A CC~§§1EEESECG§€D«ORDER §’ER.TURBAT?GN CALCUEATLON OF 'IHE QumDQUf-‘QH HY? EEC- {3&3 STRUCTURE 0? THE RQYANCRAL SPECTRUM O“? ETHYE. BRO$3DE “1933-; {in :‘E'az: beams Cs? M. S. m‘scxrutéxazx; 1:»: Emma r: in: any: i: s .1; fin 11's; 13" {"94 :hzi-‘tu‘fi «nth-5.1.” , * 3, A -”,_.__ MOW-“AA THLL‘SIS LIBRA R y Michigan State University MICHIGAN STATE uwvmsnv ABSTRACT A COMPLETE SECOND-ORDER PERTURBATION CALCULATION OF THE QUADRUPOLE HYPERFINE STRUCTURE OF THE ROTATIONAL SPECTRUM OF ETHYL BROMIDE by Paul Francis Dougher A procedure is described for a complete second-order perturbation calculation of the quadrupole hyperfine structure in the pure-rotational spectrum of an asymmetric molecule. Characteristics are summarized and equations are given for the matrices of the products of direction cosines required in the theory. Short tables (J 5 6) of the numerical values of the functions needed are also given. The theory is applied to the spectra of the 798r and 81Br isotopes of CH3CHZBr, CH313CH2Br, 13CH3CHZBr, CD3CHZBr and CH3CDZBr using the experimental data of Pierce and Flanagan (J. Chem. Phys. ;§, 2963 (1963)). The quadrupole parameters were reevaluated and are reported. An ex- tension to a partial third-order perturbation treatment was found necessary to predict the experimental spectra of the 79Br and 81Br species of CH3CDZBr. The new quadrupole constants are used to show that the assumption of cylindrical charge distribution about the C—Br bond is most nearly the approximation to the experimental case. A COMPLETE SECOND-ORDER PERTURBATION CALCULATION OF THE QUADRUPOLE HYPERFINE STRUCTURE OF THE ROTATIONAL SPECTRUM OF ETHYL BROMIDE By Paul Francis Dougher A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Master of Science Department of Chemistry l96h T o my parents and grandparents Ho Ho ACKNOWLEDGMENTS I wish to tender great appreciation to Professor R. H. Schwendeman for the kind and fruitful guidance in the assimila- tion of this thesis. I also wish to thank Mr. James H. Hand for his discussions of the computer programs used. I wish to thank the Petroleum.Research FUnd of the American Chemical Society for partial support of the research. iii TABLE OF CONTENTS Page I. HISTORICAL BACKGROUND . . . . . . . . . . i'. . . . . . ‘ 1 II. THEORY OF QUADRUPOLE INTERACTION . . . . . . . . . . . 3 2.1 Introduction . . . . . 3 2. 2 Causality and Effect of Nuclear Quadrupole Interaction . . . . . . . . . b 2.3 Rotational and Asymmetry Dependence of <33Wv715 . . . . . . . . . . . . . . 12 2.h First-order Quadrupole Perturbation . . . . . 16 2.5 Second-order Quadrupole Perturbation . . . . . 19 2.6 Summary . . . . . . . . . . . . . . . . . . . 21 III. APPLICATION OF NUCLEAR ELECTRIC QUADRUPOLE INTERACTION OF’ETHYL BROMIDE . . . . . . . . . . . . . . . . . . . 22 3.1 Introduction . . . . . . . . . 22 3.2 Analysis of the Spectra in Terms of Rotational and Quadrupole Parameters . . . . 2A 3.3 Extension to Partial Third- order Perturbation. 28 3.h Summary . . . . . . . . . . . . . . . . . . . 29 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . ho APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . he iv Table II. IIIa-e. IIIf. IV. LIST OF TABLES Rotational Parameters of the Ethyl Bromide Species . Quadrupole Coupling Constants . . . . . Observed and Calculated Frequencies (MC) of Ethyl Bromide . . . . . . . . . . . . . . . . . . . . . Third-order Corrections to the Calculated Spectra Of GISCDZ79Br and W3CDzelBroo o o o o o o o o 0 Experimental and Calculated Hypothetical Unsplit Frequencies of the Ehtyl Bromide Species . Transformation Angles Derived from Quadrupole Theory and Experimentation . . . . . . . . . . . . Page 31 32 33+> 37 38 39 I. HISTORICAL BACKGROUND Prior to l9h5, nuclear quadrupole investigations were limited to the atomic level Optical spectra (1) and to molecular beam radio-frequen— cy spectra of diatomic molecules (2). Unfortunately, only the latter - method provided the necessary precision to investigate, with accuracy, nuclear moments of atoms. The declassification and sale of governmental electronic systems after the Second World War led to a tremendous output of research literature in the fields of radio frequency atomic and molec— ular experimentation and theory. Microwave Spectrosc0py, a child of radar electronics, allowed unprecedented accuracy in the determinations of molecular structures and parameters. Because of the high resolution inherent in microwave spectroscopy, Good (3), in l9h6, discovered a nuclear quadrupole effect in the inversion spectra of 14NH3. The effect appeared as a group of weak symmetrical satellites bracketing a strong absorbtion line. This Spectrum was studied more thoroughly in the same year by Bailey, Kyhl, Strandberg, van Vleck and Wilson (b). The initial theoretical discourse on quadrupole interaction was given in the classic work by Casimir (5). As the experimental tech- niques allowed the analysis of more complex systems, the theory was extended to cover linear molecules by Nordsieck (6) and by Feld and Lamb (7), symmetric-top molecules by Coles and Good (8) and by Van Vleck (9). It was this later extension of the theory which was used to account for the hyperfine spectra of 14NH3. 2 In l9h8, quadrupole interaction was extended to include asym- metric rotors by Bragg (10) and by Knight and Feld (ll). The unsym- metrical nature of the systems which could now be investigated al- lowed for a greater extension into the theory of molecular structure and electronic cloud distributions. II. THEORY OF QUADRUPOLE INTERACTION 2.1 Introduction As the definition of a species further removes the multiplicity or degeneracy of a genus and allows a fuller exploration of the particular object under discussion, so the hyperfine structure perturbing the pure rotational levels yields more information concerning the nuclear and electronic composition of a molecule than pure rotational Spectra. This thesis is particularly concerned with but one of the many forms of hyperfine structure, that arising from electric quadrupole interac- tion. It neglects those which do not generally contribute to molecules, as magnetic hyperfine structure; or those which are not easily measur- able and are called "isotOpe effects". Magnetic hyperfine interaction is usually neglected because most molecules have Spin-paired electrons and are therefore in a 11:: ground state. A few molecules such as NO and C102, which have large molecular megnetic movements due to unpaired electrons, do exhibit ex- tensive magnetic hyperfine structure which greatly overshadows the nuclear quadrupole interactions. The "isotope effects" which are measured as 1 small shifts of the spectra due to a finite nuclear mass.and the dis- continuity of the electron density within the nuclear radius, are ex- tremely small and may be noticeable only by comparing various.isotopic species, hence the name of this effect. This thesis is concerned uniquelywith the removal of the 21+l degeneracy by the nuclear quadru- pole interaction and the aforementioned effects are neglected in the treatment. )1 2.2 Causality and Effect of Nuclear Quadrupole Interaction Nuclear moment theory is based upon several experimentally provable, but not theoretically derivable statements. Nuclei with an odd mass number, ($.3. the sum of the number of protons and neutrons.is odd), have half-integral Spin values and obey Fermi statistics. Nuclei with an even mass number have integral units of’fi* for their spin values and obey Bose statistical theory. The spin of the nucleus, I, is then defined as the maximum component of the quantum-mechanical vector I>, a dimensionless vector related proportionally to the spin angular momentum of the nucleus. It will be apparent below that nuclei with spins of zero of'l/2 do not exhibit nuclear quadrupole spectra. Hence the discussion will be restricted to nuclei of spin greater than 1/2. Nuclear quadrupole energy arises out of the interaction