 1 produces M « (-1)J-M(_1)J-M M[J(J + 1)(2J + 1)]-1 / 2<J|| T 1 (J) ||J> (1.45) Since J and M both have integral or half-integral values, (-1)2(J-M) is 1, which reduces equation (1.45) to: <J ||T1(J)|| J> = [J(J + 1)(2J + 1)]i/2 (1.46) An important reduced matrix element is that of the rotation matrix element D.q(k)(apy) (cf. equations 1.29 and 1.30): <J ,K ,||D.qk*(apy)||JK> = (-1 )J'"K'[(2J + 1)(2J' + 1 ) ] 1 / 2 / J' k J \ (1.47) \ - K ' q Kj in which the dot replacing the p indicates that no reduction has been performed with respect to space-fixed axes, so there is no dependence on the M quantum number. Another useful formula gives the matrix elements of the scalar product of two commuting tensor operators (that is, ones which act on different parts of the system) in a coupled basis: 1 8 <Y'J1 ,J2 ,J'M ,| Tk(1)-Uk(2) |yjiJ2JM> = ( -1)J1 +J2 ' +J 5 J M 5 M - M / J 12 j l ' l l <yjl'll Tk(1) || YMJ1> < Y V | | Uk(2) ||Yj2> {k h j 2 J Y " (1.48) in which T k acts on ji and U k on j 2 . The term in curly brackets is a Wigner 6-j symbol, a coefficient which arises in the coupling of three angular momenta, as compared to two in the 3-j symbol. 1 9 11 I.C. Selection Rules and Hund's Coupling Cases. An electronic transition can occur in a molecule only if there are non-zero matrix elements of the electric dipole moment operator M which allow interaction with electromagnetic radiat ion. 2 0 The probability of such a transition occurring between electronic states n and m is proportional to the square of the transition moment, R n m : Rnm = J^n'M^mdT , (1.49) where and are the eigenfunctions of states n and m . 2 0 The electric dipole moment M for a total of N particles (electrons and nuclei) i s 2 1 N M=Ze in (1.50) i=1 where e\ is the charge on particle i which has coordinates rj. In the general case the transition moment integral vanishes unless the change in total angular momentum, J , is zero or unity, o r 2 2 AJ = 0, ±1 (1.51) Changes in J of -1, 0 and +1 are denoted by the letters P, Q and R, respectively. The specific selection rules vary depending on the manner in which the spin, orbital and rotational angular momenta are coupled to one another and to the internuclear axis. The angular momentum coupling schemes in diatomic molecules are distinguished by sets of molecule-fixed basis functions called the Hund's coupling cases. The main property differentiating the four coupling cases described below is the number of angular momenta which have well-defined components (quantum numbers) along the internuclear axis. The 12 appropriate coupling case is the one which produces the smallest off-diagonal matrix elements for the rotational Hamiltonian, or diagonal elements which most closely reproduce the observed spectral pattern. The most common cases by far in molecules with no very heavy atoms are cases (a) and (b). Hund's case (a) coupling has the maximum number of well-defined quantum numbers, such that the relations given in equations (1.1), (1.2) and (1.3) for a non-rotating molecule remain va l id . 1 - 2 3 The basis function for a case (a) coupling scheme is therefore |(L)A>|SZ>|Jft>, or |riA;SI,;JftM>, where A, X and ft are the eigenvalues of the z components of L, S and J , with M being the space-fixed analog of ft, and ft = A + X . 1 The semicolon separators indicate products of component wavefunctions. L is incorporated into the label TI for the vibronic state, as it is not a good quantum number (cf. Section I.A). The case (a) representation is a good working approximation when there are no strong interactions in the Hamiltonian which uncouple these angular momenta from the axis. Case (a) occurs where there is a non-zero orbital angular momentum and fairly small spin-orbit coupling, where the coupling of L and S to each other is less important than the coupling of L to the ax is . 2 4 The vector diagram for case (a) coupling is given in Fig. 1.2. In case (b) coupling, S is coupled only weakly to the axis, but L remains strongly coupled. Given a large enough value of J , any case (a) state uncouples toward case (b) because as J increases the rotational and spin magnetic moments must ultimately be coupled more strongly to one another than L and S are. Formally it can be said that the rotation (R) has increased to the point where it couples 13 Fig. 1.2. Vector diagram of Hund's coupling case (a). 2 4 14 to the orbital angular momentum to form a resultant N, causing S to uncouple from L, and therefore from the molecular axis. The effects of rotation become important when BJ becomes large compared to the separations between the spin-orbit components. 1 T h e transformation of a case (a) situation to case (b) occurs by way of the spin-uncoupling operator, -B(J+S. + J-S+). With its selection rules AS and AA = 0, and AO. = AX = ± 1 , this operator most commonly mixes spin-orbit components of a given 2 S + 1 A state, which is consistent with the physical case (b) phenomenon of uncoupling L from S. 2 3 The case (b) representation also arises for X states in which there is no orbital angular momentum to couple the spin to the axis. The total angular momentum J in case (b) is thus obtained a s : 2 * R + L = N; N + S = J (1.52) instead of the case (a) situation R + L + S = J (1.53) The case (b) basis function, |r i ;NASJ>, is the more physically realistic representation in those cases where the rotational angular momentum N is quantized about the axis, with electron spin providing only minor corrections to the total energy. Its vector diagram appears in Fig. 1.3. When nuclear spin is included in the basis set describing angular momentum coupling in diatomic molecules, the Hund's coupling cases (a) and (b) must be further subdivided. In the majority of diatomic molecules, including those considered in the current work, I is coupled so loosely to the internuclear axis or to S that the dominant coupling is to the rotational angular momentum J, or 15 16 J + I = F (1.54) By analogy with Hund's case (b), those coupling schemes following equation (1.54) are denoted by p* subscripts. The extended Hund's coupling cases are called ap and bpj , corresponding to basis functions |ASXJQIF> and |NASJIF>, respectively.25,26 Coupling schemes in which I is not coupled to J are a a , bpN and bps. In the a a case, nuclear spin is coupled to the molecular axis with the projection quantum number l z , though molecules exhibiting case (a a) coupling have never been observed. 2 7 This is expected since nuclear magnetic moments are on the order of a thousand times smaller than that of the electron, making it unlikely that the dominant nuclear spin coupling will be to the internuclear axis by a magnetic interaction with the electronic and orbital angular momenta. In the bpN and bps cases I is coupled to N and S, respectively, rather than to J as in case (bpj). Case (bpN) coupling is not expected to be observed, as the magnetic moment of N (composed of R + L) is normally considerably less than that of either J or S, as S has a large magnetic moment and J is the sum of S and L. 2 7 In Hund's case (bps), I couples to S to form a vector G, which couples to N to form the total angular momentum F: I + S = G G + N = F In a nonrotating molecule, where any rotationally induced angular momenta are absent, case (bps) will be the dominant case (b) coupling scheme. In a rotating case (b) molecule, however, the coupling case that occurs depends on the relative sizes of the coupling of S to I and N: if the I S coupling dominates, the (bps) 17 case occurs. The best condition for a case (bps) molecule is a X state which originates nearly completely from an atomic s orbital. Case (bps) coupling is therefore rather rare, though it has been extensively described in the ground 2X state of scandium oxide, S c O . 2 8 - 2 9 ' 3 0 This molecule is ideal because the transition metal ion and closed shell oxygen have widely differing ionization potentials. This leaves the S c 2 + uncontaminated by contributions from O 2 " , and the 2X state far removed from the closed state of non- spherical symmetry with which it could mix . 2 7 Other molecules that have been observed to conform to case (bps) coupling are the b 3X and c 3X states of AIF 3 1 , and the ground 2X+ state of L a O 3 2 . Note that both of these molecules also adhere to the conditions required for the bps coupling case. Case (c) coupling occurs in molecules containing an atom sufficiently heavy that the spin-orbit interaction which results is so large that electron motion can no longer be defined in either the L or S representations; one of the consequences is that spin multiplicity is no longer defined. This phenomenon is expressed as an axial J (J a) equal to the sum of L and S, which is then coupled to R to form the resultant J, as illustrated in Fig. 1.4. 2 4 The basis function for case (c) is therefore |rtJ a ;JQM>, where the only well- defined axial component is fl.1 Case(c) molecules observed so far are 209BiO (X 2 I l i / 2 s ta te) 3 3 . 3 4 and InH ( 3Ili state) 3 5 . Case (d) coupling is normally only found in molecules where an electron has been promoted to a Rydberg orbital with higher principal quantum number n. The effect of the long distance between 18 Fig. 1.4. Vector diagram for Hund's case (c). 2 4 19 the electron and the nuclei is that the electron orbital motion is coupled only weakly to the internuclear axis, but can instead couple more strongly to the rotational angular momentum, R . 2 1 > 2 4 Case (d) is equivalent to case (b) but with the difference that L is uncoupled from the axis rather than S ; the transition from case (a) A A A A is made by the L-uncoupling operator, -B(J+L. + J.L+) rather than via the S-uncoupling operator. 2 3 While still in the case (a) or (b) limits, the L-uncoupling operator may induce A-doubling, which lifts the degeneracy of the ±A states. The selection rules for interactions by this operator are AQ = A A = ±1 and A S = 0 . 2 3 The phenomenon of A-doubling is discussed in more detail in the last section of this chapter. Case (d) becomes the appropriate representation when - 2 B J L makes a contribution to the energy levels that is large with respect to the separation of states with differing A . The Hund's coupling cases corresponding to the niobium nitride (NbN) and cobalt oxide (CoO) molecules in this work are most appropriately described by the case (a) and, with higher rotation, case (b) coupling schemes. As A and S are defined in both of these cases, the following selection rules can be stated for cases (a) and (b): 2 4 For case (a), with £ and Cl as good quantum numbers, there are the more specific rules: A A = 0, ±1 (1.55) A S = 0 (1.56) A Q = 0, ±1 (1.57) A l = 0 (1.58) 20 where equation (1.57) follows from equations (1.55) and (1.56). 2 4 The A S = 0 and A X = 0 rules become less strict as the spin-orbit interaction increases, because the selection rules for the spin-orbit interaction are AQ = 0 with either AA = AX = 0 or AA = -AX =±1. 2 4-36 In case (b) neither X nor Q. are well-defined, so the 'rotational' selection rule becomes AN = 0, ±1 (1.59) 21 I.D. The Hamiltonian. I.D.1. Nuclear rotational Hamiltonian. From equation (1.53) it follows that the nuclear rotational A _ A . Hamiltonian B R 2 - D R 4 should be written in the form appropriate for case (a) as: Hrot = B(J - L - S ) 2 - D(J - L - S ) 4 (1.60) where B is the rotational constant, and D is the centrifugal distortion constant representing the influence of centrifugal force due to rotation on bond length. Expansion of the B term of equation (1.60) gives A A A A A A A A - A A H = B(J2 + L2 + S2 - 2 J L - 2 J S + 2 L S ) (1.61) Because the x and y components of L are not defined in a non- spherical system, their effects are omitted in subsequent calculations 1 . Equation (1.61) therefore simplifies to: H = B[J2 + L.2 + S2 - 2J Z L Z - 2 J Z S Z - (J+S. + J .S + ) + 2L Z S Z ] (1.62) The off-diagonal term, -(J+S. + J.S + ), is the spin-uncoupling operator discussed in Section 1.C. The diagonal and off-diagonal rotational matrix elements are calculated by applying equations (1.1) through (1.10) and equation (1.15) to equation (1.61): <JQLASI|H|JnLASX> = B[J(J + 1) - Q2 + S(S +1) - X(Z + 1)]1'2 (1.63) and <JS, Q±1 ,X±1 |H|JSQX> = -B{[(J(J + 1) - Q(Q ± 1)] x[S(S + 1 ) - X ( X ± 1)]}1/2 (1.64) The D terms are obtained by squaring the matrix of the coefficients of the B terms. 22 I.D.2. Spin Hamiltonian. Spin-orbit coupling can be expressed as the scalar product of the many-electron electronic spin and orbital angular momentum operators, S and L, which (using equations 1.8 and 1.15) is represented in Cartesian form as: H L-S = A[(LX + iL y ) (S x - iSy)/2 + L Z S Z + (Lx - iL y ) (S x + iSy)/2] = A L Z S Z + A ( L + S . + LS+)/2 (1.65) where A is the spin-orbit coupling constant. Neglecting the terms off-diagonal in L, equation (1.65) can be shortened to: 3 1 H L .s = A L z S z (1.66) which has the selection rule AS = 0, and produces diagonal matrix elements of A A S . The dipolar spin-spin interaction can be represented by the classical Hamiltonian for two bar magnets, or dipoles, n : 3 7 H = a i i u 2 ) / ( r 1 2 ) 3 - 3(m-r 1 2)(H2-ri2)/(ri2) 5 ( 1- 6 7) in which ri2 is the vector between dipoles u i and u 2 , or ri - T2- The magnetic dipole of spin S is u = - g u B S (1.68) where g is the dimensionless electronic g factor and LIB is the Bohr magneton (the unit on an electronic magnetic moment, equal to efV2m where e and m are the charge and mass of the electron, respec t ive ly ) . 3 8 The dipolar interaction in terms of two electron spin vectors separated by vector r is therefore: H s - s = ( g 2P 2 / r 3 ) { S i - S 2 - 3(si-r)(s2r)/r2} (1.69) Considering only the q = 0 terms (i.e., neglecting the components q = ±1 and ±2) , the interaction reduces to : 3 7 H s - s = (g 2P 2 /r 3){S z(i)S z(2)(3cos20 1 2 - 1) 23 - (S.(DS + (2) + S +(1)S.(2))(3 C O S2ei2 - 1)/4} (1.70) Averaging over all orientations of ri and xi and expressed in terms of a total spin S, equation (1.70) becomes: Hs-s= ( g W / r 3 ) [ 3 S z2 - S z2 - (S.S+ + S+S.)/2] = (9 2 | iB 2 / r 3 ) [2S z - (S x + iS y ) (S x - iSy)] = ( g W / r 3 ) ( 3 S z - S2) (1.71) or in terms of the spin-spin coupling constant X (or zero-field splitting parameter 2 A 3 8 ) , H s-s = 2X(3S Z - §2)/3 (1.72) The spin-spin interaction originates from two mechanisms: the primary contribution to X is from the dipolar interaction of two unpaired spins, but there is also an effect due to second order spin- orbit coupling, which may in fact be considerably larger: 3 9 x = ass + aso Ci .73) Second order perturbation theory applied to the spin-orbit interaction produces a spin-spin interaction as follows. The second order contribution of the spin-orbit interaction in single particle terms is: E s o ( 2 ) = I [ E l l A S - E T l - A ' s ,r 1 X^Alai l i ln'A^ X<T1,A ,|ajfj|TiA> TI'A'S' i j x X<SX|Si |ST><ST|Si |SX> (1.74) The term summing over X ' produces the dipolar spin-spin term <SX | s rSj |SX>, as well as other matrix elements not of interest here because they are off-diagonal in A . The dipolar spin-spin interaction matrix elements are obtained by applying equations (1.2) through (1.5) to equation (1.72): 24 <jni_ASI| Hs-s |Jfll_ASI> = 2X[X2 - S(S + 1 )/3] (1.75) The states they mix have AX (=AA) and AS = 0, ± 1 , ±2 4 0 Centrifugal distortion corrections to the spin-orbit and spin-spin interactions—Ao and XQ, respectively-must also be considered. Terms containing the parameters AQ and XD are therefore added to the rotational Hamiltonian (equation 1.60) as follows: 4 1 Hrot = BR2 - D R 4 + A D R 2 L Z S Z + 2X DR2(3S Z - S 2 ) /3 (1.76) Since the products of the operators in the AQ and XJJ terms are not Hermitian, a Hermitian average must be taken by symmetrizing the products with the anticommutator. The diagonal matrix elements for the AD and Xo parameters therefore follow the rotational constant B, but are multiplied by the elements for the spin-orbit and spin-spin interactions, respectively. The off-diagonal elements do likewise, except that since there are no off-diagonal terms in A or X, the factor for these interactions becomes the average of the two A diagonal elements. As before, the operator R 2 is simplified by omission of the x and y components of -2J-L + 2L-S + L 2 . The spin-rotation operator, the dot product of the spin and rotational angular momenta, is written in Cartesian form a s : 3 1 A- A- A . A- H S -R = y(J "L - S ) S (1.77) Neglecting L+terms, equation (1.77) produces the expanded Hamiltonian: H S -R = Y[J 2 S Z - L Z S Z - S z 2 + ( J + S. + J-S +)/2] (1.78) with diagonal elements: <jni_ASX| Hs-s |Jnl_ASX> = y[X2 - S(S + 1)] (1.79) and off-diagonal elements equal to those given in equation (1.64), but replacing B with -y/2. 25 I.D.3. Magnetic hyperfine interactions. The magnetic hyperfine interactions include all interactions of the nuclear spin, I, with the other angular momenta in the basis set, which for the case(a) basis are J , L and S . Nuclear magnetic moments interact weakly with the rotational magnetic moment giving rise to a scalar interaction term written: 2 5 H|.j = cil-J (1.80) where ci denotes the interaction constant. From equation (1.54), F 2 = J 2 + 21-J + I2 (1.81) so that the IJ interaction can be expressed in terms of F as: H|.j - q ( F 2 - J 2 - i 2)/2 (1.82) The matrix elements can be obtained directly from equation (1.4) as: <ASIJQIF| HI .J |ASIJQIF> = C|[F(F + 1) - J(J + 1) - l(l + 1)]/2 (1.83) The interactions of electronic and nuclear spins are represented by the Hamiltonian: 2 6 Hi.s = b lS + c l z S z (1.84) with b = aF - c/3 (1.85) where aF and c are the isotropic (Fermi-contact) and dipolar hyperfine constants, respectively. The former interaction is directly proportional to the quantity of electron density at the spinning nucleus, while the dipolar, or bar magnet, interaction between l z and S z is the same as given in equation (1.67). The interaction of nuclear spin with the electronic orbital magnetic moment is a scalar product of I and L which is treated in the same manner as the L S interaction described by equations (1.65) and (1.66). The resulting Hamiltonian is therefore: 2 6 - 3 1 26 HI .L -aizLz (1.8.6) in which a is the interaction constant. The b term of equation (1.84) is expressed in spherical tensor form as: H i s = bTl(l)Tl(S) (1.87) To derive the matrix elements of the interaction, I is first uncoupled from J by application of equation (1.48): <nASIJQIF| T1(I)T1(S) h , ASTJ , Q , I F> = [l(l + 1)(2I + 1)]1'2 <nASLJQ| | V(S) I h ' A S T ' J ' ^ ^ (1.88) where the [(l(l + 1)(2I + 1)] 1 / 2 term is the reduced matrix element of T 1(l) according to equation (1.46). By projecting the reduced matrix element in equation (1.87) from the space-fixed axis system to the molecule-fixed system, using Wigner rotation matrices as in equation (1.47), the general matrix element can be expressed a s : 3 1 <iiASUOIF| HT .S |ri ,ASTJ ,n ,IF> = (-1)>+J'+F/F J |) [l(l + 1)(2I + 1)(2J + 1)(2J' + 1)]i/2 I ( -1 )J-«/J 1 J ' \ \ l I J'J q \-Q q Q'J X ( -1 )S - I /S 1 S ' \ Z <S||Tl(S)||S'><TiAS|bih,AS,> (1.89) The c l 2 S z and a l z L z Hamiltonians are treated by the same method. Evaluation of the 3-j and 6-j symbols with the appropriate A- A A A A A f o r m u l a e 5 - 4 2 , yields the matrix elements for bl-S, c l z S z and a l z L z , except that the only matrix elements written for the a and c constants are those diagonal in A and X, respectively. The resulting matrix elements employed in the hyperfine analysis of NbN are as follows: 27 <JIFQIM| Hut |JIFQIM> . Qh R(J)/[2J(J + 1)] (1.90) <JIFftXM| Hhf |J-1,IFflIM> = -h(j2-Q2)1/2p (J )Q(J) / [ 2 J(4j2 -1)1/2] (-|.91) <JIFQXM| Hhf |JIFQ±1,X±1,M> = b[(J+Q)(J±Q+1 )]1/2R(J)V(S)/[4J(J+1)] (1.92) <JIFQXM| Hhf_|J-1,IFQ±1,I±1,M> = +b[(J*Q)(J+n+1 )]1/2P(J)Q(J)V(S)/[4J(4J2-1 )1/2] (1.93) where the following abbreviations have been used: R(J) = F(F + 1) - J(J + 1) - l(l + 1) (1.94) P(J) = [(F - I + J)(F + J + I + 1)]l/2 (1.95) Q(J) = [(J + I - F)(F - J + I + 1)]i/2 (1.96) V(S) - [S(S + 1 ) - I ( I ± 1 ) ] l / 2 (1.97) The constant b is that given in equation (1.84), while h is used in the diagonal elements in order to incorporate the a, b and c constants into one: h = aA + (b + c)X (1.98) I.D.3.a. The sign of nuclear hyperfine coupling constants in transition metal complexes. I.D.3.a.i. The sign of the Fermi contact interaction. For an isotropic (Fermi contact) interaction involving only pure s electrons, the isotropic hyperfine constant aF is positive because the magnetic field generated at the nucleus by the interaction is in the same direction as the electronic spin. However, negative contributions to the isotropic hyperfine interaction occur when there are open shell d or p electrons which polarize s electrons in inner (filled) orbitals via an exchange interaction which promotes an 28 electron from an inner s orbital to an outer empty o n e . 4 3 For example, a ground electronic configuration with a single unpaired 3d electron, ¥ 0 = (3s+)(3s-)(3d+) can mix with excited states resulting from the promotion of an electron from a 3s to 4s orbital to produce the three functions: 4 3 ¥1 = (4s+)(3s-)(3d+) ¥ 2 = (3s+)(4s-)(3d+) ¥ 3 = (3s+)(4s+)(3d+) This is known as a configuration interaction, in which the ground and excited states possess different spin distributions yet form the basis for the same irreducible representation, in keeping with the requirement that the energy of the system remains constant. 4 4 First order perturbation theory is applied to describe the mixing, yielding an expression for the hyperfine contribution due to configuration interaction that is a function of the product of the ns and ms orbitals evaluated at the nucleus [ns(0)ms(0)], times an exchange integral J(ms,3d,3d,ns), divided by the energy separation between the ms and ns orbitals: 3 °° X=8TCS X [ns(0)ms(0) x J(ms,3d,3d,ns)]/(Em-E n) (1-99) n-1 m=4 The quantity x ' s independent of c h a r g e 4 3 and is related to the isotropic Fermi contact coupling constant, aF, by : 4 4 a F = (2/3)geLiBgnHnX (1.100) where g e and g n are the electronic and nuclear g factors and LIB and L i n are the Bohr and nuclear magnetons. The quantity [ns(0)4s(0)]/(E4- 29 E n ) for the n • 1, 2, 3 s orbitals of the neutral atoms of the first row transition metals from V to Cu was found to increase by about 20% across the series. The exchange integrals varied in the opposite sense, though more gradually, decreasing by an overall 14% from V to C u . 4 3 An alternative approach to the configuration interaction (CI) is core (or spin) polarization, a treatment which may be easier to conceptualize but is not as theoretically s o u n d . 4 4 This theory differs from CI in that the orbitals involved belong to a single configuration which originates from spin-dependent one-electron orbitals. The resulting hyperfine interaction is therefore a function of the amount of spin density of each sign. CI requires two spin- independent configurations to represent the wavefunction. The wavefunction for the core polarization model is a spin-polarized unrestricted Hartree-Fock function (UHF) where UHF differs from the conventional, or restricted, Hartree-Fock function in that the trial one-electron wavefunctions are not required to be independent of the orientation of the s p i n . 4 4 The radial functions whose spins are being polarized, corresponding to spin up and spin down, differ from one another because they couple differently with the unpaired d or p electrons. The resulting hyperfine interaction is negative because the polarized spin has the opposite sense to the unpaired electron which induces the polarization. 4 4 30 I.D.3.a.ii. The sign of the dipolar nuclear hyperfine interaction. The sign and magnitude of the dipolar hyperfine interaction depends on the number and type of open shell d and p electrons. The interaction constant for such an electron in orbital r\ i s 4 5 Cj = 3geUBgnun<il|r-3(3cos2e - 1)/2fo> (1.101) where 6 is the angle between the nucleus and the ith unpaired electron at a distance r; closed shell electrons do not contribute to <3cos29 - 1>. Using for sake of illustration the ground electronic 4 X " state of VO, with the configuration ( a 27i 4 a n 1 5 2 ) , there are three non-bonding o n 1 8 2 open shell electrons contributing to the IS interaction. If the assumption is made that the interacting electrons are metal centered, the hyperfine constants are : 4 6 ( A i s 0 ) V 0 « (1/3)(A|8 0)4so (1-102) (A dip)vo= (2/3)(Adip)3d6 (1-103) where these A parameters are related to aF, b and c by: Aiso = A i + Adip = aF (1.104) A ± = b = a F - c/3 (1.105) Adip = c/3 (1.106) A|| = b + c (1.107) Combining equations (1.101), (1.103) and (1.106), the expression for c becomes: c = 3geUBgnM2/3)<3d5|r-3.(3cos28 - 1)/2|3d8> (1.108) Using the algebreic expression for the spherical harmonic Y20 (see Section I.B) 4 7, the matrix element portion of equation (1.108) can be written in terms of the n, I and m quantum numbers as: <nlm)r-3.(3cos2e - 1)/2|nlm> = (1/2)<lm|3cos2e - 1 |lm><nl|r-3|nl> 31 3m2- 1(1+1) <r-3> n| (1.109) (2l-1)(2l+3) For a 8 orbital, equation (1.109) reduces to (2/7)<r3>ni, producing a value for c (in cnr 1 ) o f 4 6 c = -(4/7)geLiBgn^n<r-3>3d/hc (1-110) When an electron is promoted from the 4so to 4pa orbital to produce the C 4 Z * excited state, all three electrons contribute to the dipolar term and c becomes (in cm- 1): c = 3geUBgnM(2/3)<r-3-(3cos2e - 1)/2>3d8 + (1/3)<r-3(3cos2e-1)/2>4pa]/hc c = g e^BgnM-(4/7)<r- 3>3dS + (2/5)<r- 3> 4 p a]/hc (1.111) Using this method the different values for c corresponding to the various possible electron configurations of an electronic state can be estimated, which assists in the assignment of an electronic state. I.D.4. The nuclear electric quadrupole interaction. The nuclear electric quadrupole interaction involves two second rank tensors, representing the electric field gradient and the nuclear quadrupole moment. A simple method by which to derive the quadrupolar Hamiltonian is with the use of spherical harmonics and Legendre polynomials. To obtain the Hamiltonian for the electrostatic interaction of the nuclear quadrupole moment with the electric field gradient at the nucleus, a multipole expansion is made for the scalar coupling of the charges of the nucleons with those of the electrons. A multipole 32 expansion is a spherical harmonic expansion (or Legendre polynomial expansion) where the values of I in the spherical harmonic Y | m are referred to as monopole, dipole, quadrupole and octopole for I = 0, 1, 2 and 3 . 