ANALYSiS OF THE $FECTRA OF P‘LAW ASYMMETRKC MOLECULES, WITH AP’PLECAHQN TO HYDROGEN SERENEDE Thosls {or the Deqroe of DH. D. MICHEGAN STATE UNIVERSITY Ronald Ames Hill 1963 ---- mWWW LIBRARY Michigan State University This is to certify that the thesis entitled Analysis of the Spectra of Planar Asymmetric Molecules, with Application to Hydrogen Selenide presented by Ronald Ames Hill has been accepted towards fulfillment % of the requirements for ALL degree in‘ Phys ic s . 7 zfl‘f/f/é Major professor ABSTRACT ANALYSIS OF THE SPECTRA OF PLANAR ASYMMETRIC MOLECULES, WITH APPLICATION TO HYDROGEN SELENIDE by Ronald Ames Hill Specific analytical expressions for the centrifugal distortion energy, and for the sum rules, are derived for the planar asymmetric molecule in terms of the four centri- fugal distortion constants Taaaa’ Tbbbb3. Taabb’ and Tababo These equations are applied in a least squares analysis, by digital computer, of the ground state combination differences of the near infrared absorption bands 2 pl , 2/1 + L/3, 80Se and 2 U l + v 2 , and b’l + y 2+ 2/3 for H2 H27BSe. Due to a Coriolis interaction between the states 2 ”Ll and 1’1 + 1/3 and between the states 2 1/1 + 1/2 and V11 + LIE + v 3, the first order energy expression is not able to fit either set of upper state rotational energy levels. However, certain combinations of the upper state molecular constants, and the band centers were obtained for 2 El , and V]_+- 2/3 through the use of coupled sum rules for the interacting submatrices, E+ with E‘ and 0+ with O“ for the two states. The analysis of the two states was completed, and a value of the perturbation coefficient GC was calculated, through the diagonalization of three, 2 by 2, coupled sub- matrices. Ronald Ames Hill Band center separations for the various isotopic species, H2808e - H2828e, H2788e - HgaoSe, H277Se - H2 76 80 Se - H2 Se are obtained from the spectra for these 808e, and H2 four absorption bands° Copyright by RONALD AMES HILL 1964 ANALYSIS OF THE SPECTRA OF PLANAR ASYMMETRIC MOLECULES, WITH APPLICATION TO HYDROGEN SELENIDE By Ronald Ames Hill A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1963 ACKNOWLEDGMENT I wish to express my sincere appreciation to Professor T. H. Edwards for his help and encouragement throughout the course of this research and for his support through a special Graduate Research Assistantship financed from a grant from The National Science Foundation. I am also grateful to The National Science Foundation for Summer Fellowships in 1961 and 1962. I especially wish to thank Professor R. H. Schwendeman of the Department of Chemistry for several helpful discus- sions, for the use of several I.B.M. card decks and the use of several tables of Coefficients. I wish to thank Professor C. D. Hause, Professor P. M. Parker, Dr. J. w. Boyd, and Mr. w. E. Blass for their help and advice; and Miss Mary E. Douglas for her help in measuring many of the lines, for checking the derivations of the various equations, and for preparing many of the figures for this thesis. I want to thank the staff of the computer laboratory for their help. I am deeply indebted to my wife for her encouragement and support throughout the course of this work. 11 TABLE OF CONTENTS ACKNOWLEDGMENT LIST OF TABLES LIST OF FIGURES LIST OF APPENDIXES . . . . . . . INTRODUCTION L Chapter I. THEORY OF THE ASYMMETRIC MOLECULE . The Rigid Rotor Hamiltonian . . Symmetry PrOperties of the Energy Ievels. Trace of the Rigid Rotor Hamiltonian . . First Order Centrifugal Distortion Terms. Trace of the Hamiltonian, Including First Order Centrifugal Distortion Terms The Rotation-Vibration Hamiltonian. The Asymmetric Rotor Selection Rules II. THEORY OF THE PLANAR ASYMMETRIC MOLECULE. Effect of Planarity on the Rigid Rotor Hamiltonian. . . Effect of Planarity on the Centrifugal Distortion Terms . . Effect of Planarity on the Trace of the Hamiltonian. . Simplification of the Selection Rules for Planar Triatomic Molecules. . III. THE ANALYSIS SCHEME. Computer Programs . . IV. THE NEAR INFRARED SPECTRUM OF HQSe. Preparation of H Se. . The Near Infrare Absorption Spectrum of H288 iii Page ii vii viii ix l" 13 13 2O 21 2A 29 3O 3O 39 45 47 as 59 62 66 Chapter V. ANALYSIS OF THE NEAR INFRARED SPECTRA. Symmetric Rotor Approximation . The 5600 cm 1 region. . - The 4600 cm 1 region. . . . The Asymmetric Rotor Analysis of the Ground State . The 5600 cm 1 region. . . . The 4600 cm 1 region. . . . Simultaneous analysis of the ground st§8e combinat on differences of Se and H27 Se . Asymmetric Rotor Analysis of the Upper State Energy Levels Evidence of a Coriolis Interaction. The 6800 cm 1 Region . . . . . . IsotOpic Species of H28e . . . . . VI. SUMMARY. REFERENCES. APPENDIXES. iv Page 73 73 82 BA 84 86 87 92 93 118 119 125 127 131 Table II. III. IV. VI. VII. VIII. IX. XI. XII. XIII. XIV. XVI. XVII. LIST OF TABLES The Six Representations of the Reduced Energy Matrix . . . . . . . V(a,b,c) Character Table Correlation of Representations of V(a,b,c) with V(x,y,z) . . . . . . . . Symmetry Classification of the Submatrices EH’, E‘,(Th 0’ . . . . . . . Symmetry Classification of RI, E‘, 0+, 0' by the Parity of K-l, K+1 . . . . . Selection Rules on K_l, K+l Overall Selection Rule AK 2 AK-1 + AK+1 Molecular Constants of H288 Symmetry Species for the Point Group 02v 788a 0 Experimental Conditions Ground State Constants of H2808e and H2 Rotational Constants of H2808e for the States 2 V1, V1 + V3, 2 V 1+ V‘3, and yl+y2+y3o o 0 Comparison of Observed and Calculated Coupled Sums for 2 V1 and U1 + v3 ”Coupled” Rotational Constants for the States 22/1 and V1 + b’3 . . Perturbed Energy Expressions for J = A Band Center Separations for the Istopic Species of H28e . . Comparison of Calculated and Observed Band Center Separations . . . . . Page 10 IO 12 26 26 60 63 69 89 94 110 111 113 I21 124 Table Page XVIII. Calc lated Ground State Energy Levels of H2 OSe . . . . . . . . . . . . 13A WO : Rigid Rotor Energy Levels from E(K ) W1 : With Distortion, No XIX. Calculated Ground State Energy Levels of HEBOSe . . . . . . . . . . . 139 WO : Rigid Rotor Energy Levels from cn W1 : With Distortion, >2 >frgm theucn w2 : With Distortion, (<8?Z <13Z >r1 vi Figure 10. ll. l2. 13. 14. 15. LIST OF FIGURES Page Energy Level Diagram . 14 Geometry of HESe . 62 Symmetry Classification of the Rotations of HgSe. . . . . . . . . . . . 64 Normal Modes of H288. and Symmetry Classifi- cation . . . . . 64 Apparatus forthe Production of H28e 65 Absorption Spectra of H28e near 6800, 5600, and 4600 cm 1.. . . . . . . 68 Isotopic Absorption Lines of HQSe. 72 Calculated Spectra for the Symmetric Rigid Rotor . . . . . . 75 Graph of Upper State Combination Differences 79 Expanded Graph of Upper State Combination Differences . . . . . . . 81 Elements of lay-when Factored into the BI, E‘, 0+ , 0‘ Submatrices . . . . . 99 Elements of (El+ + E3’) Submatrix Including the Perturbation Coefficient, GC . . . 101 Elements of the (01+ + 0 Submatrix Including the Perturbation Coefgicient,G GC . . 102 Energy Level Diagram for J = 4. . . 114 Grap of AV for the ”zero”-series lines of I20 H2 0Se and H2788e vs. Frequency for 211 vii Appendix I. II. III. IV. LIST OF APPENDIXES 0n the Approximations ()2 = and w(b) = K+12 + Z cnboncn‘ [(2 0.79241 gnments and Observed Frequencies Line Asgi OfHQOSEfOF2V1+ ye and V1+V2+ V3 Line As§6gnments and Observed Frequencies Of H2 SE for 2V1 and V]. + U3 Comparison of Observed and Calculated Ground State Combination Differences . . viii 131 144 156 174 INTRODUCTION A polyatomic molecule is an example of the many body prOblem, i.e., the number of particles is 2'3. As such, there is no exact solution to this problem in either classical or quantum mechanics so that one can never obtain exactly the motions or the energies of the particles. Thus the approach taken is to begin with simplified approximate descriptions of the molecules for which we can obtain solu- tions, and then use perturbation methods to improve upon these solutions. The starting assumptions used in classical mechanics are (a) that vibrational motions occur only as small oscillations and thus the theory of normal vibrations may be applied and (b) we separate at the center of mass, so that the simplest description of the rotational motion of the rigid body is in terms of its moment of inertia ellipsoid. Those molecules with principal moments Ia = Ib = IC have a spherical moment of inertia ellipsoid and are called spherical- tOp molecules (e.g. CH4, SiHu, and 0014). Those molecules with two and only two equal principal moments have an inertia ellipsoid which is an ellipse of revolution and are called symmetric-top or axially-symmetric molecules. These are further classified qualitatively as prolate if Ia‘< Ib = IC ix (e.g. CH3Cl, CH3D, etc.) or oblate if Ia = Ib < Ic(e.s. CD3H or the planar molecule BF3). A Special type of prolate symmetric molecule is the linear molecule (e.g. 002, N20, etc.). Finally, those molecules with principal moments Iai< Ib> Evib + Erot so that the electronic and rotation—vibration problems may be separated, b. that the theory of normal vibrations may be applied, and c. by separation at the center of mass, the trans— lational motion is separated from the rotational motion of the rigid body which is described in terms of its moment of inertia ellipsoid. In the case of axially-symmetric molecules, explicit expressions for the rotational energy and wave functions may be found. For nearly symmetric molecules, Ia< Ibg IC or Ia g’Ib < IC, approximate solutions may be obtained in terms of the solutions of the corresponding symmetric molecules. Also, the absorption spectra of the nearly symmetric molecule will have, for the most part, the regular appearance of the spectra of the corresponding symmetric molecule. In the more general case, however, the absorption spectra of asymmetric molecules have been found to be very difficult to analyze. This is due in part to the apparent irregular structure of the Spectra, to the lack of a closed expression for the rotational energy levels, and to certain perturbations which can occur between particular vibrational states. Our purpose in Chapter I will be to record and discuss the results of asymmetric rotor theory as available from the literature. In Chapter II, we obtain various new expres- sions pertaining to the Special case of the planar asymmetric molecule with particular emphasis on obtaining explicit expressions which may be used directly in an analysis (e.g. by the method of least squares) of the spectra of planar asymmetric molecules. In the subsequent chapters these expressions are employed in an attempt to analyze the near infrared absorption spectra of HQSe. xi CHAPTER I THEORY OF THE ASYMMETRIC MOLECULE The Rigid Rotor Hamiltonian The rotational Hamiltonian for a rigid asymmetric rotor is given by the expression 2 2 2 X0 - Pa /2Ia + Pb /2Ib + PC /2IC (1) where Pa, Pb, and PC are the components of the total ang- ular momentum along the body fixed principal axes a, b, and c and Iag< Ib‘< IC are the principal moments of inertia. The 33 ways a right-handed cartesian system of coordinates (x,y, 2) may be attached to the rotor will be considered later; however, the origin is placed at the center of mass and the z axis is normally chosen as that axis which becomes unique in the nearest symmetric rotor limit. Since no two principal moments of inertia are equal, Pz does not commute with lfo, i.e., )(o is nondiagonal if P2 is diagonal. The result is that no closed expression has been found which will represent all the energy eigenvalues and thus one must use approximation or perturbation methods. The well-known commutation rules for the Pi's are Py Pz - Ezry: - .ian Psz-PxPz: -in1=y, 2 These matrix equations have the solution (J,KIPX|J,KI1)= :itR/2)[(J;K)(JiK+1fl L/E (J,K'Py'J,Ki1)= (h/2)[(J;K)(J:K+l)] 1/2 (3) (J,K[PZ[J,K) = hK where J is the total angular momentum quantum number and K is the projection of J along the z axis of the moment of inertia ellipsoid, so that J 2 K. We choose that repre— sentation which simultaneously diagonalizes PZ and P2 = PX2 + Py2 1+ P22. The phase factor chosen is such that Py is real and positive and PX is imaginary. The squares of the components of the angular momenta may be obtained from Eqs. (3): J’K'Py2 (J K+2) 2/4)[(J+K)(J+K+1)(JIK-l)(JiK+2)] 1/2 J,K'P:2 (J,Ki2) J KIPf lJ K+2) (‘h -( (n 2/2)[J (J+1)—K2] ( n J,KIPy2IJ,K) J KIPX 2|J, K) 2K2 (A) ll ( ( (J,KIPX2'J,K) ( ( J,K)P22fJ,K) 2 and (J7,K'P2'J,K) =71 J(J+1)- If we define the reciprocal moments of inertia, A = h/47TcIa -1 . etc., where A is in units of cm , the matrix elements of the Hamiltonian may be written (for this example, let a=x, b=y, and c=z) (J3K19Q1J3K) = 5;? J(J+1)+(c-5§§)K2 (5) (J.foglJ,K:2)=- A‘B[}J:K)(J:K+i)(J:K_i)(J:K+2y]1/2 (6) Thus we see that the rigid rotor Hamiltonian is not diagonal in K, i.e., there are matrix elements in its second off— diagonal. For the symmetric rotor with A22 B, these off-diagonal terms vanish and 3(0 becomes diagonal, reducing to (J,nyO]J,K) = B J(J+l)+(C-B)K2, (7) the usual expression for the symmetric rotor. For nearly symmetric rotors, A E B and the effects of the off-diagonal elements on the energy eigenvalues are small. The solution to the symmetric rotor problem is then used as a zero-order solution, and perturbation methods are used to approximate the energies. In this case, the rigid rotor energies are given by 1,2a WO =-£§E J(J+1)+(C- ééglnu (8) (- Where the u; are the Wang energies, which may be written as an expansion w: K2+ Z Cnbon (9) n A- B in the near symmetric rotor limit. bong:K:g iS the oblate case asymmetry parameter. The coefficients cr1 have been 2b tabulated to n = 5 for J = 0 to 12 in Townes and Schawlow and for J = 0 to 40 by Schwendeman3 and to n : 7 for J = 0 A to 50 by Davis and Beam.z1L For the corresponding prolate example, A and C should be interchanged in all the foregoing C-B dA—C-B ° Another method of obtaining a rigid rotor energy expressions as then a = z, and b — expression, which is particularly useful for greater asym- R f O O O 0 O metries, is that due to Ray./’ ConSIder the expreSSIon EQ7A+pITB+pITC+p) = 2 2 2 2 (10) [(O'A +/O)Pa +(O”B+/0)Pb +(0'C+/D)PC 1 /‘h where 0': .K35 and I): (A+C)/(C—A), so 2B-A—C = 11 JA-l-Pr-l, 0B+10="7_\TC—__ —K < ) and (TO +,0 = -l where K , called Ray's asymmetry parameter, has the range — ISI< S 1. A prolate symmetric rotor is represented by K = - 1 and an oblate symmetric rotor by K = +1. By rearranging Eq. (10), 2 2 2 2 2 2 2 E(1,K,—1)=[O‘(APa +BPb +CPC )+p(Pa +Pb +PC )]/h (1'2) =axo+p J(J+1). Thus the Hamiltonian is written A—C lfo = Egg J(J+1)+ —2‘ EQ<) (13) where E (K') = E(1, K , -1) is called the reduced energy matrix. ‘The eigenvalues of E(K ) satisfy the relation U") Em) = E, («L (14) where T' is a pseudo quantum number taking on (2J + 1) values from —J to +J, so that once the reduced energies are calculated for values of K’ from 0 to l, the energy levels for any asymmetric rotor may be determined from them. Eq. (13) is the most commonly used expression for the rigid rotor energy. The matrix elements of E(/< ) are 9 (J,K| E(K’)IJ,K) = E‘[J (J+l)-K‘] + GK2 (J,K 'E(K')'J,K + 2) = (J,K + 2 'E(K')I JyK) =(H/2)[J(J+I)—(K+i)(x+2)]1/2 [J(J+l)-(K+1)K]l/2 (12) where the values ony G-and H depend upon the 33 possible ways of identifying Pa, Pb, PC with PX, Py, PZ. These are given in Table I where the superscript r signifies right- handed and E signifies left—handed (a,b,c) axes. All the. types of coordinates give the same secular equations and energy levels. However, each is best suited to a particular range of /< . The only difference produced by using a right— handed rather than a left-handed system or vice—versa is a change in the sign of H in Eq. (15). The sign of H has no Effect on the roots of the secular equation, but is used to determine the symmetry prOperties of the energy levels. For the limiting oblate rotor, A and B are equal, or Z . ‘ _ f<= l and with z = c (type IIIr coordinates), H _ O Siva}: Siva} T. H Stem} Stem}: m T T H a e SILK} 2.0%} o o 9.2va :63} m o o n n w m N m n o a Q o z n m w o o 9 x QHHH LHHH eHH LHH QH EH ease xflmumz hmgmcm poozpom one mo mcoflpmpcomopgom xflm OLE .H oHQwB 7 and the reduced energy matrix becomes diagonal with values (J, K] E(K')] J, K) = J (J+i) - 2K2 and thus w, = B J(J+1) + (C-B) K2, the usual expression for an oblate symmetric rotor. For the limiting prolate rotor, B and C are equal, K'= ~ 1 and or I with z = a (type Ir coordinates), the reduced energy matrix again becomes diagonal with values (J, K 'E(K’)'J,K) = — Jm h poo b co>m h 660 h co>m h Ugo h Co>o. h poo h Cm>o b coo h co>m QHHH EHHH QHH AHH eH AH a+x 2-x co seated on» so 0 . o . m . m co coaoooaaammofio stoogesm .> ofioae 13 and the highest JJ. The labeling of the levels for J = 0, l, and 2 is given in Figure 1. Trace of the Rigid Rotor Hamiltonian fifi. The trace of the Hamiltonian, J Tr R0: 2 (J,K'}(’O,J,K) (16) —J remains a constant under any unitary transformation. Since this is true in any representation, from Eq. (5) J , A B A B 2 Trig: §[4§—J(J+1)+(c_—g_) K] = 1/3 (A+B+C)J(J+l)(2J+l), (17) which is known as Meckels Sum Rule.ll This expression is useful for checking the calculation of rigid rotor energies. In addition, the individual submatrices of types IT, I2 . III3 with a given symmetry have the same roots and hence the same trace. These:mib-&nnru1es are given in Table X of reference 6. First Order Centrifugal Distortion Terms A first-order perturbation treatment of the centri- fugal distortion correction to the energy levels has been worked out by Kivelson and Wilson.12 The Hamiltonian is written,13 Fig.1. Energy Level Diagram J K " J1. JK-IKH K“ #2 .——-———-—*'220 0‘ 2 —-O #22/ 02 l f 0*, I 4 ll / l0 \ '0 \lll‘ l I ___________, O ’_____.....——-—- Ll / Cl ._0 OO 000 O—— -l, Prolote K _. Ohmic-2:4 15 where I 2 ~ 2 , 2 HO a'PZ “tblPX +y Py (18) and ){I 1 R174: Twig 1115/1ng 3C) is the Hamiltonian of the rigid rotor with d' , etc. = .A, B, or C depending upon the representation chosen. if; contains the centrifugal distortion terms, with the taus representing a set of distortion constants. It turns out in many cases that due to symmetry, most of the 81 taus vanish or are related to each other. For molecules which in crystallographic notation would have orthorhombic sym— metry, only 9 distinct taus are left.14 For the monoclinic and triclinic groups there are 13 and 21 independent taus, respectively; however, these additional quantities affect 12, 121 the energy levels only in second order, and so will be neglected here. Employing the commutation relations [Eq. (23 Kivelson and Wilson12 put the Hamiltonian into the slightly different form x: x .11, M Q "U R) 1% rd >< [\D + \< "U [\D oH KO with )(O and w, = (1/4)E1/ Mm, f, , where 4 _ 1 _ ; T h /4 a _ a' +(3Tnyy 2szzx 2 yzyz) ,, 4 18:3! +(3Tyzyz ’2Txyxy -2szzx) h fli 16 = t 2 ‘ _ 4 y y + (DTZXZX 2Txyxy —2Txyxy) h /)I (20) and 4 I _ T zzzz _ T2222 4 l .. T xxxx — Txxxx h 4 l = h T yyyy Tyyyy T' = (T + 27’ ) h4 xxyy xxyy xyxy (21) T' = (r + 2? ) n4 yyzz yyzz yzyz 1+ l _ T zzxx _ (Tzzxx + gtzxzx) Thus Txyxy, szzx: and Tyzyz have been ”absorbed into” the rotational constants. These "corrected” constants are the ones which are normally determined in practice (vibrational corrections will be considered later), however, the (2’ , [3’ , and 7” define the moments of inertia of the rigid rotor considered in Eq. (1). We note that 'Kl depends on Only 6 of the T‘s, i.e., there are only 6 linearly indepen— dent distortion constants (to first—order) for the general asymmetric molecule. The first order energy expression as derived by Kivelson and Wilson13 is written 2 2 2 , w = WO+A wO +A2WOJ(J+1)+A3J (J+1) 1 A 1 P 2 A

+A w

