A STUDY OF CORfOLiS-TYPE RESONANCE IN HYDROGEN SULFIDE ‘z‘hesés fer the Degree 0f Ph. D. h‘ziCHEGAfi STATE UE‘EWERSETY LEE/‘JES E. 3mm 1.98? Tubui This is to certify that the thesis entitled A STUDY OF CORIOLIS-TYPE RESONANCE IN HYDROGEN SULFIDE presented by LEWIS E . SNYDER has been accepted towards fulfillment of the requirements for Major professor Dme Auqust 4.L967 0-169‘ ABSTRACT A STUDY OF CORIOLIS-TYPE RESONANCE IN HYDROGEN SULFIDE by Lewis B. Snyder The second-order quantum-mechanical molecular Ham- iltonian expression for the nonlinear XYX asymmetric molecule is modified to include Coriolis-type perturbations between pairs of mutually interacting infrared absorption bands. As a result, expressions are found for the coupled Wang energy matrices which include the centrifugal stretch- ing terms and the perturbation coefficients. The modified Hamiltonian is used to analyze simultaneously the (110) and (011) absorption bands of H28 near 3800 cm-1. Ground state combination differences from these two bands to- gether with those from the (210) and (111) bands (near 6300 cm-1) were combined with seven microwave lines to obtain values for the ground state parameters A, B, C, four taus, and H for H328. Using these ground state values, K the (110) and (011) bands of H2328 were reanalyzed to obtain final values for the upper state parameters (v0, A, B, C, four taus, and HK) of each band and the parameters (ny and 62) coupling the two bands. A STUDY OF CORIOLIS-TYPE RESONANCE IN HYDROGEN SULFIDE BY ;\ Lewis ED Snyder A THESIS Submitted to Michigan State.University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1967 K1“) TO MY WIFE- DORIS JEAN ii ACKNOWLEDGMENT I wish to thank Dr. T. H. Edwards for suggesting and encouraging this research and for his support through a Re- "" search Assistantship financed by the Air Force Cambridge Research Laboratories. I am also grateful to the National Science Foundation for two Summer Fellowships. vaarticularly wish to thank N. Kent Moncur for the use of his data analysis program and for checking many of the calculations necessary for this work.» I am grateful to Melvin Olman for all of his programming assistance. The averaging program-written by Lamar Bullock and Don Keck was very help- ful to me. I appreciate the advice and help given by Dr. Thomas Barnett when beginning my research. I'am grateful to the M.S.U. Computer Laboratory for the use of theirfacilities. The molecular spectrosc0py classes of Dr. C. D. Hause and Dr. P. M.-Parker were instrumental in developing an understanding for the theoretical aspects of this work. I am very grateful to Dr. T. H. Edwards, Dr. P. M. Parker,. Dr. J. A. Cowen and Dr. P. A. Schroeder for leaving their special research interests long enough to serve on my exami— nation committee. Finally, I am deeply indebted to my wife, Doris Jean Snyder, for her support and understanding throughout this research. iii TABLE OF CONTENTS ACKNOWLEDGMENT. . . . . . . . . . . . . . . . LIST OF TABLES O O O O O O O O O O O O O O O 0 LIST OF FIGURES . . . . . . . . . . . . . . . LIST OF APPENDICES O O O O O O O O O O O O O 0 INTRODUCTION 0 O O O O O O O O O O O O O O O 0 CHAPTER I. II. III. GENERAL CONSIDERATIONS FOR AN ASYMMETRIC TOP MOLECULE. . . . . . The Rigid Asymmetric Rotor Model . The Wave Function. . . . . . . . . Rotational Symmetry and Selection Rules. Relative Intensities . . . . . . . Coriolis-Type Perturbations. . . . THE HAMILTONIAN OPERATOR . . . . . . . The Untransformed General Hamiltonian. The Untransformed Hamiltonian Operator for the Nonlinear XYX Molecule. Examination of the Untransformed Hamiltonian Operator. . . . . . A Numerical Investigation of the Resonance Terms . . . . . . . . APPLICATION OF THE MODIFIED CONTACT TRANSFORMATION . . . . . . . . . . The Vibrational Representation . . Finding the Transformed Hamiltonian Hl' Finding the Transformed Hamiltonian H2' iv Page iii vi vii viii 10 16 19 26 26 31 37 4O 45 45 46 50 CHAPTER The Diagonal Elements in the Vibrational Representation . . . . The Off-Diagonal Elements in the Vibrational Representation . . . . Possible Experimental Applications . IV. THE ROTATIONAL REPRESENTATION. . . . . . Operator Matrix Elements . . . . . . The Wang Transformation. .‘. . . . . V. OBSERVED SPECTRA . . .,. . . . . .e. . . History. . .t. . . . . . . . . .~. . Procurement and Measurement of Data. Further Experimental Technique . ... VI. DATA ANALYSIS. . . . .'. . . . . . . .-. Introductory Remarks .,. .*. . . . . Components of»a Typical Frequency Fitting Program . . . . .,. .t. . Programming the Multiple Regression Subroutine. . . . .,. . . . . ._. Method of Analysis . . . . . .~. . . The (210) and (111) Bands. . . . . . The-(110) and (011) Bands. . . . . . Reanalysis of the Ground State . . . VII. CONCLUSION . . . . . . . . . . . . . . . REFERENCES. 0 o o 0'. o_o 0'. o o 0’. o o o 0‘. APPENDICESOO o o o 0'. o o 0'. o o 0'... o 0‘.- Page 52 57 58 62 62 66 77 77 78 81‘ 85 85 86 89 92 94 95 99 103 105 108 Table II. III. IV. V. VI. VII. VIII. IX. X. XI. XII. XIII. LIST OF TABLES The D2 Point Group Character Table . Summary of Selection Rules . . . . The C2v Character Table. . . . . . . . . . . Direct Product Multiplication Table for sz. The Nonvanishing Coefficients a3“B . . . . . The Hamiltonian Coefficients for H1. Contact Transformation Components. The a's. . . . . . . . . . . . . . Identification of a b c with x y z The Coupled Wang Submatrices . . . Experimental Conditions. . . . . . Molecular Parameters of H2328 for Ground State Parameters of H2328 . vi Page 11 15 20 23 30 47 49 S4 64 72 84 97 101 Figure 10. ll. 12. LIST OF FIGURES Rigid Asymmetric T0p Energy Levels . . Eulerian Angles. . . . . . . . . . . . Geometry of the H28 Molecule . . . . . Normal Modes of H23 . . . . . . . . . Classical Coriolis Perturbation. . . . The Interaction Matrix for the (110) and (011) States in the Vibrational Representation . . . . . . . . . . . The Three Band Interaction Matrix for the (210), (111) and (012) States. . Form of the Untransformed Hamiltonian for J = l O O O O O O O O O O O O O O The Double Wang Transformation Matrix for J = l. I O O O O O O O O I O O O The Transformed Hamiltonian for J = 1. Form of the Wang Transformed Hamiltonian for J = 5. . . . . . . . Energy Level Shifts Due to the Perturbations. . . . . . . . . . . . vii Page 11 20 22 23 59 61 67 69 70 76 98 Appendix I. 9 éézae for the Nonlinear XYX Molecule. II. Using IsotOpic Substitution to Find Force III. IV. V. VI. VII. VIII. IX. XI. XII. LIST OF APPENDICES Constant Estimates . . . . . . . The Vanishing Components of Hl'. . The Commutators in HZ' . . . . . . The Purely Vibrational Contributions 0 ll Ev and Ev I O O O O I O O O O O Angular Momentum Matrix Elements . Listing of Program G.S.Ex.D. . . . Listing of Program U.S.Ex.D. . . . Frequency Assignments for the32 (210) and (111) Bands of H2 S . Frequency Assignments for the32 (110) and (011) Bands of H2 S . Listing of Program CORIKORR. . . . Ground State Combination Differences S Of H2 0 O O O O O O O O O O 0 viii Page 108 111 114 115 120 122 123 129 137 144 160 169 INTRODUCTION The purpose of this study is to derive a general quantum—mechanical expression for the perturbed energy levels of the nonlinear XYX asymmetric tOp molecule, while demonstrating the mathematics involved. The derived energy expression will then be used to predict-the spectral ab-. sorption lines of two interacting hydrogen sulfide bands. The predicted absorption lines will be compared with those of spectra observed under resolution high enough that the effects of the perturbation may be clearly observed. The basic theoretical groundwork leading to the quantum-mechanical Hamiltonian is found in the literature - many times in a form that the researcher in the laboratory finds to be totally unrelated to experimental research. In addition, the literature contains many "effective" or empirical Hamiltonian expressions which are postulated with little or no theoretical justification.‘ Hamiltonian expressions of this type are often quite successful in explaining spectra as long as no unexpected perturbations appear or the spectrometer resolution is low enough to cover any anomalies between the observed and predicted values of-the spectral lines. The infrared spectra of H28 have been observed and 2 analyzed sporadically for the past thirty years yet very little quantitative information_is known about the pertur- bations which affect the upper-states of nearly all the- observable vibration-rotation bands. With improved high resolution spectroscoPy, a theoretical approach to the- problem with experimental verification is now possible. CHAPTER I GENERAL CONSIDERATIONS FOR AN ASYMMETRIC TOP MOLECULE 2-32 m Asmetric 3.212.: 22912.1. If a molecule has no threefold or higher axis of symmetry it is an asymmetric rotor since in general the three principal moments of inertia are different. Common examples of asymmetric rotors include water (H20), hydrogen sulfide (H28), hydrogen selenide (HZSe), and acetic acid (CH3COOH). Even though the classical motion of an asymmetric rotor or top is well known (ya), its behavior is not simple and the quantum - mechanical asymmetric rotor Hamiltonian is generally much more difficult to manipulate than that for either the symmetric or the linear rotor. Unlike the usual case for the linear and symmetric rotors, the energy levels of the asymmetric top can not be represented by an explicit formula for all values of its angular momentum; therefore the quantum - mechanical matrix formulation will be relied on heavily. Fortunately, much information about the general asym- metric rotor may be obtained by examining the behavior of a hypothetical rigid asymmetric rotor as it moves from the oblate to the prolate top limit. The use of a rigid top model is justified for the general discussion which follows on symmetry and selection rules since wave function symmetry must be independent of the actual values of the moments of inertia and the vibration - rotation interactions, to a first approximation, bring about only a small correction to the effective moments of inertia (2a). The rigid rotor Hamiltonian operator is given by HR = A a; + B Eh: + c BE: 1’12 1'12 f: (1-1) where A = h/ancIa etc., in cm.-1 The range of values for B between A and C customarily is used to describe the various stages of asymmetry. Using this model, a prolate symmetric t0p is formed in the limit B = C < A where a is the unique top axis and an oblate sym- metric top is formed in the opposite limit A = B >'C where c is the unique top axis. Some authors prefer to always keep c as the unique top axis in which case the oblate de- finition is unchanged but the prolate definition becomes A = B <:C. Fig. 1 shows qualitatively how the energy levels vary as B varies from C to A for a rigid rotor with C = 1.0 cm“ 1 in the prolate limit and A = 2.0 cm"1 in the oblate limit. In practice the energy levels for the intermediate cases must be found by matrix diagonalization but for illus- tration it is sufficient to join the prolate and oblate limits by continuous lines. Much more detailed drawings of the energy level scheme may be found in the literature (3a). Several parameters have been constructed to indicate Fig. 5 1. Rigid Asymmetric Top Energy Levels Prolate Limit B=C Tau J K 3 =5==::: Oblate Limit B=A K J + JK_K+ o 330/ .1 1________________....110 #0:] 1 . .____..— 1 _===::() 4_._i% 1 1 E0 ...——-—""_1"—' J IF I C=l.0 cm"l ( B 4 A=2.0 cm the degree of asymmetry (value of B with respect to A and C) but the most pOpular one is probably Ray's asymmetry param- eter kappa where K =2B-A-C A - C (I-2) In the prolate limit K=-1 and in the oblate limit K=+l. If the kappa value is known for a given molecule, the energy levels can be found from published interpolation tables with- out the use of a computer. However, with the increasing availability of high speed digital computers, rigid rotor interpolation tables have lost much of their importance because an entire energy matrix can now be diagonalized in a few minutes using an order of approximation much higher than the rigid rotor. The interested reader is referred to the literature for a discussion of E (K) tables (4 ). Also, many good approximate solutions have been devel- oped for cases of small asymmetry near either the oblate or the prolate limit in Fig. l. Townes and Schawlow have sum- marized many of the approximate solutions and have a general discussion on their range of usefulness (5a). In the case of no applied external field, the total angular momentum J and its projection on a space - fixed axis, M, are both constants of the motion for an asymmetric top; therefore J and M are called good quantum numbers. However, whereas the projection K of the total angular mo- mentum J on the symmetry axis of a symmetric t0p is a constant of the motion, this is no longer true for the asymmetric t0p and there is no good quantum number which uniquely spec- ifies a rotational energy level. The quantum number K is still used but it is no longer a constant of the motion as is illustrated in the matrix elements given in Chapter IV. Since K is not a good quantum number, various param- eters have been constructed to uniquely specify a rotational energy level. One method is to give the J value along with the corresponding K values for the limiting prolate (K-) and the limiting oblate (K+) symmetric top energy levels. Thus in Fig. l, the highest energy level is designated by 330 using the JK-K+ notation. This notation will be used ex- tensively in the experimental aspects of this work. An older method of description using the parameter tau is use- ful for computer programming purposes. Tau is an ordering index which starts with the highest rotational energy level for a given J and goes to the lowest in order from +J to -J. Thus in Fig. l the values of tau for the J = 3 levels are 3,2,l,0,-l,-2,-3 and each level may also be labeled by JT. The relationship I = K_ - K+ (1-3) may be used to determine the value K- when working in an ob- late representation and vice-versa. For example, in an oblate representation, the quantum number K is K+ in the oblate limit. By inspecting the energy levels of known J and K, tau may be readily determined for a given level and K- for that level follows immediately. Tau is also of interest because the highest rotational level'F=J can be thought of as in- volving a rotation about the axis of least moment of inertia for a set of levels with given J. Likewise, the lowest level T=-J may be considered as due to a rotation about the axis of largest moment of inertia (la). EEE.EEXE Function In order to understand the selection rules it is helpful to examine the molecular wavefunction in some de- tail. Disregarding the translational motion of the center of mass of the molecule, the total wavefunction may be approximated as (5b) “’total 2 we V’VR “’n z 11’e “’v “’R ll’n 0-“ where the electronic wave function,‘Pe, is factorable by the Born-Oppenheimer approximation. The molecular Hamiltonian remaining (after the electronic energy is removed) is mainly the sum of vibrational and rotational energies; hence the vibration - rotation wave function,‘%VR, may be factored into the product of a vibrational and a rotational contribution. The nuclear spin wave function,din, contributes little to the energy and is neglected until the relative intensities of absorption lines are discussed. The rotational wave functionswJKM of either a prolate or an oblate symmetric rotor form a complete and orthogonal set of functions by which the asymmetric rotor wavefunctions may be expanded (5c) wJK_K+M = E aJ'KM'wJ'KM' I J'KM (1-5) where the aJ'KM' terms are the probability amplitudes repre- senting the contributions of each symmetric rotor state. Since J is still a good quantum number for the asymmetric rotor, no more than one J value may be represented in a given state. Furthermore, the M label may be dropped because the energy is independent of molecular orientation in the field- free case. Thus the sum in (I-S) may be reduced to one over K alone. The symmetric rotor basis functions in (I-S) are not convenient to use for symmetry reasons which will be discussed in the next section. Instead, it is more convenient to form the Wang linear combinations of symmetric rotor functions (4 ) -J. Y s wJ,-K (I-5) where y'may be either odd or even (for convenience, l or 0) and the absolute value of K is used. For K = 0, only even y exists, and the wave function becomes S(J,0,0) = 0J0 (I-7) Using the Wang wave function combinations, the asymmetric tOp wave functions may be written in a manner analogous to 10 (I-S) (3a) wJK_K+ = g (a ) S(J,K, ) which will be useful for symmetry considerations. The symmetric rotor wave functions in (I-S), (I-6) and (I-7) are given by (6 ) _ iM¢ in wJKM(e.