of the spin property of the nucleus with all of the external charge distribu- tion. This particular treatment is based on the premise that there is only one nucleus in the system which has a spin greater than 1/2, or as assumed in Section 3, the interaction with other nuclei with spins greater than l/2 is so weak as to be neglected. Therefore the only extra-nuclear charge distribution is that of the electrons surrounding the nuclei, and point charges at the nuclear placement. The theoretical discussion leading to nuclear quadrupole inter- action is based on the well known theory of electrostatic interaction of a charge distribution with another surrounding charge distribution. * ‘fi = h/2n where h = Planck's Constant, 6.6252 x 10-27 erg—sec. ’fi = 1.05hh x 10—27 erg-sec. S This classical treatment of electrostatic interaction, discussed in terms of multipole eXpansion potential theory, is given below with the thought of deriving a "good" quantum-mechanical Hamiltonian for a nuclear quadrupole interaction. The discussion follows that given by Ramsey (13). By fixing an axis system at the mass center of a nucleus of charge density P1152), the‘interaction between a volume element, d'fn of the nuclear distribution at a distance, B:, from the axis center and a volume element, dT:ef of the external charge distribution at a distance, -> O C re, from the ax1s center may be written as: K 1 = // Fe Ge) PHiD>dT€d Tn (2_1> ’[e fiIn where Mel is the contribution to the potential energy of a molecule from electrostatic interaction. It may be more easily viewed as the average of all the interactions between two points defined by the volume elements dqfe and d'fg separated by a distance R. To calculate Mel R is written as a function of the two measurable vectors. '1: and F2} Therefore from the simpletrigonometric cosine law: 1 1 ' R = (2-2) 2 2 _ I V re + rn 2rerncos Gen 6 - a le b t th t t '> d "> en - ng € ween e U0 VZC OI‘S re an I‘m. The solution is,however, not soluble unless an assumption is made that re is always greater than rn, that is, the electrons are never allowed to exist inside the nucleus. As mentioned before, this dis- continuity of the wave functions at the nuclear position yields a very 6 small but measurable "isotopic effect". Assuming then that rn < re, equation (2-2) may be rewritten and expanded in a power series in rn/re (12) by the use of the MacLaurin series expansion, m . (1 - 2W + 3’2)'l/Z = Z P100112 (2-3) 190 In this equation the Pl(x) aux: Legendre polynomials of order 1. Re— writing equation (2-2) as: 1" 2 1‘ -1/2 121' = :1,— [1 + (E2) - 209m sen] , (241) e e e and expanding using equation (2-3)‘ 03 r 1 % = 3%: Z: Pn(cos een)(-r'2) . (2-5) e e 1=o Equation (2-5) has meaning, however, only when the series expression converges to a finite value. Since [cos een\ 51, Pl(cos Gen) has ex- treme limits of il and therefore)IPn(cos Gen). < 1. Therefore the coef- ficients of (rm/re)1'are in absolute value never greater than one, and convergence will be assured if the value of (rm/re); is always less than 1. This can only occur if rn < re, that is, when the extra-nuclear charge distribution is never allowed to permeate the nuclear charge dis— tribution. Expansion of equation (2-5) and substitution into equation (2-1) will now yield F ?> r r M e, =// 68‘ 9):“ “)[Poeos emu-52f + P1) (?>) K .1 =f/ 98 if“: “ mm <2—9> This monopole contribution is included in the normal atomic theory to make up the molecular electronic states. Since it is the same in both rotational states it will be neglected in this discussion. The second term of the series is the dipole expression and repre— sents energy due to the nuclear electric dipole moment. This condition arises due to a consideration of the parity operator which commutes with 8 the Hamiltonian of the nuclear system. The parity operator, acting on the functions for which 1 is odd in equation (2-5), yields an inversion of coordinates system of the Hamiltonian. This inversion of the coordin- ates of the Legendre Polynomial cancels the contribution from the nuclear charge distribution and the integral vanishes. A more complete theore- tical discussion of the restrictions on the orders of electrical multi- poles may be found in reference (13). This reference also contains the proof that the requirement for the non—vanishing of the integral con- taining fl is I Z. %, and therefore, as previously mentioned, spins of zero of 1/2 do not exhibit nuclear quadrupole interaction (1.3. I = 2). The proof has been experimentally verified in that no nuclear electric dipole interaction has been observed. The third and last term of the truncated eXpansion under the integral is the nuclear electric quadrupole addition to the energy of the molecule and is expressed as quadrupole (F’) (;>) r H el #38 erfn n (i)2(%c05268n- %)d’[ed’l‘n, (2-11) All other terms of the expansion are deleted for one of two reasons: l) The order of the expansion is odd, 1.3.‘1 is odd, and for reasons mentioned above, they do not contribute to the nuclear electric interaction, or: 2) The even power terms have weak components as measured experiment- ally. Only the hexadecapole term, I = b, has ever been measured. As mentioned before, I must be 2.2 to allow the hexadecapole term to enter into the Hamiltonian of the system. The nuclear electric quadrupole term may now be rewritten in the 9 form given by Strandberg (l6) in which the physical significance may be _ 2 more easily seen. Noting that the vector product (EzorZ) = rnzrezcoszee n) the inner part of equation (2-11) is written: >—>2 . _ 2 2 (:2)2(%COSZO - %) = 3(En re) rn re re en 2re 5 _ 3F F 'F F ' - (En Fe )2 " 252 ”n 88 (2-12) 2r5 e where Fn = x,y,z components of F: Fe = x,y,z components of F: Equation (2-12) may be further factored into two terms, one dependent on the nuclear coordinates and one dependent on the electronic coordinates: (242) = L— E Z (3F F ' 6r 5 , n n e F F JTFF' grr'rnZHFeFe' ' JFF're2)° (2‘13) o,F,¥F' ll H The nuclear quadrupole Hamiltonian may now be written KQI- éZF a V F}?! QFF' : (2-111) where using a tensor definition, (3F F ' — a; , r 2) fFeGZ) 8e 5 FF 8 dTe’ (2'15) e re QFF' ‘ ffnGZ><3F.Fn' - Sam Mr. <2-16> n The tensor elements are simply the charge weighted quadrupole moments 10 of the nucleus. The term VFF' may be further Simplified by noting that the values of the tensor are simply the second derivatives of the po— tential of all the extranuclear charges evaluated at the nucleus. Also utilizing a classical mechanical expression relating the potential to a force, equation (2-lh) may be rewritten as bf Q = -% z}: gQFF' (v?)ppn (2‘17) where (V E>) FF' = electric field gradient tensor of the field evaluated at the nucleus. A further appraisal of equations (2-15) and (2-16) will yield two very important properties of both V and Q FF' FF“; 1) Since the components of a vector, either E: or F2, commute among themselves, that is FF' = F'F, (2-18) both tensors are symmetric about the diagonal. 2) The traces of both tensors are equal to zero. The most useful form of HQ was first derived by Kellogg, Rabi, Ramsey, and Zacharias (2) using the methods of Casimir (S). This form was extended to include asymmetric tops by Bragg and Golden (lO,lL). It is derived by a lengthy procedure by first noting that QFF' is a function of the nuclear spin-Operator alone and that (ifEiFF, is a function of the rotational operator, only. Thus the operator form of QFF' is (1.? (I7). + (I7 (T313 'I'> 2 J 2 l ' t 1 3 _ 9 QFF!’I'C('ZI'1T)'[3 F F2 F '5”: ( ) where _ 1 Q: E J/%%(ml=l)[3zn2 - rn2]d1fn = < 32“2 - rn2 TAV 5 nuclear quadrupole moment. ll The Operator form of (€7E>)FF, is — > > > (VB )FF" ’ J(2J——)-l 3 2 - FF. , where 1f BZez - 1flez ‘9 ° 2 Paw [ r 5 1 at. (m e By rewriting equation (2-17) using the new forms written in equations (2-19) and 2-20) and using the usual commutation relations for angular momentum operators, an equation vividly showing the concept of f>.j> coupling may be written as a”? _ quQ +>1+> 2 3 -> -> 4>2e>2 Q- 1 1-1 J4 [3(14) +T,,-I-J-I J 1. (2-22) The quantity qJ may also be written as 2 - ,v 2 2 . _ qJ =fP€(mj=J) Beige Udle‘ <5 v/a ZJ >AV (2 23) Therefore the Hamiltonian for nuclear electric quadrupole inter- action for a one-quadrupole system, (3.3. atom or molecule) may be written €Q . 