+ 4J(J+ )< Z >+ 5 Z 6 0 Z (22) 17 where W0 is the rigid rotor energy, < PZ2 > and < qu > are the average values of P22 and PZA, respectively, for the given state, and the Ai’s are functions of the principal moments of inertia and the taus. An equivalent form of Eq. (22) is obtained by writing the Ai's in terms of the centri— fugal diStOI’thl’l COI’ISJCal’lI/S$6 DJ, DJK’ DK’ R5, R6, and 8}},12’15 viz., , 2 2 a 1 2 w = NO — (DJ+2R6)J (J+1) -(DJK— R6)J(J+ )< PZ > 2 P I — 4R <8 ‘ (DK+ R6) < z > 6 1 .28J82 Jtn1)—lm553 (23) where 2 2 51=(2w--w)/b 82=(<222>—w)/b 2 4 b 53=(w—)/ (2A) b = (738) / (20-8-7) *Hereafter DJ, DJK, DK, R5, R6, and SJ W111 be referred to as the six D’s. 18 and a; is the Wang energy. For slightly asymmetric molecules the expansion for u; in terms of the asymmetry parameter b [Eq. (9H is very useful. In the general case, however, from Eqs. (4) and (19) we have WO = (1/2)(B+)/)J(J+l)+ [a- (1/2)(B+7)]w and from Eq. (13) A+C A—C N0 = ‘2‘ J(J+l)+ j—ETM). By equating Eqs. (25) and (26) one arrives at a}: (A+C—13-—7”) J (J+1) + (A—C) ET(K’)_ 20—5-7 /“\ R) \T‘: v (26) /\ R) .\J and thus u; may also be obtained from the E(/< ) tables. The relations between the set ofsix D's and nine first 12 order T's are as follows: 19 4 DJ = (-1/32) (3 T + 3 T 2 T xxxx yyyy + xxyy + 4 Ticyxy)?l D = D - 1 4 T - K J ( / ) ( zzzz Tzzxx - Tyyzz )‘fii - 2 T xzxz yzyz = _ _ _ 4 DJK DJ DK (1/4) Tzzzz T] R5 = (’1/32) [ T _ T - xxxx yyyy 2( Txxzz 2 szxz) 4 + 2(1-yyzz + 2 Tyzyz)] h R = 1 64 T T - T 6 ( / )[ xxxx + yyyy 2( xxyy 2ITXYXY SJ: (”l/16) (T - T xxxx yyyy 20 The values of < PZ2 > may be obtained from awO/ga .;16 however, in general, numerical values of < PZ > must be calculated separately by continued fractions or by some other approximate method, for various values of K’.12 Values of < Pzi1L > have been tabulated in increments of 0.1 in /< from J = l to 12 by Professor Schwendeman.l7 We note that in the symmetric rotor limit, b, R5, R6, and ESJ all = 0,12 er: < P22 > = K2, < qu > 2: K4 and the 81, 52, and 83 defined in Eq. (21L) remain finite (these 8 's were chosen with this in mind). Thus Eq. (23) reduces directly to l U KC) v 2 2 4 w = wO — DJJ (J+l)2 — DJK J (J+l)K - DKK , ( the usual energy expression to this order for symmetric— tOp molecules. Trace of the Hamiltonian, Including First Order gentrifugal Distortion Terms The sub—sumrmles, i.e., the trace of the submatrices E+, E‘, 0+, and 0‘, including centrifugal distortion effects have only recently been derived.18 There are eight expres— sions representing the sums for odd J and even J for the four submatrices. These sums depend only on the inverse principal moments of inertia A, B, and C (the molecular con- stants determined in practice, e.g., from analysis of Spectra, see Eq. (20)), on the total angular momentum quantum number J, and on the six D's. Since the sum rules are independent 21 of the methods used in obtaining the individual energy levels, i.e., no dependence on ET(K') or < PZ4 > , such rules provide an excellent check on the values of the energy levels as obtained from Eqs. (2$ and (2&). The Rotation-Vibration Hamiltonian The rotation-vibration Hamiltonianlg’20 for the asym— metric molecule is of the form 3’? = X vib + X rot+ Xrot-vib where D? vib contains the terms representing the harmonic vibrational operators and anharmonic corrections, >( rot contains terms representing the rotational Operators and centrifugal distortion corrections, and D( rot—vib contains the terms representing the rotation-vibration interaction. The vibrational part of the energy of the rotating- vibrating asymmetric molecule is given in cm'1 by21a h G(V1V2°°°°Vn) = §Iui (vi + 1/2) m 30 +Z:§: Xik (vi + 1/2)(VK + l/c) ( > 1 k i2k where the Vi are the vibrational quantum numbers, (”i are the normal frequencies of the normal modes of vibration and the Xik are the vibrational anharmonicity constants. The sum extends over the 3n—6 normal modes of vibration where n is the number of atoms comprising the asymmetric molecule in question. The terms of kerot have already been discussed in detail, cf. Eqs. (23-26). However, the terms of yfrot—vib 22 modify A, B, and C so that for a given vibrational state (V1, V2, . . Vn) ” A A (v) :Ae “:01 (v1+l/2) l n B B (v) = Be — ZCIi (vl + l/2) i n C (31) C (V) 2 Ce _’Z:a:i (Vi + 1/2) . 1 If the vi are not all zero, these are the so-called excited or upper state molecular constants and are given the symbols A', B', C’. The ground state molecular constants are given by n A A0 = A" : Ae '(1/2) .201 l n Bo :. B” = Be —(1/2) :0 B (32) l n CO : c" : Ce —(l/2) 2:qu l where Ae, Be, and C6 are the equilibrium moments of inertia. These (1's which represent the vibrational correc— tions to the inverse moments of inertia are due mainly to three effects,21b Qi :: (11(harmoniC)-t ai(anharmonic)—+ 01(Coriolis). The (11(harmonic) results from the inequality of l/le and l/I when averaged over a harmonic vibration and (21 (an— harmonic) results from the change of the equilibrium length of the oscillator when anharmonicity is present. The effect 23 of the Coriolis interaction of different vibrations is given by (11(Coriolis). These quantities are likely to be large only when vibrations of certain symmetry are quite close in 22 frequency. The Coriolis effect is considered in more detail when the Specific example of H28e is discussed. The difference between the ground and excited state energies is given by the relation ix: G'(vlv2....)-G"(O,O,....)+W'(A',B‘,C')-W”(A”,B”,C”). (33) The quantity (G’—G”) is given the symbol U0 and is called the band center, band origin, or vibrational frequency of the rotation—vibration band. The quantity (W'—W”) is the difference between the ground state and excited state rota- tional energy levels discussed earlier. The selection rules on the vi are found by evaluating the matrix elements of the electric dipole moment. If there is a change in the dipole moment of a molecule during a vibration, then this vibration is said to be infrared active. In the harmonic oscillator approximation, only the fundamental rotation- vibration bands are allowed, i.e., only one zfiVi : l in a given band. When the anharmonicity of the vibrations and/or the non-linearity of the change in dipole moment is taken into account, overtone and combination rotation-vibration bands are allowed, i.e., the vi may change by O, l, 2, Thus the spectrum consists of the fundamentals v1, V2, V3, .. Vn: the overtones 2V1, 2V2 EVH, 3V1: 3V2: .... 3Vn: etc., and the combination bands (mll/l + m2 V2 + + mnyn) 24 where the mi = O, l, 2 Since the anharmonicities are in general small, the overtone and combination bands are much weaker than the fundamentals. Hot bands (those bands which originate from a state other than the ground state) are normally expected to be very weak due to the Boltzman factor, i.e., the lower state of the hot hand has a very small pOpulation compared to the ground state. In the absence of vibrational resonances or perturba- tions, the Xik may be found from Eq. (30). Thus, for example, if one knows the band center for vi and 2Ul, then X11 2 2(V1) - (eul). The Asymmetric Rotor Selection Rules In order that a transition from one rotation-vibration state \fln” to another ¢%' , be accompanied by the emission or absorption of electric dipole radiation, dipole moment matrix element ft/J:,#\/Jn.dv must be The allowed changes in J are A.J = -l, 0, +1 the electric nonvanishing. which correSpond to the P, Q, and R branches of the Spectrum, respectively. These rules on J are identical with the rules for symmetric and Spherical—top molecules. The selection rules on K-l and K+1 can be obtained by using the fact that the components of the electric moment fL along the a, b, and C axes belong, respectively, to the representations Ba, Bb: Bc of the Four Group.23 In order that the electric dipole moment matrix element be nonvanishing, the product of the characters of the representations of the initial state, 25 final state, and of the electric moment must be +1 for each group operation. Thus, if the electric moment has the repre- sentation Bi, 1 = a, b, or c, and one state has the represen- tation A, then the other state must have the representation Bi' However, if neither state has the representation A, then the states must have different representations BJ and Bk where i ¢ j # k. The permitted changes in representation from the initial to final state are listed in Table VI. From Table VI, we see that for type A bands, only the parity of K+lchanges, i.e., AK_1 O, i.2: + 4, .... and AK+1 =.i1:.: 3, For type C bands, only the parity of K—l changes, AK_1 = :l, :3, .... and AK+l = O, + 2, and for type B bands, the parity of both K-l and K+l change, AK_1 = +1, i3, .... and AK+1 =_il,.i3, The rules in Table VI are in accord with the symmetric-top selection rules of AK = o for a parallel band (dipole moment parallel to the unique axis) and AK —.i1 for a perpendicular band (dipole moment perpendicular to the unique axis). For example, in the prolate limit, a type A band becomes a parallel band with AK 2 0, whereas in the oblate limit it would be a perpendicular band with AK = il. Thus the symmetric rotor selection rules may be obtained as a Special case of the more general asymmetric rotor rules. (“11/ CO Table VI. Selection Rules on Kfl, K+l State Direction of Band Type Electric Moment A, ee (9—9 Ba’ eo a A A, ee <—+> Bb’ 00 b B A, ee <——> Bc’ oe c C Ba,eo <—+> Bb, 00 c C Ba,eo <——> Bc’ oe b B ‘ Bb,oo <——> Bc’ oe a A _w Table VII. Overall Selection Rule AK = AK_'+ AK+l Type A Band Type B Band Type C Band AJ AK AK AK +1 +1 +2, 0 +1 0 +1 0 .il 27 Since both K-l and K+l must be‘g J and K-l + K+1 = J for even levels and K_l + K+1 = J + 1 for odd levels (even- ness or oddness is defined by the parity of :7 which is given by the parity of J + K-l + K+l ), not all combinations of _AK_1 and .AK+1 are allowed. Those values of .AK = AK_l +- AK+1 which are allowed are given in Table VII, which is derived from Table VI. We note that for K'> O, the transitions with IAK+1I > 1 are normally much weaker than those with AK+1 = O, :1, and they are forbidden for K? = +1. For K < O, the transitions with. lAKfll > 1 are normally much weaker than those with. .AK_1 = 0, i1, and they are forbidden for K’ = -1. Thus, near the oblate limit, the strongest transitions in a type A band will have AK_l = O, .AK+1 = :1; the next strongest AKélz j;2, .AK+1 = .i 13 etc. For a B type band, the strongest transitions have AK_1 2.:1, AK+1 2.: 1; the next strongest AK_1 =.i 3, AK+1 =.: 13 etc. The calculation of transition intensities may be carried out with the relation 2 2 nhC 4 I a g ‘#(K)l D" n, GXPGW TI?) (3) _W”hc) is the Boltzman factor. Values of the line 2 1,111,111 Of 0.1 in K? by Professor Schwendeman for all the allowed where exp ( , have been tabulated in steps strengths, ';L(K ) 2 . lL(K')| n”,n' contains . 2 asymmetric rotor tranSltions. 4 The the factor (2J + 1) which is due to the (2J + l) - fold 28 degeneracy of the total angular momentum in the absence of an external field. The statistical weight factor g results from the presence of identical nuclei in the molecule, e.g., the two H atoms in H2Se, and the restriction that the total eigenfunction must be symmetric or antisymmetric with respect to an exchange of these nuclei. Referring to Table II, if the two—fold axis is the b axis, then the A and Bb levels are symmetric and the BC and Ba levels are antisymmetric. The statistical weight factors for the symmetric and antisym— metric levels of a given molecule depend on the spins of the identical atoms and on the number of two-fold axes. These factors have been tabulated for a number of asymmetric 2d molecules by Herzberg210 and by Townes and Schawlow. CHAPTER [I THEORY OF THE PLANAE ASYMMETRIC MOLECULE In terms of crystallographic notation four groups of planar asymmetric molecules exist.14 The orthorhombic point group contains the group C2v (H20, H2Se) and the group Vh = D2h (planar CEHQ). The monoclinic point group contains the group CS 2 Clh (HDO, HDSe) and the group Cgh (planar trans C2H2C12). The triclinic groups are not con— sidered here as they do not admit the reflection plane these molecules possess. The convention A > B > C, requires that all planar molecules lie in the ab plane. Since the z axis is chosen as that axis which becomes unique in the nearest symmetric rotor limit, only the following two cases are considered. 3 \ A ’ \ For those molecules having .figi ) B > b (prolate case,, the Ir representation is used (a : z, b = x, c = y) and for A > B > -A:9 (oblate case), the lIIZ representation is 2 used (c = z, b 2 x, a = y). For planar molecules, the sym~ metric rotor limit of the prolate case is the linear molecule while the symmetric-top limit of the oblate case is the oblate symmetric rotor, 6.2-, BF3- 29 30 Effect of Planarity on the Rigid Rotor Hamiltonian For the case of the plane, rigid, infinitely thin body, the principal moments of inertia are related by Ia + Ib = Ic- However, this relation does not hold exactly for the vibrating planar molecule, i.e., there is an inertia defect, AzI-II—I (.39) which is due to Coriolis effects and the fact that A # l/(Ia)ave, etc.21d Expressions for the inertia defect have been derived by Oka and Morino and successfully applied to several planar 25,28 molecules. Thus the only Simplification brought into the first order Hamiltonian by the planarity condition, is that the equilibrium constants satisfy the relation, 1+1_l (36) --.='- _7_ de C e “— Ae Effect of Planarity on the Centrifugal Egstortion Terms l2 2 A It was mentioned by Kivelson and Wilson ’ 7 for the . . 28 non—linear XYX Molecule and subsequently by Dowling and Oka and Morino,29 for the general planar asymmetric rotor, that only four centrifugal distortion constants are indepen- dent. The resulting linear dependence between the six D'S makes Eq. (20) inappropriate for the analysis of spectra of Planar asymmetric molecules especially if the method of least squaresijsused because linearly dependent constants must not be permitted to vary independently. 31 The relations among the taus (which are identical in both Ir and III2 representations) may be written28’29 Tbcbc Z Tcaca : O T — 2 T + 2 t 2 cccc r r aaaa S bbbb + rs I.aabb (37) chaa : r Taaaa + STaabb chbb = I" prpp + S Taabb 2 2 2, 2 . . where r = Ce /Ae and S 2 Ce /Be In practice, the equil- ibrium constants may have to be replaced by the ground state constants. Fortunately this is permissible (to first—order) 2 as the differences between Ce2/Ae and CHE/A"2 are of second order. Substituting EqS. (37) into Eq. (28), we obtained the following expression for the D's: In the prolate case,lra A 2 = - 2 Q 2 DJ ( l/j ) [or Taaaa + (3 + 28 + 38 ) Tbbbb + 2T(1 + 38) Taabb] fl 4 .- 2 aaaa + (j + 23 + 38 ) Tbbbb DK = (-1/32) [(8 .. 8r + 3r2) T —2()+ - I" + 45 " 3TS) Taabb - 16Tabab]‘hu DJK = (1/16) [‘I‘(4 - 31‘) Taaaa + (3 + 28 + 382) Tbbbb -2 (2 - I’ + 23 " 3PS) Taabb - 81.813813]th R5 = (-l/32) [r(2 “ I) Taaaa + (l - S2)‘Tbbbb _2(1 _ s + rs) Taabb ‘ 4 Tabab] ha 2 2 R6 = (1/64) [r Taaaa + (l - 28 + S ).rbbbb h - 2r(l ’ S) Taabb] fi 4 2 aaaa ‘ (l - S ) Tbbbb + 2 rs T aabb] T) ’ SJ = (1/16) [r2 r 33 In the oblate case, IIIfl, 4 DJ = (‘1/32) [3 Taaaa + 3 Tbbbb + 2 Taabb + 4 Tabab 11" 2 2 DK = (41/32) [(3 - 8r + 8r ) Taaaa + (3 - 8s. + 8s ) rbbbb 1; +2 (1 - 4r _ its + 8rs) Taabb + 4Tabab]f‘ DJK 2 (1/16) [(3 _ 4r) Taaaa + (3 " 1+3) Tbbbb 2 l 2 2 ) r 4 r ]1'\ 1* + ( - r - S aabb + abab (:9) R5 = (”l/32) [‘(1 " 21”) Taaaa + (l ‘ 25) Tbbbb 4 +2 (8 - r) Taabth ’ ' at ]‘n 4 R6 = (1/04) [Taaaa + 7"'bbbb " 2Taabb " abab 8, = (1/16) (Taaaa — Tbbbb) h“ 34 In both cases, the D's are a function of only four taus, Taaaa’ Tbbbb’ Taabb’ and Tabab: these being an apprOpriate set to choose because the molecule resides in the ab plane. Our attempt to write two of the above D's in terms of the remaining four D's resulted in very unwieldy expres— sions. For this reason the expressions for the energy and me sum rules are written in terms of the four taus, rather than in terms of four of the D's. Substituting Eqs.(38) and then (39) into Eq. (23), we obtained the expressions for the first order energy of r; a planar asymmetric molecule as follows: ’ in the piolutc case, I , W = W o +(1/16) {r2 [J2(J + l)2 -81 - 2J(J + 1) 3 2] '(I’ - 2)[21”J(J + l) + 2r 83 " (r ' 2)<1324>]}.‘-aaa2L +(1/16){(1 + s)2 [ J2(J + l)2 - 2J(J + 1') + ] -(1 - S)2 81 + 2(1 - 82) [J(J + l) 82 + 83]} Tbbbb +(l/8)«Jf(l + s) [rJ L 2(J + l)2 — 2(r - l)J(J + l) +(r — 2)] +r[(l _ s) 81 _ 2SJ(J + l) 82] -2[1 + 8(1 ’ fl] 83}Taabb 4 +(l/2) {J(J + l) - (PZ > ' 83}Tabab In the oblate case, IIIg, w = W o r 2 2 +(1/1o){J (J + 1) -81 - 2J(J + 1) 82 — (l — 2r) [mm + l)

+ 28 - (l - 2r)

]}r z 3 z aaaa 2 2 +(1/16){J (J + l) - 8 1 + 2J(J + 1)82 - (1 - 2s) [2J(J + l)

-2(r - s) 8 3 + (1 ' 21") (l " 2S‘)'}Taabb 2 4 +(l/4) {Jew + 1)2 + 8 1 - 2W + l) + } Tabab. / ‘\‘\ Iv' 37 The quantum dependences 81, 82, and 82 were defined in Eqs.(24). For cases where the asymmetry is Slight, the approx- mation