¢.x) - eJKM(e)e e (I-9) where C)JKM(9) is a hypergeometric function. The Eulerian angles are shown in Fig. 2 to avoid confusion with the other sets in the literature. Rotations about the top axis are described by x while 9 and ¢ measure rotation with respect to the space-fixed axis system XYZ. Rotational Symmetry and Selection Rules For any asymmetric rotor the selection rules are complicated by the large number of distinct rotational levels and the arbitrary direction of the permanent dipole moment. The parity of the rotational eigenfunctions can be used to classify the energy levels since it is conserved as the rotor moves from the oblate to the prolate limit as shown in Fig. 1. Since a small change in the moments of inertia corresponds to a perturbation which cannot change the symmetry of the eigenfunction, the general selection rules may be derived from the symmetry relations which must be true for the 11 Fig. 2. Eulerian Angles z X e I I : + Y ¢ \ I \\\\\ I x \I Table I. The D2 Point Group Character Table K K . c b a - + SpeCie E C2 C2 C2 e e A l l l o e BC 1 l -l -1 Tc 0 o Bb l -l l -l Tb e 0 Ba ]. -l -1 1 Ta 12 symmetric rotor eigenfunctions in both the oblate and prolate limits. The rotational Hamiltonian Operator (I-l) remains unchanged when Pa, Pb and PC are replaced by their negatives; this is equivalent to performing a two-fold rotation about each of the principal axes (a, b, and c) of the molecule. These three rotations, Cza, C2b and C20, along with the identity Operator E constitute the point group Q2 (also called V) which requires three mutually perpendicular axes (3a). Each of the nondegenerate asymmetric top eigenfunc- tions in (I-8) constructed from the Wang linear combinations of symmetric rotor eigenfunctions forms the basis of an irreducible representation for D2 while the linear combi- nation in (I-S) does not (4 ).1 Thus using the Mulliken notation, each of the rotational levels can be assigned rotational symmetry species A, Ba, Bb, or BC; A is used for the totally symmetric representation and the subscripts indicate the particular rotational axis which gives a character of £1 (7 ). In the limiting prolate symmetric rotor case, C2a about the symmetry axis a increases x in (I-9) by in, Thus S (J,K,y) is unchanged if K- is even and changes sign if K- is odd, giving the parity of the wJK_K+ under the operation C23. In the limiting oblate rotor case, CZC about c gives a character of +1 for K+ even and.—l for K+ odd. The four irreducible representations found in this manner may be written as (++), (+-), (-+) and (--) using Dennison's 13 notation (2a). They may also be labeled by the King, Hainer and Cross notation in terms of the evenness or oddness of K+ and K-. Table I gives the character table for the D2 point group and correlates the various notations that are used to label the four irreducible representations (3a). The sym- metry about the intermediate axis b may be determined from the symmetry about a and c since successive rotations of u about each of the three axes must return the molecule to its original configuration. Hence the wave function is symmetric for a rotation of n about b if K- and K+ are both odd or both even; otherwise it is antisymmetric. The usual selection rules AJ ==0, $1 for dipole radiation of a rotating body also apply to asymmetric rotors. Further selection rules are found from the symmetry prOperties of the wJK-K+ listed in Table I and the form of the matrix element governing intensities. For a transition between two rotational states 1 and Z the form of this matrix element is <1|Mi|2>= f ¢*(1) Miw(2) dT JK_K+ JK_K+ (I-lO) where M1 is the component of the permanent dipole moment of the molecule projected onto the space-fixed axis system. The component of the dipole moment along any principal axis of the molecule is unchanged by a rotation about that axis but changes sign under rotation about either of the other two axes. Such a component thus transforms in the same manner as a translation along that axis when the Operations Q23, 92b! 14 C2c or E are applied. The symmetry properties of the possible translations Ta, Tb, and TC are listed in Table I. As indicated in Table I, Tar Tb or TC is unchanged by a rotation about the translational axis but changes sign under the other two rotation Operations. It is well known from group theory that an integral of the form (I-lO) must be equal to zero unless a represen- tation to which Mi belongs is contained in the direct product of the representations of the eigenfunctions ¢(:)JK_K+ and W(2)JK_K+ (2b). That is, the representation generated by the integrand must contain the totally symmetric represen- tation if (I-10) is to be nonzero. In simpler terms, the integral (I-10) over coordinate space must give a purely numerical result. If the component of the dipole moment lies along the a axis, it will reverse direction if rotated by u about b or c. But the integral must be invariant under any of the group symmetry Operations such as Czc. Thus if Mi changes sign under Czc then the product of the eigen- functions in (I-10) must also change sign in order to keep the integral invariant under such a coordinate transformation. Otherwise, the integral must vanish. The direct products of D2 which contain a representation of Ta, Tb, or Tc are listed in Table II along with the selection rules which follow from them. In all cases AJ = 0,il. Table II may be easily verified by inspecting Table I. Experimentally, absorption bands are often labeled as type A, B or C bands depending upon whether the dipole moment lies along a, b or 15 om ++ 00 m x m . m x m o o U m 9 O0 ++ mm c xom uom x d e O .O o oo ++ co m x m . m x m m am he 00 ++ 0m 4 Xflm «Qm K ¢ 00 ++ O0 0m Xflm «gm XUM m m d m B om ++ on < xmm “mm x s mm>9 mcofluflmcmue mmfloomm mcoflumamcmua comm unscaafl ucwfioz cmsoaad mundconm DOOHHQ maoafla mmasm cofiuowawm mo mumEEsm .HH magma 16 c, as shown in Table II. Relative Intensities Knowing the relative intensities of spectral absorp- tion lines in a given vibration band is an invaluable aid in making proper assignments of rotational transitions. The intensity of a transition from state n" to a state n' is given by (8 ) 8n2ng "ll-exp(-hv/kT)]exp(-E "/kT I = n n l|2 n",n' 3hc2n Gn exp(-En/kT) (I-ll) whereV is the frequency of the absorption line, N is the molecular density and n represents the quantum numbers of the states involved. En" is the rotational energy of the lower state and gnu its statistical weight factor. The fractional number of molecules in the ground vibrational state is [l-exp(-hv/kT)]. The summation term represents the rotational partition function and the last term is the square of the magnitude of the dipole vector matrix element. The dipole matrix element in (I-II) separates into the product of vibrational and rotational matrix elements. For band analysis, it is only necessary to know that the vibrational part is not zero and observation of the band being analyzed confirms this. The remaining rotational part II2 is defined as the line strength and con- tains the degeneracy factor (2J+l) due to the (2J+l)-fold 17 degeneracy in the absence of an external field. For a given vibration-rotation band, many of the terms in (I-11) are approximately constant. This allows the relative intensities to be found by reducing (I-11) to a product of the statis- tical weight, Boltzmann factor and line strength. The resulting relative intensity expression may be written I = gnu eXp(-En../kT)||2 n",n' (I-12) Tables of line strengths as a function of kappa may be found in the literature (9 ,10). The nuclear statistical weight factor 9n" for hy- drogen sulfide is easily determined. Since the two hydrogen nuclei are equivalent, i.e. identical isotopes of the same element with the same molecular environment, H28 must have a twofold axis of symmetry. The twofold axis is the b axis and a rotation of n about b interchanges the position of the two hydrogen nuclei. Since H1 is a Fermion, it must always occur in antisymmetric wave functions; hence any true wave function wJK-K+wn for H28 changes sign when the two hydrogen nuclei are interchanged by the operation Czb. The spin function of each hydrogen nucleus may be written as Si(+) or Si(-) where + or - indicates the pro- jection on the Space-fixed axis to be either +% or -%. The four combinations of the two spin functions of each nucleus are 51(+)S2(+)' Sl(')52(')' Sl(+)SZ(-) and Sl(-)Sz(+). The first two combinations are clearly symmetric with respect 18 to an interchange of spin coordinates, but the last two are neither antisymmetric nor symmetric. To remedy this, sym- metric and antisymmetric combinations may be formed from the last two sets of spin functions. The four prOper spin functions are then of the form S1(+)Sz(+) (Symmetric) S1(’)Sz(‘) (Symmetric) Sl(+)SZ(-)+Sl(-)Sz(+) (Symmetric) S1(+)52(‘)'31(')52(+) (Antisymmetric) (I-l3) The number of symmetric (ortho) Spin functions is three and the number of antisymmetric (para) is one in (I-13). To satisfy Fermi-Dirac statistics, the symmetric spin func- tions must be combined with the antisymmetric rotational functions wJK-K+ and the antisymmetric spin functions with the symmetric rotational functions. Table I shows that the es and 00 states of w are symmetric while the oe and so JK_K+ states are antisymmetric under the Operation Czb. Since there are three times as many symmetric spin states as antisymmetric spin states, the statistical weight of the antisymmetric rotational levels is three times greater than the statistical weight of the symmetric rotational levels. Thus, for a transition originating from the ground vibra- tional state, if (K-+K+) is an odd number in the ground rotational state its relative gn" factor is three. 19 Experimentally, it is confirmed that the Odd component of an absorption line usually has a greater intensity than its even component. Coriolis—Type Perturbations Coriolis—type perturbations occur in hydrogen sulfide between vibrational levels of different symmetry species as a consequence of molecular rotation. This effect is experi- mentally observed as a irregular deviation of the spectral absorption lines from those predicted by a Hamiltonian which assumes that the excited vibrational states are uncoupled (ll). Justification for the existence of Coriolis-type perturbations usually comes from group theoretical consideration; however, for a pure Coriolis perturbation a simple classical picture may be used for visualizing the general effect. Both ap- proaches will be discussed. In order to correlate this discussion with that of others, the coordinates of the hydrogen sulfide molecule are chosen such that x = a, y = b and z = c as shown in Fig. 3. The a and b axes lie in the plane formed by the molecule while c is perpendicular to this plane. The C2V character table for.the point group of the molecule is constructed in Table III to be consistent with the axis labeling chosen. The symmetry species of the rotations about and the trans- lations along each axis are also given in Table III. By convention, A denotes species that are symmetric with re- spect to rotation about the symmetry axis b and B denotes 20 Fig. 3. Geometry of the H28 Molecule Table III. The C2V Character Table Species E C2(b) o(bc) o(ab) Al 1 1 1 1 Th A2 1 1 -1 -1 Rb El 1 -1 1 -1 T R C a 132 1 -1 -1 1 T R a C 21 those that are antisymmetric. Thus three rotations R , Rb a and RC have symmetry species B1, A and B2 respectively. 2 The three approximate normal modes of H28 are indicated in Fig. 4 along with the apprOpriate normal frequencies (12). Each normal mode is represented by the attached set of dis- placement vectors and may be assigned a symmetry species by examination of the manner in which the vectors transform under the Operations of the C2V point group. By one conven- tion, lower case letters are used for vibrations while capital letters are reserved for representations generated by vibrational, vibronic and electronic wave functions (3b). Thus Ql, the symmetrical bond stretching mode, and.QZ, the deformation mode, both have symmetry species a1 while Q3, the asymmetric bond stretching mode, has species b2. The discussion by Herzberg is quite helpful for visualizing a pure Coriolis perturbation from a classical approach but due to a different choice of axis labeling, his symmetry species for normal mode Q3 is b1 instead of b2 (lb). The effect of the rotation RC on the molecule as it vibrates in the normal mode Q3 is shown in Fig. 5. The Coriolis force 2 m X x‘g acting on each nucleus mainly tends to excite normal mode 02 with frequency\g. To a slight extent the mode Q1 is also excited since the displacement vectors for Q1 and Q2 (which have the same symmetry species) are not strictly orthogonal to one another. Since the hydrogen sulfide normal frequency v1 is very close to v3, the effect of the 22 Fig. 4. Normal Modes of H28 Ql(al) 01:2721.92cm-1 Q2(al) v2=1214.51cm'l Q3(b2) \)3=2733.36cm_l 23 Fig. 5. Classical Coriolis Perturbation V3 V3 \/7 Table IV. Direct Product Multiplication Table for C2V A1 A2 Bl 32 A1 A1 A2 B1 B2 A2 A2 A1 32 B1 B1 Bl 32 A1 A2 32 32 B1 A2 A1 24 Coriolis interaction between Q1 and Q3 is actually greater than that between Q2 and Q3. Thus the Coriolis resonance occurs between Q1 and Q3. Jahn's rule is often mentioned in the literature as a test for predicting which pairs of absorption bands are likely to be subject to Coriolis perturbation (13). One statement of Jahn's rule is that a Coriolis perturbation between two vibrational states is possible if the direct product of the two vibrational species involved contains the symmetry species of a rotation Ra' Rb or Rc' The possible direct products for the C2V symmetry species are given in Table IV; they may be confirmed from Table III (3c). It is readily seen from Table IV that the vibrational species a1 and b2 have only one direct product containing the species of a rotation. That is, the direct product of al and b2 contains B2 which is the species of Rc‘ Thus, by Jahn's rule the Coriolis perturbation shown in Fig. 5 is possible. Jahn's rule is a specific application of the general .group theoretical result used to evaluate (I-lO) since per- turbations between a pair of excited vibrational states Vi and v. can occur when vibrational representation.matrix 3 elements of the type [w* H w d1 Vi 0P Vj (1-14) 25 are nonzero. Jahn's rule describes the special case when HOp in (I-l4) is an angular momentum operator Pi because Pi has the species of a rotation Ri (1c). From a mathematical VieWpoint, the nonvanishing of the coupling matrix element (I-l4) is a necessary but not sufficient condition for a perturbation to occur. In addition, several other conditions must be satisfied. The two vibrational states involved must be close to each other in energy and differ from one another in vibrational quantum number vk and vn such that vk in Vi changes to vkil in Vj and vn in Vi changes to vnrl in Vj in the harmonic approximation (14). This condition will be discussed in some detail in Chapter II. Also, there is a strictly dynamic condition in that the motion of the molecule must be such that the coupling matrix element (I—l4) is not too small for a noticeable effect to occur. Along with a pure Coriolis perturbation, a term of operator form PxPy has been found to contribute to the per- turbation in hydrogen sulfide. This operator may be loosely classified as a Coriolis-type term because it causes a rotation-vibration interaction which has the same location in the vibrational array as the pure Coriolis interaction term. The effect of this term is discussed in Chapter II. CHAPTER II THE HAMILTONIAN OPERATOR The Untransformed General Hamiltonian The general molecular quantum - mechanical Hamiltonian is well known and may be found in the literature (15). 1 1 1 = 1+- .. - _ Ti- ZH “ g (Pa pahJaBu 7 (PB p8)p B * 1 _1 u 7 ps uTr + V(QS) 1 * + Wisp. (II-l) It is called the Darling - Dennison Hamiltonian although an earlier version originally was derived by Wilson and Howard (16). This is an exact expression involving only the Born - Oppenheimer approximation in its derivation. The components of total angular momentum are Rx==.§3 and 0) a the components of internal angular momentum (angular momen- a tum due to only vibration) are pa = X C Qsp* ss' 55' s'. The QS represent mass adjusted normal coordinates and p5* = -ifi(%§ ) the conjugate momentum. The C23. are Coriolis 5 coupling coefficients and will be discussed in detail for the nonlinear XYX molecule. The u and “a depend on the normal 8 coordinates and are given by (15) 26 = I I I I ua8 (I ny a8 + I ayI yB)u = I I _ I 2 “a“ (I 881 YY I BY )u I' -I' -I' T xx xy xz “-1 = _II II -I XY YY yz _II _II I! .1 X2 Yz zzi (II-2) where the moments of inertia in the principal axis system are given by . _ e aa ,aa Iaa - Iaa + Zsas Qs +2ss'A ss'Qst' I = _ a8 _ '08 IaB Zsas Q5 2ss' 55' Q5 Q5' (II-3) Since the a's and A's will be treated later, they will not be discussed here. Finally, V = V(Qs) represents the po- tential energyo In general, the Schroedinger equation of the Darling - Dennison Hamiltonian operator can not be solved exactly, so an approximate solution must be found. In order to do this, the Hamiltonian was ordered in the following equivalent form by Goldsmith, Amat, and Nielsen (15): 28 2H =2a8uaBPaPB -Zc8 (pcuaB +ua8pa)PB £- ‘-%- 11.