3{Q a 2I(2I-I)J(2J-I7AV [3(T>'3>)2 + #1739) - T>Z§ml (2-211) The Hamiltonian of (2-2h) is still not in the most useful form because the average value of DZV/B 2J2 is with respect to an arbitrary space-fixed axis system. Because qj changes with molecular rotation it is more convenient to eXpress qJ with respect to a molecule-fixed axis system. This is discussed in Section 2.3 with relation to direction cosine matrix elements. It is possible to discuss the eigenvalues of the operator l2 .-e'> -,-> 2 3->—> -'>z-.'>2 . M -i [3(1 -J ) + §(I ‘J ) - I J J, and hence the eigenvalues of Q’ w1th- out regard to either the rotational part of the problem or the form of the rest of the quadrupole Hamiltonian. This is possible because the operators in the brackets are dependent only on the magnitude of J and I. Casimir (5) determined the eigenvalues to be %c(c + 1) - I(I + 1)J(J + 1), (2-25) where C = F(F + l) - I(I + l) - J(J + l), and F = J + I, J + J — 1, - - ° , J - I . Here F is a new quantum number and is required because I is coupled to J and therefore only the total angular momentum is a constant of motion, i.e. a "goOd quantum number". 2.3 Rotational and Asymmetry Dependence of <32M6 22 J>AV Equation (2—2h) is the general form used to describe the nuclear quadrupole interaction in one-quadrupole systems, but it is not the best form for describing rotating molecules because the term, qJ, is dependent upon the magnitude of J. It is possible to rewrite qJ in terms of a constant for all molecules times a function which is dependent on the asymmetry of the rotating molecule. This is accomplished by writing the < BZV/b z2 in terms of an axis system defined by the J>AV moments of inertia. The principal axes of inertia a,b,c, are defined in such a way that the moments of inertia are Ia‘f Ibif IC. By writing <32V/b 2J2) in terms of the molecule—fixed axis system a transformation using direction cosines yields BZV/b 2J2 = dzaz(32V/a a3) + azb2(bZV/bb2) + dZC2(BZV/3c2) + 2dzacczb(a 2V/b a3 b)+2cczaazc( bZV/b ab c) + (2-26) 13 2azbazc(bZV/bb5c), where azi = direction cosine between the z—Space fixed axis and the "i" molecule fixed axis (i = a,b,c). Before progressing further, a brief description of the rotational wave functions for aSymmetric rigid rOtators must be given. The HamiltOpian for theérigid asymmetric rotor may be written as -' a 2 2 2 - bfr 1101138 + pr + c1=>c ) (2 27) where A, B, C = rotational constants, and P P P = components of angular momentum in the inertial axis a’ b’ c system of the molecule. It is possible to introduce a basis set ‘QJK such that P2 V’Jx 2 2 2 :3 (Pa + Pb + Pc ) ¢’JK J(J + l) v.1K and P2 wa = K JK K = J, J—l, . . -, —J (2-28) This basis set will yield the3’6r matrix in diagonal block form, 1.3., diagonal in J. Each of the diagonal blocks may be further factored into four independent submatrices by the Wang transformation (15). The Wang functions are symmetric and antisymmetric linear combinations of each.q’J"K‘ and the correspond1ng VAJ,_‘ There now eX1sts an orth- KI' ogonal matrix which diagonalizes the energy matrix calculated using the Wang basis functions. That is, there exists a transforming matrix T such that . N wR= 1‘er NJ where T = transpose of T, and W = the diagonal matrix of the rigid R. rotor eigenvalues. 1h The transforming matrix T rotates the Wang linear combination of symmetric rotor functions into the asymmetric basis set in which the asymmetric rigid rotor Hamiltonian is diagonal. These transformed basis fUnctions will be used and referred to throughout this thesis as the zeroth-order basis functions. Since I and J are coupled, requiring a new quantum number F to describe the state of a system, these basis functions must be combined with nuclear spin functions and the products transformed into a system which is an eigenfunction of an operator which is a function of F as well as J and I. This transformation leads to the matrix elements fora'eQ given in equations (2-31 a,b,c). Only the elements diagonal in I are given since the extreme separation of nuclear energy states causes the elements off-diagonal in I to be negligible. eQ < JTM =J|32V/BZZ\J 1' M =J > ~ 3 3 J J 3 3 J ” 81(2I-l)J(2J-l) X[3C(C+l) - bI(I+l)J(J+1)] (2-318) eQ < J 't M =J\ a 2V/312\J+1 13' M==J > t = 3 3 J] ) ) J < Ir’J’T’MMQIF’J+13? ’M > BI(21-1)J(2J+1)1/2 X[F(F+l) - I(I+1> - J(J+2)] (2—31b) X[(I+J+F+2)(I-J+F)(J-I+F+1)(J+1-F+1)]1/2 e0 «waxes/31x > 161(21-1)[(2J+1)(J+1)]1/2 < F,J,T,M|3{Q|F,J+2,‘Y',M > x[(I+J+F+2)(I+J+F+3)(I-J+F-l)(I-J+F)(J—I+F+l) x(J-I+F+2)(I+J-F+l)(I+J-F+2)]1/2 (2-3lc) where C = F(F+l) - I(J+l) - J(J+l). 15 It is noted that the matrix elements of B 2V/BZJ2 need be evaluated only for MJ because the energy cannot depend on the projection of J(i.e. MJ) on an = J, since the matrix elements are diagonal in MJ. This is arbitrary direction in the space—fixed system. As before, the problem reduces to the evaluation of bZV/OZJZ in terms of the inertial frame of reference, (323' the experimentally measureable axis system). The introduction of a new nomenclature al— lows the writing of equation (2-26) in the condensed form: 3211/3sz = Z 2 ¢zg gzg: sz/ngg' (2-32) 9 9' where g,g' = a,b,c, the inertial axis system. Therefore, in the zeroth order representation, the transformed basis set in which the rotational energy is.diagonal: aim/32,2 = 2:: ‘7 $529 9129, T BZV/bgbg' (2-33) 9 9' It is known (17) that the direction cosines may be written as a product of three terms. ¢FQ(J,'I.’.MJ;J','£',MJ') = ¢F9(J,J')¢FQ(J,MJ;J',MJ')¢FQ(J;I;J',I')(2-3u) F = x,y,z space-fixed axis. g = a,b,c molecule fixed axis. J' a J, J i l and as explained previously, when F = Z M' = M = J. If now a function ) G(J',J") is defined as G(J':J") = ¢zg(J',J")¢zg(~]',M=J3J":M'=J): (2-35) then 16 ¢zg(J,”£,M=J;J',’i',M'=J) = G(J,J')¢ZQ(J,T;J'T') (2-36) The possible values of G(J',J") required for later calculations are tabulated in Appendix A, as well as the products G(J,J')G(J',J") also required later. The matrix elements of ¢zg are ordered according to the symmetry of the transformed rotational energy eigenfunctions, that is accord- ing to E+, 0+, 0- and E_, the indices corresponding to the symmetry of the Wang linear combinations. The arrangement of the matrices in this thesis is that used by Schwendeman (17,18) and is summarized in Table I and Figure 1 in the Appendix. The matrix elements of a ZV/a 2J2 may now be written as < J,‘[’,MJ=JI BZV/BZJZ‘J',T',MJ' =J > =22: T1 2 [G(J,J")G(J",J') 9 9' J" x: 91290 :r;J",t">¢Zg, (J",'t"sJ',’t')JT a zv/o 939' (2-37) t! Using the formulation of references 17 and 18, the direction cosine products will yield 108 individual products consisting. of JIJ, J|J+1, J‘J+2 elements which are blocked by symmetry in terms of aa, bb, cc, ab, ac and bc products. Equations for the matrix elements of the 108 products and tables of the numbers required for their evaluation are given in Appendices A, B, C and D. 2.h First-order Quadrupole Perturbation. Using the equations defined in the previous section and the matrix elements given in the Appendix it is possible to evaluate the quadru— pole interaction and its perturbing effect on the zeroth—order rigid rotor energy levels. This effect causes a small shift of the rigid 17 rotor energies and splits them into 21+1 components. Since the quadru- pole effect is very much smaller than the pure rotational effect, the Hamiltonian describing the quadrupole effect may be treated as a small perturbation and its expectation value taken over the zeroth-order wave functions, i.e. a first-order perturbation treatment may be used. Com- bining equations (2-31a) and the diagonal elements (1.5. g = g') of (2-37) the first-order quadrupole energy may be written: 1 wQ”= = eQ x [f(I,J,J,F)] x 2 < J,T,MJ=JI?'