Z 12 has been suggested. The relations for 8 l’ 82, and 8 3 then reduce to 2 2 O 51=- (< PZ >—w) /bd 52=(—w)/b (“3) 2 . (S3 = - < PZ2 > (< PZ I>-UI) /o or in terms of 82, 8 -82 d8——8- l " 2 a“ 3 ‘ z 2 The first order energy expressions, Eqs. (40) and (#1), can then be factored into the following much simpler expressions. In the prolate €839. IP~ wawo 2 +(l/16) [rJ(J + l) - r82 - (r' - 2) ] Taaaa 2 +(l/l6) [(1 + s)J(J + l)+(l - S)82 - (l + S)] Tbbbb +(1/8) [mm + 1) .. r82 .. (r - 2)] x [(1 + s)J(J +. .1) + (l - S)32 - (1 + S)] Taabb +(1/2) [J(J + l) - +82] Tabab . In the oblate case, Illt, W = W0 2 +(l/16) [J(J + l). -82 -' (1 ' 2T)] Taaaa 2 +(1/16) [J(J + 1) +82 — (l - 28>] Tbbbb +(1/8) [J(J + 1) -82 - (l - 2r)] X [J(J + 1) +82 " (1 ' 23)] Taabb +(1/4) [J(J + 1) -82 - ] X 2 ,: [J(J + 1) +82 ' ] Tabab » ‘3?) ?9 Effect of Planarity on the Trace of the Hamiltonian We obtained the sum rules in terms of four distortion constants for the prolate case by substituting Eqs.(38) into the sum rules of Allen and Olson18 which were derived in the Ir representation. The sum rules for the oblate case were then obtained by using Table VI to determine the sym- metry of the levels in a given submatrix. The calculations were checked by substituting Eqs.(39) into the Allen and Olson sum rules, while interchanging A and C to get into the prOper representation. In each case, the same results were obtained.30 In order to save space, we call f1 = 3J2 - 7 f2 = 2J2 + 5J + 7 f3 = 3J2 + 6J - 4 f4 2 2J2 - J + 4 f = 2J3 + 3J2 + 2J — 22 f6 = 2J3 + 3J2 + 2J + 23. The sum rules for planar asymmetric molecules are as follows: “IO . I" ._ + g + even J, ee, I (prolate)E , or III (oblate) E (1/6)J(J+1)(J+2){(A+B+C) + (1/20) [ (1 + r2) £3 + r f4] Taaaa (1/20) [ (l + s2)f3 + sfq] Tbbbb + + (IL/2’0) [(1 +_r + 8) f4 + 2rsf3] Taabb + (1/10) f4 Tabab} (1+7) even J, eo; Ir E’, or III?’ O+ (1/6)J(J+ 1) {(J+ 2) A + (J - l) (B + C) + (1/20) [ (J +2) £3 + r (J + 2) f4 + r2 <3 - 1) fJJTaaaa 2 + (1/20) [(1 + S ) (J ' 1) f1 + “5] Tbbbb + (1/20) [(1 + S) (J + 2) 1‘4 + ij5 + 2” (J ‘ 1) f1] Taabb + (1/10) (J + 2) r4 Tabab} (48> 41 even J, oe; Ir 0*, or 1112 E' (1/6) J(J + 1) {(J+ 2) C + (J - 1) (A + B) + (1/20)[ (J - 1) £1 + r (J + 2) £4 + r2 (J + 2) r3] raaaa + (1/20)[ (J - 1) 11 + s (J + 2) 1‘4 + 82 (J + 2) f3] Tbbbb + <1/20)[f + (r + s) (J + 2) 1‘1 + 2m (J + 2) f3]Taabb 5 + (1/10) f5 Tabab} (1:9) even J, 00; Ir 0', or 1113 o (1/6)J (J + 1) {(J+ 2) B + (J -l) (A + C) + <1/2o>[ (1 + r2) (J - 1) £1 + M5] Taaaa +(1/2o) [(J + 2) £3 + s (J + 2)f4 + 82 (J - 1) f1] Tbbbb +(1/2o) [(1+ r) (J + 2) f), + sf5 + 2rs (J - 1) f1] Taabb + (1/10) (J + 2) f4 Tabab} . (50) odd J, ee; Ir E“, or IIIE E” (1/6) J(J+1) (J-l){(A+B+C) + (l/EO) [ (l + r2) f + rf] T l 2 aaaa + (1/20) [ (1 + 32) r + sf ] ‘r ‘ 1 2 bbbb + (1/20) [ (l + P + 5) f2 + 2I’Sfi] ‘Taabb + (1/10) :2 Tebeb} odd J, eo; Ir E+, or IIIE o' (1/6)J (J + 1) {(J — 1) A + (J + 2) (B + c) +(1/2o) [(J - 1) fl + r(J - 1) f2 + r2 (J + 2) f3] ‘Taaaa + (1/20)[ (1 + 82)(J + 2) f3 + Sf6] Tbbbb + (1/2o)[ (1 + s)(J - 1) £2 + rf6 + 2rs (J + 2) f3] Taabb + (1/10) (J ' 1) f2 .rabab}' (52> 43 odd J, oe; Ir 0', or III2 3“ (1/6) J(J+ 1){ (J- 1) 0+ (J+2) (A+B) +(1/2o) [(J + 2) £3 + r(J - 1) £2 + r2(J - 1) r1] 'Teaee +(1/2o) [ (J + 2) £3 + s (J - 1) r2 + s2 (J - 1) fl]‘¢'bbbb +(1/2o) [f6 + (r + s) (J - 1) f2 + 2rs (J - 1) f1] Taabb +(1/10) f6 Tabab} (53) odd J, 00; Ir 0*, or III2 o+ (1/6) J(J+l){(J- 1) 13+ (J+2) (A+C) + (1/20)[ (1 + r2) (J + 2) 1‘3 + 1%] Taaaa +(1/20) [(J - 1) 11 + s (J - 1) f2 + 52 (J + 2) f1] ’bbbb +(l/2o) [ (l + r) (J - 1) f2 + Sf6 + 21’s (‘7 + 2) f3] Taabb + (1/10) (J - 1) f2 Tabab} (5”) AM where A, B, and C are the molecular constants for the vibra— tional state in question. The values of r and s are obtained from the ground state constants, or preferably from the equi- librium constants if they are available. For some low J values, only one energy level is present in a given sum. The resulting expression must be the same as that obtained for the exact solution of the eigenvalue problem for the rotational Hamiltonian. For J 2 1: odd J, oe W(1io) = A+B+(1/4)[ Taaaa + Tbbbb + 2Taabb 5:5: + uTabab] (22) odd J, 00 2 w(111) = A+C + (1/4) [(1+2r + r ) Taaaa + 52 Tbbbb + 2s (1+r) raabb)] (56) odd J, eo , 2 W(lOl) = 3+0 + (1/4) [r Taaaa + (1+2s + 52)'rbbbb + 2r (1+S)'Taabb] (57) For J = 2: even J, eo 2 + (1/4)[ (16+8r+ r2) raaaa + (1+2 s+s )rbbbb + (M + Us + T + I’S)Taabb + 16‘Tabab] (58) even J, oo )((2 =A+LLB+C II) + (l/A)[(1+2r + r2) Taaaa + (16+8s+52) Tbbbb + 2 (n+4r + s+rs) Taabb + 16 T /"" \ U7 KO ’V abab] even J, oe W(212) = A + B + “C + (1/4) [(1+8r + 16r2) T + (1+88 + 1532) Tbbbb aaaa + 2 (1+Ar+us+16rs)‘raabb + uTabab] (60) For J = 3: odd J, ee W(322) : )4 [A + B + C + (l+21"+l”2) Taaaa + (1+2s+32)-rbbbb + 2 (l+r+s+rs)I‘aabb + urabab](6l) These relations agree with those obtained by Oka and Morino31 when one specializes their equations to a planar asymmetric molecule. §implification of the Selection Rules for Planar Triatomic Molecules The selection rules derived in Chapter I are simplified in the case of the triatomic planar asymmetric molecule. Since this molecule can have no vibrations out of the ab plane, there can be no component of the electric moment along the c axis and thus type C bands are not allowed. For those molecules with CS symmetry, the change of dipole moment for a given vibrational state can have components along both the 46 A and B axes and thus hybrid bands can occur, i.e., both type A and type B transitions will appear in a single vibra— tion band. For those molecules with 02v symmetry, however, the change of dipole moment for a given vibrational state must be along either the A axis or the B axis and thus only type A transitions or type B transitions, respectively, will occur. In the prolate case, a type A band will resemble more nearly a parallel band, and a type B band will resemble more nearly a perpendicular band of a prolate symmetric molecule. In the oblate case, both type A and B bands will resemble perpendicular bands of an oblate symmetric molecule. For those transitions between rotational energy levels which are not disturbed appreciably by the asymmetry, the line positions follow very closely the positions of symmetric tOp lines. However, the Q branch lines will, in general, be somewhat more gathered in the type A band as compared to a type B band for the low K+l values. Since all these effects depend strongly on. K and on differences between the upper and ground state constants,32 whenever possible a trial spectrum should be calculated with approximate constants before one attempts to make many assignments of lines in the observed spectrum. CHAPTER III THE ANALYSIS SCHEME In the past, one of the difficulties in an analysis of the infrared spectrum of an asymmetric molecule has been the many tedious calculations necessary to obtain the rigid rotor energy levels. Another difficulty was that of obtaining centrifugal distortion corrections to these energy levels. Often workers in the field used classical centri- fugal distortion corrections,33-36 or used Eq. (23) with the semi-classical approximationl2’37’38 (< PZ2 >)2 = < P24 > in the analysis of the spectra of planar asymmetric molecules. In the analysis of the microwave transitions JT‘—* JT+1 of HDS39 and HDSe,40 some difficulty was encountered in obtain— ing a consistent set of distortion constants, which was probably the result of allowing linearly dependent constants to vary independently in the least squares analysis. These difficulties have been cleared up in part by our derivation of the explicit centrifugal distortion expres- sions [ Eqs. (40) and (Ml)] and by the availability of the table of values of < PZL‘L > 17. The difficulty in carrying out the large volume of numerical calculations has been reduced by the advent of the high speed digital computer 47 A8 which has made possible accurate and rapid interpolation of the E(K ) and ‘< qu > tables as well as the rapid calcula— tion of the many terms in the new distortion expressions. Thus if computer programs are available, the energy levels may now be calculated easily and accurately for any /< The analysis scheme chosen involves calculating the coefficients of Egg, fléé’ C, and the taus for the ground state and upper state energy levels by using the best avail— H able approximate values of I< and K'. The resulting ex- pressions are fit to the Spectrum by the method of least squares to get improved values of A, B, C, and the taus. The scheme also provides for fitting the upper state or ground state energy levels to combination differences which are taken from the Spectra. The latter is a very useful way to obtain the ground state constants because there will then be no effect from perturbations which may occur in the upper state energy levels and because the ground state combination differences from all bands can then be analyzed simultaneously (this does not include hot bands which originate from a state where some Vi ¢ 0). Computer Programs The specific computer programs written for the Michigan State University digital computer MISTIC are based on the oblate case (IIIz representation) planar asymmetric rotor energy expressions and sum rules given in Chapter II. The rigid rotor portion of the energy levels was calculated from either of the expressions (25)] = (A?) J (J+1) + (c - A? )w (62> = (£39) J (J+1) + (559‘) ET ~ . (63) energy uris obtained from [see Eq. (27% [See Eq. W o and W0 The Wang DJ: (C—B) J (J+l) + (A—C) ET-(K ) (6A) (20 _ A - B) ' or from the approximation OJ where b0 2 n K+1 + Z cnbo (65) (A-B)/(2C-A-B) as discussed earlier. The expressions used in the analysis of the Spectra are written where 41 i (it? 7) (A?) + C0 X1 Taaaa +X2 Tbbbb +X3 Taabb +X4 Tabab ('56) (END (’67) (1/2)[J(J + 1) + ET (K) (K 1) M ] dl< de MIX—733) dEI.( ) SO and awo ac (1/2)[J(J + l)-ET(K) + (K - 1) ___1'____ In terms of the coefficients c these derivatives are n) written J (J+1) — K31 - 2% er0 cm bon (7o) 5 —1 7):: E n cnbo(n ) (71) 2 n Q = K+l +- ; (l-n) cnbo (72) The coefficients of the taus are given in Eq. (Al). However, using the identities J(J+1) == g + g < P22> = C u; = g + '0 b0 81 = (g2 -772 b0? - < p,“ >)/b.,2 <73) 82 = —n 83‘ (C2+C’7bO-)/bo thelxi's may also be written in terms of the variables é', T) enui C . In this manner, the entire energy expression has been written in terms of the variables g , 77, Q , and < qu > 51 In order to fit the energy levels to the spectral lines, a predicted spectrum is calculated from Eq. (33), Vcalc : yo + W' (A', B', C') _ W” (All, 5”, C") where W' and V” are calculated using the best available estimates of the molecular constants. These constants may have been obtained from previous analyses or from an approx— imate symmetric tOp analysis of the spectrum. The differ- ences between the observed frequencies and calculated fre- quencies are then fit to the expression . A‘+B’ . A7—B7 . T Vobs' Vcalc = A ”0+ i A(“2‘4“ 77A9‘—2——)+ CIAC ' II A" B" ll An-B” +ZXEATi-éAF%—+-UM ) II 1' II (7“) — 2; AC" 2X1 A T 1 where .A v0 = yo _ Vo , A'+B' A'+B' * A'+B' ( ) = ( ) ( ——2f—J, etc., and the starred quantities are the revised values of the constants. This new set of constants is then used to calcu— late a new set of spectral lines, and a new set of g', 7), Q and)(i. The fit is then repeated until a stable set of It is normally necessary to refit (b and constants is obtained. at least once as the values of the ”slopes,” g, ’0, Xi: have been obtained from the original set of constants and are assumed to be the same for the new set of constants, which is only approximately true. Thus we recalculate the g, 52 n , Q , anlei for the new constants and repeat the fit. The ground state energy levels may be fit to the ob- served ground state combination differences in a similar fashion. A set of ground state combination differences is calculated from the expression A ycalc : [WM (AH ,B” ,C” )1 _ [MIN (AII’BH ,C" )] I where the difference is taken between those ground state energy levels which have allowed transitions to the same upper state energy level. The differences between observed and calculated combination differences are fit to the expres- sion —Al/ = (5'1— €"2)A-%1?1)+(n1-17"2)A(——2——) A V calc obs +(‘CI 2;"2) )AC"+ 2(X1— X2)iATi where A_(§:§§:) = (fl:;§:)* _ (fllggl ), etc., and the starred quantities are the revised values. AS mentioned earlier, this method of obtaining the ground state constants is very useful for several reasons: (2) Since Eq. (75) is independent of the upper state energy H H . II levels, there will be no effect on A', B , C , and T1 due to perturbations in the upper state levels. (b) All bands of a given molecule may be analyzed simultan- eously as they will have the same ground state combina— tion differences. 53 (c) All the ground state combination differences of all these bands may be fit simultaneously with Eq. (75). The upper state combination differences may be handled in the same manner. Each band may be fit individ- ually in a determination of A', B', and C' or several bands fit simultaneously in a determination of the (1’s of Eq. (31). Perturbations in the upper state energy levels can, however, cause trouble in this type of fit. Three separate programs were written for MISTIC. 1. ASP3 (a) calculates wO with Eqs. (62) and (65) for J = O to 12. (b) calculates w with Eq. (45). (a) same as ASP3. (b) calculates w with Eq. (41). (a) calculates wO with Eq. (63) for J = o to 9. (b) calculates W0 with Eq. (62) and (65) for J = 10 to 12. (c) calculates W with Eq. (41). It was soon found that for an asymmetry of K — 0.79 (that of Hgse), Eqs. (45) and (65) do not give a suffici- ently accurate representation of the energy levels (see APPEHGiX I for a comparison of the energy levels as calcu- lated with the various approximations) so that ASP3 and A were not used further. The final form of the program 54 ASP5 was governed by the limited memory capacity of MISTIC, and by the availability of certain subroutines, programs, and IBM card decks. which follows is not intended complete Thus the description of ASP5 to be a guide for any future programs for more versatile computers. A fifth order interpolation routine is used to ob— tain ET( K'”) and dET( K'”)/dK of six ET(K') values in steps 42 K”. tives g'”, ’0”, and C” are (67), (68), and (69), respectively, for J stored in memory. energy level table is continued to J from a table consisting of 0.01 in I< centered about A table of rigid rotor energy levels W0 and deriva- calculated with Eqs. (63), = 0 to 9 and The above is then repeated for /<'. The = 12 with Eqs. (62), (55), (70), (71), and (72) using the coefficients (cn, n = 1 a to 7) of Davis and Beam These levels for J 10, 11, cm—1 and 12 are good to for K+12:5, 5, and 6, respectively, at for both ground and upper states. N N 0.001 ._~ _. K 0.8 which is sufficient here as transitions with large J—K+l values are too weak to be observed. routine is used to obtain << PZ from a table consisting of the 0.8, and 0.7 from J l to 12. a > for two sets of and < Pz 12 have been stored (requires 4095 available). Mecke's sum as a check on the calculation the energy). A second order interpolation A->” and '< qu >' < qu'> values for K = 0.9, 17 Thus WO’ g ,9 77) g .9 energy levels from J ~ 0 to 1690 memory locations out of rules [Eq. (17” are employed of WO (rigid rotor portion of 55 Depending upon the problem Specification, the program will perform the following operations. (61) Energy level calculations: The energy levels, including the effects of centrifugal distortion, may be calculated and output along with their quantum designation J,K_1,K+1. For each J, the levels belonging to the four submatrices E+, E‘, 0+, 0” are summed and output so that the sum rules Eqs. (47-54) can be employed to check the calculations. Spectrum calculator: IBM cards containing allowed pairs of transitions, e.g., u,2,3....5,2,u and line strengths which are tabulatedin steps of 0.1 in K ‘were kindly provided by 24 Professor Schwendeman. The quantum numbers designat- ing the transitions are used in a counting scheme to locate the parameters in memory for the two energy levels in question. The centrifugal distortion cor- rections are calculated for these two levels and the transition quantum numbers are output along with v = 220 + W' (5,2,4) — W" (4,2,3) . The intensity is calculated with the expression 2 H I (I g/J.(K")l exp ( - w hc/kT) where g is the statistical weight factor (for HESe, 3:1 2 for antisymmetric: symmetric levels), bL(K'")| is 56 the line strength obtained by interpolation between the bi(l<)|2 from the transition card and exp(-W”hc/kT) is the Boltzman factor. The |IL(K')|2 contains the factor (2J+l) which is due to the (2J+l)-fold degeneracy of the total angular momentum in the absence of an external field. The inverse transition (I 1/ = ”o + w' (4,2,3) - W" (5.2.4) is calculated and output in the same manner. If there are observed frequencies to be compared with the calcu— lated frequencies, the transition card is followed by a card containing this observed frequency. To determine molecular constants from the spectra: The transition deck including observed frequencies is used in the same manner as in (b). However, in this case the output consists of £7, 'n, g, X1, X2,.X3, Xu, and (”obs — VCalc.) for the states involved. This data is then fit with a least squares program43 to Eq. (74) to determine revised values for the molecular constants. This program is such that any particular constant or set of constants may be varied in a given fit. Ground state combination differences: The energy levels for the ground state are calcu- lated with a set of assumed molecular constants and stored as both upper and ground state levels. A ”com— bination difference" deck of IBM cards was constructed 57 by considering all sets of transitions having the same upper state. Thus the combination difference card 4,2,3-—-6,2,5 may be constructed from the type A trans— itions 4,2,3 -—- 5,2,4 and 6,2,5 —+ 5,2,4, or from the type B transitions 4,2,3—e-5,l,4 and 6,2,5 —* 5,1,4. This combination difference card is followed by a card containing the value of this combination difference as found in the spectrum. The quantum numbers designating the combination difference are used to find the para— )1 meters for the energy levels in question. The calcu- lated combination difference may be output as in (b) or a data tape output containing ( .AVobs — (Abbalc) and g , 77, g, X1, X2, X3, and X4 for the two ground states involved. This data is then fit with the least squares program to Eq. (75) to determine revised ground state constants. Upper state combination differences: The upper state energy levels, which are calculated with a set of assumed molecular constants, are stored as both upper and ground state levels. A combination difference deck is employed in the same manner as (d). Since the quantum numbers on the cards now specify the upper state energy levels, care must be taken in asSembling such a deck as transition frequencies are identified by the quantum numbers of their ground state energy levels. For example, the combination difference card 4,2,3 - 6,2,5 represents the difference between 58 the type A transitions 5,2,4 -—. 4,2,3 and 5,2,4 ——» 6,2,5 or the type B transitions 5,1,4 ——. 4,2,3 and 5,1,4 __. 6,2,5. So this combination difference card 4,2,3 — 6,2,5 is used for transitions having the ground state quantum numbers 5,2,4 for a type A band and 5,1,4 for a type B band. \ CHAPTER IV THE NEAR INFRARED SPECTRUM 0F HgSe The fundamental rotation-vibration bands of Hgse were first studied in detail by Cameron, Sears, and Nielsen in 1939ML Their values for the fundamental frequencies of vibration and of the rotational constants, which were based on the harmonic oscillator approximation and on the sym- metric rotor approximation A = B = 20, are recorded in Table VIII. In 1955, Jache, Moser, and Gordy45 measured three pure rotational lines for each of the five molecular Species H2828e, H2803e, H2788e, H277Se, and H276Se in the millimeter microwave region; however, their analysis was incomplete due to insufficient data. The pure rotational spectrum in the far infrared region was analyzed in 1957 by Palik and 0etjen.46 The centrifugal distortion effect was taken into account by finding empirical energy differences between rigid rotor and distorted energy levels. Due to insufficient resolution, the individual absorption lines for the five isotOpic species or Seleniumwere not observed. Thus the rotational constants, Which are listed in Table VIII, are some sort of average constants for the five species. 59 Table VIII. Ground State Molecular Constants of H28e Cameron, Palik, Palik Oka and Morino et. al. Oetjen IsotOpic avg. avg. avg. H2808e H278Se Species A—C 4.2689 4.2731 A~B 0.4436 0.4486 A 7'78 8.16 8.165 8.173 8.179 ’B 7.78 7.71 7.712 7.729 7.730 c 3.84 3.91 3.915 3.904 3.906 Téaaa -0.004863 ~0.oo4869 Tbbbb -0.004069 —0.004069 Iéabb +0.003003 +0.003005 Tabab —0.000714 -0.000714 v]_ 2260 2344.50 212 1074 1034.21 1 V3 2350 2357.80 H-Se—H 900 910 (910 04‘ 900 52: 90° 52: 61 The results of an analysis of the fundamental rotation- 47 vibration bands of H2Se were reported by Palik in 1959 and are recorded in Table VIII. The centrifugal distortion cor— rections obtained in the 1957 paper were used in this analysis and again, due to insufficient resolution, the rotational constants are "average” constants for the five molecular species. This analysis yielded values for the fundamental vibration frequencies and the vibrational corrections to the molecular constants, i.e., the (21's of Eq. (31). In 1962, Oka and Morino31 published values for the four distortion constants Taaaa: Tbbbb: Taabb: and Tabab which were obtained from a reanalysis of Palik‘s data}1L7 These distortion constants were employed in an analysis of Jache's data45 to obtain values for (A”—B”) and (A"—C”) for the five molecular Species. A value for C” was calculated from inertia defect considerations and thus the analysis was completed. However, an error was discovered in their orig— inal analysis of Jache's data and thus the values entered in Table VIII are the corrected values.u8 The results of these early studies indicate that the molecule is non-linear having an H—Se—H bond angle ofvv9l°. Since A 3 B, the C axis is the unique axis in the nearest symmetric tOp limit and thus H28e is an oblate molecule. From the molecular model shown in Fig. 2, it is evident that the B axis is a two~fold axis of rotation, or in the language Of group theory, a 02 axis. -Since there are also two reflec— tion planes, represented by (7(ab) and (7(bc), this 62 molecule belongs to the point group 02v with symmetry species given in Table IX. The rotations of the molecule Ra, Rb, and RC, about the a, b, and c axes and the normal modes of vibra- tion V1, V2, and V3 are classified according to the symmetry species of Table IX. These classifications are given in Figs. 3 and 4. Preparation of Hydrogen Selenide The hydrogen selenide gas was prepared by reacting aluminum selenide with distilled water, the reaction being represented by A12 Se3 + 61120 —+- 3 H28e + 2A1(0H)3 The procedure used in the production of H2Se is that due to Moser and Doctor49 and Moser and Ertl.50 The main consid— erations in handling H28e are that it is very poisonous5l (fortunately it has a strong odor), and that it readily de— composes in the presence of H20, 02 or ultraviolet light. The chemical setup is shown in Fig. 5. The only unusual glassware is the valve system used to admit small quantities of powdered AlgSe3 into the H20 reservoir. This is quite useful as the large quantity of H20 tends to keep the re- action cool, with little resultant evaporation of H20. If H20 is added to the AlgSe3, the heat of the reaction evapor- ates much of the H20 which in turn, tends to decompose the H288 . At the start, dry N2 gas is used to replace the air in the system. The AlgSe3 and H20 are then placed in their Table IX. Symmetry Species for the Point Group C2v I C2(b) 0(ab) O'(bc) Al +1 +1 +1 +1 A2 +l +1 —1 -1 Bl +1 -1 +1 -1 B2 +1 —1 -1 +1 Fig. 2. Geometry of H28e Fig. 3. Symmetry Classification of the Rotations of H258 Rotation: R Rb R Species: 82 A2 8, Fig. 4. Normal Modes of H28e and Symmetry Classification Mode: 2/ Species: 11' A. B i ill' mac: Sou 3K /D no. octim 3.28 :83 2 (all. A: I \ ON: 28 (V amazon 4|er J... F (.1 1) (B ~58 (IV I X r 1.... 