- +Za8pcuc8p8 + u Z018(pcua8u (pen )) l * ”L * E *2 +uuzs (pS u 2(ps u )) + XSPS + 2V (II-4) The first term in (II—4) represents pure rotational energy, the next four represent Coriolis coupling energy, followed by the vibrational energy and the potential energy. Goldsmith, Amat and Nielsen have expanded u and ”c8 in power series as functions of the normal coordinates. Also, they have expanded the potential function V(Qi) about an equi- librium configuration in a Taylor's series. Next the ex- pansions for ua8,u , and V are substituted into (II-4) and terms of like order are collected according to H = HO + 7H1 + y2H2 + - - - The following general results are found (to second order): N P 2 41 l p 2 :1. C1 .— 7 S + H0 2201a + 2 ESAS (hz qs ) (II-5) (1)a8 2 1 Q h F H (——) q P P 1 7 SOLB I I XS S 0L 8 a B papa - + hCZ II k II q q q H 2a Ia s,s',s 55's 5 s' s (II-6) (2)08 LI 1 1 985' ‘h I? H2 = 3255.208 ( ) qsqs'PaPB IaIB AsASI 29 1 paz + 7‘3 Ia + hcz SSISNSH Iksslsflsll lqsqslqsflqs" I (II-7) Goldsmith, Amat, and Nielsen have carried this expan- sion through H“, but for this work only terms through H2 will be considered. The notation introduced in (II-5), (II-6) and (II-7) includes the dimensionless normal coordinate qS =i§§1E 08, pS = -ifi%§§, and As = (2wcws)2 where ms is the frequency (in cm-l) of the 5 th normal mode. The k's are molecular force constants. The Q“s used in (II—6) and (II-7) are defined in a principal axis system as Q(1)a8 = —aQB s S 05 85 2 n( )as = -AQB. + 2 Ca "CB. u + 2 a1 as SS 83 SII SS 5 S (Szx'y'z <3 (II-8) To simplify the notation, the conventional "e" superscript on the equilibrium moments of inertia has been drOpped. Also, since there are no essential degeneracies for the nonlinear XYX molecule, the degeneracy index 0 has been eliminated. 30 Table V. The Nonvanishing Coefficients aso‘B xx = i . XX - T al 21X SlnY a2 — 2IX cosY YY= % yy=_Jz. al 21y cosy a2 21y SlnY azz = axx+ayy = -21%§ azz = axx+ayy = 21%; l l 1 z 23 2 2 2 z 13 xy = _ f a3 2(IXIy/Iz) Where: . -% (kll' R22) % Siny = 2 1- 2 2 % [(kll-k22) +4k12] z z «% .1 . = - = I I cos - I I Sln C13 C31 ( x/ z) _ Y ( y/ 2) Y z _ _ z _ _ % . _ % C23 - C32- (IX/Iz) Siny (Iy/Iz) cosy 2 2 (cz)+(cz)=l 13 23 31 The Untransformed Hamiltonian Operator for the Nonlinear xyx Molecule In this section, the results of Chung and Parker (17) from Table V will be used to apply the expansions for H1 and H2 to the nonlinear XYX molecule. Since there are only three components of the total angular momentum and three normal modes of vibration, Ho can be written immediately as 2 H = 12 P” + o 2 _ a to”! 3 1 p 2 + 2 Z is? (53— qs) S 1 (II-9) Ho gives the familiar rigid rotator and harmonic oscillator energies. The development of H1 may be done by separately considering the three sums of (II-6) and by (1)a8 = _aSaB from (II-8). Table V includes noting that as the nonvanishing coefficients aSOLB as listed by Chung and Parker for the nonlinear XYX molecule. The first sum in (II-6) may be written - a8 2 1 1 Lia-J- 1"... 1: 72a,828 (A ) quOPB I I S a B 1 3 xx YY ZZ xy = - fiY‘z qs [ES-EX: + as PY2 _§_EE: + as [p p 1+] 7’ 5:1 A i Ix2 Iy2 122 Iny X' Y s 32 2 q cosY q siny P 2 1 1 X3 __h%»(_l_T__ — _Z_T__) _¥; A1? A2? 1x7 X12‘ A2” Iy ' 1 I 1 A " A 1H 2? X7 A3? (InyIz)§f (II-10) The 2 superscript has been drOpped from the Coriolis term + . and [Px’Py] — PXPy + Pny in (II-10). The second sum in (II-6) may be simplified using Table V and the definition of pa: -2——=--———= .c.(Qp*.)1—z a Ia Iz SS SS S S 2 A I 1 P - - -§- -£ ‘ ZSSICSSI (AS )? qspsI ( Z A 1 A 1 P .L .1- P = -c H—l-Wq p - (.3312; p 1—2 -c [(11) L+q p 41-2-1 up q 1—3 31 A 3 1 A 1 3 I 23 A 2 3 2 3 I 3 1 z 2 A3 2 (II-ll) The last sum in (II-6) is reduced by requiring that V(q .q .q > = V(q ,q I - q > (18). 1 2 3 1 2 3 ho Ess's" kss's" qsqs'qs" 3 2 2 = hck111q13 + th222q2 + 3th112q1 q2 + 3th122q1q2 33 2 2 + 3th133q1q3 + 3th233q2q3 (II-12) where the convention k + k + k = 3k , etc., is 211 112 121 112 used. The development of H2 is more tedious. When the results of Table V and the A's from Appendix I are applied to the Qéézas in (II-8) the first sum in (II-7) becomes 11“ L» (21 9 a8 85 ) qsqs'PaPB l ( zzss'zaB IaIB AsAs' . 2 2 _ 35 qISiny qzcosY q3 IX sz 2 A}? Azh Agzlz Ix - 2 3fi (qlcosY qZSinY + q32Iy] P22 +_[____-_____ 2 L L L 2 A14 AZH A32Iz Iy + fig [Q1C23 q2C13]2 P22 2 A L L 2 1L+ A2L+ I2 12' I2 ) + 3g (q3 ) q2( x SlnY - Y cosY 2 L L AL} Au 2 + I—l L . (Ix2cosY +IIxzsinY)] [Px,Py 1 1 :- -- 2 IXIyIz 1 (II-l3) pz The 34 The second sum in H2 may be evaluated by using pa = and Table V since Q;1)a8 = -agB. (1)a8 2'l .L 25.....‘§_.u = 2 2$2018 I I (A ) (paqs + qspa) PB a B s __ L 2 __. 4 q q 2 C C + ' + 2C13C23A3 [-2— — ‘lTlp3+—l3I%3([p1’q1] -[p2,q2] )q3 % 3A2% A1? )‘34 'fi P2 11 I 2 A l. L L z 3 2 2 2 A2 2 A1 +2(_____)1.( - ) + [ (-—)‘+ - (—)" __ A112 is 23 q1q2p3ijI-53 11 qlpz i3 A2 plqg ?: (II-14) third sum of H2 is given by 2 2 41221. = 12. 32. aid 12 'c 2 .1. 13 A3 2 A1 2 [6") q12p32 + (—_) q32p12 ' q1p1p3q3 ' p1q1q3p3] 212 A1 A3 L 5232 A 2 A2 2 2 [(—-3-) q22p32 + (--) pzzq3 - q2p2p3q3 - P2q2q3P3] 21 1 A z 2 3 c c A 2 1 A A i —-——1323[(3)rqqp2+(12)”ppq21 1 2 3 x 2 1 2 3 -——-—— [(——) plq2 + (——) q1p2][q3,p3] 212 .12 11 (II-15) 35 For completeness, the last sum in H2 may be written (18) hCZSS'Snsn' kSS'Snsn' qsqslqsuqsnl - hck u + hck u + hck q u 1111q1 2222q2 3333 3 3 2 2 + 4hck1112q1 q2 + 4hck1222q1q2 + 6th1122q1 q2 2 2 2 2 2 + 6th1133q1 q3 + 6hck2233q2 q3 + 12hck1233q1q2q3 (II—16) The convention (k1112 + k1121 + k1211 + k2111) = 4k1112 has been used. Eqns. (II—5) through (II-7) will now be specialized to the nonlinear XYX molecule. 1 2 p 2 p22 H = "—}£— + l— + — ° 2(1 I I ) x y z 2 2 «§ 2 7 P2 2 +21111 (EJ— + q12)+A2 1—2— + q2 m3 (3%— + q32)] ‘HZ 4'1 ‘11 (II-l7) % qlsinY qzcosY P 2 % qlcosY qzsinY Py2 HI = ‘fi (____—— + ‘——fI‘L—£§ -h (—__—:f -'__——__)'_53 Alf Azu 1x2 A11+ Azf Iyz 2 + l-q1C23 _ q2513 Pz h&q3 [Px'P ] +m<—i" L) 3+ L A“ A“ I2 1*(III)2 1 2 z 3 x y z 36 A u A P A l A P -C31[0_L) p1q3—(_i) q1p3] -£'- C23[(_i)“q2p3—(_£J p2q3l—5 A3 A1 Iz A I 2 3 z +hk 3 +hk 3 + 311k 2 C lllql C 222q2 C 112q1 q2 + 3h k 2 + 3h k 2 + 3h k 2 c 122q1q2 C 133 q1q3 c 233q2q3 (II-18) ' Y Y 2 2 2 H = 2guqlsnm + qzcos ) + q3 (lx)] px 2 2 I '_‘—T" ——r -—5 11% A 3 A3? z X cosY sinY 2 2 I p 2 + 321(qu - ———q2 1 + ———q3 (J11 1— 2 L I I 2 Al“: A2* A3 2 y + 22431523 _ q2513 )2 P22 2 x1% A2 I2 1% J- q J- L [p P 1* %? 31: [q2L(IX sinY-I 2cosY)-__—1£(Ixzcosv+1 23inY)] X' Y A31, A21. y A11. Y I I I 2 X y z i q 2 q 2 " 1 C C + + 2£13 23A39( 2L _‘_i:)p3+—L%I21([pllq1] '[pzrqz] )q3 fi2P A22 A12 >‘3“ Z *3" L 2 2 1 L +2( ) (g3-53)q1q2p3+-—T(ga(:?) qlpz-g30——) p1q2)q3 AIAZ A3Tr 2 37 2 1. C .A3 2 2 2 A 2 2 2 + 3::- “A? can P3 + (7:) qs P1 ‘ q1P1paq3 ' p1q1q393] 2 g A 2 2 A 2 2 + _21 [1—11232 pa + (-£)%b2 q3 ‘ q2P2P3q3 ‘ quzqaps] 212 A2 *3 21 1- C C A 2 u n #1323 3 A A . [( )qlq2p3+(—L—§)plp2q§+h0k1111‘11 +th2222q2 Iz AlA2 3 $323 A f 11% + h 3 - 21 [(-*)qu2+( )q1p2][q3,p3] +th3333q3 +4th1112q1 qz 2 A2 A1 3 2 2 2 2 2 2 +4th1222q1q2 +6th1122q1 qz +6th1133q1 q3 +6th2233q2 q3 +12hck1233q1q2q32 (II-l9) Examination 2: the Untransformed Hamiltonian Qperator When a vibration-rotation resonance is present, the conventional methods for diagonalizing the Hamiltonian in the vibrational representation are usually no longer valid. The two common methods of diagonalizing the Hamiltonian to second order require the use of either a conventional contact transformation, e.g. the Herman—Shaffer Operator, or the tedious but straightforward application of nondegenerate perturbation theory (19) to obtain the equivalent results. It is useful to use second order nondegenerate perturbation theory to examine the Hamiltonian operator (II-l7) through (II-l9) for the case of a Coriolis resonance. The general solution for an eigenvalue in the vibrational representation 38 is given by ' ll2 = + , o 2 v (E _ E ) v v' (II-20) since H1 has no diagonal elements. The usual approximations have been made in (II-20), namely that any rotational con- tributions in the energy denominator will be small compared to the vibrational energy contributions and that EV =‘fizi (vi + %)Ai2 .(II-Zl) In Chapter I, the general conditions for the exist- ence of Coriolis resonance have been outlined; the specific application to hydrogen sulfide will be used here. Since A13 = A;% for hydrogen sulfide, it follows immediately from (II-20) and (II-21) that a resonance condition exists When the set of vibrational quantum numbers satisfies the con- dition o + -' (II-22) L That is, if the conditions (II-22) are satisfied and A12 2 A37: then the energy denominator of (II-20) is given by 39 (II-23) In this case, nondegenerate perturbation theory may no longer be valid and some form of degenerate perturbation theory must be applied to diagonalize the Hamiltonian. In essence, this means that the terms affected by the resonance must be diagonalized exactly. The off-diagonal vibrational matrix elements of H [(II-l7), (II-18) and (II-19)] must be searched to determine which of them contribute to the resonance. Of the one hundred and fifty-two off-diagonal elements, only those of the form (v1, v2, v3] H| vdil, v2, V3$l) can contribute for the hydrogen sulfide molecule. To evaluate these elements in the vibrational representation, harmonic oscillator matrix elements are used (20). (V1IV2IV3IH1 + Hzlvl + 1: VZrV3‘l) L .1. L L 2 2 P 2 2 P P + v +1 v 2 =fi{-i§ (A1+A3)IE - (2)(Ixcosy+I sinY)[ x: y] }[( 1 )1] 3 Z 2 Y I I I ~12- 2(A A )4: x y z 1 3 (II-24) (v1,v2,v3|H1 + H2|vl - l, v2,v3 + l) L + 1 2 -n{'z; (Ail-ffz (3) (1% Y+IJ§ ' Y) [PXIPY] [V1(V3+ H _ 113 1 3 I— 3 Xcos ySin % £5 z IXIyIz 2(A1A3) (II-25) 40 Note that (II-24) and (II-25) will have a denominator of the form (II—23) if the non-degenerate perturbation di- agonalization (II-20) is used. In general, for the case A1? = A;%the submatrices containing (II-24) will have to be diagonalized exactly using degenerate perturbation theory. It is interesting to note that Nielsen (21) modified the conventional Herman-Shaffer operator so that thel;13 terms are left in an untransformed submatrix after the transformation is applied. That submatrix must then be diagonalized exactly. However, for asymmetric top molecules such as ozone (22) and hydrogen sulfide, this procedure does not seem to be sufficient as Nielsen tacitly assumed that the magnitude of the P2 term in (II-24) and (II-25) is much greater than the [PX,Py]+ term and the latter could therefore be neglected. The next section utilizes some numerical estimates to show that this assumption is probably not true for hydrogen sulfide. Thus the [PX,Py]+ term must be taken into account after Nielsen's modified contact transformation has been applied. 5 Numerical Investigation of the Resonance Terms To evaluate the terms in (II-24) and (II—25), it is necessary to get a rough estimate for the values of the force constants and use this to calculate sinY. The necessary force constants can be estimated by using isotOpic substi- tution as discussed in Appendix II. From Table V, 41 1 1 — 7. C13 = ($3)2 cosy - (£2) Sln Y I2 2 (II-26) and .L 1 (kll ' k22) 2 SinY=7_ l- .L 2 2 [(k11 ‘ k22) + 4k122]2 .___ . ___. (II-27) The squares of the normal frequencies (in radians per second) are given by the following equations (17): 2 1 _ 1 1 _ 2 7 x1 ’ 7(k11 + k22) + ?[(k11 kzz) + 4k12 ] 2 l A _ 1 _ 1 _ 2 2 2 ' 7(k11 + k22) §[(k11 k22) + 4k12 ] A3 = k33 (II—28) l -1 where l 2 = 2"cw with w in cm. s s s From (II-28) 2 2 % (A1 —x2) = [(k11 - k22) + 4k12 1 (II-29) To complete the evaluation of sinY, isotOpic substi- tution between hydrogen sulfide and deuterium sulfide is used to estimate (k11 - k22). From AppendixILI 42 (2mH + Ms)[mH(A1 +AZ)H - mD(A1 +121D] k22 _ 2 mH[mH - mD] (II-30) and H D [mH(2mD+MS)(A1+A2) - mD(2mH+MS)(A1+A2) ] kll = 2 [m - m ] mH D H (II-31) As a check on (II-30) and (II—31) we note that H (A1 + *2) = k11 + k22 (II-32) in agreement with the results found when equations (II-28) are added together. There is enough information to find an estimate for sin Y(II-27) from experimental values since combining (II-29), (II-30) and (II-31) gives: (kll-kzz) 1=[(A1+A2)HmH(mD+mH+MS)-(A1+A2)DmD(2mH+MS)] 2 2 2' k22) +4k12] [(k - H 11 mH(mD - mH)(Al-A2) (II-33) The masses (23), in atomic mass units, are: mHydrogen = 1.00782522 = 2. 4 2 mDeuterium 01 102 = 31.972074 MSulfur 43 The values of the normal frequencies of hydrogen sul- fide (12) are 2721.92 cm“1 001 1214.51 cm"1 00 2 and the normal frequencies of deuterium sulfide (24) are 1952.98 cm“1 1 “1 mg = 872.17 cm- From (II-33) it follows that: sin y 5 .6984 cos Y é .7157 Y é 44°18' (11‘34) Using our published values of the equilibrium moments of inertia (25), IX = 2.7046 x 10’40 gm. - cm.2 1y = 3.0954 x 10‘40 gm. - cm.2 12 = 5.8000 x 10‘40 gm. - cm.2 and (II-34) may be used to find Cl3from (II-26). 1 é 2 cos.Y ; -.0214 1 c (.53368)7 sin‘Y-(.46632) 13 (II-35) Finally, numerical estimates for the coefficients of the rotational operators in (II-24) and (II-25) may be found. In the following, the coefficients of P2 and [Px,Py]+ have been divided by he to give term values in cm-1 and the 44 2 2 2 2 substitutions As = 4“ c ms and IX = h/(8N cA), etc., have been made: P 1(-) +<—>%1C‘—)*’vj é 44.2075) (3%.) /_"(vi+11v"‘31' C13 “3 ”1 (II-36) 1 1 +/_——————— [P P ] (v +l)v. ELL [(C) cosY+(§) 2311-17] _._"__¥._ _____1 _ (w1w3)2 62 2 + [P ,P ] Av.+l)v. x l 3 (.1028) —-—2—-Y— (II-37) h. where i=1, j=3 in (II-24) and i=3, j=l in (II-25). So these calculations indicate that the [Px'Py]+ term should not be neglected in an accurate treatment of hydrogen sulfide. This was also found to be the case for ozone (22). Therefore both the pure Coriolis term (term containing P2) and the higher order stretching term (term containing [Px'Py]+) in (II-24) and (II-25) will be taken into account in the computations. Also, it should be noted that the value of C depends greatly on the values found for Y,A, 13 B, and C in contrast to the higher order stretching term which is less sensitive to small change in Y,A,B, or C. CHAPTER III APPLICATION OF THE MODIFIED CONTACT TRANSFORMATION The Vibrational Representation Much of the difficult mathematics associated with a Coriolis-type vibrational resonance may be circumvented by use of Nielsen's modified contact transformation.(21) which is also called the modified Herman-Shaffer Operator (26). This particular use of the Operator simplifies the matrix diagonalization in the vibrational representation by removing most of the important off-diagonal terms in H1 to diagonal positions in H5. There are no diagonal elements in H1 or Hl'. The matrix elements coupling interacting vibrational energy matrices are left unchanged by the mod- ified contact transformation and must be treated by an exact diagonalization routine rather than by perturbation theory. After the contact transformation 8 has been applied, the components of the new Hamiltonian, H', are related to those of the untransformed Hamiltonian, H, by the following equations (26): H'=HO D: II 1 H1 - 1(HdS - SHO) 45 46 ' = .j; + I _ + 0 H2 H2 + (2)[S(H1 H1 ) (H1 H1 )8] (III-l) As usual, the rotational operators are treated as con- stants for purposes of the transformation in the vibration- al representation but their commutator properties are strictly observed in the rotational representation. Finding the Transformed Hamiltonian HL: To facilitate computation of the transformed Hamil- tonian, (II-18) is written as 3 2 3 = . 2 H1 zn=la1nqn + Zn=l(a3nq n + asinzqnqlqz + a5n33qnq3 ) A '31. A h 42%: +a71[(-—J-) p q -(—3-> q p 1+a 1(——> p q 451) q p 1 13 13 A 13 72 1 ,2 3 2 3 . 1 3 2 ' (III-2) where the a's are listed in Table VI. This notation is similar to that of Herman and Shaffer but these a's are not exactly the same as theirs. It should be noted that the a's do not necessarily commute with one another. Hl' may be found by evaluating Hl' = Hl - i[HO,Zpsp] (III-3) where the components of S needed to perform the transformation are 47 Table VI. The Hamiltonian Coefficients for H1 all- ——-[ SlnY ——%+ cosy - c23.—_% ] A1 IX Iy Z 912-;-(6051;74'SWI4”13';‘9) 2 X Y Z 2‘ a13 -———3L——?% [Px'P 1+ I I I A Y X1723 p a -132 a =C2§Z 71 72 I I z Z a31= th111 332: th222 a5112= 3th112 a5122= 3th122 a5133= 3th133 a5233= 3th233 48 3 * = . + . ZS g Sin g 1(S3n + SSinz + SSn33) + S71 + S72 (III-4) The values for the transformation components in (III-4) are listed in Table VII. Using (III-2) and Ujjk4), H may be decomposed as I 1 3 2 u _ _- 3_' H1 _ Zn=l(ainqn l[Ho'sin]) +2n=l(a3nqn 1[Ho’san]) 2 2 . +2n=l(a5in2qnq1q2 + aSn33qnq3 -1[Ho’(ssin2+ssn33)]) A 4 A3” . * +{a71[(;L) p1q3-(;—) q1p3] - 1 [HO,S71]} 3 1 l l +1.7211121”p q -“q p 1 - 118 ,s 1} = I I I I I H1 '1+ H1 '3 + H1 '5 + H1 '71 + H1 '72 (III-5) It is shown in Appendix III that all components of H1' vanish except H1'r717 this was the reason for using the mod- ified contact transformation. The commutator in Hl',71is given by 1 .4 i[H0,s:1] = - 311 1(.:..§.)‘r - 1111“] (p1q3+ q1p3) 2 1 A3 (III-6) 49 Table VII. Contact Transformation Components 3 s _ aln s _ 2aan pn qnpnqn] ‘ *7——T — - 1 n n 1 J. 2 2 a + A p p _ SD33 _ 2 .2- 2 ‘3 3 n 551133: 2 .L [(2)3 An)q3pn+>‘3)‘n[p3rq3]qn+T] TY 1%(4X37'Ar) a ‘i' 'T' + 2A pzp - 5112 2 1 1 2 = A -A A 1 +_ S5112 2 l [(2 1 2>q1p2+ 1 2[p1,q11q2 -h2 1 ‘P 12(41 -1 ) 2 1 2 a5122 2 P P + +212p: 1 =' A -A A A 85122 2 2 [(2 2 1)q2p1+ 1 2[p2,q2]q1 2 1 —h 11(412-11) J- 4 2 * a710‘3 -A1 ) p193] 5“” 1 J.