4 8 By Coulomb's law 4 9 , the electrostatic Hamiltonian is H = I e q n / R n (1-112) n which describes the interaction between n nucleons with charge q n and an electron with charge e, with an electron-nucleon separation of R n . The electrostatic potential at the electron is V = I q n / R n (1.113) The distance R n is the resultant of the two vectors originating from the nuclear center to the nth nucleon (rn) and to the electron (R), with the angle between vectors rn and R denoted by 9 n . The law of c o s i n e s 5 0 gives the relation between R n , rn, R and 0 n : Rn = (R 2 + r n 2 - 2 R r n c o s e n ) 1 / 2 = R[1 + (r n /R) 2 - 2 ( r n /R)cose n ] 1 / 2 (1.114) By the generating function for Legendre polynomials 5 1, [1 - 2(r n /R)cos9 n + ( r n /R ) 2 ] 1 / 2 = I P|(cos0 n)(r n/R) 1 (1.115) equation (1.113) can be written in terms of a Legendre polynomial a s : 5 2 V = X I P|COS(0n)qnrn l/R'+1 (1.116) l=0 n Each Legendre polynomial represents the scalar product of electronic and nuclear tensor operators (from the spherical harmonic addition theorem), producing from equations (1.112) and (1.116) the multipole expansion: 4 8 - 5 2 33 A Hmultipole = e v * = H ( -1) m [I (e/R' + 1) C|m(eei(pe) x I qni-n' C|,.M(0ni(pn)] ( 1 -117 ) 1=0 m e n where the summations over e electrons and n nucleons represent terms in electronic (8e,(pe) and nuclear (0n,<pn) angular coordinates, respectively. The first term in this expansion which is non-vanishing describes the quadrupolar interaction. The I = 0 term can be represented by Z e V 0 , or the Coulombic interaction between the nuclear charge and the electrons, and is included in the electronic Hamiltonian. 5 3 The dipole term, I = 1, is the product of the electric dipole moment of the nucleus, which is zero, and the electrostatic field of the electrons, which is invariant over the nuclear volume and therefore produces no interaction. 5 3 The I = 2 quadrupole term, however, is the interaction of the nuclear electric quadrupole moment, Q , with the electric field gradient ( V E ) experienced by the nucleus due to the charge distribution of the electrons. For those nuclei possessing a quadrupole moment, then, the quadrupolar Hamiltonian is the scalar product of these two tensor quantities: 5 4 H Q = -T2(VE)T2(Q) ( 1 . 1 1 8 ) where the minus sign is present due to the negative charge of the electron. The quadrupole moment is a measure of how spherical the nucleus is, as indicated by the value of the nuclear spin, I. The deviation of nuclear charge distribution from spherical symmetry is given by: 3 z 2 a v e - (x2 + y2 + z 2 ) a v e or 3cos 2e n - 1 (where 8 N is the nuclear angular coordinate) . 5 5 This value is non-zero if I is greater than 34 1/2, which is dictated by the number of odd nucleons (i.e, differences in the number of neutrons with respect to protons). The mechanism giving rise to specific values of I is imperfectly understood, though it seems to approximate the same shell model that applies to electrons. Thus, zero spin results from spin-pairing if the number of protons (Z) equals the number of neutrons (N), and predictions for I can usually be made for nuclei possessing odd N or Z based on the number of particles occupying open shel ls . 5 5 By convention, the nuclear electric quadrupole moment is defined classically a s 1 1 Q = ej(3z2 - r2)p(r)dx (1.119) where p(r) is the nuclear charge density, and dx denotes integration over the nuclear volume. Quantum mechanically the definition becomes: 5 2 Q = e-lXq n r n2(3cos20 n - 1) (1.120) n The quantum mechanical observable corresponding to equation (1.120) is the nuclear quadrupole moment, Q, defined by convention a s 5 4 Q - <l,mi=l| Q |l,mi=l> (1-121) A. The definition of Q was made prior to the invention of spherical tensors and therefore lacks the factor of 1/2 needed for the expressions P2(cos9) - T 2 0 (X ) = (3cos 2e - 1)/2; Q was also defined without the electron charge e. The spherical tensor definition is therefore T20(Q) = eQ/2 (1.122) with the corresponding scalar quantity 35 eQ/2 = <l ,mi=l|T2 0 (Q)| | ,mi=l> (1.123) The quadrupole tensor, from equation (1.117), is of the form T2(Q) = Iqnr2nC2(en,(pn) (1-124) n The electric field gradient (EFG) evaluated at the nucleus, ( 3 2 V / 3 z 2 ) 0 l has the spherical tensor form (from equation 1.117) of: -T2(VE) = ZeR-3C2(ee,cpe) (1 -125) e with the corresponding field gradient coupling constant defined as q = <j,mj=J|(a2V/az2)0|J,mj=J> (1.126) where (d2\lldz2)0 = eR-3(3cos6 e - 1). Thus, with the factor of 1/2 required by the spherical harmonic definition of the quadrupole moment, the EFG tensor can be expressed as: -T20(VE) = q/2 (1-127) To derive the matrix elements for the quadrupolar interaction (equation 1.116), equation (1.48) is applied to evaluate the scalar coupling of two commuting tensor operators in a coupled basis (I must be unravelled from J): ^'A'lSTiJ'n' IFI HQ |nA;SX;JQIF> = ( - 1 ) J + I + F 5 F F / F I J'^Tl 'A ' jJ 'Q 'H -T2(VE) | |r iA;JO><l|| T2(Q) |||> (1.128) 12 J 1/ Then project T 2 ( V E ) from space- to molecule-fixed axes with equation (1.29): ^ • A ' j J ' Q ' H ^ V E J I h A i J ^ = X<J'n'|| D2.q*(apY) ||J«><Ti'A ,||-T2 q (VE)||TiA> q = X(-1)J'-Q'[(2J+1)(2J'+1)]1/2/j' 2 J W A ' H -T2q(VE) |hA> (1.129) q \a' q a) The last term of equation (1.128) is evaluated with the Wigner- Eckart theorem, in conjunction with equation (1.123): 36 <l,mi=l| T2 0 (Q) ||l,mi=l> = eQ/2 « (-1)'-'/ I 2 l \<l | | T2(Q) ||l> (1.130) V - l 0 \) Substituting for the 3-j symbol 5 7 and solving for the reduced matrix element gives <l|| T2(Q) ||l> = eQ!2( I 2 V-IO I/ = eQ/2 [(2l+1)(2l+2)(2l+3)/2l(2l-1)]l/2 (1.131) In terms of the molecule-fixed T 2 ( V E ) tensor in equation (1.129), the coupling constant q is defined by the diagonal reduced element of T2(VE): <A||-T20(VE) ||A> = q/2 (1.132) A first order approximation was made in the current study to neglect the ±1 and ± 2 components of T 2 ( V E ) , that is, to exclude quadrupole matrix elements off-diagonal in Q . Appropriate combination of equations (1.128), (1.129), (1.131) and (1.132) therefore yields the matrix elements ^ 'A' jSTlJ 'n ' IF I -T2(VE)T2(Q) |riA;SI;JftlF> = (1/4)eqQ(-1)J+'+F ff I J , Nj [(2l+1)(2l+2)(2l+3)/2l(2l-1)]1/2 \ 2 J I / x l ( -1)J' -« , [ (2J+1)(2J , +1)] l/2/ J' 2 J \ (1.133) q \-Q' q ClJ From the triangle condition for a 3-j symbol, which states that the third J value must not lie outside the sum and difference of the first two J va lues 1 8 , the 3-j symbol in equation (1.133) requires AJ to be 0, ±1 or ± 2 . From equation (1.133) and these selection rules, the specific matrix elements employed in this work are as follows: 37 <JIFQIM| HQ |JIFQIM> = eQq[3fl2-J(J+1 )]{3R(J)[R(J)+1 ]-4J(J+1 )l(l+1)} 81(21-1 )J(J+1 )(2J-1 )(2J+3) (1.134) <JIFnlM| HQ |J-1,IFQIM> = -eQq3n[R(J)+J+1](j2-fl2)l/2P(J)Q(J) 2J(2J-2)(2J+2)(2I-1 )(4J2-1) 112 (1.135) <JIFQIM| HQ |J-2,IFQIM> = eQq3[(J-1)2-fl2]1/2(j2-fl2)l/2p(J)Q(J)P(J-1)Q(J-1) 41(21-1 )4J(J-1 )(2J-1 )[(2J-3)(2J+1 )]1 /2 (1.136) The terms R(J), P(J), Q(J), P(J-1) and Q(J-1) are as in equations (1.93), (1.94) and (1.95). I.D.5. A -Doubl ing. The phenomenon of A-doubling results from the breakdown of the Born-Oppenheimer approximation, which allows the separation of electronic and nuclear motion.26 it is the lifting of +A degeneracy which occurs when molecular rotation interferes with the well- defined quantization of the z component of electronic orbital angular momentum about the molecular axis. The operators in the spin and rotational Hamiltonian responsible for A-doubling are the x and y components of the electronic orbital angular momentum operators which produce matrix elements off-diagonal in A . In the rotational A A Hamiltonian, this is the L-uncoupling operator, - 2 B J L . Among the spin-interaction terms of the Hamiltonian, the spin-orbit operator is used, yielding the complete A-doubling Hamiltonian: 5 7 V = - 2 B J L + Xaji-Si (1-137) The A-doubling interaction is treated by degenerate perturbation t h e o r y 5 8 , which for A states must be taken to fourth order in order 38 to connect |A = 2> to |A = -2> via states with A = 1 and 0 (i.e., IT. and X states). For this reason the interaction is smaller than that in n states, since the mixing of |A = 1> and |A = -1> states requires only second order perturbation theory. 5 7 The unperturbed Hamiltonian contains those terms adhering to the Born-Oppenheimer approximation which are diagonal in A and independent of the orbital degeneracy. The perturbation can be treated through the use of a fourth-order effective Hamiltonian, which is obtained by subtracting out the unperturbed energy from the complete Hamiltonian expression to leave an effective Hamiltonian which operates only on the vibronic state of interest, | l 0 k > 5 7 ' 5 9 Heff<4> = P 0 V(Qo/a)V(Q 0 /a )V(Qo/a)VP 0 - P 0 V(Qo / a 2 )VP 0 V(Qo/a )VPo - P 0 V(Qo / a 2 )V (Qo/a )VP 0 VPo - PoV(Qo/a)V(Qo /a2)VPoVPo + P 0 V(Qo /a3) V P 0 V P 0 V P 0 (1.138) The operator P 0 , extending over the k-fold degeneracy of l 0 , is defined as Po = I | l 0 k x l 0 k | (1.139) k while (Qo/a n)= I I | l k x l k | / ( E 0 - E n ) n (1.140) l=l 0k where I denotes any vibronic state with energy E|, E 0 is the energy of state <l0k|, and k labels all rotational, spin and electronic quantum numbers in a vibronic state l 0 or I. The Hamiltonian in equation (1.137) has 2A+1 terms of the form ( H r o t ) 2 A - n ( H s . 0 . ) n , where n ranges from zero to 2A . In the case (a) form it is written: 5 7 39 HL.D..A = m A (S + 4 + S.4)/2 - n A (S +3j+ + S.3J.)/2 + 0 A ( S + 2 j + 2 + §.2j.2)/2 - p A ( S + J + 3 + SJ.3)/2 + q A ( j + 4 + j.4) ( 1 .141) where the factors of 1/2 are included to be consistent with the notation of Mulliken and Chr is ty 6 0 for II states. Thus the q A parameter accompanies ( H r o t ) 2 A . PA is with (H r o t ) 2 A " 1 ( H s.o. ) and so on to m A with (H S . 0 . ) 2 A . The number of those parameters that can be determined equals the spin multiplicity up to a maximum of 5. In a 4 A state, for example, only four of the five parameters are included in the A-doubling matrix elements, with m A excluded because the spin-orbit interaction need not be extended to fourth order. In a 4 A state where there are four Q, substates, the terms appear in the 4 x 4 matrix as + terms which split a given level into two levels of different parity, labelled e and f. By convention, the e levels have parity +(-1)J-k and / levels have parity -(-1)i-k, where k is 1/2 and 0 for half-integer and integer values of spin, respectively. 6 1 ' 6 2 The magnitude of the A-doubling observed in this work in the 4 A 7 / 2 - 4 A 7 / 2 transition of CoO ranged from 0.2 to 1.2 c m - 1 , while that in the 3 n 0 state of NbN is on the order of six wavenumbers. CHAPTER II THE COMPUTERIZED LASER-INDUCED FLUORESCENCE EXPERIMENTS 40 II.A. Experimental Details. The laser excitation experiments were performed using a Coherent Radiation model CR-599-21 scanning single frequency (standing wave) dye laser, pumped by a Coherent Radiation model lnnova-18 argon ion laser operated at a wavelength of 514 nm and a power of 2.0 to 3.5 W. Output power from the dye laser was normally 100 to 150 mW. The tunability of the laser comes from selecting portions of the broad fluorescence band of an organic d y e . 6 3 Two dyes were employed for both the cobalt oxide (CoO) and niobium nitride (NbN) studies. For maximum output at 590 nm (ranging from 570 to 620 nm or 17540 to 16130 cm- 1), the dye used was rhodamine 6G (Exciton Chemical Co.), with the structure 6 3 made to a concentration of 2 x 10 - 3 M in ethylene glycol. To reach the lower energy regions, the dye DCM (4-dicyanomethylene-2- methyl-6-p-dimethylaminostyryl-4H-pyran, from Exciton Chemical Co.) was dissolved in 3:7 benzyl alcohol to ethylene glycol to form nearly saturated 2.5 x 10"3 M solutions. At a pump wavelength of 514 nm, DCM's maximum output power occurs at 640 nm, and - O C 2 H 5 ^ v - C H j ^ C 2 H 5 41 broadband laser operation occurs over the range 600 to 695 nm (16670 to 14390 cm" 1). The benzyl alcohol required to dissolve the DCM leads to bubble formation, so the solution was cooled to minimize bubbling by running the dye tubing through a -30 ° C slush of dry ice mixed with a 1:3 solution of water to CaCl2- All chemicals were used as obtained. A small fraction of the output beam was diverted to an iodine absorption or emission cell for absolute frequency calibration. Another fraction was sent to a Tropel fixed-length semiconfocal Fabry-Perot interferometer with a 299 MHz free-spectral range, providing a common ladder of frequency markers against which the sample and iodine spectra could be referenced. The beam containing the majority of the output power was passed down the longitudinal axis of the stream of sample molecules, with the laser-induced fluorescence (LIF) detected at right angles to the beam with a photomultiplier tube equipped with a high transmittance low pass optical filter to reduce scattered light, and powered by 300 to 500 V from a high voltage power supply. Phase-sensitive detection was acheived with a Princeton Applied Research (PAR) model 128A lock- in amplifier receiving chopped sample and reference signals, with the reference beam supplied by a Spectra-Physics model 132 Lablite He- Ne gas laser. The resolved fluorescence experiments were performed with a 0.7 m Spex Industries model 1702 spectrometer which dispersed the spectrum onto the detector elements of a microchannel-plate intensified array detector (PAR model 1461), mounted at the output end of the spectrometer. The spectral window of the array detector was calibrated with a Burleigh model WA-20VIS wavemeter. 43 II.B. Intermodulated Fluorescence. A laser-induced fluorescence transition has a Gaussian velocity population profile forming an inhomogeneously broadened line, because of the Doppler effect, the freqeuncy absorbed by molecules moving away from the light source appears to be lower than that absorbed by molecules moving toward it. At the center of the profile (zero velocity) the transition frequency Q is not Doppler- shifted; that is, the molecules have zero velocity with respect to the light wave with which the molecules interact . 6 5 ' 6 6 In Doppler- free (or "sub-Doppler") spectroscopy, two travelling waves (laser beams) with frequency w propagate in opposite directions through the sample gas molecules. Molecules moving with velocity v along the axis of the laser beams absorbs radiation from one beam at a frequency ft = co(1 + v/c), and from the other beam at ft = co(1 - v/c). These opposite Doppler shifts cause each beam to depopulate a portion of the lower state velocity profile symmetrically about the profile center at v = c(ft ± co)/ft (see Fig. 2.1). This depletion is termed "burning a Bennet hole", creating a homogeneous profile in the lower state. 6 6 As the laser is scanned, and the laser frequency approaches the non-Doppler-shifted resonance frequency, the two Bennet holes converge until they meet at the center, or zero velocity (see Fig. 2.1). The resulting lower state population depletion causes a corresponding depletion in the intensity profile of the fluorescence, called a "Lamb dip". Intermodulated fluorescence (IMF) is a technique which enables relatively small Lamb dips to be detected against the large Doppler- broadened profile so that they are directly measured as spectral 44 Fig. 2.1. Gauss ian inhomogeneously Doppler-broadened veloci ty (v z ) populat ion (n) profile, showing two Bennet holes (solid lines) which converge at zero velocity (dotted line) to form a Lamb dip in the profile of intensity versus laser tuning f r e q u e n c y . 6 6 45 peaks. The two laser beams are modulated (i.e., chopped to produce certain phase trains) with frequencies fi and f 2. The lock-in amplifier, with the phase sensitive detector referenced to a frequency of fi + f2, passes only (fi + f2)-modulated input signals, such as those occurring when two Bennet holes meet 6 6 A schematic diagram of the IMF experiment used to obtain the niobium nitride sub-Doppler spectra is illustrated in Fig. 2.2. In practice, the two counterpropagating laser beams must be slightly misaligned from one another to avoid feedback into the laser. A LIF signal normally arises from Bennet holes caused by allowed AF=AJ transitions meeting at the velocity profile center. However, Lamb dips also originate from holes burned by "forbidden" A F # A J transitions meeting at the center. Since the selection rules 2 4 on F and J are AF = 0, ±1 and AJ = 0, ± 1 , transitions with A F = AJ ±1 and ±2 are also possible. For a Q transition, with AJ = 0, the F selection rule requiring that AF = 0, +1 allows the transitions rQ (AF = AJ + 1), qQ (AF = AJ) and pQ (AF = AJ - 1). If AJ = +1, A F = +1, 0 and -1 corresponds to the transitions rR, qR and pR (or AF = AJ, AJ - 1 and AJ - 2). The same occurs for P branches where AF = AJ, AJ + 1 and AJ + 2 lines (pP, qP and rP) occur. These satellite branches are observed only at low values of J because the intensity of the transitions is proportional to the angle between the vectors J and F . 6 7 When J and F are large with respect to I this angle approaches zero, and only A F = AJ transitions are observed. The large value of 9/2 for the nuclear spin of Nb allows A F * A J transitions to be seen at higher values of J than is normally possible. Accompanying a pair of AF = AJ and AF = AJ ± 1 transitions, or a 46 Discharge in flow system PMT Refe 'ence Lock-in spectrum Chopper Tunable dye laser signal Fabry - - - Perot PDP-11/23 Micro- computer \ , 11 3-pen chart recorder c <^ ' PMT y calibration Interpolation markers Fig. 2.2. Schematic drawing of the intermodulated fluorescence experiment used in this laboratory. The discharge cube where the sample and laser light are combined is shown in the top left corner. 47 A F = AJ ± 1 and AF = AJ ± 2 pair, may be a "crossover resonance" occurring exactly mid-way between the two. Such a phenomenon requires that the two transitions sharing a common level lie within the same Doppler profile. Crossover resonances occur in the IMF spectra of the nearly coincident transitions of closely spaced hyperfine components. The means by which crossover resonances are generated is depicted in Fig. 2.3, with a schematic stick drawing of the resulting spectrum. b) AF=AJ=0 X X AF= AJ-I 1 I A J + | 1 1 1 1 Fig. 2.3. a) The formation of crossover resonances (Fi + A 2 and F 2 + A-i) as the result of allowed AJ = AF transitions (Ai and A 2) occurring within the same Doppler-broadened velocity profile as forbidden AJ * AF transitions (Fi and F 2). The diagram shows the laser scanning toward the non- Doppler-shifted AF = AJ transition (occurring at Ai + A 2) and beyond toward higher frequency to the AF = AJ + 1 transition (Fi + F 2) . If the F's and A's are exchanged, the first central Lamb dip is the A F = AJ - 1 transition, b) Stick diagram of the spectrum of the four forbidden transitions that can accompany a AF = AJ = 0 Q transition (X denotes a crossover). With an R line, the AF = 0 and A F - -1 transitions and the associated crossovers occur to the red of the A F = AJ + 1 transition, while with a P line the forbidden transitions lie to the blue to the A F = A J - 1 transition. 49 II. C. Computerization. Part of the work for this thesis involved computerizing all stages of the Doppler-limited and intermodulated fluorescence (sub- Doppler) LIF experiments on a PDP-11/23 microcomputer with an RSX-11M operating system. These stages included: 1) laser scanning, and data acquisition and storage; 2) peak finding; and 3) frequency calibration. Each stage comprises a separate program. All of the software was written with FORTRAN-77 except for the laser scanning and data acquisition, programmed in MACRO. The PDP-11 computer is structured such that space for executable code is quite limited. This constraint required that the three programs be overlaid. Overlaying is a method of memory management which allows the sum of the individual subroutines to far exceed the memory limitations of the computer. When an overlaid program is executed, only a portion of the subroutines are sent into memory, while the remainder resides in the relatively limitless disk space. A set of overlay directives is written which describes the program in terms of a calling "root" segment and any number of subprogram "branch" segments; the branches may themselves call "subbranches". The computer uses these directives to build the task file such that during program execution the memory space at any given time is occupied only by the root segment and the branch being called at that time. Since the main responsibility of the root is to call subroutines in the branches, the root is made as short as possible to allow most of the software to remain disk-resident throughout program execution. 50 The heart of the first program is the MACRO routine which orchestrates laser scanning and data acquisition via its control of the following hardware peripheral devices: • The 16-bit digital-to-analog converter (D/A), which sends a voltage ramp to the laser so that it scans a range of up to 1.4 cm- 1 . • The 4-channel, 12-bit analog-to-digital converter (A/D), containing two registers to process incoming data. The control status register (CSR) receives the voltages (data points) from the sample, iodine and interferometer detectors. The buffer preset register receives the point from the CSR, stores it temporarily, then delivers it both to the 12-bit output D/A and to the appropriate storage buffer for transfer to disk. The sample spectrum is signal averaged over four points prior to transfer to the buffer. • Three 12-bit D/A's, which send the three data points to the chart recorder for a hardcopy of the spectra. • The real-time (crystal-oscillator) clock, by which the above peripheral devices are interrupt-driven to operate at a user-chosen rate producing a resolution compatible with the lock-in time constant and the frequency range scanned by the laser. The use of interrupts ensures that the task will be serviced by the computer's central processing unit exactly as dictated by the clock. The interfacing of computer and experiment is illustrated schematically in Fig. 2.4. Upon return from the MACRO routine after scanning is complete, the three spectral vectors are stored in unformatted files with the first record of the sample file serving as a housekeeping record containing spectral identification and experimental parameters. 51 PROGRAM 12-bit A/D: 16-bit D/A Interface Chan 1 Chan 2 12-bit D/A 0 1 2 BNC Connections Fabry-Perot Interferometer Si-diode Detector Fig. 2.4. Schematic diagram of the laser-induced fluorescence experiment and how it is interfaced to the PDP-11/23 micro- computer. 52 CHAPTER III ROTATIONAL ANALYSIS OF THE RED SYSTEM OF COBALT OXIDE III.A. Introduction. In German-occupied Belgium during World War II, Malet and Rosen observed a number of electronic bands of gaseous cobalt oxide (CoO) between 5000 and 10000 A using the exploding wire technique. 6 9 The lower state vibrational frequency (i.e., the separation of the v"=0 and v"=1 vibrational states) was found to be 840 c n r 1 , and this state was assumed to be the ground electronic state. The next spectroscopic experiments on CoO came years later, in 1979. The first was a low resolution infrared spectrum of CoO (with ±0.2 cm- 1 line precision) obtained with a microwave discharge source, giving a vibrational frequency of 842.2 c m - 1 , an equilibrium rotational constant (B e) of 0.522, and an equilibrium bond length (re) of 1.60 A.70 The absence of a Q branch in the spectrum led to the tentative assignment of a X ground state 2 4 , though the possibility was not ruled out that the spectrum was that of a low-lying excited state. 7 0 A matrix isolation infrared study followed shortly afterwards 7 1 , in which cobalt from a cobalt cathode sputtering source and oxygen were codeposited at low temperature (14 K) into a solid matrix of argon. The ground state vibrational frequency was measured in this work to be 846.4 cm" 1 . In the next year, matrix isolation electron spin resonance (ESR) studies of a large group of transition metal- containing molecules with high spin multiplicities were reported, including C o O . 7 2 In spite of high concentrations of CoO within the matrices and the expertise of the laboratory in conducting 53 experiments of this type, no CoO ESR signal was observed. CoO was therefore concluded to possess an orbitally degenerate ground state, because orbital degeneracy in linear molecules (in matrices of low enough temperature that only the ground state is populated) causes a g tensor anisotropy so large that the spectrum is spread out over such a large magnetic field that it cannot be observed. The ESR spectrum of a paramagnetic 1 state, on the other hand, will possess little or no g anisotropy and will exhibit only a small deviation from the free electron value, g e = 2.0023, due to the spherical symmetry of the overall orbital angular m o m e n t u m . 