6mm: Io cozosuocd 9: s8 3.8534 .m .9... 66 respective containers, and dry N2 is passed through the system continuously. The powdered AlgSe3 is then added in small quantities to the H20 and the resultant H28e gas is carried by the N2 through the CaCl2 drying tube into the liquid air cold traps. Any H28e which is not condensed in either cold trap is decomposed in the wash bottles which then turn a rusty red color due to the metallic selenium. The by—pass tube around the first cold-trap is very useful in case its center tube freezes up. A 37% yield of 16 grams of H288 was obtained from 50 grams of AlgSe3 and 100 ml. of H20. The Near Infrared Absorption Spectra of H28e The near infrared (l to 3fl) absorption spectra of H28e were obtained with the Michigan State University high resolution infrared spectrometer. A small-volume coolable multiple-traverse cell of the J. U. White type52’53 was employed to hold the gas sample. The Spectrometer employs an f/5 Littrow—Pfund type monochromator with a 600 line/mm certified precision Bausch and Lomb echelette grating for the dispersing element. The ruled area of the grating is 212 x 158 mm. A type P Eastman Kodak lead sulfide photo— conductor, cooled with a circulating dry ice acetone mixture and Operated at 90 cycles per second, served for the detector. Frequency standards were obtained by interposing accurately known absorption lines of HCN, N20, and C0,?”55 67 obtained using a second multiple-traverse cell, or emission 6 lines of Neon and Argon,5 onto the infrared spectra of Hgse. Calibration of the spectra is achieved by simultaneously recording Edser—Butler bands from a Fabry—Perot etalon of «a3 mm Spacing. The frequencies of the standard lines are used to calibrate these ”fringes” of constant frequency separation, and then the frequencies of the H28e lines are ,1 calculated from their positions relative to the fringes. Strong absorption bands of H2Se were found near 3600 cm"1 and 4600 cm‘l, two weaker bands were found near 5600 cm"1 and 6800cm"l,and two very weak bands were found 1 and 8900 om‘l. The bands at 4600 cm'l, near 4350 cm- 5600 cm-1, and 6800 cm‘1 are shown in Fig. 6. These records of the 4600 cm'1 and 5600 cm“1 regions were recorded on a slow speed slave recorder Simultaneously with the recording of the high resolution spectra and so exhibit most of the detail observed. The spectral records from which the meas- urements were taken are approximately 60 feet and 40 feet long, respectively. The conditions under which the high resolution Spectra were obtained are listed in Table X. The Specific calibration techniques employed for some of the above bands are as follows. (a) 4600 cm‘zL region. Calibration of this region was obtained by re— cording the (2—0) band of 00 at 4260 0646 om‘l, followed by an uninterrupted record of the H2Se Spectrum from 4450 cm‘1 to 4750 cm“l. A second 68 come 83 08¢ She 8? I) _ a a _ _ see ,. .2225: OOhm 000m 000m Ommm OOmm _ _ _ I ._ I ._ 00mm Omww OOwQ Omkm OONm Omww _ _ _ _ _ 5 18 come see .88 .88 See ewe: t £88 8:283 6.9.1 Table X. Experimental Conditions Region, om‘l 4600 5600 6800 H28e Pressure, cm Hg 5.0 10.0 20.0 Path Length, m 5.3 12.8 8.0 Spectral Slit width, cm—1 0.042 0.063 0.070 Fringe separation, cm- 0.4018 0.4018 '0.5358 Approximate-fringe Separation on chart, mm 25 18 12 Calibration Standards CO,N2O NCN,Ne,A A 70 record of H28e was recorded from 4600 cm’1 to 4750 cm”1 with 23 absorption lines from (V1 + 2U? + V3 ) at 4630.1659 cm‘1 and 25 absorption lines from (21/l + V3) at 4730.8274 cm‘1 of N20 interposed on the spectrum. The frequencies of the N20 lines55 were fit to the fringes by least squares; the standard deviation of the frequencies was 0.004 cm‘l. This calibration was used to obtain the frequencies of 52 lines of HQSe, which were chosen for their sharp unblended character. The frequencies of these 52 lines were fit to the fringes of the first record along with 21 lines of the co hand,51+ the standard deviation being 0.006 cm'l. The remaining H28e line positions were determined from the first record. 5600 cm"1 region. The calibration was obtained by recording (V1 + V3) of HCN at 5393.698 cm”l followed by the HgSe spectrum in which one Argon and four Neon emission lines were interposed. The frequencies of the five emission lines56 and of 18 HCN lines54 were fit to the fringes with a standard deviation of o 006 cm‘l. 6800 cm‘1 region. The calibration was obtained by interposing four emission lines of Argon onto the H28e spectrum. These lines were fit to the fringes with a standard deviation of .001 cm‘l. The absorption lines have been well resolved in portion of the spectrum of hibits this isotope effect The "fringes” Shown on the 71 due to the five molecular Species all the bands recorded. A small the 6800 cm”1 region which ex- clearly is reproduced in Fig. 7. lower part of Fig. 7 have a fre— OI quency separation of 0.5358 cm‘i. "b- Fig.7. Isotopic Absorption Lines of Selenium isotOpe 82 80 78 77 76 Percent abundance 9 5O 24 8 9 CHAPTER V ANALYSIS OF THE NEAR INFRARED SPECTRA Symmetric Rotor Approximation As a first step in gaining an understanding of the general appearance of the spectra of oblate, planar, slightly asymmetric molecules, a computer program57 was written to calculate line positions and relative inten- sities using the symmetric rotor approximations _ A+B . _ A48 2 and I a g llL exp (WO‘i h c/kT). (77) 2 J,K The perpendicular band selection rules and corresponding line strengths [1 2 are218 J,K 2 _ LI +22+-K)_tI-+1.+ K) AJ=+1’ It‘lJm ' (J+1)(2J+1) 2 (J + 1 + K) (J - K) [SJ = O ’ |(‘IJ,K = J (J + 1) (78) 2 (J - 1 - K) (J - K) AJ = ‘1 ’ |#|J,K '3 J (2J + 1) 2 for AK = +1. The factors ly'J K for AK = -l are ob- ) tained by changing the Sign of K. For K = 0, Eqs. (78) are 73 74 multiplied by 2. The g-factor is given by (2J + l) for K = 0 and by 2(2J +-1) for K.> 0. The rotational constants used were those predicted by Palik37 for the band (2 V1 +~ V3 ) of H282, i.e., the (2's in Eq. (31) were used to find the upper state constants. In order to gain some understanding of the dependence of the relative line positions on changes in the rotational constants, spectra were calculated for four different combinations of these constants as shown in Fig. 8. The notation for an individual line is AKAJKH (J") where the symbols P, Q, and R represent changes of -l, 0, and +1 in the quantum numbers respectively. The low frequency side of a perpendicular band for an oblate molecule consists of the PPK(J), RPK(J), and RQK(J) lines and the high frequency RRK(J), PRK(J), and PQK(J) lines. However, K(J) or PRK(J) lines are included in Fig. 8 as R QK(J) and PQK(J) lines have been omitted for clarity in all but the uppermost spectrum. side contains the none of the RP they are relatively very weak. The The strongest series of lines on each Side of the band center have K" = J” (the "zero" series), the next strongest have K" - J" - l (the ”first"series), the next have K" = J” - 2 (the "second" series), etc. Because 2 C 3 (A+B)/2, the first line in the nth series lies near the position of the third line in the (n-l)th series. The result is that the clusters of lines, especially noticeable on the left hand side of the spectra, contain the lines 75 '0 in ”Mind: _ _ . I _ 3:313 its"? 88m 29m 218683 2.: a: 2.8wa 363200 .wdE A PPl (J—n), odd J PP J , PP (J-l) , PP (J-2)....4 J - -4 (J 2) (J ) PP2 (J-n), even J k where n is the series number. We note that the lines in these groups may degrade to the right or left or form a "head" (third spectrum from the tap) depending on the values of the upper state constants. We also note that the several series of the right hand side of the spectra are quite over- lapped due to the general convergence on that side of the band. This type of behavior normally makes identification of lines more difficult on the high frequency side than on the low frequency side where the divergence of the band tends to separate the lines. The general resemblance between these predicted spectra and the observed spectra may be seen by comparing Figures 8 and 6. The 5600 cm'1 region.-—The initial line identifications l (i.e., assignment of transitions) in the 5600 cm' region were made using the rigid symmetric rotor approximation to the ground state combination differences. If for the zero- series we write the difference RRJ(J)'PP(J+2)(J+2) law: (J+1)-WSW] -[”o+W$(J*1)'W3(J+24 wg (J + 2) - w; (J), “J '\l we obtain RR (J) - PP (J+2) - A" + B" + 4 c” (T J J+2 ~ .. + 1)- (79) Thus, insofar as centrifugal distortion effects and asym- metry may be neglected, the above expression should predict R P the differences between the RJ(J) and P (J+2) lines of J+2 the "zero"-series. Values for Eq. (79) were calculated using Oka and Morino's ground state constants.31 The lines of the strongest series on the low frequency side of the band center were chosen as the PPJ(J) transitions. The values of the ground state combination differences were added to the PPJ+2(J+2) lines to predict the positions of the RRJ(J) lines. A series of lines was found on the high frequency side of the band center for only one choice of the set of J values with the PPJ(J) series. A small discrepancy, which increased with increasing J (to about 0.5 cm‘1 at J = 10) was noted between the calculated and observed line pasitions, the observed lines being at a lower frequency. This is mostly due to the effects of centrifugal distortion, which are so far neglected. The ground state combination differences for the "first”-series were calculated from RR.J(J+l) - PPJ+2(J+3) = 3 (A” + B") + 4C" (J + l) (80) 31 The lines of the using the constants of Oka and Morino. second strongest series on the low frequency side were chosen 78 P as the PJ(J+1) transitions; the quantum numbers were direct- ly assigned by assuming that the line PP (J+3) was to the J+2 right of the line PPJ+4(J+4). The calculated combination differences of Eq. (80) were then used to predict the positions of the RRJ(J+1) lines which again were found at a slightly lower frequency than predicted. The rigid rotor approximation to the upper state com- bination differences for the zero-series is given by RRJ(J) - PPJ(J) = A' + B' + 4 c' J. (81) Using the lines identified through the use of ground state combination differences, a graph of RRJ(J) - PPJ(J) vs. J was constructed (Fig. 9). To the scale shown, a straight line may be passed through the points with a slope giving 0' = 3.74 cm"1 and intercept giving A' + B' = 15.7 cm'l. The symmetric top analysis was carried one step further by taking into account the effects of centrifugal distortion. Employing Eq. (29), the upper state combination differences for the"zerolseries are given by RRJ(J) - PPJ(J) = A' + B' + 4 C'.I-—AE3 (82) where AE =10I [(J+1)2 (J+2)2-(J-1f3J2] +-I3K.[(J+2) (J+1)3 - (J-l)3 J] + DK [(J+1)LL - (J—l)u] . (83) Fig.9. Graph of Upper State Combination Differences 1 ISO (ao- RRJtJ) - PPJU) in cm“ l20” l00" 80” R -P = ' W4C...)- 40 RJU) PJiJ) A + B A. + B. )5.7 Cm” C' 3.74 cm‘l 20 80 The approximate values for the DJ = +7.39 x 10'” DJK a —13.04 x 10‘” and 0K = +6.07 x 10"LL were calculated with Eqs. (39) Marina.31 Graphs of RRJ(J) con- nt 0 D . and D sta s SJ, JK’ K’ cm‘*, —1 cm , cmfll, using the taus of Oka and PPJ(J) - 151-+ AEIvs. J and of RRJ(J) - PPJ(J) - l5J vs. J were constructed (Fig. 10) to show the effect of the centrifugal distortion terms. (The Slope was altered somewhat so that the vertical scale could be expanded.) A straight line for which 15) gave C' = 3.754 cm”1 15.85 cm-l. and its inte-cept gave A3 was passed through the points AE was taken into account; its lepe of (40‘ - + B' = An approximate band center was calculated using the expressions 1]: 0 An average of five values gave In this preliminary analysis, PPJ(J) - W‘ (J—l. J—l) + w“ (J,J). % = 5613.74 cmPl. several series of lines were observed on the right hand side which were not accounted for. ground state combination differences employed in a search for the PPK(J) I a 'Iirst"-, and "second"-series on the R RK(J) lines. Assuming these to belong to another band, The only lines found were of the the observed from the first band were H H lines, using the zero -, right hand side as the "zero“—series, Fig.l0. Expanded Graph of Upper State Combination I6.I I60 I58 l5.7 I56 15.5 I Differences RFilo) -PPJ(J) - )5 J + AE in cm“ A A A AE added A'+e' =I5.854 cm" 0' =- 3.754 cm“ 0 AF. not added 0 These points off curves 0 8} due to asymmetry o RRJ(J)—PPJ(J)—I5J+AE = A'+e'+ (40—15).) 1 l l l l l l l l l I 2 3 4 5 6 7 8 9 l0 82 i.e., the PPJ(J) lines, and these were very weak in comparison to the RRJ(J) lines. A graph of the upper state combination differences was constructed to obtain approximate values for the upper state rotational constants. The values are given for both bands as follows: Band A‘+B' 0' V0 2 Vli+ V 2 , 15.80 cm“1 3.748 cm'1 5612.82 cm-l V 1+ 222 + v3. 15.854cm-l 3.754 om'l 5613.74 on.‘1 5 (The band identifications will be discussed later.) Th 4600 cm‘1 region.--The preliminary analysis of this (D region was carried out in the same manner as in the 5600 cm“1 region. The observed ground state combination differences 1 from the 5600 cm— region were used to make the initial line assignments. In both of the two bands observed here, the PPJ_1(J) ("first"-series) lines were not resolved from the PPJJ ("zero”~series) lines for J = 2 through 9. For J > 9, the lines of the l'first”-~series are to the left of the lines of the "zero”-series, howe er, those of the ”second”-series are to the right of the ”zero”—series and those of the "third”- series are to the right of the "second”—series, etc. On the right hand side of both bands, the lines of the ”first”-series are to the right of the lines of the ”zero"-series, etc., how- ever, this spacing is much less than in both of the bands found at 5600 chl. Graphs of the upper state combination differences were constructed for both bands, and approximate band centers were 83 calculated. The values obtaine for these bands are: Band AI+BI C' yo 2 V1 15.44 mil 3.792. (3.3-1 4615.33 o...“1 2’1 “’3 15.43 cm‘1 3.800 on.‘1 4617.37 Cm-l These four rotation-vibration bands were assigned by considering the various sums of the fundamental vibration frequencies. Thus it is expected that the bands 2 V 1’ V1 + V3, and 2 V3 will be near 4700 arm-1 and the bands ,2/ V V V . V i 2V1+V2 1+ 2+ 3,and 2+2 3wllbenear 5700 cm'l. The differences between these sums of the observed fundamental frequencies and the observed band centers are attributed to anharmonicities in the vibrations and to a 58.36 vibrational resonance which can exist between 2 V3 and 2 V and b ee- ' L’ + V and Z/ + 2 V . In analo 3 etw n 2 l 2 2 3 SY with H28, where the bands 2 V l and V1 +~V3 were separated by 2.24 cm“1 (2 V2 was not observed)35 and the bands m+2é+L3and 2LI*V were separated by 0.98 cm‘1 ( V2 + 2 V3 was not 2 observed),34 we have identified the following bands of HQSe: 3222. Symmetric Tgp M, Typg 2 V 1 4615.33 om'l B V l + V3 4517.37 A 2 V l + V 2 5612.82 B Vl+ V2+V3 5613.74 A 84 The Asymmetric Rotor Analysis of the Ground State 1 5600 cm~ region.~-Employing the ground state constants 80 of Oka and Morino31 for H2 Se, and the values of (A' + B'), C' and 2% obtained from the symmetric rotor approximation for V1 + 1/2 + 2/3, spectra were calculated for (A' - B') = 0.15 cm'l, 0.37 cm’l, and 0.60 cm”1 using type A selection rules. The taus of Oka and Morino31 were used in both the ground and upper state calculations. The positions of the spectral lines were plotted on a roll of chart paper as a function of (A‘—B;), and the points for a given spectral line were Joined by a smooth curve. Thus each curve represents the position of a given spectral line as a function of (A‘ - B‘). These curves were very useful in obtaining the approximate value for (A' - B') needed to account for the line positions and as a result were very useful in assigning lines. Due to the large K, (for H289, K208) the splitting of many of the energy levels having the same K+l value and having J a; K+1 is very small. The result is that many of the symmetric and antisymmetric transitions are superimposed. However, those pairs that were resolved exhibited approxi— mately the 3:1 intensity ratio predicted by the nuclear spin statistical weight factor g. Since the b axis is the two- fold axis of the molecule, from Table II, the A = ee and Bb = 00 levels are symmetric and the Ba = eo and BC 2 0e levels are antisymmetric. Thus if K_1 + K+1 is even, the level is symmetric with g = l and if K—l + K+l is odd, the level is antisymmetric with g = 3. 85 A copy of the spectrum is shown in some detail in Appendix II together with the line assignments and the ob- served frequencies for the H2808e isotopic species. The observed ground state combination differences were evaluated by taking the differences of the appropriate line frequencies and these were fit numerically to the ground state energy levels by least squares in the manner described in Chapter III. However, in this procedure, the microwave values for (A" - s") and (A” — c”) of Oka and Morino3l were held constant, i.e., in Eq. (75) we require that A(M") 50 2 H H and A (A EB ) 5 AC"; thus the combination differences were fit with the expressions i Av... - Mala=<£;'—£;+c.;'-c;>ac" ”'2‘ X... “'9'.“ TI ' (84) The revised values of the constants are: A" = 8.1704 cm-1 B" = 7.7265 0" = 3.9015 :_0.0013 15,3, = 0.00478 :_0.00082 Tbbbb = 0 00402 :.0«00041 Taabb = 0.00306 :_0.00037 Tabab = 0.00090 :_0.00027 where the standard deviation in the 46 combination differ- ences was 0.023 cm'l. The ranges given for the constants are the simultaneous confidence intervals for a confidence 86 coefficient of 95%59 (limits on A" and B” will be considered later). These revised values of the ground state constants were used to calculate a set of ground state combination differences which were used in a further assignment of lines in the type B band, 2211+ V2. In the manner previously des- cribed, spectra were computed (type B selection rules were used) for several values of (A‘ — B‘) and a graph of the line positions vs. (A' — B') was constructed for use in making the line assignments. The RRK(J) and RQK(J) lines were among the strongest found. The only lines found for which A1f+l = -l were of the PP J) "zero”—series and these were J( very weak in comparison to the RRJ(J) lines. The line assign- 80 ments and observed frequencies for H2 Se are given in Appendix II. The #600 cm-1 region.