- 1““3 2 2(A113) (11 +13 ) L a 2p p S = ’ 72 [(4 +1 )q q +2(1 A ) 2 3] 72 1 2 3 2 3 2 3 2 (A213)F(A3-12) 'h 50 From (III-5) and (III-6) the final expression for the trans- formed first order Hamiltonian H1' is found to be 1 L H' H' 3.71 (A17+A32) ( ) = I = p1q3 - qlps 1 l 71 2 (A1}.3)%; (III-7) Finding the Transformed Hamiltonian H;' Hz' may be evaluated by extending (III-l) as . 3 v - i H2 — H2 + 2[2n=lsin,(H1 + H1.)] , 2 1 + _ I 2[Zn=l(83n + Ssinz +S5n33)' (H1 + H1 )1 + [(831 + S72), (H1 + Hl')] (III-8) The quantity (H1 + Hl') in (III-8) is given by 3 2 H + H' = a + . 2 + . 2 1 1 £21 1qu Z 1(a33q J a5132q1q2 + a5333q3 )3] J- + a [N p q M q p ] + a [N p q - M 72 2 2 3 1 2 3 7 g P 1 1 1 3 2 1 3 (III-9) where the notation 51 has been introduced for convenience. The commutator Oper- ations in (III-8) are worked out in Appendix IV; When vibra- tional representation matrix elements are formed (20) the commutators in (III-8) make a diagonal contribution of the form 2 = l; _ 3 2 in an n zfi,n=l T 1“ n=1 l n Anz 1 _ _l_§ . 1 DnjainaSIjzwj +7) [1 = 41 J A 2 n 2 § 2 2 -(v3 +_ 2){ i2 a5n33ain _ a71(l3%-)1 )’fi + a72h(3x3+A2)1 n=1 “—7—"— T"? l ’ An?“ 4(1113)%(MZH37) 213203-12) l. 2 % % 2 2 _ (V1+2)a7{fi(x3 -11 ) - (v2+%)a72‘fl(312+ka) L l L L 2 _ 4(1113)2(11+132) 2122(13 AZ) I + EV (III-10) 52 The contribution from H2 may be found in a similar manner to be 2 P 2 2 P 2 C 2P 2 = (v1+ g><3fi )[sin Y _i. + cos y .1. +._Zl_E_ 1 2A § 2 2 2 . 1 I I I x y z 1 3fi 2 x2 2 P 2 2 P 2 . z + (V2+ '5) (T—T) [COS Y -—-2- + 5111 Y -L2 +C13 -—-2- ] y z 1 3h Px 2 P 2 + (v3+ -)( )[ -+-JK——] + E u 2 21 12* I I - 3 X Z IyIz (III-11) ._ I N The symbol Dnj — (lqsnj), EV and EV are purely vibrational contributions listed in Appendix V. The Diagonal Elements in the Vibrational Representation Since HO = H ', the diagonal elements of H0' in the O vibrational representation give the harmonic oscillator energies 2 1 Pg 3 E 1 + Z hxn (Vn+Y) = 2 o 1 2=x,y,z 2 n=1 (III-12) 2 2 3 By combining (III-10), (III-ll) and (III-12) the diagonal elements of the Hamiltonian in the vibrational representation may now be found. Division by he puts the entire energy 53 1 expression in cm . HO +g2 hc X2 P 2 P 2 = A(v) -— + B(v) —Y— + c*(v) l— A2 62 62 + _l_. T P 1 + T P “ + T P “ + T (P P + P P )2-+ 453 1 X 2 Y 3 Z 9 X Y Y X 2 2 + 2 2 + 2 2 + rS[PX ,Pz ] +16[Px ,Py ] + TH[PY ,Pz ] + E (v v v ) 1 2 3 (III-13) A(v), B(v) and C*(v) are the vibrationally corrected recip- rocal moments of inertia which are customarily written 3 _ _ A i A(V) — Ae 2 ai (vi + 2) 1:1 3 B 1 B(V) = Be - 2 mi (vi + §> 1:1 3 C* * = _ 1 C (V) Ce L (Xi (Vi + 7) 1:1 (III-l4) h . . In (III-l4) A =‘"'§-“‘, etc. and the<1's are listed in e 8n cIX Table VIII. It is worth pointing out that the Coriolis 54 Table VIII. The a's CIA: 6A2(sin y Affluslm + k112°~°SY]) l e w Ae w1%_ “2% 2 % - A__ _6A2(COS Y + 2 kZZZCOSY kIZZSlnY 0‘2“ e T X“ [*3— + 3 ]) e (02.2. ml? 1 a3- "6Aece(-J; + Ce [ a. + ._T_]) 2 ”1 ”2 cos ' aB= -6 2( Y + [§_]2 L[k11:COSY - k11281ny]) 1 e ”1 e & w 2 B 2 sin Y 2 % k122cosY kzzzsiny a2: -GBe[ E” ' ) “2 e g m l 2 (ZB ) k cos k s' .3: _GBC[.1_+__. ) a = - ---—-- e ‘9 i l 1 ”22 ”I? J 6w1w3(w1+w3) 2 k222c13 k122523 Cz§3w2+w3) C*_ 6C2(513 + [2 _ w 2 w 2 3w (w -w ) 2 1 2 3 2 2 - 2 2 C*- -6C: 23 2 32 2 +[é_)lX233C13 _ k133C23J-C13(w3"w1) ) l 1 e w22 w12 6w1w3(w1+w3) L 2 2 2 2[k112C13 _ k111C23] + §13(3w1+w3) —"—" 2 2 ) 3w1(w2 -m 3) - .3. L02 2 (012 2 w2 2 C= -6C2(C23(3 3+w2)+[2[__e[E233:13..k133C23]+C13(3:3+:21)) a a 3 e 2 2 3w (w - ) 3 3 w2 ”2? “12 3w3(w§ -w 2) 55 . c c . . c* and perturbation causes a1 anda3 to be modified to<11 * C C ). In the unperturbed case, the expression (III—l3) would be the entire Hamiltonian in the vibrational . . * * c c representation if ale and 030 were replaced by a1 and a3 , since any off-diagonal elements should be ignorable. The main difference between the starred and unstarred ac's listed in Table VIII is that the resonant denominator ((1132 -w12) is 'k * no longer present in ale and a3 ’ The 1's in (III-l3) are centrifugal distortion con- stants given in cm —1 and are closely related to the equi- librium 1's (T etc.) of Chung and Parker (17). - I T I XXXX yyyy For experimental work it is convenient to use the relations T ‘5“ T 3 SinZY COSZY 1 =-HE xxxx = - 16Ae ( 2 +-———;—) ml “2 u €052 . 2 T _ ‘15 T = - 16B (—__Y. + Ell—I) - ~‘ YYYY e 2 2 2 hc w w 1 2 2 3 = —— 2222 e 2 2 he ml w2 q % C13sinY C23cosY T = 1‘— Tszz = + 16 <--——— + —————> a be m 2 w 2 2 1 i C cosY C sinY L1 13 23 = _ C 2 -——-———-——— — _........__._ T5 =‘E— Txxzz l6(Ae e) ( 2 2 ) hC w w 2 1 56 L1 .3. _ 41 _ _ 2 . 1 _ 1 T6 _ FE TXXYY — 16(AeBe) SlnY COSY ('w—l—f 0022) 1+ 16A B C e e e r = §—-Txyxy = - 2 9 hc 111 (III-15) In (III-15), frequency mi is in cm"1 using the conversion XI? = 27TQ),C. i i The vibrational and constant terms remaining are lumped together in E(v1v2v3) which will contribute to the bandcenter only. The diagonal term (III-l3) is usually simplified by using the well-known commutator relation 2 2 (P P +P P )2=2(P 2P 2+P 2P 2)-2fi2P 2-2h2P 2+3fi P X X X X X Z y y y y y (III_16) and the Dowling relations which apply to the equilibrium taus for planar molecules (27). c 2 C 2 = _e .2 T5 (B) T6 + (Ae) T1 e 2 2 I“ = (4%) T6 + (99‘) T2 e Be C 2 Ce 2 13 (—9) r5 + (—— I“ (III-l7) Ae Be 57 Ce 2 Ce 2 When the substitutions r = K— , s = E— , (III-l6) e e and (III-l7) are made in (III—l3) the final result for the diagonal elements of the Hamiltonian in the vibrational representation may be written I 2 o 2 T PX T9 P 2 ._...__.___| v v v > = (A(V) .. .2) 2 + (3w)- .._..)_Y_. 2 62 ={2iGz——+ny-jEEJL }'———;——*- ‘ hc -fi (III-19) H +H P u> P I+/v (v +f$ _ . ' 1 3 (vlvzvsl 1 2lvl-l,v2,v3+l’-{-21ng-z-+ny-§%—2L "7“" ho (III-20) 58 where Z 1 (w1w3)3 (III-21) and 6A B c 12* G = _ __S_E_E_ [EQ§1 + SinY] XY .437— §- .3 1 3 A B (III-22) e e -1 ny and G2 are given in cm in order to facilitate experimental evaluation of their magnitudes. Possible Experimental Applications It is informative to examine the matrix elements for several sets of interacting excited vibrational levels of hydrogen sulfide. For the states (011) and (110), i.e. v1 = 0 and v2 = v3 = l, the off-diagonal interaction matrix element is + H +H P [P P ] "— <011| 1 2I110> = {21G l - G ill—Lil. hc 25 XY fl? 2 (III-23) and its hermitian conjugateo The diagonal matrix elements are found by specializing (III—18) to the (110) and (011) states. The general form of the Hamiltonian matrix is shown in Fig. 6 for this two band interaction. Fortunately both the (110) and (011) bands are infrared active so the Coriolis interaction between them can be studied experimentally. 59 Fig. 6. The Interaction Matrix for the (110) and (011) States in the Vibrational Representation (110) (011) EROT.(llO) - 1Gz Pz fi— + (110) + E (110) G [P ,P ] VIB. - xx x y 2 it iGz :3 EROT.(011) 1T + (011) + _ GX [Px'Py] EVIB.(011) 2 ’52 60 For a three band interaction, such as found for the (210), (111) and (012) states, the coupling matrix elements are found from (III-l9) and (III-20) to be H +H G P G [P P 1+ <.111| 1 2| 012) = { -21 z z - XY X' 1 fig hc an 452 2 (III-24) and H1+H2 <3sz Gx [P ,P 1+5 <111| | 210> = {221—— - y X Y } 3 hc ’5 4T2 (III-25) along with their hermitian conjugates. The complete form of the Hamiltonian matrix for these three states is shown in Fig. 7. Unfortunately, the (012) band of hydrogen sulfide has not been found to be infrared active. As noted by Wilson (14), even though a vibrational state may not give an infra- red band, it may perturb the upper state of another band. Experimentally, it appears that the infrared inactive vibra- tional state (012) does perturb the infrared active states (210) and (111), so the analysis may be more fruitfully applied to the (110) and (011) bands. Figo (210) (111) (012) 61 The Three Band Interaction Matrix for the (210), (111) and (012) States (210) (111) (012) EROT (210) -iGz P2 /3 + -H E (210) c; [P P 1+ VIB. x x’ 2 .hz . /- . 1G2 Pz v2 EROT°(lll) 1G2 P2 /3 4?” ‘5’ + + + -GX [PX,P ] EVIB°(111) _ GX [PX,P ] c: ‘nz :2 +1 1G P2 /5 EROT (012) if‘ + + GX [PX,P ] EVIB (012) CHAPTER IV THE ROTATIONAL REPRESENTATION Operator Matrix Elements Each matrix element in the vibrational represen- tation shown in Fig. 6 and Fig. 7 is a complete submatrix of dimension (2J + l) in the rotational representation. Thus to complete the mathematical description of the Hamil- tonian matrix an appropriate rotational representation must be chosen. King, Hainer and Cross have covered the general details of the various possible choicescflfrotational repre- sentations so only the points pertinent to this problem will be recapitulated here ( 4). The operators Pa' Pb and Pc in (I-l) may be identified with a right—handed system of Cartesian body—fixed axes with origin at the center of mass. The operator commutator rules are [P.,P.] = - ifiP 1 k 3 (IV-l) where the ijk stand for xyz and cyclic permutations. Sup- pressing the J and M notation in the matrix elements, a well known solution to the set (IV-l) in a representation diag- onalizing P2 and P2 is 62 63 I = -i = §[J(J+1) - K(K+1)]2 =‘fiK (IV-2) All necessary rotational representation matrix elements have been calculated by Posener from (IV-2) using the same phase factor chosen by King, Hainer and Cross (19). The 3! ways in which Pa’ Pb and PC may be identified with x, y, and z are listed by King, Hainer and Cross and shown in Table IX. Thus the choice x = a, y = b, 2 = c, made in Chapter I amounts to choosing a IIIr representation and allows the direct use of Posener's operator matrix elements. The operator matrix elements used in this work are listed in Appendix VI using the substitution f = J(J+l). When the proper substitutions from Appendix VI are made, the elements of the submatrix for the Hamiltonian Operator (III-l8) become A(v) + B(v) --r KZR (VIKIVIK> = [f'KZ] 2 9 + T +. [f2-2(K2—l)f + K2(K2- Sui-(g + EN) + 3.1% K2 1+ +14:— + [3f2 - 2(3K2 + 1)f + I F(K11)F(Ki2) + (2f-K2-(Ki2)2]§ + [K2 + P(Ki1)F(Ki2)P(Ki3)P(Ki4) U (IV-3) In (IV-3), the symbol V indicates either V1 (v1v2v3) or V2 (v1v2v3) for a two band interaction. The following sub— stitutions were used in (IV-3) to save space: [f—K(Kil)]% F(Kil) = F(Ki2) = [f-(Kil)(Ki2)]% etC. f = J(J+l) R = r(rl+16) + 5(T2+T6) S = 16(r—s) +TZS‘T1r T = (2rST6 + r211 +8212) W = T -T 2 1 X = T1+T2 Y = 16+219 (Iv—4) 66 The vibration-rotation matrix elements coupling the upper vibrational states of the interacting bands are found from (III-19) and (III-20). 1 - _ . 1 , 7 ' 1 1 2 3 itiyF(Ki1)F(Ki2)[(v1+1)v312 . - 1 itiyF(Ki1)F(Ki2)[v1(v3+1)]7 (Iv-5) The form of the Hamiltonian matrix for J = 1 generated by (IV-3) and (IV-5) is shown in Fig. 8 for the case when the interacting vibrational states are (110) and (011); i.e. v1 = v23= l and V3== 0. It is easily seen from Fig. 8 that the elements (IV-3) and (IV-5) describe a matrix of dimen- sion 2(2J+l) by 2(2J+l). The size of the Hamiltonian matrix rapidly becomes very large with increasing J value when left in the form described by (IV-3) and (IV-5); thus an undue amount of computer memory space for an accurate numerical diagonalization process is required if the Hamil- tonian if left in this form. The Wang Transformation The symmetry requirements discussed in Chapter I can be satisfied by applying a modified Wang transformation (28) to the Hamiltonian matrix described by (IV-3) and (IV-5). 67 Fig. 8. Form of the Untransformed Hamiltonian for J = l '{——(110|110)-———> +——(110|011)————> Block Block 0 -l l 0 -l o o // / . ' / r—(011I110)-————>l4-—(011|011)——§ Block Block § 68 The transformation reduces the Hamiltonian matrix to four uncoupled submatrices which are easier to diagonalize than the cumbersome form shown in Fig. 8. Due to the off-diagonal coupling terms G2 and ny a "double" Wang transformation is necessary to partition the untransformed Hamiltonian into four submatrices. The double Wang transformation W is of dimensions 2(2J+l) x 2(2J+l) and is shown in Fig. 9 in a form suitable for operating on the Hamiltonian in Fig. 8. Since W is unitary, (IV-6) is a similarity transformation which leaves the trace of the matrix unchanged. Fortunately, odd and even values of K are not connected in the untransformed Hamiltonian. Therefore, (IV-6) brings about a final matrix form very similar to that given by the ordinary Wang transformation acting on the vibrational state described by (IV-3). The transformed matrix Htrs is shown in Fig. 10 for J = 1. Using (IV-6) rapidly becomes very tedious as J grows larger, since determinant manipulations performed with insight are required to achieve the neat form shown in Fig. 10. Usually it is much easier to go to the set of modified Wang linear combination wave functions constructed by Wilson and calculate the transformed elements directly (14). The combination wave functions are given for an interaction between vibrational states V1 and V2. 69 Fig. 9. The Double Wang Transformation for J = l W Sui-a 70 Fig. 10. The Transformed Hamiltonian for J = l A0 F0 A1 Q1 I‘1 (D1 + .. (E +E ) A0 V1 V2 F0 (E-+E;) l 2 A1 + _ (0 +0 ) // Vivz r 1 _ + /L (OViOYg: / 71 AK = 2 2(wJKM + l"J.KM)“’V (K>0) 1 -% rK = 2 (wJKM + wJ-KM)wV2 (K>0) P0 = wJOMwVZ ‘% ¢K = 2 (wJKM ‘ q’J-KMWVI (K>0) '% QK ’ 2 (wJKM ' wJ—KM)wv (K>O) 2 (Iv-7) The use of (IV-7) to calculate the matrix elements of the Hamiltonian is entirely equivalent to calculating the untrans- formed matrix (IV-3) and (IV-5) and then applying the double Wang transformation as in (IV-6). Wilson originally used the wave functions (IV-7) to calculate the effect of a pure Coriolis perturbation on a rigid asymmetric rotor. The four submatrices generated by the wavefunctions applied to (III-18), (III-l9) and (III-20) are classified in Table X for the interacting vibrational states V1 and V2. By the convention used for the ordinary Wang submatrix, E or 0 gives the oddness or evenness of K and no negative values of 72 m> as Ae_ec . Ae_av Aa_av .....m.m.H +0 + no N a 2:8 . 2:5 2:: 225$; wo + wo m a i._.c . i._rv iu_ac .......N.o wm + »m m H Ae_ev . Ae_ mmOHHumEQsm mcmz pmamsoo one .x magma 73 K are used. The + or - over E or 0 gives the sign joining the wave functions in (IV—7). The combination wave functions from (IV—7) are indicated in Fig. 10. Since Fig. 10 dis- plays the J = 1 case, the EV2 and EVT contributions are zero even though their positions are indicated. The general Wang submatrices may be generated from the elements (IV-3) and (IV-5). The diagonal matrix elements are given by 2 KZR [fSK ] {ATV} + B(V) - 19 + —7—] Y + [f2-2(K2-l)f + K2(K2-5)] + [3f2-2(3K2+l)f + K2(3K2 + 5)] é? a 3T + 2%—-+ [C*(V) + —13] K2 + E(V) t (5K'l) §IB‘§, + g] 1 (6K 2)f(f—2) U (IV-8) In (IV-8) the elements (AK(V1)IAK(VI)) and (PK(VZ)'PK(VP))' which include the K = 0 case, are given when the positive sign is chosen for the delta functions. The elements (¢K(v1)|¢K(v1)) and (QK(v2)|QK(v2)), which exclude the K = 0' case, are given when the negative sign is chosen. As usual, 74 V in (IV-8) stands for either set of quantum numbers V1(v1v2v3) or V2(v1v2v3). The off-diagonal terms for either state V1 or state V2 are given by (IV-9) and (IV-10). 1 (1+5 )7 4 W K'0 F(K+l)F(K+2)[B(V)-A(V)+(f-K2-2K-2)I + (K2+2K+2)%] - 1 i (GK'1)f[(f-2)(f-6)]7 U (Iv-9) In (IV-9) the elements (AK(V1)|A (V1)) and (I‘K(V2)lr (V2): K +2 K+2 which include the K = 0 case, are given when the positive sign is chosen for 6K,l‘ The elements (¢K(Vj)|¢K+2(V1)) and (QK(V2)IQ (V2)) are given when the negative sign is K+2 chosen. The remaining elements within a vibrational block V1 or V2 are (AKIAK+4) = (FKIFK+4) = (¢KI¢K+4) = (QKIQK+4) 1 = (1+5K O)TF(K+1)F(K+2)F(K+3)F(K+4)U ’ (Iv-10) The elements coupling the two vibrational states are due to the higher-order stretching term GXy (AKIQK+2) = (FK|¢K+2) = ' (FK+2|¢K) = " (AK+2|QK) G 1 = ' _EX 7 l 4 (1+5KO) F(K+l)F(K+2) (IV-11) 75 and to the mixing of the pure Coriolis term G2 with ny _ . ‘ f (AKIQK) ‘ " 1[GzK ' (5&1) SW 71'] (K750) — . f (I‘K|K) - HGZK + (5191) ny 71'] (Iv—12) The equations (IV-8) through (IV-12) describe the Hamiltonian in terms of the four Wang submatrices for the case when one vibrational state interacts with another. These equations, with some small empirical modifications, were used to analyze the (110) and (011) absorptions bands of hydrogen sulfide. The form of the array generated by these equations is shown in Fig. 11 for the J = 5 case. However, Fig. ll may be used as a model for generating an array using other values of J and equations (IV-8) through (IV-12). It should be noted that if G2 = ny = 0, the 0 and E blocks stand alone and Fig. 11 then describes the unperturbed case. Fig. 11. 