W ANALYSIS OF ASYMMETRIC ROTOR -M0 LECULAR ”94 PUTER PECTRA AND DATA ACQUTSITION A 'S T TMENT BY A MI'NICOM TR Thesis for the Degree of :Ph. D‘ HIGAN STATE UNIVERSITY MIC PAUL DANIEL WILLSON 19:73 4.4... it: v1.1, ’vn 41;..o...‘.t< . '11,... a. I... v...“ t. ' gnu ,— X‘TH‘ \T. a I: . IT , . . 1 AT‘ T l... T, .143 . v |\..‘W. ‘N..¥J§. n“ . \L.\Q.A.?4.!1 .— x“ “thzu. .'\ 11:9 B 115 ABSTRACT 'ANALYSIS OF ASYMMETRIC ROTOR MOLECULAR SPECTRA AND DATA ACQUISITION AND TREATMENT BY A MINICOMPUTER BY Paul Daniel Willson The various Hamiltonians now available for calcu- lating the vibration—rotation energy levels of asym- metric rotor molecules are compared and their appli— cation is outlined. The means of forming "reduced" Hamiltonians and the restrictions on their use are briefly described. A means of fitting several mole- cular isotopes simultaneously is described. Computer programs were written to apply the results of the above work to infrared spectra. Ground state combination differences from the Zvl band of HD13OTe, which is a prolate XYZ molecule were fit to the planar Hamiltonian and to the KFC Hamiltonian to second-order with T6 restrained to be zero. The Watson Hamiltonian to second-order was used to fit upper and ground states band of the molecular species HDl3oTe, HDlZBTe, l HD126Te, HDlste, and HD124Te simultaneously. The planar of the 2v Paul D. Willson Hamiltonian was used to fit the upper states of the Coriolis coupled bands v1 and v3 of H28. The rotational and second-order constants and the perturbation constants required for the above fits are listed. Also described in this work are methods and prin- ciples involved in the acquisition of spectral data, baseline adjusting, smoothing, and deconvolution by means of a minicomputer. A means of deconvolUtion based on successive approximations has been further developed and programmed on a minicomputer. The need for variable relaxation, normalization, and a point simultaneous method for the deconvolution process is explained. These methods of treating data were applied to several HDS and D S Vibration-rotation bands. The results for the 2 Q—branch of the v +v l 3 band of HDS are shown. ANALYSIS OF ASYMMETRIC ROTOR MOLECULAR SPECTRA AND DATA ACQUISITION AND TREATMENT BY A MINICOMPUTER BY Paul Daniel Willson A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1973 TO MY DAD ACKNOWLEDGMENTS I am very grateful to Dr. T. H. Edwards for encour- aging and directing this research. I wish to thank him for assigning the transitions of the spectra used in this work. He has not only helped with the research, but has personally helped my family. I also wish to thank Dr. S. C. Hurlock for the many contributions he has made towards this work. I wish to thank Dr. C. D. Hause for his help and advice in the operation of the spectrometer and Dr. P. M. Parker for his helpful advice. I am grateful to Jim Hanratty who contributed much of the computer programming work and was helpful as a consul- tant on that work. Many other graduate students and under- graduates have contributed to this work. Dr.D. F. Eggers from the Chemistry Department at the University of Washington furnished the HDS—D28 sample used in this work and is jointly analyzing that spectra with us. The analysis of the spectra was done on the CDC 3600 computer at the M.S.U. Computer Laboratory. The National Science Foundation has supported this work. I am very grateful to my wife and children who have daily loved and cared for me during this time. I am also grateful for the many people who have been praying for me and the completion of this work. iii LIST OF TABLES . . . . . . . . . . . LIST OF FIGURES . . . . . . . . . . INTRODUCTION . . . . . . . . . . . CHAPTER I. THE VIBRATION-ROTATION HAMILTONIAN FOR AN ASYMMETRIC MOLECULE . . . . . Second-Order Planar Hamiltonian . . Fourth-Order Hamiltonian . . . . Coriolis Coupling . . . . . . II. APPLYING THE HAMILTONIAN . . . . . The Wang Transformation . . . . Identification of Levels . . . . Evaluation of Coefficients . . . Isotopic Substitution . . . . . Computer Programs . . . . . . III. REDUCTION OF SPECTRAL DATA . . . . TABLE OF CONTENTS Acquisition of Data . . . . Noise or Error in the Data . Smoothing of Data . . . . Base Line Adjusting . . . . Deconvolution . . . . . . iv Page vi vii I4 16 18 22 25 27 28 32 34 35 38 39 41 Measuring Line Centers . . . . . . IV. DECONVOLUTION OF SPECTRA . . . . . . Van Cittert Method . . . . . . . . Maximum Resolution Enhancement . . . . Point Simultaneous vs. Point Successive Deconvolution . . . . . Quadratic Relaxation . . . . . . . Normalization . . . . . . . . . . Smoothing During the Time of Deconvolution Conclusion . . . . . ,. . . . . . V. DATA ANALYSIS . . . . . . . . . The (2,0,0) Band of HDTe . . . . . . The (1,0,0) and (0,0,1) Bands of H28 . . VI. CONCLUSION . . . . . . . . . . REFERENCES . . . . . . . . . . . . APPENDICES I. NONVANISHING ANGULAR MOMENTUM MATRIX ELEMENTS II. JACOBI METHOD OF DIAGONALIZATION . . . . III. LISTING OF PROGRAM ISPECFIT . . . . . . IV. POLYNOMIAL FITTING BY LEAST SQUARES . . . V. FREQUENCY ASSIGNMENTS FOR THE (2,0,0) BAND OF HDTe . . . . . . . . VI. FREQUENCY ASSIGNMENTS FOR THE (1,0,0) AND (0,0,1) BANDS OF H28 . . . . 42 45 46 51 53 55 58 59 60 62 64 68 72 75 78 82 90 123 128 147 LIST OF TABLES Table Page I. Molecular Axes Identification . . . . . 4 II. Three Allowed Ways of Setting a Ti Equal to Zero . . . . . . . . . . 10 III. The 20 Allowed Ways of Setting Three Coefficients $1 Equal to Zero . . . . . 10 IV. Classification of the Submatrices E+, E-, 0+, and 0' . . . . . . . . . 23 V. Experimental Conditions . . . . . . . 65 VI. Ground State Constants of HD13OTe . . . . 67 VII. Molecular Constants for 201 of HDTe . . . 68 VIII. Molecular Constants for V1 and v3 of H28 . 70 vi Figure II. III. IV. V. VI. VII. VIII. IX. X. XI. XII. XIII. XIV. LIST OF FIGURES Normal Modes of Vibration For An XYX Molecule. Elements of the Submatrix E+ + E— for l 2 the Hamiltonian Hl + H2 + HI' . . . . Elements of the Submatrix 0: + 0; for the Hamiltonian Hl + H2 + HI . . . . Energy Level Diagram . . . . . . . . Smoothing Function For a 13 Point Smooth . Measuring Line Centers by Line Reversal . . 1’ X2, and X5 for = A = exp(-x ) . . . . . . . . The Functions X xi Results of Deconvolution by van Cittert's Me thOd O O O O I O O O I O O O Quadratic and Triangular Relaxation Functions. Maximum Resolution Enhancement for Gaussian Lines . . . . . . . . . Point Successive vs. Point Simultaneous Deconvolution of Triplets . . . . . A Plot of S, W, and [A W for Gaussian Lines. 1 The Q—Branch of the vl+v3 Band of HDS Deconvoluted . . . . . . . and v of H S . . . Survey Spectra of v1 3 2 vii Page 17 20 21 24 38 44 48 48 50 52 55 58 63 71 INTRODUCTION In the last ten years a number of new developments have occurred in the theory for molecular asymmetric—top vibration-rotation Hamiltonians. This work includes a comparison of various Hamiltonians now available, espe- cially in their application to the analysis of the vibration-rotation spectra obtained in our laboratory of nonlinear triatomic molecules. During this same period the field of digital elec- tronics has grown rapidly and is changing the tools available to the experimentalist. With the addition of a minicomputer to our laboratory, many means of data processing formerly prohibited could now be done. Therefore, much of this work has been devoted to the development of ways of digitally processing the raw spectral data, preparing it for measurement,viz. the application of digital methods to baseline adjusting, digital smoothing and deconvolution of high-resolution infrared spectra of molecules. CHAPTER I THE VIBRATION-ROTATION HAMILTONIAN FOR AN ASYMMETRIC MOLECULE In this study, we will be considering asymmetric—top triatomic molecules. In the past ten years many theore- tical advances have been made in the Hamiltonian for such molecules. Let us begin with the general form of the vibration-rotation Hamiltonian as given by Chung and Parker1 in 1963. Second-Order Planar Hamiltonian To second order of approximation their result is H = h + XP2 + YP2 + ZP2 + %T P4 + JET P4 + $3 P4 v x y z xxxx x yyyy y 2222 2 2 2 2 2 2 2 2 2 3w 3v 4ryyzz(Psz + PzPy) + 4TXXZZ(PXPZ + psz) ax (P2P2 + P2P2) + gr (p p + p P )2 xxyy x y y x yzyz y z z y 2 2 IV 1v — ‘szxz(PxPz + Psz) + ‘Txyxy(PxPy + Pny) (I 1) where Px’ Py, and P2 are components of the total angular momentum; hv is essentially a constant for a given vibra- tional state; X, Y, and Z are the inverse moments of inertia, e.g. X = h/8n2cIX, where IX is the principal moment of inertia about the x axis; and the taus are centrifugal distortion constants. The Hamiltonian angular momentum operators will be evaluated in W(J,K) space, where w(J,K) arethe rigid symmetric rotor wave functions,such that < J,K | P2 | J,K > = J(J+1) =K 2 2 2 2 _ . . where P = P + P + P and P P -P P - -iP , With x y z a b b a c a,b,c and x,y,z being cyclic. Reductions of the Hamiltonian due to angular momentum commutation rules are based on the above space. It is convenient to attach to the molecule a rectan~ gular coordinate system along the molecular principal axes, (a,b,c), such that the molecule lies in the ab plane. Then using Ae' B , and C8 as the equilibrium e inverse moments of inertia, it follows that C8 = AeBe/(Ae+Be), which implies that Ce will be less than Ae or Be' The choice of Ae > Be then yields the conventional ordering A > B > C . e e e The above conditions led to the planarity relations as given by Dowling,2 and by Oka and Morino? These are a consequence of the molecule being planar and lying in the ab plane and are independent of the space the Hamil- tonian is evaluated in. The relationscan be expressed by e e 2 _ e e 2 e e 2 (Ice/Ida) Taacc - (Iaa/Iaa) Taaaa + (Ibb/Iaa) Tddbb and Tacac - Tbcbc = 0 (1'2) where 12a, Igb’ and 12c are the equilibrium moments of inertia along the a, b, and c aXES respectively and a may take on the values a,b, and c. There are six possible ways the molecular axes (a,b,c) may be related to the body-fixed axes (x,y,z) which are listed in Table I. The Hamiltonian (I-l) may be evaluated in W(J,K) space using any of the six ways or representations. But for a near symmetric-top molecule, two of them will yield diagonal matrix elements of the Hamiltonian which are close to the eigenvalues of that matrix. Since these two ways will be the most likely to diagonalize with small round-off error, one of them should normally be used. There- fore for a near prolate asymmetric—top we choose the so- L called Ir representation (although I would do as well); Table I. Molecular Axes Identifications Body-Fixed Axes Molecular Axes x b c c a a b Y c b a c b a Z a a b b c c r z r R r l Type I I II II III III and for a near oblate asymmetric-top, we use the IIIr representation as found in Table I. After choosing the representation,one may apply the planarity conditions as well as the commutation relation (Pan + PbPa)2 = 2(P2P: + pgfi ) + 5p2 — 2p2 so that the Hamiltonian (I-l) simplifies to the planar form H = h + AP2 + BP2 + CP2 + T 0 v a b c aaaa aaaa +Tbbbbobbbb +Taabboaabb +Tababoabab (1‘3) where o = %[P4 + r2P4 + r(P2P 2 +p2p 2)] aaaa a c a c c a 4 2 4 2 P2 2 2 = y Obbbb 4[Pb + 5 Pc + s(PbP c + PCP b)] 4 P2 2 2 2 = L Oaabb .[2rsPC + (P2P b + PbPa 2) + s(PaP C + P: i) + r(P2P2 + P2P2)] _ P2 2 2 _ 2 2 _ Oabab - 4:P[2(P b + PbP a) 2P + SPC] (I 4) The constants retain the same a,b subscripts for both the oblate and the prolate case, but one must evaluate the angular momentum operators in the (x,y,z) coordinates using the Ir relations for the prolate case and the IIIr relations for the oblate case as found in Table I. The terms r and s in Eq.(I—4) are defined as C/A: 2 e C/B (DMNTDN Since the equilibrium constants are not generally known, the ground state constants are often used, i.e. , ‘ 2 r B /(AO+ BO) A (0 ll ON ON 2 /(AO+ BO) Moncur4 and Snyder5 used the Hamiltonian as given by Eq.(I-B), but included an empirical fourth-order term, 6 HKPz’ when fitting their H2Te and H28 data respectively. Fourth-Order Hamiltonian Although all of the terms in the planar Hamiltonian are determinable from experimental data, not all of the terms in Eq.