—-Spectra were calculated for both the type A and type B bands using the revised ground state constants and the upper state constants from the symmetric top approximations. Graphs of the line positions vs. (A‘ - B') were constructed in the manner previously des- ) plots and the calculated ground state combination differences. w cribed. Many lines were assigned using both the (A' — The spectrum is shown in Appendix III along with the line 0 assignments and observed frequencies of H2 8 Se for both type A and B bands. 87 Simultaneous analysis of the_ground state combination differences of HQBOSe and H2788e.-—As many observed ground state combination differences as possible were formed by taking the differences between the apprOpriate line frequen- cies for the four bands of H2808e previously discussed. However, those combination differences which depended on lines that ap- peared to be a blend of several unresolved transitions were omitted from consideration. Those differences which were avail- able from two or more bands were averaged and this average dif- erence was given a weight in the least squares analysis equal to the number of differences averaged. In this manner, 115 distinct combination differences were obtained from 146 observed combination differences as taken from the four bands. These were fit by least squares to Eq. (75) and to Eq. (84). Before using Eq. (84), however, it was necessary to reanalyze the “5 employing our microwave data of Jache, Moser, and Gordy revised values of the taus to obtain revised values for (A"—B”) and (A” - C"). Since the three observed microwave transitions (110 ‘— lOl), (220 ‘— 211), and (330 "" 331) occur between ground state energy levels for which exact solutions to the rotational Hamiltonian exist,31 we have obtained exact expressions for the above transitions in terms of the rota- tional constants and the taus. The value for (A" — C”) is obtained from the transition (llO '*- 101) while the other two transitions give values for (A” - B”) which are averaged. 88 The three expressions used are given in Appendix IV. It was found that more precise values for (A" — B") and (A" - C") were obtained by using the microwave data and Eq. (84) than by using our infrared data alone with Eq. (75) where all quantities are allowed to vary in the least squares analysis. The constants obtained in this manner for H280 Se are given in Table XI. The ranges given for C" and the four taus are the simultaneous confidence intervals for a confidence limit of 95%. The range given for A” is the same as that for C” because we have assumed the limits on the microwave value of (A” - C”) to be quite small in comparison to the limits on the infrared value of C”. However, the range on B” is given by the square root of the sum of the squares of the deviations for A" and for (A" - B"), where the deviation for (A” - B”) was obtained as the average deviation of the two values for (A" — B”). The standard deviation in the combination differences is given by 0' and is 0.0134 cm—l. The inertia defect 13 compares quite favorably with the value of 0.1235 i.O-0008 x 10.40 gm cm2 06 as calculated by Oka and Morino.‘ A comparison of the ob— served and calculated ground state combination differences is given in Appendix IV, and the calculated ground state energy levels are given in Appendix I for J = 1 through 9. 78 The ground state combination differences for H2 Se 80 were handled in the same manner as those for H2 Se. The line identifications were easily established by picking the appropriate line to the right of the corresponding H280Se line. Eo Em I So em I m -3 x mood + 810 m -2“ x 80.0 + @910 < o: o: moomed Hemmsd X m: m: : Tea $8.0 Ts... $8.0 .me b . I . . I . swam 880 o + mmooo o- moooo o + $80 0.. s I. I. Mm mmoood + 880.3 1188.0 + ommoo.o+ an t. omoood H $80.0. $08.0 H @386- gangs C686 H $1186- omoood H 3118.0- seems moood H mmom.m moood H mfiomfi 0 0.80.0 H Shed. 88.0 H memsé m also 0806 H Sid Tao moood H 8st a N O m mwem: mow m emwwmm Gem mmommm wo museumcoo muwpm ecsomm .Hx magma 90 However, since many H2788e lines were blended with other transitions, only 48 distinct combination differences were obtained from 69 observed combination differences taken from the four bands. The final constants obtained from the analysis of the microwave and infrared data together using Eq. (84) are given in Table XI. The ranges on the constants are somewhat wider than those for HQBOSe. This is due mostly to the smaller number of observed combination differences employed 78 in the least squares analysis of H2 Se. A comparison of the observed and calculated ground state combination differ— ences is given in Appendix IV. The six centrifugal distortion constants DJ, DJK’ DK, R5, R6, and SJ [see Eq. (23)], which are the distortion constants most often used, were calculated from the taus of 80 H2 Se with Eq. (39): DJ = + 7.46 x lqu cm"l . —4 —1 = ~12.'- 1 cm DJK 13 42 x 0 , m DF = + 6 32 x 10'4 cv‘l ~5 .: R5 = - 2.16 x 10 cm R6 = - 1.84 x 10¢L cm‘1 85 = - 3.38 x 10‘5 cm“1 . It is interesting to note that the constants DJ, DJK, and DK satisfy (to 2 x 10*6 cm‘l) the relation DJK = *(2/3) (DJ + 2 DK) 91 which has been obtained by Silver and Shaffer60 and by Dowling28 for the axially symmetric planar molecule, eg., BF3. Here, how-ever, R5, R6, and SJ are not at all vanishingly small as they would be for a true (axially) symmetric top molecule. To check on the validity of using Eq. (23) in a least squares analysis for a planar molecule, these six D's were used as starting values in a simultaneous analysis of the ob served ground s a combination differences of HQSe. If Eq. (23) is safe to use in the analysis of the spectra of planar asymmetric top molecules, then the six Dis as obtained from the taus, should remain virtually un- changed in thi least squares analysis However, in fact, the D‘s were found to change by large amounts. These changes were not reliable, though, because the simultaneous confidence intervals which resulted were very wide. Our conclusion is that due to the known linear dependence between the six D‘s for the planar asymmetric molecule, Eq. (23) must not be used in a lea st squares analysis. How~ ever, once values of the six Dis are available, Eq. (23) can certainly be used to calculate ene gy levels. This same point must be applied to the sum rules. One can safely use Allen and Olson‘s ru les18 to check the energy level calcula- tions for a planar asymmetric molecule but must not use them in an analysis where the six D‘s are allowed to vary. We conclude that the various expressions containing the six D's should be referred to as applying to noneplanar asymmetric molecules. 92 Asymmetric Rotor Analysis of the Upper State Energy Levels The observed upper state combination differences for 2 V 7/1 + V3, and V1 + V2 + V» were evaluated and 1’ d were fit to their respective upper state energy levels by least squares. The initial values of (A‘ + B') and C‘ chosen for each band were those determined by the symmetric rotor analysis. The initial approximate values of (A' - B') were obtained from the various graphs of line positions vs. (A' - B') as described earlier. Since no microwave informa- tion was available for these upper levels, Eq. (75) was used in terms of the upper state parameters, £4, 7f, C|: and I :X i . ,For each band, the taus were held constant in one II least squares fit, i.e., T ' a T1 and in a second fit 1 I I l__! were allowed to vary simultaneously with A—§§—, A 2B , and C'. The agreement between the observed and calculated combination differences in each of these cases was poorer than for the ground state, i.e., here the standard deviation of the combination differences was larger than 0.10 cm‘l. Also, when they were allowed to vary, the taus changed by large amounts, in some instances even changing sign. Again, the simultaneous confidence intervals were so wide that these new values of the taus were not considered significant. The only least squares fits for which the standard deviation Of the combination differences was 1< 0.03 cm'l, were ob- tained by omitting the observed combination differences which 93 depend strongly on (A' ~ B‘), i.e., on the asymmetry. Because the lines in these upper state combination differ- ences had been used successfully in several ground state combination differences, their assignment was believed correct. In a second approach to obtaining the upper state con- stants, the spectral line frequencies were fit to the expres- sion AZ/o + EIA(A3+BF)+ 77A (A-e-Bi) +€IACfl ( V 2 obs - Vcalc) = which is the form of Eq. (74) when the taus and ground state constants (of Table XI) are not allowed to vary. The results of these fits are given in Table XII. It is observed that these least squares fits are also rather poor in comparison with the fit of the ground state combination differences. Because of these poor fits and also because of the close proximity of the band centers (in both the 5600 cm.1 region and the 4600 cm‘1 region) the possibility of a Coriolis interaction between the states 2 V 1 and V l + l/3 and between the states 2 V1 + 1/2 and l/l + 1/2 + 1/3 was investigated. Eyidence of a Coriolis Interaction In 1936, Wilson61 wrote on the possibility of a Coriolis interaction in asymmetric molecules, indicating that this could occur between two closely lying vibrational states V1 and V3 provided that: mmsé mms.o mmm.o ems.o x Haas m:H.o also mmo.o H-so smfi.o H.so mmH.o mud.o Hma am am mmfi c mo.ons.mem ofi.ons.mfimm no.0Hms.sHme wo.oHsm.mHms oa mfioo.oHesms.m mfioo.lomss.m mfioo.QHooow.m saoo.oH6mms.m o mHo.onHe.o eso.oHome.o smo.lome.o emo.onmm.o m-< moo.onsm.mH moo.QHmmm.mfi soo.oHoms.mH soo.onss.mH m+< e m a m ease ms+ms+fis ms+fism ma+aa Ham .mA + m; + HA new NA + HA m qu + HA «HA mmopmpm exp sow mm mm .8 3.53980 Hmeoafimpom .HHX ofipme ow 95 (a) 49'493 has the same symmetry properties as one or more of the components of the vibrational angular momentum and, (b) that v1 and v3 differ in two and only two of their vibrational quantum numbers and that these two quantum numbers change by one unit from V1 to V3. Condition (b) is satisfiedfkn1H28e by the states 2 V1 and 1/1 + 1/3 and also by the states 2 1/1 + 2/2 and L/l + L’2 + 2/3. We now turn to an investigation of condition (a). Consider the HéSe molecule in the vibrational mode 1’3, shown in Fig. 4. If the molecule is also in rotation about the z = c axis, its individual particles will be sub- Ject to a Coriolis force which will tend to excite 2/2, but with the frequency 0J3 of 2/3. Since the normal frequencies “’2 and (03 are quite widely separated, V3 is not very I successful in generating V2. According to Jahn,22 however, V1 will involve* some of V2 and since “’1 3 (U3, the rotational energy levels of these two modes may be perturbed. The rotational Hamiltonian 3C is now written in the form (III! coords.) * toms in 2/ are not Classically the motions of the a 1 exactly orthogonaljto the motions of the same atoms in l/g. Quantum mechanically, the two states l/ and 1/2 have t e same symmetry and thus can interact weak y. 96 2 A 2 X0: “Pa "‘ pa) '+ H91: ~ pt) + W. - p.>2 (85> where pa, pb, and pC are the components of the vibrational angular momentum, and Pa, Pb, and PC are the components of the total angular momentum along the three moving axes a, b, and c of the molecule. Since the vibrational angular momenta must transform like the rotations Ra? Rb, and Rc922 referring to Fig. 3, we have pa -*'Z82, pb 'r' A2, and pC -*' Bl° The possible product representations of the vibrations as obtained from Fig. 4 and Table IX are Bl x El 1 A1, Al X A1 = A1 and Al x Bl 2 Bl. Thus, according to cone dition (a), pC = pZ is the only possible component of the vibrational angular momentum which can cause a Coriolis pertubation in H2Se. For this example, according to Wilson,61 the non-zero matrix elements of the reduced Hamiltonian 3?? in the Symmetric rotor basis are A‘+B 2 l l l A +B (J,K,V J.K.V )=K2+ W + J(J+1) i 3 ) [ l ) ) Kr (J.K.Vl MI, J,K i 2,v1)=(bO My 1 ’8' (86) . .. , 1/2 (J.~K,V3 J,K:2,V3)=(bo)3 f (J,Kil) K '= —(J K.v K J.K.v) and (J,K,vl r J,K,V3) , , 3 r , , 1 i G K = i G K H 97 where the subscripts l and 3 refer to the two interacting states, e.g., Vl = 2 l/l and V3 = 1/1 + L/3 , wV1 is the vibrational energy of the state V1, (b0) is the asymmetry parameter for an oblate molecule, GC is the only non-zero perturbation coefficient and l r (J,K 1 1) = f (J,n) = E [J(J+l) - n(n—1)] [J(J+l)-n(n+l)]. (87) Centrifugal distortion effects have been neglected. We point out that Wilson obtained these elements in the IIIr represen- tation with PX assigned as real and Py as imaginary. However, these elements have the same form in the III! representation providing PX is taken as imaginary and Py as real which is the convention used by King, Hainer, and Cross and has also been adOpted here. This equivalence may be shown by assuming GC = 0; then the reduced Hamiltonian for one state becomes (J,K{)r€rlJ,K) = K2 + [WV+J (J+1) (£3? ][F715__I‘2r_:] (J,Kl>f’r ‘J,Ki2) = b0 f (J,Kil) which can be written — AB (0%) XI. : wV + %§J(J+1)+( C - 421—)Xw , (88) WV + X0 3 where the matrix elements of the Wang reduced energy XL! are 2 = K 98 1/2 and (J,K J,K:2) = b0 r ‘ (J,K:l), (89) Min and ){O is the rigid rotor Hamiltonian discussed in Chapter I. The matrix elements of the reduced energy E(I< ) are written [Eq° (15)]: (J,K lE(K)l J,K) = F J(J+1) + (G~F)K2 and (J,K |E(K) J,Ki2) _ H 51/2 (J,K:1) Since H / (G ~ F) 2 b0 for III! coordinates, we write (J,K ’E(K) J,K:2) = (G~F)bo fl/2 (J,Kil) and then from Eq. (13), }( = £i9.J(J+1) + £§9.[F J(J+l) + (G-F))fiu] o 2 ’ which in the case of III! coordinates becomes X0: EQEJUH) + (CZ-gig)“ w J which is the same as that in Eq. (88). The analogous expres- sion for If coordinates (prolate case) is written M; 92—3 J(J+1) + (xx-953mm (90) where bO is replaced by bp in )QU. In the non—perturbed case, by application of the Wang transformation (cf. page $3) each J block of )Qd (which has 2J + 1 rows and columns) can be factored into the four sub- matrices E+, E', 0+, and O' as shown in Fig. ll. In the Fig. 11. Elements of Fir when Factored into the E+, E”, 0+ and O- Submatrices O J? fl/2(J,l)b o o a? fl/2(J,l)b A fl/2(J,3)b o o f (J,3)b 16 f (J,5)b o o fl/2(J,5)b 36 Order = l/2(J + 2) for J even, = l/2(J + l) for J odd +). (E—) is obtained.by omitting the first row and column of (E Order = 1/2 J for J even, = 1/2(J - l) for J odd 1_: 1/2 J(J+l)b fl/2(J,2)b o o fl/2(J,2)b 9 f1/2(J,4)b o (0:) = O fl/2(J’4)b 25 fl/2(J,6)b o o fl/2(J,6)b 49 Order 1/2 J for J even, = l/2(J + l) for J odd 100 D p. K rturbed case, with GC £ O, each J block of the reduced Hamiltonian (which has 2(2J + 1) rows and columns) can also be factored into four submatrice‘: U1 + i (El + E3 )9 + - . . E1 + E3 ) for J = 4 is shown in Fig. 12. /‘\ The form of Note that GC connects energy levels with the same K+1 value and same J value, so that ee levels interact with oe levels . - -7 .- ’A + . and 00 levels interact with eo levels. The form of KU1 + +— 03“) for J = 4 is shown in Fig. 13. The matrices (E3 + E1“) '4' u-\ w and (03' + 01 ) are obtained by interchanging the subscripts l and 3 in Figs. 12 and 13. In general then there are four submatrices of the reduced Hamiltonian to be considered. For even J, the subm + - + n .- " v3 3’: f‘ A matrices (01 + 03 ) and (03 + Cl ) are of olde- J and the submatrices (El+ + E3“) and (E3+ + El”) are of order (J+1). / + . For odd J, the submatrices (01‘ + . + _ m . of order (J+1) and the submatrices E1" + E3 ) and (E3+ + El") are of order J. An analytical expression for the eigenvalues of these submatrices has not been found and thus one must resort to calculating the various elements for given values of the asymmetry parameter K’ and of the perturm bation coefficient GO. m - H mm + ma - H - ham + AMEN)” I HOW— _Hllll.mlll S + hvh + >3 u H> omens H Hm + Ha - m m N 0 HI 0 O > + on pAm :vm\fie ea. m a m + GU HI 0 nAm :Vm\em >. e o m o H H « men 0 > + we nflm avm\ae o O H n H + H 6 o 92 a? imhe > e p: imhemk. o o o anfla.:v e w». e> + o m\fi A: u ev .ou pcmaoaeemoo coeumnmSQme esp mcHUSHQCH Xflsumansmnumm + +Hmvm£p mo mucoEon .mfi .wflm M357. _. Hu .0 coepwpgzppdm AH hv U psofiofimmooo at 0 mm. - Sp CHUSHOCH stme95m A mo + a - + ovmnp Ho mpCoEon m > + m m D. a mflmi .H m o m\H > +mon - H O 0 0H- 0 omH o H Kr + m HQANA v O o .2 m\HcH. eH H HAN. avm\He H>_+AHOH + H .ma .mHm 103 In order to determine the effect of Go on the individ- ual rotational energy levels of the states V1 and V3, the reduced Hamiltonian was transformed from the Wang symmetric rotor basis to an asymmetric rotor basis, and then second order perturbation theory was used in an attempt to approxi~ mate the disturbed positions of the energy levels.62 The transformation matrices T, which are available tabulated in steps of 0.1 in I< for’J S 12,63 diagonalize the Wang reduced energy matrix.