7 2 ' 7 3 The value of g is deduced from the relation hv = giiBH, where v is the resonance frequency, n B the Bohr magneton, H the applied magnetic field, and h Planck's constant; the g anisotropy is taken as g_i_ - gn. No further work has been published on CoO since this ESR study, leaving the ground state of the molecule to be the only one of the first row transition metal oxides yet to be established. Field-free atomic orbitals of a diatomic transition metal molecule are split by the axial field of the other atom, as shown in Fig. 3.1. From the electron configurations of manganese, iron and nickel monoxides, MnO (4SO)1 (3d5)2 (3d7i)2 FeO (4SC)1 (3d5)3 (3d7i)2 NiO (4sa) 2 (3d8)4 (3dn)2 , it can be seen that there are two possible candidates for the ground electronic state of CoO. If the seventh valence electron occupies the 4 s o orbital, the spin multiplicity and direct products given by the resulting o 2 8 3 r c 2 configuration produce a 4 A j electronic state 7 4 ; if 54 M orbitals MO orbitals 0 orbitals 4po- 4p \ \ \ v \ \ . \ \ \ \ \_ 4ptr ~7 "*"= / / / 3 d * 3 d / 3 d r r / 3d* N / / --- \ / / \ _4j j,' 4s<* \ V , x \x \ \ \ \ 2pff \ \ 2 pa \ A 2p == 2s<J 2s Fig. 3.1. Relative orbital energies of a diatomic 3d transition metal oxide. 9 2 The ordering of the 3d5 and 4sa molecular orbitals is variable. 55 instead it fills up the 8 orbital, a 4 X " state results. However, the uncertain ordering of the 4so orbital with respect to the 3d8 orb i ta l 7 5 left the problem in the hands of the theoreticians. Multi- configuration self-consistent-field complete active space (CAS MCSCF) calculations on FeO were extrapolated to CoO to predict a 4 X ~ ground state. 7 6 Weltner, however, first predicted a state of A symmetry based on trends in the other TM o x i d e s 7 7 , then later predicted a 4 0 ground state based on ESR experiments 7 2 . It was from this stage of development that the current study proceeded. 56 III.B. Experimental III.B.1. Synthesis of gaseous cobalt oxide. Cobalt oxide was made in a Broida-type oven assembly 7 4 as follows: an alumina crucible containing cobalt metal powder (Fisher Scientific Co.; 0.14% Ni, 0.11% Fe) was heated resistively in a tungsten basket. The basket was enclosed in a radiation shield comprising an inner ceramic sleeve enveloped by an outer copper sleeve and fitted lid, with zirconia felt packed very tightly around the basket. To produce cobalt oxide (CoO) in quantities sufficient for measurable fluorescence, temperatures approaching the melting point of the alumina crucible (1920 ° C ) were required, well in excess of cobalt's melting point of 1495 ° C . CoO was formed in the gaseous stream of vaporized cobalt atoms, argon carrier gas and molecular oxygen at a pressure of roughly 1 Torr, with a ratio of approximately 150(±15):1 argon to oxygen. Fluorescence, however, occurs only in the presence of laser excitation, which is as with NiO in which only the ground state is populated by the reaction of metal and O2. 7 8 Unlike the production of C u O 7 9 , which is more efficient with N2O than O2, no CoO fluorescence was observed using N2O as the oxidant. The requirement of high temperature drastically hampered the efficiency of CoO synthesis in two ways. First, there was extensive formation of Thenard's B l u e 8 0 (cobalt aluminate) deposits on the crucible and on the surface of the liquid cobalt; this phenomenon was also reported in 1966 by Grimely and coworkers who heated solid CoO in an alumina cell to high temperatures 8 1. Second, the reaction of cobalt vapor with the tungsten basket produces an alloy that renders the basket very susceptible to 57 cracking, with breakage occurring after at most three heatings of a basket assembly. III.B.2. The spectrum. The laser excitation spectrum of gaseous CoO was investigated over the range of 7000 to 5800 A at Doppler-limited resolution, as described in Section II.A. It is evident that the system extends further to both higher and lower energies. The bands observed by Malet and Rosen with the exploding wire technique 6 9 correspond in frequency to those we have measured, though the intensities sometimes varied dramatically between the two techniques. With the superior sensitivity provided by the LIF method, a number of additional bands were observed. The most prominent ones, as measured from a broadband laser spectrum (i.e., one obtained without the intracavity assembly), are listed in Table 3.1. The portion of the spectrum rotationally analyzed thus far covers the range from 15450 to 15790 c n r 1 (6470 to 6335 A), which includes three red-degraded bands whose heads lie at 15778 cm- 1 (6338 A), 15598 cm- 1 (6411 A) and 15538 cm- 1 (6436 A). The broadband spectrum of this region is shown in Fig. 3.2. 58 Table 3.1. The most prominent bandheads in the 7000 to 5800 A broadband emission spectrum of gaseous CoO. Values are accurate to roughly ± 3 c m - 1 , with band strength denoted by: s = strong, m = medium, w = weak. Wavelength group 5920 A 6120 A 6320 A 6650 A 6900 A Wavenumber 16916 m 16366 w 15832 w 15296 vw 14704 w and intensity 16846 s 16322 s 15778 s 15228 w 14477 m 16256 m. 15597 w 15036 m 14469 s 16088 w 15538 m 15004 s 0"= 7 / 2 5 / 2 7 / 2 Fig. 3.2. Broadband laser excitation spectrum of the three bands of gaseous CoO analyzed in this work (linewidths are on the order of 1 cm- 1 ) . 60 III.C. Analysis. III.C.1. Rotational analysis of the 6338 A subband. I l l . d . a . Rotational constants and hyperfine structure. The strongest band, at 6338 A, was the only band of the three for which a complete analysis was possible, given the available data. The line assignments, listed in Table 3.II, were made using lower state combination differences. For added assurance, some wavelength-resolved fluorescence experiments were performed to verify that certain lines possessed common upper levels. For example, if a pair of lines with a common upper level, such as Q(J") and R(J"-1), are excited, the fluorescence pattern produced as a result of the R line excitation will be identical to that obtained from the Q line, barring changes in the scattered laser light at the excitation wavelength. Lower state combination differences measure differences between lines with a common upper state that differ in their J" value, thereby providing information on the lower state energy structure: 1 AiF"(J) = R(J) - Q(J +1) = Q(J) - P(J + 1) (3.1) A2F"(J) = R(J - 1) - P(J + 1) (3.2) From the definitions of R, Q and P, and from the energy level expressions, it can be shown that 8 3 AiF"(J) = 2B"(J + 1) - 4D"(J + 1) 3 (3.6) A 2F"(J) = (4B" - 6D")(J + 1/2) - 8D"(J + 1/2)3 (3.7) The lowest Q line of this band was assigned as J' = J" = 7/2, using the average of the A-|F" combination differences from the first R and P lines, and a rough estimate of 0.5 c m - 1 for the value of B. The 61 possible electronic states corresponding to a value of ft of 7/2 are 4 A and 2 0 , but only the 4 A state has an electronic configuration that can reasonably be expected to belong to the ground state. The three subbands analyzed in the current work demonstrate that the most intense CoO transitions are those with ft" = 7/2. Presumably these must come from the lowest spin-orbit component of the ground state. Since the spin-orbit manifold must be inverted for its lowest energy component to be 7/2, the electronic state is assigned as 4 Aj. The relatively low intensity of the Q lines1 (see Fig. 3.3), identifies the transition as parallel, or ft' = ft" = 7/2. The lower state vibrational level can definitely be assigned as v" = 0, based on resolved fluorescence experiments where the Q(3.5) line was excited: strong fluorescence was observed 851.7 cm- 1 to the red of the Q(3.5) transition, but nothing to the blue. In the absence of isotopic labelling studies, such as with C o 1 8 0 , no information is available on the upper state quantum number. 8 3 Extensive structure to the red of the 6338 A band indicates that there are lower lying vibrational levels; on this basis, the upper state vibrational quantum number is suggested to be at least two. The lower state rotational constants B and D were calculated by least squares from the A 2 F " combination difference formula in equation (3.7). The A 2 F"(J) combination differences are given in Table 3.II along with the assigned lines of the 6338 A band. Using these B" and D" values to calculate the lower state energy levels, the upper state energy levels were calculated; then a least squares fit to the expression E(J) = T 0 + BJ(J + 1) - DJ2(J + 1)2 (3.8) to «sr' m U 3.5 6.5 8.5 5.5 7.5 4.5 LAW II I 10.5 12.5 I i 3.5 5.5 I 4.5 8.5 15.5 10.5 6.5 R Q Fig. 3.3. Bandhead of the Q' = Q" = 7/2 transition at 6338 A, exhibiting the broadening due to hyperfine interactions and the weak Q branch signifying a parallel transition. The weak background is reproducible. 63 Table 3.11. Assigned lines from the 6338 A band ( 4 A 7 / 2 - 4 A 7 / 2 ) of CoO with the lower state combination differences, A 2 F H , in cnv 1 . An asterisk denotes a blended line. J" R Q P A 2 F " i 103 O-C 3.5 157747* 15771.060 45 15774.7* 15770.2* 15766.64* 5.5 15774509 15769.2* 15764737 6.5 15774110 15768.002 15762.704 14.015 I 7.5 15773.526 15766.64* 15760.494 16.014 -2 8.5 15772.760 15765.024 15758.096 18.011 -6 9.5 15771.810 15763.259 15775.515 20.013 -5 10.5 15770.668 15761.291 15752.747 22.015 -3 11.5 15769.343 15759.158 15749.791 24021 2 12.5 15767.828 15756.829 15746.647 26.024 6 13.5 15766.126 15754316 15743.319 28.017 - l 145 15764236 15751.615 15739.811 30.019 2 15.5 15762.159 15748.715 15736.107 32.018 2 16.5 15759.892 15732.219 34011 -3 17.5 15757.445 15728.147 36.012 1 18.5 15754806 15723.880 38.006 -2 19.5 15751.994 15719.439 40.005 1 20.5 15749.350 15747.947 15714.801 42.000 0 15748.821 15749.039 21.5 15745.600 15745.906 15709.994 43.995 15745.720 22.5 15742.273 15742.345 15705.347 15703.946 45.987 -3 15704829 15705.045 23.5 15738.696 15738.769 15699.615 15699.914 47.984 1 15699.735 245 15734923 15735.005 15694290 15694360 49.980 4 25.5 15730.951 15731.051 15688.720 15688.786 51.975 6 26.5 15726.783 15726.911 15682.948 15683.030 53.968 8 27.5 15722.428 15722.592 15767.986 15677.080 55.956 6 285 15717.883 15718.106 15670.826 15670.954 57.938 -2 64 Table 3.11. continued. J" R 1 P A 2 F " 103 O-C 29.5 15713.130 15713.500 15664491 15664654 59.929 0 30.5 15709.131 15708.185 15657.951 15658.178 61.908 -9 15707.960 31.5 15703.047 15703.110 15651.226 15651.589 63.903 0 32.5 15697.705 15697.888 15645.226 15644.282 65.890 l 15644.057 33.5 15692.171 15692.441 15637.158 15637.220 67.870 -4 34.5 15686424 15686.786 15629.837 15630.015 69.861 3 35.5 15680.487 15680.945 15622.311 15622.580 71.836 -5 36.5 15674.251 15674893 15614.591 15614.950 73.820 -2 37.5 15667.933 15668.678 15606.670 15607.122 75.787 -15a 38.5 15661.347 15662.251 15598.464 15599.106 77.783 1 39.5 15654.547 15665.638 15590.150 15590.896 79.752 -8 40.5 15647.498 15648.844 15581,594 15582.500 81.738 2 41.5 15640.453* 15641.831 15572.810 15573.900 83.713 1 42.5 15634643 15563.785 15565.131 85.684 -2 43.5 15627.272 15554769 15556.159 87.641 • -183 44.5 15619.712 15547.002 89.615 -158 45.5 15611.954 15537.657 91.591 -98 46.5 15604.010 15528.121 47.5 15518.405* af\lot included in the least squares fit. 65 was used to obtain B' and D' from the unperturbed levels with J' = 5.5 to 19.5. The results appear in Table 3.III. Note that since the upper state B value is only 81% of the lower state, by the relation r"/r' = (B7B") 1/ 2 the CoO bond length increases by a full 10% upon electronic excitation. The hyperfine structure in CoO arising from the 5 9 C o nuclear spin of 7/2 follows the case (ap) pattern where the hyperfine widths decrease with increasing rotation, described to a first approximation by equation (1.90):8 5 E hfs - ft[aA + (b + c)X](1/J){[F(F+1) - J(J+1) - 1(1+1 )]/2(J+1)} (3.9) (In case (bpj) coupling the hyperfine widths are independent of N for each of the spin components.) The hyperfine splitting in the P lines is found to be wider than that in R lines of the same J", while P and R lines possessing the same upper state J are of comparable widths. Since a comparison of P and R lines of the same J" demonstrate upper state properties, while those with equal J ' represent the lower state, it can be seen that the hyperfine interactions produce larger splittings in the upper state than in the lower state. From equation (3.9) it can also be seen that the eight hyperfine components of a rotational line will be more widely spaced at higher values of F. Partially resolved hyperfine splittings in some low J lines, for instance P(5.5), show that the highest F value component is on the high frequency side. This ordering of the hyperfine components shows that the change in the Fermi contact parameter, b' - b", is positive. 8 4 66 Table 3.III. Rotational constants for the analyzed bands of the red system of C o 0 . a B 10?D Upper Levels: 6338 A, Q = 7/2 15772.513 + 3 0.40531 ± 9 6.4 ± 19 0.0038 6411 A, ft = 5/2 a+15594.974 ± 2 0.42503 ± 24 27 ± 7 0.0049 6436 A, Q= 7/2 15535.77 0.422 4 b Lower Levels (X4Aj): Q = 5/2 a Q = 7/2 0 0.50266 ± 9 3.6 ± 14 0.0024 0.50058 ± 4 6.50 + 15 0.0031 a Values in c m * 1 , with error limits of three standard deviations in units of the last significant figure, a • AA - 244 c m - 1 . b No least squares fit; see text. 67 111.0.1 .b. Perturbations. A plot of the upper state energy levels as a function of J(J + 1) illustrates the perturbations in the upper states. The lower state appears to be free of A-doubling (cf. Section 1.B.6) and other perturbations, since the lower state combination differences are entirely regular: the two A2F"(J) values, given by the two A-doubling components, are equal to within experimental error. Figure 3.4 shows that upper state A-doublings begin at J ' = 21.5, and that in some places extra transitions occur; the section of spectrum in Fig. 3.5 illustrates these perturbations. The extra lines could be securely identified because they give exactly the same A2F" (J ) combination differences as the main lines and their relative intensities are in the same ratio. Two avoided crossings can be seen in Fig. 3.4: a strong one, where both of the A components are perturbed, at J' = 30.5 - 31.5, and a weaker one where the lower A component is mildly perturbed at J' = 37.5. Since the avoided crossings affect the A -components differently, the perturbing state is orbitally non-degenerate, or alternatively has a very large A-doubling of its own. The state perturbing the J' = 22.5 level appears to have a relatively small A - or Q-doubling. (A-doubling exhibited by a X state is referred to here as Q-doubling 8 5 . ) The state responsible for all of the above perturbations could conceivably be a single case (a) 4 X state. The small Q-doubling occurring near J ' = 22.5 could arise from the 4X3/2 component, while the considerably larger Q-type splitting associated with the 4 X i / 2 c o m p o n e n t 8 5 is capable of affecting upper state levels that are 1 5 7 7 3 • 4 0 0 8 0 0 1 2 0 0 J(J +1) Fig. 3.4. Upper state energy levels of the 4 A z / 2 - 4Aj/2 6338 A band, scaled by subtracting the quantity 0.405J(J + 1) - 6.4x10- 7J 2(J + 1)2, plotted against J(J + 1). o Fig. 3.5. A section of the spectrum of the 6338 A band containing A- doubling, two avoided crossings, and extra lines. The extra R(30.5) line to the blue of the A-doubled R(30.5) lines corresponds to the anomalous point in Fig. 3.4 near J(J + 1) = 1000. All lines, down to the weakest, are reproducible, though relative intensities between lines at either end of the spectrum may not be accurate since the spectrum is compiled from several laser scans. 70 widely spaced in J , analogous to the situation observed in Fig. 3.4. III.C.2. Rotational analysis of the 6436 A subband. The fairly intense 6436 A band is another Q' = Q" = 7/2 parallel transition whose lower state is the same as that of the 6338 A band, as the lower state combination differences of the two bands are equal to within experimental error. Because the upper level lies only 237 cm" 1 below that of the 6338 A upper state, and the frequency separating the strong groups of subbands (cf. Table 3.1) is on the order of 600 c m - 1 , it cannot belong to the same upper electronic state as the 6338 A band. Also, the hyperfine structure is considerably wider than in the 6338 A subband, which also points to a different upper electronic state. There is not enough information available to say what this other electronic state is. Although its high intensity suggests that it is another 4A7/2 - 4A7/ 2 transition, there are other channels through which intensity can be derived. In the very dense, perturbed "orange" system of FeO, for example, transitions to the high vibrational levels of various lower electronic states acquire considerable intensity by interacting with the upper state of the system. 8 6 The upper state energy levels are plotted as a function of J(J + 1) in Fig. 3.6, up to the limit of our analysis thus far at J ' = 26.5. A- doubling is first observed at J' = 20.5, very much like the 6338 A band upper levels which are first seen to split at J ' = 21.5. Perturbations in the upper state have scattered the levels to such a degree that a good least squares fit to the upper state constants was not possible, though a value of B' could be estimated (see Table 1 5 5 3 8 E-red c m " 1 1 5 5 3 7 1 5 5 3 6 - i r + + + -J 1 I l_ 2 0 0 4 0 0 6 0 0 J ( J + 1 ) Fig. 3.6. Upper state energy levels of the 4 A z / 2 - 4 A 7 / 2 6436 A band, scaled by subtracting the quantity 0.42J(J + 1), plotted against J(J + 1). 72 3.III). The lines assigned in the 6436 A band are compiled in Table 3.1V. III.C.3. Rotational analysis of the 6411 A subband. The ft1 = ft" = 5/2 subband whose head lies at 6411 A is much weaker than the other two subbands, and is also badly perturbed, which has precluded analysis beyond J ' = 20.5. All the lines assigned so far are listed in Table 3.V. The transition was assigned as ft' = ft" = 5/2 by the methods used previously for the 6338 A subband, and it appears that the lower state is the ft = 5/2 spin-orbit component of the ground electronic state. The crowded head region of the band is shown in Fig. 3.7. The perturbations in the ft' = 5/2 upper state are illustrated by the plot of the scaled upper state energy levels as a function of J(J + 1) in Fig. 3.8. The A-doubling is much larger than in the upper levels of the ft = 7/2 bands, with the splitting first discernible at Doppler- limited resolution at J' = 10.5. At J' = 16.5 one of the A -components is drastically pushed to lower energy, and no further J' levels could be assigned. The other component also disappears abruptly at J' = 20.5. The suddenness with which the branches break off is surprising, because there is no appreciable loss of intensity before the rotational structure ceases. This fragmentary behavior has been observed before, for example in the 5866 A band of FeO where the structure disappears suddenly at J' = 15, and then reappears 12 cm- 1 to the b lue . 8 6 The 6411 A upper level in CoO has obviously suffered a massive perturbation near J ' = 20.5. To find where the branches resume will require extensive wavelength resolved 73 Table 3.1V. Assigned lines from the 6436 A (4A7/2-4A7/2) band of CoO, in cm- 1 . J" R P 7.5 15537.998 15524.45 8.5 15537.505 15522.347 9.5 15536.866 15519.996 10.5 15536.061 15517.490 11.5 15535.096 15514.839 12.5 15533.976 15512.039 13.5 15532.692 15509.082 14.5 15531.241 15505.950 15.5 15529.644 15502.682 16.5 15527.897 15499.226 17.5 15525.955 15495.625 18.5 15523.820 15491.877 19.5 15521.589 15521.967 15487.959 20.5 15519.589 15519.536 15483.815 21.5 15516.412 15517.029 15479.586 15479.959 22.5 15513.383 15514.393 15475.130 15475.539 23.5 15511.448 15511.650 15470.420 15471.035 24.5 15508.438 15508.784 15465.395 15466.408 25.5 15461.468 15461.670 26.5 15456.463 15456.809 27.5 15451.345 15451.840 74 Table 3.V. Assigned lines from the 6411 A band (4A5/2-4As/2) of CoO, in c n r 1 . An asterisk denotes a blended line. J" R Q P 2.5 15597.270* 15594.293 3.5 15597.577* 15593.751 4.5 15597.730* 15593.067 15589.2* 5.5 15597.730* 15592.23* 15587.557 6.5 15597.577* 15591.183 15585.670 7.5 15597.270* 15590.039* 15583.654 8.5 15596.194 15588.739 15581.492 9.5 15596.194 15596.240 15579.076 10.5 15595.432 15595.512 15576.718 11.5 15594.523 15594.647 15574.076* 15574.125 12.5 15593.461 15593.618 15571.307 15571.389 13.5 15592.254 15592.708 15568.389 15568.512 14.5 15590.896 15591.441 15565.323 15565.493 15.5 15589.390 15588.739 15562.106 15562.559 16.5 15587.729 15582.734 15559.270 17.5 15585.958 15555.222 15554.561 18.5 15583.958 15551.558 19.5 15581.845 15547.736 20.5 - 15543.770 21.5 15539.646 P(39.5) 6338 A band III i g i III III 1 1 1 1 1 I U II II 55 85 45 105 1 1251 145 i 155 165 • 1 i 175 25 35 55 85 R Q 45 6 5 Fig. 3.7. Bandhead of the ft' = ft" = 5/2 transition at 6411 A. The band is extensively overlapped by other bands, as evidenced by the dense collection of unassigned lines. 1 5 5 9 8 1 5 5 9 5 1 0 0 2 0 0 3 0 0 4 0 0 J(J + 1) Fig. 3.8. Upper state energy levels of the 4As/2 - 4 A s / 2 6411 A scaled by subtracting the quantity 0.42J(J + 1), plotted against J(J + 1). 77 fluorescence measurements in the surrounding region. Such studies must be postponed until we develop a less cumbersome method by which to synthesize gaseous CoO. The upper and lower state B and D rotational constants, calculated in the same manner as for the 6338 A band, are given in Table 3.III. Kratzer's relationship 8 3, D e = 4Be 3/co e 2 (3.10) for the equilibrium values of the rotational constants and the vibrational frequency (coe) can be approximated for the v = 0 level by D 0 = 4 B 0 3 / A G i / 2 2 (3.11) Using equation (3.11) to calculate an approximate value for D", it is found to be about 60% larger than the observed value. 78 III. D. Discussion. Of the two possible ground electronic state configurations for CoO, 4 £ - (a7t 25 4) or 4 A(o 2 J t 2 5 3 ) , evidence has been presented in the rotational analysis of the excitation spectrum of gaseous CoO which strongly supports that the ground state is 4 A j . The fundamental vibrational frequency of 846.4 c m - 1 measured by infrared spectroscopy in low-temperature (14 K) matrix isolation 7 1 closely matches the value of 851.7 cm- 1 obtained from this laser induced fluorescence work. Since the ground electronic state should be the only one populated at 14 K, and a 5.3 c n r 1 shift from the solid to gas phase is not unreasonable, this suggests that the lower electronic state of the three bands studied here is the ground state. The matrix isolation electron spin resonance s tudy 7 2 which could not produce a signal from CoO eliminates the possibility for 4 £ - as the ground state, taking this absence of a result as valid. The only condition under which an orbitally non-degenerate electronic state with case (a) coupling can produce no ESR signal when isolated in a low-temperature matrix is if it possesses an odd spin multiplicity with the ft = 0 level the only one populated. The band intensities support an inverted order for the spin-orbit manifold since the ft' = ft" = 7/2 bands are strongest, followed by ft' = ft" = 5/2. The rotational analysis of two ft spin-orbit components of the same electronic state provides the information required to determine the true B value and an estimate for the spin-orbit interval, A A . For molecules in which spin uncoupling is small because the spin-orbit interaction is very large, the effective B value for a given spin-orbit component differs from the true B value 79 A A by an amount that depends on the spin-uncoupling operator, - 2 B J S . A second order perturbation treatment of two O substates separated by AA and connected by this operator produces the relation: 2 4 Beff.ft = B(1 +2BI/AA) (3.12) Solving equation (3.12) simultaneously for both AA and the true B value for the v" = 0 level, using the effective BQ=7/2 and BQ=5/2 values in Table 3.Ill, gives B = 0.5037; AA - -244 cnr 1 (3.13) The spin-orbit coupling interval AA is not expected to be accurate to better than 10%, as equation (3.12) does not take into account the centrifugal distortion corrections to A and X, called AD and Xo (cf. Section I.B.3). For example, the initial estimate of |AA| made for F e O 7 5 was 180 cm- 1 , based on the approximation in equation (3.12), yet the value was later found 8 7 to be 190 cm" 1 . The definition of B, as a function of the mean value of the bond length r during the vibration, i s 8 3 B = (h/87i 2cu)<r 2> (3.14) where | i is the reduced mass of the molecule. With the B value in equation (3.13), the bond length in the zero point vibrational level is calculated from equation (3.14) to be: r 0(X 4Aj) = 1.631 (±0.001) A (3.15) The 10% increase in bond length to 1.80 A upon electronic excitation to the upper 4 Aj state is quite large compared to transitions in the other first row diatomic transition metal oxides. The A 4 n <- X 4 X " transition of VO produces a 7% increase 4 5 ; A 5!, <- X 5 FIr and B 5 I I r<- X 5 n r in CrO give 2-1/2 and 5-1/2% increases 9 0 ; the 8 £ + <_ 6 £ + parallel transition of MnO at 6500 A shows a 4% increase 9 1 ; but 80 various subbands of the orange system of FeO do show bond length increases of up to as m u c h a s 1 1 % 8 7 , and a state perturbing the MnO A 6 1+ state has a bond 10% longer than that of the ground state 9 1 . The magnetic hyperfine structure and spin-orbit coupling constant can be used to give information about the excited states as well as the ground state. The insignificant hyperfine structure in the ground state is consistent with the lack of unpaired s electron density in the 4 A o 2 r c 2 8 3 configuration. The upper state configuration can be assigned as a7i 2 5 3 a* for three reasons: 1) the large, positive hyperfine splittings in the upper state indicate a strong Fermi contact interaction due to open shell s electrons (cf. Section I.B.3). When an unpaired s electron is present in a diatomic transition metal oxide it usually shows up clearly in the Fermi contact parameter. Most states with unpaired s electrons have positive values for ap: aF for S c O 7 3 a 2 L + = +0.0667 cm- 1 ; aF for VG-45 o8 2 4 £ - - +0.02593 cm* 1 ; a F for M n O 9 0 o 8 2 n 2 6£+ .» +0.0151 cm-" 1. An exception is the ground state of CuO, which has a large, negative Fermi contact parameter in spite of the presence of open shell so electrons. 