¢ZQ mm, M =J > o Zv/agz (2-38) 9 The f(I,J,J,F) are functions introduced by Casimir and are tabulated in Reference (19). They are related to the C(I,J,J',F) functions described in the Appendix. Bragg (lb) makes use of the following expression for BZV/a 2J2; = (IT+1)€2J+37% azv/aa2[ J(J+1)+E in) dis" (x) 2 <15" (x) 2 dEJ (X) ‘(x*1)'?11><—]* 53:2 d7), +3, aa—‘C’ZLI (J+i)—E‘;(x)+(x-1)—(§TT,—J (2-39) where Biol.) == rigid asymmetric rotor energy levels, and E§£SEZ = derivative of the energy with respect to the X asymmetry parameter,)£. Another useful form of writing equation (2-39) is written by first recognizing that 3 w <35 TOO Ar §[J(J+1>+B§(x>- 0w) T: —] 3 W ( ) £3511- (ET; (2-bo) 18 J awr h J “157”) -5—C- ' §[J(J+1) ' ETUC) + (K --1) -—d-—x—-], where W = rigid-rotor energy levels, I‘ A,B,C = rotational constants. Therefore MgJ‘aZ/ Z __ _ 2 [aZW/ war+ “ h(ZJ+3)(J+1) aa 311— 2w/ b2 Owl“ + my 2 3er (2411) a a DB be ac By defining the quadrupole coupling constants ' = 52V 3 = 2- 2 X99 <30 /59 20 qgg ( h) and the substituting (2-b2) and (2-hl) into (2-33), the first order quadrupole interaction energy expression in an easily computable form is given by wQ‘" = [r/ azv/a a2 = (Xbb-XCCana The first-order perturbation energy is written: “om = mm + Wm where a = [f(I,J,J,F)] f1(J,)() and ‘ (2-h6) B = [f(I,J,J,F)] f2(J,)() 2.5 Second-order Quadrupole Perturbation Normally only the first-order quadrupole correction is needed to fit the obServed spectra. However, under certain conditions it may be necessary to include higher-order perturbation to account for observed differences from the first-order predictions. The conditions listed below are given in terms of relative magnitudes, and therefore the real criterion for applying higher order perturbations is how well the lower order theory predicts the spectra. In general the second-order pertur- bation is applied under the conditions that 1) There must be a quadrupolar nucleflswith spin I 3 1. 2a) The quadrupole coupling constant is "large", and/or 2b) There exist“ near-degenerate rigid-rotor energy levels such that the denominator of the second-order eXpression, equation (2-h7), becomes sufficiently small. The general form of the second—order quadrupole contribution to a given level is given as 2O 2 _ (2-b7) my wowxr) - wowm me (mm where w0(F,J,'t) the zeroth-order rotational energy levels, and beg = Quadrupole Hamiltonian after it has been properly transformed by the same transformation which diagon- alized the zeroth—order problem. A convenient form for 36(3 is )6 Q = [C(I’J’J"F)]1/2{Xaa’Xaa +XabXab +xacxac +K bcxbc} (24:8) where X 33 = “C; +7 Mn: xq Mg. - (91;, + gag/2, X7 ($521, - ¢:C)/2 ’71 (’Xbb " ch)/Xaa’ and 99' 2¢Zg 29' (g g 9') 3f Equations for the C(I,J,J',F) and tables of their numerical values are given in the Appendix. Of primary importance,it is to be noted that the second-order con- tribution to a given level is calculated by summing over all possible J's and'T"s for a given F value, as shown by equation (2-h7). Since for the representation used (17,18), the various blocks -- aa, ab, ac, and bc -- are independent of each other they may be transformed separately and evaluated in any order desired. Using the equations derived in Sections 2.3, 2.h and 2.5 and the tabulated matrix elements given in the Appendix, a complete second- 21 order perturbation calculation may be made for any one-quadrupole molecular system. 2.6 Summary The rotational energy of a one—quadrupole molecular system may be treated as a zeroth-order problem perturbed by a higher order quad- rupole interaction term. The addition (or subtraction) of the extra energy due to the interaction of a spinning nucleus with the electro- static potential of an unsymmetrical charge distribution splits the asymmetric rigid rotor energy into 2I+l levels, or 2J+l whichever is the smaller. = (1) (2) _ wtotal wR + wQ + wQ (2 L9) The orders of magnitude of the various energies are such that (1) (2) WR >>> WQ >> No However, within the limit of accuracy of microwave spectrosc0py, terms as high as the second-order terms are of great importance in the analysis of the experimental spectra. In the next section it will be shown that in fact a partial third-order analysis is required to ac- count for descrepancies not accounted for by the second-order terms. III. APPLICATION OF‘NUCLEAR ELECTRIC QUADRUPOLE INTERACTION TO ETHYL BROMIDE 3.1 Introduction Ethyl bromide is an important example to utilize in the applica— tion of quadrupole analysis for many reasons. 1) Both 79Br and 81Br have large nuclear quadrupole moments, 0.33 x 1024 cm2 and 0.28 x 1024 cm2 respectively (19), thus insuring a rather large first and second-order quadrupole effect if other require— ments are satisfied. 2) 79Br and 81Br have nuclear spins of 3/2 therefore allowing the quadrupole effect as mentioned in the discussion of the existence of multipole expansion in terms of nuclear spin included in Section II. 3) Accidental near degeneracy of rotational levels of CHSCD279Br and CHSCDzelBr is the predominate reason for the large second-order splittings. h) The primary purpose of using ethyl bromide is to evaluate the angle required to diagonalize the experimental quadrupole tensor and to compare this angle to those calculated using the two predominate methods of estimating the off-diagonal element. The two approximations are a) By simply calculating the angle between the "a" inertial axis and the C-Br bond and using this angle to evaluate theX ab trial term, or b) By assuming a cylindrical charge distribution about the C-Br bond and using the relations of cylindrical symmetry, 52v _ 32v _ 1 2v (3;?)Av " (WNW " " (’2 'gfihw 22 23 where by the symmetry of the system, (3131’) =2( 3 -—‘°’—")AV 622“ OC 2A and since °§}§§) (igz 225AV'(2£2%_—__-)AV then xaa a _ Xaa = (3cos2 6—1) we 27;.- 5 ‘ The predominant near degenerate pairs occur between the 110 and 202 levels, the 211 and 303 levels and the 312 and 504 levels. Therefore, as shown by equation.0}h7), as the running sum is carried over the J states and connects any pair of these near degenerate levels; the denominator of the second—order correction becomes quite small yielding a large second-order term to both connected levels. This effect is so large in the CH3CD279Br and CH3CD251Br species that it becomes neces- sary to use a partial third-order perturbation correction to predict the experimental spectra. This is discussed more fully in the follow— ing section. A microwave analysis of the parent species CH3CH279Br and CH3CH251Br had been attempted in 1957 by Wagner, Bailey and Solimene (20). They assigned several "a"-type (R-branch) transitions for the two species and were able to evaluate the B and C rotational constants, as well as the diagonal components of the quadrupole tensor in the principal inertial axis system. Certain transitions, however, were indicated as being poorly fit, and the discrepancies could not be removed by includ- ing second-order quadrupole corrections. This was later disproved by 2h Flanagan and Pierce (21), who showed the Xab terms connect states which are extremely dependent on the rotational constant A. Flanagan and Pierce succeeded not only in determining the A rotational constant but were also able to establish the off diagonal element,jxlab of the quadrupole tensor for the ten independent isotopes of ethyl bromide, 133° the 79Br and 81Br species of CH3CHZBr, CHSCDZBr, CD3CH2Br, CH313CH2Br, and 13CH3CH2Br. They performed a partial second-order analysis of the quadrupole coupling using a symmetric rotor basis. Agreement with experimental results was quite good except for the 79Br and 81Br isotopic species of CHSCDZBr which, as will be shown in Section 3.2, requires an extension to a third—order perturbation treatment of the quadrupole Hamiltonian. 