76 Form of the Wang Transformed Hamiltonian for J A0 A2 AL, 92 an EV+ l + - (EV + EV )Block E - l 2 V 2 F0 P2 Th¢2 CPL, F0 E + r2 v2 I” + - q (Ev + EV )Block - 2 1 «>2 EV l ¢u A1A3A5 Q19395 A1 0 + A, 3 V1 - A ( O + O ) Block 5 V1 V2 91 o - Q3 V2 95 F11‘3Ts ¢1¢3¢5 r1 + T3 OV2 ( 0V- + 0V+ ) Block F5 1 2 ¢1 0 ' V (D3 1 ¢5 5 CHAPTER V OBSERVED SPECTRA History One of the first detailed rotational analyses of hydrogen sulfide using an early quantum-mechanical energy expression was published by Cross in 1935 (29). Cross ana- lyzed the (301) absorption band of H28 at about 10,000 cm-1 and was able to identify 84 absorption lines. His values for ground state energy levels and inertial constants were used successfully by infrared spectroscopists in analyzing many other H28 bands. A summary of these analyses was published by Allen and Plyler in 1956 (12). In recent years, additional information about the ground state of H28 has been obtained from pure rotational transitions measured by microwave spectroscopy. Frequencies for two H2832 microwave lines were first published by Burrus and Gordy in 1953 (30). Since then several other microwave lines of H28 have been found and measured by various researchers. Because microwave transitions can be measured very accurately, they are a useful supplement to infrared data for determining ground state structural parameters. The main bands studied in this work are the (110) band (type B) and the (011) band (type A), both at about 77 78 3800 cm—1. The first prOper analysis of these two bands was published by Savage and Edwards in 1957 (31). They correctly determined the presence of two absorption bands in this region rather than only one as was reported by earlier investigators. Two other bands were investigated to supplement this work. These are the (210) band (type B) and the (111) band 1 (type A) at about 6300 cm- . The first rotational study of these two bands was published by Allen and Plyler in 1954 (32, Procurement and Measurement of Data The hydrogen sulfide spectra were obtained on the Michigan State University high-resolution infrared vacuum recording spectrometer. The basic components of this instru— ment were described in 1964 by Aubel (33) and subsequent improvements will be described by Keck (34). In the initial work a 300 watt commercial zirconium arc was used as the infrared radiation source. Later, a water-cooled carbon rod source (35) with greater energy was successfully utilized to obtain improved spectra. Two different multiple traverse Cells were used during the course of this work. The 80 cm cell was useful for low pressure runs but was initially limited because the infrared beam had to pass through several inches of air (containing H20 vapor) before entering it. This problem was later eliminated by the construction of an all-vacuum path for the beam (35). The other cell is a small volume coolable 79 multiple-traverse cell usually designated as a J. U. White type cell (36,37). Since the air space around this cell could be completely evacuated, it was used for initial data runs. Two Bausch and Lomb echelette gratings are mounted on a turntable in the main tank of the spectrometer so that either may be readily used. One has 600 rulings per mm and a ruled area of 212 x 158 mm; the other has 300 rulings per mm and a ruled area of 254 x 128 mm. An Eastman-Kodak N—type lead sulfide cell operated at liquid nitrogen temperature was used as a detector for the hydrogen sulfide spectra. The calibration and data processing used in our laboratory have been reported in Wavelength Standards in the Infrared (38) and also described in detail by Barnett (39); so only the main points will be described here. Calibration lines were recorded both immediately before and immediately after the hydrogen sulfide trace on each chart. The cali- bration gases used were HCN, CO, HCl and.N20 Which have frequencies measured so accurately that they may be used as standards. The H28 spectra were calibrated by means of Edser- Butler bands (or fringes) which were recorded simultaneously with both calibration gases and the H28 trace. Edser-Butler bands are used because the consecutive positions of their peaks vary linearly with frequency; thus there is a constant frequency separation between adjacent peaks. After running a spectrum, the fringes were numbered 80 and the entire chart photographed for measurement on a Hydel digitized film reader connected to an IBM 526 card punch. A Hydel operator scanning the filmstrip of the chart measured the appropriate coordinates of the fringes and calibration gases for punched-card input to the computer program SCAN (40). The first SCAN computer output gave the relative fringe number of each calibration line. Next, the known frequency and fringe number of each calibration line on the chart was input to the computer in a least-squares routine, CALFIT (40). The CALFIT output gave the slope and intercept parameters necessary to establish the linear equation between the fringe number and frequency of an individual calibration line. These two parameters along with the measured fringe number (from the Hydel) were input to the computer in SCAN. The resulting output gave the frequency of each H28 line from its fringe number through the linear equation between frequency and fringe number. The entire procedure was repeated several times for each chart and the resulting measurements averaged together in order to minimize Operator error. The averaging of several hundred data points per chart was done using ASHAFT (41), another computer program developed in this laboratory. Since several runs were made in each region, it was possible to combine the average frequencies from each run by using weighted averaging. The weights assigned to each frequency measurement on a given chart were based on the quality of 81 the least-squares calibration fit between fringe number and calibration frequency for that chart and on the estimated reproducibility of the frequency measurement for that partic- ular line. Further Experimental Technique More details about the experimental techniques used in each region are outlined as follows: (a) 6300 cm.1 region This region was calibrated using the (301) band of N20 (42) with bandcenter at 5974,8504 cm‘1 and the (002) band of HCN (43) with bandcenter 6519.6145 cm-l. Two runs, H28 1 and H23 2, were made in this region using the 600 line/mm grating in lst order in a double-pass configuration. A zirconium arc was used as the infrared energy source and the 80 cm multiple traverse cell as the sample holder on both runs. The gas sample, C.P. grade (99.5% purity), was purchased from the Matheson Company and used without further purification. The standard deviation of the calibration fits was about 0.004 cm-l. The ex- perimental conditions are summarized in Table XI. (b) 3800 cm-1 region Five data runs were made in this region with run numbers 8, 9, 10 15-I and 15-II. Calibration for the first three was obtained using the (101) 82 band of HCN (43) in 3rd order with bandcenter at 5393.698 cm‘1 and the (210)! band of co (43) in 2nd order with bandcenter at 4260.0646 cm-l. A zirconium arc was used as the infrared source and the small coolable multiple traverse cell as a sample holder on all three runs (8, 9, and 10). The H28 sample for these runs was C.P. grade (99.5% purity), purchased from Matheson and used without further purification. The next run in this region was divided into two over- lapping parts. Run 15-I covered the first half or low frequency side of the absorption region and was calibrated between the (101) band of HCN in 3rd order and the (20) band of HCl (44) in 3rd order with bandcenter at 5667.9841 cm-l. Run number lS-II overlapped the end of lS-I and covered the second half of the region. It was calibrated between the (20) band of HCl in 3rd order and the (20) band of CO in 2nd order. Both a C02 impurity in the sample and H20 background lines were a problem in this region. For these runs (lS-I and lS—II) the H28 sample was C.P. grade (99.6% purity) with a C02 impurity content certified to be less than 500 ppm. It was purchased from Air Products and Chemicals Inc. The standard deviation of the calibration fits on runs 8, 9, 10 and lS-I was about .003 cm-1. On 83 run lS-II, the standard deviation was about .0015 cm- On all five runs the 300 line/mm grating was used and the H28 trace was in 2nd order. The experimental conditions are summarized in Table XI. In addition to the data runs, one C02 background run and several vacuum background runs were made in this region to identify the absorption lines due to impurities in the sample and H20 vapor in the spectrometer. l 84 OU.HUm Hum.zum OU . zom 20m . Omz mcumccmum soflumunflamo mmo.o meo.o oeo.o auso.uflseq.eoflusH0mmm .mbo mamm.o mamm.o mHmm.o oamm.o mamm.o mmmm.o mmmm.o Hugo..mmm mmceum .m>< com com EE\mmcHH .mcwumnw com conumu oufi Enacoonflm monsom cmnmumcH mm om: mm o. .mhsumummsme was so om when: so om demo coehmhomna me.m e.e m.me ms.me s.ehmemq game e m m e e on em on as .mhsmmmue mmm HHImH Hlma 0H m m N H Hwnfidz cam comm comm 180.coflmwm H mc0fluflccoo Hmucmsfluwmxm .HX OHQMB CHAPTER VI DATA ANALYS I 8 Introductory Remarks In the past few years, many of the procedures for data analysis in the literature have almost become obsolete because they were not designed for use on the computer. In— cluded among these are interpolation tables and the many other routines that have been published specifically for hand calculator manipulation. In addition, many of the computer techniques used successfully on early machines have been replaced by more accurate methods ideally suited to a high-speed digital computer with large memory capacity. For example, first order perturbation theory is no longer nec- essary to describe the centrifugal stretching since digital computers can now accurately perform the exact diagonaliza- tion of an entire Hamiltonian matrix in seconds. Computer technology is evolving so rapidly that the beginning re- searcher often spends hours scanning obsolete techniques for data analysis before finding the best method available for a given analysis problem. In addition, an author's description of the solution to a pertinent problem often assumes a complete knowledge of the steps leading from a matrix formulation of the molecular 85 86 Hamiltonian to the analysis of experimental data. Many times this is an unwarranted assumption because constructing an analysis program usually requires a considerable amount of insight and hard work. To help clarify the analysis pro- cedure, simple examples will be used to illustrate much of the discussion that follows and references to specific applications will be made when possible. The general remarks made in the next two sections are by no means limited to asymmetric molecule analysis. For example, an analysis program for treating perturbations between interacting symmetric top energy levels might be designed using the general outline discussed. Components 9: a Typical Frequengy Fitting Program It is informative to examine the components of a typ- ical frequency fitting program. To simplify the discussion, in this section it will be assumed that the ground state molecular parameters are well known, starting values for the upper state molecular parameters have been found, and a few absorption lines have been correctly identified. A later discussion will treat the case where this information is not known for the molecule. The internal operations are outlined as follows: (a) The values for the ground state molecular parameters (A(O), B(0), C(O), taus, etC.), and starting values for the upper state molecular parameters (A(V), B(V), C(V), taus, perturbation (b) (C) 87 terms, etc.) are input and stored in memory followed by the quantum numbers, frequency, and weight of each identified transition. If a perturbation con- nects two or more excited vibrational states as in Fig. 6, Fig. 7 or Fig. 11, a set of parameters for each upper state will be input along with the ground state parameters. A matrix array for each J value is formed using the apprOpriate Hamiltonian for the ground state and the ground state molecular parameters. For example, a suitable array for the ground state may be generated from (IV-8), (IV-9) and (IV-10) when the ground state molecular parameters are used and E(V) is set equal to zero. The ground state array is then input to a suitable exact diagonaliza- tion subroutine which finds the eigenvalues. Then a labeling scheme (such as the one described in Chapter I) is used to assign quantum numbers JK—K+ to each ground state energy level or eigenvalue. The ground state levels and their identifying quantum numbers are then stored in memory for future use. An example of a program that performs these functions is G.S.Ex.D. listed in Appendix VII. The exact diagonalization routine used is JHERMX (45 ) . Next an upper state Hamiltonian matrix array for each J value is formed using the input starting (d) (e) 88 values. For example, a suitable array for two interacting bands may be generated from (IV-8), (IV-9), (IV-10), (IV-ll) and (IV-12). The upper state array is then diagonalized preferably using the same exact diagonalization routine used for the ground state. Each energy level is labeled with the appropriate quantum numbers JK_K+ and stored in memory. The set of eigenvectors diagonalizing the array for each J value is computed and stored in memory for future use. As an example, the program U.S.Ex.D. listed in Appendix VIII will perform all of these functions for the interacting (110) and (011) bands except calculate eigenvectors. However, it can be modified to calculate eigenvectors, too. Using the selection rules, appropriate energy levels from the ground state are subtracted from each upper state level to give a set of calculated fre— quencies which are stored in memory. This set of frequencies would represent the observed spectrum of the molecule if the input upper state parameters were the true values for the molecule. However, this is rarely the case at the beginning of an analysis. At this point a least squares subroutine may be used to compare calculated and observed frequencies and make adjustments to minimize the differences between them. The transition from a matrix array to 89 the least squares subroutine is done by means of the stored eigenvectors and will be treated separately in the next section. (f) The new upper state parameters found from least squares may be input again and the steps b, c, and d repeated to find the set of frequencies pre- dicted from them. Using frequencies predicted from the new parameters, more observed frequencies may be input and the whole process repeated. In this general manner all of the absorption lines in a given region may hopefully be identified. Programming the Multiple Regression Subroutine The main problem in using a least squares subroutine to vary upper state parameters and predict the resulting frequencies (multiple regression analysis) is to get from the N x N matrix array containing these parameters into a system of N equations suitable for a least squares treatment. zuu understanding of this method is very important for using theoretical Hamiltonian arrays to analyze experimental data. In general, the difference between an observed fre- + B

+ C

a c 1 b (VI-6) At this point, the eigenvalue equation (VI-5) is in a form suitable for entry into a least squares subroutine. If (VI-6) describes the upper state of this hypo- thetical rigid rotor, then the frequency due to a transition from a vibrational ground state energy level Eg is 2 2 2 f = El - E = A

+ B

+ C

+V - E g a b c o (VI-7) 92 where \6 is the bandcenter and it is assumed that E9 is very close to its true value. If the "observed" frequency fo is close to the calculated frequency f1 then to a good approx- imation. 2 2 f = (A+AA) + (B+AB)

+ (C+AC) o b + (“0+Avo) ' Eg (VI-8) where the difference between f0 and f1 is assumed to be due to small changes in A,B,C and the bandcenter which don't affect the expection values of the operators. Thus (VI-7) and (VI-8) are combined to give 2 2 2 f — f = AA

+ AB

+ AC