7 9 Three configurations are believed to make significant contributions to the 2Ti\ ground state: C u + ( 3 d 1 ° ) 0-(2p5), Cu(3d 1 °4s) 0(2p 4), and Cu(3d94s4p) 0(2p 4) Only the last one has open shell metal-centered orbitals which will participate significantly in the hyperfine interactions. In terms of molecular orbitals, this configuration is proposed to b e : 7 9 3da1847i4(Cu), 4sa(Cu) + 2pa(0), PTC(CU) + 2p7i(0) 81 The wavefunction can therefore be expressed as a linear combination of Slater determinants (showing only the unpaired electrons for clarity): V(2rii) = (1A/6){2|do(a) po(a) pit(P)| - |do(a) pa(P) pw(o)| -|do(P) pa(a) pn(a)\} (3.16) The authors propose that the negative terms in the wavefunction are responsible for the negative value for aF of -0.0139 cm- 1 . The C 4 X " state of VO, with a 3d8 2 o* configuration, is an example where the promotion of an electron from sa to a non-s type a orbital produces a negative value for the IS interaction constant of -0.00881 c m - 1 , as a result of spin polarization. 4 5 The a* orbital is believed to be a linear combination of 3do, 4sa and 0(2pa). 2) the fact that the Q. = 7/2 and Q. = 5/2 subbands lie very close in the spectrum shows that the spin-orbit intervals AA" and AA' are nearly equal. The 4 A states of the configurations C 2 T C 2 8 3 and o 7 t 2 6 3 a * will have orbital angular momentum coming only from the '6' hole, so that they should have roughly the same spin-orbit couplings. 3) Following from 2), the negative sign of A also suggests a 8 hole, or 8 3 configuration. The o - 7 t 2 8 3 c * configuration can give rise to 19 electronic states from the different arrangements of the electrons within the orb i ta ls . 7 4 The result will be a dense collection of states ranging up to S = 5/2 and A = 4, among which are, for example, a 4 r state with the configuration a (T )n ( t i )8 (T 11 )a * (T ), and a a(T)jc(TT)8(TiT)a*(T) 6 A state. As the states comprising such a melange are expected to interact strongly with one another, this 82 could explain the extensive perturbations experienced by the upper states of CoO investigated here. As discussed in Section III.C.I.c, the only perturbing state for which we have clear evidence appears to be a 4 Z state, arising possibly from a a 7 i 2 8 3 a * configuration, or 2 I x l A x 2 A x 2 l = 4 £ . Now that the ground state configuration of CoO has been determined in this work, the entire series of first row diatomic transition metal oxide ground states is now established. The ground states and some major molecular constants of the 3d transition metal monoxides appear in Table 3.V. Although many more excited states of cobalt oxide remain to be discovered, the most interesting results for the immediate future would be the direct measurements of the spin-orbit coupling intervals, and sub-Doppler measurements of the hyperfine structure. However, the experiments would require a more efficient means of generating CoO than has been used so far. 83 Table 3.VI. Ground states and configurations of the first row diatomic transition metal oxides, with the fundamental vibrational frequency A G 1/2, B and r for the v" = 0 state, and the spin-orbit interval AA for the orbitally degenerate electronic states. The AA value for CoO has not been established with certainty. Ground Electron AG-| / 2 B 0 r0 state configuration (cm-1) (cm-1) (A) AA Ref ScO 2£+ a 964.65 0.51343 1.668 - 29,30 TiO 3A r oS 1000.02 0.53384 1.623 101.30 89 VO 4 I - o82 1001.81 0.54638 1.592 - 45,88 CrO 5 p i r o827t 884.98 0.52443 1.621 63.22 90 MnO 6£+ 0 § 2 n 2 832.41 0.50122 1.648 - 91 FeO 5AJ 00H2 871.15 0.51681 1.619 -189.89 87 CoO 4 Aj O283TC2 851.7 0.50370 1.631 (-240) this work NiO 3£- c28 47c 2 825.4 0.5058 1.631 - 78 CuO 2 U \ a 2 8 47i3 629.39 0.44208 1.729 -277.04 92,93 84 CHAPTER IV HYPERFINE ANALYSIS OF NIOBIUM NITRIDE IV.A. Introduction. Niobium nitride (NbN) is an exemplary molecule in which to study hyperfine interactions in diatomic molecules, because the nuclear magnetic moment (JIN) of 9 3 N b exceeds that of any other non- radioactive atom. The magnetic hyperfine structure which results is proportionately large and well-resolved, allowing precise, informative analysis. Following the initial observation of NbN in 1969 by Dunn and R a o 9 4 , the first low resolution hyperfine analysis of the 3<X>-3A system was performed in 1975 by Femenias e j . a i 9 5 with a grating spectrograph. The study produced values for the magnetic hyperfine constants a, b and c which suggested that the excited 3 < X > state makes a non-negligible contribution to the hyperfine structure. The spectra also exhibited line broadening at very high J values, indicating either A -doubling in the 3 A state or a transition from case (ap) to (bpj) coupling with increasing rotation. In the meantime, the fundamental frequencies of the ground states of N b 1 4 N and N b 1 5 N were measured to be 1002.5 cm- 1 and 974 cm"1 by IR spectroscopy in a 14 K argon matrix. 9 6 A Russian group published a number of papers on the 3 0 - 3 A s y s t e m 9 7 . 9 8 - 9 9 , culminating in the 1986 publication by Pazyuk e _ L a i 1 0 0 , in which they proposed a set of rotational, centrifugal distortion and spin- orbit coupling constants (B, D and A), and an energy level scheme for the system. However, the spin-orbit splittings for both states were drastically miscalculated, and the ordering of the spin-orbit 85 manifolds was inverted, due to their interpretation of bands they observed near 5600 A as 3 $ 3 - 3 A 3 and 3 o 2 - 3 A 2 sp in -orb i t satellites, rather than as parts of the n - A system to which they actually belong. In 1979, an optical emission study measured eight subbands belonging to five systems, including 3 < E > - 3 A , and determined the upper and lower state B values for e a c h . 1 0 1 Most recent was a grating spectrograph analysis of the 3 o - 3 A system performed by the same investigators involved in the preliminary 1975 study, but at a higher resolution (±0.01 cm- 1 line position), and up to J" = 8 8 . 1 0 2 Their work produced the following set of molecular constants for the (0,0) band, in units of cm" 1 with the uncertainty in the last digit given in parentheses: T 0 A 8 B 10 7D 105A D X 3 A fixed to 0 183.0(2) -33.1(2) 0.50144(4) 4.56(6) =-4 A3o 16504.938(3) 241.6(1) 7.39(2) 0.49578(4) 4.88(6) =-4 The central shift parameter 8 accounts for the shift in the 3<j>3- 3 A subband because of second order spin-orbit effects. The investigations described in the current work mark the first high resolution laser spectroscopy performed on NbN. 86 IV.B. Experimental. IV.B.1. Synthesis of gaseous niobium nitride. Niobium nitride was formed in a flow system by reacting the vapor from a sample of warmed niobium (V) chloride (=80 °C) with nitrogen. The nitrogen was entrained with argon in a ratio of approximately 1:18 (v/v) at 1 Torr pressure. A few centimeters upstream from the fluorescnce cell, the vapor was passed through a 2450 MHz microwave discharge (powered by a Microtron model 200 microwave generator). To obtain intermodulated fluorescence spectra, two nearly coincident laser beams were passed in opposite directions across the lavender-colored flame of the discharge, with the fluorescence detected at right angles to the beams through a deep red low pass filter to the photomultiplier tube, as described in Section II.A. IV.B.2. Description of the 3 0 - 3 A spectrum. Broadband spectra of the three subbands of the 3G> -3A system of NbN are illustrated in Fig. 4.1. The middle spin-orbit component, 3<E>3-3A2, is shifted to higher energy rather than being equidistant between the outer subbands, and is also considerably weaker, presumably due to intensity stealing by an unseen state. The vibrational sequences are plainly visible, up to (v',v") = (5,5) in the 30>4-3A3 subband. At sub-Doppler resolution, the variation in hyperfine structure between the three subbands is apparent from the Q head regions shown in Fig. 4.2. The hyperfine interaction in the 3<E>3-3A2 subband is much less pronounced than that in the other two because 3fl> 2- 3Ai 50 cm* 1 i 1 3<I>3-3A2 16145 cm-1 16543 c m 1 Fig. 4.1. Broadband spectrum of the 3<J>-3A system of NbN, obtained with the intracavity assembly removed, using the dye rhodamine 6G. Note that the vibrational sequence of the 3 O 4 - 3 A 3 subband is visible up to (v\v") = (5,5). a) Fig. 4.2. The Q heads of the a) 3 < D 2 - 3 A i , b) 3 < D 3 - 3 A 2 , and c) 3 < D 4 - 3 A 3 subbands of NbN. CD 89 the value of X in both states is zero. In the 3<X>4-3A3 subband the hyperfine splitting is considerably larger than that in 3<J>2-3A-|, since Q is three times as large in the former subband (cf. equations 1.90 and 1.98). The assignment of the densely overlapped 3<j>2-3Ai Q head is shown in Figs. 4.3 and Fig. 4.4. The low-J R branches of the 3<x>2-3Ai subband, illustrated in Fig. 4.5, are exemplary for their completely resolved A F * AJ transitions and crossover resonances (cf. Section II.B for a discussion of these transitions). The hyperfine pattern is quite different in the central subband: at J" = 2 the high F component is on the low frequency side, but at J" = 3 the hyperfine structure reverses order and continues on at higher J values with the highest F component at high frequency. The development of this 3<X>3-3A2 R branch hyperfine structure is shown in Fig. 4.6. As the rotation of the molecule increases, spin-uncoupling is observed in the Q branches of the outer two subbands as a reversal in the hyperfine structure: the hyperfine splitting narrows with increasing J until the components collapse into a spike; then they reverse their order and widen with increasing rotation (see Fig. 4.7). Therefore hyperfine structure which begins with its components increasing in F toward increasing frequency reverse to an order in which the F values decrease with frequency. The reversal in the Q branches occurs at J = 27 and J = 38 in the 3<l>2-3Ai and 3<I>4-3A3 subbands, respectively. The hyperfine structure in the 3<E>3-3A2 transition is less sensitive to the effects of rotation, since its diagonal matrix elements are independent b and c. The Q branch of o -»• 3 2 Fig. 4.3. The beginning of the Q head of the 3<D2-3Ai subband. Each AF = 0 line is connected to the AF = +1 lines with the same F" value by a thick horizontal line. Components of the Q(7) and Q(8) lines are also present in this region, but are not labelled. CO o Fig. 4.4. The higher J portion of the 30>2 - 3 Ai Q head, and the first resolved Q lines. The crossover resonances are not labelled. CO 92 a) 4.5 6.5 b) 5.5 0.05 cm-1 R(1) 4.5 4 * 3.5 • • • 2.5 7.5 c) 6.5 5.5 4.5 3.5 r r r I ic r C C C CC c Fig. 4.5. a) R1, b) R2, and c) R3 lines of the 3 o 2 - 3 A i subband, illustrating the "forbidden" AF * AJ transitions (• for qR, * for pR) and the crossover resonances (c) between the rR and qR lines. Each A F - AJ transition (•) is labelled with the lower state F value, with the corresponding satellite transitions following it to the red (right) in the order: c (if seen), • , * (if seen). The scale shown in (b) is the same for all spectra. 93 Fig. 4.6. a) R2, b) R3 and c) R4 lines of the 3<D3-3A2 subband of NbN, showing the rR, qR and pR transitions and the crossover resonances associated with the rR and qR lines (denoted by c.o.). 94 qR d) 7 ;5 2.5 rRII l l l l l | l CO. 0.01 cnr1 I 1 9 5 5 5 -is e) rR| | | | | 0.01 cnr1 Fig. 4.6. d) R5 and e) R6 lines of the 3d>3-3A2 subband of NbN; the labelling follows that of Fig. 4.6 a, b and c. Q(29) Q(31) Q(33) Q(35) P(8) tliJlL Q(37) Q(39) P(9) 0.3 cm* 1 l 1 Q(41) P(11) Q(43) 16848.5275 cm" 1 I 16851.5933 cm"1 Fig. 4.7. The reversal of hyperfine structure at high J in the 3<I>4-3A3 Q branch, caused by the effects of spin-uncoupling. Actual reversal occurs in the line of maximum intensity, Q(38). CO cn 96 this subband therefore narrows up to about J = 12, and then remains nearly constant in width up to the limit of our data at J = 27. 97 IV. C. Non-Linear Least Squares Fitting of Spectroscopic Data. In order to acquire the best set of molecular constants in a Hamiltonian, one must iteratively improve an estimated set of constants until a satisfactory fit of the observed data is obtained. In approaching the non-linear type of Hamiltonian typically describing a spectroscopic problem, the Hamiltonian is divided into its two constituents: the coefficients containing the quantum number dependence, and the molecular constants, o r 1 0 3 H = X X m H m m=1 (4.1) X m is the mth parameter (or molecular constant) out of a total of p parameters, and H m is the "skeleton matrix" containing the quantum number dependence of the mth parameter. For example, a simple 2n Hamiltonian may be expressed a s : 1 0 3 1 / 2 0" H - T r 1 0 0 1 0 -1/2 + B (J + 1/2)2 - 2 -[(J + 1/2)2 _1]1/2 -[(J + 1/2)2 - -|]1/2 (J + 1/2)2 The matrix of eigenvalues (or energy levels) E Of the Hamiltonian is obtained by diagonalization with the eigenvectors U: UtHU = E (4.2) U is a unitary matrix such that the adjoint of U (U^, or the conjugate of the transpose U T ) equals the inverse of U (U- 1). The combination of equations (4.1) and (4.2) allows the Hellmann- Feynman theorem to be employed, which states: 1 0 4 a E m / 3 X = fam*(dWdX)*¥mdx (4.3) 98 For a single matrix element ii of parameter m, the Hellmann- Feynman theorem becomes: 1 0 3 [UT(aH/3Xm)U]ii = 3Ei/aXm = Bj (4.4) Using equation (4.1), equation (4.4) can also be written as: Bim - [UTHmU]ii (4.5) The Hellmann-Feynman derivatives B j m form the derivatives matrix, B, which give the dependence of the energy on variations in the parameters. To apply this relation to an iterative solution of unknown molecular parameters, equation (4.2) is expressed in terms of a single energy level, E j c a l c : Ejcalc = (UtHU)n (4.6) Substituting equation (4.1) into equation (4.6) gives P E p i c = X X m (UtH m U)ii (4.7) m=1 With the relations in equations (4.4) and (4.5), the energy can be written: P E p i c = I X m B i m (4.8) m=1 To express equation (4.8) in terms of transitions rather than energy levels, the upper and lower state eigenvalue vectors (E 1 and E") are subtracted to give y, and B' and -B" are combined into one derivatives matrix B. Equation (4.8) therefore transforms t o : 1 0 5 y = BX (4.9) where y is the vector of calculated transitions, B is the matrix of known derivatives, and X is the vector of estimated parameters. If 99 there are N transitions and p parameters to be determined, y has length N, B is a matrix of size N by p, and X has length p. To obtain X , both sides of equation (4.9) are multiplied by ( B T B ) - 1 B T : (BTB ) -1 (BTB)X = ( B T B ) " 1 B T y X = ( B T B ) _ 1 B T y (4.10) In a problem where the estimated parameters X are iteratively improved, we calculate parameter changes A X . rather than X itself. Equation (4.10) is therefore expressed a s : 1 0 6 A X = ( B T B ) " 1 B T A y (4.11) where A y is the vector of residuals (i.e., the observed transitions minus the calculated). The fitting process begins with a set of estimates for the molecular constants, which are used to generate calculated transitions (ycaic) a n c j {heir residuals (Ay). The set of corrections to the constants, given by equation (4.11), is added to the initial estimates to provide improved constants for the next iteration. The process is repeated, iteratively producing improved sets of calculated transitions, residuals and constants until the magnitude of the residuals is reduced to a satisfactory level, for example, to the vicinity of the experimental precision. The least squares program for the 3<X> - 3 A system of NbN was written in FORTRAN 77 by the author, except for UBC Amdahl library routines for diagonalizing and inverting matrices, and calculating parameter changes from the Hellman-Feynman derivatives. The Hamiltonian matrices for the 3<D and 3 A states have a maximum dimension of (21 + 1)(2S + 1), or 30. The 30 x 30 matrices (one for each F) were diagonalized in two steps. In the first step, only the rotational part of the Hamiltonian was diagonalized, in ten separate 100 J submatrices. In the second step, the entire matrix (rotational and hyperfine) was diagonalized. Two steps were employed because the ordering of eigenvalues from step one was used in the second diagonalization to preserve the matching of eigenvalues with the original basis functions. This is possible because the separation of the spin-orbit components is large compared to the perturbation made by the hyperfine interactions. Analogous to the common formula for the standard deviation, s = [I(xobs . xcalc)2/ n]l/2 (4.1 2) the weighted least squares standard deviation is obtained f r o m : 1 0 5 n a = [ I (yjQbs . y jcalc)2Wjj/(n-m)]1/2 (4.13) i=1 where n is the number of independent measurements, m the number of unknowns to be estimated, n-m the degrees of freedom, and Wjj the diagonal element of the weight matrix for point i . 1 0 5 To determine estimates of the precision of the estimated constants, a variance- covariance matrix @ is calculated b y : 1 0 5 0 = c 2 (BTB)- 1 (4.14) A diagonal element 0jj is called the variance (not to be confused with the variance that is the square of the standard deviation, G2). The square root of &\\ gives the standard error, or precision, of estimated molecular constant i. The off-diagonal elements ©jj are covariances. Both the variances and covariances are only estimated values, because they depend on the precision of the measurements, a 2 . The goodness of the structure of the model lies in ( B T B ) - 1 . 101 Normalization of the variance-covariance matrix gives the correlation matrix, C, with elements cjj = eij/(eiiejj)i/2 ( 4 . 1 5 ) where Cjj = 1 for i ~ j, and (-1 < Cjj < +1) for i * j. C is independent of the precision of the measurements since o 2 has been cancelled out. Therefore the off-diagonal elements represent the interdependence of the molecular constants on one another, for a given data set. A value for CJJ that closely approaches unity indicates that constants i and j cannot be determined independently. 102 IV.D. Results and Discussion. Initial line assignments were facilitated by the unpublished grating spectrograph work of Dunn e l s i 1 0 2 , who listed the positions of the P, Q and R rotational lines. Initial attempts to obtain a least squares fit to the hyperfine constants in a case (a) basis (i.e., as they were presented in Sections I.D.3 and I.D.4) did not succeed, because the hyperfine constants required to fit the three subbands are not consistent with one another. In the light of this observation, and the unequal first order spin-orbit spacings, it was concluded that the various substates are perturbed differently by second order spin-orbit interactions. According to the AQ = 0 selection rule for this interaction 1 0 9 , the electronic states perturbing the 3<J> substate include ^3,4, 1T4, 10>3, 3A2,3 and 1A2. The 3 A substates can interact with 3 0 2 , 3 , 1<J>3, 1 A 2 , 3T11,2 and 1 n 1. The 1 0 and 1 A states isoconfigurational with 3G> and 3 A are expected to be the closest of these states to 3d> and 3 A , and therefore the ones most responsible for the perturbations (see Fig. 4.8). The effect would be to shift the central spin-orbit components, 3<X>3 and 3 A 2 , to lower energy. However, the hyperfine constants suggest that there could also be second order spin-orbit interactions occurring with the other members of the manifolds, though we can say nothing about their relative sizes. The 3 o - 3 A system of NbN is the first observed instance of a molecule represented by Hund's case (a) which requires modifications to the Hamiltonian because of extensive second order spin-orbit interactions. This phenomenon can be considered a slight tendency toward the case (c) coupling s c h e m e . 1 0 9 The molecular constants obtained for the 3 o - 3 A system of NbN 103 817*1 — \ C 2 a i 5 i • 1 0 in 3n . 3 0 — < •in -3n Hs .oO ) H s.o.( 2) .4 --- o CD 00 CD 1L+ 1A 2 - ^ 3A , 3 _ . _ . 1 — 742.2 CO i n CD i n CD $383.4 Fig. 4.8. Partial energy level diagram for NbN. The figure is not to scale, but illustrates the relative ordering of states, except in the case of the low-lying configurations o 2 and 0 8 where the ordering is uncertain. 104 are given in Table 4.1. The unequal perturbations in the 3<D and 3 A spin-orbit manifolds means that, in the magnetic hyperfine structure, only the h constants in the matrix elements diagonal in ft and £ can be determined, rather than individual a, b and c constants (cf. equations 1.90 and 1.98). The h constants, subscripted by their X values, are as follows: In an unperturbed system, the average of h-i and h+i equals ho; that is, (b + c)-i and (b + c)+i in equations (4.16) and (4.18) are equal. This is far from the case in the 3<x>-3A system of NbN, where (b + c)+i is 39% smaller than (b + c)-i in the 3 A state, and 10% larger in 30>. It was also found, in the 3 A state, that two distinct b constants are required in the <X=-1|X=0> and <X=0|Z=+1> matrix elements (referred to here as b.-i/o and brj/+i, respectively). Therefore, a total of five magnetic hyperfine constants are required to fit the data, rather than the usual three: h-i, ho, h+i, b-1/0 and bo/+i replace a, b and c. It is clear that the perturbations in the 3 A state are much more pronounced than those in 3G>. The 3 A b+i/o value is 34% smaller than bo/ - i , comparable to the 39% difference between the 3 A (b + c)-i and (b + c)+i constants. In the upper state, however, two distinct b values off-diagonal in X are not necessary: attempts to distinguish two 3 0 b constants produced values that were very highly correlated (-0.998) and with standard errors so high that the constants were indeterminable. It is evident, then, that the 1 A state lies closer to h-i = aA - b - c = aA - (b + c)-i ho = aA h+i = aA + b + c = aA+(b + c)+i (4.16) (4.17) (4.18) 105 Table 4.1. Molecular constants for the 3C>-3A system of NbN.a O A To 16518.509(1) 0 A 247.4116(5) 191.7038(8) B 0.495814(4) 0.501465(4) D 0.4943(4) x 10-6 0.4622(2) x 10-6 X -16.817(2) 3.430(2) y 0.011(2) -0.0217(6) A D -0.58(2) x 10-4 -0.105(3) x 10-3 XD -0.150(6) x 10-4 -0.1314(6) x 10-3 h-1 0.0633(2) -0.0616(3) ho 0.0411(4) 0.0458(5) h+1 0.0168(2) 0.1112(3) b -0.02(1) - b-1/0 - 0.085(5) bo/+i - 0.056(5) e 2 Q q o -0.39(8) x 10-2 fixed to zero Derived constants: (b+c).i -0.0222(4) 0.1074(6) (b+c)+i -0.0246(5) 0.0654(6) a 0.000547 3 Values are in cm- 1 . The numbers in parentheses are three times the standard errors of the constants, in units of the last significant figure. The standard deviation of the transition measurements is given by a. The magnetic hyperfine constants, h, (b + c) and b, are explained in the text. 106 the 3 A state than 10> does to 3<X>. Note from Fig. 4.8 that the ordering of states in the 8rc manifold is contrary to that dictated by Hund's r u l e 1 1 0 , which would place the higher multiplicity 1<I> state below T I (and therefore closer to the 3 0 state). The dipolar hyperfine constant c cannot be extracted since separate b constants are required for the three substates. The (b + c) and b constants clearly support the 5 s a 1 4 d 6 1 and 4drc 1 5 1 configurations for the 3 A and 3 0 states, respectively. The 3 A (b + c) and b values are large and positive, indicating that the dominant mechanism for the coupling of electronic and nuclear spins is the Fermi contact interaction. This is consistent with the presence of an unpaired sa electron, as in the s a 1 d 8 1 configuration of 3 A . The 3 0 (b + c) and b constants are negative, and small compared to those in 3 A . This is characteristic of a hyperfine interaction which occurs because of spin polarization by electrons in orbitals having nodes at the nucleus, such as T C 1 5 1 . The difference between the Fermi contact and spin polarization hyperfine constants in NbN is similar to that found in the VO states 4 5 4 s a 1 3 d 8 2 X 4 Z " and 4 p o 1 3 d 8 2 C 4 I \ The ratio of 3 A(b + c ) a v e / 3 ^(b + c ) a v e = -3.7, while b(X 4 !" ) / b(C 4Z-) = -3.1. The quadrupole coupling constant for the lower state is -3.9 (+.8) x 10"3 c m - 1 , while that of the upper state was fixed to zero after it was found to be too small to be determined. The sign of the 3<J> state e 2 Q q 0 is consistent with the quadrupole moment for 9 3 N b of -2 x 10- 2 4 e c m 2 . The upper and lower state constants for the interaction of nuclear spin and rotation (ci) were fixed to zero, as they were found to be on the order of -10 - 5 to -10 - 6 cm- 1 , almost completely 1 0 7 correlated ( .999), and with standard errors as large as the values themselves. It is the usual case for diatomics containing a transition metal for ci to be too small to be determined (see for example references 3 1 , 4 5 , 7 9 and 91). In the rotational part of the Hamiltonian, the A, B and D constants are very well determined in spite of the high correlations between A' and A" ( .9985) and B' and B" (.995). The high rotational lines carrying information about the spin-uncoupling operator, - 2 B J S , allow B and D to be determined individually, rather than simply determining their differences, B' - B" and D' - D". Since all three subbands were fitted simultaneously, and B was extracted with good precision, A could also be determined. This is possible since A, B and the effective B values for each subband are related by: 2 4 Beff,n = B(1 + 2 B I / A A ) ( 4 .19 ) From the B values, the bond lengths are calculated to be: r0 ( 3 A ) = 1 .6618 A r0 ( 3 < D ) = 1 6 7 1 2 A There have been very few rotational studies of transition metal mononitrides. Aside from the current work, the known bond lengths (r0, in A) are: T i N l n X 2 I 1.583 A 2 n r 1.597 B 2 I 1.646 Z r N ^ 2 X 2 I 1.696 B 2 I 1.740 A 2 n 1.702 M O N 1 1 3 x4I" 1.634 108 A * n 1 -654 The 3d transition metal monoxide series isovalent with ZrN (and TiN), NbN, MoN is ScO, TiO, VO, whose ground state bond lengths go as 1.668 A3<\ 1.623 A89 and 1.592 A*5. Here the bond length decreases with each additon of a bonding 8 electron. The 3 A and 3 0 NbN bond lengths show that the nitrides are consistent with this trend, with values intermediate between those of ZrN and MoN. The very large spin-spin interaction constants X (equations 1.72 and 1.73) are caused by contributions from the second order spin- orbit interactions which induce the substantial shift of the 3 G > 3 - 3 A 2 subband. The centrifugal distortion correction to X, however, is considerably larger than its expected value of Xo - X(Ao/A). The reason for this probably lies in the fact that we have not yet made direct measurements of the spin-orbit intervals. In this context, then, the centrifugal distortion correction constants A D and XQ are essentially fudge factors which enable the least squares fit to converge to a minimum lying within a broad minimum which contains the true molecular constants. So although this set of constants is an internally consistent one which fits the data, once the derived A values are replaced by direct measurements the constants may change slightly to enable the fit to converge to the true, nearby minimum. With the data we now possess, however, the A D and Xo values given in Table 4.1 are necessary to obtain a fit. To demonstrate this fact, a fit of the rotational constants was made in which Xo and y were fixed to zero, and all hyperfine constants were fixed at the values determined in this work. The initial values for the floated constants were taken from the grating 109 spectrograph work of Dunn e_tai 1 0 2 (see p. 85), with the exception of AQ which was given an initial value of zero; the parameter 8 in their work is equal to -2X. The fit converged to a standard deviation of 0 . 