(I-l) are determinable. Since Chung and Parker's 1963 paper, many changes in the Hamiltonian have been made. Olson and Allen6 produced an important simpli- fication of the Hamiltonian for the orthorhombic point groups through wise use of angular momentum commutation relations. Chung and Parker extended their previous work by publishing a study7 of the fourth-order centrifugal distortion effect. It was then shown by Kneizys, Freedman, and Clough8 that the Hamiltonian for the orthorhombic point groups, could be given in much simplified form through extensive rearrangement, again based on angular momentum relations. A The form of the resulting Hamiltonian is, for a given vibrational state, a power series in the angular momentum components which needs for its specification, to fourth- order of approximation, three coefficients, X, Y, and Z, of terms of the second-power in the body-fixed angular momentum components (these being the three effectiVe rotational constants); six coefficients Ti of fourth-power angular momentum terms (these being the second-order centrifugal distortion constants Ii,plus fourth-order,pi, corrections to them); and ten coefficients of sixth-power angular momentum terms (these being the fourth-order centri- fugal distortion coefficents @i). Later Watsonll showed that the Kneizys, Freedman, and Clough's Hamiltonian, Ref.(8), Eq.(3) or Eq.(6), could be used for asymmetric- top molecules of any point group whatsoever, and thus its use need not be restricted to molecules of the orthorhombic point group. The Hamiltonian just described, but slightly modified by Yallabandi and Parkerlo, [the operator associated with ¢ 0' which Kneizys et.al. took as (PiPiP: + P2P2P2) 1 z y x 2 2 2 2 2 2 . was replaced by (PXPZPy + PszPx)]’ which we shall refer to as the KFC Hamiltonian, takes the form H = hv + H + H + H6 (I—5) 2 4 with H = XP2 + YP2 + ZP2 2 x y z _ 4 4 4 2 2 2 P2 H4 — TlPx + szy + T3Pz + T4(Psz + PZP y) + T (P2P2 + P2P2) + T (P2P2 + P2P2) 5 z x x z 6 x y y x _ 6 6 6 2 4 4 2 H6 — lPx + 2Py + 3Pz + 4(PXPy + PyPX) + 65 (P2P 4 + P4 P2) + P6 (P2P 4 + P: P2) Y X ”X Y Y Z Y + Q7 (P2P 4 + P4 P2) + ¢8 (P2P 4 + P: P2) z y Py z z x z + ¢9 (P2P 4 + P4P i) + 610(P2P2P2 + P2P 2P i) x 2 P2P x z y y zP This Hamiltonian, as stated by Kneizys et al., although developed in the IIIr representation, can be generalized to any representation by the appropriate identification of the rotational constants with the x,y,z axes(see Table I). The KFC Hamiltonian still contains experimentally indeterminable coefficients. But Watsonll'12 succeeded in showing how one can obtain from the KFC Hamiltonian a "reduced" Hamiltonian devoid of redundant or experimentally indeterminable coefficients. The reduction of the Hamil- tonian can be carried out in an infinite number of ways, but not entirely without restrictive conditions. Watson's theory requires one constraint to be imposed on the coef- ficients Ti’ thereby reducing them to five independent ones, and three constraints to be imposed on the coefficients @i, therby reducing them to seven independent ones. Mathematically the reductions are achieved by applying a unitary transformation of the form U = exp(iSB)exp(iSS) with S3 = Slll(PxPzPy + PszPx) and _ 3 3 3 3 S5 - 5311(PXPZPy + PszPx) + 5131(PXPZPy + PszPx) 3 3 + 5113(PXPZPY + PszPx) to the Hamiltonian given by Eq.(I-S). The constants S311, 5131, and 5113 are chosen such that the appropriate ¢ terms are zero; and S111 is chosen such that the appro— priate T term is zero. From the unitary transformation a new set of constants, i.e. experimentally determinable coefficients, which have the same angular momentum dependence as the old set, but different theoretical interpretations in terms of mole- cular parameters, is obtained. The theoretical interpre- tations depend on the exact reduction used and will vary considerably. Henceforth, we will denote those coefficients following the transformation, which are experimentally determinable, by tildes, e.g. T1. The details of Watson's theory, as Ford, Yallabandi, 13 and Parker show, allows one to set either T4, T5, or T 6 equal to zero with the restrictive conditions stated in 10 Table II. Three Allowed Ways of Setting a Ti Equal to Zero Case Constraint Ti=0, Condition where i = 4 f g 2 5 z ¢ 2 6 x g 2 Table III. The 20 Allowed Ways of Setting Three Coefficients 5i Equal to Zero. Constraint ¢i=$j=fifi=0 Case k Condition where i,j,k,= 1 4,5,10 x g f 2 6,7,10 f g 2 3 8,9,10 