}fld(in the Wang basis) t _l_ T >QU T u w where W is a diagonal matrix whose elements are the eigen- values of )flu- The matrix elements of T were originally prepared for the Ir representation; however, they may be used as written in the III! representation. Thus for the (01+ + 03“) submatrix of the reduced ( Hamiltonian shown in Fig. 13 for J = 4), we consider the transformation + - _l T (01 + 03 ) T or + o iG |T 0 T1 0 1 1 (92.) _ o ' O T 0 T3 1G 3 3 where T T G, O T T11 T12 T = 33 34 G a C l ' 3 T O GC 104 + “J - ~ and T1 (01 ) T1 and T3 (03 ) T3 are diagonal and give the rigid rotor energy levels. This resulting transformed sub- matrix of the reduced Hamiltonian is written wl(uul) + le C O I G Cl -(Al+Bl)/2 13 14 W 4 W O l( 23) + V1 C23 G24 Cl ~(A1+Bl)/2 w (4 ) + w 3 31 V G31 G32 ‘3 O . (93) C3 -(A3+B3)/2 w (4 ) + w G41 G42 0 3 13 13 C3 -(A3+B3)/2 where Wl (441) is the rigid rotor energy for the rotational‘ state (441), WVl is the vibrational energy for the vibra- tional state V1, and G13 = i[ T11 T33 + 3 T12 T34 ] GC G etc. = e* 31 13’ An approximate description can now be obtained through the use of second order perturbation theory. The approximate expressions for the perturbed energy levels are written, for example, as 105 1 l |G |2 + (c _ 51+Bl) < _ 13 W1(441) + Wvl W3 (431) + an LC;-(A1+Bl)/2 03 ~(A3+B3)/2 o '2 l _ + 1'“ ,, L . (94) W1 (4&1) + Wvl W3K4i3) + Wv3 cl -(A1+B1)/2 C3» (A3+B3)/2 J For this example of J a L, there will be 2(2J + l) z 18. such equations. Since an accurate calculation depends on having ”unperturbed" values of the upper state rotational constants, we turn our attention to the sum rules previously discussed in Chapter II [Eqs. (27-54)]. As has been shown in Figs. 12 and 13, under the influ— ence of the perturbation element Go, the energy levels belonging to one submatrix, say El+, can interact with the levels of the submatrix 33-. Thus the individual traces of the E1+ and E3" submatrices are no longer invariant. How- ever, the trace of (El+ + E3“) is invariant. Expressions for the coupled sum rules were obtained by adding together the appropriate sum rules, Eqs. (47‘5“) given in Chapter II. These are as follows; 108 E coordinates even J, 31+ + E3’, III (1/6)J(J+I) [(J42) (Al+Bl) + (J+2) (01+C3) + (J-l) (A3+B3f] + <1/2) [(J+2) ( yo), + J( m3] (95) even J, E3+ + El‘ (l/6)J(J+l) [(J—I) (A1431) + (J+2) (014—03) + (J+2) (A3+B3)] + <1/2) [(J+2) < v0), + J( v.91] <96) even J, 01+ + 03’ (1/6)J(J+l) [(J+2) (Al-+133) + (J—l) (01+C3) + (J— 1) ( A3+31)] + (1/2) J [( Vo)l + ( ”(93] (97) even J, 03% + 01‘ )1 (I/6)J(J+1) [(J—l) (Al+B3) + (J-l) (cl+c3) + (J+2) ( A3+Bl +(1/2) J [< 110)., + < v.91] <98) odd J, E + + E3”, 1112 l coordinates (1/6)J(J+I) [(J+2) (A1+Bl) + (J-l) (01+C3) + (J-l) (A3+B3)] + (IL/2) [(J+1) < v0), + (J-l) ( u0)3] (99) odd J, 33+ + El" (1/6)J(J+1)[ (J-l) (A1+Bl) + (J-l) (01+C3) + (J+2) (A3+B3fl + <1/2) [(J+1) < yo), + (J-l) ( v.91] (100) odd J, 01+ + 03’ (l/6)J(J+l)[(J+2) (A1+B3) + (J+2) (cl+c3) + (J-l) (A3+Bl)] + (1/2) (J+1) [( Vo)1 + ( u0)3] (101) odd J, o3+ + 01‘ (1/6)J(J+l) [(J—I) (Al+B3) + (J+2) (01+C3) + (J+2) )A3+Bl)] + (1/2) (J41) [( Vo)1 + ( Vo)3] (132) 108 The J dependences of the band origins ( 1% ) and ( z, )3 l were obtained from the order of the submatrices containing ( Lg )1 and ( g )3 respectively. The centrifugal distortion terms have been omitted from these expressions. An attempt to obtain the upper state constants and the band centensfrom these expressions was carried out in the following manner. The observed transitions of 2 v1 and 1’1 + V3 (given in Appendix II and III) were tabulated according to their upper state quantum numbers and only those transitions were retained for which there existed a complete submatrix (eg. El+ + E3—) of upper state energy levels. The appropriate calculated ground state energy (given in Appen— dix I) was added to each transition to obtain an empirical value for the upper state energy level. Because the asymmetry parameters of the two upper states will be nearly the same as that of the ground state, the values of <1p22> : and E(I< ) used in Eqs. (24) will be approximately equal for ground and excited states, particularly for the low J energy levels considered here. Thus, approximate values for the upper state rigid rotor energy levels, which may be treated with the coupled sum rules, were obtained by adding the appro- priate ground state centrifugal distortion energy to the empirical levels. Nine equations were obtained in terms of the six com- binations ofunknowns Al + Bl, A3 + B3, A1 + B3, Cl + C3, ( v )1 and (15 )3. The term in A3 + B1 is not independent 0 of the terms in Al + B1, A3 + B3, and A1 + B3 so that it is 109 necessary to modify Eqs. (95-102) before performing a least squares analysis. The specific equations obtained are given in Table XIII. The weights chosen are the square roots of the number of transitions used to obtain a given sum. The constants obtained from the least squares analysis are given in Table XIV along with the values of the constants obtained with Palikis 0'84? and the values from the least squares analysis of the line frequencies. It is interesting that the values of Al + B1 and Cl + C3 are fairly consistent in all three cases and that for the two least squares analyses, only the values of Al + B3 are significantly different. The band centers were also determined by adding the calculated ground state energies to the transitions PPl(l) for the two states. For 2 v1, ( M. )1 a (000 +— 111) + W"(lll) 4615.33 cm'1 and for 1/1 + 113, l ( pg )3 = (000 ._.101) + W”(lOl) 4617.40 cm‘ We believe the ground state energies W"(lll) and w”(101) (given in Appendix I and also given by Eqs. (56) and (57) -1 respectively) to be correct within 0.001 cm , so that the error in these values of the band centers depends almost solely on the accuracy of the frequencies of the two trans- Since the greatest deviation in the fre- .011 itions involved. Quencies of the H28e lines used for calibration was 0 l l 0 Table XIII. Comparison of Observed and Calculated Coupled Sums for 22/1 and V1 + V3 J Equation Obs. Sum Assigned 0bs.-Calc. Weight 1 El+ 4630.772 cm’1 1.4 +o.oo3 cm‘1 1 03+ + 01“ 9255.771 2.0 —o.005 2 E3+ + El’ 13957.728 2.2 +o.oo3 2 03+ + 01- 9317.569 1.7 +o.o33 2 01+ + 03‘ 9317.583 1.7 +o.o28 3 33+ + 31' 14096.656 2.2 -0.013 3 31+ + E3” 14094.701 3.2 -o.008 3 03+ + 01‘ 18757.615 3.3 -o.009 5 03+ + 01’ 28813.398 4.1 +o.oo3 g-pts = 0.024 cm-1 111 Table XIV. "Coupled” Rotational constants for the states 2 V1 and V1 + V3 Coupled sum rules Line Frequencies Palik's Q‘s 15.455io.o34em‘l 15.449io.007cm‘l 15.464c:m"'l 1 1 A3 + 33 15.428:o.o3o 15.436io.007 15.474 A1 + B3 15.445io.o22 15.481 15.494 01 + c3 7.5984p.o27 7.594ip.002 7.593 ( yo)l 4615.3li0.16 4615.3710.08 ( y0)3 4617.43:o.15 4617.45io.o7 112 -1 cm (the standard deviation in the frequencies was 0.006 -1 cm as given in Chapter IV), we believe the frequencies of these two transitions to be good to better than 0.02 cm-l. Thus ( v0 )21’1 = 4615.33 i0.02 cm'1 and __ . -1 ( K: )V1+'V3 _ 4617.40 : 0.02 cm . The ranges on these values of ( V5 )2%- and ( Z, ) V1+'l’3 come within 0.01 cm'1 of the values obtained with the coupled sum rules. In order to observe the effect of the perturbation on the individual energy levels of the states 2 v'l and 2/1 + 2/3, the transformation matrices for I< = 0.8 were used to obtain values for the perturbed energy levels using second order perturbation theory as already discussed. Because this already assumes that K'l = K’3 = 0.8, Eq. (11) is used to obtain the constants Al’ B1, C1, A3, B3, and 03 from the values of the coupled constants given in Table XIV. The resulting expressions for the perturbed energy levels are given in Table XV for J = 4. A plot of some rigid rotor energy levels for J = 4 is shown in Fig. 14. The first column contains those levels- . 47' calculated with constants obtained from Palik's (1's for 2 1’1. The levels in the second column were calculated with the constants obtained from the line frequency fit of 2 v.1. 113 Table XV. Parturbed Energy Expressions for J - 4. Level 2 V1 v1 + y3 2 “40’E+ 31:477o.37+ 0.126c E3=4772.15+ 0.10Gc 2 2 4u1,0+ El=4768.10+ 6.68GC E3=4769.92+ 2.62Gc 431,6- 31:4763.94- 2.66Gc2 E3=4765.85_ 6.54Gc2 - 2 432,E 31:4754.07- 46.96302 E3=4756.18+ 26.3200 2 422,E+ 31:4753.59- 26.98G02 33:4755.71+ 45.926c #23:0+ E1=4734.l5- 79.45Gc2 E3=4736.66+ 75.88602 - 2 I 2 13,0 E1=4734.l4— 78.48Gc E3=4736.65+ 77.830.C - 2 2 414,3 El=4706.47-14o.82Gc E3=4709.55+137.9BGc 2 119? Fig. l4. Energy Level Diagram for J=4. w _.t_.. E A __ __L_ - 4770— 44o _ ,——’—— 0 —— _:__ ’ 44, \ \ I /\ .. // ~__‘_ 0 I 43: i 4760 - ___L_ E: /’_L_ E Cm" - I 7’, 432 —— —T——T—’ 4750 - L.— 4740- -*-{E'+ / E / / F423} .—-—- “‘1' I 4'3 Calc. Colc. Obs. Colc. Colc. K=082 K=O.72 ”080 ”0'80 ' V+ V 4730.. 2v. 2V: 2” 22" ' 3 115 The third column contains the"observed‘rigid rotor energy levels for 2 v41 and the fourth and fifth columns contain the energy levels for 2 v'l and 2/l+ 2/3 as calculated with the constants obtained from the coupled sum rules and I< a 0.80. The directions in which the various levels are moved by the Coriolis interaction are given by the expressions in Table XV and are indicated by the arrows in Fig. 14. The dashed lines connect those levels which interact most strongly. It is noted that the observed levels of 2 V 1 (column 3) have been moved in the irection predicted by the arrows on the calculated levels of 22/1 (column 4). Unfor- tunately, the corresponding levels of V1 + :13 were not observable due to badly blended lines. We conclude that in this particular case it is not possible to solve the bands using the expressions in Table XV because the off-diagonal matrix elements used to obtain them are not small in comparison with the diagonal elements of the Hamiltonian and thus the second order perturbation techniques are not applicable. Fortunately, however, a method of completing the analysis of these two states is + - __ available through the submatrices (03 + 01 ) for J — 1, and (03+ + 01' ) and (01+ + 03' ) for J = 2. The secular equations for these 2 by 2 matrices are written w(1 )]+02—0 (103i [A3+ C3 + (V. )3“ W3(111) ] [131+Cl + (16 )1- 1 01 -— . 2 [113+ 4B3+C3+ (210)3-w3(211) ][441+B1+cl+( 1!. >1-w1(221)] +G = o (104) 116 [4A3+B3+C3+(ZQ )3- W3(221):][A1+4Bl+01+(2g )1-W1(211)] + GB: 0 (105) where G2 = [c3 —(l/2)(A +33)] [01 ~(l/2)(Al + 31)] 002. (106 in Using the values of the coupled constants and the band origins obtained from the coupled sum rules, and the observed values of the energy levels, these expressions were put into the form (A3 - cl - 4.097) (A3 - cl - 4.102) - G2 = 0 (3AQ + c1 - 27.214) (3113 + cl - 27.187) - G2 = 0 J .J. (3A3 - cl - 20.083) (3A3 - cl - 20.049) - 02 = 0 By using the average values of the fractions, these expres- sions were written A3 — C1 - G = 4.100 cm'l, 343+ 01 - G = 27 200, and 3A3- 01 — G = 20.066. which have the solution A3 = 7.983 cm"1 Cl = 3°567 cm-1 and G = 0.316 om‘l. Substituting these solutions back into Eqs. (103-105), the resulting errors are less than 0.0005 cm- Using these 11? solutions and the values of the coupled constants from Table XIV, the following set of constants was obtained: Constants 2 V l Vl+- V3 A' 8.000orr.‘l 7.983 cm-1 3‘ 7.455 7.445 C‘ 3.567 4.031 K' 0.754 0.728 A“ + 0.594x10-L‘Logmcm2 -0.322x10‘40gmcm2 The perturbation element G0 was then found from Eq. (106): G = 0.255 (dimensionless) C Intuitively, the values of the constants may be suspect because 03 = 4.031 > C" = 3.9013 cm'l. 0n the other hand, the observed values of the energy levels used in Eqs. (103- 105) were also used successfully in the coupled sum rules. Thus, for example, if w3(211) were larger. then W1(221) would have to be smaller by the same amount to satisfy the sum rules. To produce a noticeable change in 03 an improbably large shift in these levels would be required. In the case of the bands 2 V1 + V2 and Vl+ V2+ V3: an analysis based on the coupled sum rules was not possible as an insufficient number of transitions were observed for the band 2 V:L + V 2. Finally, accurate values for Va . , P _ could not be obtained for either band from the Pl(1) trans itions as these two lines were also not observed. 118 The 6800 cm'1 Region 5.) From the fundamental vibrational frequencies, the bands expected near 6800 cm-1 are 2 L/l + V 3 and V1 + 21/3. The line assignments were made with the use of the calculated ground state combination differences. The P PJ(J) lines are overlapped by the PPJ_1(J) ines as in the 4600 cm“1 region. The RRJ(J) lines are badly blended with a second, stronger series of lines which are approximately 0.1 cm"1 higher in frequency. It is believed that this stronger series of lines is the RRJ PPJ(J) transitions as in the example of 2 V (J) (zero) series for another band which has very weak + V 2. .1 _L. Thus the band center separation of 2 L/l +-V3 and V1 + 2 1/3 appears to be approximately 0.1 chl. It is interesting that again in the case of HQS,64 the 2 V l + 2/ 3 and l/l+ 2 V 3 bands were believed to be coincident. Since this pair of bands also satisfies the criteria for a Coriolis interaction, and because of the many blended lines, an asymmetric rotor analysis was not carried out. However, a graph of the upper state combination differences for the band at the lower frequency gave 1 A8 + B3 15.19 cm- II . -1 c1 = 3.745 cm and an approximate band center was calculated to be 14 = 6798.16 cm'l; Isotopic Species of H28e We have already discussed the ground state constants 80 7 of H2 Se and HQ‘BSe. In the case of the isotopic species 1128238 , H277 Se, and H2768e (having relative abundances of 9%, 8%, and 9% respectively) it was not practicable to carry out an analysis because most of their lines are badly overlapped and/or too weak. However, it was possible to calculate approximate band center separations using the few lines available in the "zero”-serie . The isotopic line separa- tions, H2788e - HESOSe, for the "zero"-series of the state 2 V1 15. Thus if K) for 2 V1 of H280 Se is at 4615.33 cm‘l, l have been plotted as a function of frequency in Fig. then K, for 2 V1 of H2788e is at 4615.99 cm_ The separations of the band centers were calculated from a least squares analysis of the isotopic separations Ad! as a linear function of frequency. These separations are listed in Table XVI for 2 V1 , V]_-t V'3, 2 V l + V 2, and V1 + L’2 + L’3. The confidence intervals are for a con- fidence limit of 95%. A complete analysis for 2 Lil +- V 2 was not possible due to the many ”missing” lines. 65 The isotopic frequency rule of Krimm, J i 1 ___:li_ =\fl-( :ATJ / p Tk ) (107) wk was used to predict the ratios of the isotopic frequencies. _ th In this expression, 01k is the frequency of the k—— normal 0.68 0.67 P .0 0.65 0.64 0.63 0. 62 0.6l 0. 60 O Graph of AV for Zero—Series Lines of HZBOSe and H278'Se vs. Frequency for 2V:- Fig. l5. AV in cm" (Hz-"BSe-HZBOSe)J=K V0 for 2Vl of H28°Se / 4500 4550 4600 4650 4700 . -| Frequency m cm 121 mao.oflfios.fi moo.oflmmm.e mHo.oHsmm.H omowmmiomosmm meo.oflmmm.e sHo.onmo.H moo.oflwmm.o omommm1omssmm soo.oflomw.o H-anmo.oflmom.o aoo.oflmso.o woo.oflomm.o omowmm-omwsmm Cl O .01 C .III 0 ml! m H-EoHHo o+oms o H-2omfio o+Hsm o H.soeoo o+amo o ommm m omow m ms+ms+as ms+Hsm ms+HA Him ,§AV mofioemm mm mm go mofioomm OHQOpomH exp pom mCOHpmemmom sopcmo Unmm H>x oHQmE 122 '2 mode, Tk = (1/2)§:m£qkl is the total kinetic energy associated with the kEQ-vibrational mode, p = mi/m where the 1 refers to the isotopically substituted atom, and .AT. = (1/2) Ami o2 J J kJ th is the change in kinetic energy in the k——-mode associated with the substituted atoms J when a mass change Am:J is made. If symmetry coordinates are used, however, in place of normal coordinates, then the conservation of momentum uniquely defines the amplitude of motion of the atoms, and the ratios qu1/ “’k may be easily calculated. Assuming the H-Se-H bond angle to be 91.00 and using the following values for the 66 isotOpic masses, 768e, m = 75.9459 amu, 788e, m = 77.9421 amu, 808e, m = 79.9420 amu, 828e, m = 81.9426 amu, the following ratios were calculated; i i _ (All/(1)1 (102/(4)2 4113/0)3 H282Se/H2803e 8 0.999851 0.999845 HQBOSe/H2788e 0.999843 0.999838 H2788e/H276Se 0.999835 0.999829 From Eq. (30), the band center of 2 V 1 is written 2V1: 2wl+6Xll +Xl2 +Xl3 123 where “’1 is the normal frequency, and the XJk are the an- harmonic constants. If we assume that X jk = Xjk’ a fairly good approximation for H2808e -*- H2788e (but not good for H288 -*- DQSe), then i he i (2 V1)'(2 V1) = 2(011- (“1) AU i (Ave) 2 2001(1-(01/001) (108) 1.. 22/1 The analogous expression for V1 + 1/3 is ; (Av) zwil-wi/wwwu-wi/w) (109) ° an-V3 l l l 3 3 3 and for V1 + V2 + 1/3 is i _ i u) (A’/chip“,§w1(l’w1/w1)+w2(l 002/ 2) l 2 3 1 +0) 1—0.) /w) 110) 3( 3 3 ( Using the calculated ratios and w 53 V — 2345 cm-1 (1) g 1/ = 1034 cm-1, and l “ l " ’ 2 2 a) g l/ = 2358 cm-1, 3 3 Eqs. (108-110) were evaluated and the XVII. results listed in Table It is noted that the calculated separations are system— atically approximately 0.075 cm’1 larger than the observed separations. .This may be due to the use of symmetry coordin- ates in place of the normal coordinates in Eq. (107). Hem.o som.o was.o oms.o mmo.o mse.o ommsmm-omosmm omm.o mfim.o mso.o oms.o 088.0 ome.o omowmm-omwsmm Hisoome.o H11:85.0 H-soaem.o Hisomes.o Hisofimod H1233.0 ommwmmuomowmm N4. .86 N4 .360 :4. .86 .34 .oamo S4 .mpo A4 .360 mA .+ NA + HA m; HA A m moflomgm .mcofipmgmaom howsoo ecmm ©o>smmno 0cm mopedSOHmo mo somHngEoo .HH>X mfinme CHAPTER VI SUMMARY We have investigated the problem of analyzing the rotation-vibration bands of planar asymmetric molecules, and in particular, the near infrared bands of HQSe. Beginning with the symmetric t0p approximation, approx- imate values of the rotational constants and of the band centers were found. We have derived specific energy expres- sions and sum rules for planar asymmetric molecules. These have been employed to obtain good values of the ground state constants for H2808e and H2788e through a simultaneous analysis of the ground state combination differences. We have demon- strated that the energy expression in terms of the six centri- fugal distortion constants DJ, DJK’ DK, R5, m5,énk1 SJ-should not be used to analyze the spectra of planar asymmetric mole- cules. Our attempt to use the planar asymmetric top energy expression in an analysis of the upper states indicated the presence of a Coriolis interaction between the upper state energy levels. Through the use of coupled sum rules, we were able to evaluate certain combinations of the upper state constants and also to find values for the band centers. By solving the secular equations for three low J, 2 by 2, coupled 125 126 submatrices, we arrived at values for the upper state rota- tional constants. Finally, the observed isotopic band separations were shown to be in agreement with that predicted. 10. 11. 12. 13. 14. 15. REFERENCES 8. c. Wang, Phys. Rev. 34, 243 (1929). C. H. Townes and A. L. Schawlow, "Microwave Spectro- s00py," McGraw-Hill, New York, 1955. a page 85,86. b Appendix III, page 522. c Appendix IV, page 527. d Table 4-7, page 104. R H. Schwendeman, J. Chem. Phys. 27, 986 (1957). H. L. Davis and J. E. Beam, J. Mol. Spectros00py 6, 312 (1961). ’ s. 8. Ray, 2. Physik. 78, 74 (1932)° G. W. King, R. M. Hainer, and P. C. Cross, J. Chem. Phys. 11, 27 (1943). T. E. Turner, B. L. Hicks, and G. Reitwiesner, Ballistic Research Laboratories, Report 878, Aberdeen Proving Ground, Maryland, 1953. G. Erlandsson, Arkiv Fysik lg, 65 (1955)- J. W. Blaker, M. Sidran, and A. Kaercher, Gruman Aircraft Engineering Corporation, Research Report RE—l55, Bethpage, New York, 1962. R. s. Mulliken, Phys. Rev. 59, 873 (1941). R. Mecke, Z. Physik. 81, 313 (1933). D. Kivelson and E. B. Wilson, Jr., J. Chem. Phys. 29, 1575 (1952)~ E. B. Wilson, Jr. and J. B. Howard, J. Chem° Phys. 4. 260 (1936). P. M. Parker, J. Chem. Phys. 37, 1596 (1962)- R. A. Hill and T. H. Edwards, J. Mol. Spectroscopy 9, 494 (1962). ‘ 7 127 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 128 J. K. Bragg and S. Golden, Phys. Rev. 16, 735 (1949). Tables of <:P “>1 are available from Professor R. H. Schwendeman, Department of Chemistry, Michigan State University. H. C. Allen, Jr. and W. B. Olson, J. Chem. Phys. 37, 212 (1962). " G. Amat and H. H. Nielsen, J. Chem. Phys. 36, 1859 (1962). 7" K. T. Chung and P. M. Parker, J. Chem. Phys. 38, 8 (1963). G. Herzberg. ”Molecular Spectra and Molecular Structure 11. Infrared and Raman Spectra of Poly- atomic Molecules," Van Nostrand, New York, 1951. a page 205-208. b page 460. c Table 10, page 53. d page 461. e page 426. H. A. Jahn, Phys. Rev. 66, 680 (1939). P 0. Cross, R. M. Hainer, and G. W. King, J. Chem. Phys. 12. 210 (1944). R. H. Schwendeman and V. W. Laurie, ”Tables of Line Strengths for Asymmetric Rotator Molecules," Pergamon Press, London, 1958. T. Oka and Y. Morino, J. Mol. Spectros00py 6, 472 (1961). T. Oka and Y. Morino, J. M01. Spectroscopy 6, 9 (1962). D. Kivelson and E. B. Wilson, Jr., J. Chem. Phys. 2;, 1229 (1953). J. M. Dowling, J. Mol. Spectros00py 6, 550 (1961). T. Oka and Y. Morino, J. Phys. Soc. Japan 66, 1235 (1961). R. A. Hill and T. H. Edwards (to be published in J.M.S.) T. Oka and Y. Morino, J. Mol. Spectroscopy 8, 300 (1962). H. C. Allen, Jr., Phil. Trans. Roy. Soc. London 253 A, 41 (1961). 129 33. P. C. Cross, Phys. Rev. 47, 7 (1935). 34. H. C. Allen and E. K. Plyler, J. Research Nat'l. Bur. Standards 62, 205 (1954). 35. H. C. Allen and E. K. Plyler, J. Chem. Phys. 22, 1104 (1954). 36. H. C. Allen and E. K. Plyler, J. Chem. Phys. 26, 1132 (1956). 37. R. B. Laurence and M. W. P. Strandberg, Phys. Rev. 83, 363 (1951). "‘ 38. G. A. Crosby, E. J. Bair, and P. 0. Cross, J. Chem. Phys. 23, 1660 (1955). 39. R. E. Hillger and M. W. P. Strandberg, Phys. Rev. 63, 575 (1951). 40. V. G. Veselago, Optics and Spectroscopy 6, 286 (1959). 41. Suggested by Professor T. H. Edwards. 42. The E( K ) Interpolation program was prepared by Dr. J. W. Boyd. 43. The double precision, least squares program was pre- pared by Dr. J. W. Boyd and Mr. W. E. Blass. 44. D. M. Cameron, W. C. Sears, and H. H. Nielsen, J. Chem. Phys- 1. 994 (1939)- 45. A. W. Jache, P. W. Moser, and W. Gordy, J. Chem. Phys. 25. 209 (1956). 