0 0 1 3 8 c m - 1 , which is about 2 . 5 times higher than the fit which incorporates XD and y. As expected, the final set of constants (Table 4.II) is very similar to those determined by the grating spectrograph analysis, with the exception of To, which was found from LIF data to be 13.5 cm- 1 higher than that from the grating work. The residuals contain systematic errors in the positions of the rotational lines, as compared to the random residuals generated by the full set of constants. The systematic errors and higher standard deviation reflect the inability of the model to fit the data without XD, AD and y. However, as stated above, the resulting rotational constants, other than B and D, are only effective ones. Another important feature of this fit is that the first order spin-orbit coupling constants A' and A" are 1 0 0 % correlated, as are the second order spin-orbit parameters X' and X." (see the correlation matrix in Table 4.II). This is a direct reflection of the fact that the spin-orbit coupling constants are derived rather than measured. As a result, only the difference A X can be determined, rather than separate X' and X" values. For these reasons, a fit excluding XD and y may produce a set of rotational constants that more accurately represents the real situation, though the addition of XD and y creates a model which is able to fit the data. It is worth noting that in a purely case (a) basis, y, AD and XD are correlated such that only two of the three can be determined. 4 1 In the 3 O - 3 A system of NbN this correlation is broken 110 Table 4.II. Rotational constants obtained for the 3d>.3A system of NbN with the XD and y parameters fixed to zero, and the hyperfine constants fixed to the values in Table 4. l . a The correlation matrix follows the constants. O A To 16518.4653(2) 0 A 242.59(8) 184.5(1) B 0.495796(8) 0.501447(8) D 0.5005(7) x 10-6 0.4685(4) x 10-6 X -3.70(8) 16.53(8) A D -0.484(5) x 10- 4 -0.793(8) x 10-4 a 0.00138 a T h e format of the table follows that of Table 4.I. Correlation Matrix To A' B' D' X AD To 1.0000 0.0996 0.0820 -0.4333 0.3401 0.1538 A' 1.0000 -0.0473 0.1235 0.5124 -0.0645 B' 1.0000 -0.0973 0.0934 -0.3204 D' 1.0000 0.0366 -0.4503 X' 1.0000 -0.0664 A D ' 1.0000 A" B" D" X" A D " To 0.0995 0.1580 -0.2227 0.3408 0.1715 A' 1.0000 -0.0335 0.3977 0.5115 0.3233 B' -0.0474 0.9936 -0.0789 0.0928 -0.3322 D* 0.1233 -0.1874 0.4224 0.0360 -0.2099 X,' 0.5124 0.1025 -0.2270 1.0000 0.0874 A D ' -0.0643 -0.2737 -0.0883 • •0.0659 0.8660 A" 1.0000 -0.0336 0.3976 0.5116 0.3233 B" 1.0000 -0.0839 0.1020 -0.2929 D" 1.0000 -0.2275 0.0982 X" 1.0000 0.0875 A D " 1.0000 111 to some extent by the high J data where there as a distinct tendency towards case (b) (see the correlation matrices in Appendix I and Table 4.II). For the future, a direct measurement of the spin-orbit intervals must be made. The most likely method for doing this is to locate forbidden "spin-orbit satellite" transitions which disobey the case (a) selection rule A £ = 0 (equation 1.57). Since these transitions are very weak, resolved fluorescence experiments can be performed to enhance the signal. To record the spectrum of a 3<X> 2-3A 2 line, for example, an allowed 3 < D 2 - 3 A I transition is excited. The resulting emission spectrum of the satellite transition is recorded over a long exposure time using the microchannel-plate intensified array detector. The lines which hold the most promise for producing spin- orbit satellites are high J lines affected by spin-uncoupling, since the A L = 0 selection rule weakens with increasing rotation. However, it is also important that the excited line be strong, so a compromise must be made between high J and line strength when choosing lines for excitation. Other important tasks are to locate the singlet states which interact with the 3A2 and 3<x>3 spin-orbit components, and to search for the expected a 2 1 Z + state to determine if the ground state is 3 A or 1 5> . The ordering of the 0 8 states ( 3 A and 1 A ) and the o 2 state ( 1 X + ) depends on the relative ordering of the 4 S C T and 3d5 metal- centered molecular orbitals (see Fig. 3.1). Diatomic transition metal oxides and fluorides isoelectronic with NbN demonstrate that these orbitals lie very close to one another. Therefore one cannot readily predict in NbN whether the 3A R or 1 Z + state will be lower in energy. 112 For example, the d2-transition metal monoxide series, consisting of titanium oxide (TiO), zirconium oxide (ZrO) and hafnium oxide (HfO), is variable in this respect. TiO has a 3 A r ground s t a t e 1 1 4 , with the 1 A state lying 3500 cm" 1 above tha t 1 1 5 . However, ZrO has a 1X + ground s t a t e 1 1 3 which lies 1650 cm- 1 below the 3 A r s t a t e 1 1 6 . HfO is believed to have a 1X+ ground state also, but with the 08 states further removed from the ground state than those in ZrO due to the greater ligand field splitting between the o and 8 orbitals in H f O . 1 1 4 In the d1-transition metal monofluoride series, comprising scandium fluoride (ScF), yttrium fluoride (YF) and lanthanum fluoride (LaF), S c F 1 1 5 and YF have 1 Z + ground states, while the ordering of 1 Z + and 3 A r in LaF is not k n o w n 1 1 8 . Tantalum nitride (TaN), the 5d counterpart of NbN, is predicted from matrix isolation studies to have a 1 £ ground state, though the possibility of 3 A has not been entirely ruled o u t . 1 1 9 To identify the ground state of NbN securely, then, the relative position of the 1 X + and 3 A r states must be determined experimentally. 113 CHAPTER V ROTATIONAL ANALYSIS OF THE Vy-FUNDAMENTAL OF AMINOBORANE, NH 2BH 2 V.A. Background. This work examines the BH2 out-of-plane wagging fundamental of aminoborane (NH2BH2), the simplest alkene in the B=N homologues of the hydrocarbons. Long before N H 2 B H 2 was studied experimentally, its small size and the interest in B-N compounds led to extensive theoretical studies of it. In particular, the donor- acceptor nature of the B-N bond atttracted attention, as Huckel theory calculat ions 1 2 0 done in 1964 predicted that the bond moment was in the direction B to N rather than the reverse, as required by formal valence theory. These preliminary calculations, covering charge distributions, electronic structures and geometries for a number of B-N compounds, were followed by CNDO (complete neglect of differential over lap ) 1 2 1 and ab i n i t i o 1 2 2 ' 1 2 3 - 1 2 4 calculations predicting these and other properties such as the dipole moment, force constants, barriers to rotation and stabilities. Aminoborane's extreme instability at room temperature, however, imposed practical difficulties for experimentalists to verify or refute the theoreticians' predictions. It's first synthesis was in 1966 from the symmetrical cleavage of vacuum sublimed cycloborazine pyrolyzed at 135°C, where NH2BH2 and other decomposition products could be trapped in a liquid nitrogen cold trap, and then identified by mass s p e c t r o s c o p y 1 2 5 . The aminoborane was found to have decomposed spontaneously after warming to room temperature. In 114 1968, gaseous aminoborane and diborane (B2H6) were observed by molecular beam mass spectroscopy as products of the spontaneous decomposition of solid ammonia borane ( N H 3 B H 3 ) at room temperature. 1 2 6 When Kwon and McGee performed both pyrolysis and radiofrequency discharge experiments on borazine (the BN analog of benzene), N H 2 B H 2 a n d B 2 H 6 w e r e again the products. 1 2 7 They were recovered in a -168 ° C trap, then separated by vacuum distillation of diborane from aminoborane at -155 ° C . At this temperature, small amounts of both evaporation and polymerization of NH2BH2 were observed. Polymerization becomes the dominant process at temperatures above this, and is fairly significant at -130 ° C . 1 2 7 The pronounced instability of monomeric aminoborane led Pusatcioglu et a l 1 2 6 in 1977 to investigate the possibility of using N H 2 B H 2 to build thermally stable inorganic polymers. They pyrolyzed gaseous ammonia borane, condensed the monomeric NH2BH2 product at 77 K, then allowed it to polymerize as it warmed. In 1979 a microwave spectrum of NH2BH2 was obtained, using a sample formed from the reaction of 5-10 mTorr each of ammonia and diborane at 500 ° C . 1 2 9 Molecular constants calculated by a least-squares fit were consistent with a planar configuration, thereby establishing the symmetrical structure N H 2 = B H 2 for aminoborane, rather than the asymmetrical NH3BH. Perhaps the most important outcome of this work was the determination of the dipole moment to be 1.844 D in the direction from N to B, as opposed to the theoretical predictions of B to N . 1 2 0 ' 1 2 1 The assumption of an N B direction for the dipole moment was based on the observation that the dipole moment of NH2BH2 is 0.751 D smaller than that in BH2BF2. 115 The same group recently reported microwave spectra of five isotopic species of NH2BH2, improving the constants and geometric parameters obtained in the previous s tudy . 1 3 0 Recently, at the University of British Columbia, the first gas phase Fourier transform infrared spectrum of aminoborane was m e a s u r e d . 1 3 1 The synthesis combined the solid-state and vapor- phase ammonia borane pyrolysis techniques. Solid N H 3 B H 3 w a s heated to about 70 ° C in a flow system maintained at approximately 200 microns, and the vapors produced were passed through a furnace at about 400 ° C , to pyrolyze unreacted sublimed sample. Nine of aminoborane's eleven infrared (IR) active fundamental vibrations were recorded at medium resolution (0.05 cm- 1) , with the V4 A-type band at 1337 cm- 1 being also recorded at very high resolution (0.004 c m - 1 ) . Since that time the bands of all of the IR active fundamentals have been recorded at UBC at 0.004 cm- 1 resolution (see Table 5.1), though V5 is vanishingly weak because its dipole derivative appears to be very small. Some analysis has been c o m p l e t e d 1 3 2 - 1 3 3 ' 1 3 4 , with the remainder currently underway. The present work is a contribution to the high resolution Fourier transform IR study of aminoborane, being the rotational analysis of the C-type V7 fundamental whose origin is at 1004.7 cm- 1 . 116 Table 5.1. Vibrational fundamentals of gaseous NH2 1 1BH2- Symmetry cm- 1 Type of motion Ai V1 3451 NH symmetric stretch v 2 2495 BH symmetric stretch V3 1617 NH2 symmetric bend v 4 1337.4741 BN stretch V5 1145 BH2 symmetric bend A 2 V6 837 Torsion (twist) Bi V7 1004.6842 BH2 wag V8 612.19872 NH2 wag B 2 V9 3533.8 NH asymmetric stretch vio 25643 BH asymmetric stretch V11 1122.2 N H 2 rock V12 742 B H 2 rock 1 Reference (131) 2Reference (133): vs (1,0) band; reference (132): vs (2,0) band 3Reference (134) 117 V.B. The Michelson Interferometer and Fourier Transform Spectroscopy. The infrared interferogram was recorded and Fourier transformed with a BOMEM DA3.002 Michelson interferometer and associated software (version 3.1). Three sources of infrared light are available depending on the wavelength region desired: a quartz- halogen lamp for the near IR and visible regions, a globar for the mid-IR, and a mercury-xenon lamp for the far IR. After first being filtered and focused at an aperature, the infrared light passes to a collimating mirror and is reflected as a parallel beam to a beamsplitter, where it is divided in two. One beam continues through to a fixed mirror, while the other is reflected onto a mirror moving at constant velocity. As one of the beams has a fixed path length and the other a constantly varying one, the recombination of the beams at the beamsplitter produces a resultant of sinusoidal waves that are out of p h a s e . 1 3 5 The portion of the resultant not absorbed by the sample is measured at the detector as the interferogram. The point along the moving mirror's travel at which the fixed and moving mirrors are exactly equidistant-called the zero path difference (ZPD)--should in principle bring all the sinusoidal waves into phase, with constructive interference producing a maximum in the amplitude. 1 3 5 Because the interference patterns producing the infrared interferogram result from the optical path difference between the two light beams, it is essential that signal sampling occur at constant intervals of mirror displacement. This is achieved in the BOMEM DA3 spectrophotometer by a He-Ne laser. Operating at 632.8 118 nm, or 15796 cm" 1 , the laser provides an extremely precise time base of 31,592 cycles per cm of mirror travel . 1 3 6 The cycles, called fringes, trigger spectral sampling at a frequency normally equal to one sample/laser fringe, though the rate can be increased to up to eight times the laser fringe frequency. The phase coherence provided by this laser is excellent: its single-mode operation prevents destructive interference by two other closely lying transition frequencies, and its thermal stabilization removes temperature dependent fluctuations in the laser optics. The resulting uncertainty in the mirror's position is 0.0025 fringes per cm of mirror travel, which even at the maximum translation of 125 cm amounts to a variation of only 0.3 fringes over the length of the mirror's s c a n . 1 3 6 The interferogram not only requires that its points be sampled at precise intervals, but also that one of these points occurs at an origin that is exactly reproducible from scan to scan. The BOM EM DA3 spectrophotometer acheives this by triggering the commencement of each scan at the ZPD of an interferogram of white light. The beams from the white light source follow the same optical path as that of the radiation of interest, with the incoherent nature of the white light producing an interferogram characterized by an intense pulse at ZPD (the WLZPD), and low intensity amplitudes at non-zero mirror translations. The occurrence of the pulse is precise to well within one laser fringe, so the actual WLZPD trigger is marked as the laser fringe immediately following the pulse. The result is a synchronization signal which references the points in the 119 IR interferogram to a constant position along the scanning mirror's path .136,137 A Fourier transform infrared experiment is therefore the process of obtaining the infrared interferogram in conjunction with the white light reference interferogram and the time base generated by the He-Ne laser. These data are processed by Fourier transformation from the IR interferogram time domain to an IR spectrum in the frequency domain. The integrals of the Fourier transformation can be understood in terms of the phase differences between the IR beams split by the beamsplitter. When a wave with angular frequency co reflects off a mirror moving with velocity v, the frequency is Doppler shifted by an amount 138 Aco = 4JIV/X (5.1) Expressed as a function of the speed of light and the incident frequency, using the relation X= 2TIC/CO, the phase shift becomes 138 Aco = (v/c)2co (5.2) The magnitude of Aco is on the order of 1 kHz to 100 kHz, a frequency that can be processed easily as compared to the 1 0 1 3 to 1 0 1 5 Hz frequencies of IR radiation itself. The time-averaged beat intensity, I, produced by the combination of two waves out of phase by Aco i s 1 3 8 I = l0(1 + cosAcot)cos2[(co + co')t/2] = (l0/2)(1 + cosAcot) (5.3) where l 0 is the signal intensity when Aco = 0. Represented in terms of amplitude or electric field strength [E0(co)], phase difference [5(co) = Acot], and the reflectivity (R) and transmittance (T) of the beamsplitter, equation (5.3) becomes 1 38; l(co,8) = cec-RT |E0(co)2| [1 + cos8(co)] (5.4) 120 where c is the speed of light and e 0 is the vacuum permittivity 1 3 9, equal to 8.85 x 1 0 - 1 2 C 2 J - 1 r r r 1 . Integrating over all frequencies of the spectral components, l(8) = J l(co,5)dco = ceoRT[J |E0(co)|2dco + j [E0(co)|2cos8dco] (5.5) At zero path difference, or 8 = 0, the two terms in brackets in equation (5.5) are equal, so the ZPD intensity is given by: l 0 = 2ce 0RTJ |E0(co)|2dco (5.6) The time-averaged signal intensity as a function of phase difference, l(8), is the quantity measured at the detector. The interferogram points themselves are taken to be the oscillations of these intensities about l 0 / 2 : 1 3 8 |l(8) - l 0/2| = C £ 0 R T J |E0(co)|2cos8dco (5.7) The cosine Fourier transform of an interferogram of the form of equation (5.7) yields a spectral intensity distribution function l(co) in which intensity is a function of discrete frequencies: l(co) = ( 1 / K R T ) J [l(8) - l 0/2]cos8d8 (5.8) However, since imperfections in manufacture do not produce equivalent reflectivities in the fixed and moving mirrors, sine components as well as cosine are introduced into the interferogram. The actual Fourier transform therefore employs the complex form of the e x p r e s s i o n 1 3 8 - 1 4 0 ' 1 4 1 l(v) = C J [l(8) - l 0/2] e - i 2 * v 5 d8 (5.9) In general form, the Fourier transform of function f(x) i s 1 4 2 3{f(x)} = F(a) = Jf(x)e-'«xdx (5.10) The inverse Fourier transform of F(a) is therefore 3"1{Fa)} = f(x) = (1/2TI)J F ( a ) e ! « x d a (5.11) 121 Likewise, the spectrum expressed in equation (5.9) is one member of a Fourier pair, which consists of two non-periodic functions related by the Fourier integral transforms 1 4 1 : g(v) = J f(5)e'2"v6ds (5.12) f(8) = 1 g(v)e-'2*v8dv (5.13) A Fourier pair is illustrated graphically in Fig. 5.1. Fourier transform spectroscopy is able to exploit the Fourier pair relationship between the time domain (phase, 5) and the frequency domain (co or v), because frequency can be obtained with greater accuracy, resolution and speed by measuring and transforming phase differences rather than by directly measuring relative frequency. With the Michelson interferometer the integration cannot be performed over all space (-°° to +°°) but is limited to the range 0 - L where L is the total mirror displacement. As the distance travelled by the mirror increases, the number of terms included in the integration increases, extending the amount of information available for extraction into the spectrum l (v ) . 1 4 1 The theoretical maximum spectral resolution of an interferometer is therefore inversely proportional to the maximum optical path difference between the fixed and moving m i r r o r s . 1 4 2 Defining resolution as the full width at half height, the maximum unapodized resolution i s : 1 4 4 Avi/2 = 1/(2L) (5.14) Imposing the 0 to L limits on an interferogram is known as a "boxcar" truncation (see Fig. 5 . 2 ) . 1 4 4 When a boxcar-truncated interferogram is Fourier transformed, the spectral line shape contains the sine function [sine z = (sin z ) / z ] : 1 4 6 « 1 4 7 122 Fig. 5.1. A polychromatic signal in the frequency domain (above) Fourier transformed into the time domain (below).141 123 F{D(x)} = 2L(sincz) (5.15) where z = 2ji(a-a 0)L. The half-width of the center spike of this form is very narrow: A a = 1.207/2L, or about 20% wider than the theoretical resolution of 1/2L. However, the sidelobes next to the central peak have about 21% of its intensity, and the amplitudes of subsequent lobes are slow to die a w a y . 1 4 5 In order to approximate more closely the true frequency domain spectrum, an apodization function is often included in the data processing. This process dampens the effects caused by truncating the interferogram at a definite mirror displacement of L. Though there are many forms of apodization functions, the effect is to give decreasing weight to the data points recorded at large mirror d isp lacements . 1 4 5 - 1 4 6 One of the simplest is the triangular function in Fig. 5.3, in which all sidelobes are positive and the largest is only about 4.5% that of the center spike; the linewidth is increased by almost 50% over the boxcar c a s e . 1 4 6 - 1 4 7 The apodization applied to the aminoborane experiment in this work was a cosine function referred to as "Hamming" or "Happ-Genzel". It produces spectral lines with negative sidelobes of only 0.0071 the height of the maximum peak, and lines about 2% broader than those from the triangular apodiza t ion . 1 4 5 124 ^ V Fig. 5.2. A boxcar function D(x) (above). The Fourier transform of a boxcar truncated interferogram is a spectrum with the line shape function F{D(x)} = 2Lsin(27ivL)/27tvL (where L denotes the maximum mirror displacement.) The full width at half-height (Avi/2) is 1.207/2L, and the strongest sidelobe has 21% the intensity of the m a x i m u m . 1 4 5 125 Triangular D(x){1 - |x|/L} Fig. 5.3. The triangular apodization function D(x) (above) produces a spectrum with the line shape function F{D(x)} = 2Lsin(27tvL)/(27cvL)2 (below). The full width at half-height (Avi/2) is 1.772/2L, with the strongest sidelobe only 4.5% of the maximum intensity. 1 4 5 126 V.C. Experimental. The aminoborane was prepared by pyrolysis of borane ammonia (BH3NH3, Alfa Products) according to the procedure of Gerry and c o w o r k e r s 1 3 1 , except that in the present work the temperature of the solid N H 3 B H 3 was raised to only 67 ° C - 68 ° C for the first several hours, then lowered to 63 ° C - 65 ° C for the remainder of the experiment. The 70 ° C pyrolysis temperature employed in reference 131 was found to be unnecessarily close to the temperature of uncontrolled thermal decomposition, which initiates violently at approximately 71 ° C . At the time the interferogram was measured, the temperature of the solid ammonia borane was 63.5 (±0.5) °C . The sample absorption cell, set to an optical path of 9.75 m, was maintained at a pressure of IOOJJ. during data acquisition. The BOMEM DA3.002 interferometer was fitted with a potassium chloride beam splitter and a liquid nitrogen-cooled HgCdTe detector. 127 V.D. The Asymmetric Rotor. A vibrational fundamental is infrared active if the dipole moment JI changes as a result of motion along the normal coordinate Qk, or in other words if the derivative (3|i /3Qk)o in the Taylor series expansion of the dipole moment u = uo + I (3u/3Q k ) 0 Q k (5.16) is non-zero . 1 4 8 The linear character of the dipole operator means that its components transform as translations along the principal axes, and therefore so do the various (3^i/3Qk)o Qk's. Aminoborane is a prolate asymmetric top molecule belonging to the point group C 2 v , whose character table is given in Table 5.II. The irreducible representations of the normal vibrations are: 5Ai + A 2 + 2 B i + 2 B 2 , for a total of twelve fundamental vibrations. The B H 2 out-of-plane wagging vibration is antisymmetric with respect to reflection in the yz plane, and therefore transforms as the Bi representation (see Fig. 5.4). Thus the V7 vibration represents translation along the c inertial axis and generates a C-type infrared band. Accompanying any molecular vibration are the rotational transitions involving changes in the total angular momentum, J . In order to understand the rotational selection rules for an asymmetric top molecule, one must write down asymmetric top rotational wave functions which are eigenfunctions of the symmetry operations of the molecular point group, in this case C 2 v . We begin by examining the effects of the C 2 v symmetry operations on the symmetric top wave functions, YJK(6,<|>). From equation (1.23) we know that: YjK(e,<>) - NPjK(cos 6)e'K<|> (5.17) 128 Table 5.II. Character table for the C 2 v point group, and the correlation of the axes of translation to infrared band type. The molecule-fixed axes x, y, z given here are related to the inertial axes a, b, c by the l r representation. Rotation (R) and C 2 v E c 2 CTv(XZ) oV(yz) Translation (T) axes A i 1 1 1 1 T a A 2 1 1 -1 -1 Ra (Rz) Bi 1 -1 1 -1 T o Rb (Ry) B 2 1 -1 -1 1 T b . Rc (Rx) 129 where N is a normalization factor, P j K ( cos 6) is an associated Legendre polynomial, and the spherical polar angles 9 and $ are shown in Fig. 5.4. A C 2 rotation about the a inertia! axis (C2<a)) adds an amount n to <|>, but does not change the 6 coordinate: C2(a)YjK(8,(t)) = NPjK(cos e)e'KM>+«) (5.18a) = NPjK(cos e)eiK<J)eiK« (5.18b) = eiK*YjK(e,<|>) (5.18c) where (- 1 for even K e'K* \ (5.19) I. = -1 for odd K Note that the operation of C2 on YJK(9,<)>) gives a multiple of the original spherical harmonic, YJK(9,<1>). C2 rotations about the b and c inertial axes are not symmetry operations of the CZM point group. Unlike CzW, the a v a c and a v a b operators reverse the directions of the angles 6 and (>. Both reflections change e into -9, causing the associated Legendre polynomial to become Pj K ( -cos 9). By the Rodrigues formula 1 4 9 Pj K ( -cos 9) = (-1)J+KpjK( C 0 S 0) (5.20) The operation of o v a c changes <|> to -<|>. o v a b projects the c axis in the opoosite direction and changes <J> ton -<J>. The overall effects of the reflections are therefore: avacYjK(9,<)>) = (-1)J + KNPjK(cos 9)e-iK<|> (5.21) and o-vabYjK(9,(|>) = (-1)J + KNPjK(cos 9)e'K^e-'K(t> (5.22) Clearly the spherical harmonics themselves are not eigenfunctions of the reflection operators, though the linear combinations obtained 130 a(z) >b(y) Fig. 5.4. Schematic drawing of the C 2 v NH2BH2 molecule in the x, y, z principal axis system and the a, b, c inertial axis system, showing the C2 o v reflection planes. 