3.2 Analysis of the Spectra in Terms of Rotational and Quadrupole Parameters The experimental "a" and "b" - type transitions observed and re- ported by Pierce and Flanagan for the ethyl bromide series are used as the basis for establishing the molecular parameters. The experimental _values of Pierce and Flanagan as well as the calculated values are entered in Tables IIIa,b,c,d,e. A simple but extremely effective method was used to evaluate the parameters. The method discussed below was used for all species except the 79Br and 81Br isotopes of CHSCDZBr which required a modification to include a partial third-order perturbation analysis. The parameters, A, B, C, X aa’ 11 Xaa and Xab evaluated by Pierce and Flanagan were used as a first estimate to the problems A FORTRAN language program developed at the Chemistry Department of Michigan State University, using only these parameters, was run on a Control Data Corporation 3600 25 Computer for each of the ten species of ethyl bromide. The program generates the Wang symmetric rotor basis set and transforms them to the correct asymmetry. It also generates the direction cosine matrix elements and their appropriate multiplicative factors used in evalu- ating the quadrupole interaction terms. All constants mentioned in the following discussion are calculated and generated as part of the output of this prOQram. It is possible to write the transition frequency as a simple sum of differences between levels; Vt = woup - wolow + Wu)up - Wuglow + M22113 — Wmlow (3-1) where Vt = experimental frequency, W0 = rigid rotor energy (upper and lower levels of a given transition), Wu) = first-order quadrupole interaction energy, and We) second-order quadrupole interaction energy. By using equation (3-1) it is possible to isolate the energy contri- bution from either the zeroth, first or second—order problem. The procedure used to evaluate the final parameters listed is given below and was used for all species except the 79Br and 81Br iso- topes of CH3CD2Br which will be discussed separately. 1. Since the contribution from the second-order perturbation is the smallest of the components a small error in evaluating this part would not greatly disturb any further evaluation of the other parameters. Several programs were run with variable values of xab to give several sets of calculated spectra. Use of equations (3-1) and (2-h8) allowed 26 the derivation of equation (3—2). exp exp exp exp 2 2 z 2 ’X 2. (’Xabznv‘l 'V3 ”V2 +V1 ‘V4aa+y3aa+V2aa'y1aa] b — _—=— Z Z 2 2 (3‘2) a [V4ab'ysab‘vzab+VIab] where 2 7(en3 = is a trial value of the square of the off diagonal quadru- --- pole elements, and where for example 2 1 up 1 low v lab H ab X ab The subscript integers, 1,2,3, and A label the four experimental compo- nents of a given transition in order of decreasing frequency. By using several values of 'Xab, it is possible to evaluate many Xab's from which a value of flab—may be obtained for the rest of the calculations. It is interesting to note that the first-order terms do not occur in equation (3-2). This is because of the symmetrical nature of first-order theory which predicts that: vim-V. Wimp-V that is, first-order theory predicts that four transitions will appear exp exp 1 near the hypothetical unsplit frequency as two doublets, each of the same splitting. 2. The second step in the quadrupole analysis utilizes equations (3-1), (24m) and 2415): (Z) (2) (1X 1} vtm - y (F) - v.32") + v a“) = v (F) - y‘ (1“) (3-b) where v(1‘(f) = Ad (F) )7 aa + Ammqbb - ”XCC) , 27 (1) = I + I _ y (In) AMP ) Xaa M30“ )()(1,}, 7“,) and F, F' indicate the various F components of a given transition. Using a least squares program, the parameters 'X and (X - X ) aa bb cc may be evaluated from a set of linear equations calculated using the experimental spectra. The individual diagonal 'X 's are evaluated using the relation X aa + Xbb + ch = 0' (3‘5) 3. The last set of parameters to be fitted to the experimental spectra are the rotational constants A, B and C. Use may be made of another least squares calculation using a set of equations written in three variables of the form v0 -v0 =915+11?- <3-o> obs calc obs calc 3A obs calc 3B Bwr + (Cobs Ccalc) 5C where n 2 vob = experimental frequency - y< calculated - V( ) calculated. 0 s 0 = calculated rigid-rotor frequency. calc (A,B,C)Obs = A, B and C rotational constants which will give the 0 best possible fit to the v obs spectra. 0 calc. (A,B,C)Calc= A, B and C rotational constants used to evaluate v 3 wr/a i = The change of rotational energy with respect to the i = A, B, C rotational constants. A second iteration of the three steps above did not appreciably change the values of the constants. As a check on the quadrupole 28 constants, a term known as the hypothetical unsplit frequency, defined as: m (2) -)/O -n tht' 1 l'tf yexp vcalc - V calc exp _ ypo e 1ca unsp 1 reg??? was calculated for all experimental frequencies listed and are tabulated with the calculated zeroth-order transitions in Table IV. 3.3 Extension to Partial Third-Order Perturbation A noticeable discrepancy was noted when the procedures of Section 3.2 were applied to the 798r and 81Br species of CH3CD2Br. This becomes immediately apparent when a fitting of X ab using equation (3—2) on the 202 - 303 and 211 - 312 transitions yielded values Ofxab differing by 30-h0 Mc. It was impossible to use the other reported transitions to calculate furtherXab,S because they were too insensitive to the changes in Xab' A simple average of the two Xab's yielded values similar to those reported by Pierce and Flanagan (21). Since the most nearly degenerate pair of energy levels, 211 - 303, contributes to both transitions, it was thought that a partial third- order perturbation calculation would account for the discrepancy. This reasoning may more easily be seen by writing the partial third-order term. w1(3)(1:) = _w,(3)(p) = M12 XZZ-Mll (343) o o o 0 W1 ‘W2 w1 ‘wz where (3) l3) _ W1 (F), W2 (F) = third-order energy of levels (1) and (2), the near degenerate pairs. “22, X1, = the diagonal components of the first-order quadru- pole Hamiltonian of the respective levels. 29 W01 , W02 = zeroth—order rigid rotor energy levels. Equation (3—8) may be written more explicitly by noting that the first term in brackets is simply the second-order term connecting the two states in question, and that the numerator of the second bracketed term is simply the difference of the first-order quadrupole energies of the two states. 0’ (fl (3) (3) (2) w303(F) - W211(F) w211(F) = - W303(F) = w211'303(F) w - w (3-9) R211 R303 The significant partial third-order terms evaluated are tabulated in Table IIIf where they are added as perturbations to the frequencies calculated to second-order. It is to be noted that the 5/2‘—+> 7/2 and 3/2 -—> 5/2 assignments for the transition 211l—e> 312 must be inverted from that reported by Pierce and Flanagan for both the CH3CDZ79Br and CH3CD281Br species. The lack of more low J transitions hindered a more precise evaluation of the quadrupole constants. 3.h Summary The application of second-order perturbation to systems defined by the conditions discussed in Section 3.1 will allow for excellent correlation between observed and calculated spectra. And, as shown, when conditions are such that near degenerate levels are connected by the quadrupole offjdiagonal elements then it becomes necessary to ex- tend the theory to include a third-order correction. Primarily, however, the most useful information derived from the accurate evaluation of the off-diagonal element of the quadrupole tensor is that it allows a further insight into the electronic structure of 30 the C-Br bond. The primary purpose of this thesis is to show from a complete second-order analysis which of the two commonly used methods is the more accurate for estimation of the off—diagonal element of the quadrupole tensor. Table V lists in columns I and III the results of the two most used methods of estimation, and in column II, the experimental results from the second-order analysis. As can be seen from the angles tabu- lated, the approximation of assuming cylindrical charge distribution about the C—Br bond seems to be the more accurate. The uncertainty in the listed values of')(ab is i 10 Mc; therefore the values of 0 in column II are accurate to within 1 30'. This would account for the inversions in the orders of magnitudes of some of the angles listed, and also brings the values in column I close enough to the eXperimental limits of the values in column II to prohibit ahy positive statements concerning the existence of slightly bent bonds. 