+ Av O 1 0 Systems of equations of this form may now be input to a least square subroutine since AA,AB,AC, andAvO will be common to all equations. The changes found from the fit then are added to get the new values of A,Eh C, and the bandcenter. Several detailed examples applying equations of the form (VI-9) to more complicated Hamiltonian matrix equations have been given by Moncur (46). Method 2: Analysis So far, for purposes of illustration it has been assumed that good estimates for the molecular parameters were available to aid in identifying transitions. If this 93 is not true the method of Hill and Edwards (47) may be used to start the analysis. By this method the nearly symmetric rigid rotor approximation E = %(A+B)J(J+l) + [C-%(A+B)]K2 o (VI-10) is used where K is the quantum number corresponding to K+ in the oblate limit. Individual absorption lines are identified by the notation AKAJK(J) where the changes AK and AJ are represented by P, Q, and R which stand for —l, 0, and 1 respectively. Using (VI-10) the ground state com- bination differences formed from the "zero series" lines, (RR and PP lines for which J" = K") may be written RRJ(J) - PP (J+2) = A"+ B"+ 4c" (J+l) (VI-ll) J+2 where the double prime is used for the ground state. The zero series lines are used because they are least affected by the asymmetry and are among the strongest observed. If the ground state combination differences are plotted, against J(J+l), (A"+B") is the intercept and 4C" the slope. To get an estimate for the upper state A, B, and C, the upper state combination differences are found from the zero series as R _ P .. I I I RJ(J) P(J) ‘ A + B + 4C J (VI-12) 94 and the plotting procedure repeated. The starting values thus obtained are cycled through the least squares fit of the Hamiltonian to help predict more lines and obtain better values of the molecular parameters. Finally the molecular parameters should converge to a final set of values when all possible lines have been assigned. The observed absorptions bands in the 6300 cm-1 region are the (210) and the (111) which are type B and type A, respectively. It is believed that a three band Coriolis-type interaction (such as the one illustrated in Fig. 7) is present between the (210), (111), and (012) upper state energy levels. Unfortunately the (012) band is not observed to be infrared active; thus so far only the least perturbed lines have been fit to a Hamiltonian expression with no perturbation terms included. The details of this analysis are given by us (25) and the weighted lines used in the spectrum fitting are listed in Appendix IX. The values of A8 and Be found from this analysis were used in the analysis of the two band interaction between the (110) and (011) bands. Using A 10.3491 cm"1 e -1 B 9.0426 cm 95 the r and s values for (IV-8) and (IV-9) were calculated. This is thought to be a better estimate than using the ground state values. At present, these two bands are being reanalyzed for "effective" perturbation terms but the values found for Ae and Be are not expected to change significantly. The ground state molecular parameters found from these bands were used as starting values in the fit of the (110) and (011) bands. The (110) and (011) Bands The two bands located in the 3800 cm-1 region are the (110) and (011) bands which mutually interact through a Coriolis-type perturbation as shown in Fig. 6. Initial frequency assignments were made from the lines reported by Savage and Edwards (31). The ground state parameters from the (110) and (011) bands were used to begin the anal- ysis. An unperturbed fit of each band assuming no inter- action was tried but was unsuccessful, since many lines remained unassigned. A successful fit of both bands was obtained by taking into account the perturbations between them. This was done using SPECFIT, a program written by Moncur (46) which inputs equations (IV-8), (IV-9), (IV-10), (IV-ll) and (IV-12) into an exact diagonalization subroutine and a least squares subroutine in the manner previously discussed. This program and the details for using it are given by Moncur. 96 Two slight empirical modifications were added to the equations derived in Chapter IV. These were the addition of 6 a small term, H K , and permitting the centrifugal stretching K terms to change from their ground state values in order to get a better fit of the data. The HKK6 is one of several empirical terms suggested in the literature (48); the fourth order Hamiltonian of Chung and Parker as treated by Kneizys, Freedman, and Clough (49) also contains a term with this quantum depend- ence. The fourth order correction to the equilibrium centrifugal stretching terms as listed by Chung and Parker also shows a vibrational dependence. Thus there is some justification for these empirical modifications. In the (110) band, 356 lines were identified of which 279 were given nonzero weights. Zero weights were assigned when the shape of a line was very poor, two or more transitions were unresolved, or the line was masked by either an impurity or the isotOpe HZBQS. In the (011) band 391 lines were identified of which 316 were given nonzero weights. After all the assignments were made, all possible ground state combination differences were formed and com- bined with those from the (210) and (111) bands and seven microwave lines to reanalyze the ground state. Then the new ground state parameters were used to refit the upper state of the (011) and (110) bands. The observed and calculated values of the line identifications are listed in Appendix X 97 3 Table XII. Molecular Parameters of H2 2S for the States ' (110) and (011) State (110) (011) 00 3779.176j0.011 cm“l 3789.279:0.0ll cm-1 A 10.5469:0.0023 10.4828:0.0019 B 9.1057:0.0020 9.1515:0.0018 c* 4.6049:0.0014 4.6109:0.0014 ‘raaaa -0.00971:0.00014 -o.00973:o.00012 Tbbbb -0.00575:0.00015 -0.00579:0.00010 Taabb +0.00678:0.00017 +0.00676:0.00010 rabab -0.00213i0.00010 -0.00227:0.00009 1 HK -204x10'9: 24x10“9 -118X10-9: 12x10‘9 1— 62 0.261 t 0.028 cm“1 GXY -0.3603 1 0.0079 cm'1 $121.2: 595 3:322:13. Fig. -1 12. (110) State (011) State Energy Level Shifts Due to the Perturbations )3 \L Energy Levels in cm 4514.4 880 881 5 2 1 4508.1 880 881 d “ 4434.1 863 853 4429.3 Pert. Pert. Unpert. JK-K Value Value IJK-K+ {F’ “ 99 and the results of the analysis in Table XII. The tol- erance of each parameter is for a confidence interval of 95% which is about 6 times the standard error for this fit. The shift of several of the more perturbed upper state energy levels from the position predicted by an un- perturbed Hamiltonian is shown in Fig. 12. These shifts were calculated using the program NEW CORIKORR listed in Appendix XI. Reanalysis of the Ground State When the assignments from the (110) and (011) bands were completed the transitions within each band were used to form as many observed ground state combination difference as possible by taking the differences between the appropri- ate frequencies. Whenever the same combination difference was found within a band or from several bands a weighted average was formed. In all 317 different ground state combination differences were formed from the four bands. In addition, the following seven microwave lines found by Gallagher (50) were included with large weights approxi- mately in prOportion to the relative precision of measure- ment with respect to the infrared combination differences. 4 + 4 369,126.912 MHz 0 2 2 2+ 2 393,450.40 - o 3 + 3 300,505.56 100 3_1+ 3l 369,101.56 1-17 1l 168,762.762 20 + 22 216,710.46 42 + 44 424,315.45 -5 The conversion used is lMc/sec = 3.3356382 x 10 cm Using CD-FIT, a program written by Moncur (46), a set of combination differences Af (the appropriate differ- calc ences between the calculated energy levels of the ground state) was formed using the molecular parameters found in our previous analysis. The equation input to the least squares subroutine is AfObS — Afcalc -_— (alfl _ 02") AA" + (81" _82") AB" + (YIN _ Y1") Ac" L, + 2.1:]. [(Xi")1 - (Xiu)2]ATi" + [(XB")1 - (X8")2] AHK" (VI-13) where Afobs stands for the observed ground state combination difference and the coefficients of the molecular parameters are the average values of the operators for the two rota- tional states involved. The results of the analysis are given in Table XIII. The tolerance of each parameters is for a 95% simultaneous confidence interval which in this case is about 4 times the standard error. 101 32 Table XIII. Ground-State Parameters of H2 S A 10.3601 1 0.0003cm’l B 9.0156 1 0.0003 C 4.7318 1 0.0002 T -0.00811 + 0.00002 aaaa - Tbbbb -0.00485 1 0.00002 Taabb +0.00473 1 0.00004 Tabab -0.00138 1 0.00002 HK -73x10‘9 1 12x10’9 No. of Points 324 Standard 0 006 Excludes Deviation ‘ Microwave Lines 102 These values lie within the confidence intervals of those reported earlier from our analysis of the (210) and (111) bands. The observed and calculated values of the ground state combination differences are listed in Appendix XII. CHAPTER VII CONCLUSION The second-order quantum mechanical Hamiltonian expression for the nonlinear XYX asymmetric molecule has been extended to include Coriolis-type perturbations between pairs of mutually interacting absorption bands. New expressions have been found for the coupled Wang en- ergy matrices which include the perturbation coefficients and centrifugal stretchihg terms. The results from the analysis of two calibrated high resolution spectra of the (210) and (111) bands of H2328 (near 6300 cm-1) were used to evaluate terms in the Hamiltonian expression and make order of magnitude estimates. Five calibrated high resolution spectra of the (110) and (011) bands of H2328 (which mutually interact through a Coriolis-type perturbation) were run near 3800 cm-1. In order to minimize measurement error, measured frequencies from each recording were combined using a weighted averaging technique which took into account the precision of measure- ment of each line and the quality of calibration of each record. The (110) and (011) bands of H2328 were simultane- ously analyzed using the derived Hamiltonian expression 103 104 which takes into account their mutual interaction. It was possible to identify most of the Observed H2328 absorption lines in the 3800 cm.1 region by means of this analysis. Using data from the (210), (111), (110) and (011) bands along with seven microwave lines, the ground state of H2328 was refit by means of ground state combination differences.' Improved values were obtained-for the ground state A, B, C, taus and H term. Then the upper statesof K the (110) and (011) bands were fit again using the new ground state values.‘ This fit yielded final upper state values of A, B, C, four taus, H and the perturbation K coefficients Gx and G2' The revised values of the ground Y state parameters make no appreciable change in the values for the equilibrium molecular constants and structure published earlier by us (25). The favorable comparison of the standard deviation of the ground state combination difference fit and of the upper state fit with the standard deviation of the calibra- tion fit is interpreted to mean that the equations used are essentially correct. However, the improved results given by the slight empirical modifications of allowing the taus to differ in different vibrational states and adding the HK term indicate that a higher-order expression might be useful for analyzing the 3800 cm.1 region. The molecular parameters found in this analysis should be directly applicable to such an extension of the energy equations. 10. ll. 12. REFERENCES G. Herzberg. Infrared and Raman S ectra of Polyatomic Molecules, D. Van Nostr-d Co., New YorE,— 1945. a) pp. 42- 44 b) p. 466 c) p. 376 D. R. Bates (ed. ). Quantum Theory II. Aggregates of Particles, Academic Press, New York and LondOn, 1962. a) D. M. Dennison and K. T. Hecht. "Molecular Spectra", pp. 309-310. b) S. L. Altman.. "Group Theory", p. 152. G. W. King. Spectrosggpy and Molecular Structure, Holt, Rinehart and Winston, Inc., New York, 1964— a) pp. 307- 310 b) p. 335 c) p. 279 G. W. King, R. M. Hainer, and P. C. Cross, J. Chem.' Phys. 11, 27 (1943). C. H. Townes and A. L. Schawlow. Microwave SpectrOSCOpy, McGraw-Hill, New York, 1955. a) PP- 83-92 b) p. 64 c) p. 95 L. Pauling and E. B. Wilson,.Jr. Introduction Eg_Quantum Mechanics, McGraw-Hill, New York, I935, p. 277. R. S. Mulliken, Phys. Rev. 29, 873 (1941). H. C. Allen, Jr. and P. C. Cross.’ MoleCular Vib-Rotors, John Wiley and Sons, New York, 1963, p. 110. P. C. 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 MolecuIes, PErgamon Press, London ,71958. D. Kivelson and E. B. Wilson, Jr., J. Chem. Phys. 22, 1575 (1952). H. C. Allen, Jr., and E. K. Plyler, J. Chem. Phys. 25, 1132 (1956). 105 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25- 26. 27. 28. 29. 30. 31. 32. 33. 106 H. A. Jahn, Phys. Rev. 56, 680 (1939). E. B. Wilson, Jr., J. Chem. Phys. 4, 313 (1936). M. Goldsmith, G. Amat, and-H. H. Nielsen, J. Chem. Phys. 21, 1178 (1956). E. B. Wilson, Jr., and J. B. Howard, J. Chem. Phys. 4, 260 (1936). K. T. Chung and P. M. Parker, J. Chem. Phys. 43, 3869 (1965). _ B. T. Darling and D. M. Dennison, Phys. Rev. 51, 128 (1940). D. W. Posener, Thesis, Massachusetts Institute of Technology (1953). W. H. Shaffer, Rev..Mod. Phys. 16, 245 (1944). H. H. Nielsen, Phys. Rev. 68, 181 (1945). S. A. Clough and F. X. Kneizys, J. Chem. Phys. 44, 1855 (1966). W. Finkelnburg. Structure QEJMatter, Academic Press, New York, 1964, p. 34. T. Oka and Y. Morino, J.~Mol. Spectry. g, 9 (1962). T. H. Edwards, N. K. Moncur and L. E. Snyder, J. Chem. Phys. 46, 2139 (1967). R. C. Herman and W. H. Shaffer,.J. Chem. Phys..16, 453 (1948) . '— J. M. Dowling, J. Mol. Spectry. 6, 550 (1961). S. C. Wang, Phys. Rev. 32, 243 (1929).- P. C. Cross, Phys, Rev. 41, 7 (1935). C. A. Burris and W. Gordy, Phys. Rev. 22, 274 (1953). C. M. Savage and T. H. Edwards, J. Chem. Phys. 27, 179 (1957) . _ H. C. Allen and E. K. Plyler,.J. Research Natl. Bur. Standards. 53, 205 (1954). J. L. Aubel, Thesis, Michigan State University (1964). 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 107 D. B. Keck, forthcoming thesis, Michigan State Univer-. sity (1967). D. B.-Keck, J. L. Aubel, T. H. Edwards, and C. D. Hause, 1966 Symposium on Molecular Spectroscopy and Structure, Columbus, Ohio. J. U. White, J. Opt. Soc. Am. 22, 285 (1942). T. H. Edwards, J. Opt. Soc. Am. 51, 98 (1961). K. N. Rao, C. J. Humphreys, and D. H. Rank. Wavelength Standards in_ the Infrared, Academic Press, New York, 1966). T. L. Barnett, Thesis, Michigan State University (1967). M. D. Olman, Thesis, Michigan State University (1967). Program ASHAFT was written for the Michigan State Univer- sity C.D.C. 3600 Computer by L. E. Bullock and D. B. Keck. D. H. Rank, D. P. Eastman, B. S. Rao, and T. A. Wiggins, Jo Opt. SOC. AI“. 23:, 929 (1961) o D. H. Rank, G. Skorinko, D. P. Eastman, and T. A. Wiggins, J. Mol. Spectry. 4, 518 (1960). D. H. Rank, B. S. Rao, and T. A. Wiggins, J. Mol. Spectry. 11, 122 (1965). Subroutine JHERMX was written by N. W. Naugle, N.A.S.A. Manned Spacecraft Center, Houston, Texas. N. K. Moncur, Thesis, Michigan State University (1967). R. A. Hill and T. H. Edwards, J. Mol. Spectry. 14, 203 (1964). L. Pierce. N. DiCianni and R. H. Jackson, J. Chem. Phys. 38, 730 (1963). F. X. Kneizys, J. N. Freedman and S. A. Clough, J. Chem. Phys. 44, 2552 (1966). J. Gallagher, Martin Co., Orlando Division, Orlando, Florida; private communication. P. M. Parker, private communication. APPENDIX I 2)a8 ( st‘ FOR THE NONLINEAR xyx MOLECULE From the definition in (II—8) ( ) B ad5a86 2 0! l s s sto = -A:gl + ZS" 5:5" C218" + 5=X,er I6 The A22, are defined as (15) ca _ B B Y Y Ass' — Zi (lislis' + lislis') AO‘B _ Z a 8 55' i is is' When the values for the lin (17) are substituted into this definition, the following non-zero terms result: xx _ y 2 y 2 y 2 _ . 2 A — (111) + (121) + (131) — Sln-y AXX = (131,?) 2 + (1%2)2+ (11:2)2 = coszy _ I xx _ y 2 .y 2 1 A . ‘ (123) + (133) ' "x 33 I 2 xx 1 Y Y Y Y Y Y _ - A21 — 112111 + 122121 + 132131 — cosY SinY yy _ x 2 x 2 _ ‘2 A11 — (121) + (131) - COb Y Ayy 2 (1X )2 + (lX )2 = sin2y 22 22 32 108 I YY _ X 2 X 2 X 2 = X A33 ' (113) + (123) + (133) I2 AYY - 1X 1X + 1X 1X = -sinY cosY 21 22 21 32 31 22 _ x x y 2 _ A11 ‘ £1 [111111 (111) 1 1 A22 = 2 [(1X )2 + (1y )2] = 1 22 i 12 12 ‘ (See p° 110 for 22 _ v X 2 Y 2 _ nonzero x terms A33 _ L1 [(113) + (113) 1 ‘ l y ) All other A22, terms were found to be zero. It was nec- essary to carry out this calculation since many of the terms in similar calculations found in the literature con- tain typographical errorso It was heartening to find that the above values agreed with those found earlier by Parker (51). By using Table V the possible values of Qéizas are adéaB? .§__§. 16 (2)a8 = _AaB L a B + r ss' 55' Zs"Css"'s By carrying out the sum over the as components in the first sum of (II-7) the nonzero values are found as follows: (2)xx xx agxa§¥ a§Ya§¥ st' 2 “Ass' + 1_____ +" I I X Y (2) xy xy 2:X Z): agyag st' = -Ass' + _-—-'“ + '"‘-"" I I 110 YX XX YY XY a a . a a , Qé:)yx = 'AZ:' + s s + s s I I X Y x x (2)yy yy a: a:' agya§¥ a . = —A , + + ss 55 I I X Y 22 zz (2)22 _ zz Z 2 as as' RSS, — ASS' isncssucstsu + I 2 Therefore the first sum in (II-7) is: (2)xx 2 (2)xy (2)yx ‘E Z , qsqs' st' x + 955' Pxpy + st' Pny 55 I 1 2 J» _ (Asks ). IX IXIy Iylx 2 2 2 zz 2 Qéslyypl 9;." P. + + I 2 I 2 y z Q(2)018 4 % = 1.2 2 55' (JH ) q q P P 2 ss' OB I I Asks, s s B a B Nonzero xy terms: AXy - -c Y (I ’I )2‘ — Ayx 13 ' OS x/ z ‘ 31 AXY — sinY (I /I )2 = AyX 23 — x z 32 Axy - -(I /I )% sinY - AYX 31 — y z — 13 AXY = -(I /I )% cosY = Ayx 32 y z 23 APPENDIX II USING ISOTOPIC SUBSTITUTION TO FIND FORCE CONSTANT ESTIMATES IsotOpic substitution is useful for estimating the force constants in (II-28) since the three normal fre- quencies of hydrogen sulfide do not constitute enough infor- mation. By including the normal frequencies of deuterium sulfide, (II-28) may be solved for estimates of the force constants. The method of isotopic substitution is detailed in the literature (19) so only the important steps are covered here. Using the same geometry and mass-adjusted symmetry coordinates as Chung and Parker (17), the symmetry coordi- nates without the mass adjustment are denoted by primes on u, v, and w. Then the harmonic potential energy is 2 2V = k u2 + k ,\7 + k w2 22 33 11 + 2k12uv '2+k 'v'2+k 'w'2+2k 'u'v' _. I _ k11 u 22 33 12 111 112 The harmonic force constants in this system are given by — m - k11' ‘ 7 k11 k33' ‘ “3 k33 _ _ mu k22' - u k22 klz' - ‘3‘ k12 From (II-28) and the force constants, it is found that >’ + >2 II W + W ll 5H0 W To find k22 from hydrogen sulfide and deuterium sulfide data, respectively, the expression for Al +12 is written m H H _ , m H 7— (Al + 12) — k11 + (7—) k22' m D D _ , m_ D , 2 (11 + 12) — k11 + (Zn) k22 Solving for kzz', k may be immediately found. 