131 by taking Wang sum and difference funct ions 1 5 0 are eigenfunctions of these operators: *FJK± = ( 1 / V 2 ) ( Y J K ± Yj, . K ) (5.23) In equation (5.23) the sums and differences (JK+ and JK., respectively) correspond to the upper and lower asymmetry components of a JK level. The effects of the C 2 v symmetry operations, performed on asymmetric top rotational wavefunctions, follow from equations (5.18c), (5.19), (5.21), (5.22) and (5.23): C 2( a) ^ J K t = ( - 1 ) K ¥ J K ± (5.24) C v a c ^ J K t = (1/V2)(-1)J-K Y j , . K ± (1/V2)(-1)J+K Y J K - ± ( - 1 ) J + K(i /V2) ( Y J K ± Y j , . K ) = + ( - 1 ) M f j K ± (5.25) o v a b x F j K ± = ( "1) K ( -1 ) J + K , PJK± - ± ( - 1 ) J * j K ± (5.26) For even and odd values of K, and the + and - asymmetry components, the result of each operation can be tabulated using (5.24) through (5.26), as given in the first two sections of Table 5.III. The irreducible representations in the third section of Table 5.Ill are obtained by substituting even and odd values for J into section 2. The quantum numbers K a and K c in section 3 denote the projections of the angular momentum components J a and J c along the axes of lowest and highest inertia. The values of K a and K c corresponding to each irreducible representation are derived from the rule that K c = J - K a and K c = J - K a + 1, for the + and - asymmetry components, respectively. For example, for even J , even K a and the - asymmetry component, K c must be odd, giving K a K c = eo. The eo notation 132 Table 5.III. Character sets for an asymmetric top rotational wavefunction in the C 2 v point group. Wang sum & Irred. represen- difference tations ( K a K c ) E ± / 0 ± J K a functions E c 2 a v a c o v a b Jeven Jodd notation J Keven + 1 1 (-1)J (-1)J A i (ee) A 2(eo) E+ J Keven 1 1 -(-1)J -(-1)J A 2(eo) A i (ee) E - J Kodd + 1 -1 -(-1)J H) J B 2(oe) Bi(oo) o + J Kodd 1 -1 (-1)J -(-1)J Bi (oo) B 2(oe) o- 133 indicates that the rotational wavefunction is symmetric with respect to rotation about the a inertial axis and antisymmetric with respect to rotation about the c inertial a x i s . 1 5 1 The E±/0± notation given in the last column of Table 5.Ill is explained in Section V .E . From Table 5.Ill, the selection rules for a C-type band are: Ai <=> B 2 and A 2 <=> Bi (5.27) or in K a K c notation: ee <=> oe and eo <=* oo (5.28) The restrictions on changes in K a and K c are therefore: A K a = ± 1 , ±3, ±5,... and A K C = 0, ±2, ±4,... (5.29) so that C-type bands consist of the following branches, in A K a A J notation: Branch AJ. AKa Intensitv +1 +1 0,0 strong PP -1 -1 0,0 strong r Q 0 +1 0,-2 intermediate PQ 0 -1 0,+2 intermediate rp -1 +1 -2,-2 weak PR +1 -1 +2,+2 weak 134 V.E. The Rotational Hamiltonian. V.E.1. The Hamiltonian without vibration interaction. The rotational Hamiltonian representing the purely kinetic energy, T, of a freely rotating rigid asymmetric top molecule is: Hrigid = (B x + B y) J 2 / 2 + [B z - (B x + By)/2] J z 2 / 2 + (Bx - B y ) (J + 2 + J.2)/4 (5.20) A A A A A A where J+ 2 + J . 2 = (J x + U y ) 2 + (J x - i J y ) 2 , and the quantities B a = h / 8 7 i 2 c l a (in cm- 1) are the rotational constants . 1 5 2 B X ) B y and B z are to be identified with the rigid-rotor rotational constants B, C and A, respectively, for the l r representation which is appropriate for a near-prolate asymmetric top molecule. The third term of equation (5.20) (which vanishes in a symmetric A- . top) produces a matrix representation for H r |9 l d that contains off- diagonal matrix elements with AK ± 2: <J,K±2|J ±2 |JK> = (f|2/4)[J(J + 1) - K(K ± 1)] 1 / 2 x [J(J + 1) - ( K ± 1 ) ( K ± 2)]1'2 (5.21) A . The matrix of H r |9 | d can be factorized at once into blocks containing only odd or even values of K in the basis set (because no matrix elements of the type AK = ±1 arise from (5.20). These submatrices can be further factorized by taking sums and differences of the original symmetric top basis functions by means of a Wang similarity t ransformat ion 1 5 0 : |J,0+> = |J,0> |J,K±> = (1Al2){|J,K> ± |J,-K>} , (K > 0) (5.22) The four submatrices constructed from the basis functions |J,K ±> are designated E±and Q± for even and odd K, respectively. 135 To obtain a more accurate description of the rotational structure of an asymmetric top, centrifugal distortion must be considered. Centrifugal forces cause expansion (or stretching) and distortion in a rotating molecule, which lead to deviations from the rigid rotor Hamiltonian that increase with increasing angular momentum. The distortion Hamiltonian, H'd, is therefore treated as a power series which adds higher degree angular momentum terms to the rigid rotor Hamiltonian: H'd = (f>4/4) I T „ P y 8 JaJpJrJs (5.23) afJyS where T a p Y s is the centrifugal distortion constant and a, (3, y and 5 = x, y or z . 1 5 3 The number of terms in the general power series of equation (5.23) is 81. However, symmetry constraints reduce the number to 6 for an orthorhombic molecule (i.e., one which possesses at least two perpendicular planes of symmetry), since all terms vanish which are antisymmetric with respect to one or more of the symmetry operations. All of the remaining terms have only even powers of J , since those with odd powers change sign under the operation of Hermitian conjugation and time r e v e r s a l . 1 5 2 ' 1 5 4 Further reduction of the orthorhombic Hamiltonian follows one of two routes: the "asymmetric top reduction" for the general asymmetric top, or the "symmetric top reduction" for asymmetric tops that are nearly symmetric. In the A-reduction the J+ 4 + J . 4 term is eliminated, leaving only terms of the type A K = 0, ± 2 , whereas the "S" reduced Hamiltonian retains AK = ±4 , ±6, . . . terms. Aminoborane was treated using Watson's "A" reduced 136 H a m i l t o n i a n . 1 5 2 Written out completely up to terms in J 8 , this i s : 1 5 4 I W A ) = B X ( A ) J X 2 + B Y ( A ) j y 2 + B Z ( A ) J 2 2 - A J J 4 - A j K J 2 J Z 2 - A K J Z 4 - 2 5 j J 2 ( J x 2 - J y 2 ) - 8 K [ J 2 2 ( J x 2 - Jy 2 ) + (Jx 2 - J y 2 ) J z 2 ] + <X>J J 6 + O j K J 4 j z 2 + 0 K j J 2 J Z 4 + 0 K J Z 6 + 2 ( j)jj4 (J X 2 - j y 2 ) + <!>JKJ2tfz2(Jx2 - Jy 2 ) + (Jx 2 " Jy 2 )Jz 2 ] + <MJz 4(Jx 2 - Jy 2 ) + (3x2 - J y 2 ) J z 4 ] + L j J 8 + L j J K J 6 J z 2 + L J K J 4 J z 4 + L K K j J 2 J z 6 + L K J z 8 + 2 l j j 6 ( j x 2 . j y 2 ) + | J K J 4 [ j z 2 ( j x 2 . j y 2 ) + ( j x 2 . j y 2 ) j z 2 ] + l K j J 2 [ J z 4 ( J x 2 " Jy 2 ) + (Jx 2 - J y 2 ) J z 4 ] + I K [ J Z 6 ( J X 2 " J y 2 ) + (Jx 2 - Jy 2)JAz 6] (5.24) The fitting program employed in this work to analyze the aminoborane vj band included all matrix elements through to the off-diagonal sextic terms (J6), plus the diagonal elements from the octic terms: E K I K = <J,K | H R O T ( A ) | J , K > = [ B X ( A ) + B Y ( A ) ] J ( J + 1)/2 + ( B Z ( A ) + [ B X ( A ) + B Y ( A ) ] / 2 } K 2 - A j J 2 ( J + 1) 2 - Aj K J(J + 1 ) K 2 - A K K 4 + O J J 3 ( J + 1)3 + <*>JKJ 2 (J + 1 ) 2 K 2 + 0 K j J ( J + 1)K4 + 0 K K 6 + LJJ4(J + 1)4 + LjJKJ 3 (J + 1 ) 3 K 2 + L J K J 2 ( J + 1 ) 2 K 4 + L K KJJ (J + 1 ) K 8 + L K K 8 (5.25) EK±2 ,K = <J ,K±2 | Hrot<A> | J , K > = { [B X (A ) - B Y (A)] /4 - 8jJ(J + 1) - 8K[(K ± 2 ) 2 + K 2 ] / 2 + + (J)jJ2(J + 1) 2+ 4> J K J(J + 1)[(K ± 2 ) 2 + K 2 ] /2 + <J)K[(K ± 2)4 + K4]/2} {[J(J + 1- K ( K ± 1)][J(J + 1) - ( K ± 1)(K ± 2)]}1/2 (5.26) 137 V.E.2. Coriolis interaction. The only perturbation present in the \-? fundamental, up to the limit of this analysis at K a ' =11, is a Coriolis interaction globally affecting all levels to an extent which increases quadratically with the rotational quantum number K. A Coriolis interaction is the coupling of two vibrations by the rotation of the molecule. Put simply, certain combinations of vibrations generate an internal angular momentum which is part of the total angular momentum of the molecu le . 1 5 5 In other words, rotational and vibrational motion are not separable. This internal, or vibrational, angular momentum is a vector, written n , whose components are Tlx, Ily and TIz- To obtain the rotational Hamiltonian, the vibrational angular momentum must be subtracted from the total angular momentum, P, to give the rotational angular momentum. Instead of the simple form H - f |2 ( j x 2/ | x + J y 2 / | y + J z 2/ | z ) /2 + H v i b (5.27) the rotation-vibration Hamiltonian (in joules) b e c o m e s 1 5 5 ' 1 5 6 ' 1 5 7 : H = fi2{ [(Px - ftx)]2/lx + [(Py - n y )]2/ |y + [(P2 - f h ) ] 2 / | 2 }/2 + H v i b (5.28) H = f j 2 ( P x 2 / | x + p y 2 / | y + P 2 2 / | z ) / 2 - f i 2 ( n x P x / l x + ftyPy/ly + n z P z / l z ) + f i 2 (n x 2/ iy+ n y 2 / i y + n z 2 / i z ) + H v i b (5.29) The first term in equation (5.29), independent of the vibrational angular momentum, is the rigid rotor Hamiltonian, while the third term, independent of rotational angular momentum, affects only the vibrational energy. The second term, a function of both the vibrational and the total angular momenta, represents Coriolis coupling. The Coriolis interaction can therefore be considered as the scalar product of the rotational and vibrational angular momenta, 138 the magnitude of which increases the faster the molecule rotates and the nearer the vibrations approach degeneracy. According to Jahn's rule two normal coordinates Qk and Q| are coupled via an a- axis Coriolis interaction only if the product of their irreducible representations is of the same symmetry as P a - 1 5 7 Thus the V 7 (Bi) fundamental at 1005 cm- 1 undergoes an a-axis Coriolis interaction with the nearby v n (B 2) fundamental at 1122 c m - 1 , since Bi x B 2 gives the A 2 symmetry species (corresponding to rotation around the a-axis). The v n vibration in turn interacts with vs (A-|) at 1145 c m " 1 (the B H 2 symmetric bending vibration) by a c-axis Coriolis interaction, while the direct product of the V 7 and vs symmetries produces Bi symmetry for a b-axis Coriolis interaction. Each of these three vibrations is therefore affected by the other two. The vibrational angular momentum, in units of fi, is defined a s 1 5 5 : I I « - I Ck|(«)qkPl(co|/cok)1 / 2 (5.30) k.l where the normal coordinate Q and its momentum conjugate, P = -ih3/3Q, are expressed in the dimensionless forms, q and p: Qk = Y k 1 / 2 Q k (5.31) Pk = Pk/Yk 1 / 2h (5.32) Yk = 27iCG)k/h (5.33) The Coriolis coupling constant, Ck|(°0, is a measure of the angular momentum about the a-axis induced by the interaction of two normal vibrational modes, Qk and Q|, having frequencies (in cm- 1) of cok and co|. 139 V.F. Band Analysis and Discussion. Aminoborane's BH2-wag forms a C-type band whose appearance is characterized by a central spike, due to the asymmetry of the molecule causing low-K Q branches to pile up about the band origin (see Fig. 5 .5) . 1 5 8 At high resolution (Fig. 5.6), it can be seen that the spike is composed largely of the two lowest Q branches, PQi and rQo (using the notation A K aAJKa")- The | i n e s of the 1 1 B form of NH2BH2 were assigned by a process of successive refinement of the upper state constants. The ground state constants were held fixed at the best values available so f a r , 1 3 3 and the structure of the band was calculated using a prediction program. As the upper state constants were improved the prediction became more accurate so that more lines could be assigned. The assignments were limited to a maximum upper state value of K a equal to 11, as a result of the Boltzmann distribution at room temperature. Lines of ammonia, present as an impurity in the spectrum, were used as an internal standard for absolute frequency calibration. The NH3 frequencies were taken from the diode laser study by Job et a l . 1 5 9 A complete set of molecular constants cannot be given at this time because the V 5 fundamental has not yet been observed directly since its dipole derivative is very small. Without lines from V 5 , it is extremely difficult to analyze the V 5 - V 7 - V 1 1 Coriolis interactions. However, it is hoped that a sufficient portion of the V 5 band can be assigned in the near future to allow a fit to be made. The data were fitted to the matrix elements in equation (5.25) and (5.26) by means of a least-squares program written by Dr. Wyn Lewis-Bevan. In this program the Hellmann-Feynman theorem is used to calculate the 140 Fig. 5.5. NH2BH2 spectrum of the v7 band and the vs and v n bands with which it undergoes Coriolis interactions. 141 Fig. 5.6. Center of the V 7 band of NH2 1 1BH2. 142 derivatives of the energy levels with respect to the parameters (see Section IV.C). The computations were performed on the University of British Columbia Computing Centre Amdahl 470 V/8 mainframe computer. Two sets of molecular constants appear in Table 5.IV. Both were obtained by ignoring the Coriolis perturbation, but one was produced from a reduced data set of 606 transitions With a maximum K A ' of 6. In the excited state, all constants were floated except the off- diagonal sextics, namely §j, < | > J K , and <J>K- Eliminating all K A ' values above six reduces the standard deviation in the line positions from 0.001 cnrr1 to 0.0003 c n r 1 . This is expected from the K dependence of the Coriolis coupling. The standard errors of most constants improved when the data set was reduced, except for very small ones (= 10"8 cm- 1) and with matrix elements dependent on K . Note in particular that O K J , L K K J and I _ K , which accompany the variables K 6 , J ( J + 1 ) K 6 and K 8 , are very poorly determined in the reduced data set. This reflects the importance of a wide range of K values in determining terms containing high powers of K . Without including Coriolis terms in the Hamiltonian, the constants in Table 5.IV are not true values. Rather, they comprise internally consistent sets which have incorporated the effects of the Coriolis interactions in order to fit the data. This is particularly evident in that the A K and 8 K constants are negative, rather than positive as they should be. Since A K and 8 K accompany the variables - K 4 and - [ ( K ± 2 ) 2 + K 2 ] , these terms are the most sensitive to Coriolis interactions. An estimate of -0.406 was made for the V7-V11 a-axis Coriolis coupling constant (£7,11) from the V7 1 4 3 Table 5.IV. Molecular constants of the \j band of N H 2 1 1 B H 2 (in cm- 1) , for both the full and reduced (K a ' ^ 6) data sets. The numbers in parentheses denote one standard deviation in units of the last significant figures. Where a ground state constant is blank, it was fixed to zero. EXCITED STATE GROUND STATE Reduced Full T 0 1004.68420(5) 1 0 0 4 . 6 8 3 1 ( 2 ) A 4.51446(2) 4 .51512(3) 4 .610569(8 ) B 0.9060531(8) 0.90605(2) 0 .916897(2) C 0.7646658(7) 0.76467(2) 0 .763137(2) Aj 1.173(1) x 10-6 1.161(3) x 10-6 1.542(2) x 10-6 AjK 1.04(1) x 10-5 1.15(2) x 10-5 9.87(3) x 10-6 A K -1.17(3) x 10-4 -0.68(1) x 10-4 8.692(8) x 10-5 § J 1.116(6) x 10-7 1.06(2) x 10-7 2.86(3) x 1 0 - 7 8 K -1 .197(6) x 10-5 -1.89(2) x 10-5 1.016(2) x 10-5 <E>JK 6.7(2.2) x 10-10 -3.3(2) x 10-10 <E>KJ -4.3(7) x 10-8 5.7(4) x 10-8 7.0(32) x 10-11 <DK -6.7(1.6) x 10-7 3.7(2) x 10-7 5.94(30) x 1 0 - 9 LjK -2.5(7) x 10-11 6.4(3) x 10-11 L-KKJ 4.6(1.3) x 10-10 L K 3.8(2.3) x 10-9 o 0 . 0 0 0 3 0.001 144 and v n d a t a . 1 6 0 This is in good agreement with a force field estimate of -0 .40 . 1 6 1 Appendix I. NbN 3<D_3A Correlation Matrix B' A D ' X D' A' X' i U h.i' h 0 ' h + 1 ' b' e2qQ' B' 1 -0.462 0.105 0.028-0.107 0.034 0.055-0.002 0.039 0.022 0.025 0.013 A D ' 1 -0.159-0.258-0.158 0.333 0.157-0.032 0.022 0.047-0.319-0.004 XD' 1 0.248-0.273 0.004 0.049-0.072 0.242-0.143 0.133-0.005 A' 1 0.241-0.515-0.235 0.012 -0.073-0.004 0.671 0.049 X' 1 -0.573 -0.187 0.124-0.391 0.151 0.392 0.011 i 1 0.517 -0.1 17 0.121 -0 .011-0.706-0.013 D' 1 0.002 0.066 0.151-0.119-0.016 h.i' I -0.027 -0.004 -0.002 -0.068 h 0 ' 1 -0.027 -0.075 -0.002 h+i' 1 0.101 -0.018 b' 1 0.041 e2qQ' I Ul T 0 B" A D " XD" A" X" f D" h . r h 0 M h + 1 " b.i/o" b 0 / + r B' -0.038 0.995-0.462-0.017 0.026-0.113 0.011 -0.131 -0.025 0.032 0.021 0.027 0.023 A D ' -0 .278-0.464 0.974-0.181 -0.275 -0.100 0.272 0.232 0.013 0.030 0.027-0.318-0.318 XD' -0.108 0.091-0.206 0.884 0 .250-0.255-0.168-0.257-0.068 0.195-0.106 0.127 0.146 A' 0.152 0.026 -0.355 0.396 0.999 0 .235-0 .882-0 .455-0 .026-0 .078 0.041 0.670 0.672 X' 0 .592-0.091-0.060-0.114 0.242 0.965-0.268-0.067 0.084-0.345 0.150 0.398 0.377 i -0.902-0.004 0.227-0.267 -0.520-0.382 0.680 0.284-0.063 0.123-0.050-0.701 -0.701 D' -0.512-0.022 0.089-0.145 -0.247-0.053 0.246 0.518 0.017 0.077 0.134-0.116-0.012 h-i ' 0.111 0.008-0.017-0.031 0.021 0.106-0.060 0.152 0.848-0.016 0.003 0.010 0.002 h 0 ' -0.116 0.036 0.005 0.221 -0.073-0.415 0.072 0.105-0.011 0 .910-0 .012-0 .082-0 .062 h+i' 0.034 0.034 0.073-0.110 -0.020 0.151 0.056 0.356 0.000-0.006 0.933 0.096 0.081 b' 0.408 0.025-0.350 0.312 0.670 0.330-0.877-0.191-0.050-0.078 0.153 0.996 0.994 e2qQ'-0.016 0.015-0.017-0.003 0.047 0 .018-0.058-0.008-0.047 0.001-0.012 0.043 0.036 T 0 1 0.007-0.097 0.147 0.159 0.368-0.301-0.110 0.074-0.112 0.050 0.401 0.400 B" 1 -0.453-0.024 0.025-0.108 0.012 -0.130-0.017 0.028 0.031 0.027 0.022 A D " 1 -0.197-0.364-0.045 0.346 0.269 0.028 0.014 0.045-0.350 -0.352 X\f 1 0 .397-0.176-0.351-0.308-0.042 0.187-0.069 0.307 0.324 A" I 0.234-0.883 -0 .460-0 .021-0 .078 0.028 0.669 0.671 X" 1 -0.221-0.048 0.069-0.365 0.148 0.339 0.316 y" 1 0.442-0.005 0 .078-0.008-0.877-0.880 D" I 0.149 0.121 0.318-0.186-0.200 h_i" 1 -0 .010-0.004 -0.051-0.051 h 0 " 1 -0.01 1 -0.079 -0.078 h + 1 " 1 0.153 0.150 b.1/0" 1 0-993 b o / + i " APPENDIX II. Transitions Appendix II.A. 3 0 2 - 3 A i . E J " F" J " F ft 1 4 .5 pR 16146 .4540 2 5 5 1 4 .5 qR 1 6 1 4 6 .5474 2 6 5 1 4 .5 r R 1 6146 .6629 3 5 .5 1 5 .5 pR 16146 .7154 3 6 .5 1 5 .5 qR 16146 .8310 3 7 .5 1 5 . 5 r R 16146 .9692 4 5 .5 2 2 .5 pR 16147 .3250 4 6 .5 2 2 .5 qR 16147 .3517 4 8 .5 2 2 .5 CO 16147 .3705 5 4 .5 2 2 .5 r R 16147 .3890 5 5 .5 2 3 . 5 PR 16147 . 3 8 3 3 5 6 .5 2 3 .5 qR 16147 .4205 5 7 .5 2 3 .5 CO 1 6 1 4 7 .4442 5 8 5 2 3 .5 r R 16147 .4682 5 9 5 2 4 .5 pR 16147 .4643 6 5 5 2 4 5 qR 16147 .5121 6 6 5 2 4 5 r R 16147 .5698 6 7 5 2 5 5 qR 16147 6 2 7 2 6 9 5 2 5 5 r R 16147 6 9 4 8 6 10 5 2 6 5 qR 16147 7 6 6 9 7 2 5 2 6 5 r R 16147 8 4 3 6 7 3 5 3 1 5 pR 16148 3 5 0 6 7 4 5 3 1 5 qR 16148 3 6 0 4 * 7 5 5 3 1 5 CO 16148 3 6 8 4 * 7 6 5 3 1 5 r R 16148 3 7 6 3 7 7 5 3 2 5 PR 16148 3 7 1 2 * 7 8 5 3 2 5 qR 16148 3 8 7 3 7 9 5 3 2 5 CO 16148 3 9 8 5 7 10 5 3 2 5 r R 16148 4 0 9 8 7 1 1 5 3 3 5 pR 16148 4 0 3 2 8 3 5 3 3 5 qR 16148 4 2 5 7 8 4 5 3 3 5 CO 16 148 4 3 9 9 8 5 5 3 3 5 r R 16148 4541 8 6 5 3 4 5 pR 16148 4 4 6 9 8 7 5 3 4 5 qR 16148 4 7 5 5 8 8 5 3 4 5 CO 16148 4 9 3 0 8 9 5 3 4 5 r R 16148 5 1 0 3 8 10 5 .3 5 5 qR 16148 5 3 7 5 8 1 1 5 3 5 5 CO 16148 5 5 8 2 8 12 5 3 5. 5 r R 16148 5 7 8 2 9 4 5 3 6. 5 qR 16148 61 18 9 5 5 3 6. 5 r R 16148 6 5 8 3 9 6 5 3 7 . 5 qR 16148 6 9 9 6 9 6 5 3 7 . 5 r R 16148 7 5 1 2 9 6 5 4 O. 5 r R 1 6 149 3518 9 7 5 4 1 . 5 qR 1 6 1 4 9 3 5 5 3 9 7 5 4 1 . 5 CO 1 6 1 4 9 . 3 6 0 7 9 8 5 4 1 . 5 r R 1 6 1 4 9 . 3661 9 8 5 4 2 . 5 qR 1 6 1 4 9 . 3 7 2 5 9 9 5 4 2 . 5 C O 16149 3801 9 9 5 4 2 . 5 r R 1 6 1 4 9 . 3 8 7 5 9 9 5 4 3. 5 qR 1 6 1 4 9 . 3 9 6 9 9 10 5 4 3. 5 CO 1 6 1 4 9 . 4 0 6 4 9 10. 5 4 3. 5 r R 1 6 1 4 9 . 4 1 5 9 9 10. 5 4 4 . 5 qR 1 6 1 4 9 . 4 2 8 2 9 1 1 . 5 4 4 . 5 CO 1 6 1 4 9 . 4 3 9 7 9 1 1 . 5 4 4 . 5 r R 1 6 1 4 9 . 4 5 1 4 9 1 1 . 5 4 5. 5 qR 1 6 1 4 9 . 4 6 7 0 9 12. 5 4 5. 5 CO 1 6 1 4 9 . 4 8 0 7 9 12. 5 147 of the 3 < D _ 3 A System of NbN.a Q £ J" F" qO 16144 7 0 7 4 3 6. 5 PP 1 6 1 4 1 . 71 14 qO 16144 9 1 7 9 3 7 . 5 pP 1 6 1 4 1 . 8 9 3 0 qO 16144 5 9 7 5 4 3. 5 pP 1 6 1 4 0 . 4 1 5 5 qO 16144 6 9 8 2 4 4 . 5 PP 1 6 1 4 0 . 4 6 5 3 qO 16144 .8162 4 5. 5 PP 1 6 1 4 0 . 5 2 9 3 qO 16144 5 2 5 5 5 4 . 5 PP 1 6 1 3 9 . 4 4 7 4 qO 16144 .5854 5 5. 5 PP 1 6 1 3 9 . 4 8 5 8 qO 16144 .7337 5 6. 5 PP 1 6 1 3 9 . 5321 qO 16144 .4281 5 7 . 5 pP 1 6 1 3 9 . 5 8 7 5 qO 16144 .4622* 5 8 . 5 pP 1 6 1 3 9 . 6 5 0 6 qO 16144 .5010 5 9. 5 PP 1 6 1 3 9 . 7 2 1 6 qO 16144 .5462 6 3. 5 PP 1 6 1 3 8 . 3 8 8 4 * qO 16144 5 9 7 5 6 4 . 5 pP 1 6 1 3 8 . 4 0 8 3 * qO 16144 6 5 5 5 6 5 . 5 PP 1 6 1 3 8 . 4 3 3 9 qO 16144 3 9 6 1 * 6 6 5 pP 16138 4 6 4 8 qO 16144 4 2 3 2 * 6 7 5 PP 16138 5 0 1 5 qO 16144 4 5 5 6 * 6 8 5 pP 16138 5 4 3 5 qO 16144 5 3 1 4 6 9 5 pP 16138 5 9 0 9 qO 16144 5764 6 10 5 PP 16138 6441 qO 16144 2 8 2 9 7 2 5 pP 16137 3 3 4 7 qO 16144 2937 7 3 5 PP 16137 3 4 4 9 qO 16144 3 0 6 9 * 7 4 5 pP 16137 3 5 9 2 qO 16144 3241 7 5 5 PP 16137 3 7 7 3 qO 16144 3 4 4 5 * 7 6 5 PP 16137 3992 qO 16144 3674 7 7 5 pP 16137 4251 qO 16144 3 9 3 8 7 6 5 p P 16137 4 5 4 8 qO 16144 4 2 3 2 7 9 5 PP 16137 4 8 8 6 qO 16144 4 5 5 6 7 10 5 pP 16137 5 2 6 3 qO 16144 4 9 1 5 7 1 1 5 PP 16137 5 6 7 8 qO 16144 2 2 1 6 8 3 .5 pP 16136 .2870 qO 16144 2 3 2 5 8 4 . 5 PP 16136 .2976 qO 16144 2 4 5 0 8 5 .5 pP 16136 .3108 qO 16 144 2 GOO 8 6 .5 PP 16136 . 3 2 7 0 qO 16144 2 7 8 1 * 8 7 .5 PP 16136 .3461 qO 16144 2 9 7 6 8 8 .5 PP 16136 . 3 6 8 2 qO 16144 3198 8 9 .5 PP 16136 .3931 qo 16144 3 4 4 5 8 10 .5 PP 16136 .421 1 qO 16144 3 7 1 9 8 1 1 .5 PP 16136 .4520 qO 16144 4 0 2 4 8 12 .5 pP 16136 .4858 qO 16144 1476* 9 4 .5 pP 16135 .2249 qO 16144 1575* 9 5 .5 PP 1 6 1 3 5 .2350 CO 16144 1647* 9 6 .5 pP 16135 .2472 qO 16144 1 6 9 5 * 9 7 .5 PP 16135 .2616 CO 16144 1750* 9 8 .5 pP 1 6 1 3 5 .2784 qQ 16144 1830 9 9 .5 PP 16135 .2974 CO 16144 1888* 9 10 .5 p P 1 6 1 3 5 .3187 qO 16144 1989 9 1 1 .5 PP 16135 . 3422 CO 16144 2 0 4 9 * 9 12 .5 PP 16135 .3681 CO 16144 2 0 9 2 * 9 13 .5 pP 1 6 1 3 5 .3961 qO 16144 2 1 5 7 CO 16144 2 2 2 6 * CO 16144 2 2 7 9 * qO 16144 2 3 5 0 C O 1 6 1 4 4 . 2 4 3 1 * CO 1 6 1 4 4 . 2 4 8 5 * qO 1 6 1 4 4 . 2 5 6 2 CO 1 6 1 4 4 . 2 6 4 5 * CO 16144 2 7 1 0 * qO 1 6 1 4 4 . 2 7 9 4 148 Appendix II.A, continued. 3®2m3&i. H Q P J" F n J" F It 4 5 .5 rR 16149 4943 9 13 5 C O 16144 2947* 4 6 .5 qR 16149 5134 9 13 5 qO 16144 3047* 4 6 .5 C O 16149 5292 10 5 5 qO 16144 0616* 4 6 5 rR 16149 5446 10 6 5 qO 16144 0707* 4 7 5 qR 16149 5678 10 6 5 CO 16144 0749* 4 7 5 rR 16149 6026 10 7 5 qO 16144 0815* 4 8 5 qR 16149 6298 10 8 5 CO 16144 0888* 4 8 5 rR 16149 6681 10 8 5 qO 16144 0937 5 1 5 CO 16150 3267 10 8 5 CO 16144 0989* 5 1 5 rR 16150 3306 10 9 5 CO 16144 1018* 5 2 5 qR 16150 3348 10 9 5 qO 16144 1075 5 2 5 CO 16150 3401 10 9 5 CO 16144 1 132* 5 2 5 rR 16150 3454 10 10 5 CO 16144 1 167* 5 3 5 qR 16150 3513 10 10 5 qO 16144 1226 5 3 5 C O 16150 3581 10 10 5 C O 16144 1289* 5 3 5 rR 16150 3649 10 1 1 5 CO 16144 1328* 5 4 5 qR 16150 3726 10 11 5 qO 16144 1393 5 4 5 CO 16150 3808 10 12 5 CO 16144 1507* 5 4 5 rR 16150 3891 10 12 5 qO 16144 1575 5 5 5 qR 16150 3989 10 13 5 qO 16144 1775 5 5 5 CO 16150 4086 10 14 5 qO 16144 1989 5 5 5 rR 16150 4 184 1 1 7 5 CO 16143 9767 5 6 5 qR 16150 4302 11 8 5 CO 16143 9783 5 6 5 CO 16150 4414 11 8 5 qO 16143 9824 5 6 5 rR 16150 4528 11 8 5 CO 16143 9869 5 7 5 qR 16150 4669 11 9 5 CO 16143 9889 5 7 5 CO 16150 4795 11 9 5 qO 16143 9933 5 7 5 rR 16150 4921 11 9 5 CO 16143 9982 5 8 5 qR 16150 5088 11 10 5 CO 16144 0004 5 8 5 C O 16150 5228 11 10 5 qo 16144 0053 5 8 5 rR 16150 5364 1 1 10 5 CO 16144 0106 5 9 5 rR 16150 5862 11 11 5 CO 16144 0134 6 1 5 rR 16151 2772 11 11 5 qO 16144 0188 e 2 5 CO 16151 2838 11 11 5 CO 16144 0246 6 2 5 rR 16151 2878 11 12 5 CO 16144 0278 6 3 5 qR 16151 2917 11 12 5 qO 16144 0334 6 3 5 CO 16151 2968 11 12 5 CO 16144 0396 6 3 5 rR 16151 3019 11 13 5 CO 16144 0431 6 4 5 qR 16151 3071 11 13 5 qO 16144 0494* 6 4 5 CO 16151 3133 11 14 5 qO 16144 0666* 6 4 5 rR 16151 3195 11 15 5 qO 16144 0852 6 5 5 qR 16151 3260 12 7 5 CO 16143 8591 6 5 5 CO 16151 3332 12 8 5 CO 16143 8609 6 5 5 rR 16151 3407 12 8 5 CO 16143 8671 6 e 5 qR 16151 3485 12 9 5 CO 16143 8684 6 6. 5 CO 16151 3569 12 9 5 qO 16143 8722 6 6 5 rR 16151 3652 12 9 5 CO 16143 8763 S 7 5 qR 16151 3748 12 10 5 CO 16143 8778 6 7. 5 CO 16151. 3843 12 10 5 qO 16143 8819 6 7. 5 rR 16151 3937 12 10 5 CO 16143 8864 6 8 5 qR 16151 4046 12 1 1 5 CO 16143 8882 € 8 5 C O 16151 4153 12 1 1 5 qO 16143 8928 € 8 . 5 rR 16151. 4256 12 1 1 5 CO 16143 8977 6 9. 5 qR 16151. 4385 12 12 5 CO 16143 8997 6 9. 5 C O 16151 . 4501 12 12 5 qO 16143 9046 6 9. 5 rR 16151 4613 12 12 5 CO 16143 9097 e 10. 5 rR 16151. 5009 12 13 5 CO 16143 9121 7 2. 5 rR 16152. 2180 12 13 5 qO 16143 9175 7 3. 5 C O 16152. 2246 12 13 5 CO 16143 9230 7 3. 5 rR 16152 2287 12 14 5 C O 16143 9257 7 4 . 5 CO 16152. 2370 12 14 5 qO 16143 9315 7 4 . 5 rR 16152. 2419 12 14 5 CO 16143 9374 7 5. 5 qR 16152. 2461 12 15 5 CO 16143 9405 7 5. 5 C O 16152. 2521 12 15 5 qO 16143 9464 7 5. b rR 16152. 2578 12 15 5 CO 16143 9525 149 Appendix II.A, continued. 