Z ¢ 8 4 4,6,10 x g f g Z 5 5,7,10 2 g f g 2 6 6,8,10 f g 2 g x 7 7,9,10 f g 2 g x 8 4,8,10 2 ¢ 2 ¢ 2 9 5,9,10 Z x i g f 10 4,9,10 x g f g 2 g x 11 5,6,10 x y f g i g x 12 7,8,10 x g f y 2 ¢ 2 13 5,6,7 2 g f g i 14 4,5,6 x ¢ f g 2 15 6,7,8 2 g 2 ¢ 2 16 7,8,9 2 g 2 ¢ 2 17 4,5,9 2 g 2 ¢ f 18 4,8,9 2 g x g f 19 4,6,8 2 ¢ 2 ¢ 2 g x 20 5,7,9 2 ¢ f g 5 ¢ 2 11 Table II. The restrictive conditions for the values of the rotational constants, e.g. X # Y, might be thought to be of little consequence except for accidentally symmetric rotors. However, even if two rotational constants are different by an amount of the same order of magnitude as 11 indicates that the the centrifugal constants, Watson reduction breaks down, and another reduction should be used. The restrictive conditions given in Tables II and III result from the expressions obtained for the "s" constants, i.e. the "s" terms are functions of the differences of two rotational constants to a negative power, so that when the constants are equal, the value of U becomes infinite, viz. Eq.(32), Ref.lO. The simplest way to obtain constraints on the coef— ~ ficients @i, is to set three of them equal to zero. Ford et al.13 have found that this can be done in 20 ways. Twelve of these take 510 equal to zero and thereby give rise to somewhat more symmetrical appearance of Hamiltonian than the remaining eight ways. In no case may ¢l’ ¢2, 53 be taken equal to zero. Table III lists the 20 ways of or reduction described. Watson has given another entirely different reduction of the KFC Hamiltonian, based on Eq.(6) of Ref.8. His 13 reduced Hamiltonian, as written by Ford et al. , is given by 12 H = hV + H2 + H4 + H6 (I-6) with H2 = X*P2 + Y*P2 + z*P2 x y z _ 4 ~ 2 2 ~ 4 2 2 2 H4 - DIP + DZP Pz + D3Pz + D4P (Px - Py) 2 2 2 2 2 2 + D5[P (Px - Py) + (PX - Py)Pz] _ ~ 6 ~ 4 2 ~ 2 4 ~ 6 ~ 4 2 _ 2 H6 — HlP + HZP Pz + H3P Pz + H4Pz + HSP (Px Py) + H P2[P2(P2 - P2) + (P2 - P z x y x 2 7 )Pz] Nth t = [21GZPz +ny(PXPy % + PyPX)][(vl+l)v3] /2 and = [-21GZPz + ny(PxPy 8 + PYPX)][V1(V3+1)] /2 (I-8) (I-9) (I-lO) CHAPTER II APPLYING THE HAMILTONIAN The Hamiltonians described in Chapter I are used to calculate the energy levels for a given vibrational state of an asymmetric-top molecule. But in near infrared spectroscopy the energy levels themselves are not obser- ved, but rather transitions between energy levels are observed. That is, the observed absorption lines are transitions from an energy level in the ground vibrational state to some energy level in an excited vibrational state. Most often, several absorption lines will have the same upper state energy level, or the same ground state energy level, so that the difference between two line frequencies is a difference between energy levels within one vibrational state. When the value of that difference involves only the energy levels of the ground state, it is commonly called a ground state combination difference,vGSCD. The absorption lines are clustered in bands with all of the transitions in the same band occurring between the same two vibrational states. Since there are three inde— pendent normal modes of vibration for a planar triatomic 16 17 Fig. I. Normal Modes of Vibration For An XYX Molecule. lb-axis Type B Type B 18 asymmetric—top molecule, three vibrational quantum numbers, (vl,v2,v3), are required to designate the state (see Figure I). The transitions originate from dipole moment changes which occur for a molecule lying in the ab plane along the a-axis for type A bands and along the b-axis for type B bands. The Wang Transformation A Hamiltonian, when evaluated in angular momentum space, i.e. w(J,K) space, yields an independent matrix, E, for each J value. Because of the cross diagonal symmetry within each such matrix and because nonzero terms appear alWaySnin a checker board pattern, each matrix can be factored into four noninteracting sub- matrices, which are denoted by E+, E- , 0+, and 0-. The factoring is mathematically done by applying the Wang Transformation, W, i.e. (E+,E-,O+,O-) = W-lE w . (II-1) where ... ... —1 1 —1 1 w = w'1 — 2-;5 2l5 1 1 1 1 19 This is equivalent to choosing a new basis W(J,K,y), given by 9(J.K.v) = 2‘*[ w(J,K) + <-1)Y4 It should be noted that all of the E terms are real and that in the figures use has been made of the relations E E E ; etc. E —K,-K-2= -K-2,-K E ; E K,K= -K,-K K,K+2= K+2,K= Also all of the I terms are pure imaginary, but the K,K' whole coupled submatrix is Hermitian. Appendix I contains a list of the angular momentum Operators evaluated in W(J,K) space which are necessary to calculate the elements of the energy matrices. 