46. E. D. Palik and R. A. Oetjen, J. Mol. Spectroscopy l: 223 (1957)- 47. E. D. Palik, J. M01. Spectroscopy 3, 259 (1959). 48. T. Oka and Y. Morino, J. Mol. SpectroscwyLfiErratum, 152 (1963). 49- L. Moser and E. Doctor, Z. Anorg. u. Allgem. Chem. 118, 284 (1921). 50- L. Moser and K. Ertl, Z. Anorg. u. Allgem. Chem. 118, 269 (1921). 51- N. I. Sax, ”Handbook of Dangerous Materials," p. 200. Reinhold, New York, 1951» 52. 53. 54. 55- 56. 57- 58. 59- 60. 61. 62. 63. 64. 65. 66. 130 J. U. White, J. Opt. Soc. Am. 32, 285 (1942). T. H. Edwards, J. Opt. Soc. Am. 6;, 98 (1961). D. H. Rank, G. Skorinko, D. P. Eastman, and T. A. Wiggins, J. Mol. Spectros00py 4, 518 (1960). D. H. Rank, D. P. Eastman, B. S. Rao, and T. A. Wiggins, J. Opt. Soc. Am. 22, 929 (1961). American Institute of Physics Handbook, p. 7—42, D. E. Gray, Editor, McGraw-Hill, New York, 1957. This program was written for M100, the MISTIC compiler. B. T. Darling and D. M. Dennison, Phys. Rev. 57, 128 (1940). ‘— J. W. Boyd, Thesis, Michigan State University (1963). S. Silver and W. H. Shaffer, J. Chem. Phys. 9, 599 (1941). E. B. Wilson, Jr., J. Chem. Phys. 4, 313 (1936). The author is indebted to Professor R. H. Schwendman for this suggestion. R. H. Schwendeman and V. W. Laurie, Tables of trans- formation coefficients are available, see reference 17. G. L. 0rdway, P. C. Cross, E. J. Bair, J. Chem. Phys. 33. 541 (1955). s. Krimm, J. Chem. Phys. 32, 1780 (1960). W. F. Meggers, ”Key to the Welch Periodic Chart of the Atoms,” No. 4858, 1959, Welch Scientific Company, 1515 Sedgwick Street, Chicago 10, Illinois. APPENDIX I )2 4 0n the Approximations ( < P22) > and )2 = was investigated by altering the program ASP5 slightly to output J, K_l, K+l’ W0, W1, and W2 where the rigid rotor energies WO are calcu- lated with Eq. (63) using fifth order interpolation in a table of E(/< ) which is tabulated in steps of 0.01 in I< .7 The energies W1 are calculated with the correct first order centrifugal distortion expression [Eq. (41)] where the values Of < qu>' are obtained from second order interpolation in a table of <2qu> which is tabulated in steps of 0.1 in K .17 The energies W2 are also calculated with Eq. (41), however, the values of < qu> are obtained from ( Q )2 = ( < P22>' )2 where the g are given by Eq. (69). The energies are given in Table XVIII for K = 0.79241 (that for the ground state of H280Se). The sums of the appropriate levels for the sum rules are given for W1 and W2 in the first and second rows respectively at the end of each block of J values. The first value in each row is the sum of the levels having the symmetry of the first energy level in that block, the second value is the sum of the levels having the symmetry Of the second level, etc. The sums of the individual levels 131 132 W1 are the same as those predicted by the sum rules, Eqs. (47- 54). The only energy levels We which are within 0.001 cm-1 of the levels W1 are the three J = 1 levels, the three inter- mediate J = 2 levels, and the middlemost J = 3 level. These are among the levels for which an exact solution of the rotational Hamiltonian exists and the only levels for which < P22>> = K2 and <1qu>» = K“, with the result that for them ( ' )2 = <.qu>r is exact. We also note that the sums of the individual levels indicate the discrep- ancy between the levels W1 and W2. Thus if energy levels 1 )2 _ . . 2 accurate to 0.001 cm are required, the approx1matlon( = i should not be used for (K15 0.80. The approximation hf(b) = K+f + :: Cnbon was investi- gated by using the program ASP4. The output is identical in format with that of ASP5 (see above). The rigid rotor energies WO are calculated with Eqs. (62) and (65) where the coeffici- ents Cl through C7 are used.u The energies W1 are calculated with the correct centrifugal distortion expression [Eq. (41)] using the interpolated values of The values of 2 * are determined by Eq.(72), 7 n C: (122% = K + 2|:(1-n)CnbO The energies W2 are calculated with Eq. (41) using the approxi- mation 2 4 £2: ( ) E 133 These energies are given in Table XIX for the same value of K as above. The rigid rotor energy levels which are the least well‘ represented by this approximation to ur have the quantum and J numbers J as may be observed by comparing the J,0 J-2,2 '1eve1s 99 0 and 97 2 in Tables XVIII and XIX. Unfortunately, 3 J Mecke's sum rule [Eq. (17)], which applies to all the rigid rotor energy levels of a given J does not indicate this dis- crepancy, i.e., Mecke‘s sum rule predicts that sum obtained by adding up all the rigid rotor energy levels. This is because in many cases the deviations of the levels from the positions predicted by the E( K ) tables cancel one another. 0n the other hand, those rigid rotor energy levels with K+1 1 nearly equal to J are within 0.001 cm- of the levels as calculated with the E( K ) tables and so may be used for asymmetries as large as I< = 0.79. Another discrepancy appears in the calculation of W1, this being that Eq. (72) is a somewhat worse approximation to < P 23> than Eq. (65) is to hf. This is due to the z 2 multiplicative factor n in the expansion for . How- ever, this discrepancy is noticeable only for those levels with the larger values of J—K+1 and thus the energy levels with small J—K+l values may safely be used. 134 Table XVIII. Calculated Ground State Energy Levels,H2808e. W0: Rigid Rotor Energy Levels From E(K). W1: With Distortion, No Approximations 2 2 W2: With Distortion, () = J K-l K+1 W0 w1 W2 1 l 0 15.898 15.896 15.896 1 1 1 12.072 12.070 12,070 1 0 1 11.629 11.627 11.627 15.896 12.070 11.627 0.000 15.896 12.070 11.627 0.000 2 2 0 47.729 47.706 47.715 2 2 1 44.510 44.292 44.292 2 1 1 42.981 42.965 42.965 2 l 2 31.503 31.500 31.500 2 0 2 31.467 51.454 51.463 79.161 44.292 42.965 31.500 79.178 44.292 42.965 31.500 3 3 0 95.565 95.466 95.508 3 3 1 92.689 92.599 92.609 3 2 1 90.032 89.952 89.964 5 2 2 79.195 79.166 79.166 3 1 2 79.016 78.942 78.985 3 i 3 58.936 58.916 58.926 3 0 5 58.935 58.912 58.924 174.408 151.514 148.855 79.166 151.535 148.889 79.166 174.492 Table XVIII. (contd) J K-l K+1 W0 .wl W2 4 4 0 159.504 159.216 159,533 4 4 1 157.262 156.954 156.994 4 6 1 152.815 152.574 152,527 4 3 2 142.814 142.686 142.707 4 2 2 142.285 142.060 142.167 4 2 6 122.459 122.655 122.696 4 1 6 122.446 122.660 122.682 4 1 4 94.188 94.146 94.166 4 0 4 94.188 94.142 94.166 695.688 279.610 274.904 266.829 695.666 279.690 275.009 266.870 5 0 269.654 268.984 269.224 5 1 267.968 267.606 267.402 5 4 1 261.674 260.816 260.965 5 4 2 222.682 222.060 222.109 5 6 2 221.191 220.556 220.871 5 6 6 201.841 201.510 201.642 5 2 6 201.789 201.409 201.589 5 2 4 176.595 176.699 176.478 5 1 4 176.594 176.697 176.477 5 1 5 167.246 167.159 167.192 5 0 5 167.246 167.159 167.192 662.966 575.972 569.684 695.460 666.572 576.266 569.746 695.587 Table XVIII. (contd) J K -1 K+1 W W W o l 2 6 6 0 556.107 554.752 555.154 6 6 1 554.925 555.566 555.760 6 5 1 525.775 524.676 525.012 6 2 517.950 517.158 517.555 6 4 2 515.674 514.562 514.952 6 4 5 297.074 296.299 296.614 6 5 5 296.921 295.999 296.455 6 5 4 268.858 268.287 268.550 6 2 268.854 268.275 268.526 6 2 252.557 252.220 252.541 6 1 5 252.557 252.220 252.541 6 1 6 188.100 187.958 188.007 6 0 6 188.100 187.958 188.007 1105.547 862.084 852.895 775.404 1106.659 862.715 855.805 775.889 7 7 0 448.920 446.457 447.021 7 7 1 448.151 445.652 445.960 7 6 1 456.099 454.168 454.827 7 6 2 429.492 428.020 428.409 7 5 2 425.715 425.545 424.504 7 5 5 408.160 406.659 407.257 7 4 3 407.786 405.904 406.844 7 4 4 579.965 578.689 579.251 7 5 4 579.950 578.647 579.257 7 5 5 545.691 542.850 545.187 Table XVIII. (contd) J K_1 K+l W0 W1 W2 7 2 5 545.690 542.829 545.187 7 2 6 299.281 298.802 298.974 7 1 6 299.281 298.832 298.974 7 1 7 246.761 246.556 246.602 7 0 7 246.761 246.556 246.602 1547.229 1441.637 1429.457 1105.512 1549.535 1445.006 1451.461 1106.654 8 8 0 578.117 576.892 574.668 8 ‘ 1 577.610 575.561 575.861 8 7 1 562.456 559.525 560.477 8 7 2 557.106 554.551 555.254 8 S 2 551.559 547.425 548.874 8 6 5 555.105 552.446 555.515 8 5 5 554.509 550.881 552.587 8 5 4 596.899 504.477 505.562 8 4 4 506.860 504.550 505.519 8 4 5 470.695 468.854 469.656 8 3 5 472.692 468.850 469.655 8 5 6 426.524 425.102 425.591 8 2 6 426.524 425.102 425.591 8 2 7 575.827 575.159 575.569 8 1 7 375.827 575.159 575.569 8 1 8 515.225 512.884 512.972 8 0 8 313.223 312.884 312.972 2553.521 1947.800 1952.195 1797.014 2567.624 1950.402 1956.088 1799.559 Table XVIII. (contd) J K-l K+1 W0 W1 W2 9 9 0 726.698 716.923 717.896 9 9 1 723.382 716.577 717.286 9 8 1 704.963 700.170 701.999 9 8 2 700.814 696.667 697.761 9 7 2 692.604 686.562 688.681 9 7 6 677.926 673.639 675.365 9 6 3 676.499 670.662 675.445 9 6 4 649.653 645.511 647.376 9 5 4 649.554 645.184 647.259 9 5 5 613.532 610.167 611.661 9 4 5 615.529 610.121 611.658 9 4 6 569.221 566.706 567.777 9 3 6 569.221 566.706 567.777 9 6 7 516.758 515.091 515.732 9 2 7 516.758 515.091 515.732 9 2 8 456.176 455.221 455.519 9 1 8 456.176 455.221 455.519 9 1 9 347.489 386.996 387.108 9 0 9 387.489 386.996 387.108 3070.595 2902.441 2885.041 2364.105 3077.162 2907.122 2889.940 2568.455 139 Table XIX. Calculated Ground State Energy Levels,H280Se. we: Rigid Rotor Energy Levels From the on 1: With Distortion, (P22) From the on . 2 w2. With Distortion, ()2= J K_l K+1 W0 W1 W2 1 1 0 15.898 15.896 15,895 1 1 1 12.072 12.070 12.070 1 0 1 11.629 11.627 11.627 15.896 12.070 11.627 0.000 15.896 12.070 11.627 0.000 2 2 0 47.729 47.706 47.715 2 2 1 44.310 44.292 44.292 2 1 1 42.981 42.965 42.965 2 1 2 31.503 31.500 61.500 2 0 2 31.467 31.454 31.466 79.161 44.292 42.965 31.500 79.178 4.292 42.965 31.500 3 6 0 95.565 95.466 95.508 . 3 1 92.689 92.599 92.609 3 2 1 90.032 89.952 89.964 3 2 2 79.195 79.166 79.166 3 1 2 79.016 78.942 78.985 3 1 3 58.966 58.916 58.926 3 0 3 58.935 58.912 58.924 174.408 151.514 148.865 79.166 174.492 151.535 148.889 79.166 Table XIX. (contd) 140 J K- 1 K+1 W0 W1 W2 4 4 0 159.504 159.216 159.663 4 4 1 157.232 156.954 155.994 4 6 1 152.815 152.574 152.627 4 3 2 142.814 142.686 142.707 4 2 2 142.285 142.029 142.167 4 2 3 122.459 122.355 122.696 4 1 3 122.446 122.360 122.382 4 1 4 94.188 94.143 94.166 4 0 4 ' 94.188 94.142 94.163 695.687 279.310 274.904 266.829 395.663 279.390 275.009 236.870 5 5 0 239.657 258.985 239.225 5 5 1 237.968 237.303 267.402 5 4 1 231.374 230.816 260.965 5 4 2 222.382 222.060 222.109 5 . 2 221.189 220.546 220.867 5 6 3 201.841 201.510 201.642 5 2 3 201.789 201.409 201.589 8 2 4 176.595 176.399 173.478 5 1 4 176.594 173.397 173.477 5 1 5 137.243 137.159 167.192 5 0 5 167.243 137.159 137.192 632.925 575.972 569.384 595.450 633.569 576.236 569.746 595.587 Table XIX. (contd) J K-l K+l W0 W1 W2 6 6 0 336.136 334.763 665.154 6 6 1 334.925 333.566 363.760 6 5 1 325.773 324.676 625.012 6 5 2 317.930 317.158 317.353 6 4 2 315.645 314.229 314.901 6 4 3 297.074 296.299 296.614 6 3 3 296.921 295.999 296.453 6 3 4 268.858 268.287 268.560 6 2 4 268.854 268.275 268.526 6 2 5 232.537 232.220 232.641 6 1 5 232.537 232.220 232.341 6 1 6 188.100 187.958 188.007 6 0 6 188 100 187.958 188.007 1105.225 862.084 2.895 773.404 1106.588 862.715 3.805 776.889 7 7 0 449.164 446.463 446.862 7 7 1 448.131 445.632 445.960 7 6 1 436.099 434.167 434.826 7 6 2 429.492 428.020 428.409 7 5 2 425.471 422.186 423.749 7 5 3 408.160 406.640 407.257 7 4 3 407.786 405.907 406.845 7 4 4 379.963 378.689 379.251 7 3 4 379.950 378.647 679.267 7 3 343.691 342.860 345.187 Table XIX. (contd) J K 142 -1 +1 WC W1 W2 7 2 5 643.690 342.829 646.187 7 2 6 299.281 298.802 298.974 7 1 6 299.281 298.802 298.974 7 1 7 246.761 246.536 246.602 7 0 7 246.761 246.536 246.595 1546.068 1441.638 1429.439 1105.512 1548.821 1443.007 1431.461 1106.634 8 8 0 579.577 572.890 571.608 8 8 1 577.609 573.361 576.863 8 7 1 562.453 559.314 560.472 8 7 2 557.106 554.551 555.264 8 6 2 549.879 539.905 546.776 8 6 3 535.106 532.454 536.516 8 5 3 534.312 530.908 532.592 8 5 4 506.899 504.477 505.562 8 4 4 506.860 504.350 505.519 3 4 5 479.693 463.954 469.656 8 3 5 470.692 468.850 469.655 8 3 6 426.324 425.102 425.591 8 2 6 426.324 425.102 425.591 8 2 7 673.827 373.139 373.369 8 1 7 376.827 373.139 676.669 8 1 3 313.226 312.884 312.972 g 0 8 313.223 312.884 612.972 2355.131 1947.808 1932.210 1797.014 2659.466 1950.405 1936.088 1799.559 Table XIX. (contd) J K_l K+1 wo W1 W2 9 9 0 730.473 702.103 675.268 9 9 1 726.677 716.582 717.295 9 8 1 704.944 700.110 701.972 9 8 2 700.814 696.667 697.761 9 7 2 685.829 646.497 668.750 9 7 3 677.931 673.675 575.541 9 6 6 676.428 670.839 '676.486 9 6 4 649.652 645.509 647.376 9 5 4 649.554 645.182 647.258 9 5 5 613.562 610.137 611.661 9 4 5 613.529 610.122 611.658 9 4 6 569.221 566.706 567.777 9 3 6 569.221 566.706 567.777 9 6 7 516.758 515.091 515.762 9 2 7 516.758 515.091 515.732 9 2 8 456.176 455.221 455.519 9 1 8 456.176 455.221 455.519 9 1 9 387.489 386.996 387.108 9 0 9 387.489 386.996 687.108 3012.709 2902.481 2883.157 2664.106 2984.572 2907.167 2889.955 2368.463 APPENDIX II The Line Assignments and Observed Frequencies of 80 H2 Se for 22/1 + LI? and Ill + 1/2 + v:3. The spectrum of the 5600 cm-1 region of Hgse is shown on the following four pages. The spectrum continues from page to page from low to high frequency without overlapped or omitted sections. The line assignments above the spectrum are for 2L/1 + 1/2 and below the spectrum are for 1’1 + 1/2 + 2/3. The symmetric transitions are iden- tified by a short line and the antisymmetric transitions by a tall line. 144 ; ’ _ _L 5500 T P (J) K 107.! l 1 .1- 11 3 3" ' K's!" 1 l L l ; I .14- 12 * l0 ‘ 1% K'J-Z I l l 1 3 J8 ll 9 ; Kg!!-: I I I L . Jr IO 3 L Kurd-4 1 l I J {L J- 9 7 146 p fidfl 1 K-J (molt) 1 1 1 1 1 I a: 7 5 5 R001 3 K K: i l 1 i 1! r I I I l1 - i 1'! I In" ’ l 1 . * 5550 1 R 1 014mm 111 m 7 K-5 3 1 7 5 3 8 I , 6 '«4 I 2 1 1 4 l #4 7 5 . 3 I l ____1 6 .y- . 4 I i 21" + 1’2 1 Type B IL__J_4_. l ' é ' 1|“ 1 [I l I l PQKU) (not obumd) $— 11' 1 @1111 1 1 1'1 1 J I 1 1 5600 565i 11.11 1 1 11 1141111114:th .11 111. 111 m 1 2 1 o I 1 2 3 4 5 |.___1___._1 y 1 1 1 1 1 1 ° 0 2 1 1 1 1 1 14+u2+ v3 1 Type IX 1 IV IVWI 1: _2 4 . 5 e .1 10 , ‘ II I l K3J'L 3 5 7 9 1" ll 1 l K'J 149 The obs d erve frequencies, yobs , of V1 + v2 + V3 are listed in the following table. The observed frequencies are compared with the frequencies calculated with Eq. (41) using the constants from the line frequency fit (see Table XII). Those frequencies marked with an asterisk were not used in any analysis as they are from blended lines. The relative intensities, I, were calculated with Eq. (34) for T = 3000K. J" Kt, K2| J: KilKil I ”606 yobs—Lbalc 1 1 1, 2 1 2 1.4 5632.566 +0.007 2 1 2, 1 1 1 3.8 5594.197 -0.012 2 1 2, 3 1 3 6.5 5639.818 +0.067 6 0 6, 2 0 2 5.7 5585.684 +0.078 3 0 3, 4 0 4 7.9 5646.559 +0.066 4 1 4, 5 1 5 8.6 5656.007 +0.019 5 0 5, 4 0 4 7.0 5568.310 +0.031 8 + .0 8 5 “i: Z 1 2 2:2 222311;? -861. 7 0 7, 6 0 6 6.0 5549.785 +0.006 8 1 8, 7 1 7 5.0 5540.086 -0.025 7 0 7, 8 0 8 6.9 5670.564 -0.055 9 0 9, 8 0 8 6.9 5530.114 -0.044 9 0 9, 10 0 10 4.4 5680.768 -0.129 11 0 11, 10 0 10 2.1 5509.665 -0.082 10 1 10, 11 1 11 6.3 5685.436 -0.156 12 1 12, 11 1 11 1.4 5498.527 -0.106 11 0 11, 12 O 12 2.3 5689.764 -O.?24 150 Observed frequencies of 111 + v2 + y3 (contd) J" K'.'|K;'l J' K1,K;, I bobs ”bbs'LEalc 1 1 0, 2 1 1 4.1 5640.056* -0.212 2 2 1, 6 2 2 . 5647.766 -0.059 2 1, 6 1 2 . 5649.015 +0.240 3 1 2, 2 1 1 3.9 5577.121* -0.099 3 1 2, 4 1 3 5.6 5655.545 +0.094 3 2 2, 4 2 3 1.8 5655.290 +0.016 4 2 3, 6 2 2 4.5 5569.693 -0.070 4 2 3, 5 2 4 6.1 5662.095 +0.066 5 1 4, 4 1 3 4.8 5561.093 +0.096 5 1 4, 6 1 5 6.1 5668.488 +0.066 6 2 5, 5 2 4.6 5552.235 +0.070 6 2 5, 7 2 5.5 5674.586 +0.072 7 1 6, 6 1 5 4.0 5543.081 +0.061 7 1 6, 8 1 7 4.7 5680.652 +0.042 8 2 7, 7 2 6 6.3 5533.666 +0.070 8 2 7, 9 2 8 3.8 5685.862 +0.053 9 1 8, 8 1 7 2.5 5523.954 +0.063 9 1 -, 10 1 9 2.9 5691.076 +0.061 10 2 9, 9 2 8 1.8 5513.971 +0.062 10 2 9, 11 2 10 2.1 5695.978 +0.062 11 1 10, 10 1 9 1.3 5506.711 +0.063 11 1 10, 12 1 11 1.4 5700.569 +0.048 12 2 11,11 2 10 0.8 5493.186 +0.072 2 2 0, 3 2 1 1.5 5654.968 -0.094 3 2 1, 2 2 0 6.7 5571.426 +0.065 4 3 2, 6 3 1 2.8 5566.734 -0.026 Observed frequencies of U1 + U2 + V3 (contd) J" K K" 151 4 H J! K; K1] I z’obs yobs'zbalc 4 2 2, 3 2 1 1.2 5560.660 -0.079 4 3 2, 5 3 3 4.4 5670.564* -0.o3o 6 3 4, 5 3 3 6.2 5544.991* -0.002 5 2 6, 6 2 4 4.4 5677.532 +0.088 7 2 5, 6 2 4 2.8 5566.125 +0.102 6 3 4, 7 6 4.0 5686.781 +0.086 8 6 6, 7 6 2.2 5526.964 +0.084 7 2 5, 8 2 6 3.3 5689.764 +0.066 9 2 7, 8 2 6 1.7 5517.511 +0.076 8 3 6, 9 3 7 2.6 5695.455 +0.057 10 6 8, 9 3 7 1.2 5507.792 +0.079 9 2 7, 10 2 8 1.9 5700.851 +0.049 11 2 9, 10 2 8 0.8 5497.780 +0.066 10 6 8, 11 6 9 1.3 5705.965 +0.059 3 3 0, 4 3 1 4.3 5669.376* -0.085 4 6 1, 6 3 0 1.1 5556.560 +0.082 5 3 2, 4 3 1 3.1 5544.307 -0.067 5 4 2, 4 4 1 0.6 5549.005” +0.028 5 3 2, 6 3 3 3.5 5686.951 +0.391 6 4 3, 5 4 2 2.3 5537.974 -0.371 6 4 3, 7 4 4 6.0 5692.485 -0.016 7 6 4, 6 3 3 2.0 5528.871 +0.406 7 6 4, 8 3 5 2.4 5698.799* -0.004 8 4 5, 7 4 4 1.6 5519.900* -0.045 8 4 5, 9 4 6 1.9 5704.662 -0.019 9 3 6, 8 3 5 1.2 5510.796 +0.052 152 Observed frequencies of 2/1 + [/2 + 2,3 (contd) J" K2.K2I J! K; K;, I ”663 yobs’vcalc 9 3 6, 10 6 7 1.3 5710.258 -0.020 10 4 7, 11 4 8 0.9 5715.496’t -0.077 11 6 8, 10 3 7 0.5 5491.578 -0.006 6 5 2, 5 5 1 1.4 5564.464 -0.062 5 4 1, 6 4 2 6.5 5694.225 -0.145 7 4 3, 6 4 2 1.6 5519.186* -0.097 8 5 4, 7 5 3 1.1 5512.471* -0.648 7 4 6, 8 4 4 1.9 5707.922 +0.097 9 4 5, 8 4 4 0.8 5503.711* +0.103 1 1 0, 1 1 1 4.1 5609.805 -0.008 1 1 1, 1 1 0 1.4 5617.533 +0.066 2 2 0, 2 2 1 2.0 5610.258‘ -0.042 2 2 1, 2 2 0 6.3 5617.082 +0.061 3 3 0, 3 3 1 .5 5610.960 -0.020 4 4 0, 4 4 1 2.5 5611.792* -0.000 4 4 1, 4 4 0 7.7 5616.011 +0.072 6 0, 5 5 1 6.9 5612.658 -0.013 5 5 1, 5 5 0 2.3 5615.646* +0.062 6 6 0, 6 6 1 1.8 5616.509‘ -0.058 7 7 0, 7 7 1 6.9 5614.507 +0.046 2 0 2, 2 2 1 0.6 5626.544* .-0.008 2 2 1, 2 0 2 1.7 5600.283 +0.056 2 1 2, 2 1 1 2.1 5624.486 -0.176 6 1 2, 3 3 1 2.1 5627.463 -0.041 3 2 2, 3 2 1 1.0 5626.498 -0.104 3 2 1, 3 2 2 2.9 5602.112* -0.056 153 Observed frequencies of v1 + v2 + v3 (contd) J" Kr'Kfi J' Kg Ki' I yobs z’obs'd/calc 4 4 1, 4 2 2 1.6 5597.547 +0.126 4 3 2, 4 3 1 3.3 5622.171 -0.069 5 3 2, 5 5 1 1.1 5661.087 -0.015 5 4 2, 5 4 1 1.0 5620.538 -0.106 5 4 1, 5 4 2 6.0 5603.467* -0.360 6 6 1, 6 4 2 0.6 5591.595 -0.026 6 5 2, 6 5 1 2.6 5618.719 -0.204 4 3 2, 4 1 3 2.1 5591.846 +0.138 5 3 2, 5 3 3 2.0 5592.682* -0.046 5 2 3, 5 4 2 2.1 5632.940 -0.295 6 4 3, 6 4 2 1.8 5628.794 -0.094 6 2, 6 3 3 1.4 5590 386 +0.429 4 3, 0 4 1.4 5586.106 +0.023 4 1 4, 4 1 3 1.6 5640.35 +0.105 5 1 4, 5 3 3 1.9 5639.818* -0.065 6 4 6, 6 2 4 1.5 5582.607 +0.053 s 3 4, 5 3 3 1.7 5639.265 +0.409 7 3 4, 7 5 3 1.2 5638.462 -0.517 c 1 4, 4 1 5 1.1 5576.714 -0.019 5 0 5, 5 2 4 1.3 5647.369* +0.143 5 2 5, 5 2 4 1.5 5646.559* -0.074 6 3 4, 6 1 5 1.6 5573.628 +0.093 7 3 4, 7 3 5 1.0 5573.421 +0.086 7 2 5, 7 4 4 1.2 5645.964 -0.037 8 4 5, 8 4 4 0.8 5644.951 +0.076 5576.010 +0.067 154 The observed frequencies, Lgbs , of 2L3 + V2 are listed in the following table. The observed frequencies are compared with the frequencies calculated with Eq. (41) using the constants from the line frequency fit (see Table XII). Those frequencies marked with an asterisk were not used in any analysis as they are from blended lines. The relative intensities, I are claculated with Eq. (34) for T = 3000K. J" K" K" J! K! K, I yobs ”obs'vcalc l 0 1, 2 l 2 4.2 5631.926 -0.0?5 2 1 2, 3 0 3 6.3 5638.689 +0.012 6 0 6, 2 1 2 5.5 5584.619 -0.047 3 0 6, 4 1 4 7.8 5645.388 -0.012 4 1 4, 3 0 3 6.6 5576.045 +0.01? 4 1 4, 5 0 5 8.5 5651.767 -0.017 5 0 5, 4 l 4 6.9 5567.136 -0.018 5 0 5, 6 1 6 8.4 5657.839 -0.0?1 6 1 6, 5 0 5 6.6 5557.961 -0.007 6 l 6, 7 O 7 7.8 5663.599 -0.0?5 0 7, 6 1 6 5.9 5548.489 +0.006 7 0 7, 8 1 8 6.8 5669.046 -0.030 a 1 8, 9 0 9 5.6 5674.176 -0.039 9 o 9, g 1 8 3.9 5528.618 +0.00? 