3<D2- 3Ai. H fl £ J" F n 0" F 7 6 5 qR 16152 2632 12 16 5 CO 16143 9559 7 6 5 CO 16152 2697 12 16 5 qO 16143 .9626» 7 6 5 rR 16152 2762 13 9 5 qO 16143 7435 7 7 5 qR 16152 2827 13 10 5 qO 16143 7524 7 7 5 CO 16152 2900 13 10 5 CO 16143 7553 7 7 5 rR 16152 2972 13 1 1 5 CO 16143 7562 7 8 5 qR 16152 3047 13 1 1 .5 qO 16143 .7602 7 8 5 CO 16152 3130 13 12 .5 qO 16143 .7698 7 8 5 rR 16152 3210 13 12 . 5 CO 16143 .7744 7 9 5 qR 16152 3298 13 13 . 5 CO 16143 . 7756 7 9 5 CO 16152 3388 13 13 . 5 qO 16143 .7801 7 9 5 rR 16152 3477 13 13 .5 CO 16143 .7851 7 10 5 qR 16152 3578 13 14 . 5 CO 16143 . 7870 7 10 5 CO 16152 3676 13 14 5 qO 16143 7916 7 10 5 rR 16152 3772 13 14 5 CO 16143 7966 7 11 5 CO 16152 3995 13 15 5 CO 16143 .7986 7 11 5 rR 16152 4096 13 15 5 qO 16143 8036 8 3 5 rR 16153 1443 13 15 5 CO 16143 .8099 8 4 5 CO 16153 1507 13 16 5 CO 16143 8113 8 4 5 rR 16153 1546 13 16 5 qO 16143 .8168 8 5 5 CO 16153 1621 13 17 5 qO 16143 .8307 8 5 5 rR 16153 1668 14 10 5 qO 16143 6131 8 6 5 CO 16153 1758 14 1 1 5 qO 16143 6199 8 6 5 rR 16153 1810 14 12 5 qo 16143 6279 8 7 5 CO 16 153 1914 14 13 5 qO 16143 6365 8 7 5 rR 16153 1973 14 14 5 qO 16143 6459 8 8 5 CO 16153 2091 14 15 5 qO 16143 6557 8 8 5 rR 16153 2157 14 16 5 qO 16143 6662 8 9 5 CO 16153 2290 14 17 5 qO 16143 6778 8 9 5 rR 16153 2362 14 18 5 qO 16143 6901 8 10 5 qR 16153 2432 15 10 5 qO 16143 4662» 8 10 5 CO 16 153 251 1 15 1 1 5 qO 16143 4721» 8 10 5 rR 16153 2589 15 12 5 qO 16143 4787 8 1 1 5 CO 16153 2754 15 13 5 qO 16143 4854 15 14 5 qO 16 143 4931 15 15 5 qO 16143 501 1 15 16 5 qO 16143 5100 15 17 5 qO 16143 5193 15 18 5 qO 16143 5293 15 19 5 qo 16143 5401 16 12 5 qO 16143 321 1* 16 13 5 qO 16143 3267 16 14 5 qO 16143 3331 16 15 5 qO 16143 3393 16 16 5 qO 16143 3465 16 17 5 qO 16143 3544 16 18 5 qO 16143 3625 16 19 5 qO 16143 3713 16 20 5 qO 16143 3805 17 12 5 qO 16143 1555 17 13 5 qO 16143 1602 17 14 5 qO 16143 1650 17 15 5 qO 16143 1706 17 16 5 qO 16143 1763 17 17 5 qO 16143 1824 17 18 5 qO 16143 1890 17 19 5 qO 16143 1963 17 20. 5 qO 16143 2037 17 21 . 5 qO 16143 2118 18 13. 5 qO 16142 9849 18 14 . 5 qO 16142 9887 18 15. 5 qO 16142 9932 18 16 . 5 qO 16142. 9978 18 17. 5 qO 16143. 0028 18 18. 5 qO 16143. 0081 Appendix II.A, continued. 3 0 2 - 3 A i . R a 0" F M 16 19 5 qO 16143 0140 18 20 5 QO 16143 0200 18 21 5 qO 16143 0266 18 22 5 qO 16143 0336 19 15 5 qQ 16142 8074 19 22 5 qO 16142 8404 19 23 5 qO 16142 8462 20 15 5 qQ 16142 6139 20 16 5 qO 16142 6166 20 17 5 qO 16142 6194 20 18 5 qQ 16142 6230 20 19 5 qQ 16142 6262 20 20 5 qQ 16142 6301 20 21 5 qQ 16142 6341 20 22 5 qQ 16142 6387 20 23 5 qQ 16142 6433 20 24 5 qO 16142 6486 21 16 5 qQ 16142 4132 21 17 5 qo 16142 4152 21 18 5 qQ 16142 4182 21 19 5 qQ 16142 4207 21 20 5 qO 16142 4233 21 21 5 qQ 16142 4261 21 22 5 qO 16142 4299 21 23 5 qQ 16142 4337 21 24 5 qO 16142 4378 21 25 5 qQ 16142 4417 31 27 5 qQ 16139 8508 31 28 5 qQ 16139 8488 31 29 5 qO 16139 8467 31 30 5 qQ 16139 8423 31 31 5 qO 16139 8408 31 32 5 qO 16139 8382 31 33 5 qQ 16139 8366 32 28 5 qQ 16139 5394 32 29 5 qQ 16139 5368 32 31 5 qQ 16139 5321 32 32 5 qQ 16139 5293 32 33 5 qQ 16139 5268 32 34 5 qO 16139 5242 32 35 5 qQ 16139 5217 33 28 5 qQ 16139 2199 33 29 5 qQ 16139 2171 33 30 5 qQ 16139 2143 33 31 5 qQ 16139 2113 33 32 5 qQ 16139 2083 33 33 5 qQ 16139 2053 33 34 5 qQ 16139 2023 33 35 5 qQ 16139 1994 33 36 5 qQ 16139 1966 33 37 5 qQ 16139 1938 34 29 5 qQ 16138 8863 34 30 5 qQ 16138 8832 34 31 5 qQ 16138 8798 34 32 5 qQ 16138 8764 34 33 5 qQ 16138 8730 34 34 5 qQ 16138 8696 34 35 5 qO 16138 8662 34 36 5 qQ 16138 8628 34 37 5 qQ 16138 8596 34 38 5 qQ 16138 8562 35 32 5 qQ 16138 5376 35 34 5 qO 16138 5299 35 35 5 qO 16138 5261 35 36 5 qQ 16138 5222 Appendix II.A, continued. 3 C>2- 3 Ai . R Q. J " F It 35 37 .5 qo 16 138 5184 35 38 5 qO 16138 5147 35 39 .5 qo 16138 51 10 36 31 .5 qO 16138 1934 36 32 5 qo 16138 1893 36 33 5 qO 16138 1853 36 34 5 qO 16138 1812 36 35 5 qO 16138 1773 36 36 5 qO 16138 1729 36 37 5 qO 16138 1686 36 38 5 qo 16138 1642 36 39 5 qO 16138 1601 36 40 5 qO 16138 1559 37 32 5 qO 16137 8315 37 33 5 qO 16137 8270 37 34 5 qO 16137 8226 37 35 5 qO 16137 8181 37 36 5 qO 16137 8136 37 37 5 qO 16137 8087 37 38 5 qo 16137 8044 37 39 5 qO 16137 7995 37 40 5 qO 16137 7950 37 41 5 qO 16137 7904 38 33 5 qO 16137 4595 38 34 5 qO 16137 4548* 38 35 5 qO 16137 4505 38 36 5 qO 16137 4449 38 37 5 qo 16 137 4403 38 38 5 qO 16137 4352 38 39 5 qO 16137 4305 38 40 5 qO 16137 4251* 38 41 5 qo 16137 4199 38 42 5 qO 16137 4150 39 34 5 qO 16137 077 1 39 35 5 qo 16137 0720 39 36 5 q O 16137 0669 39 37 5 q O 16137 0617 39 38 5 qo 16137 0563 39 39 5 qO 16137 05 10 39 40 5 qo 16137 0455 39 4 1 5 qO 16137 0403 39 42 5 qO 16137 0346 39 43 5 qO 16137 0293 40 35 5 qo 16136 6841 40 36 5 qo 16136 6787 40 37 5 qO 16136 6732 40 38 5 qO 16136 6677 40 39 5 qo 16136 6619 40 40 5 qO 16136 6562 40 41 5 qO 16136 6505 40 42 5 qO 16136 6444 40 43 5 qO 16136 6387 40 44 5 qO 16136 6329 41 36 5 qO 16136 2810 41 37 5 qO 16136 2749 41 38 5 qO 16136 2689 41 39 5 qO 16136 2631 41 40 5 qO 16136 2569 41 41 5 qO 16136 2510 41 42 5 qO 16136 2448 41 43 5 q O 16136 2387 41 44 5 qO 16136 2324 41 45 5 qO 16136 2261 42 37 5 qO 16135 8672 Appendix II.A, continued. E 3<j>2-3Ai. F 4 2 38 .5 qO 16135 8 6 1 2 42 39 .5 qO 16135 8 5 4 9 42 4 0 5 qO 16135 .8485 42 41 5 qO 16135 8421 42 42 5 qO 16135 8 3 5 6 42 43 5 qO 16135 8 2 9 3 42 44 5 qO 16135 8 2 2 7 42 45 5 qO 16135 8 1 5 8 42 4 6 5 qO 16135 8091 4 3 38 5 qO 16135 4 4 2 7 4 3 39 5 qO 16135 4 3 6 2 43 4 0 5 qO 16135 4 2 9 8 43 41 5 qO 16135 4 2 3 0 43 42 5 qO 16135 4 1 6 4 43 43 5 qO 16135 4 0 9 3 43 44 5 qO 16135 4 0 2 4 43 45 5 qO 16135 3961 43 46 5 qO 16135 3 8 8 9 43 47 5 qO 16135 3 8 1 6 44 39 5 qO 16135 0 0 8 1 44 4 0 5 qO 16135 0 0 1 4 44 41 5 qO 16134 9 9 4 7 44 42 5 qO 16134 9 8 7 5 44 43 5 qO 16134 9 8 0 4 44 44 5 qO 16134 9 7 3 3 44 45 5 qO 16134 9 6 6 4 44 4 6 . 5 qO 16134 9 5 8 8 44 47 . 5 qO 16134 9 5 1 5 44 48 5 qO 16134 9 4 4 2 Appendix II.B. 3 d > 3 - 3 A 2 . J " F J " F 2 2 5 r R 16545 9 6 8 0 * 3 5 5 qO 16542 8 9 9 2 2 3 5 PR 16545 8 8 1 0 3 5 5 r O 16542 9 6 5 4 2 3 5 r R 1 6 5 4 5 9621 3 6 5 qO 16542 9 1 8 0 2 4 5 pR 16545 8 4 9 5 3 6 5 r O 16542 9 9 4 4 2 4 5 qR 16545 8 9 5 4 3 7 5 qO 16542 9 4 0 3 2 4 5 rR 16545 951 1* 4 1 5 qO 16542 8 1 8 2 2 5 5 qR 16545 8 6 8 4 4 3 5 qO 16542 8 2 8 9 2 5 5 rR 1 6 5 4 5 9 3 5 0 * 4 4 5 qO 16542 8 3 7 9 2 6 5 r R 16545 9 1 3 3 * 4 5 5 qO 16542 8 4 6 5 3 2 5 qR 16546 8 6 0 8 4 6 5 qO 16542 8 5 7 4 3 3 5 qR 16546 8 5 5 5 4 6 5 r O 16542 9 0 3 4 3 4 5 PR 16546 8 2 3 3 * 4 7 5 qo 16542 8 7 0 7 3 4 5 qR 16546 8 4 9 2 * 4 7 5 r O 16542 9 2 2 4 3 5 5 PR 16546 8 0 8 7 * 4 8 5 qO 16542 8 8 5 8 3 5 5 qR 16546 8 4 2 2 * . 5 1 5 CO 16542 7 6 9 0 3 6 5 qR 1 6 5 4 6 8 3 3 9 5 2 5 CO 16542 7 7 2 3 3 6 5 r R 16546 8 7 9 2 * 6 2 5 qO 16542 6 9 7 6 3 7 5 qR 16546 8 2 4 6 * 6 2 5 CO 16542 7 0 2 8 3 7 5 r R 16546 8 7 6 3 * 6 3 5 qO 16542 7 0 0 9 4 2 5 qR 16547 7 9 8 1 * 6 4 5 qO 16542 7 0 5 4 4 2 5 r R 16547 8 1 0 9 * 6 5 5 qO 16542 7 1 0 7 4 3 5 qR 16547 7 9 6 5 * 6 6 5 qO 16542 7 1 6 5 4 3 5 r R 16547 8 1 3 9 * 6 7 5 qO 16542 7 2 2 7 4 4 .5 qR 16547 7 9 4 8 * 6 8 5 qO 16542 7 2 9 8 4 4 .5 r R 16547 8 1 5 8 * 6 9 5 qO 16542 7 3 8 2 4 5 . 5 qR 16547 7 9 3 1 * 6 10 5 qO 16542 7 4 7 3 4 5 .5 r R 16547 8 1 7 8 * 7 4 5 qO 16542 6 2 2 8 4 6 . 5 qR 16547 7 9 1 0 * 7 5. 5 qO 16542 6 2 6 5 4 6 .5 r R 16547 8 1 9 7 * 7 6 . 5 qO 16542 6 3 1 2 4 7 .5 qR 16547 .7888* 7 7 . 5 qO 16542 6 3 5 4 4 7 .5 r R 16547 . 8 2 0 9 * 7 8. 5 qO 16542 6 4 0 8 Appendix II.B, continued. 3 < E > 3 - 3 A 2 . E F n 0" F II 4 8 5 qR 16547 7 8 6 4 * 7 9 5 qQ 4 8 5 r R 16547 8 2 2 1 * 7 10 5 qQ 15 10 5 r R 16557 3 9 6 3 7 1 1 5 qQ 15 11 5 r R 16557 3 9 8 0 8 4 5 qQ 15 12 5 r R 16557 4 0 0 3 8 5 5 qQ 15 13 5 r R 16557 4 0 2 7 8 6 5 qQ 15 14 5 r R 16557 4 0 5 3 8 7 5 qQ 15 15 5 r R 16557 4 0 7 6 8 8 5 qQ 15 16 5 r R 16557 41 10 8 9 5 qQ 15 17 5 r R 16557 4 1 4 6 8 10 5 qQ 15 18 5 r R 16557 4 1 7 6 8 1 1 5 qQ 15 19 5 r R 16557 4 2 1 9 8 12 5 qQ 16 1 1 5 r R 16558 1 9 6 5 * 9 7 5 qQ 16 12 5 r R 16558 1986 9 8 .5 qQ 16 13 5 r R 1 6 558 2 0 0 8 9 10 5 qQ 16 14 5 r R 16558 2 0 2 9 * 9 1 1 5 qQ 16 15 5 r R 16558 2 0 5 6 9 12 5 qQ 16 16 5 r R 1 6 558 2 0 8 6 9 13 5 qQ 16 17 5 r R 16558 21 16 10 8 5 qQ 16 18 5 r R 16558 2 1 5 0 10 9 5 qQ 16 19 5 r R 16558 2 1 8 6 10 10 5 qQ 16 2 0 5 r R 16558 2 2 2 2 10 1 1 5 qQ 17 12 5 r R 16558 9 8 5 3 * 10 12 5 qQ 17 13 5 r R 16558 9 8 6 9 10 13 5 qQ 17 14 5 r R 16558 9 8 9 2 10 14 5 qQ 17 15 5 rR 16558 9 9 1 7 11 8 5 qQ 17 16 5 r R 16558 9 9 4 3 11 9 5 qQ 17 17 5 r R 16558 9 9 7 2 11 10 5 qQ 17 18 5 r R 1 6559 0 0 0 4 11 11 5 qQ 17 19 5 r R 1 6 5 5 9 0 0 3 6 11 12 5 qQ 17 2 0 5 r R 1 6 5 5 9 0 0 7 1 11 13 5 qQ 17 21 5 r R 1 6 5 5 9 0 1 0 9 11 14 5 qQ 18 13 5 r R 1 6 5 5 9 7 6 0 5 * 11 15 5 qQ 18 14 5 r R 1 6559 7 6 3 5 * 12 9 5 qQ 18 15 5 r R 1 6 5 5 9 7 6 5 2 * 12 10 5 qQ 18 16 5 r R 1 6559 7 6 7 9 12 1 1 5 qQ 18 17 5 rR 1 6 5 5 9 7 7 0 2 12 12 5 qQ 18 18 5 rR 1 6 5 5 9 7 7 3 2 12 13 5 qQ 18 19 5 r R 1 6559 7 7 6 5 12 14 5 qQ 18 2 0 5 r R 1 6 5 5 9 7 7 9 8 12 15 5 qQ 18 2 1 5 r R 1 6 5 5 9 7 8 3 2 12 16 5 qQ 18 22 5 r R 1 6 5 5 9 7872 13 10 5 qQ 19 14 5 r R 1 6 5 6 0 5 2 4 8 13 1 1 5 qQ 19 15 5 r R 1 6 5 6 0 5 2 6 8 13 12 5 qQ 19 16 5 r R 1 6 5 6 0 5291 13 13 5 qQ 19 17 5 r R 1 6 5 6 0 5 3 1 4 13 14 5 qQ 19 18 5 r R 1 6 5 6 0 5 3 4 4 13 15 5 qQ 19 19 5 r R 1 6 5 6 0 5371 13 16 5 qQ 19 2 0 5 r R 1 6 5 6 0 5 4 0 4 13 17 5 qQ 19 21 5 rR 1 6 5 6 0 5 4 3 7 14 10 5 qQ 19 22 5 r R 1 6 5 6 0 5 4 7 4 14 1 1 5 qQ 19 23 5 r R 1 6 5 6 0 5 5 1 4 14 12 5 qQ 2 0 16 5 r R 16561 2 7 8 2 14 13 5 qQ 2 0 17 5 r R 16561 2 8 0 8 14 14. 5 qQ 2 0 18 5 r R 16561 2 8 3 4 14 15 5 qQ 2 0 19 5 r R 16561 2 8 6 3 14 16 5 qQ 2 0 2 0 5 r R 16561 2891 14 17. 5 qQ 2 0 21 5 r R 16561 2 9 2 2 14 18. 5 qQ 2 0 22 5 r R 16561 .2957 15 12. 5 qQ 2 0 23 5 r R 16561 2991 15 13. 5 qQ 2 0 24 5 r R 16561 3 0 3 2 15 14. 5 qQ 21 16 5 r R 1 6562 .0156 15 15 5 qQ 21 17 5 r R 16562 .0181 15 16. 5 qQ 21 18 .5 r R 1 6562 .0203 15 17. 5 qQ Appendix II.B, continued. 3<J> 3-3A2. B d " F" 21 19 5 r R 16562 0 2 2 8 21 2 0 . 5 r R 1 6 5 6 2 . 0 2 5 6 21 21 . 5 r R 16562 0 2 8 7 21 2 2 . 5 r R 1 6 5 6 2 . 0 3 2 1 21 2 3 . 5 r R 1 6 5 6 2 . 0 3 5 5 21 24 5 r R 16562 0 3 8 9 21 2 5 . 5 r R 16562 0 4 2 5 22 17. 5 r R 16562 7 4 3 3 22 18 5 r R 16562 7 4 5 6 22 19 5 r R 1 6562 7 4 8 3 22 2 0 5 r R 16562 7 5 0 6 22 21 5 r R 1 6562 7 5 3 6 22 22 5 r R 16562 7 5 6 9 22 23 5 r R 16562 7 5 9 9 22 24 5 r R 1 6562 7 6 3 3 22 25 5 r R 16562 7 6 6 9 22 26 5 r R 1 6562 7 7 0 6 2 3 18 5 r R 16563 4 5 8 5 2 3 19 5 r R 16563 4 6 0 8 23 2 0 5 r R 16563 4 6 3 5 23 21 5 r R 16563 4 6 6 0 23 22 5 r R 16563 4 6 8 9 2 3 23 5 r R 1 6 5 6 3 4 7 2 2 2 3 24 5 r R 16563 4 7 4 8 2 3 25 5 r R 1 6 5 6 3 4 7 8 5 2 3 26 5 r R 16563 4821 2 3 27 5 r R 16563 4 8 6 4 24 19 5 r R 16564 1609 24 2 0 5 r R 16564 1637 24 21 5 rR 16564 1668 24 22 5 r R 16564 1690 24 23 5 r R 16564 1724 24 24 5 r R 16564 1752 24 25 5 r R 16564 1785 24 26 5 r R 16564 1818 24 27 5 r R 16564 1857 24 28 5 r R 16564 1897 2 5 2 0 5 r R 16564 8 5 1 4 25 21 5 r R 16564 8 5 4 3 25 22 5 r R 16564 8 5 6 8 25 23 5 r R 16564 8 5 9 7 25 24 5 r R 16564 8 6 2 7 2 5 25 5 r R 16564 8 6 6 0 25 26 5 r R 16564 .8687 2 5 27 5 r R 16564 8 7 2 9 25 28 5 r R 16564 .8765 2 5 2 9 5 r R 16564 8 8 0 3 26 21 5 r R 1 6 5 6 5 .5304 26 22 .5 r R 16565 .5326 26 23 .5 r R 16565 .5349 26 24 .5 r R 16565 .5381 2 6 2 5 .5 r R 1 6 5 6 5 . 54 1 1 26 26 .5 r R 16565 . 544 1 26 27 .5 r R 1 6 5 6 5 .5474 26 28 .5 r R 16565 .5510 26 29 .5 r R 16565 .5547 26 3 0 .5 r R 1 6 5 6 5 .5590 27 22 .5 r R 16566 . 1951 27 23 .5 r R 16566 . 1975 27 24 5 r R 16566 . 2 0 0 6 27 25 .5 r R 16566 .2034 27 26 .5 r R 1 6 5 6 6 .2068 27 27 5 r R 1 6 5 6 6 .2103 27 28 .5 r R 16566 .2138 2 7 2 9 .5 r R 16566 .2171 d " F H 15 18 5 qO 16541 5 6 1 6 15 19 5 qO 16541 5 6 6 2 16 13 5 qO 16541 3521 16 14 5 qO 16541 3 5 4 7 16 15 5 qO 16541 3 5 7 7 16 16 5 qO 16541 3 6 0 8 16 17 5 qO 16541 3644 16 18 5 qO 16541 3 6 8 3 16 19 5 qO 16541 3 7 2 5 16 2 0 5 qO 16541 3 7 6 7 17 14 5 qO 1654 1 1503 17 15 5 qo 1654 1 1529 17 16 5 qO 16541 1558 17 17 5 q O 1654 1 1591 17 18 5 qO 16541 1628 17 19 5 qO 16541 1664 17 2 0 5 qO 16541 1707 17 21 5 qO 16541 1748 18 14 5 qO 1 6 5 4 0 9 3 4 6 18 15 5 qO 1 6 5 4 0 9 3 7 0 18 16 5 q O 1 6 5 4 0 9 3 9 7 18 17 5 qO 1 6 5 4 0 9 4 2 8 18 18 5 q O 1 6 5 4 0 9 4 6 0 18 19 5 qO 1 6 5 4 0 9 4 9 4 18 2 0 5 qO 1 6 5 4 0 9 5 3 0 18 2 1 5 q O 1 6 5 4 0 9571 18 22 5 qO 1 6 5 4 0 961 1 19 15 5 qO 1 6 5 4 0 7098 19 16 5 qO 1 6 5 4 0 7 1 2 3 19 17 5 q O 1 6 5 4 0 7 150 19 18 5 q O 1 6 5 4 0 7 180 19 19 5 qO 1 6 5 4 0 7214 19 2 0 5 qO 1 6 5 4 0 7 2 4 7 19 21 5 qO 1 6 5 4 0 7 2 8 3 19 22 5 qO 1 6 5 4 0 7322 19 23 5 qO 1 6 5 4 0 7 3 6 5 2 0 16 5 qO 1 6 5 4 0 4 7 2 8 2 0 17 5 qO 1 6 5 4 0 4 7 5 5 2 0 18 5 qO 1 6 5 4 0 4781 2 0 19 5 qo 1 6 5 4 0 4 8 1 0 2 0 2 0 5 qO 1 6 5 4 0 4 8 4 3 2 0 21 5 qO 1 6 5 4 0 4 8 7 7 2 0 22 5 qO 1 6 5 4 0 4 9 1 2 2 0 23 5 q O 1 6 5 4 0 4 9 5 3 2 0 24 5 q O 1 6 5 4 0 4 9 9 3 21 17 5 qO 1 6 5 4 0 2 2 4 5 21 18 5 qO 1 6 5 4 0 2 2 6 9 21 19 5 qO 1 6 5 4 0 2 2 9 7 21 2 0 5 qO 1 6 5 4 0 2327 21 21 5 qO 1 6 5 4 0 2 3 5 7 21 22 5 qO 1 6 5 4 0 2391 21 23 5 qo 1 6 5 4 0 2 4 2 7 21 24 5 qO 1 6 5 4 0 2 4 6 6 21 25 5 qO 1 6 5 4 0 2507 22 18 5 qO 1 6 5 3 9 9 6 3 3 22 19 5 qO 16539 9 6 6 0 22 2 0 5 qO 16539 9 6 8 8 22 21 5 qO 16539 97 17 22 22 5 qO 1 6 5 3 9 9751 22 23 5 qO 16539 9 7 8 4 22 24 5 qO 16539 9 8 2 2 22 25 5 qO 16539 9 8 5 8 22 26 5 qO 16539 9901 23 19 5 qO 16539 6 9 0 3 2 3 2 0 5 qO 16539 6 9 2 8 155 Appendix II.B, continued. 3d>3- 3A2. R Q. J " F " 27 3 0 . 5 r R 27 3 1 . 5 r R 1 6 5 6 6 . 2 2 1 1 1 6 5 6 6 . 2 2 5 4 J " F " 23 21 . 5 qQ 16539 6 9 5 6 23 22 5 qQ 16539 6 9 8 8 23 23 5 qQ 1 6 5 3 9 7021 23 24 5 qQ 16539 7054 23 25 5 qQ 1 6 5 3 9 7 0 9 2 2 3 26 5 qQ 16539 7 1 3 0 23 27 5 qQ 16539 7172 24 21 5 qQ 16539 4 0 7 4 24 22 5 qQ 16539 4 1 0 0 24 23 5 qQ 1 6 5 3 9 4131 24 24 5 qQ 1 6 5 3 9 4 1 6 5 24 25 5 qQ 16539 4 2 0 0 24 26 5 qQ 16539 4 2 3 6 24 27 5 qQ 16539 4 2 7 5 24 28 5 qQ 16539 4 3 1 7 25 21 5 qQ 16539 1073 25 22 5 qQ 16539 1099 25 23 5 qQ 16539 1 129 25 24 5 qQ 16539 1 158 25 25 5 qQ 16539 1 191 25 26 5 qQ 16539 1227 25 27 5 qQ 16539 1264 25 28 5 qQ 16539 1307 25 29 5 qQ 16539 1344 26 22 5 qQ 16538 7 9 6 8 26 23 5 qQ 16538 8 0 1 4 26 24 5 qQ 16538 8 0 4 3 26 25 5 qQ 16538 8 0 7 7 26 26 5 qQ 16538 8 1 0 7 26 27 5 qQ 16538 8 1 4 6 26 28 5 qQ 16538 8 1 8 3 26 29 5 qQ 16538 8 2 2 5 26 3 0 5 qQ 16538 8 2 6 4 27 23 5 qQ 16538 4771 27 24 5 qQ 16538 4 8 0 2 27 25 5 qQ 16538 4 8 2 9 27 26 5 qQ 16538 4 8 5 9 27 27 5 qQ 16538 4 8 9 5 27 28 5 qQ 16538 4931 27 29 5 qQ 16538 4 9 6 9 27 3 0 5 qQ 16538 5 0 1 0 27 31 5 qQ 16538 5 0 5 0 Appendix II.C. 3<X>4-3A3. d " F » d " F M 3 3 5 pR 16864 5 1 2 3 7 2 5 3 3 5 qR 16864 5 2 4 0 7 3 5 3 3 5 C O 16864 5 3 1 2 7 4 5 3 3 5 r R 16864 5 3 8 5 7 5 5 3 4 5 pR 16864 4 0 1 6 7 6 5 3 4 5 qR 16864 4 1 6 7 7 8 5 3 4 5 CO 16864 4 2 5 5 7 10 5 3 4 5 r R 16864 4 3 4 8 8 5 5 3 5 5 pR 16864 2 6 5 7 9 4 5 3 5 5 qR 16664 284 1 9 6 5 3 5 5 c o 16864 2 9 5 0 9 8 5 3 5 5 r R 16864 3 0 5 8 9 9 5 3 6 5 pR 16864 1039 9 1 1 5 3 6 5 qR 16864 1256 9 12 5 3 6 5 CO 16864 1383 10 6 5 3 6 5 r R 16864 1509 10 7 5 3 7 5 qR 1 6 8 6 3 9 3 9 6 10 8 5 <J" F " qQ 1 6 8 6 0 1 9 9 5 * 6 7 5 pP 16854 1702 qQ 1 6 8 6 0 1 8 1 1 * 6 8 5 pP 16854 1221 qQ 1 6 8 6 0 1 6 1 3 * 6 9 5 pP 16854 0 6 8 9 qQ 1 6 8 6 0 1358* 6 10 5 pP 16854 0 1 1 7 qQ 1 6 8 6 0 10 6 1 * 7 3 5 pP 16853 2 2 8 6 qQ 1 6 8 6 0 0 3 2 9 * 7 4 5 pP 16853 2077 qQ 16859 9 4 3 6 * 7 5 5 p P 16853 1823 qQ 1 6 8 6 0 0 4 1 2 * 7 6 5 pP 16853 1531 qQ 16859 9 4 7 6 * 7 7 5 pP 16853 1202 qQ 16859 9 1 3 4 * 7 8 5 PP 16853 0 8 3 6 qQ 16859 8 6 8 5 * 7 9 5 pP 16853 0 4 3 1 qQ 16859 8 4 2 0 * 7 10 5 PP 16852 9 9 9 3 qQ 1 6 8 5 9 7 8 1 9 * 7 1 1 5 PP 16852 9 5 1 8 * qQ 16859 7 4 8 2 * 8 3 5 pP 16852 1317 qQ 16859 7 9 3 8 * 8 4 5 pP 16852 1151 qQ 16859 7 7 7 0 * 8 5 5 pP 16852 0 9 5 8 qQ 1 6 8 5 9 7 5 7 2 8 6 5 PP 16852 0 7 2 6 * 156 Appendix II.C, continued. 3<I>4-3A3. E Q d " F " J " F n 3 7 .5 C O 16863 .9543 10 9 5 qO 1 6 8 5 9 7 3 6 0 3 7 .5 r R 16863 .9686 10 10 5 qO 16859 7 1 2 4 * 4 0 .5 r R 1 6 8 6 5 . 5 2 4 3 * 10 11 5 qO 16859 6 8 7 2 4 2 .5 qR 16865 .4658* 10 12 5 qO 1 6 8 5 9 6 5 9 8 4 2 .5 CO 1 6 8 6 5 .4690* 10 13 5 qO 16859 6 3 0 1 4 2 .5 r R 1 6 8 6 5 . 4 7 2 9 * 10 14 5 qO 16859 5991 4 3 .5 PR 16865 .4072 11 7 5 qO 16859 6 4 6 1 4 3 .5 qR 16865 .4154 11 8 5 qO 16859 6 3 0 1 * 4 3 .5 C O 16865 .4201 11 9 5 qO 16859 6 1 2 4 4 3 .5 r R 16865 .4250* 11 10 5 qO 16859 5931 4 4 .5 pR 16865 .3407 11 11 5 qO 16859 5 7 1 9 4 4 .5 qR 16865 .3509 11 12 5 qO 16859 5 4 9 4 4 4 .5 C O 1 6 8 6 5 .3568 11 13 5 qO 1 6 8 5 9 5251 4 4 .5 r R 1 6 8 6 5 .3630 11 14 5 qO 1 6 8 5 9 4 9 9 9 4 5 .5 pR 16865 .2600 11 15 5 qO 16859 4 7 2 5 * 4 5 .5 qR 16865 .2726 12 8 5 qO 1 6 8 5 9 4 8 7 4 4 5 .5 CO 16865 .2796 12 9 5 qO 16859 4 7 2 5 * 4 5 .5 r R 1 6 8 6 5 2 8 7 0 12 10 5 qO 16859 4 5 6 7 4 6 .5 pR 16865 1656 12 1 1 5 qQ 16859 4 3 8 9 4 6 .5 qR 16865 1802 12 12 5 qO 16859 4 2 0 2 4 6 5 C O 16865 1886 12 13 5 qQ 16859 3 9 9 8 4 6 5 r R 16865 1970 12 14 5 qQ 16859 3781 4 7 5 PR 16865 0 5 7 7 12 15 5 qQ 16859 3 5 5 2 4 7 5 qR 1 6 8 6 5 0 7 4 3 12 16 5 qO 1 6 8 5 9 3 3 1 0 4 7 5 C O 16865 0 8 4 1 13 8 5 qQ 16859 3 3 1 0 * 4 7 5 r R 1 6 8 6 5 0 9 3 6 13 9 5 qQ 16859 3 1 8 3 4 8 5 qR 16864 9 5 5 7 13 10 5 qQ 16859 3 0 4 6 4 8 5 C O 16864 9 6 6 4 13 11 5 qQ 16859 2 8 9 6 4 8 5 r R 16864 9 7 7 3 13 12 5 qQ 16859 2 7 3 6 5 3 5 qR 1 6 8 6 6 3 2 1 8 13 13 5 qQ 1 6 8 5 9 2 5 6 4 5 3 5 C O 16866 3 2 4 7 13 14 5 qQ 16859 2 3 8 3 5 3 5 r R 16866 3 2 8 3 13 15 5 qQ 16859 2 1 8 7 5 4 5 PR 16866 2 7 1 7 13 16 5 qQ 16859 1983 5 4 5 qR 16866 2 7 9 2 13 17 5 qQ 16859 1771 5 4 5 CO 16866 2 8 3 3 14 9 5 qQ 16859 1494 5 4 5 r R 16866 2 8 7 6 14 10 5 qQ 16859 1377 5 5 5 PR 16866 2 183 14 1 1 5 qQ 16859 1248 5 5 5 qR 16866 2 2 7 4 14 12 5 qQ 16859 1112 5 5 5 C O 1 6 8 6 6 2 3 2 6 14 13 5 qQ 16859 0 9 6 4 5 5 5 r R 16866 2 3 7 8 14 14 5 qQ 16B59 0 8 0 8 5 6 5 PR 16866 1563 14 15 5 qQ 16859 0 6 4 2 5 6 5 qR 16866 1667 14 16 5 qQ 16859 0 4 6 8 5 6 5 C O 16866 1726 14 17 5 qQ 16859 0 2 8 3 5 6 5 r R 16866 1785 14 18 5 qQ 16859 0 0 9 2 5 7 5 pR 1 6 8 6 6 0 8 5 2 15 10 5 qQ 16858 9 5 6 2 5 7 5 qR 1 6 8 6 6 0 9 7 5 15 1 1 5 qQ 16858 9451 5 7 5 C O 16866 1043 15 12 5 qQ 16858 9 3 3 2 5 7 5 r R 1 6 8 6 6 1 109 15 13 5 qQ 16858 9 2 0 5 5 8. 5 qR 1 6 8 6 6 . 0 1 9 8 15 14 5 qQ 16858 9 0 7 0 5 8. 5 C O 1 6 8 6 6 . 0 2 7 3 15 15 5 qQ 16858 8 9 2 8 5 8 5 r R 1 6 8 6 6 . 0 3 5 0 15 16 5 qQ 16858 8 7 7 7 5 9 5 qR 1 6 8 6 5 . 9 3 3 8 15 17 5 qQ 16B58 8 6 1 7 5 9 5 r R 1 6 8 6 5 . 9 5 0 8 15 18 5 qQ 16858 8 4 5 3 6 1 . 5 r R 1 6 8 6 7 . 2 7 0 2 15 19 5 qQ 16858 8 2 7 9 6 2. 5 r R 1 6 8 6 7 . 2 5 4 6 16 1 1 5 qQ 16858 7 5 1 2 6 4 . 5 qR 1 6 8 6 7 . 1976 16 12 5 qQ 16858 7407 6 4 . 5 C O 1 6 8 6 7 . 2 0 0 7 16 13 5 qQ 16858 7 2 9 B 6 4 . 5 r R 1 6 8 6 7 . 2 0 3 7 16 14 5 qQ 16858 7 1 8 2 6 5. 5 qR 1 6 8 6 7 . 1611 16 15 5 qQ 16858 7 0 5 7 6 5. 5 C O 1 6 8 6 7 . 1647 16 16 5 qQ 16858 6 9 2 6 6 5. 5 r R 1 6 8 6 7 . 1687 16 17 5 qQ 16858 6 7 8 7 6 6. 5 qR 1 6 8 6 7 . 1 182 16 18 5 qQ 16858 6 6 4 4 6 6. 5 C O 1 6 8 6 7 . 1225 16 19 5 qQ 16858 6 4 9 9 6 6. 5 r R 1 6 8 6 7 . 1270 16 2 0 5 qQ 16858 6 3 3 8 6 7 . 5 qR 1 6 8 6 7 . 0 6 8 9 17 12 5 qQ 16858 5 3 4 4 p J - F" 8 8 5 pP 16852 0 1 8 4 8 9 5 pP 16851 9 8 6 5 8 10 5 pP 16851 9 5 2 0 8 11 5 PP 16851 9 1 4 7 8 12 5 pP 16851 8 7 4 8 9 4 5 PP 16851 0 1 0 2 9 5 5 pP 1 6 8 5 0 9 9 4 6 9 6 5 PP 1 6 8 5 0 9 7 6 4 9 7 5 PP 1 6 8 5 0 9 5 5 8 9 8 5 PP 1 6 8 5 0 9 3 2 5 9 9 5 PP 1 6 8 5 0 9 0 7 2 9 10 5 pP 1 6 8 5 0 8 7 9 3 9 11 5 pP 1 6 8 5 0 8 4 9 4 9 12 5 pP 1 6 8 5 0 8171 9 13 5 pP 1 6 8 5 0 7 8 2 7 10 5 5 pP 16849 8 8 0 2 10 6 5 PP 16849 8 6 5 6 10 7 5 pP 16849 8 4 8 8 10 8 5 pP 16849 8 2 9 8 10 9 5 PP 16849 8 0 8 9 10 10 5 pP 16849 7861 10 1 1 5 PP 16849 7 6 1 5 10 12 5 PP 16849 7 3 5 2 10 13 5 pP 16849 7 0 7 0 10 14 5 PP 16849 6 7 7 1 11 6 5 PP 16848 7 4 0 2 11 7 5 pP 16848 7261 11 8 5 pP 16848 7 1 0 6 11 9 5 pP 16848 6 9 3 4 11 10 5 PP 16848 6 7 4 5 11 1 1 5 PP 16848 6 5 3 4 11 12 5 pP 16848 6 3 1 6 11 13 5 PP 16848 .6086 11 14 .5 PP 16848 .5835 11 15 .5 pP 16848 .5575 Appendix II.C, continued. 3 0 4 - 3 A 3 . B. Q. J" , F m 6 7 5 CO 16867 0742 6 7 5 r R 16867 0792 6 8 5 qR 16867 0141 6 8 5 CO 16867 0197 6 8 5 r R 16867 0254 6 9 5 qR 16866 9532 6 9 5 C O 16866 9597 e 9 5 r R 16866 9659 6 io 5 qR 16866 8867 6 10 5 C O 16866 8940 6 10 5 r R 16866 9008 7 2 5 r R 16868 1433 7 3 5 r R 16868 1266 7 5 5 qR 16868 0739 7 5 5 CO 16868 0768 7 5 5 r R 16868 0795 7 6 5 qR 16868 0420 7 6 5 CO 16868 0456 7 6 5 r R 16868 0488 7 7 5 qR 16868 O057 7 7 5 CO 16868 O097 7 7 5 r R 16868 0136 7 8 5 qR 16867 9650 7 8 5 CO 16867 9695 7 8 5 r R 16867 9738 7 9 5 qR 16867 9200 7 9 5 C O 16867 9247 7 9 5 r R 16867 9297 7 10 5 qR 16B67 8706 7 10 5 C O 16867 8760 7 10 5 r R 16867 8815 7 1 1 5 qR 16867 8172 7 1 1 5 CO 16867 8234 7 1 1 5 r R 16867 8292 8 3 5 r R 16869 0122 8 4 5 r R 16868 9957 8 6 5 qR 16868 9467* 8 6 5 C O 16868 9496* 8 6 5 r R 16868 9521 8 7 5 qR 16868 9187* 8 7 5 C O 16868 9219* 8 7 5 r R 16868 9250 8 8 5 qR 16868 8873 8 8 5 C O 16868 8909 8 8 5 r R 16868 8943 8 9 5 qR 16868 8526 8 9 5 C O 16868 8566 8 9 5 r R 16868 8604 8 10 5 qR 16868 8144 8 10 5 CO 16868 8188 8 10 5 r R 16868 8231 8 1 1 5 qR 16868 7731 8 11 5 C O 16868 7779 8 1 1 5 r R 16868 7826 8 12 5 qR 16868 7289 8 12 5 C O 16868 7341 8 12 5 r R 16868 7390 9 4 5 r R 16869 8738 9 5 5 r R 16869 8579 9 6. 5 qR 16869 8346 9 6. 5 C O 16869 8369 9 6. 5 r R 16869 8391 9 7 . 5 qR 16869 8123 9 7 . 5 C O 16869 8152 9 7. 5 r R 16869 8177 J " F N 17 13 .5 qO 16858 5253 17 14 .5 qO 16858 5158 17 15 .5 qO 16858 5037 17 16 .5 qO 16858 4922 17 17 5 qQ 16858 4801 17 18 .5 qO 16858 4676 17 19 5 qO 16858 4546 17 20 5 qo 16858 44 12 17 21 5 qO 16858 4269 18 13 5 qO 16858 3054 18 14 5 qO 16858 2967 18 15 5 qO 16858 2868 18 16 5 qO 16858 2771 18 17 5 qO 16858 2665 18 18 5 qQ 16858 2556 18 19 5 qQ 16858 2439 18 20 5 qQ 16858 2325 18 21 5 qQ 16858 2197 18 22 5 qO 16858 2072 19 15 5 qQ 16858 0564 19 16 5 qO 16858 0475 19 17 5 qO 16858 0383 19 18 5 qO 16858 0284 19 19 5 qQ 16858 0186 19 20 5 qQ 16858 0081 19 2 1 5 qO 16857 9973 19 22 5 qQ 16857 9862 Appendix II.C, continued. 3<J>4-3A3. R J " F n 9 8 .5 qR 1 6 8 6 9 .7877 9 8 .5 C O 1 6 8 6 9 .7907 9 8 .5 r R 1 6 8 6 9 .7936 9 9 .5 qR 1 6 8 6 9 .7600 9 9 .5 C O 1 6 8 6 9 .7635 9 9 .5 r R 1 6 8 6 9 .7668 9 10 .5 qR 1 6 8 6 9 .7299 9 10 .5 C O 1 6 8 6 9 .7335 9 10 .5 r R 1 6 8 6 9 .7369 9 1 1 .5 qR 1 6 8 6 9 .6975 9 1 1 .5 C O 1 6 8 6 9 7 0 1 3 9 1 1 5 r R 1 6 8 6 9 .7051 9 12 5 qR 1 6 8 6 9 6 6 2 1 9 12 5 C O 1 6 8 6 9 6 6 6 3 9 12 5 r R 1 6 8 6 9 6 7 0 4 9 13 5 qR 1 6 8 6 9 6 2 4 6 * 9 13 5 CO 1 6 8 6 9 6 2 9 2 9 13 5 r R 1 6 8 6 9 6 3 3 4 22 17 5 r R 1 6 8 8 0 0 7 6 8 22 18 5 r R 1 6 8 8 0 0 6 9 9 22 19 5 r R 1 6 8 8 0 0 6 2 8 22 2 0 5 r R 1 6 8 8 0 0 5 5 7 22 21 5 r R 1 6 8 8 0 0 4 7 6 22 22 5 r R 1 6 8 8 0 0 3 9 8 22 23 5 r R 1 6 8 8 0 0 3 1 3 22 24 5 r R 1 6 8 8 0 0 2 2 8 22 2 5 5 r R 1 6 8 8 0 0 1 5 1 22 26 5 r R 1 6 8 8 0 0 0 6 3 24 19 5 r R 16881 4 6 2 7 24 2 0 5 r R 16881 4 5 6 9 24 21 5 r R 16881 4 5 1 2 24 22 5 r R 16881 4 4 5 0 24 23 5 r R 16881 4 3 8 7 24 24 5 r R 16881 4 3 2 0 24 25 5 r R 16881 4 2 5 1 24 26 5 r R 16881 4 181 24 27 5 r R 16881 4 1 1 3 24 28 5 r R 16881 4 0 4 3 25 2 0 . 5 r R 1 6882 1360 2 5 21 . 5 r R 1 6882 1305 25 22 . 5 r R 1 6 8 8 2 . 1254 25 23 5 r R 1 6882 1 193 25 24 . 5 r R 1 6882 1 138 25 2 5 . 5 r R 1 6 8 8 2 . 1085 25 26 . 5 r R 1 6 8 8 2 . 1 0 2 0 2 5 2 7 . 5 r R 1 6 8 8 2 . 0 9 5 8 25 2 8 . 5 r R 1 6 8 8 2 . 0 8 9 7 25 2 9 . 5 r R 1 6 8 8 2 . 0 8 3 2 26 21 . 5 r R 1 6 8 8 2 . 7961 26 2 2 . 5 r R 1 6 8 8 2 . 7 9 1 9 26 2 3 . 5 r R 1 6 8 8 2 . 7 8 6 2 26 2 4 . 5 r R 1 6 8 8 2 . 7 8 1 5 26 25 . 5 r R 1 6 8 8 2 . 7 7 6 6 26 2 6 . 5 r R 1 6882 . 7 7 1 2 26 27 5 r R 1 6 8 8 2 7 6 6 0 26 28 5 r R 16882 7 6 0 3 26 29 5 r R 16882 7 5 4 8 26 3 0 5 r R 16882 7491 F n 19 23 .5 qO 16857 9 7 4 5 2 0 15 .5 qQ 16857 8 1 1 6 2 0 16 .5 qQ 16857 .8037 2 0 17 .5 qQ 16857 .7960 2 0 18 .5 qQ 16857 .7872 2 0 19 .5 qQ 16857 7 7 8 2 2 0 2 0 5 qQ 16857 7 6 8 9 2 0 21 5 qQ 16857 7 5 9 3 2 0 22 5 qQ 16857 7 4 9 2 2 0 23 .5 qQ 16857 7 3 9 0 2 0 24 5 qQ 16857 7 2 8 6 21 16 5 qQ 16857 5 4 6 3 21 17 5 qQ 16857 5 3 9 3 21 18 5 qQ 16857 5 3 1 8 21 19 5 qQ 16857 5 2 3 6 21 2 0 5 qQ 16857 5 1 5 5 21 21 5 qQ 16857 5 0 7 0 21 22 5 qQ 16857 4 9 8 3 21 23 5 qQ 16857 4 8 9 4 21 24 5 qQ 16857 4 8 0 0 21 25 5 qQ 16857 4 7 0 6 22 17 5 qQ 16857 2691 22 18 5 qQ 16857 2621 22 19 5 qQ 16857 2 5 5 0 22 2 0 5 qQ 16857 2 4 8 0 22 21 5 qQ 16857 2 4 1 0 22 22 5 qQ 16857 2 3 3 0 22 23 5 qQ 16857 2 2 5 0 22 24 5 qQ 16857 2 1 6 7 22 25 5 qQ 16857 2084 22 26 5 qQ 16857 1997 23 18 5 qQ 16856 9 7 8 9 23 19 5 qQ 16856 9 7 2 7 2 3 2 0 5 qQ 16856 9 6 6 4 23 21 5 qQ 16856 9 5 9 8 23 22 5 qQ 16856 9 5 2 8 23 23 5 qQ 16856 9 4 5 7 23 24 5 qQ 16856 9 3 8 6 23 25 5 qQ 16856 9 3 0 9 23 26 5 qQ 16856 9 2 3 3 23 27 5 qQ 16856 9 1 5 7 24 19 5 qQ 16856 6761 24 2 0 5 qQ 16856 6 7 0 6 24 21 5 qQ 16856 6 6 4 5 24 22 5 qQ 16856 6 5 8 5 24 23 5 qQ 16856 6 5 2 2 24 24 5 qQ 16856 6461 24 25 5 qQ 16856 6391 24 26 5 qQ 16856 6 3 2 4 24 27 5 qQ 16856 6 2 5 6 24 28 5 qQ 16856 6 1 8 5 2 5 2 0 5 qQ 16856 361 1 25 21 5 qQ 16856 3 5 6 0 2 5 22 5 qQ 16856 3 5 0 9 25 2 3 5 qQ 16856 3 4 5 4 25 24 5 qQ 16856 3 3 9 6 25 25 5 qQ 16856 3 3 4 2 2 5 26 5 qQ 1 6 8 5 6 3 2 7 7 2 5 27 5 qQ 1 6 8 5 6 3 2 1 7 25 28 5 qQ 16856 3 1 5 4 25 29 5 qQ 16856 3 0 9 3 26 21 5 qQ 16856 0 3 3 0 26 22 5 qQ 16856 0 2 8 4 26 23 5 qQ 16856 0 2 3 4 26 24 5 qQ 16856 0 1 8 8 Appendix II.C, continued. 3 3>4 - 3 a 3 . R Q. £ d " F 26 25 5 qQ 26 26 5 qO 26 27 5 qQ 26 28 5 qQ 26 29 5 qQ 26 30 5 qQ 27 22 5 PQ 27 24 5 q Q 27 25 5 qQ 27 26 5 qQ 27 27 5 qQ 27 28 5 qQ 27 29 5 qQ 27 30 5 qQ 27 31 5 qQ 28 23 5 qQ 28 24 5 q Q 28 25 5 qQ 28 26 5 qQ 28 27 5 qQ 28 28 5 qQ 28 30 5 qO 28 31 5 qQ 28 32 5 qQ 29 24 5 q Q 29 25 5 qQ 29 26 5 q Q 29 27 5 qQ 29 28 5 qQ 29 29 5 qQ 29 30 5 qQ 29 31 5 qo 29 32 5 qQ 29 33 5 qO 16856.0135 16856.0079 16856.0028 16855.9973 16855.9916 16855.9862 16855.6942 16855.6846 16855.6800 16855.6754 16855.6705 16855.6663 16855 .6610 16855 .6560 16855.6509 16855.3426* 16855.3366 16855.3329 16855.3288 16855.3251 16855.3199 16855.3117 16855.3064 16855.3030 16854.9763 16854.9724 16854.9690 16854.9656 16854.9620 16854.9575 16854.9542 16854.9505 16854.9466 16854.9423 aTransitions in units of crrr 1. Blended lines are denoted by asterisk. 