22 H D (2mH+M)[mH(11+12) - mD(11+12) ] k22 = 2mH(mH-mD) To find k11' arrange the equation for Al +12 as 113 H 2 H u (11+12) n N TE W + x D D A u ( 1+12) II N TE W + 7?“ Solving for kll', k11 may be found. H D k = mH(2mD+M)(A1+12) - mD(2mH+M)(Al+12) 11 2mH(mD - mH) ,1 = - n=11 2 D —' 2 2_—_ ll 2 , .21:31§E_. (21 _A_) a1j+a51n2qn+a5j33q3 qz 24h nlj=1 .L n J n 12 2 (41D -1 _ + Aj j) a 7j' an3 3a 3jqj +2a51j2qnqj -Za7nqunqu3 2 a + 2 a . q.— + 1n 5132 3 i qnqj 2 (A X-) n 3 a7nMjp3+a5n33q3 2 3 2 + 6a3nqnqj+4a51n2qjqn + ‘a7ijlpn,qn] q3 -2a7nannqjq3 117 2 2 [a1.+a5.33q3-a .M p ]p + -l 2 DnjaSInz ZAn J J 73 n 3 n +.__ n ._ ( ) Zfi 'j_1'l 'fi2 2[ -+ 2 ] A%(4A -x.) aln a5n33q3-a 7n Mjps p npj J n 3 2 2 nn[p ,qn 2+] pj +3 asjpnqj + 2 2 +2 . a51]2[pn quj +qn p2 “q 1 I (Eig')[§n=1sl+sn33’(H )] 1 ' 2 2——-—2_i 2 D . + . +8. =(:£)Z ' nlasnas (21 _A ) (aln a51j2q3a5n33q3)q3 2h n,3=1 1 3 n 2 2 An(423-An) +3a a3nqnq:+ +2a a51n2qnqjq3 +2a7jannqjq3 2 +a7nMj(2pnqnq3-p3q3 ) + ‘L2(aq+aqu)q (AA)2 13n7jjn'j3 n 3 +4( + ) 2 a5133q1qn a5233q2qn q3 -a7n Nn(p3 q :+q 3p3q3- 2p nq “q ) +2(a72M1q2+a71M2q1)qnp3 118 2 2 2A (a1n+a51j2qj-a7nMj)p3 _ 5 3 +(-—-1)2 -1 n1 n3 ( 3) 2h n 3‘ A%(4A -x ) fiz +3 2 2+2( ) 2 n 3 n aBnqnpa a51n2 qnqu3 i l (§)[S;1,(H1+H1 )] 2% a71(A3-A1)‘h F l 2 +A 3) A q 3)(A v—aN 4(A1A + + 2 +a5n33[2[p3:q3] ann+q3 2 p31 +2(a13pn+a71N1p1pn+a72sznpz)p3 ‘ 2 2 2 2 1 +(a11p3+55122q2P3'a71M2p3)(£30 2 1 +(a13p1+a72N2p1p2+a71N1P1)(a) P + 2a q1q2p3 5112 432 a5133 2 333333 + ———- [2p q q +q p 1 + p 5? 1 1 3 3 3 fl? 1 qzq 3 i I (.5)[s72,(H1+H1 )1 -a 72 % 4(A2A3)(A3-A2) 2 2 A A - ( 2+ 3) a72(M1q2 N2q3) +a71M2q1q _“_——— ______i 119 i. 2 2(A2A3) A 2 2 a1293+a5112qlp3'a72M1p3 + + N 2+ N a13p2 a72 292 a71 1p1p2 2 +3a32q2p3+2a5122q1q2p3 +2a5133q192q3+ 2 2 +a5233( p2q2q3+q393) APPENDIX V THE PURELY VIBRATIONAL CONTRIBUTIONS EV' AND EV" n L 2 I1 15 2 SE n 2 J. 2 + a3 n(vn 2) . + 5132 a 5333 +(3a r______ a . (v 5132 .1. +(3a T 3n L._________ 120 a _él2£)(v 2 y +l + (v3 2) a51n2 4 % (Anxj) .+ a51 3] (4An-Aj) (v ,+l + 3 2) a5n33 3 4(Anx3)2a X3-An) 5n33 (4 (vn+%) L L j+2)(vn+2) (vn+%) n2)(vj+%9 _____J +%) (v3+%) )(vn+%) 2 [(vi+J2-) +5 J. ‘11 C13 A 2 - 2 —-————-—— Hi) + (ii) 1 (v1+§) 1 (v2+12-)(v3+12-) 21 A2 3 2 3 «n 2 2 3hc -—-— (€13 +c23 ) +-———-§._ k.._. 41 2 1-1 1111 Z 3. 3hc 2. (v.+l)(Vj+%) ‘ D‘.k.... l,j=1 13 1133 l 2 APPENDIX VI ANGULAR MOMENTUM MATRIX ELEMENTS Listed here are the angular momentum matrix elements from Posener (l9) needed to evaluate (III-18), (III-l9) and (III-20) to arrive at (IV-3) and (IV-4) using f = J (J+1)o 2. 2 2 u u u =hK =h K =fi K . 2 \..I 2 .412 2 — B-——[3f -2(3K +l)f+K (3K +5)] 2 2 + 4h“ 2 2 2 2 —z-[f -:(K gl)f+K (f -5): 2 + y ,pZZ] |K>=fi [f-K ]K % =¥%+4[f-K(Ki1)][f-(Kil)(Ki2)]] 2 2+l ==- X y 53 2 2 2 =-§{[fFK(Kil)][f-(Kil)(KI2)]] [254<-(K+2) 1 2 ,pz ]+|Ki2>=_ L 2111+ 2 2 =-Z—[[f-K(Kil)][f-(Kil)(Ki2)]] [K +(Ki2 =ii§-[[f-K(K_l)][f-(Kil)(K-2)]]2 2 2 + “th4>=w% = IFIK0.NE.1IOP(K.K)IFIIAoBaCo JaKOI IF¢K0.EQ.1)OM(K.K>IF1(A,B.C. J.K0)-F2(Ao3oJ) IF¢K0,NE.1)0M(K.K)sF1(A.B.c. J.K0) CONTINUE c FORMING HERHITIAN CONJUGATES 19 109 DO 9 NIloKHAX ININ+1 DO 9 MIIN,KMAX EPIM.N)IEP(N0M) Kx-J/Z DO 19 NligKX 1N8N*1 D0 19 H3INJKX EM!H.N)' EM‘NaM’ DO 109 NalaKIX ININ*1 DO 109 HEINaKIX 0P!M,N)=OP(NvMI OMIH.N)‘OM(NJMI 126 IFIIFMATI 517.516 516 CONTINUE THIS SECTION PRINTS THE MATRIX ELEMENTS PRINT 518gJ 518 FORMATI/ICEPLUS FOR JavI3) DO 14 K31.KMAX PRINT 10a( EPIKaLIaL=1aKHAXI 10 FORMATI/II12(F10.4)II 14 CONTINUE PRINT 519aJ 519 FORMATIIItEHINUS FOR JstISI DO 15 K31JKX PRINT 16.¢EM¢K;LI.L:1,KX) 16 FORMAT¢//(12(F10.4>)) 15 CONTINUE PRINT 520.J 520 FORMATI/ltOPLUS FOR J=¢ISI DO 114 K31.le PRINT 1100I0PIK0L)0L311KIX) 110 FORMATI/lIlZIF10.4)I) 114 CONTINUE PRINT SZiaJ 521 FORMATIIItOMINUS FOR JatIS) DO 115 KflioKIX PRINT 116.(0H(K.LI.L 116 FORMATI/l‘iZIF10.4II 115 CONTINUE 517 CONTINUE EIGENVALUES ARE FOUND IN THIS SECTION CALL JHERMX IEP,11.KMAX) CALL JHERMX (EM.10.KX) CALL JHERMX (OP.10.KIXI CALL JHERMX IOH,10.KIX) . PRINTING EIGENVALUES HITH J;Kc.Kt LABELS PRINT 3120J ' 312 FORMATIIIOGROUND STATE ENERGY LEVELS FOR J=*I3//I DO 313 I'iaKMAX KPLUS=2tIIle TAUPIJIZ‘KPLUS KMINU53TAUPOKPLUS PRINT 314.J0KMINUS.KPLUS.EP(I.I) 314 FORMATISI3.F15.4) 313 CONTINUE DO 320 1'1.KX KPLUS 12*! TAUPBJ-ZtKPLUSo1 KMINUSITAUPtKPLUS . PRINT 314oJaKMINUSaKPLUS.EMIIoI) 320 CONTINUE DO 330 Il1oKIX KPLUS 82*I-1 TAUPnJu2tKPLUS =1.KIX) I 340 11 ACM 127 KMINUSITAUPOKPLUS PRINT 314,JpKMINUS.KPLUS.OP(I.I) CONTINUE D0 340 I81,KIX KPLUS IZ‘Icl TAUP:J-2*KPLUS¢1 KMINUSITAUP‘KPLUS PRINT 314.J¢KMINUS,KPLUS.OM(I.I) CONTINUE CONTINUE END SUBROUTINE JHERMXIAcNM;NI COMMON IOPTgRHOpIER DIMENSION A(NMINM). 5(21321) COMPLEX A: 5: V1: V2. V3. CSNTa SNT, TEMP IOC a O IER 8 O EPI2=0.000000000001 EN I FLOATFIN) DO 1 IaloN DO 1 J=1oN S‘IoJ) ' ‘0|°00.0) IFII,EO.J)S(I.J) 8(1.000.0) CONTINUE IFIN.LE.1)GO TO 12 SOFFD I 0.0 NM1 3 Nil 00 2 II1;NM1 IJ = 1&1 DO 2 JilJaN XIIAIIaJ) XZIAIIgJIRIOuOI'1.0I SOFFD=SOFFD¢2.0i(X1*X1*X2tX2) IFISOFFD ,LT. 0.1E010)GO TO 12 THR a SQRTFISOPFDI FTHR I RHOaTHR / EN IND I O THR : THR / EN DO 10 L320N LM1 : Lv1 DO 10 K310LM1 XllAILaK’ XZ'AIL.KIi(D.0}1o0) IFISORTIX1*X1*X2*X2).LT.THRI GO TO 10 IND : 1 V1 ' A(KgK) V2 I CONJG( A(LnK) I V3 8 AILIL’ .AIMAGI A(L.K) I REALI A(LIK’ I PEALI A(LnLI I PEALI A(KgKI I UFWUJO III. O 128 THET 3 0.0 IFIAESFICI.LT, EPIZIGO TO 13 THET 8 1.5707963 IF(ABSF(BI.LT, EPIQIGO TO 13 T I C / B THET a ATANFITI 13 CONTINUE PHI E 0978539818 ED 3 ABSFIE-DI IFIED .LT, EPIZIGO TO 15 TAO I 2.0'IB'COSFITHETI * CiSINFITNETII / It”DI PHI 3 0.5 t ATANFITAO) 15 CONTINUE CSNT =COSF(PHII * (0.010.0I CS I COSFITHETI * SINFIPHII SS : SINFITHETI * SINFIPHII SNT a CS 0 (0.0.1.0) t SS DO 8 I'laN TEMP : A(I.K)ccsNT - A(InL)*C0NJG<5NTI A(IoLI s A(IIKI*SNT t A(I.L)¢CSNT A(I.K) 8 TEMP IFIIOPT .EO. OIGO TO 8 TEMP 8 SII.KI*CSNT v SIIILI*CONJG(SNT) SIIaLI 3 SIIJKI*SNT * 5(IaLI'CSNT SIIaKI ' TEMP 8 CONTINUE DO 9 I311N A(K.II : CONJGI A(IoKI I A(LnII 3 CONJGI A(IaLI I 9 CONTINUE A(ng) 3 SNTtCONJGISNTI*V3 * CSNT*CSNT*V1~SNTtCSNTCC0NJGIV2) 1 BCSNTtCONJGISNTItVZ A(LoLI 3 CSNTtCSNTtVS t CONJG¢SNTI¢SNT¢V1 * SNT‘CSNT* 2 CONJGIVZI * CONJGISNTI'CSNTtvz A(KoLI l CSNTtCSNTiVZ I SNTtSNthoNJGIVZI * CSNT*SNT*V1 1 uCSNTtSNTtVS A(LaKI 3 CONJGI A(KaLI I 10 CONTINUE IOC I IOC t 1 IFIIOC .GE. 100IGO TO 21 IFIIND .LE. OIGO TO 11 IND : 0 GO TO 4 11 IFITHR-FTHRI12I12p3 12 RETURN 21 IER 8 1 RETURN END RUN0501200 APPENDIX VIII LISTING OF PROGRAM U.S.Ex.D. 129 F r‘ 130 PROGRAM HSCXO DIRENSION FPVIIZI.71):PPV2t21o21I.OPV1(20.20I.OPV2(20.20I. 1 A(.7).BIZI.CS(2IoE(32.3hTA3(2IoTATIZI.TAS(?)nTAM2IoEN(2).HKIZI CO”PLEY PPVinEPV?:DPV1.OPV2 COMMON 10PT.RHO.IER ARITHMFTIC STATEMENT FHNrTIOVS D FiIAanCS,FN.TAN.R.Y,T,X,H=retJ.Kn) «n2 OPVQIK.K¢23=F8¢J.KO) *UZ OPV2(KAT.KAT+2):F8(J.KO) tu1 IFIKO.EO.1) OPVlCKaK+1I=F6(A(1):B(1I.JIKO.V1aSI)#F7(JIiU1 IFIK0.NE.1) DPV1IK.K+1)=F6(A(1)18(1I:J.KO.V1oSII IFIKO.¢O.1IOPV1(KAT,KAT+1)2F6(A(2).B(2).J.K0ov2.52)-F7(J)*U2 IFIKO.NE.1IOPVI(KATaKAT+1)¢F6(A(Z).B(2)oJoKOgV2:82I Opv1¢K,KAT+1)=¢0..-1.)tF10(JaKO) OPV1IK¢1.KAT)=q0PV1(K.KAT+1I IFIKD.EO.1309V?(K.K+1I:F6(A(2).BIZIoJ.KO:V2152I+F7(J)tU2 IFIKO.NE.110PV7.8<2>.J.Ko.V2.82) IF!KD.FO.130PV?(KAT.KAT+1):F6(AI1).811):JoKO:V1.SlI-F7(JI*U1 IFIKO.NE.1IOPV?(“ATgKAT*1I=F6IA(1I38(1I1J1K00V1351I OPVQIK.KAT¢1I:flPV1IK.KAT+1I ' OPVQIKt1.KATI=-OPV1(KaKAT+II loonIFIKO.FO.1IOPV1(K.K)=F1(A(inBI1IaCS(1):ENI1I.TANIIIaRInY1:T1a 1 xlvHKIII3J1K0I¢F?(A(1Io8f1I:JaV1351I nIFIKO.NE.1IOPV1(KoKIBF1(A(1Io8¢1IoCSI1IoENtlIaTANIII1R11Y12T10 1 xlnHKIIIpJOKOI nIFIKO.FO.1IOPV1(KATaKATI3F4(AIZIoBIZI:CS(2IoENI2IpTAN(2I:R2pY2: 1 T?.X7,HK(2I.J:KOI~ F?(A(7Io8(2I:J9V2.32I nIFIKO.NE.1IOPV1(KAT.KATI=F1(A(ZI:B(2I.CS(2I;EN(2IaTANIZIgRZpYZ. 1 TZnX’aHKf2I9J1K0I IFIKO.FO.1IOEVI(KvaTI=(O.n1-I*(G*KO'.25*N*J*IJ+1II 133 IF¢K0.~E.1)OPV1OPVD.CS(2).EN(2>.TAN(?).R2.Y2.T2.X2: 1 HK(25,J.KO)+F2(A(2)oP(?).JaV?352) IF¢k0.NE.1SOPV9(K.K)=F1(A(9)18(2),CS(2):ENt2).YAN(2):R2.Y2.T2.X2. 1 HK(2).J.K0) nIFtKn.Eo.1wopvackAT,KAY)=F1(A<1>.B¢1).CS(1S;EN(13.TAN(1).R1,Y1. 1 YInX1aHK'1’oJ9KO) '72(A(1)09(1)1J0V1381) 0IF(K0.NE.1509V?(KAT.KAY)IF1(A(1)p8¢1),CS(1)oENC1)oTAN(1):R1:Y1: 1 71pX1.HK(1’.J.KO) IF¢K0.FQ.1)09V?(KoKAT):(0..“1-)*(G*K0*.25*U*J*(J*1)) IF¢K0.NE.1)OPV?(K:KAT):(n..-1.)t(G'K0) 101 CONTINUE C FOQMING HEQMITIAN COKJHGATFS 0r EPVIOEPV210PV190PV2 n0 9 N:1,JCMF IN=N*1 U0 9 MsIM.JEMD FPV2(M.N):n0NJG(FPV2(N.M)) a EPV1(M.N)=EONJG(FPV1(N.M)) no 109 N31.IMD INBN¢1 "O 109 Halunan OPV2(M.N5=r0NJa(fiPv2(N.M)) 1n? 0Pv1(M.N)=CONJG(CPV1(N.M)) IF¢IFMAT5 $17.61‘ 514 CONTINUE 6 THIS SFCTIHN PRINTS MAVRIX ELEMEVIS .SFT IFMAT =1 IF NOT DESIRED pRlNT 518.J 519 FDRMATtlltFV1PLU9+FV2MYNHS FDR J=*!3) DO 14 Ks1nJEMD pRINT floi‘FPVI‘KDL)'L=‘OJEMD) 1a FORMATc/(66(F1n.4.rlo.4)3) 14 CONTINUE pRINT Siqu 519 FORMATtlltEV2PLUS+EV1MINUS FDR J=*!3) no 15 K:1.JEHD PRINT 16.(FPV2(K}L).L=1.JEHD) 14 FORMAT(/(6P(Flfl.4p710.4)3) 1% CONTINUE PRINT 620.4 52* FORMATtlltfiV1PLUS+OV2MINNS FDR J=*!3) no 114 K=1.1HD PRINT 119:(OPV1(K.L).L81.XMD) 11w FORMAT(/{66(F1n.4.F10.4))) 114 CUTTINUE pRINT 5211J 524 FORMATt/ItfiV?PLU°+OV1MINHS FOR Jaoys) 00 115 K81.IMD FRINT 116.!0Pv7(K.L)aL=1.IM0) 114 FORMAT(/(6F(F1n.a,rlg,4))) 115 CONTINUE S17 COWTINUE n TH'S SFCTION CALLS JFEQMY 134 CALL JHEQMY(EPV1121:JE”D) CALL JHEQMY(FPV21210JE“D) CALL JHERMY(0PV1,20.IMH) CALL JflEQMY(OPV202011Mn) F DRYNTING EIGENVALUfiS WITH J:K9,K* LABELS 5?‘ 314 31‘ 31% 529 114 117 11 pR’NT 523 FORMAT(liEleLUS+EV2MIVUS CoLEVELS EVZDLUS+EV1HINUS 1LS*/) ”0 313 I=1gKMAY KPLUS=?t(I-13 TAUPBJ-ZthLUS KMINUS=TAUP+KPLUS PRINT 314;J,KMYNLS.KPLHS:EPV1(1oI).JoKMINUSoKPLUSo€PV2(InI) FORMATf2(3Y3oF15.5910X)) CONTINUE YMYNiaKMAX+1 DO 315 IBI”IN1oJFM“ KPLUS=9¢(I-KVAY) TAUPSJ32*KPLUS+1 KMINHS=TAUP+VPLU9 DRINT 314:J:“M!NlSaKDL”$.E°V1(Int):JpKMINUS:KPLUSaFPV2(I:I) COMTtNUE pRINT 52? FD”MAT(/*0V1PLHS¢OV2M[NUS C.LEVELS OVZDLUS+0V1MINUS 1L3'l) no 316 1:1.K1x KPLUS=901~1 TgupaJ-2.KOLUS KMINU53TAUD+KPLUQ pRINT 314oJ:VMYNhS.KPLHSp0°V1(I:I)anKMINUQaKPLUSonPV2(InI) CONTINUE YMIN7=KIX+1 no 317 I=IHI~2.I"D KPLUS:?0¢I-KIX$'1 TAUP=J92*KPLUS*1 ‘M!NUS=TAUP+KPLU9 pRINT 3149J:KMTN‘SaKpLHSoODV1(IDI)nJaKMINngKPLUSIOPV2(InI) F0”T!NHE CONTINUE FN” SURROUTINE JHFR"X!A.NM,N) CO“MUN [HPT,QanYER DI”ENSYOM A(NM,N"). S(?1n21) COMPLEx A: S; V1. V2. V3. HSVT. SNT: TFMP 108 a n IEH : n 997230.030"0“000”01 EN = FLOATF(N) D0 1 1311N 00 1 J311N S('1J’ 3 (fi.0,fi.fl) YF‘IoEQ.J)§(Y.J) =(1001000, E.LEVE E.LEVE 11 135 C0“TINHE IF(N.LF.1)C0 IO 12 QOFFD a n.n NM1 = ”-1 n0 2 1:11NM1 IJ = 1+1 ”0 2 ngth x1=A(loJ) ‘23A(11J)*(0909’1-C) SOFFD:§0FFU+?.nt(X1iX1+X?tY2) IF(SnFVD .LT. n.1E-10)G0 TH 12 THH = 909TF(90FFP) FTHR = RH0*THR I EN IN“ a n THH : THQ / EN ”0 13 L=?:* LM1 3 L—1 DO 10 KBIpLMj x1‘A‘LnK) x2=AcL1K,*(0001100) [FtsnRT(X1cx1+x2tX2).LT.TH°)GU IO 10 IN? a 1 V1 3 AthK, V2 = CONJG( A(L,K) ) V3 8 A(LnL’ C t sAtMAGt A(L:“) ) R = REAL! A(L.K) ) F 3 PEAL( A(L1L) ) U I pEAL! A(‘.K) ) THFT B 0.0 IF‘AQSF(C).LY. E91?)GO T0 13 THET = 1.5707963 IFCADSF(R).LT. EPI?)GO To 13 T a C / u THFT = ATANFtT) CONTINUE PHI 3 n.78839818 FD 2 AQSF(F~B) IFfEn .LTo E917)GO T0 15 TA” 3 2.”*(B*COSF(THFT) + CtSIMF(THET)) / (E'D) 9H1 8 fl.5 * ATAN‘WTAO) “OVTINUE CSNT IfiOSF(PHI) * (0.030.03 CS 3 CGSF(THFT) * SINFprI) SS : SYNF(THET) ‘ SINF(PH13 SNT 3 CS * (0.01103) * SS ”0 8 I‘ION TEMP = A(I.K)*CSKT - AcI.LstcovJG01>4>uho;n\a)gna.a-¢:yo:n~aoAra-u\a9‘ HF‘F‘H 4>deH*3~O ODVJ‘J‘bl‘O IIIIIIOI'IIIIIF'IIJIIIIJIIIJQIIIIIIIOIIC GDOUNU .AH...‘ 1...... 03‘s.; \OmVO‘mle-‘(NNO‘OGVombCANMHhOOVVb-HOOGNONMHOOGVOWAOAN p.533”... 3 ~Axr4xga:4\Jpcnusacnwwah‘osicao#fiflrAna “.54. muomaupouMQomwomAummr-‘HVommmoomwompmpoomwomaum OnSfiRvED 6?63.356 69530924 6743.639 6937.067 6921.717 6910.193 6190.301 6186.045 6173.425 6160.424 6147.098 6952.360 6911.476 6907.128 6190.R10 6178.937 6166.490 6154.311 6220.038 6194.149 6191.067 6182.929 6171.408 6909.085 6‘10.986 6319.818 6319.016 6127.783 6335.471 614?.n10 6340.749 6156.111 6K6?.487 6369.949 6170.648 6183.950 6187.457 6323.901 6127.104 6346.040 6354.459 6361.696 6360.:94 6178.103 6181.926 DIFFFRENCE ’00005 0.0?4 0.004 '00003 0.018 0.030 0.039 0.045 0.043 0.013 0.008 0.000 0.019 0.002 0.074 0.047 0.053 0.077 '00018 “00065 0.043 0.030 0.003 '00023 0.014 0.005 “00023 '000n6 0.002 0.014 0.015 0.0?3 0.032 0.015 0.028 0.058 0.097 0.039 0.039 '090?9 '00004 0.011 00031 0.050 0.067 WEIGHT 0.05 0.08 0.70 0.90 1.00 0.60 0.30 0.40 0.10 0.02 0.03 0.50 0.50 0.70 0.30 0.01 0.01 0.03 0.00 0.20 0.10 0.04 0.10 0.50 0.20 1.00 0.40 0.80 1.00 1.00 1.00 1.00 1.00 0.80 1.00 0.10 0.02 1.00 0.70 0.12 0.70 1.00 0.?0 0.30 0.?0 9.4 J‘Q.fl\’fi4‘\3‘\3‘ fli)fi‘i\4)‘4) AAJD~JfimfiJledJ 0~J}£>A‘OJ‘QJBJID,-fiab‘.J\JQ¢>A 0 H UHUOJM-b\JPCdN3HF‘NJMCDH‘flwhOJH:3\JmkflbWNNJHF‘CDQKnOdMP‘CDAKHUN5CdabbFOOJU"URJU“U 3:3B¥JH54-33T33 D\IV‘J}C)Qtd€d&fU\)MfiDNHUN)M‘OHW‘d‘ur‘JWAN)%KflUIUF‘OiDUVfi\J”F* II I II I II I II I II I‘II I ll I Id I II I II I II I II I II I IUF‘&CRUWMRJOHUP‘GMOODV‘JO‘QOKbCNODVC>OWmJ>UPORJVC>QJHRJHCIUNA‘QUL50dO\OONOCNNH4 J Vr‘I>$ZAU‘D'UH*3"DDWML‘\)M\N&IM*‘I!)I)U‘bidfi%&%‘\4).bbdflf‘d‘b.bblb(dbiu‘0\fuid\)4 P‘F‘DCDKJNFUF‘HFJ‘JOCFOHW\DKJNFURJHF‘F‘HE‘F‘PW‘F*CHDC3CIOCDADHCDJhNP‘CDW‘Mth##‘CDO 139 6392.316 6345.179 6341.993 6373.069 6373.301 6393.510 6399.481 63668629 6375.428 6370.868 6392.110 6387.813 6397.406 6282.270 62806535 6277.507 6273.110 6261.790 6256.123 6274.236 6270.437 6274.895 6274.570 6272.800 6969.152 6258.268 6264.256 62500924‘ 6262.752 6264.054 6266.536 6266.164 6227.:60 6227.926 6215.343 6215.648 6215.477 6902.790 6993.750 6295.124 6999.180 6300.980 6999.950 6300.400 6225.975 6206.752 6331.152 6350.173 0.104 0.024 00039 0.014 ‘000?8 0.047 0.023 0.012 0.094 0.044 “00091 '00061 0.029 0.021 '00011 "09000 '09022 '00034 '09065 ’00159 0.013 "00075 0.036 0.024 '00012 "00035 0.015 '00030 '00068 -0.003 0.034 0.038 0.000 0.022 0.018 0.014 0.034 -0.013 0.026 .00073 '00037 '000?1 '0.008 ”00015 “00032 0.031 0.029 -0.058 “09034 0.02 0.60 0.10 0.60 0.03 0.20 0.10 1.00 0.30 0.30 0.06 0.40 0.70 0.40 0.90 0.30 1.00 0.25 0.20 0.50 0.15 0.10 0.80 0.60 1.00 0.14 1.00 0.60 0.03 0.30 0.10 0.20 1.00 0.15 0.08 0.03 0.03 0.01 0.02 0.50 0.06 0.90 0.03 0.50 0.25 0.90 0.10 0.20 1.00 140 NSSIGNFD FQETU‘NCICS FOR THE (111) BAND upPEH 11%).”.34400 HAM-- 01:04.30 .44 J15‘JO’dbbd‘JOBVJthA HE‘ AbubuuMUMUO-IMU‘UMHMv—JMr‘M‘r—‘NJMMHMHDHDuAHQSHoH-Av-é 01455:qu ’DJ‘JIb’JJUr‘ z-Ar—I- u—Is—B “V' H l 3 A .A964 gamuxuw-‘DomuamuA—ba \J‘IDOIIV)J15.DJ?\IMA iII'III'IIIIIII‘IIIITIIII1*II IIJI‘lIiIIIIIIII-II'IIIJIIIJ 4.3.; y—J.JJ GDOUND 2 3 4 5 6 7 9 ‘ 1 1 1 1 1 L-‘Lu‘ an; L4 J‘bbfdhhideAD-I’DCAVQJQMJMMUUJ\Jr‘V‘VVMi-‘NJHDH‘343M33HQ-‘r‘r45 NHomvamwpocmw’oumruHumo—toomwombumm Hmb-CANHO‘OVO‘U‘ILOAM OQSERVED 6265.785 6955.910 6244.722 6233.065 6222.849 6211.379 6187.365 6174.824 6161.037 6148.685 6135.103 6121.173 6106.005 6957.388 6947.782 6944.797 6936.905 6925.369 6214.566 6203.458 6191.080 6180.159 6167.089 6155.486 6149.608 6129.445 6115.025 6240.111 6230.434 6925.568 6217.642 6184.032 6172.575 6160.764 6148.610 6136.912 6223.941 6213.671 6907.175 6187.908 6153.113 6141.323 6129.190 6208.576 6197.403 DIFFFRFNCE “00006 '00006 0.007 0.028 0.030 0.037 0.033 0.019 0.003 '00033 ”00053 '00070 '00066 0.021 0.001 ”00066 '00038 .00032 0.017 0.051 0.048 0.052 0.050 0.054 0.018 0.031 0.021 0.025 0.019 “00030 '00035 0.018 0.044 0.046 0.048 0.138 0.037 0.011 -0.021 ’00076 '00021 'OOOnl 0.005 0.020 ~0.037 NEifiqT 0.60 0.80 001.0 0.80 0.80 0.60 0.60 0.80 1.00 0.50 0.60 0.?0 0.30 0.80 1.00 0.02 1.00 0.10 0.50 0.70 1.00 1.00 0.90 0.70 0.02 0.30 0.08 1.00 1.00 0.60 0.30 0.90 1.00 0.30 0.10 0.06 0.70 0.40 1.00 0.10 0.60 0.02 0.02 1.00 0.80 03M)J}$SJJAU).D~J)DV1551 r-‘r—A 9.4 D'VI’fi‘JJODJfiJ‘A-‘Jonflibb b\flULbJIqu&hb£bbCNOJMZdNJMCdOIU004‘Mr‘AJHWONJH6*N)V-‘:IHF‘CDHCDHWDCDH5*O\mChUIbkflU1b OLD\J)~££>WCA\)DLJ\JH’hid\)9 hCfl\)4%D\JQC>J0M00H‘O‘D\JOKIUL§RJ‘ IIIIIIIIIIIII‘IIIIIIIII‘I‘IIIIIIIIIIII‘IIIIIIIIIIIIIIII 4.A_A ‘0330\fl#b#CMNH‘CDGTVC>U3AC£RJNF4\JOCD\JOCL\JO Apaa~¢ NCNRJHID \JOKflJsO\OODO‘QU1#(NCJ€HD‘JOChUHN .Jr‘)‘.n.)d'b\n\r3 .-\ - 4v bkfixflasbfiubub-bibasuCANJifiumJOMuiflNJH*M~‘\3‘FUNJA+JAQ\JA.394d-394:1” 3 (NR3PCD‘JOKMUM5NH‘CJQ‘QOWMJMAGQHCDNHJCDOCD\HJJ>UCNAHVP*DCD\JOkfl#&NfiJNF‘H‘bcdm\IJSUTU MAE‘ 141 6189.877 6182.123 61683A43 6155.283 6173.756 6165.160 6144.283 6170.421 6311.565 6320.600 6320.868 6320.076 6336.622 6344.002 6351.001 6357.630 6363.895 6375.247 6380.349 6385.093 6339.248 6347.743 6348.016 6355.568 6362.876 6376.399 638?.602 6388.388 6393.986 6398.974 6403.685 6337.522 6346.657 6366.791 6374.180 63746404 6381.529 6388.124 6394.791 6400.989 6353.519 6363.313 6374.753 6392.420 6399.703 6390.695 6406.584 6412.783 6368.477 6379.395 6391.493 6405.996 0.013 0.012 '09054 0.003 0.029 0.021 0.030 '00166 “00023 0.008 0.035 0.021 0.033 0.028 0.018 0.011 0.019 .00002 0.007 0.068 '00023 0.010 0.049 0.024 0.044 0.045 0.045 0.003 0.054 0.085 0.145 '00042 '00070 '00024 0.010 0.042 0.052 0.034 0.025 0.119 .00025 ’00010 .0.094 '00060 '00022 0.012 0.089 “00010 “00016 0.018 '00041 0.007 0.50 0.20 0.20 0.20 0.80 0.50 0.08 0.50 0.40 0.30 0.40 0.30 0.30 1.00 0.70 0.80 1.00 1.00 1.00 0.60 1.00 0.70 1.00 0.60 1.00 0.90 1.00 0.06 1.00 0.30 0.20 0.40 0.60 0.50 1.00 0.70 0.20 0.10 1.00 0.50 1.00 1.00 0.30 0.12 0.60 0.04 0.04 0.40 0.10 0.10 0.60 0.40 V.h£au\u) 9113.3289‘4J1344B‘d)~fiabfl.leh.d\3* D)|J 3 A H a AJUJ‘OJla N.)4 2.)) 