31 Table I. Rotational parameters of the ethyl bromide species. Species A (Mc) B (Mc)' 0 (Mc) x; CH3CH27QBr 29 955.82 3 803.95 3 522.89 -O.978735 CH3CHzelBr 29 9b7.80 3 781.07 3 503.17 -O.978982 €8313CH279Br 29 187.70 3 765.b9 3 A78.75 -0 977659 CH313CHzelBr 29 120.93 3 7h2.b2 3 h58.85 -O.977899 13CH3CH27gBr 29 665.00 3 669.87 3 20L.02 -0.979752 13CH3CH281Br 29 659.10 3 6b7.50 3 382.38 —0.979971 CD3CH279Br 28 663.62 3 297.91 3 076.20 -0.979h60 CD3CH231Br 2b 657.79 3 276.99 3 057.95 -0.979718 CH3CD279Br 23 213.13 3 675.70 3 373.55 -0.969Sbi CHSCDzelBr 23 206.6u 3 652.78 3 35h.06 -0.969906 32 Table II. Quadrupole coupling constants (No). x' 79 a Species aa ’X aa Xbb ch Yab 42Xaa 81 7C... CH3CHZ79Br L17.2 -1h3.7 -273.5 299.8 129.9 1.1959 CHSCstlBr 3u8.9 -120.5 -228.L 227.7 107.9 CH313CH279Br L21.0 -1h6.9 ~27h.1 289.6 127.2 1.2010 CH313CH281Br 350.6 -123.0 -227.6 2u0.6 102.6 13CH3CHZ79Br L11.h -137.7 -273.7 298.3 136.0 1.1918 13CH3CH281Br 385.2 -117.2 -227.8 259.7 110.3 CD3CH279Br 397.1 -123.8 -273.3 312.8 189.u 1.1933 CDSCHzelBr 332.8 -10u.6 -228.2 259.3 123.6 CH3CDZ79Br A36.u -162.2 -27h.2 272.0 112.1 1.1963 CH3CDzelBr 36b.8 -135.2 -229.6 226.0 98.8 aThe ratio of quadrupole moments of 79Br and 81Br is 1.19707 (Appendix VII, Reference (19)). 33 Table IIIa. Observed and calculated frequencies (Mc) of ethyl bromide. Transition CH3CH27QBF CH§CH231Br Observeda Calculated Observed Calculated 2oz ‘ 303 7/2 - 9/2 21 966.25 21 966.51 21 839.67 21 839.72 5/2 - 7/2 21 966.25 21 966.30 21 839.67 21 839.63 3/2 - 5/2 21 992.22 21 992.51 21 861.55 21 861.27 1/2 - 3/2 21 991.60 21 991.66 21 860.97 21 860.89 212 ‘ 313 7/2 - 9/2 21 525.67 21 525.71 21 222.66 21 222.60 5/2 - 7/2 21 571.56 21 571.61 21 226.21 21 226.29 3/2 - 5/2 21 576.25 21 576.36 21 250.30 21 250.22 1/2 - 3/2 21 550.02 21 550.06 21 228.29 21 228.27 211 ‘ 312 7/2 - 9/2 22 388.13 22 388.09 22 258.07 22 257.91 5/2 - 7/2 22 215.50 22 215.50 22 280.80 22 280.70 3/2 - 5/2 22 209.10 22 208.07 22 275.75 22 275.71 1/2 - 3/2 22 386.72 22 386.66 22 256.63 22 256.53 505 - 514 13/2 - 13/2 28 273.63 28 273.62 28 260.03 28 260.11 11/2 - 11/2 28 225.75 28 225.83 28 236.96 28 236.99 9/2 - 9/2 28 253.58 28 253.28 28 223.22 28 223.38 7/2 - 7/2 28 280.95 28 280.95 28 266.19 28 266.27 707 ' 716 17/2 - 17/2 30 255.39 30 255.23 30 218.10 30 218.03 15/2 - 15/2 30 231.79 30 231.76 30 398.22. 30 398.36 13/2 - 13/2 30 236.59 30 236.61 30 202.28 30 202.39 11/2 — 11/2 30 260.07 30 260.07 30 222.02 30 221.91 aAs reported by Pierce and Flanagan, reference (21). 32 Table IIIb. Observed and calculated frequencies (Me) of ethyl bromide. Transition CH3CHZ79Br CHSCHzelBr Observed Calculated Observed Calculated 202 ‘ 303 7/2 - 9/2 19 110.67 19 110.69 18 992.08 18 992.13 5/2 - 7/2 19 109.86 19 109.99 18 993.67 18 993.72 3/2 - 5/2 19 135.33 19 135.30 19 012.75 19 012.80 1/2 - 3/2 19 133.69 19 133.67 19 013.66 19 013.69 212 ‘ 313 7/2 - 9/2 18 777.32 18 777.29 18 665.63 18 665.66 5/2 - 7/2 18 801.98 18 801.93 18 686.28 18 686.32 3/2 - 5/2 18 807.27 18 807.22 18 690.79 18 690.72 1/2 - 3/2 18 782.26 18 782.19 18 669.75 18 669.77 211 ' 313 7/2 - 9/2 19 221. 68 19 221.71 19 322.63 19 322.55 5/2 - 7/2 19 267.83 19 267.79 19 322.35 19 322.23 3/2 - 5/2 19 260.56 19 260.52 19 338.73 19 338.69 1/2 - 3/2 19 239.72 19 239.76 19 320.66 19 320.66 606 ‘ 615 15/2 - 15/2 23 909.10 23 909.12 23 890.79 23 890.81 13/2 - 13/2 23 881.79 23 881.77 23 868.13 23 868.12 11/2 — 11/2 23 888.35 23 888.31 23 873.61 23 873.52 9/2 - 9/2 23 915.29 23 915.38 23 896.07 23 896.02 707 ' 716 17/2 - 17/2 22 757.70 V22 757.66 23 728.53 22 728.26 15/2 - 15/2 22 731.52 22 731.56 22 706.90 22 706.65 13/2 — 13/2 22 736.91 22 736.93 22 711.31 22 711.09 11/2 - 11/2 22 762.80 22 762.71 22 731.62 ,. 22.732.28 aAs reported by Pierce and Flanagan, reference (21). 35 Table 1110. Observed and calculated frequencies (No) of ethyl bromide. Transition CH313CH279Br CH313CHzelBr Observeda Calculated Observeda Calculated 202 ' 303 7/2 — 9/2 21 718.20 21 718.02 21 590.10 21 590.15 5/2 - 7/2 21 717.79 21 590.02 3/2 - 5/2 21 722.70 21 722.22 21 612.01 21 611.96 1 2 - 3/2 21 723.35 21 611.35 212 ' 313 7/2 - 9/2 21 289.31 21 289.21 21 166.97 21 167.02 5/2 - 7/2 21 315.28 21 315.35 21 188.89 21 188.83 3/2 - 5/2 21 320.07 21 320.03 21 192.78 21 192.68 1/2 - 3/2 21 293.52 21 293.28 21 170.52 21 170.60 211 ‘ 312 7/2 - 9/2 22 129.21 22 128.92 22 017.62 22 017.58 5/2 - 7/2 22 176.82 22 176.32 22 020.21 22 020.28 3/2 - 5/2 22 170.88 22 170.22! 22 035.68 22 035.73 1/2 - 3/2 22 127.69 22 127.27 22 016.12 22 016.05 Table IIId. Observed and calculated frequencies (Mc) of ethyl bromide. Transition 13CH3CH27QBF 13CH3CH251Br Observeda Calculated Observeda Calculated 202 ' 303 . 7/2 — 9/2 21 208.90 21 208.62 21 083.35 21 083.56 5/2 - 7/2 21 208.29 21 083.27 3/2 - 5/2 21 233.95 21 232.32 21 102.27 21 105.12 1/2 - 3/2 21 233.57 21 102.53 212 ' 313 7/2 - 9/2 , 20 809.92 20 809.92 20 689.88 20 689.90 5/2 - 7/2 20 835.57 20 835.28 20 711.20 20 711.36 3/2 - 5/2 20 820.22 20 820.21 20 715.21 20 715.35 1/2 - 3/2 20 812.35 20 812.27 20 693.57 20 693.63 211 ' 312 7/2 - 9/2 21 606.77 21 605.91 21 277.98 21 277.90 5/2 - 7 2 21 632.32 21 633.61 21 501.20 21 501.19 3/2 - 5/2 21 626.26 21 625.82 21 292.67 21 292.76 1/2 -3/2 21 605.90 21 605.13 21.277-22 aAs reported by Pierce and Flanagan, Reference (21). 36 Table IIIe. Observed and calculated frequencies (Mc) of ethyl bromide. 7 9 8 1 Transition CH3SD2 Br b C2130) 2 BIT b Observed Calculated Observed Calculated 202 ' 303 7/2 — 9/2 21 128.81 21 128.61 21 002.59 21 002.55 5/2 - 7/2 21 132.01 21 131.92 21 005.19 21 005.19 3/2 - 5/2 21 157.81 21 157.72 21 026.68 21 026.77 1/2 — 3/2 21 162.20 21 161.28 21 030.18 21 029.71 212 ' 313 7/2 — 9/2 20 680.26 20 680.11 20 560.17 20 560.07 5/2 - 7/2 20 707.16 20 707.21 20 582.62 20 582.75 3/2 - 5/2 20 711.23 20 711.22 20 586.21 20 586.30 1/2 - 3/2 20 682.07 20 683.89 20 563.36 20 563.31 C 211 ' 312 7/2 - 9/2 21 590.35 21 590.12 21 259.19 21 259.18 5/2 - 7/2 21 616.26 21 826.27 21 281.07 211281.06 3/2 — 5/2 21 617.29 21 616.56 21 281.72 21 281.23 1/2.; 3/2 21 582.83 21 582.67 21 252.68 21 252.70 Sos - 514 .13/2 - 13/2 22 061.39 22 061.20 22 026.32 22 026.56 11/2 - 11/2 22 035.23 22 035.22 22 022.26 22 022.62 9/2 - 9/2 ' 22 022.53 22 022.61 22 030.70 22 030.81 7/2 - 7 2 22 068.60 22 068.57 22 052.26 22 052.61 606 ' 615 15/2 - 15/2 23 060.18 23 059.79 23 032.66 23 032.72 13/2 - 13/2 23 036.91 23 036.37 23 012.88 23 013.00 11/2 - 11/2 23 021.97 23 021.95 23 017.22 23 017.67 9/2 - 9/2 23 065.27 23 065.20 23 036.91 23 037.21 707 ‘ 716 17/2 - 17/2 22 263.26 22 263.13 22 220.70 22 220.65 15/2 - 15/2 22 220.92 22 220.86 22 201.97 22 202.02 13/2 - 13/2 22 225.82 22 225.57 22 205.82 22 205.92 11/2 - 11/2 22 267.19 22 267.15 22 222.11 22 222.11 8As reported by Pierce and Flanagan, Reference (21). bAll transitions calcualted only to second-order. The 202'303 and 211-312 transitions are calculated to third-order in Table IIIf. CThe reported values of the 5/2-7/2 and 3/2-5/2.components of both Species have been interchanged. 