20 mm mm m: mm . mm mm m m m m .. H H o o 0 mm mm m: mm mm mm m: m m m m .. H: H H o o o: m: :5 :N... :N m: :5 :N m m m A mu me .. o H: H H o mm mm :NI :N «NI. Nu :N «N No m m A m: mvx Hi my .. o o H: H Hmm mm mm mm mm m: cm H: H o o .. m m m m o mm mm m: mm mm m... wN mo H: H: H o .. m m m m mmm w: .... :N m: w: 4.: :NI :N :o o H: H- H .. m m m A m+ my mmm :N «N am mm :Nl :N «N! Na No 0 o H: H: .. m m A m+ mvx m+ my mxm o o o NoH m o mom .om Nam com m mm mm Pm m NH H N H um +mH . m m + m :chouHHsmm map How xHHumsnsm 65» Ho mucmsmHm .HH.mHm NHH Hm 21 5A. um ...m 5H NH. hm ... m m m m ... H H o 0 nm mm mm mHl mH um mm mm ... m m m A MI E ... HI H H 0 mm mm mml mm mHl mH mm mm mH ... m m A m: mVA mu my ... 0 HI H H RH mHl nH mHl mH HHI HH mH HHI HA ... m A mI HVA mu mCA w: my ... o o HI A H+ He 5A. um hm. mm hm 5H ... HI H O O ... m m m m um mm mm mm mm mm mHl mA ... HI HI H o ... m m m A m+ my mm mm mH mm mm mml mm mHl mA 000 0 HI! H- H 000 m m A m+ my“ WIT mv mH HHI HH 5H mHl mH mHI. mH HHI HH ... o 0 HI A H+ HVI ... m A m+ mVA m+ mVA m+ my mo NHH H N H I +0 . m + m + m cMHcouHHEmm may Mom xHuumeasm may Ho mucmamHm .HHH.6HH NH H oH +0 22 Identification of Levels The eigenvalues of these submatrices constitute the energy levels of the Hamiltonian. They are found expli- citly by applying the unitary transformation matrix, S, to the energy matrix, E, which diagonalizes it, i.e. E' = S_1E S. In the transformed space K is not a good quantum number. But the energy levels may be identified uniquely by assigning three quantum numbers, (J,K_,K+), to each, where K = K_ in the limit as the molecule ap— proaches a prolate symmetric-top,and K = K+ in the limit as the molecule approaches an oblate symmetric-top. To evaluate the angular momentum operators, one must choose the arbitrary phase factor for Px or for Py. We have chosen this factor such that PY has real and pos- itive matrix elements, hence PX has pure imaginary matrix elements, viz. < J,K | Px | J,K+l > = (i/2)[J(J+l) - K(K+1)]%. (II-2) For this phase convention all of the symmetry arguments concerning the assigning of energy levels as given by King, Hainer, and Cross14 apply to this work. Therefore the evenness or oddness of the K_ and K+ values within a submatrix must agree with those given in Table IV, which agrees with Table VII in Ref.l4. The assigning of the levels within each submatrix must be such that 23 A ~\AH+hv1 m\h o o o m m o o 0 IO m\AH+bv N\h o m o o o o m 0 +0 «\AHInV m\h m m m o m m o 6 Im N\AH+hv H + «\h m o m m o m m m +m h 660 n cm>o +x as s IH +M Ix +m Ix coHumucmmmummu HHHH coHumucmmmHmmH HH xHHumEQSm mo mNHm h poo h cm>m h woo h cm>m xHHumEQSm . O cam . o . m .+m mmOHHumEnsm on» NO coHHMOHHHmmmHU .>H mHnma + I. 24 Fig.IV. Energy Level Diagram J K=K_ Ir JK K IIIr K=K+ Submatrix — + Submatrix E+ ———-—-'—’0 “‘ o" 330/0‘ 1‘7 +,/__ ##- r—3———--—=o * 331 +—/— ’///////'O 321 _ 2___ ’///////’312 _ O / + #3 ————'—O+'——-—-—-3-_ -—l O+—v 313..__———----""'"O ——-——-—"""" o3 L_0-———————'E E+——————"O"' 2 E+___fl_A——- 20 .—”—’::O ::::=="1-db _ _____.‘ __ E 21/071“ 211 1 t: + 01 l ‘ A--- fijvl i a V I I I r7 T ‘T"‘ ' r K —l.0 '0.5 0.0 0.5 1.0 Prolate Oblate Limit Limit B=C B=A 25 for increasing energy, the value of K_ increases and the value of K+ decreases in an even fashion. Figure IV shows the assignments of energy levels for J equal to 1,2, and 3. Evaluation of Coefficients If we ignore the vibrational term, hv’ we can write each of the Hamiltonians in the form H = z xlxl . (II-3) where x1 are the coefficients and x1 are the angular momentum operators. The equation for the energy matrix elements then becomes Ers = z xlxis (II-4) The ath_eigenvalue of the energy matrix is given by _ -1 - Ea — 22 (s )arErsssa (II 5) In terms of Eq.