0110 ”\40 fl MINNJMH‘ULbCer‘h‘H00NJDCAHWJJ>MCHAJDCH&JNNJ&PAKJOh‘O\fiOJMP‘\H’UMbOiMCFUWbCNNJHCh01& I)3~O()D;DJJV‘d\JJI’J‘3i)07mkfl01#lsh>blbblwtd34M‘0\PU‘U\)0"4”4F‘49J\JJ-) 3 I I II I II I II I II I II I II I II I II I II I II I-II ACNOJNPUULbCNNiJCDOtb(ImVOGWHIDQCD\JOCDODVOhmkflUMbJim‘QULbOHG‘dO\MJLUCIUL504N04\JW‘0 “\Hb-MN)‘:MH‘5\JM2NUH‘ILbIAN3V\Fhdfl\)JCA\JM?‘)‘midkr“d31flhdfl\))kfihfi~\)4C)UlA D u«‘\J NFOhJNHUF*POJF‘H(DODV‘Q\JOC)O‘m\fl01m\flJbbwaMU(d04U(NAJNHURJMI‘F‘HIJF‘HWDCDDCDC300*CJU‘ 142 6418.n96 6389.438 6414.011 6984.710 6985.312 6287.081 6988.:96 6989.171 6977.942 6979.379 6981.623 6983.597 6984.050 6985.439 6963.948 6265.999 6267.647 6973.404 6276.653 6959.177 6951.485 6952.919 6951.741 6246.671 6940.485 6941.444 6942.640 6940.331 6928.749 6928.991 6929.991 6929.099 6929.730 6916.A01 6916.981 6917.121 6904.119 6904.484 6904.747 6191.987 6191.771 6993.437 6992.310 6291.140 6290.947 6989.733 6301.883 6106.164 6999.618 6100.145 6296.050 0.001 0.036 0.011 '00002 0.009 0.017 0.018 ”00053 0.018 '00048 “00040 '0.023 0.006 0.008 0.006 0.013 '00098 '00079 '00064 0.030 0.032 0.022 0.013 0.026 0.007 0.037 0.007 ‘00069 0.052 0.031 0.037 0.026 0.026 “00039 0.023 0.050 0.020 0.024 0.054 0.029 0.015 0.062 0.021 0.018 0.027 0.014 '00041 -0.038 0.009 .00075 0.021 '09321 0.60 0.60 0.50 0.80 1.00 1.00 0.50 1.00 0.80 0.90 0.40 0.70 0.50 1.00 0.10 0.30 0.20 0.20 0.30 0.04 0.20 0.70 0.80 0.10 0.01 0.14 0.50 0.03 0.50 0.20 0.30 0.01 0.10 0.04 0.40 0.30 0.02 0.20 0.30 0.09 0.10 0.04 0.60 0.90 0.90 0.00 1.00 0.80 0.30 0.40 0.50 0.00 113143.111. as J‘h.fid)nb\)deJOJQDJOD‘JDVIDJIth‘Jfi \anlfiJI)‘ TA'“\)7\J>\JMP‘F“-3A-3TDD"35‘0". D*TD\I\I\J3‘)J)J':J‘J‘.Ah-CA\)\1\J\J'\)'\)F*‘Q 'IIII-IIIIIIIIIIIIII O‘NC)UHA\DJ>QHU“JUL50£HWJ£>ucai30\OODb\OOJVCD\JO(DU1&‘Q\JO\fiJHWCDULA bxfll’l\_3.fi"3'xnb\) ~JW’44AR)‘\3 DSUBNU (Nb bflnl'-\)"II(AT-*;-‘V_‘I.hffi'\) 143 6114.440 6994.900 6991.422 6113.193 6113.484 6315.624 6318.068 6298.890 6424.161 6321.485 6329.919 6325.947 6337.964 6336.078 6335.780 6345.801 6144.493 6343.165 6441.460 6352.770 6351.404 6850.110 6359.442 6158.304 6922.488 6188.981 6156.054 6333.734 6378.562 6403.901 6428.940 6357.896 6166.163 6375.968 6411.124 6415.773 6410.850 6431.484 6459.954 6453.442 6454.929 0.011 0.010 0.029 ”00042 '00044 “00037 0.003 .00099 “00002 .0000? 0.013 0.011 0.052 0.015 0.083 0.044 0.038 '00018 "09000 0.057 0.038 .0.002 0.051 0.029 '00039 “00016 0.018 0.028 0.036 '00077 0.042 '00021 0.051 “00012 “09035 0.035 “00018 “00001 0.028 0.021 ‘00046 0.01 0.60 1.00 1.00 0.80 1.00 0.10 0.30 0.10 0.30 1.00 0.04 0.40 0.40 0.02 0.70 0.70 0.10 0.04 0.15 0.20 0.01 0.20 0.20 0.20 0.20 0.04 0.90 0.20 0.04 0.20 0.01 0.20 1.00 0.130 1.00 0.40 0.04 0.50 0.40 0.30 APPENDIX X FREQUENCY ASSIGNMENTS FOR THE (110) AND (011) BANDS OF H2328 144 145 ASSIGNED ruenuENcICS FOR THE (110) BAND A» MiApA 0“) ,3 WPPEW GQUUND ORSFRVED DIFFCRENCE dEYGHT 9 0 0 - 1 1 1 3764.081 -0.006 4.81 4 0 1 - 2 1 2 3754.565 -0.007 6.25 1 1 1 a 2 0 2 3756.294 '0.015 0.00 9 1 9 e 3 0 3 3745.783 -0.001 0.69 9 0 2 - 3 1 3 3745.435 ~0.001 4.56 3 1 3 - 4 0 4 3735.661 0.006 0.02 1 0 3 . 4 1 4 3735.418 0.013 1.25 4 1 4 P 5 0 5 3725.406 0.014 0.64 5 0 5 9 6 1 6 3714.025 0.035 0.00 4 1 6 - 7 0 7 3704.174 0.030 0.00 7 0 7 - 8 1 8 3693.172 0.021 0.02 9 1 8 - 9 0 9 3681.029 0.013 5.50 9 0 9 - 10 1 10 3670.423 -0.020 0.00 7 1 10'- 11 0 11 3658.701 '0.032 0.81 1 0 11 v 12 1 12 3646.728 -0.055 1.52 1 1 0 9 2 2 1 3743.462 -0.006 1.25 9 2 1 - 3 1 2 3739.983 0.007 1.25 9 1 1 - 3 2 2 3734.780 '0.019 0.00 3 2 2 a 4 1 3 3727.947 -0.008 0.00 4 1 9 . 4 2 3 3724.924 ~0.005 7.00 4 2 3 - 5 1 4 3717.595 0.010 0.11 4 1 3 - 5 2 4 3717.744 0.004 0.83 5 2 4 - 6 1 5 3707.519 0.012 0.09 5 1 4 - 6 2 5 3707.465 0.016 0.25 4 2 5 - 7 1 6 3697.303 0.018 6.25 7 1 6 - 8 2 7 3686.860 0.028 0.64 8 2 7 - 9 1 8 3676.167 0.021 0.00 9 1 9 P 10 ? 9 3666.945 0.021 5906 2 9 - 11 1 10 3654.082 0.009 0.77 1 1n - 12 2 11 3642.683 -0.012 0.05 2 11 - 13 1 12 3631.066 '0.024 0.52 9 2 0 - 3 3 1 3723.083 -0.002 0.87 3 3 1 - 4 2 2 3725.567 0.021 0.02 1 2 1 . 4 3 2 3713OQ95 .04014 5050 4 3 2 9 5 2 3 3711.334 0.007 1.06 4 2 2 - 5 3 3 3706.371 -0.012 0.34 5 3 3 o 6 2 4 3700.081 0.003 0.55 5 2 3 o 6 3 4 3698.720 v0.003 0.64 6 3 4 - 7 2 5 3689.075 0.014 0.20 4 2 4 - 7 3 5 3689.688 0.015 0.09 7 2 5 - 8 3 6 3679.935 0.024 0.64 8 3 6 n 9 2 7 3669.824 0.021 0.19 G 2 7 ' 10 3 8 35590442 00028 0.58 0 3 8 - 11 2 9 3648.422 0.023 0.03 HLJ B4)fiQD.NAOU‘VJ\JfiJ¥D\J\J))fl£b-‘JODDQ'VIJ‘J‘fiJ'IhJI—‘ rim-AAJHFOHIMb‘\)39¢ca#«3hb364::Hcar-orAcch>u1mkflUIbkna-mknbu§OMACAOJACdasu.boabCA01M (DD‘QJ\flISbCADIU\J‘*‘\)A:J5&A3HVNJflr‘:Im\J)\flU!bleCdevh‘D O 639.4 \)A 3 g 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O‘WKFU1W\HU1AJ>A-#13asbCNOJGCHOJWCAOJMCNNHVAJNTORJN 159 3899.981 3015.425 3921.719 3939.785 3944.905 3964.922 3972.457 3889.012 3893.188 3908.113 3914.804 3924.819 3936.457 3943.484 3959.459 3963.420 3986.336 3920.143 3820.982 3938.854 3941.167 3961.713 3982.888 3989.128 3949.311 3949.474 3968.721 3969.988 3987.455 3989.436 3977.870 ”09000 ”00001 0.005 0.002 0.014 0.034 0.039 90.015 0.011 .00078 0.004 '00008 0.007 '09014 0.014 “00005 0.012 "09002 ’00008 '00013 “00004 '00013 ‘00007 0.002 0.005 0.004 0.003 -0.010 0.013 '000?1 0.002 0.06 0.62 0.00 0.08 0.14 0.00 1.27 0.58 0.06 0.03 1.56 2.31 0.16 0.14 1.33 2.31 0.14 0.00 0.00 0.00 0.05 1.33 0.14 0.02 0.12 0.06 0.03 0.14 0.37 0.02 0.03 APPENDIX XI LISTING OF PROGRAM CORIKORR 160 161 PROGRAM CORIKORR DIMENSION EPV1(21,21),EPV2(21.21),0PV1(20,20),0Pv2(20,20), 1 4(2).8(2).CS¢2).t(62.3).TAO12).T4T<2).T4512).TAN<2>.CN(2).HK(2) COMPLEX EPV1.EPV2.UPV1.0PV2 COMMON IOPT.RHO.IEH C ARITHMETIC STAYEMCNT FUNCTIONS 0 r1~2.)>* 1 (8-A+.25~V*(J*‘(K*1.)t(K*2.))t 1(Jo(J¢1.>-(K~2.>~)«v F10(J.K)=,258H.SORIF((Jt(J¢1.)-K*(K¢1))9(J*(J¢1.)-(K+1.)‘(K¢2,))) C INPUT INSTRUCTIONS FOR PcRTuRBATION TERMS,., G: -GZ AND 0: -GXY READ 12.(4111.8(1):CS(I).TAO(I).TAT(I).TAS(I).TAN(I).EN(I).HK(I). 1 181.2).G.A.AEO.BEQ.JRUN 12 FORMAT(6F15.7/4F10.7/F10.4.F15.12/3F15.7/4F10,7/F10.4,F19,12/ 1 2F10.4/2=15.7.48X.12) C ARITHMETIC STATEMtNTS RHO ‘ 1,0E‘06 IER=0 IOPT=0 CEO=18E0*BEO)/(AEQ*BEQ) RS:BEO¢¢2/(AEQ+BEO’**2 SS:AEQ**2/¢AEQ¢8EQ>**2 R1:RS*(TAO(1)¢TAS(1))+SS*(TAT<1)*YAS<1)) RZ-RS.(TAO(2).YAS(£)>4SS*(TAT(2)*TAS<2)) SlaTAS(1)9(RS~SS)*TAT(l)*SS-TAO(1)*RS 52aTAS<2)t¢RS-SS)¢IAT(2)*SS-TAO<2)*RS 7182.8RS*SS*YAS(1)*R84828TA0(1)+SS*9247AT(1> T2:2.*RS*SS*TAR(2)*RSt*2*TAO(2)6$S*tZoTAT12) V1ITAT(1)-TAO(1) V2:TAT(2)-TAO(2) x1-TAT(1).140(1) X28TAT<2>47A0(2) Y1: TAS(1)¢2.¢TAN11) 12. 14$<2>42.*14N12) U18(.50X10Y1)/32. 02.1.5.x2-12)/32. 162 PRINT 2000.A(1).B(1).C5(1).TAO!1).TAT(1).T45(1).TAN(1):EN(1).HK(1) 2000 FORMAT(//*(110) BAND PARAMETERS*// 1 .41v1)=~r15.7. u(v1)=wF15.7. cs(v1)=.F15,7/ 2*T4AAA:*F10.7* TUBBB=4F10,74 TAABBBtF10.79 TABAB=.F10,7/ 3 0(110) BAND CENTtR :tF10,4/4HK1:*F15.12//) PRINT 2001.A<2>.B.HK(?) 2001 FORMAT(//*(0111 BAND PARAMETERS'II 1 94(V2)=*F15.7¢ B(V2):*F15,7. CS FPV2IK.K*1)=F6IA(2).B(Z).J.KK.V2.82) EPVl (KM. (MAX . K) = F6 ( 4(2). 812). J. KK.v2.82) EPVZIKM.KMAX*K)=F6(A(1),B(1).J.KK,V1151) GO TO 2 251 CONTINUE EPv1 I K. (MAX 4 K I = (0,;-1. )trlo (J,KK) EPV2.tN12).TAN(2>.82.Y2.T2. 1 x2.HK(2).J.KK) -F3(J)*U2 IFIKK.NE.2) EPv1IKM.KM)ar1(AI2).B(2).c5(2).tNI2).TAN(2).82.Y2.T2. 1 x2.HK(2).J.KK) IF(KK.E0.2)EPV2.811I.Cs(1>.EN(1).TAN<1).81.Y1.11. 1 X1.HK(1).J.KK) GD 10 1 252 CONTINUE EPVl (K. K1) = (0.:1.)*G*KK EPV21K,KM)=-EPv1(K.KM) 1 CONTINUE c MATRIX FOR ov1PLUS+ov2MINus AND ov1MI~us*0V2PLUS le=.0Pvg(KAI,KAI.1):F6:F6IA(1).811).J.40.v1.S1)-F7IJ>.U1 IFIKo.NE.1)0PV9(KAT.KAI.1I=F6(AI1>.BI1I.J.KU.VI.51> GO TO 102 261 CONTINUE OPV1IK.K8T*1)=(0..’1-)‘F101J.KO) opv1(K.1.KAT)=.OPV1IK.KAT+1I OPVZIK.KAT¢1I=0PV1(K.KAT+1I OPVZIK.1.K4TI=.opvl(K,KAT.1I 102 IF (NTMS.E3.2) GO I0 262 0IFIKO.EQ.1)OPV1(K.K)=F1(8(1).811).CSI1I.EN(1).TANII).R1.Y1.T1. 1 X1.HK(1).J.KO)*FZ(A(1).B(1).J.V1181) OIF -F2(AI1).B(1).J.V1.51) 0!FIK0.NE.1)OPV2(KAT.K4T)=F1(A(1).B¢1).CSI1).ENI1).TAN(1)0R1.Y1. 1 T1.X1.HK(1).J.KO) GO TO 101 262 CONTINUE IF(KO.F091)OPV1(K'KAT):(0..1.).(G.K0‘.25'H‘J.(Jil)) IFIK0.NE.1)OPV1(K.64T)=(0..1.)«(GtKOI IFIKO.E0.1)0PV2(K.84T)=(0..-1.)t(G*K0¢.29tktJt(J*1)) IFIKO,NE.1)OPV2(K.‘AII=IU..-1.1*(G*K0) 101 CONTINUE C FORMING HERMITIAN CONJUGATtS OF EPV1.EPV2.0PV1.OPV2 Do 9 N=1.JEMD INsN+1 DO 9 M:IN,JEMD EPVZIM.N)=30NJGIEPV2(N.M)I 9 FPVIIM.NI=30NJG(EPV1(N.M)I DO 109 N=1.IMD IN3N+1 00 109 M31“. IND 09v2(M.N):30NJG(OPVZ(N.M)) 109 OPV1IM.N)=30NJG(OPV1(N.M)I 0 ALL MATRIX ELEMENTS HAVE BEEN FORMED 47 7815 POINT IF INTMS.E3.1) 2701250 165 270 wRITE TAPE 51.EPV1:EPV2.0PV1.0PV2 280 CONTINUE EIGENVALUES ARE FOUND IN THIS SECTION CALL JHERMX CALL JHERMX (Epv2.d1.JEMD) CALL JHERMXIOPV1.ZU.IMD) CALL JHERMXIOPv2.20.lMU) STORING EIGENVALUES IN E(NCOL.NTMSI NCOL=0 D0 301 IainJEMD NCOL=NCOL+1 301 E(NCOL,NTMS)=EPV1(1.I) DO 502 IglaJFMD NCOLaNCOL+1 302 E(NCOL.NTNSI=EPV2I1.I> DO 303 [=1IIMD NCOL:NCOL.1 303 E(NCOL.NTMS)=OPV1(1.I) DO 304 I=1.IMD NCOL=NCOL41 304 E(NcoL.NTMSI=Opv2(I.II GO TO <200.311).NTMS 311 CONTINUE PRINTING EIGENVALUES WITH J,K.,K. LABELS PRINT 312.J.J 3120FORMAT(/lt(110) STATE FOR J: «I38 V1 OH 5 RANU*23X 1*(011) STATE FOR J= .13. V2 OR A BAND.// 22(. J K- <+«4waNPE8TURaED COR.pERTURB:D CUR-UNP811XI) IMINSIJEMD DO 313 I=1.KMAX KPLUS=?*(I-1) TAUPaJ-28K9LUS KMINUS = TAUP4KPLU5 E(I.3)=E(I.2)~E(I.l) IM1N5=IMIN5+1 E(IMIN5.3)=E(IMIN5:2)-E(lMIN5.1) PRINT 314.J.KMINUS.KPLU8.(EII.K).K=1:3I.J.KM1NUSoKPLUS. 1(EIIMIN50K)4K=113) 314 FORMAT(2(313.2F15.4.F8.3.10X)) 313 CONTINUE IMIN4=KMAX IMIN1=JEMD.KMAx.1 IMAX1=28JE40 DO 320 I=IWIN1.IMAXI KPLUS=28(IuJEMD-KMAXI TAUP:J-2*K2LUS¢1 KMINUS = TAUP¢KPLU5 E(I.3)=E(I.2>-E(l.l) IMIN4=IMIN4¢1 E(IMIN4.3)=E(IMIN4.2)-E(IMIN4.1) PRINT 314.J.KMINUS:KPLUS. COMMON IOPT;RH011ER DIMENSION A A(I.KItSN1 + A(laL)*CSNT A(I.K) TEMP IFIIOPT .EJ. OIGO TO 8 TEMP = 5(InK)*CSNT - SIIpL)*CONJG(SNT) SIIaL) = SIIoKIOSNT + SII,L)*CSNT SII.K) = TEMP CONTINUE DO 9 I=1ON A(K.I) I CDNJGI A(I,K) ) A‘Lpl’ = CDNJGI AtlaL) I CONTINUE AIKgK) = SNT*CONJG(SNT>OV3 + CSNthsNT*V1'SNTtCSVT*CONJG(V2) 1 ICSNTOCONJGISNT)OV2 A(LoL) a CSNTtCSNT*V5 + CONJGISNT)tsNT'v1 o SNTtCSNTt 2 CONJGIVQ) * CONJG‘SNT)*CSNT*V2 10 1 A(K,L) = CST‘JT'CSNT'VZ -CSNT*SNT*V3 A(LaK) = CONJGI A(K,L) CONTINUE IOC = 10C 0 1 IFIIOC ,GE. 100)GO T0 IF(IND ,LE. 0)GO TU 11 IND = 0 GO TO 4 IFITHR-FTHR)12,12.6 RETURN IER = 1 RETURN END 168 — SNTtSNTRCONJGIVZI + USNT*SNT*V1 T 21 APPENDIX XII GROUND STATE COMBINATION DIFFERENCES OF H2328 169 qquAn STATE COMRIKATION DIFFEQENCES h gcsh.>:s# AIxA-¢:LA-A£bb >€>AIAJsAWAJIA AJdd dJaA Ala» u;1x.daluilu axle I I K + OJbJSDPJH‘HFUKJM’OAJVCAOJMCAOJbJiéubthF4h9uINC>DhiF3MTUNJ”FDOJGCNNJHrJNDM 4DDAAAJW'dZN‘U'UuUDUJ-‘JA-‘JDDD’JJV‘J’JIHAVVU4-34433 JV-‘H'D II I‘IJ I II I II III! l II I II I II III. I I1 I II I I] I-II I II I I I L UIUCNOGOHUIUhOOIGHURD04h¢d0dh43h30L5'UJLNHVO‘F‘QHUf*P‘KHUFOP‘OHUIUF‘QHDF‘HW‘TU K- K4 GIORJNPURJHEJF‘QHUCJNLbCDNCNF‘HI‘OIOFUKIHcDF‘QHJP‘HIVR)OIJOJHFJC>M3\OC>O\HPU\)M‘UNJGC§04UCAOJ#«deW widde‘UNJU“H‘3L)Qflwidblwld\lU20NJJF‘J1J4flh-A.hfllw-*H‘H~3233\m4nd1h15bhbilbidhiu‘ONJU I'll.Ilvllilfi'lllllil’!I‘iIII‘IIJIllil‘llill‘llllillil.IJ‘IIII O\fiUH)OKW\NOVN\nUHJJIb1>&liu1#ChULbJLm\flJbbkflOW>J>OMb\nULthbkflUHflJbb\flUnA\flU0mkflO\b UnmidbimCddiblbbb‘CI#E‘Ol5N)MVONJWF‘QH‘DH‘CJMV‘PWU£551}fithh‘fiJDNDhHUF‘PZNQIWCdR353dm UF‘OLbRHUC)Ufldth\flOCNHWJUL§AJb“)UF‘UHHOdeLOWflOIUfaHchadeW€ARHUJSARJNP‘OHNthFU 172 713.922 62.737 104.325 116.026 112.122 60.418 151.185 61.405 114.927 210.072 154.684 53.806 38.960 115.955 133.182 113.765 220.051 145.740 139.939 220.017 57.907 910.050 74.927 97.194 114.656 210.749 133.459 37.905 115.952 909.924 943.734 124.147 50.512 905.096 165.170 77.762 163.708 173.962 908.373 178.629 193.027 17.726 89.032 39.424 167.993 909.083 942.017 117.974 180.413 948.974 197.972 136.952 0.002 '04003 ”00001 0.003 900003 0.005 0.006 0.003 ~0.003 0.003 0.003 '00005 0.003 '00002 '00002 -0.005 0.020 0.012 0.015 0.016 0.011 0.000 0.004 .00005 0.001 0.003 0.006 0.003 0.003 900016 '00016 0.002 -0.003 0.000 0.000 “00004 0.003 '00004 ”00003 “00000 0.004 -0.006 0.005 0.003 0.010 0.008 0.006 0.006 0.007 90.004 0.006 ’00001 1.56 3.16 3.98 9.41 1.91 0.27 2.02 2.89 2.02 0.28 0.86 0.14 2.21 1.93 1.18 2.47 0.27 0.21 3.04 0.39 0.92 3.26 0.12 0.08 0.20 0.06 0.16 0.59 0.13 0.10 0.04 1.38 0.70 0.17 0.15 0.20 1.64 0.26 0.05 2.10 0.12 0.57 6.39 2.64 2.11 4.73 0.90 1.37 3.13 0.32 0.10 1.08 D09109DDDQQQQQVIQQQMVVVJQQQQVQHQQQVQ‘J‘JQQVVVQQVA‘JVQ‘JV444% \flbh55hQ’AiAKfl‘JWTAKAVUMDV'J‘J‘UIAhbbbéflQU’U‘4V‘J‘JJ-fl‘mmmhhbbidbididfl'fl III‘l-IIII'OIII’IIUOIIIIII'IJIII-‘IIUI‘I‘IIIIIIIIIIIIII‘II (DNOVVO‘VO‘VWNU‘U‘O‘VWO‘VOOO‘OOO‘VWOOO‘O‘VOO‘U‘U‘O‘OU‘OU‘O‘WWUTVO‘UIU‘OWVO‘ \lbvbANNF-‘(AU‘GVkflWU‘bNONU‘AU‘MO‘NO‘$5NU1HANHHU1£>O~«AUGWU‘bNO‘éOflNGI-‘U‘H 173 101.479 71.758 909.901 127.868 187.551 943.992 132.734 95.528 194.139 247.456 249.138 133.880 172.138 56.109 152.131 134.408 37.209 132.420 249.322 110.079 148.916 47.979 70.459 205.427 83.240 133.752 248.407 96.312 69.996 229.030 59.007 133.931 37.437 17.373 51.264 107.600 253.239 65.674 211.777 320.899 249.183 116.512 171.135 77.948 987.n91 119.797 222.444 104.961 152.447 985.681 149.885 115.388 “0.001 0.007 0.011 0.006 '00012 0.006 0.001 0.002 0.001 “09007 0.003 0.003 0.004 '09002 '00006 0.003 0.002 ”00012 0.011 '09008 0.007 0.006 I"'00007 ~0.001 0.003 0.000 0.001 “09003 0.003 '00002 0.000 0.001 0.001 “00002 0.037 '00006 0.003 0.001 0.001 0.013 0.004 0.010 0.003 0.001 0.004 '00006 0.011 “00006 0.007 0.008 0.005 0.006 1.40 1.19 1.43 3.25 0.31 1.84 5.35 1.44 2.84 0.60 0.34 3.39 3.58 1.98 2.82 1.31 2.55 1.60 0.19 0.79 0.62 3.62 0.19 0.10 1.07 1.65 0.15 0.05 3.48 0.10 0.31 0.7 0.04 2.97 0.54 0.10 0.06 0.15 0.04 0.03 0.12 1.66 1.79 0.04 1.51 1.33 0.47 0.37 2.71 0.44 1.78 0.76 JOOGQOOGDDDDDDDDBDDDDDDDfiDDDDJODDDXJDDDDDDDDDDDDDDJODDD NJMCAOJMCN‘bbsnOKQLbJLbkfiU1$C>O\V‘J\JVFONJMCN04WCN&I#kn013()9\0‘d3>fir*\JMfiobduCNOLblbb V‘d)‘)l>3‘fl\fltshLflgflhrbDWJ:A\IU’UNI4*‘O\)C)3\mxflJLh155£N03H‘0\JV1333fi‘q\lfl.¥)\)ibJ1wufl I0'01.llll‘l-Ili-III'IIIIIIIJIOI‘I‘IIllijllllfli‘ilIIIIQ-IIII| \JO‘QIJOCDGfflcn\J\H)\JO‘QODO€h\JV‘QO‘V‘Q‘JO<’OVVC>\JQ‘Q\J\HD\JGC%OMO‘4\H)\Jm‘q\40‘Q\JO MFUOJ”\flOJOwhOi#kfiAJQMbOJU£)flbG\fl\Jm‘4\)AfiOHW‘OiQMUhJHRJNJ#ubILUH)+5PWUKJD+*H‘U{AfiDb-h \fi\Lb00bCihHHOhGCdUHUhJWKflHWUhJNC)HW‘OMbJ>O\NUHNO\OC>OJmkflbnthO(Iokmkn\HDOMAJ>WCNOJ 174 908.512 80.634 152.199 190.148 57.415 152.945 135.889 153.521 170.996 37.017 17.058 1516449 799.932 292.582 46.430 196.524 78.574 230.981 142.439 248.372 93.267 246.710 711.177 152.170 286.105 7878351 190.370 56.422 152.934 124.780 289.787 123.434 174.704 130.848 279.762 229.882 67.706 919.885 233.130 84.220 286.282 74.332 ' 3590225 95.425 250.625 91.402 171.198 75.761 134.466 228.611 171.919 208.944 '00003 ”00019 “00005 “00008 ’00008 “00004 0.006 "00003 “00003 ‘00005 '00008 '00023 “00009 0.027 0.006 0.019 0.003 0.006 0.031 0.005 0.011 0.014 ”00003 .00003 0.002 '00004 '00006 ’00002 '00006 '000?7 0.014 ‘000?8 0.015 0.016 0.021 '00008 0.024 0.030 0.013 0.013 0.001 0.005 0.002 0.008 0.017 0.018 0.010 '06010 0.002 '00000 '00005 “00002 1.28 0.68 1.42 1.14 1.54 1.66 0.30 2.22 7.48 2.37 4.18 0.54 0.12 0.31 0.13 0.17 0.70 0.97 0.24 0.25 0.09 0.03 0.40 0.07 0.08 0.08 0.04 0.11 0.05 0.11 0.06 0.26 0.46 0.15 0.07 0.68 0.03 0.03 0.16 0.14 0.27 0.12 1.55 0.75 3.92 1.76 0.58 0.69 0.14 1.45 3.20 0.78 cJOOOOO-DJJJOOOQDUOJO‘J00000030 .s ‘OO‘D‘J=)&DO'DDLOOD‘IJDVHVV‘JTJT‘J‘IJ‘hbhbbbhfi‘fl:}-))3‘J~‘J1‘J1\HOO13.131)“ z... LA «.4 »;A HLA 313454;) 4. h.0c:ODONOGTV‘QGJOC>040CDGJV‘QODOC501V\flb#-A#‘U J htoc:c:o~0<>0c3c.oxooaocao:ocn<>ownoao~o<>N~qaawcnxJo~qa>m~40:V~q\4m(nxaocnaam‘q\4mcnaz -D\HPWHNJ38‘A)AaifiidJM3#‘flWVNJ#Cd&L&AJ3C3\3&\DU0V\flb-hls9i}\JVthNUJIhflv£bm0052#+‘&la Clil-lIIII-lIlillllilllilIIIIIO-IIIIIIIIIIIIIII-illillllll ‘4‘; *«44 A-t‘ A.JJ J40) 1:); 3 1; 3.)) 4 8.. J ‘5‘ -‘ 0 g 175 56.533 36.723 172.409 189.663 170.771 169.977 113.446 227.862 171.241 324.179 324.142 261.795 137.014 956.932 126.075 172.700 322.600 94.726 270.562 95.957 245.095 103.058 322.943 322.383 155.165 76.613 190.n53 746.590 56.359 190.838 208.407 36.487 174.380 191.652 250.559 189.298 16.461 908.791 246.183 727.118 209.496 36.280 ?10¢657 206.416 193.526 "0.009 '00001 0.002 '00010 '00018 ”00010 0.015 0.021 0.032 '00000 “00006 0.003 0.001 0.004 “00017 0.009 0.034 0.009 0.009 0.029 0.017 0.005 -0.011 0.012 '00016 0.014 0.024 0.004 ’09022 0.004 ”00001 '00011 “00024 '00010 I"00008 ‘00004 '00005 '00001 '00000 90.025 0.028 “00006 0.004 '00072 '00021 909007 0.005 ‘09017 0.003 .00002 '00015 “00016 101,6 2.44 1.97 4.58 3.03 1.46 0.11 0.19 0.12 0.21 0.21 0.07 0.02 0.14 0.03 0.06 0.05 0.11 0.09 0.11 0.02 0.06 0.05 0.10 0.05 0.03 0.68 0.22 0.39 0.65 2.38 1.13 0.22 1.06 4.90 2.24 5.08 0.10 1.28 0.06 0.23 0.87 0.09 0.31 0.36 4.56 2.44 0.68 0.25 1.23 0.04 0.04 11 11 17 13 1o 19 1') 1a 19 1“) 11 11 {AHVMMMbA—‘WCAD 1‘1 10 19 11 11 1n 11 1? 1n v.4s...‘ 0000C) 11 11 11 11 11 11 tdgcg-AMAJMbI-‘CA‘JN3 a OOHDOVO‘OVOCDO‘VO “LL—b 176 16.174 965.152 764.775 226.862 283.694 727.461 745.794 36.088 928.493 929.620 964.440 309.n24 0.003 -0.019 .00021 0.032 ~0.015 0.008 ”00001 0.006 -0.026 '00004 0.023 '000?2 0.27 0.03 0.20 1.88 0.22 0.14 2.16 0.10 0.06 0.26 0.54 0.04