9 0 9, 10 1 10 4.4 5679.005 -0.036 3 5683.550 -0.001 O 1 10 ll 0 11 3 :1 0 11: 12 1 1? 2.3 5687.770 +0.0?4 155 Observed frequencies of 2V + v2 (contd) J" K" K2. J! K1, K}, I yobs I’obs"vcalc 6 1 2, 4 2 3 5.1 5654.256 -0.189 A 4 2 3, 5 1 4 5.8 5660.939 +0.064 5 1 4, 6 2 5 5.8 5667.256 +0.022 6 2 ,, 7 1 6 5.3 5676.266 +0.035 7 1 6, 8 2 7 4.6 5679.005 +0.088 8 2 7, 9 1’ 8 6.7 5684.656 +0.069 9 1 8, 10 2 9 2.8 5689.449 +0.106 5 2 3, 6 6 4 4.0 5676.322 -0.033 6 3 4, 7 2 5 6.7 5682.541 +0.039 7 2 5, 8 3 6 3.1 5688.460 +0.068 8 3 6, 9 2 7 2.5 5694.067 +0.051 9 2 7, 10 6 8 1.9 5699.647 +0.056 10 6 -, 11 2 9 1.3 5704.639 +0.084 6 4 3, 7 3 4 2.6 5691.646 +0.401 7 4 4, 8 6 5 0.7 5697.638 +0.080 8 4 5, 9 3 6 1.7 5706.379 -0.020 9 6 6, 10 4 7 1.2 5708.809 -0.088 10 7, 11 6 8 0.9 5714.018 -0.053 6 6 0, 3 2 1 5.7 5606.688 +0.253 4 4 0, 6 1 1.5 5604.298 -0.216 APPENDIX III Line Assignments and Observed Frequencies of H2808e for 22/1 and Ill + 713 The spectrum of the 46OO cm“l region of H2 Se is shown on the following seven pages. The spectrum continues from page to page from low to high frequency without overlapped or omitted sections. The line assignments above the spectrum are for 21/1 and below the spectrum are for U l + V The symmetric transitions are 3. identified by a short line and the antisymmetric transitions by a tall line. 156 PF (.11 1 1 J- 10 8 .644 1 I ’1 J3 H 9 1 !!=:!-§ I l I 1' J: l2 IO I" :58“... z I 1 J8 l3 Ill ‘63”... | I .1- 14 l2 8 l I .1- 15 I3 1 1 1‘ “I ‘ F3 13 / 1.? 4550 1 1 3 i 1 1 . J 4 2 _ 1 1 1 1 R w 5 3 1015”“ 1 1 1 1 111 1- 1 1 1 11 11 , RQKM . 1 1 1 1 1 1_ K8 4 3 ' 1 1 1 l i I I ll 1 4 2 1 I Type B .11 1 1111 1 1 1 f , 111111 1 1 I ' I ll. . V w d y. 1 W} 1 I 4600 l . II I 1 I I1l I 1 ‘12 l O I # 11 J V. ' I u, + v3 4 6 J 8 I 3 5 7 J 9 1 1 4 6 8 .1 IO 1 1 K=1H4 5 7 9 J ll 1 1 K=J 6 8 IO J l2 1 K=J 6 8 J I0 I I 5 7 ’ 9 J= ll 1 I 1 , 4 3 ' 5 J' 9 164 The observed frequencies, bobs’ of U1 + V3 are listed in the following table. The observed frequencies are compared with the frequencies calculated with Eq. (41) using the constants from the line frequency fit (see Table XII). Those frequencies marked with an asterisk were not used in any analysis as they are from blended lines. The relative intensities were calculated from Eq. (34) with T = 3009K. J" K2 K2, J' KLIK1. I z,b V _V O S obs calc 0 0 0, 1 0 1 1.0 4628.743* +0.019 1 0 1, 0 0 0 2.8 4605.773 -0.050 1 0 1, 2 0 2 4.4 4636.385 -0.016 2 0 2, 1 0 1 1.3 4597.292 +0.023 2 1 2, 1 1 1 3.8 4597.623 -0.087 2 1 2, 3 1 3 6.5 4643.220 -0.037 3 0 3, 2 0 2 5.7 4589.094 -0.023 4 1 4, 3 1 3 6.7 4580.593 -0.021 3 0 3, 4 0 4 7.9 4650.082 -0.046 = 0 5, 4 0 4 7.0 4571.841 -0.041 4 1 4, 5 1 5 8.6 4656.744* -0.026 6 1 6, 5 1 5 6.7 4562.903* -0.052 5 0 5, 6 0 6 8.5 4663.188 -0.019 7 0 7, 6 0 6 6.0 4553.794 -0.036 6 1 6, 7 1 7 7.9 4669.427 -0.009 8 1 8, 7 1 7 5.0 4544.493 -0.017 7 0 7, 8 0 8 6.9 4675.464 +0.010 9 0 9, 8 0 8 3.9 4534.992 -0.002 165 Observed frequencies of V1 + y3 (contd) N H 11 J K4 K+' J! KL Kh I Vobs ”obs—Vealc 8 1 8, 9 1 9 5.7 4681.253 -0.010 10 1 111, 9 1 9 3.0 4525.298 +0.014 9 0 9, 10 0 10 4.4 4686.866 +0.005 11 0 11, 10 0 10 2.1 4515.428“ +0.046 10 1 10, 11 1 11 3.3 4692.255 +0.007 12 1 12, 11 1 11 1.4 4505.310*’ +0.024 11 0 11, 12 0 12 2.3 4697.446 +0.024 1 1 0, 2 1 1 4.1 4643.644 +0.445 2 1 1, 1 1 0 1.2 4589.862 -0.058 2 1 1, 3 1 2 1.5 4651.125 -0.008 3 1 2, 2 1 1 3.9 4580.593*' +0.441 2 2 1, 3 2 2 4.0 4649.874 -0.198 4 2 c, 3 2 2 4.5 4571.841* -0.168 3 1 2, 4 1 3 5.6 4657.351* -0.009 4 2 3, 5 2 4 6.1 4663.573 -0.042 5 1 4, 4 1 3 4.8 4562.903* -0.003 5 1 4, 6 1 5 6.1 4669.772* -0.013 6 2 5, 5 2 4 4.6 4553.706 -0.045 6 2 5, 7 2 6 5.5 4675.735 +0.000 7 1 6, 6 1 5 4.0 4544.358 -0.022 7 1 6, 8 1 7 4.7 4681.487 +0.012 8 2 7, 7 2 6 3.3 4534.801 -0.014 8 2 7, 9 2 8 3.8 4687.024 +0.020 9 1 8, 8 1 7 2.5 4525.058 +0.002 0 2 9, 9 2 8 1.8 4515.113 +0.009 11 1 10, 10 1 9 1.3 4505.042 +0.082 166 11 Observed frequencies of V (contd) J" K; Kfil J1 K; K; I yobs ”obs‘Vcaic 3 2 1, 2 2 0 3.7 4573.812 -0.016 4 3 2, 3 3 1 1.8 4564.847*’ -0.008 5 2 3, 4 2 2 . 4554.234 +0.378 4 3 2, 5 3 3 4.4 4670.326 -0.148 6 3 4, 5 3 3 3.2 4544.712 -0.161 5 2 3, 6 2 4 4.4 4676.600 -0.036 7 2 5, 6 2 4 2.8 4535.175 -0.040 6 3 4, 7 3 5 4.0 4682.224’ -0.019 8 3 6, ‘7 3 5 2.2 4525.412 -0.016 7 2 5, 8 2 6 3.3 4687.710 +0.011 9 2 7, 8 2 6 1.7 4515.428* -0.009 8 3 6, 9 3 7 2.6 4692.967 +0.029 3 8, 9 3 7 1.2 4505.310* +0.058 9 2 7, 10 2 8 1.9 4698.009 +0.044 2 9, 10 2 8 0.8 4494.917 +0.041 4 3 1, 3 3 0 1.1 4557.674 +0.059 5 3 2, 6 3 3 3.5 4684.374 +0.151 6 4 3, 5 4 2 2.3 4536.414 -0.387 6 4 3, 7 4 3.0 4688.771 -0.119 7 3 4, 6 3 3 2.0 4526.271 +0.141 7 3 4, 8 3 5 2.4 4694.188 +0.023 8 4 5, 7 4 4 1.6 4516.208 -0.127 8 4 5, 9 4 6 1.9 4699.100 +0.022 9 3 6, 8 3 5 1.2 4506.118 +0.013 0 4 7, 9 4 6 0.8 4495.716 +0.010 167 Observed frequencies of z/l + V3 (contd) J" KEIK; J1 K3 K1, I zxobs ”obs‘Vcaic 1 1 0, 1 1 1 4.1 4613.226* -0.087 1 1 1, 1 1 0 1.4 4620.834* +0.020 2 2 1, 2 2 0 6.3 4619.462 -0.026 3 3 0, :3 3 1 7.5 4612.087 +0.011 3 3 1, 1. 3 0 2.5 4617.640 +0.050 5 5 0, 5 5 1 6.9 4609.805* +0.460 2 0 2, 2 2 1 0.6 4629.067 -0.030 2 2 1, 2 0 2 1.7 4603.742 +0.006 2 1 2, 2 1 1 2.1 4628.088 +0.493 3 2 1, 3 2 2 2.9 4604.225 -0.187 3 1 2, 3 1 3 1.6 4595.787“ -0.027 3 2 2, 3 0 3 0.5 4595.625* +0.039 3 0 3, 3 2 2 1.8 4635.274* -0.178 4 2 3, 4 2 2 2.4 4633.278 +0.368 4 3 2, 4 1 3 2.1 4593.668* +0.052 5 3 2, 5 3 3 2.0 4592.463 -0.144 4 2 3, 4 0 4 1.4 4586.652 -0.033 4 1 4, 4 1 3 1.6 4642.116* -0.044 5 1 4, 5 3 3 1.9 4639.626 -0.136 5 2 3, 5 2 4 1.7 4584.517 -0.045 6 3 4, 6 3 3 1.7. 4636.649* +0.160 1 4, 5 1 5 1.1 4577.496 -0.021 5 0 5, 5 2 4 1.3 4648.769 -0.043 6 2 5, 6 2 4 1.5 4645.785 -0.039 s 3 4, 6 1 5 1.3 4574.891 -0.004 7 2 5, 7 1.2 4642.116* -0.244 6 2 5, 6 0 6 0.8 4568.099* -0.047 7 2 5, 7 2 6 0.9 4565.127 +0.002 168 yobs’ of 2 V1 The observed frequencies are com- The observed frequencies, are listed in the following table. pared with the frequencies calculated with Eq. (41) using the constants from the line frequency fit (see Table XII). Those frequencies marked with an asterisk were not used in any analysis as they are from blended lines. The relative intensities I were calculated with Eq. (34) for T = 3000K. J" K: Krl J1 K4 K0 I yobs LQbs'Vcalc l 1 1, 0 0 0 0.9 4605.258 -0.042 1 0 1, 2 1 2 4.2 4634.307 -0.056 2 1 2, 1 0 l 5.8 4595.142“. 40.036 1 1 1, 2 0 2 1.3 4635.837 -0.014 2 1 2, 3 0 5 6.3 4641.116 -0.008 3 0 3, 2 1 2 5.5 4587.015 -0.066 3 0 3, 4 1 4 7.8 4647.943 -0.Cl9 4 1 4, 3 0 3 6.6 4578.462 -0.019 4 1 4, 5 0 5 8.5 4654.548 -0.002 5 0 5, 4 1 4 6.9 4569.704 -0.011 5 0 5, 6 1 6 8.4 4660.915 -0.005 6 1 6, 5 0 5 6.6 4560.753 +0.019 6 l 6, 7 0 7 7.8 4667.077 +0.009 7 0 7, 6 1 6 5.9 4551.547 +0.004 7 0 7, 8 1 8 6.8 4673.018 +0.023 8 1 8, 7 0 7 4.9 4542.150 +0.007 8 1 8, 9 0 9 5.6 4678.757 +0.037 169 Observed frequencies of 2 U1 (contd) J" K" K:' J! K7 K1' I yobs z’obs’Vcalc 9 0 9, 8 1 8 3.9 4532.552 +0.017 9 0 9, 10 1 10 4.4 4684.241 +0.061 10 1 10, 9 0 9 2.9 4522.748* +0.026 10 1 10, 11 0 11 3.3 4689.493* +0.056 11 0 11, 10 1 10 2.1 4512.771 +0.070 11 0 11, 12 1 12 2.3 4694.554 +0.087 12 1 12, 11 0 11 1.4 4502.567 +0.093 1 1 3, 2 2 1 4.1 4642.116 -0.601 2 2 1, 1 1 0 3.6 4586.488 -0.038 2 2 1, 3 1 2 3.3 4647.761 +0.082 2 1 1, 3 2 2 1.3 4649.144 -0.203 3 2 2, 2 1 1 1.1 4577.846 +0.086 3 1 2, 4 2 3 5.1 4655.215*’ -0.079 4 2 3, 3 1 2 4.2 4569.704*' +0.088 3 2 2, 4 1 3 1.7 4655.007 -0.018 4 2 3, 5 1 4 5.8 4661.435 -0.041 5 1 4, 4 2 3 4.5 4560.753" -0.086 4 1 3, 5 2 4 1.9 4661.435 -0.071 5 1 4, 6 2 5 .8 4667.571 -0.034 6 2 5, 5 1 4 4.4 4551.547 -0.064 6 2 5, 7 1 6 5.3 4673.467 -0.029 7 1 6, 6 2 5 3.9 4542.150 -0.050 7 1 6, 8 2 7 4.6 4679.143 -0.023 8 2 7, 7 1 6 3.2 4532.552 -0.025 8 2 7, 9 1 8 3.7 4684.598 -0.014 9 1 g 2 7 2.5 4522.748* +0.000 170 Observed frequencies of 2 v1 (contd) J" K11 K11 J' K1. K1,. I yobs z/ -z/ obs calc 9 1 -, 10 2 9 2.8 4689.828 -0.005 10 2 , 9 1 8 1.8 4512.699* -0.012 10 2 9, 11 1 10 2.0 4694.857* +0.028 11 1 13, 10 2 9 1.3 4502.473 +0.003 11 1 10, 12 2 11 1.4 4699.642 +0.043 12 2 11, 11 1 10 0.8 4492.048 +0.024 3 3 1, 2 2 0 1.3 4569.097*’ -0.060 4 3 2, 3 2 1 2.8 4559.829 +0.111 4 3 2, 5 2 3 3.8 4668.249 +0.020 5 2 3, 4 3 2 2.9 4552.096*' -0.553 5 3 3, 4 2 2 0.9 4551.836* +0.278 2 3, 6 3 4 4.0 4674.453*' -0.093 6 3 4, 5 2 3 2.9 4542.657 +0.029 6 3 4, 7 2 5 3.7 4680.059*' -0.039 7 2 5, 6 3 4 2.6 4533.037 -0.088 7 2 5, 8 3 6 3.1 4685.455 -0.053 8 3 6, 7 2 5 2.1 4523.235 -0.048 8 3 6, 9 2 7 2.5 4690.631 -0.054 9 2 7, 8 3 6 1.6 4513.181 -0.064 9 2 7, 10 3 8 1.9 4695.600 -0.037 10 3 8, 9 2 7 1.1 4502.943 -0.056 10 3 8, 11 2 9 1.3 4700.340 -0.022 11 2 9, 10 3 8 0.8 4492.499 -0.049 11 2 9, 12 3 10 0.9 4704.852* -0.009 12 3 10, 11 2 9 0.5 4481.857 -0.036 171 Observed frequencies of 2 V1 (contd) J" K3 K3 J1 K3 K1. I ”obs ltbs‘Vealc 4 4 1, 3 3 0 3.8 4551.300 -0.008 5 3 2, 4 4 1 1.1 4548.561 +0.501 5 4 2, 4 3 1 0.7 4541.137 +0.117 5 3 2, 6 4 3 2.8 4682.224* -0.345 6 4 3, 5 3 2 1.9 4533.199* +0.468 6 4 3, 7 3 4 2.6 4686.665 -0.059 7 3 4, 6 4 3 1.7 4524.092* -0.382 7 3 4, 8 4 5 2.2 4692.033 -0.024 8 4 5, 7 3 4 1.4 4514.105 -0.064 8 4 5, 9 3 6 1.7 4696.865*’ -0.061 3 6, 8 4 5 1.1 4503.962 -0.037 3 6, 10 4 7 1.2 4701.538 -0.065 10 4 7, 9 3 6 0.7 4493.468 -0.087 10 4 7, 11 3 8 0.9 4705.990 -0.058 11 3 8, 10 4 7 0.5 4482.833 -0.075 12 4 9, 11 3 8 0.3 4471.998 -0.058 5 5 1, 4 4 0 1.1 4533.199* +0.067 6 5 2, 5 4 1 1.6 4521.798 +0.071 7 4 3, 6 5 2 1.0 4518.655 +0.802 9 4 5, 8 5 4 0.7 4494.756 -0.290 9 4 5, 10 5 6 0.9 4707.731* -0.019 10 5 6, 9 4 5 0.5 4484.396 +0.084 1 1 0, 1 0 1 4.1 4610.750* +0.040 1 0 1, 1 1 0 4.2 4619.132* -0.059 2 2 0, 2 1 1 1.8 4609.310 +0.090 2 1 1, 2 2 0 1.9 4618.735 -0.056 I72 Observed frequencies of 2 V1 (contd) J" K: K21 J1 K4 K11 I L6bs yobs-ycalc 3 3 0, 3 2 1 5.7 4607.059 +0.120 3 2 1, 3 3 0 5.8 4618.300* -0.011 4 4 0, 4 3 1 1.5 4603.948 +0.113 4 3 1, 4 4 3 1.6 4617.890* +0.029 5 5 0, 5 4 1 3.4 4600.009* +0.107 2 1 1, 2 0 2 0.7 4602.960 +0.003 2 2 1, 2 1 2 2.0 4601.612 -0.087 3 1 2, 3 2 1 3.8 4623.582 +0.119 3 2 1, 3 1 2 3.6 4602.121 +0.102 3 3 1, 3 2 2 0.8 4599.532 -0.181 4 3 1, 4 2 2 1.3 4600.776 +0.282 4 3 2, 4 4 1 2.7 4626.430 +0.504 4 4 1, 4 3 2 2.6 4596.549 -0.554 5 4 1, 5 3 2 . 4598.686 +0.472 3 1 2, 3 0 3 1. 4593.668 -0.013 3 0 3, 3 1 2 2.1 4633.163* +0.104. 4 3 2, 4 2 3 2.5 4591.477 -0.072 4 2 3, 4 3 2 2. 4631.126 -0.575 4 2 2, 4 1 3 0. 4592.141* -0.021 5 3 2, 5 2 3 2.5 4590.425* +0.062 5 2 5, a 3 2 2.8 4628.088 +0.467 5 4 2, 5 3 3 0.8 4588.899 -0.161 6 5 2, 6 4 3 1.9 4585.591 -0.372 6 4 3, 6 5 2 2.1 4628.263 +0.805 . 4 1 4, 4 2 3 1.8 4640.022 -0.071 4 2 3, 4 1 4 1.6 4584.517 -0.001 173 Observed frequencies of 2 V1 (contd) J" KSIK£| J' K; Kh I v v -v obs obs calc. 4 1 3, 4 0.5 4584.517 —0.026 1 4, 2 3 2.3 4637.538 +0.020 5 2 3, 5 1 4 2.0 4582.386 -0.036 5 3 3, 5 2 4 0.6 4582.290 -0.036 6 4 3, 6 3 4 1.8 4579.565 -0.090 4 3, 3 4 1.3 4577.071 -0.048 7 5 3, 7 4 4 0.4 4576.275* ~0.195 5 1 4, 5 0 5 1.2 4575.282 -0.012 5 0 5, 5 1 4 1.5 4646.617 -0.055 6 3 4, 6 2 5 1.5 4572.690 -0.026 6 2 5, 6 3 4 1.7 4643.644 -0.090 7 2 5, 7 3 4 1.5 4640.115 -0.079 8 5 4, 8 4 5 0.8 4566.213* -0.014 6 1 6, 6 2 1.2 4652.998 -0.046 6 2 5, 6 1 6 0.9 4565.847 -0.012 7 1 6, 7 2 5 1.3 4649.515 -0.068 8 3 6, 8 4 5 1.0 4645.569 -0.033 8 4 5, 8 3 6 0.8 4559.446‘r -0.037 9 4 5, 9 3 6 0.5 4555.599* -0.060 APPENDIX IV Comparison of Observed and Calculated Ground State Combination Differences. The quantities listed in the following tables are: The quantum numbers of the levels between which the combination differences were obtained. The calculated combination differences which are obtained from Eq. (41) and the constants of Table XI. The average combination differences which are taken from the spectra. The assigned weights, which are equal to the number of observed combination differences used to obtain the average combination difference. The comparison of the observed and calculated combin- ation differences. 174 175 The Combination Differences of HQBOSe: (3) (b) (c) (d) (e) J K_| K+I1 J K_|K+| Calc Obs Wts Obs-Calc 1 0 1, 3 0 3 47.285 47.292 2 +0.007 2 l 2, 4 1 4 62.643 62.653 1 +0.011 3 0 3, 5 0 5 78.246 78.243 3 -0.003 4 1 4, 6 1 6 3.816 93.818 1 +0.00? 5 0 5, 7 0 7 109.377 109.375 3 -0.002 6 1 6, 8 1 8 124.926 124.931 2 +0.005 7 0 7, 9 0 9 140.460 140.463 3 +0.00? 8 1 8, 10 1 10 155.979 155.955 1 -0.024 9 0 9, 11 0 11 171.480 171.452 2 -0.028 10 1 10, 12 1 12 186.962 186.909 1 -0.053 2 2 1, 4 2 3 78.064 78.074 1 +0.010 3 l 2, 5 1 4 94.455 94.452 1 -0.002 4 2 3, 6 2 5 109.864 109,861 1 -0.004 5 1 4, 7 1 6 125.405 125.407 1 +0.00? 6 2 5, 8 2 7 140.919 140.925 5 +0.004 7 1 6, 9 1 8 156.418 156.413 2 , -0.005 8 2 7, 10 2 9 171.900 171.902 2 +0.002 9 1 8, 11 1 10 187.365 187.358 2 -0.005 176 The Combination Differences of H2803e (contd): J K_|K+I , J K_'K+I Cale Obs Wts Obs-Cale 2 2 O, 4 2 2 94.323 94.308 1 -0.015 4 3 2, 6 3 4 125.601 125.603 2 +0.003 5 2 3, 7 2 5 141.420 141.416 2 -0.005 6 3 4, 8 3 6 156.815 156.817 1 +0.00? 7 2 5, 9 2 7 172.262 172.263 2 +0.001 8 3 6, 10 3 8 187.685 187.675 2 -0.010 9 2 7, 11 2 9 203.089 203.086 2 -0.003 10 3 8, 12 3 10 218.470 218.484 1 +0.014 5 3 2, 7 3 4 158.094 158.079 1 -0.015 6 4 3, 8 4 5 172.555 172.561 2 +0.006 7 3 4, 9 3 6 188.059 188.071 2 +0.01? 9 3 6, 11 3 8 218.694 218.692 2 -0.00? 10 4 7, 12 4 9 233.992 233.992 1 +0.000 7 4 3, 9 4 5 204.218 204.207 2 -0.010 2 2 1, 3 2 1 45.660 45.653 2 -0.007 3 3 1, 4 3 1 59.975 59.967 1 -0.009 4 4 l, 5 4 1 73.862 ‘73.859 1 -0.003 2 1 1, 3 3 1 49.634 49.638 1 +0.004 2 0 2, 3 2 2 47.711 47.731 1 +0.019 3 1 2,, 4 3 2 63.744 63.742 3 -0.00? 3 2 2, 4 2 2 62.864 62.838 1 -0.026 4 3 2, 5 3 2 77.866 77.865 1 ~0.00? 5 3 2, 6 5 2 96.606 96.623 1 +0.017 1 1 0, 2 1 2 15.604 15.567 1 -0.056 2 2 o, 3 2 31.460 31.470 1 +0.011 4 2 3, 5 2 3 79.053 79.037 2 -0.016 177 The Combination Differences of H280Se (contd): J'K__| K+| , J K_I K+I Calc Obs Wts Obs-Cale 5 2 3, 6 4 3 94.890 94.889 1 -0.001 6 4 3, 7 4 3 109.605 109.608 2 +0.003 2 2 1, 3 0 3 14.620 14.598 1 -0.0?? 4 1 4, 5 1 4 79.255 79.263 +0.008 5 l 4, 6 3 4 94.890 94.898 2 +0.008 6 3 4, 7 3 4 110.360 110.364 1 +0.004 3 1 2, 4 1 4 15.200 15.189 1 -0.011 4 3 2, 5 l 4 30.711 30.705 2 -0.006 5‘ 3 2, 6 3 4 47.734 47.716 1 -0.019 5 0 5, 6 2 5 95.061 95.066 2 +0.005 6 2 5, 7 2 5 110.609 110.608 2 -0.001 7 2 5, 8 4 5 126.025 126.010 1 -0.015 4 2 3, 5 0 5 14.803 14.818 1 +0.015 5 2 3, 6 2 5 30.811 30.814 1 +0.003 6 4 3, 7 2 5 46.530 46.550 2 +0.020 7 4 3, 8 4 5 62.950 62.971 1 +0.021 1 1 0, 2 1 2 15.604 15.607 1 +0.004 2 2 0, 3 2 2 31.460 31.454 1 +0.004 3 3 0, 4 3 2 47.221 47.228 2 +0.00? 4 4 0, 5 4 2 62.814 62.811 1 -0.004 5 5 0, 6 5 2 78.175 78.194 +0.020 0 1, 2 2 1 32.665 32.669 2 +0.004 1 1 1, 2 1 1 30.894 30.877 1 -0.017 2 2 1, 3 2 1 45.660 45.645 2 -0.015 2 1 1, 3 3 1 49.634 49.612 1 -0.02? 2 2 1, 3 0 3 14.620 14.615 3 -0.006 178 The Combination Differences of H28OSe (contd): JIK__| K+I’ J K_I K+| Calc Obs Wts Obs—Calc 2 1 1, 3 1 3 15.951 15.945 1 -0.006 4 4 l, 5 2 3 44.454 44.453 1 -0.001 5 4 1, 6 4 3 65.482 65.487 1 +0.004 2 1 2, 3 1 2 47.443 47.435 2 -0.008 3 1 2, 4 3 2 63.744 63.715 2 -0.0?9 4 3 2, 5 3 2 77.866 77.862 1 -0.004 3 1 2, 4 l 4 15.200 15.206 1 +0.005 4 3 2, 5 1 4 30.711 30.752 1 +0.04? 5 3 2, 6 3 4 47.734 47.721 2 -0.013 5 4 2, 6 2 4 46.245 46.24? 1 -0.003 6 5 2, 7 3 4 61.488 61.512 1 +0.023 3 0 3, 4 2 3 63.443 63.428 2 -0.015. 4 2 c, 5 2 3 79.053 79.049 1 -0.005 5 2 3, 6 4 3 94.890 94.889 2 -0.001 6 4 3, 7 4 3 109.605 109.594 1 -0.011 4 2 3, 5 0 5 14.803 14.812 2 +0.009 2 3, 6 30.811 30.825 2 +0.014 4 3, 7 2 5 46.530 46.505 2 -0.026 7 4 3, 8 4 5 62.950 62.966 1 +0.016 4 1 4, 5 1 4 79.255 79.265 1 +0.011 5 1 4, 6, 3 4 94.890 94.882 1 -0.008 6 3 4, 7 3 4 110.360 110.360 1 +0.000 5 1 4, 6 1 6 14.561 14.530 1 -0.032 6 3 4, 7 1 6 30.515 30.536 2 +0.021 7 3 4, 8 3 6 46.456 46.457 1 +0.001 5 0 5, 6 2 5 95.061 95.068 1 +0.007 179 The Combination Differences of H2808e (contd): J K_|K+', J'K; K+' Calc Obs Wts Obs-Calc 6 2 5, 7 2 5 110.609 110.608 1 -o.001 6 2 5, 7 0 7 14.516 14.300 1 -0.016 7 2 5, 8 2 7 30.510 30.526 1 +0.016 6 1 6, 7 1 6 110.844 110.837 1 -0.007 8 5 6, 9 3 6 141.603 141.596 1 -o.008 7 1 6, 8 1 8 14.082 ‘14.091 1 +0.009 9 3 6, 0 3 8 46.082 46.092 1 +0.010 3 1 2, 5 3 0 16.525 16.511 5 -0.013 5 5 2, 5 5 0 18.431 18.429 1 -0.002 3 0 3, 3 2 1 51.040 51.043 1 +0.002 5' 2 3, ‘5 4 1 29.408 29.402 1 -0.005 4 1 4, 4 3 2 48.544 48.528 2 -0.016 5 1 4, 5 5 2 47.156 47.158 2 -0.017 5 2 4, 5 4 2 48.631 48.659 1 +0.008 6 5 4, 6 5 2 48.871 48.852 1 -0.020 5 0 5, 5 2 3 64.250 64.241 2 -0.009 6 2 5, 6 4 3 64.079 64.079 1 +0.000 7 2 5, 7 4 5 63.075 63.044 1 -0.050 6 1 6, 6 3 4 80.529 80.308 1 -0.020 180 The Combination Differences of H2788e: 4* (a) (b) (c) (d) (e) J K4 K+|, J K4 K+, Calc Obs Wts Obs-Calc 1 0 1, 0 3 47.302 47.306 2 +0.005 2 1 2, 4 1 4 62.662 62.693 2 +0.051 5 0 ., 5 0 5 78.270 78.260 2 -0.010 4 1 4, 6 1 6 93.844 93.876 1 +0.051 5 0 5, 7 0 7 109.410 109.399 2 -0.011 6 1 6, 8 1 8 124.965 124.961 1 -o.002 7 0 7, 9 o 9 140.502 140.480 1 -0.025 9 0 9, 11 0 11 171.551 171.521 2 -0.011 10 1 10, 12 1 12 187.018 186.975 1 -o.045 2 2 1, 4 2 5 78.078 78.086 2 +0.008 5 1 2, 5 1 4 94.484 94.502 1 +0.018 4 2 5, 6 2 5 109.897 109.905 3 +0.006 7 1 6, 9 1 8 156.464 156.457 1 -0.007 8 2 7, 10 2 9 171.950 171.945 1 -0.005 9 1 8, 11 1 10 187.417 187.451 1 +0.015 10 2 9, 12 2 11 202.865 202.870 1 +0.006 4 3 2, 6 5 4 125.633 125.652 2 -0.001 5 2 5, 2 5 141.461 141.450 1 -0.010 6 3 4, 5 6 156.858 156.848 2 -0.010 7 2 5, 9 2 7 172.309 172.525 2 +0.014 10 5 8, 12 5 10 218.528 218.554 1 +0.026 7 5 4, 9 5 6 188.107 188.118 1 +0.012 181 The Combination Differences of H2788e (contd) J K-IK+I’ J K4 KH Calc Obs Wts Obs-Calc 2 2 1, 3 2 1 45.655 45.655 2 +0.002 5 2 1, 4 4 1 67.060 67.048 1 -0.012 3 1 2, 4 5 2 63.765 65.769 1 +0.004 4 5 2, 5 3 2 77.874 77.868 1 -o.005 5 5 2, 6 5 2 96.642 96.642 1 -0.001 4 1 4, 5 1 4 79.277 79.249 1 -0.029 1 2, 4 1 4 15.207 15.255 2 +0.028 4 3 2, 1 4‘ 50.719 50.740 2 +0.021 1 1 0, 2 1 2 15.608 15.624 2 +0.016 5 5 0, 4 5 2 47.250 47.226 2 -0.004 1 0 1, 2 2 1 52.686 32.679 2 -0.006 2 2 1, 3 2 1 45.653 45.651 2 -0.002 2 2 1, 5 0 3 14.616 14.621 1 +0.005 5 2 1, 4 2 5 52.425 52.435 2 +0.011 3 1 2, 4 5 2 65.765 65.760 1 -0.006 5 0 3, 4 2 3 65.462 63.430 1 -0.052 4 2 5, 5 2 5 79.074 79.082 1 +0.008 5 2 5, 6 2 5 50.823 50.844 1 +0.021 6 4 5, 7 2 5 46.547 46.545 2 -0.005 6 5 4, 7 3 4 110.385 110.368 1 -0.017 6 5 4, 1 6 30.527 30.528 1 +0.001 7 5 4, 8 5 6 46.473 46.482 2 +0.009 5 0 5, 6 2 5 95.088 95.090 1 +0.001 6 2 5, 7 0 7 14.322 14.503 1 -0.019 4 2 5, 4 4 1 54.635 54.636 1 +0.001 5 2 3, 5 4 1 29.594 29.599 1 +0.005 182 The equations listed on the following pages were used to obtain values for (A” - C") and (A” — B”) using the micro- wave data of Jache, Moser, and Gordy, and using the taus obtained from the near infrared spectra. From 1/(1:LO <—-lOl) = V1 2 A - C=Vl '(1/11)[(l " 1") Taaaa _ 8(2- 8) Tbbbb + 2“ ’ r " rs) Taabb + 111-abab] From z’(220<“’ 211) = ”2 2 2 2 A - B = 2 2 -11 112 + 3(1 - 2r + r ) Taaaa '1‘ 35 Tbbbb + 6s(INCH-raabb Where . 2 2 C = A _ c +(1.75 + 0.5r + 1.75r ) Taaaa -(2 + S - 1-758 )Tbbbb + 0.5(s - 21‘ + 7P8 - 2) Taabb " 2 Tabab c—A-C—r(l+r)T 4511-82)? 2 — aaaa bbbb +(l - s - 2rS) Taabb + 2 Tabab ~A C (211°+I‘2)T -(l+28+82)T _ + + aaaa bbbb + (l + s - 2r + 2P8) Taabb + 4 Tabab 184 2 2 2 2 where - 6(11 .. 0) + 16.5(82 - 111- C = 2 + S(l6.53 + 3) Tbbbb 1 88.88. V 3 + 3(1lrs + r ' l) Taabb - 6 ‘rabab 02 = A — c + (20 - 5.51“ - r2) Taaaa + 117-5 + 9-5S + $2) Tbbbb + (2.5 + 5-55 ‘ 9.5r ’ 2rs) z-aabb + 5.Tabab 2 - A - C + I‘(r' - 25) Taaaa 7112-5 ‘ 1-53 ‘ S) Tbbbb + (2.5 - 1.51" - 2058 " 2T8) Taabb + 5Tabab . 2 2 c1=1-c+(4+8r-5r>7aaaa-5<1+28+S”bbbb + 2(4 + 48 - 5P - 5rs)'raabb + l61.5-abab r(2.5r + l)‘T + 2-5(1 ' 52) urbbbb O 11 11> I O l aaaa + (l - S - 5rs)‘raabb + e‘rabab 2 — 4C1 (4C4 - 12005) 2 2 2 — - 404 + 1200 + 111C2 — 30C3 5 2 2 2 2 2 C - 'Cl - C4 ' 6005 + C2 + 15C3 “WWE‘EWLWMQWETS