160 APPENDIX III. Transitions of the V7 Fundamental of N H 2 1 1 B H 2 . a E B r a n c h J " B r a n c h J " r R O 0 1010. 1049 rOO 1 1 1011 . 9 0 7 7 2 2 1013 . 7 7 3 8 3 3 1 0 1 5 . 7 0 9 5 4 4 1017 . 7 2 4 4 * 5 5 1 0 1 9 . 8 2 8 7 * 6 6 1022 . 0 4 0 8 * 7 7 1 0 2 4 . 3 6 9 6 8 8 1026. 8 2 8 0 9 9 1029 . 4 2 1 4 * 10 10 1032. 1 5 2 5 * 1 1 1 1 1 0 3 5 . 01 15 12 12 1037 . 9 8 5 4 13 13 1041 . 0 5 3 8 14 14 1044 . 1942 15 15 1 0 4 7 . 3 8 3 0 16 16 1050. 5 9 8 3 * 17 17 1053. 8 2 0 6 * r 0 1 3 18 1057 . 0 3 4 9 2 19 1060. 2 3 2 8 * 2 2 0 1063 . 4 0 7 8 3 21 1066 . 5 5 8 2 4 22 1069 . 6 8 4 6 * 4 23 1072 . 7 8 8 5 5 r R 1 1 1019 . 0 4 0 7 * 5 1 1018 . 8 9 1 2 6 2 1020. .8460* 6 2 1020. .4058* 7 3 1021 . .8576* 7 3 1022 . .7189 8 4 1024 . .6586* 8 4 1023. .2645 9 5 1026. .6682 9 5 1024 . .6459* 10 6 1028 .7450* 10 6 1026 . 0 2 5 3 1 1 7 1030 .8941 1 1 7 1027 .4255 12 8 1033 . 1 127 12 8 1028 .8737* 13 9 1035 .4025 13 9 1030 . 3 8 9 3 14 10 1037 .7617 15 10 1031 . 9 9 2 3 16 1 1 1040 .1900* 17 1 1 1033 .7003 18 12 1042 . 6 9 9 9 * 19 12 1035 . 5 3 0 9 * 2 0 13 1045 .2437 21 13 1037 .4928 22 14 1048 . 8 6 1 6 * 23 14 1039 .6025 24 15 1 0 5 0 .5364* 25 15 104 1 .8683 26 16 1053 .2621 27 16 1044 . 2 9 6 4 * 28 17 1058 . 0 3 6 7 * 29 17 1046 . 8 8 7 3 * r 0 2 3 18 1058 . 8 4 9 7 * 3 18 1049 .6386* 4 19 1061 . 7 9 8 1 * 4 19 1052 .5335* 5 2 0 1064 .7856* 5 2 0 1055 . 5 5 5 2 * 6 a E B r a n c h J " 1008 . 2 8 4 1 * r P O 2 1005 .0696 1008 .1281 3 1003 .5316* 1007 .9039 4 1002 .0797* 1007 .6227 5 1000 .7299* 1007 . 2 9 8 3 * 6 9 9 9 . 4 9 8 5 1006 .9468 7 998, ,4016* 1006 . 5 8 7 5 * 8 997 , .4567* 1006 . 2 3 6 6 9 9 9 6 .6731 1005 . 9 0 9 4 10 9 9 6 .0551 1005 .6170 1 1 9 9 5 .6010 1005 . 3 6 4 5 12 995, . 3008 1005 . 1540 13 995, . 1399* 1004 .9840* 14 9 9 5 , .0959 1004 .8502 15 9 9 5 , .1471* 1004 . 7 4 6 8 * 16 9 9 5 , , 2 6 6 9 1004 . 6 7 1 1 * 17 9 9 5 .4321 1004 .6182* 18 9 9 5 .6203 1016 .0602 19 995 .8127 1015 . 3731 2 0 9 9 5 . 9 9 5 3 1015 .8385 21 9 9 6 .1585* 1015 . 1 179* r P 1 4 1008 . 1 2 8 1 * 1016 .3770* 5 1008 .3194* 1014 .7801* 5 1006 .0794* 1016 .8055 6 1007, .0793* 1014 .3608 7 1005, ,9205* 1017 . 3 6 5 7 7 1001 .9 174* 1013 .8658 8 9 9 9 .8510* 1018 .0803 9 1003 . 8 4 5 6 1013. .2978 9 997 . 8 3 0 0 1018 . ,9708* 10 1002 .9319 1012 . ,6643* 10 9 9 5 .8851 1020. 0551 1 1 1002 .1016* 1011. 9 7 2 6 1 1 994 .0435* 102 1 . 3 4 9 6 * 12 992 .3300* 101 1 . 2 3 2 7 13 1000 . 6831 1022 . 861 1 13 9 9 0 . 7 6 7 8 1010. 4 5 6 3 14 1001 .0897* 1024 . 5 9 6 3 14 9 8 9 . 3 7 5 3 1 0 0 9 . 6 5 9 0 15 999, .5678* 1008 . 8 5 5 6 15 9 8 8 . 1679 1026 . 5 5 3 8 16 987 , . 1582* 1008 . 0 6 4 9 17 9 8 6 . , 3492 1007 . 2 9 8 3 * 18 9 8 5 , .7379* 1006. 5 8 7 5 * 19 9 8 5 . ,3241* 1 0 0 5 . 9 3 2 3 20 9 8 5 . .0787* 1005 . 3 4 3 3 21 984 . .9990 1004 . 8 2 5 3 * r P 2 4 1015 , ,9509* 1004 . 3 7 9 1 * 5 1014 . ,2566* 1004 . 001 1* 6 1012 . ,564 1* 1003 . 6 7 8 6 7 1010. ,8849* 1003 . 4 1 1 5 8 1009. 2 2 6 6 1003 . 1891 9 1007 . 5 9 7 1 * 1003. 0 0 3 4 9 1006 . .2047* 1002 . 8 4 7 8 * 10 1006 . .0037* 1002. 7 1 7 6 10 1004 . .0011* 1002 . 6 0 6 2 * 1 1 1001 . 7142 1002 . 5 3 4 5 * 12 1002. ,9580 1022 . 6 4 5 7 * 12 9 9 9 , 3 4 7 3 * 1022 . 6 7 2 4 * 13 1001 . 5 2 0 8 1022 . 5 7 4 6 13 9 9 6 9 3 0 5 1022 . 6 4 5 7 * 14 1000. 1527* 1022 . 4 5 8 3 14 9 9 4 . 4 8 3 1 * 1022 . 6 2 2 5 * 15 9 9 8 . 8 5 6 1 * 1022 . 2841 15 992 . 0 4 2 0 Appendix III, continued. a B r a n c h J " B r a n c h U' r R 1 21 1058 .68 13* r 0 2 6 22 1061 . 8 7 0 9 * 7 23 1065 .1156* 7 r R 2 2 1027 . 7 0 6 6 * 8 2 1027 . 7 0 6 6 * 8 3 1029 . 3 5 9 9 9 3 1029 .3 3 7 7 * 9 4 1031 .0100 10 4 1030 .9418 10 5 1032 .6636 1 1 5 1032 . 5071 1 1 6 1034 .3278* 12 6 1034 . 0 2 1 2 * 12 7 1036 .0076 13 7 1035 .4699 13 8 1037 .71 10 14 8 1036 .8455 14 9 1039 . 4 4 4 5 15 9 1038 . 1416 15 10 1041 .2140 16 10 1039 .3617 16 1 1 1043 .0269 17 1 1 1040 .5153 17 12 1044 .8897* 18 12 104 1 .6205 18 13 1046 .8063 19 13 1042 .6999* 19 14 1048 .7838* 2 0 14 1043. .7824 2 0 15 1050 .8236 21 15 1044 . .8975* 22 16 1052 . .9239* 23 16 1046 . .0744 24 17 1055 , , 1066 25 17 1047 . , 3 4 0 8 26 18 1057 . 3 5 2 3 27 18 1048. ,7218* 28 19 1059 . 6674 29 19 1050. 2 3 9 6 * r 0 3 4 2 0 1062 . 0 5 1 0 4 2 0 1051 . 91 10 5 21 1064 . 5001 5 21 1053. 7 5 1 8 7 22 1067 . 0 1 2 0 * 7 23 1069. 5 8 3 5 8 23 1057 . 9821 8 24 1072 . 2 1 6 5 * 9 24 1060. 3 8 2 2 9 25 1074 . 8 8 3 6 10 25 1062 . 9 7 0 8 10 26 1077 . 7 0 3 0 * 1 1 26 1 0 6 5 . 7 3 7 2 1 1 27 1080. 3 6 1 4 * 12 r R 3 3 1036 . 2 7 4 4 * 12 3 1036 . 2 7 4 4 * 13 4 1037 . 9 0 6 0 * 13 4 1037 . 9 0 6 0 * 14 5 1039 . 5 2 6 2 * 14 5 1 0 3 9 . 5 2 6 2 * 15 6 1041 . 1 3 8 3 * 15 6 104 1 . 1 3 0 4 * 16 7 1042 . 7 3 6 3 * 16 7 1042 . 7 1 8 4 * 17 8 1044 . 3 2 3 3 17 8 1044 . 2 8 3 4 18 161 B r a n c h d " 1022 .6132* r P 2 16 997 . 6 3 7 2 1022 .0408* 16 9 8 9 . ,6367 1022 .6225* 17 9 9 6 . ,5003 1021 .6947 17 9 8 7 . 3044 1022 . 6 6 1 9 * 18 9 9 5 . ,4475* 1021 .2534 18 9 8 5 . 0 7 8 7 * 1022 . 7 4 7 3 * 19 9 8 2 . 9 9 1 2 1020 .7012* 2 0 9 9 3 . 5 9 8 4 * 1022 . 8 9 1 9 2 0 981 . 0 7 1 6 1020 .0366 21 9 9 2 . 8 0 2 9 * 1023 .1151 21 9 7 9 . 3 4 3 6 * 1019 .2616 23 9 7 6 . 5 3 8 7 * 1023 .4372* 24 9 7 5 . 4 8 1 6 * 1018 . 3 8 2 5 pP1 2 9 9 7 . 3 1 4 0 1023 .8812 3 9 9 5 . 3837 1017 .4096* 4 9 9 3 . 3 6 2 0 1024 .4698 5 991 . 2 4 2 3 1016 . 3552 6 9 8 9 . 0 1 5 1 1025 .2282 7 9 8 6 . 6 7 0 4 1015 .2320 8 984 . 1997 1026 . 1794 9 981 . ,5953 1014 .0559 10 9 7 8 . ,8544 1027 , .3419 1 1 9 7 5 , , 9 8 0 2 1012 . .8439* 12 9 7 2 , .9832 1028 . ,7324* 13 9 6 9 . .8798 101 1 , .6111 14 9 6 6 .6919 1030. . 3 5 9 4 * 15 9 6 3 .4440 1010. .3800* 16 9 6 0 . 1606 1032. .2262* 17 9 5 6 .8643 1009. . 1683* 18 9 5 3 . 5 7 3 8 1007 . 9954 pP2 2 9 8 9 .9796 1006 . .8817* 3 9 8 9 .8431* 1005. .8420* 3 9 8 8 .4041 1004 , .8886* 3 9 8 8 .0031 1004 . .0550* 4 9 8 6 . 8 6 2 6 1003. ,2674* 4 9 8 6 .0845 1002. ,5995* 5 9 8 5 . 3 387 1002 . ,0207* 5 984 .0876 1029 . 5 4 5 5 * 6 9 8 3 .8095 1029. 5 4 5 5 * 6 982 .0125 1029 . ,4912* 7 982 , .2515 1029. ,4912* 7 9 7 9 . 8 5 9 2 1029 . ,3290* 8 980, .6406* 1029 . . 3 4 7 5 8 977 .6288 1029. ,2137 9 9 7 8 . 9 5 3 0 1029. , 2551 9 9 7 5 . 3 2 2 1 * 1029. ,0685 10 977 , .1707* 1029. ,1510* 10 972 .9398 1028 . .8846 1 1 975, .2746 1029. .0362* 1 1 970, .4851 1028. .6512 12 973, .2551* 1028. ,9174 12 967 .9602 1028. , 3 5 6 8 13 971 , . 1009 1028. ,7961 13 9 6 5 . 3 6 8 6 1027 . .9869 14 9 6 8 . 7 4 4 3 * 1028. ,6796 14 9 6 2 .7151 1027. .5265 15 9 6 0 .0028 1028. ,5772 16 9 6 3 .7555 1026. .9597* 16 9 5 7 .2391 1026. ,9597* 17 9 6 0 .9989 1026 . ,2778* 17 954 .4276 1028 . ,4608 18 951 .5756 1025. .4670 19 9 5 5 .0372 1028. .4753 19 9 4 8 .6880 1024 .5247* 2 0 951 .8544 162 Appendix III, continued. R B r a n c h J " B r a n c h J " r R 3 9 1045 .8996 r 0 3 18 9 1045 .8216 19 10 1047 .4676* 2 0 10 1047 . 3 2 4 8 2 0 11 1049 .0294 21 11 1048 .7838* 21 12 1050 .5882* 22 12 1050 . 1895 22 13 1052 . 1491 23 13 1051 .5302 r Q 4 5 14 1053 .7169 5 14 1052 .7941 6 15 1055 .2980 6 15 1053 .9720* 7 16 1056 .8992 7 16 1055 .0612 8 17 1058 .5274 8 17 1056 .0591 9 18 1060 .1880* 9 18 1056 .9737 10 19 1061 . 8 9 2 3 * 10 19 1057 .8196 1 1 2 0 1063 .6434 1 1 2 0 1058 .6198* 12 21 1065 .4484 12 21 1059 , .4005 13 22 1067 . .3136* 13 22 1060. .1829* 14 24 1061 . ,9630* 14 25 1062 . 9 9 9 0 * 15 26 1064 . 1769* 15 r R 4 4 1044 . 6 7 4 1 16 4 1044 . 6 7 4 1 16 5 1046 . 2 9 6 5 17 5 1046 . 2 9 6 5 18 6 1047 . 9 0 7 6 18 6 1047 . 9 0 7 6 19 7 1049. 5 0 7 0 * 19 7 1049 . 5 0 7 0 * 2 0 8 1051 . 0 9 4 1 * 2 0 8 1051 . 0 9 4 1 * 21 9 1052 . 6 6 7 6 * 22 9 1052 . 6 6 7 6 * 23 10 1054. 2 2 6 0 * 23 10 1054 . 2 2 6 0 * 24 11 1 0 5 5 . 7 7 3 9 * 24 11 1 0 5 5 . 7 6 5 8 * 25 12 1057. 3 0 4 4 * 25 12 1057. 2 8 6 8 * 26 13 1 0 5 8 . 8 1 7 7 * 26 13 1058 . 7 8 5 8 27 14 1 0 6 0 . 3 1 5 5 27 14 1 0 6 0 . 2 5 8 6 * r 0 5 6 15 1061 . 7 9 8 1 * 6 15 1061 . 7 0 0 8 * 7 16 1 0 6 3 . 2 6 4 5 7 16 1063 . 1 060 9 17 1064 . 7 190* 9 17 1064. 4 6 7 6 10 18 1066 . 1 620 10 18 1065 . 7771 1 1 19 1067. .5976 1 1 19 1067 . ,0244* 12 2 0 1069, .0303* 12 2 0 1068. .1959* 13 Q E B r a n c h i P 1028. 5 6 1 2 * pP2 2 0 9 4 5 . 7 7 2 0 1023 . 4 5 0 6 * 21 9 4 8 . 5 6 1 6 1022 . 2521 21 942 . 8 3 0 0 * 1029. 0 3 6 2 * 22 9 4 5 . 1 8 3 9 * 1020. 9 3 8 3 22 9 3 9 . 8 6 8 3 1 0 2 9 . 4 6 7 0 23 94 1 . 7 4 7 5 1030. 0 6 3 9 23 9 3 6 . 8 7 5 5 * 1019 . 5 2 3 6 24 938 . 2 7 9 8 * 1018 . 0 0 5 6 * 25 934 . 8 0 3 2 1036. 2 6 6 9 * p P 3 4 9 7 8 . 7 0 8 2 1036. 2 6 6 9 * 4 9 7 8 . 6 8 8 4 1036 . 2 0 3 4 * 5 977 . 01 16 1036. 2 0 3 4 * 5 9 7 6 . 9 5 5 5 * 1036. 1 2 8 9 * 6 9 7 5 . 3 3 0 3 * 1036. ,1289* 6 9 7 5 . , 1961 1036. .0395* 7 9 7 3 . ,6720 1036 . 0 3 9 5 * 7 9 7 3 . ,4099 1035. ,9358* 8 972 , ,0465* 1035. ,9358* 8 971 , .5892* 1035, .8159* 9 9 7 0 .4631 1035, ,8159* 9 969, , 7 2 8 2 1035 .6753* 10 9 6 8 . 9 2 2 4 1035 .6843* 10 9 6 7 .8217 1035 .5144 1 1 967 .4219* 1035 .5309* 1 1 9 6 5 .8621 1035 . 3291 12 9 6 5 .9497 1035 . 3631 12 9 6 3 .8465 1035 .1149* 13 964 . 4 8 9 3 1035 . 1769 13 961 .7748* 1034 .8680 14 9 6 3 .0167 1034 .9734 14 9 5 9 .6248* 1034 .5800 15 961 .5062 1034 . 7 5 3 7 15 957 .4 1 4 5 * 1034 .5206 16 9 5 9 . 9 2 9 6 1033 .8482* 16 9 5 5 . 1299 1034 . 2 7 8 3 17 9 5 8 . 2 6 2 3 1033 . 3 8 2 3 17 9 5 2 .7749 1034 .0309* 18 9 5 6 . 4 8 0 8 1032 .8330 18 9 5 0 . 3 4 7 6 1033 .7838* 19 954 .5650* 1032 . 1 8 5 3 * 19 9 4 7 .8489 1031 .4235 2 0 952 . 5 0 3 2 1030 .5349 20 9 4 5 . 2 8 0 3 1033 .1511 21 9 5 0 .2831 1029 .5080 21 942 .6462* 1033 .0169 22 9 4 7 .8973 1028 . 3 3 5 4 * 22 9 3 9 .9468 1032 .9486 23 9 4 5 . 3 4 0 8 1027 .0079 23 9 3 7 . 1 8 9 7 * 1032 .9670 24 9 4 2 .6114 1025 .5505 24 934 .3056* 1033 .0 9 1 5 * 25 9 3 9 . 7 1 1 1 * 1042 .8 3 5 7 * p P 4 4 9 7 0 . 7 2 4 3 * 1042 .8 3 5 7 * 4 9 7 0 .7243* 1042 .7645 5 9 6 9 .0060* 1042 .7645 5 9 6 9 .0060* 1042 .5844 6 967 .2789* 1042 .5844 6 967 . 2 7 8 9 * 1042 .4754 7 9 6 5 .5417* 1042 .4754 7 9 6 5 . 5 4 6 7 * 1042 .3520* 8 9 6 3 .8095 1042 .3520* 8 9 6 3 .7943* 1042 .2139 9 962 .0712 1042 .2139 9 962 .0391 1042 .0604 10 9 6 0 .3347 163 Appendix III, continued. E B r a n c h J " B r a n c h J " 21 1070 .4622* r 0 5 13 21 1069 . 2 9 4 7 * 14 22 1071 .9036 14 22 1070 . 3 0 6 8 * 15 2 3 1073 . 3 2 3 9 * 15 23 1071 . 1 9 5 6 * 16 5 1052 . 9 2 3 9 * 16 e 1054 .5354* 17 7 1056 . 1 3 6 7 * 17 8 1057 .7263 18 9 1059 . 3 0 4 9 18 10 1060 .8713 19 11 1062 .4240 19 12 1063 .9636 2 0 12 1063 .9636 2 0 13 1065 .4879 21 13 1065 .4879 21 14 1067 .0112* 22 14 1067 .0112* 23 15 1068 .4896* 23 15 1068 .4896* r 0 6 7 16 1069 . 9677* 7 16 1069 .9616* 8 17 107 1 .4260* 8 17 1071 .4088* 9 18 1072 . .8650 9 18 1072 . ,8457 10 19 1074 . , 2 8 6 3 10 19 1074 . , 2 5 1 6 1 1 2 0 1075 . 6 8 6 1 * 1 1 2 0 1075 . 6 3 0 1 12 21 1077 . 0 6 9 1 12 21 1076. 9 7 7 3 13 22 1078. 4 3 1 8 * 13 22 1078 . 2 8 8 8 14 23 1079 . 5 5 7 7 14 24 1081 . 1 101* 15 24 1080. 7 7 7 2 15 25 108 1 . 9 3 6 3 * 16 6 1061 . 0 3 8 0 16 6 106 1 . 0 3 8 0 17 7 1062 . 6 3 9 6 17 7 1062. 6 3 9 6 18 8 1064 . 2 3 0 0 * 18 8 1064 . 2 3 0 0 * 19 9 1065 . 8 0 9 8 19 9 1065. 8 0 9 8 2 0 10 1067. 3781 2 0 10 1067. 3781 21 1 1 1068 . 9 3 4 8 21 1 1 1068. 9 3 4 8 22 12 1070. ,4794 22 12 1070. ,4794 23 13 1072. .0105 23 13 1072. 0 1 0 5 24 14 1073. ,5296 24 14 1073 . .5296 r 0 7 8 15 1075, .0336* 9 15 1075. .0336* 9 16 1076. .5238* 10 16 1076 .5238* 10 17 1078 . 9 8 2 3 * 1 1 17 1078 . 9 8 2 3 * 1 1 18 1079 .4562* 12 18 1079 .4562* 13 Q. E B r a n c h J " 1042 .0604 pP4 10 9 6 0 . 27 18 1041 .8900* 1 1 9 5 8 . 6 0 6 9 104 1 .8900* 1 1 9 5 8 . 4 9 2 2 * 104 1 .7013* 12 9 5 6 . 8 9 2 0 1041 .7013* 12 9 5 6 . 6 9 5 9 104 1 , :4913* 13 9 5 5 . 1994 1041 .4988* 13 954 . 8 8 0 8 104 1 , . 2 5 9 9 14 9 5 3 . 5 3 7 2 1041 .2739 14 9 5 3 . ,0431 104 1 .0057 15 951 . 9 1 3 4 * 1041 , .0288 15 951 . 1774 1040 .7239 16 9 5 0 . ,3321 1040 .7620 16 9 4 9 . , 2791 1040 .4126 17 9 4 8 . 7 9 9 4 1040 .4768 17 9 4 7 . . 3421 1040 .0675 18 947 . ,3134 1040 .17 14 18 9 4 5 , .3613 1039, .8448* 19 9 4 5 , ,8664 1039, .2502* 19 9 4 3 , ,3313 1039 .4988* 2 0 944 .4455 1049 . 2 7 2 6 2 0 941 . 2 4 5 7 1049 , , 2 7 2 6 21 9 4 3 .0298* 1049 , , 1907 21 9 3 9 . 1012 1049, . 1907 22 941 .5931 1049 , 0974 22 9 3 6 .8755* 1049 , 0 9 7 4 23 9 4 0 . 1075 1048 , .9930 23 934 .6179 1048 . .9930 24 9 3 8 .5446* 1048 .8757 p P 5 5 9 6 0 . 8 7 8 3 1048 . ,8757 5 9 6 0 .8783 1048 .7456 6 9 5 9 .1 5 0 5 * 1048 , . 7 4 5 6 6 9 5 9 .1 5 0 5 * 1048 .6023 7 957 .4 1 4 5 * 1048 . 6 0 2 3 7 957 .4 145* 1048 , .4444 8 9 5 5 .6703* 1048 , . 4 4 4 4 8 9 5 5 . 6 7 0 3 * 1048 .2726 9 953 .9166* 1048 . 2 7 2 6 9 9 5 3 .9166* 1048 .0851 10 9 5 2 .1558* 1048 .0851 10 9 5 2 .1558* 1047 .8814* 1 1 9 5 0 . 3 8 7 5 * 1047 .8814* 1 1 9 5 0 . 3 8 7 5 * 1047 .6606* 12 9 4 8 .6166* 1047 .6606* 12 9 4 8 . 6 0 9 9 * 1047 .4220 13 9 4 6 . 8 3 9 5 * 1047 .4220 13 9 4 6 . 8 2 6 2 * 1047 . 1643 14 9 4 5 .0593 1047 . 1643 14 9 4 5 .0298 1046 .8878* 15 9 4 3 . 2 6 0 8 1046 .8878* 15 9 4 3 .2365 1046 .5863* 16 941 . 5 0 4 1 * 1046 .5950* 16 941 .4301 1046 . 2 6 2 3 * 17 9 3 9 . 7 3 4 1 * 1046 . 2 7 7 3 * 17 9 3 9 .6135 1045 .9127* 18 9 3 7 .9768 1045 .9401* 18 937 .7859 1055 .5844* 19 9 3 6 . 2 3 6 3 * 1055 .5005 19 9 3 5 .9466* 1055 . 5 0 0 5 2 0 934 . 5 2 2 0 * 1055 .3983 23 9 2 8 .3993* 1055 .3983 p P 6 6 9 5 0 .8855 1055 .2833 6 9 5 0 .8855 1055 .2833 7 9 4 9 . 1497 1055 .1 5 1 5 * 7 9 4 9 . 1497 1055 .0186 8 9 4 7 . 4 0 4 9 Appendix III, continued. E B r a n c h d " B r a n c h d " r R 6 19 1080. 8 9 8 3 * r 0 7 13 19 1080. 8 9 8 3 * 14 2 0 1082 . 3 2 2 7 * 14 2 0 1082. 3 2 2 7 * 15 21 1 0 8 3 . 7 2 8 1 * 15 21 1083 . 7 2 8 1 * 16 22 1 0 8 5 . 1 118* 17 23 1086 . 4 8 5 8 * 17 23 1086 . 4 7 0 8 * 18 24 1087. 8 3 2 1 * 19 24 1087 . 8 1 7 0 * 2 0 25 1089 . 1 6 1 0 * 2 0 25 1 0 8 9 . 1333 21 26 1090. 4 2 4 8 * 21 26 1090. 3 5 7 5 * 22 r R 7 7 1 0 6 9 . 0 3 6 3 * 22 7 1069 . 0 3 6 3 * r 0 8 9 8 1070. 6 2 6 6 10 8 1070. 6 2 6 6 1 1 9 1072 . 2 0 6 7 12 9 1072 . 2 0 67 12 10 1073 . 7 7 4 9 13 10 1073 . 7 7 4 9 13 1 1 1 0 7 5 . 3 3 2 9 14 1 1 1075 . 3 3 2 9 14 12 1076 . 8 7 8 5 * 15 12 1076 . 8 7 8 5 * 16 13 1078 . 4 1 2 5 * 17 14 1 0 7 9 . 9 2 8 9 * 17 15 1081 . 4 4 3 8 18 15 1081 . 4 4 3 8 19 16 1082. 9 4 0 2 2 0 16 1082 . 9 4 0 2 2 0 17 1084 . 4 2 3 2 21 17 1084 . 4 2 3 2 2 1 18 10 8 5 . 8 8 6 7 * 22 19 1087. 3 4 8 7 23 19 1087 . . 3 487 23 2 0 1088 , ,7940* 24 21 1090. .2200* 25 21 1090. .2200* 26 23 1093 . .0102* 26 23 1093 .0102* PQ1 1 24 1094 .3885* 2 25 1095 .7295* 3 25 1095 .7295* 4 r R 8 8 1076 .9357 5 8 1076 .9357 6 9 1078 .5143 7 9 1078 .5143 8 10 1080 .0824* 9 1 1 1081 .6390 10 1 1 1081 .6390 1 1 12 1083 . 1847 12 12 1083 . 1847 13 13 1084 .7134* 14 14 1086 .2408* 15 15 1087 .7500 16 15 1087 .7500 17 16 1089 . 247 1 18 16 1089 .2471 19 17 1090 .7314 2 0 17 1090 .7314 21 18 1092 . 2 0 2 9 * 22 19 1093 .6596 23 164 a E B r a n c h d " 1055 .0186 p P 6 8 947 .4049 1054 .8669 9 9 4 5 .6509 1054 .8669 9 9 4 5 .6509 1054 . 7 0 2 3 10 9 4 3 . 8 8 8 3 1054 .7023 10 943 .8883 1054 .5236* 1 1 942 . 1 175 1054 .3329 1 1 9 4 2 . 1 175 1054 .3329 12 9 4 0 .3385 1054 . 1 2 5 9 * 12 9 4 0 . 3 3 8 5 1053 .9050* 13 9 3 8 .5518 1053 .6692 13 9 3 8 .5518 1053 .6692 14 9 3 6 . 7 5 7 3 * 1053 . 4 2 2 3 * 14 9 3 6 . 7 5 7 3 * 1053 . 4 2 2 3 * 15 934 . 9 5 6 1 * 1053 . 1 4 8 1 * 15 934 . 9 5 6 1 * 1053 . 1 4 8 1 * 18 9 2 9 .5201* 1061 .8161* 18 9 2 9 .5118* 1061 .7127* 19 927 .6994* 1061 .5989 19 927 .6 8 4 7 * 1061 .4747 20 9 2 5 .8776* 1061 .4747 2 0 9 2 5 .8 5 1 5 * 1061 . 3371 21 924 .0542* 1061 .3371 21 924 .0132* 1061 . 1880 22 922 .2312* 106 1 .1880 22 922 .1676* 1061 .0259* pP7 7 9 4 0 . 7 6 6 2 1060 .8523* 7 9 4 0 .7662 1060 .6641 8 9 3 9 .0210 1060 . 664 1 8 939 .0210 1060 .4628* 9 9 3 7 .2674* 1060 .2462* 10 9 3 5 .5052 1060 .0175 10 9 3 5 . 5052 1060 .0175 1 1 9 3 3 . 7 3 4 2 1059 .7736 1 1 9 3 3 . 7 3 4 2 1059 .7736 12 931 .9544 1059 .5139* 12 931 . .9544 1059. .2391 13 9 3 0 . . 1658 1059 . 2391 13 9 3 0 . . 1658 1058 .9472* 14 928 . 3691 1058 . .6402* 14 928 . 3691 1058 3 1 4 7 15 9 2 6 . 5 6 4 5 1058 . .3147 15 9 2 6 . 5 6 4 5 1000. .9813* 16 924 . .7520 1001 . . 1 124 16 924 . 7 5 2 0 1001 . 3 0 1 5 * 17 922 . 9 3 2 2 * l O O l . 5 4 0 5 17 922 . 9 3 2 2 * 1001 . 8 1 7 4 18 921 . 1042* 1002 . 1 190* 18 92 1 . 1042* 1002 . 4 2 8 7 2 0 917 . 4 2 8 3 * 1002 . 7 3 5 3 2 0 917 . 4 2 8 3 * 1003. 0 2 3 7 21 9 1 5 . 5 8 1 8 * 1003. 2 8 5 3 21 9 1 5 . 5 8 1 8 * 1003. 5 1 4 3 p P 8 8 9 3 0 . 5 3 8 6 1003 . 7 0 8 6 8 9 3 0 . 5 3 8 6 1003 . 8 7 0 2 9 9 2 8 . 7 8 6 0 1004. OOI 1* 9 928 . 7 8 6 0 1004 . 1081 10 927 . 0 2 4 4 1004. 1938 10 9 2 7 . 0 2 4 4 1004 . 2 6 3 4 1 1 9 2 5 . 2 5 3 5 1004 . 3 2 0 5 1 1 9 2 5 . 2 5 3 5 1004. 3 6 8 8 12 9 2 3 . 4 7 3 7 1004 . 4 1 0 4 12 9 2 3 . 4 7 3 7 1004 . 4 4 8 0 * 13 921 . 6 8 5 6 1004. 4 8 3 0 13 921 . 6 8 5 6 1004 . 5 1 6 9 14 9 1 9 . 8 8 8 4 Appendix III, continued. Q. B r a n c h J r R 8 r R 9 pR1 pR2 * B r a n c h J " 19 1093 .6596 pQ1 24 1004 . 5 5 0 3 * 2 0 1095 .1036* 25 1004 . 5 8 2 8 * 2 0 1095 .1036* 26 1004 .6182* 21 1096 .5326* 27 1004 . 6 4 9 5 * 22 1097 .9470 28 1004 .6954* 22 1097 .9470 29 1004 . 7 3 5 9 * 23 1099 .34 14* 3 0 1004 . 7 7 9 0 * 24 1100 .7295 p 0 2 2 9 9 3 .4676 24 1 100 .7295 2 9 9 3 .0378 25 1 102 .0978* 3 9 9 2 .7770* 27 1 104 .7808* 3 9 9 3 .6490 28 1106 .0987* 4 992 .4059 9 1084 . 7 4 6 8 4 9 9 3 . 8 8 9 5 * 9 1084 .7468 5 991 .9070 1 1 1087 .8486 5 994 . 1894 1 1 1087 .8486 6 991 . 2 5 8 3 13 1090 .9030 6 994 .5445 13 1090 .9030 7 9 9 0 .4366 14 1092 .4124 7 994 .9527 14 1092 .4124 8 9 8 9 . 4 2 0 0 15 1093 . 9097 8 9 9 5 .4 107 15 1093 .9097 9 9 8 8 .1900* 16 1095 . 3929* 9 9 9 5 .9129 17 1096. .8673* 10 9 8 6 .7319 17 1096. , 8 6 7 3 * 10 9 9 6 . 4 5 2 1 * 18 1098. .3274* 1 1 9 8 5 .0383 19 1099 , . 7738 1 1 997 . 0 2 0 7 * 19 1099. . 7738 12 9 8 3 . 1079 2 0 1 101 . 2 0 7 9 12 997 .6071 2 0 1101. 2 0 7 9 13 9 8 0 .9447 21 1 102 . 6 2 8 4 * 13 9 9 8 . 1994 21 1 102 . 6 2 8 4 * 14 9 9 8 . 7 8 4 3 * 22 1 104 . 0 2 2 3 * 15 9 9 9 . 3 4 7 3 * 1 1004 . . 1651 16 9 9 9 .8737 2 1005 .6467 17 1000 .3537* 3 1007 , .0216 18 1000 .7778 4 1008. . 2 7 6 6 * 19 1001 .14 14 5 1009 . . 3984 2 0 1001 , .4443* 6 1010. 3 7 4 5 * 21 1001 .6879 8 101 1 . 8422 22 1001 , .8780 9 1012 . ,3259* 23 1002 , .0207 10 1012 . .6451 24 1002 . , 1190* 1 1 1012 . 8 1 2 6 25 1002 . , 1934* 12 1012 . .8438* 26 1002 . ,2361* 13 1012 . . 7 6 8 7 * 27 1002, ,2564* 14 1012 . .6051 28 1002 . ,2851* 15 1012 . ,3836 29 1002 . ,3037* 16 1012 . 1 2 9 3 * P 0 3 3 9 8 5 . 4 1 6 4 2 9 9 7 . ,8488* 3 9 8 5 . 4 3 8 2 * 3 9 9 9 . ,1911* 4 9 8 5 . ,4283* 4 1000. 4591 4 9 8 5 . 3 6 7 1 * 5 1001 . 6 3 7 1 * 5 9 8 5 . 4 3 8 2 * 6 1006. ,3518* 5 9 8 5 . 2 9 4 5 7 1008 . ,2185* 6 9 8 5 . 4 8 1 3 * 7 1003. 7 3 3 0 6 9 8 5 . 1924* 8 1010. 01 10 7 9 8 5 . 5 6 9 5 8 1004 . 6 4 9 5 * 7 9 8 5 . 0 5 2 0 * 9 101 1 . 7 0 2 3 8 9 8 5 . 7 1 7 7 * 9 1005. 4 7 4 8 8 9 8 4 . 8 6 0 5 10 1013. 2 6 7 3 9 9 8 5 . 9 3 7 7 10 1006. 2 1 6 0 9 984 . 6 0 5 4 1 1 1014 . 6 8 4 9 10 9 8 6 . 2 3 8 0 1 1 1006. 8 7 5 0 * 10 984 . 2701 12 1015. 9 3 8 8 1 1 9 8 6 . 6 2 2 0 12 100 7 . 4 5 1 4 1 1 9 8 3 . 8 3 4 7 * B r a n c h p P8 p P 9 r P O 14 15 15 16 16 17 17 18 18 19 19 2 0 2 0 21 21 22 23 9 9 10 10 1 1 1 1 12 12 13 13 14 14 15 15 16 16 17 17 18 18 10 10 1 1 12 12 13 13 14 14 15 15 16 16 17 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 9 1 9 . 9 1 8 . 9 1 8 . 9 1 6 . 9 1 6 . 9 1 4 . 914 . 9 1 2 . 912 . 9 1 0 . 9 1 0 . 908 . 9 0 8 . 9 0 7 . 9 0 7 . 9 0 5 . 9 0 3 . 9 2 0 . 9 2 0 . 9 1 8 . 9 1 8 . 9 1 6 . 9 1 6 . 914 . 914 . 9 1 3 . 9 1 3 91 1 911 9 0 9 9 0 9 9 0 7 9 0 7 9 0 5 9 0 5 904 904 9 0 9 9 0 9 9 0 8 9 0 6 9 0 6 904 904 9 0 2 9 0 2 9 0 0 9 0 0 8 9 9 8 9 9 8 9 7 1005 1003 1002 1000 9 9 9 9 9 8 9 9 7 9 9 6 9 9 6 9 9 5 9 9 5 9 9 5 9 9 5 9 9 5 8 8 8 4 0 8 3 0 0 8 3 0 2 6 8 4 2 6 8 4 4 4 6 5 4 4 6 5 6 1 6 0 6 1 6 0 7 7 8 2 7 7 8 2 9 3 1 8 9 3 1 8 0 7 9 4 0 7 9 4 2 1 8 6 * .3517* .2284 .2284 .4668* .4668* .6970 .6970 .9177 .9177 . 1299 . 1299 . 3 3 3 0 . 3 3 3 0 .5262 .5262 .7126 .7126 .8895 .8895 .0587' .0587' .8565' .8565' .0866' .3071 . 3071 .5200 .5200' . 7215' .7215 .9151 .9151 . 1004 . 1004 . 2 7 5 8 .0696 . 5 3 1 6 .0797 .7299 .4985 .4016 .4567 .6731 .0551 .6010 .3008 . 1399 .0959 .1471 Appendix III, continued. E B r a n c h J " pR2 13 1017 .0141 13 1007 .9534 14 1017 .9005 14 1008 . 3 8 2 2 * 15 1018 .5919 15 1008 .7512* 17 1019 . 3 8 2 5 * 17 1009 .3150* 18 1019 .5032 18 1009 .5251 19 1009. .6976* 21 1018 .9620* 22 1018 .5698 pR3 5 9 9 5 . 5 7 9 1 * 6 9 9 7 . 3 5 1 6 6 9 9 6 .8616* 7 9 9 9 . 1 8 0 1 * 7 9 9 8 .3831* 8 1001 .0670* 8 9 9 9 .8603* 9 1001 .2852 10 1002 .6512 1 1 1007 .0793* 1 1 1003 .9510 12 1009 . 1494 12 1005 . 1782 13 1006 . 3 2 8 5 * 14 1013 . 2431 14 1007 . 3 9 7 2 15 1015 .2062 16 1017 .0540* 17 1018 .8015 19 102 1 .7521* pR4 7 9 9 0 .6643 8 992 .2798* 9 994 .0085 10 9 9 5 . .6927 10 9 9 5 . .4825 1 1 9 9 7 .4076* 1 1 9 9 7 . .0645 12 9 9 9 . . 1 6 2 7 * 12 9 9 8 . .6270 13 1000. .9692* 13 1000. ,1659* 14 1002. .8357 14 1001 . .6739 15 1003. , 1454 16 1006. 7 6 9 6 * 17 1008 . 8 3 5 7 * 17 1006 . 9 4 9 1 * 18 1007 . 1670* 19 1008 . 5 1 8 1 * 2 0 1009. 6 9 7 6 * 21 1010. 8 0 0 8 * P04 166 a E J " B r a n c h J " 12 9 8 7 . 0 9 0 7 * r P O 16 9 9 5 . 2 6 6 9 12 9 8 3 .2784* 17 9 9 5 . 4 3 2 1 13 9 8 7 . 6 3 8 7 * 18 9 9 5 . 6 2 0 3 13 9 8 2 .5775 19 9 9 5 . 8 1 2 7 14 9 8 8 .2622 2 0 9 9 5 . 9 9 5 3 14 981 .7067 21 9 9 6 . 1 5 8 5 * 15 9 3 8 .9500 r P I 4 1 0 0 8 . 1 2 8 1 * 15 9 8 0 .6406* 5 1 0 0 8 . 3 1 9 4 * 16 9 8 9 .6941 5 1 0 0 6 . 0 7 9 4 * 16 9 7 9 . 3 6 0 8 * 6 1 O 0 7 . 0 7 9 3 * ' 17 9 9 0 .4827 7 1 0 0 5 . 9 2 0 5 * 17 9 7 7 . 8 4 2 2 * 7 1 0 0 1 . 9 1 7 4 * 18 991 .3031 8 9 9 9 . 8 5 1 0 * 19 9 9 2 . 1436 9 1 0 0 3 . 8 4 5 6 2 0 9 9 2 .9882 9 9 9 7 . 8 3 0 0 21 9 9 3 . 8 2 2 4 10 1 0 0 2 . 9 3 19 22 994 . 6 2 9 5 * 10 9 9 5 . 8 8 5 1 23 9 9 5 . 3 947 11 1 0 0 2 . 1 0 1 6 * 24 9 9 6 . 1022 1 1 9 9 4 . 0 4 3 5 * 25 996 .7844* 12 9 9 2 . 3 3 0 0 * 26 9 9 7 .4076* 13 1 0 0 0 . 6 8 3 1 28 9 9 8 .4732* 13 9 9 0 . 7 6 7 8 4 9 7 7 . 4 1 4 1 * 14 1 0 0 1 . 0 8 9 7 * 4 9 7 7 . 4 1 4 1 * 14 9 8 9 . 3 7 5 3 5 9 7 7 .3718* 15 9 9 9 . 5 6 7 8 * 5 9 7 7 .3718* 15 9 8 8 . 1679 6 9 7 7 .3208* 16 9 8 7 . 1582* 7 9 7 7 . 2 7 7 8 17 9 8 6 . 3 4 9 2 7 977 . 2 6 2 5 18 9 8 5 . 7 3 7 9 * 8 977 .2310* 19 9 8 5 . 3 2 4 1 * 8 9 7 7 . 1963* 2 0 9 8 5 . 0 7 8 7 * 9 977 . . 1 8 9 1 * 21 9 8 4 . 9 9 9 0 9 977 . . 1 199* r P 2 4 1 0 1 5 . 9 5 0 9 * 10 9 7 7 . .1609* 5 1 0 1 4 . 2 5 6 6 * 10 977 . .0319 6 1 0 1 2 . 5 6 4 1 * 1 1 977 . . 1 5 2 6 * 7 1 0 1 0 . 8 8 4 9 * 1 1 976 , .9286 8 1 0 0 9 . 2 2 6 6 12 9 7 7 . 1707* 9 10O7.597 1* 12 9 7 6 . 8 0 4 0 9 1 0 0 6 . 2 0 4 7 * 13 9 7 7 . 2 3 9 7 * 10 1 0 0 6 . 0 0 3 7 * 13 9 7 6 . 6 5 1 7 10 1 0 0 4 . 0 0 1 1 * 14 9 7 7 . 3 6 0 3 * 1 1 1 0 0 1 . 7 1 4 2 14 9 7 6 . 4 6 2 7 12 1 0 0 2 . 9 5 8 0 15 977 . 5 5 0 2 12 9 9 9 . 3 4 7 3 * 15 9 7 6 . 2 2 6 2 13 1 0 0 1 . 5 2 0 8 16 9 7 7 . 8 2 1 2 * 13 9 9 6 . 9 3 0 5 16 9 7 5 . 9 2 7 4 14 1 0 0 0 . 1 5 2 7 * 17 978 . 1846 14 9 9 4 . 4 8 3 1 * 17 9 7 5 . 551 1 15 9 9 8 . 8 5 6 1 * 18 9 7 8 . 6 4 6 6 15 9 9 2 . 0 4 2 0 18 9 7 5 . 0 7 8 3 16 9 9 7 . 6 3 7 2 19 9 7 9 . 2 1 0 4 16 9 8 9 . 6 3 6 7 19 974 . 4 8 7 2 17 9 9 6 . 5 0 0 3 2 0 9 7 9 . 8 7 4 4 17 9 8 7 . 3 0 4 4 2 0 9 7 3 . 7 5 3 7 18 9 9 5 . 4 4 7 5 * 21 9 8 0 . 6 3 3 5 * 18 9 8 5 . 0 7 8 7 * 21 9 7 2 . 8 5 3 2 19 9 8 2 . 9 9 1 2 22 981 . 4 7 6 4 * 2 0 9 9 3 . 5 9 8 4 * 22 971 . 7 6 0 9 2 0 9 8 1 . 0 7 1 6 23 9 8 2 . 3 9 4 4 * 21 9 9 2 . 8 0 2 9 * 23 9 7 0 . 4 5 1 3 21 9 7 9 . 3 4 3 6 * 24 9 8 3 . 3 7 2 6 * 23 9 7 6 . 5 3 8 7 * 25 984 . 3 9 6 6 24 9 7 5 . 4 8 1 6 * 26 9 8 5 . 4 5 0 7 27 9 8 6 . 5 1 8 6 * 167 Appendix III, continued. R Branch U" P04 28 987 . 5992* P05 5 969. 2386* 5 969. 2386* 6 969. 1859* 6 969 . 1859* 7 969 . 1260* 7 969 . 1260* 8 969 . 0586* 8 969 . 0586* 9 968 . 9853* 9 968 . 9853* 10 968. ,9064* 10 968 . ,9064* 1 1 968 . 8264* 1 1 968. 8202* 12 968 . 7435* 12 968 .7285 13 968 .6588 13 968 .6318 14 968 .5784 14 968 .5291 15 968 .504 1 15 968 .4193 16 968 .4421* 16 968 . 3020 17 968 .3970* 17 968 . 1728 18 968 . 3793 18 968 .0280* 19 968 .3970* 19 967 .8656* 20 968 .4421* 20 967 . 6744* 21 968 .5614* p06 6 960 .9167 6 960 .9167 8 960 .7837 8 960 . 7837 9 960 .7057* 9 960 .7057* 10 960 .6200 10 960 .6200 1 1 960 .5279 1 1 960 .5279 12 960 .4291 12 960 .4291 13 960 .3237* 13 960 .3237* 14 960 .2128* 14 960 .2128* 15 960 .0969* 15 960 .0969* 16 959 .9748* 17 959 .8587* 17 959 .8487* 18 959 .7366* 18 959 .7183* 19 959 . 5839 19 959 .6133* 20 959 .4937 20 959 .4453* 21 959 .3002* 21 959 .3758* 22 959 . 1627* 23 958 .9872* 168 Appendix III, R continued. Q. E B r a n c h j " P07 P08 p09 8 952 . 3931* 8 952 . 3931* 9 952 .3136* 9 952 .3136* 10 952 .2261 10 952 .2261 11 952 . 1301 11 952. . 1301 12 952, .0267 12 952 . 0267 13 952 .9134* 13 952 .9134* 14 951 .7966 14 951 .7966 15 951 .6714 15 951 .6714 1S 951 .5399 16 951 .5399 17 951 .4016* 17 951 .4016* 18 951 .2580* 18 951 .2580* 19 951 .1094* 19 951 .1094* 20 950, .9490* 8 943 .9068 8 943 .9068 9 943, .8256 9 943 .8256 10 943 . 7367 10 943 , .7367 1 1 943, .6391 1 1 943 .6391 12 943 .5336 12 943 . 5336 13 943 .4193 13 943 .4193 14 943 . 2973 14 943, . 2973 15 943 . 1668 15 943 . 1668 16 943 .0298* 17 942 .8855* 18 942 .7328 18 942 . 7328 19 942, .5744 19 942 .5744 20 942, .4087 20 942 .4087 21 942, .2373* 21 942 , .2373* 22 942 , .0601* 22 942, .0601* 24 941 .6879* 24 941 , .6879* 12 934, .9675* 12 934 , .9675* 13 934 , .8515* 13 934 , .8515* 14 934 .7237* 14 934 , .7237* 15 934 , .5948* 15 934 .5948* 16 934 , .4536* 16 934 .4536* Appendix III, continued. a a B r a n c h J " PQ9 17 934.3056* 17 934.3056* 18 934.1575* 18 934.1575* 21 933.6237* 21 933.6237* ^Transitions in units of cm asterisk. 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