(II-4), this becomes Ea = z x1a (II-6) where = 22 (8-1) ”xi S (II-7) a ar rs sa i.e. a is the expectation value of the ith_Hamiltonian angular momentum operator. The coefficients , x1, are determined by least;squares fitting the frequencies of the observed lines to the 26 differences of the calculated energy levels, i.e. to vO-Eg+Eu, where E9 is the ground state rotational energy+, Eu is the excited state rotational energy, and 00 is the vibrational contribution, referred to as the band center, which comes from the hV term in the Hamiltonian. If Eq.(II-S) were linear in xi, the method of least squares would directly yield the values of the coeffi- cients. But since the unitary transformation matrix, S, is itself a function of the xi, Eq.(II-S) is nonlinear in xi and a longer procedure must be used. Generally one begins with an estimate of the rota- tional constants, A, B, and C obtained either from past work on the molecule or by using combination differences as described by Moncur.4 From these rotational constants the energy matrices are calculated using Eq.(II-4), and then diagonalized to obtain the energy levels and the angular momentum operator expectation values. Those expectation values, , and energies corresponding to the energy levels of the observed transitions are used to calculate the normal matrix for the least squares fit. The least squares fit yields the values 51, where _ i i _ AEa — Z 6 a (II 8) such that Ea + AEa are on the average closer to the values which are necessary for fitting the observed lines. Then i i the new constants, x + 6 , are used to calculate new +Actually the E's are term values, differing from energy by a factor of l/hc. 27 energy levels and the whole procedure is repeated until convergence is reached. Isotopic Substitution When several atomic isotopes are present simultane- :ously in the molecules under study, the absorption bands of each isotope are observed overlapping one another. The small differences in the frequencies of the lines for different isotopes can partly be accounted for by making a linear change in the Hamiltonian coefficients with atomic mass]:5 That is, Eq.(II-3) would become i H = (x + EiAm)xi (II-9) where ii is the mass dependent constant and Am is the mass of the most abundant isotope less the mass of the isotope being fit. The use of Eq.(II-9) requires many more operations than the simpler approximation __ i i i _ Ea — (x + g Am)a (II 10) which we have used successfully. Eq.(II—lO) differs from Eq.(II-9) in that is a function of (xi + giAm) in Eq.(II-9), but only of xi in Eq.(II-lO). The band center, 'V0' is also fit to the observed data and a linearfplus quadratic change in it with change in atomic mass, is sufficient to complete the compensation 28 for the differences in the frequencies of the lines for different isotopes. i.e. v0 is replaced by v0 + gmAm + gm(Am)2. Computer Programs To apply the preceding material to experimental data.three computer programs were written. The programs are similar in nature and parallel one another in oper- ation. ICDFIT was written to fit combination differences, ISPECFIT was written to fit single spectral bands, and SPECFITZ was written to fit simultaneously two bands which are Coriolis coupled. Because of the complexity of the programs, each major part was written as a subroutine. Due to the volume of data,which must be stored and sorted by the programs, core storage and superfluous calculations had to be economized. The programs were written for a batch com- puter, specifically the CDC 3600, where core storage is not shared with other users. Appendix III isiilisting of the Fortran source for ISPECFIT. The other programs were written very similarfly to ISPECFIT, using some of the same subroutines. This program and the program ICDFIT incorporate Eq.(II—lO), which allow the user to fit different isotopes simul— taneously. 29 The main part of each program reads all of the control cards and the data cards, sequences the oper- ations, sets up the data for the normal matrix, changes the Hamiltonian coefficients, and iterates the fitting procedure. The subroutine PMAKE calculates the angular momentum operators for whichever Hamiltonian the user-chooses, The planar Hamiltonian, Eq.