37 Table IIIf. Third—order corrections to the calculated Spectra (Mc) 0f CH3CD279Br and CH3CDzelBr. CH3CDZ79Br H (O+l+2) (3) (O+l+2+3) a V 0133 - Transition v calc V calc v calc )J obs y calc 202 ' 303 7/2 - 9/2 21 128.61 -- 21 128.61 21 128.81 + 0.20 5/2 - 9/2 21 131.92 - 0.23 21 131.71 21 132.01 + 0.30 3/2 - 5/2 21 157.72 + 0.01 21 157.75 21 157.81 + 0.06 1/2 - 3/2 21 161.28 + 0.22 21 161.92 21 162.20 + 0.28 b 211 ' 312 7/2 - 9/2 21 590.12 - 0.23 21 589.91 21 590.35 + 0.22 ' 5/2 - 7/2 21 616.27 + 0.01 21 616.28 21 616.26 + 0.18 3/2 - 5/2 21 616.56 + 0.22 21 617.00 21 617.29 + 0.29 1/2 - 3/2 21 582 67 -- 21 582.67 21 582.83 + 0.16 CH3CD26 181‘ 202 ‘ 303 7/2 - 9/2 21 002.55 -- 21 002.55 21 002.59 + 0.02 5/2 - 7/2 21 005.19 - 0.18 21 005.01 21 005.19 + 0.18 3/2 - 5/2 21 026.77 + 0.01 21 026.78 21 026.68 - 0.10 1/2 - 3/2 21 029.71 + 0.32 21 030.05 21 030.18 + 0.13 b 211 ‘ 312 7/2 - 9/2 21 259.18 - 0.18 21 259.00 21 259.19 + 0.19 5/2 - 7/2 21 281.06 + 0.01 21 281.07 21 281.07 0.00 3/2 - 5/2 21 281.23 + 0.32 21 281.57 21 281.72 + 0.17 1/2 - 3/2 21 252.70 -- 21 252.70 ..21 252.68 0.02 8As reported by Pierce and Flanagan, Reference (21). bThe reported values of the 5/2 - 7/2 and 3/2 - 5/2 components been inverted. have 38 Table IV. Experimental and calculated hypothetical unsplit frequencies of the ethyl bromide species. Species Species Transition Observeda Calculated Observeda Calculated CH3CHZ7 QBI‘ CH3CH28181' 2OZ - 303 21 971.25 21 971.51 21 822.01 21 823.92 212 313 21 557.52 21 557.52 21 232.57 21 232.50 2,, 3,2 22 200.73 22 200.69 22 268.30 22 268.20 50, 5,4 28 263.20 28 263.19 28 251.31 28 251.20 707 7,6 30 225.81 30 225.82 30 210.11 30 210.02 CD3CH27 QBI‘ CD3CHZBIBT 202 303 19 115.23 19 115.25 18 998.08 18 998.13 2,2 3,3 18 788.72 18 788.69 18 675.21 18 675.22 211 - 312 19 253.80 19 253.80 19 332.20 19 332.32 605 6,5 23 898.36 23 898.38 23 881.89 23 881.87 707 715 22 727339 22.727.08 22.719.22 22.719.27 CH313CH279Br CH313CH281Br 202 303 21 723.02 21 723.08 21 592.36 21 592.36 2,2 3,3 21 301.21 21 301.13 21 176.97 21 176.97 2,, 3,2 22 161.80 22 161.32 22 027.69 22 027.69 13CH3CH279Br 13CH3CH251Br 202 303 21 213.51 21 213.57 21 087.27 21 087.10 2,, 3,3 20 821.65 20 821.65 20 699.69 20 699.72 2,, 3,2 20 618.92 20 619.17 21 289.08 21 289.08 aAs reported by Pierce and Flanagan, Reference (21). 39 Table V. Transformation angles derived from quadrupole theory and experimentation. Species 18 11b - <» _ 111C CHSCH279Br 22° 15' 23° 28' 23° 26' CHSCHZBIBr 22° 13' 23° 16' 23° 23' CH313CH279Br 21° 51' 22° 27' 23° 10' CH313CH2°1Br 21° 29' 22° 23' 23° 2' 130H30H2799r 22° 21' 23° 21' 22° 1' 13CH3CH281Br 22° 39' 22° 9' 23° 22' CD3CH27QBr 22° 2' 25° 2' 25° 17' CD3CHzalBr 22° 2' 22° 55' 25° 8' CH3CD27gBr 20° 20' 21° 8' 21° 39' CH3CDzelBr 20° 38' 21° 3' 21° 22' aThe angle between the "a" inertial angle and the C-Br bond. bThe angle calculated from the experimental quadrupole coupling constants, tan 20 = ZXb/(xaa 'th)’ and 15 the angle needed to diagonalize the experimental tensor. CThe angle calculated assuming a cylindrical charge distribution about the C-Br bond. It is evaluated by using experimental quadrupole coupling constants by using the cylindrical restrictions: Zaa = 2a = (300320—1). 222 37cc 2 5a. 5b. 10. 11. 12. 13. 12. 15. 16. 17. 18. REFERENCES Schuler, H. and T. Schmidt, Z. Physik 92, 232 (1935); 98, 230 (1935). — Kellogg, J. M. B., I. I. Rabi, N. F. Ramsey, and J. R. Zacharias, Phys. Rev. 57, 677 (1920). Good, W. B., Phys. Rev; 19, 213 (1926). Bailey, B. P., R. L. Kyhl, M. W. P. Strandberg, J. H. VanVieck, and E. B. Wilson, Jr., Phys. Rev. 19, 982 (1926). Casimir, H. B. 6., On the Interaction Between Atomic Nuclei and Electrons, Teyler's Tweede Genootschap, E. F. Bohn, Haarlem,fi(1936). Reprinted by W. H. Freeman and Company, San Francisco, (1963). Nordsieck, A., Phys. Rev. 58, 310 (1920). Feld, B. T. and w. E. Lamb, Phys. Rev. §7, 15 (1925). Coles, 0. x. and w. E. Good, Phys. Rev. 19, 979 (1926). VanVieck, J. H., Phys. Rev. 11, 268 (1927). Bragg, J. H., Phys. Rev. 12, 533 (1928). Knight, 0. and B. T. Feld, Phys. Rev. 12, 352 (1928). Margenau, H., G. M. Murphy, The Mathematics of Physics and Chemisthy, D. VanNostrand Company, Inc., New York, (1956). Ramsey, N. F., Nuclear Moments, John Wiley and Sons, Inc., New York, (1953). Bragg, J. K., and S. Golden, Phys. Rev. 75, 735 (1929). Wang, S. C., Phys. Rev. 32, 223 (1929). Strandberg, M. W. P., Microwave Spectroscopy, Methuen and Company, Ltd., London, (1952). Schwendeman, R. H., J. Mol. Spect. Z, 280 (1961). Schwendeman, R. H., Technical Note: Matrix Elements of Some Products of Direction Cosines and Second-Order Quadrupole Coupling Calculations, (1962). 20 21 l9. Townes, C. H. and A. L. Schawlow, Microwave Spectroscopy, McGraw- Hill, New York, (1955). f 20. Wagner, R. 5., B. P. Dailey and N. Solimene, J. Chem. Phys. -2_6, 1593 (1957). 21 Flanagan, C. and L. Pierce, J. Chem. Phys. 38, 2963 (1963). APPENDIX The representation used in the evaluation of the matrix elements in the Appendix assumes a prolate rotor representation Ir (1.3. az, bX, ccdzy. The procedure to convert to the oblate rotor IIIr representation, (a<~>x, beDy, c<<>z) and the III1 representation (ay, bx, C2) is given below. For representation IIIr, the required changes are; 1. In the column labeled "9" in Table I, change aI-a> c, b -—> a; c -—> b. 2. In the columns labeled "d" and "d'" in Table I of Reference 18, change 0 -i> 1 and 1 -> O for all 0+ and 0' matrices. 3. Label the matrices according to Figure 7 of Reference 18. 2. Interchange K -1 and K ii in each label. For representations IIIl, all that is required is to form and label the matrices as for Ir, then interchange "a" and "c", and K_1 and K+1. 22 .H oppoa no asapoo pone“ opp op oaopaaa soapooaappaooa opp one muogapa on» «.mm one unoppoa one opp.mawbopm oomogaanompm .mNemNe mooanpoa opp mo acappom 4 .H madman .mMooap opnfi nofiponomom ii. .1 .. I W on 00 mm 00 mm 00 m pm pp p: pd, on on we op mp op mm oo. m: po om pp mp pp NH do 7 w no 4 do an 00 an 00 Pm co m: po nu pp mp op mm on pm on He op mp pp a: pm pm pm mp pp Hp do r do n on an 00 on 00 mm 00 am on so op. mm pp a: po 6: pp mp pp om op cm oo m: on an op pp pp oa dd 0 on N no t mm 00 mm 00 mm 00 no op mm on na pm pm pp NH pp an po m: on an op Pm op p: oo an pd mp pp 1 m or n do a co -m o o a m o o a pm 0 o m m+u H+a h Characteristics of the blocks of the matrices of the products of direction cosines Table I . G Multc It Post- Matri X Pre- Matri a! Matrix Block J! 33' J 0 No.9‘ 7.182 + + m. n.30 30 To To To To + + + .5 _E_AO .U .JJJJ 1-1 2-1 2-2 21212121 11111111 + 71+8292 92 To To To To ....OOEE JJJJJJJJJJJJJJJJ 10+ 10 + + .i.n7 nc_nu nc_qt 1i_qt.nu.n7.nu 72.8291 . a +E+O+O§OOEEEOOE 49.. z; z; h-l h-Z 5-1 5-2 6-1 6-2 1 2 1 a - a 7 7.. 8 8-2 9-1 1041 5.43h5‘43h5555 111111111111 7 88 10 + 88 bb bb bb bb bb bb Eb 11-1 12-1 13-1 15-2 1h-1 2121212 1111111 + + + no no nu.mU JJJJ .- 0 ..d 1h-2 1h 12 ll “4" v 0+ J+l J+l O J E+ J41 E o J 3* J+1 E J o+ J+1 o J o+ J+l o J bb bb bb bb bb bb bb 15-1 15-2 16-1 16-2 17-1 17-2 18-1 18-2 19-2 Table I. (Continued) 9. T E612: Block Pre- Post- d No. 33' J a J ' a' Matrix Matrix Mult G 20-1 1:1: f J E' J+1 3* Lu“ 15 1 3 20-2 'bb J E' J+1 E+ 12 _3: -1 h 21-1 bb J E+ J+2 E+ 11 12 -1 5 22-1 bb J o+ J+2 0+ 15 11 -1 5 25-1 bb J 0" J+2 0" 1h _1_2_ -1 5 2h-1 bb J E‘ J+2 E' 12 _1_l_1. -1 5 25-1 cc J E+ J E+ 15 15 1 2 25-2 cc J E+ J E+ .5 5+ 1 1 26-1 cc J o+ J 0+ 17 17+ 1 2 26-2 cc J 0* J 0+ 5+ 5 1 1 27-1 cc J o' J o" 18 18+ 1 2 27-2 cc J 0‘ J o' 6 6+ 1 1 28-1 cc J E" J E" 16 16+ 1 2 28-2 cc J E" J E’ 6" 6 1 1 29-1 cc J E+ J+1 E’ 5 17 1 5 29-2 cc J 13’r J+1 E‘ 15 _6_ 1 L» 50-1 cc J o+ J+1 0' 5+ 15 1 5 30-2 cc J o+ J+1 o' 17 6: 1 u 51-1 cc J o' J+1 0+ 6 16 1 3 51-2 cc J o' J+1 0+ 18 _5_ 1 11 32-1 cc J E" J+1 13+ 6" 18 1 5 52-2 cc J E' J+1 E+ 16 2: 1 h 55-1 cc J E+ J+2 3+ 15 18 1 5 511-1 cc J 0" J+2 0+ 17 16 1 5 55-1 cc J 0" J+2 o‘ 18 _1_5_ 1 5 56-1 cc J E" J+2 E’ 16 11 1 5 37-1 ab, J E+ J 0” 7 15" 1 2 37-2 ab J E J 0+ 1 h” 1 1 38-1 ab J o’ J E' 9 12+ 1 2 38-2 ab J o" J E’ 2" h -1 1 39-1 ab J E+ J+1 o' 1 12 -1 5 39-2 ab J F." J+1 0" 7 _5_ i 1* (Continued) Table I . Multc X Post- Matri X Pre- a' Matri J! Matrix Block O 68' J No.3 3h3143145555 212 -i -i -i -i J o+ .J+1 E ab ab ab ab ab ab ab ab ab ab hO-l -i 8291 J+1 E hO-2 J+1 E+ ‘J+1 E+ 0 To hl-l 0 J h1-2 J+1 0+ h2-1 h2-2 -i -i h5-1 hh-l -i 0 15-1 h6-l 718 BC 17-1 CC h7-2 BC h8-1 BC h8-2 3.45.43.43.45555 2121 111111111111 + 5857. m _1 .1 1. 7. 9..o 32 + + , . no no.mo .E .E .u _u .u .u .u JJJJJ + + + . .E #5 Av no no .JJJJJ CCCCC a a l a 8 1. 9. 1. a. 1. . . . . . o, o/ n. no 1. ulna-555 " J+1 E' ac J 0 51-2 E” J+1 o 1 ac J 52-1 10 J41 0' 'J E' SC 52-2 53-1 J E+ J+2 O J O+ J+2 E ‘0 8.0 5h-1 J+2 E+ 0 ac J 55-1 80 56-1 .1 - .1 .1 o .1 . JJJJ bc 57-1 :5.Ju.du JJJ be 57-2 be 58-1 bc 58-2 18 + be .59-1 5— ll J+1 E+ E+ be J 59-2 Table I. (Continued) a Matrix Block Pre- Post- d No. gg’ J a J ' a .' Matrix Matrix Mult G 60-1 bc J 0” J+1 o+ h 16 .1 5 60-2 bc J 0* J+1 + 15 _5_ .1 h 61-1 bc J o' J+1 0" 5+ 15 .1 3 61-2 bc J 0’ J+1 0' 11+ 6: -1 11 62-1 bc J E" J+1 E" 11* 17 1 5 62-2 bc J E' J+1 E' 12 _6_ .1 1+ 65-1 bc J E+ J+1 E' 11 1_7 -1 5 6h-1 bc J o+ J+1 o" 15 _12 -i 5 65-1 bc J o" J+1 o+ 1h 16 -i 5 66-1 bc J E" J+1 E+ 12 18 .1 5 a Number assigned for identification. b Numbers of prematrix and postmatrix refer to Table I, Ref. 1. n+ is conjugate transpose of n. 3 is matrix n for J = J+1. c Product of mult factors from Table I, Ref.~ 1. Change of sign for conjugate transpose of imaginary matrix is included. ' (1 Number of G factor is number assigned in Appendix A. Table of G factors and products of G factors APPENDIX A The G factors in the following table are defined as: 0‘ u: z (p no r4 CJIH G(J,J) ‘ §Z%:l7 G(J,J+1) = 2 1 nun (J+1)(ZJ+3) G.(J*1 J+1) = - J ' 2(J+1)(J+2) G = (J+2)(2§i;;2;:123*5)ha J C(JLJ) G(J,J+1) G(J+1,J+1) G(J+IJJ+2) o 0.50000600 0.28867513 0 0.129099uu 1 0.25000000 0.111803uo 0.08333333 0.07968191 2 0.16666667 c.06299u08 0.08333333 0.05u554u7 3 0.12500000 0.04166667 0.07500000 0.0u020151 u 0.10000000 0.03015113 0.06666667 0.03116490 5 0.08333333 0.02311251 0.05952381 0.02505880 6' 0.071u2857 0.018uu278 0.053571u3 0.020710u2 6(1) 0(2) G(3) C(u) G(5) G(J1J)3 V G(J-JJ+1)2 G(J,J)G(J,J+1) G(JJJ+1)G(J+1,J+1) G(J1J+1)G(J+1,J+2) 0.25000000 0.08333333 0.1uu33757 o 0.03726780 0.06250000 0.01250000 0.02795085 0.00931695 0.00890871 0.02777778 0.00396825’ 0.01ou9901 0.0052u951 0.00343661 0.01562500 0.00173611 0.00520833 0.00312500 0.00167506 0.01600000 0.00090909 0.00301511 0.00201008 0.00093966 0.0069uuuu 0.00053u19 0.00192604 0.0013757u 0.00057917 0.0051020u .o.oooau01u 0.0013173u 0.00098801 0.00038196 Appendix B Numerical Values of C(I,J,J',F) for I = 5/2 and I = 5/2 for J g 6. 'The values tabulated here are: [5c(c+1) - h I(;+1)J(J+1)]2 [8 1(2 I-1)J(2 J-1)]2 c(I,J,J,E) = where c = F(F+l) - I(I+1) - J(J+1) C(I,J,J+l,F) = [F(F+l) - I(I+1) - J(J+2)]2(I+J+F+2)(I-J+F) J -I+F+l (J+1-F+l I 2I-l J 2J+l C(I,J,J+2,F) = (I+J+F+2)(I+J+E45)(I-J+E+1)(I-J+F) x QJ-I+F+l)(J-I+F+2;(I+J-F+l)(I+J-F+22 [l6 I(2 I-l 2 J+1 J+1 r4 C) It; Ifi 5/2 5/2 3/2 1/2 7/2 5/2 5/2 1/2 9/2 7/2 5/2 5/2 11/2 9/2 7/2 5/2 15/2 11/2 9/2 7/2 15/2 15/2 11/2 9/2 OOOOOOOOOOOOOOOOOOOOHI—‘OO J,J .000000 .062500 .000000 .562500 .062500 .390625 .000000 .765625 .062500 .250000 .022500 .560000 .062500 . 1911106 .05858h .2h1151 .062500 .160000 .oh69uh .187778 .062500 .1ho625 .051655 .158187 5/2 L111 .oooooo .1h5855 .250000 .512500 .187500 .100000 .550000 0000000 .229167 0.05952h 0.57202h 0 0.270855 0.0h1667 0.h01oh2 0.512500 0.051818 o.h5h518 0.55h16¢ 0.0256h1 0.h70085 J,J+2 0.312500 0-583353 0.218750 0°937500 0.500000 1.575000 0-859375 1-895833 1. 500000 2.500000 1.822917 5.187500 2.h28571 IC-i HO I'd 5/2 7/2 5/2 3/2 9/2 7/2 5/2 3/2 1/2 11/2 9/2 7/2 5/2 5/2 1/2 15/2 11/2 9/2 7/2 5/2 3/2 15/2 15/2 11/2 9/2 7/2 5/2 17/2 15/2 15/2 11/2 9/2 7/2 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO .062500 .6h0000 .h90000 . 062500 .180625 .062500 .062500 .u90000 .062500 .090000 .090000 .ooohoo .108900 .560000 .062500 .056h06 .087870 .010727 .05858h .2h1151 .062500 .ohoooo .082178 .019600 .016900 .187778 .062500 .050625 .076880 .025510 .00826h .158187 OOOOOOOOOOOOO 00000 00000 00000 .112500 .060000 .210000 .137500 .002500 .121500 .169000 .070000 .162500 .001h29 .078571 .178571 -153929 .187500 .008750 .056875 .180625 .171875 .212500 .018182 .0hh182 .185782 .202682 .257500 .028269 .055962 .188h62 .250769 OOHO 0000 0000 00000000 c> F’ #4 F“ .175000 .281250 .202500 .052500 .h12500 .h12500 .202500 .ohsooo .568750 .658125 -397768 .122768 .750000 .9h5000 .657000 .227500 .956250 .275000 .920h55 .557955 .182500 .6h8929 .2h8579 .513736 APPENDIX C Table of a(J,q), b(J,q), g(J,q), and h(J,q) The elements in the following table are defined as: a(J,q) \/[(J+1) + qu — qJE - 1 b(J.q) g(J,q) = \/J(J+1) - q(q+1) h(J,q) = 2\/(J+1)2 - q2 1 g ad , 3.0+ - J+1,0‘ J+1/2 x J+2/2 (89.22) J,0‘ - J+1,0+ J+1/2 x J+2/2 (2a+,8a) J,0‘ - J+1,0+ J+1/2 xJ+2/2 (93:22f) J,E' - J+1,E+ J/2 x J+5/2 (la+,7a) J,E‘ - J+1,E+ J/2 x J+3/2 (108416?) J,E+ - J+2,E+ J+2/2 x.J+4/2 (73212) J,0+ - J+2,0+ J+1/2 x J+3/2 (86,86) 11-1, G(J,J+1)G(J+1,J+2) J,0‘ - J+2,0‘ Same as Mh = 10-1 J+1/2 x J+5/2 (9a,23) 12-1, G(J,J+1)G(J+1;J+2) 'h(J,2)h(J;2) 0 0 ‘ J,E‘ - J+2,E' 0 . h(J,4)h(J{4) .0 J/2 x J+2/2 _ o 0 - h(J,6)h(J{6)_ ‘(106,1gg) bb-ty'pe 15-1, G(J,J+1)G(J,J+l) 2.8(J,0)2 J" 2,;(J,0)b.(J,1) 0 " .féa(J,0)u(J,1) b(J,1)2+,a(J,2)2 a(J,2)b(J,3) L 0 a(J,2)b(J,3) b(J,5)2+a(J,4)é_ J,E+ - J,E+ J+2/2 x J+2/2 (116.1ib+) 15-2, G(J,J)G(J,J) ’ 28(J:0)2 a¢§é(J,o)g(J,1) o ‘ 4{§S(J:O)S(J:l) 8(J.1)2+8(J.2)2 -8(J,2)8(J,3) _ o -8(J.2)8(J.5) 8(J53)2+8(J:h)2d J,E+ - J,E+ J+2/2 x J+2/2 (5b.5b+) 14-1, G(J,J+1)G(J,J+1) "26(J,0)2+6(J,1)2 a(J,1)b(J,2) 0 ‘ a(J,l)b(J,2) b(J,2)2+a(J,5)2 6(J,5)b(J,4) 0 a(J,5)b(J,4) b(J,4)2+a(J,5)2 111-1 (Continued) J ,O+ - Jay-0+ J+1/2 x J+1/2 (15b,15b+) 14-2, G(J,J)G(J~;J) ' 8(J:1)2 -8(J,1)8(J.2) o 1 -8(J,1)8(J.2) 8(J.2)2+8(J.5)2 ,-g(J,5)g(J,h) _ o -g(J,3)g(J,h)2 8(J,h)28(J,5)?_ J,o+ - J,0+ J+1/2 x J+1/2 (415,415+) 15-1, G(J,J+1)G(J,J+l) V a(J,1)2 a(J,1)b(J,2) 0 " a(J.1)b(J,2) b(J,2)2+a(J.3)2 a(J.5)b(J.h) _ o a(J.5)b(J,h) b(J,h)2+a(J.5)2_ J,O' - J,0‘ J+1/2 x J+1/2 . (146, 1415’“) 15-2, G(J,J)G(J,J) “28(J,o)2+8(J.1)2 -8(J,1)8(J,2) 0 ‘ -8(J:1)8(J:2) g2+g(J,3)2 60,980.10 _ O -g5~c(J.s)21 15-2 (Continued) J,0‘ - J,0' 16‘1, 16‘2, 17‘1, J+1/2 x J+1/2 (5b+.5b) G(J,J+1)G(J,J+l) 0b(J,1)2+a(J,2)2 a(J,2)b(J,5) 0 a(J.2)b(J.3) b(J,5)2+a(J.h)2 a(J,h)b(J.5) _ 0 6(J,4)b(J,5) b(J,5)2+a(J,6)2_ J,E‘ - J,E‘ J/2 x J/2 (1213, 1213+) G(J,J)G(J,J) "8(J,1)2+8(J,2)2 -g(J,h)g(J.5) o “" -8(J.2)8(J,5) 8(J.5)2+8(J.h)2 -8(J,h)8(J,5) L 0 —8(J,h)g(J,5) g(J.5)2g(J.6)2_ J,E' - J,E‘ J/2 x J/2 (46+,4b) G(J,J)G(J,J+l); J,E+ - J+1,E' J+2/2 x J+1/2 (56,146) 'V§E(J,0)a(J,1) 0 T ‘8(J:1)8(J:1)+8(J:2)b(J:2) 8(J.2)a(J,3) -8(J.5)b(J;2) -8(J,5)a(J,3)+8(J,h)b(J,h) L 0 -8(J,5)b(J,h) 17-2, -G(J,J+1)G(J+1,J+l); J,E+ -J'+1,E" J+2/2xJ+1/2 (116,43) ' -{§a(J.o)s(J’.1) o " -b(J,1)g(J§1)+a(J.2)8(J’. 2) «(J,2)8(J’.5) bg(a’.2) 4081603460,106014) __ 0 b(J,-5)8(th) ' 1 18-1, G(J,5)G(J,J+l) x (—1); J,0+ - ~J+1,0‘ J+1/2 x J+2/2 (415,126) "-b(J,1)g(J,1) -8(J,l)a(J,2) 0 b(J,1)s(J.2) a(J,2)8(.J,2)-b(J.5)8(J.5) -8(J.5)a(J,h) 1. 0 _b(J,3)g(J,4) 'a(J,4)g(J,4)-b(J, 5)8(J,:52_ 18-2, G(J,J+1)G(J+1,J+l) 3: (+1); J,O+-J+l,0' J+1/2 x J+2/2 (1516,53) '2b(J,O)g(J:O)-a(J,1)g(J:l) a(J;1)8(J.’2)- I ‘ 0 " -b(J,2)8(J51) b(J.2)8(J12)-a(J,5)8(J’,5) a(J,5)8(~J,’h) o .b(J,4)g(J’, 5) b(J,h)g(J/:l*)-‘8.(J9 5)S(J:5)_ h- 19-1, G(J,J)G(J,J+1) x (-1); J,0" - J+1,0+ J+1/2 x J+2/2 (3b+,llb) ’2a(.:,o)g(cr-.o)-b-6(J.3)8(Jf5) a(J.5)8(J’,h) _ o 40.06015) 6(J.4)g-a<4,s)g .(J,2)a(ai;) o b(er)b(J.3) -a(Jf;)4(J,;)-a(J,4)4(Jf4) a(J.4)a(st) o 40’.4)40.s) -40:s)40.s)-a0.6)4016)_ ab-type 37-1, ”(54(J.o)h(J.o) a(J,1)h(J,2) 0 37'21 0 'h8(J)l) L 0 353-1, 'B(J,l)h(J,1) 3(J92)h(J23) 0 58-2, 1G(J,J+1)G(J,J+l); iG(J,J)G(J,J); 10(J,J+1)G(J,J+1); 'iG(J:J)G(J4J); J, 0 b(J,2)h(J,2) 3(J15)h(J)h) 0 0 43(J,2) 0 '88(J93) 88(J2h) J; 0 b(J.5)h(J.3) 3(J9h)h(J95) J,0' J'-,E+ - J,O + E+ - J,O J+2/2 x J+1/2 (76,154+) 0 1 0 b(J,4)h(J,4)_ + J+2/2 x J+1/2 (16,4b+) "I 0 -- J,E J+1/2 x.J/2 (9a,12b+) o ‘ 0 b4