(I-3), the KFC Hamiltonian, Eq.(I-S), and the Watson Hamiltonian, Eq.(I-7), are available in the programs and are selected by a control card. All nineteen angular momentum operators for the KFC Hamiltonian are included. Since only fifteen are determinable, the user must choose, using Tables II and III as a guide, those best for his molecule. Both the prolate and the oblate form of the planar Hamiltonian are included, hence, since the KFC Hamiltonian and the Watson Hamiltonian will calculate the energy levels for oblate or prolate molecules, any one of the Hamiltonians may be chosen to fit any type of planar molecule. The subroutine CAPPA prints the Hamiltonian chosen: along with the current coefficients. It also calculates Ray's asymmetry parameter, K, using the equation K = (ZB-A-C)/(A-C) (II-11) Note that -l S K S l 30 When K is positive, the molecule is oblate, becoming an oblate symmetric-top for K = +1. When K is negative, the molecule is prolate, becoming a prolate symmetric-top for K = —1. If is positive, the program chooses the IIIr representation and identifies the (E+, E-,O+,O-) subma- trices with the quantum numbers as listed in Table IV. If K is negative, the program chooses theIr representation and identifies the submatrices accordingly. WANGT calculates the Wang transformed energy matrices. SYMDIG diagonalizes the resulting symmetrical energy matrices (HERMDIG in SPECFIT2 diagonalizes the Hermitian energy matrices required for Coriolis coupling). Diago- nalization is done bythe Jacobi method, which is out— lined in detail in Appendix II. Both the eigenvalues and the eigenfunctions are calculated. The expectation values of the angular momentum op- erators are calculated using Eq.(II-7) in the subroutine FORMEP. FORMEP also assigns the quatum numbers to the energy levels. REGRESS is a stepwise multiple regression routine 16 It is used originally written by M. A. Efroymson. to determine the least squares "best" values for the coefficients, 61, as required to solve Eq.(II-B). The routine is different from most weighted least squares routines in that the variables are added to the solution 31 one at a time in the order in which they will make the greatest improvement in the "goodness of the fit". The routine also decides which variables are determinable and does not add the indeterminate variables to the fit. The user weights his data according to the expected 2 accuracy of his measurements,Avi, choosing wi « (Avi)_ . The program normalizes the weights so that the average weight, w., is equal to one. If for the ith line l —_ 42w 2 XOUT (z Azw.)/(N—m) i i i i where A1 is the deviation between the observed line frequency and the calculated frequency, N is the total number of nonzero weighted lines and m is the number of independent variables in the fit, that line's Weight will be set to zero during the next iteration. It has been observed that a good value for XOUT for our data is approximately 12. CHAPTER III REDUCTION OF SPECTRAL DATA Spectral data "originate" in the form of a voltage signal from the detector circuit of the spectrometer, whose relative amplitude is a measure of the absorption of infrared radiation by the gas under consideration. The voltage signal, recorded as a function of frequency, traces out the spectrum, which for asymmetric-top mole— cules consists of hundreds or thousands of "lines", many of which overlap in a complicated fashion. In the past these absorption "lines" were recorded on paper strip charts along with Fabry-Perot equal frequency spaced fringes of visible light generated §multaneously 17 in the spectrometer. The line centers were measured from the paper by utilizing a fairly complicated optical display.18 No pre-processing of the raw spectral data was done except that done by the amplifier through RC smoothing. With the recent addition of a PDP-l2, a Digital Equipment Corporation minicomputer, to our laboratory, 32 33 processing of the raw data before measuring line centers became possible. That is, the raw data, after RC smoothing,are digitized and recorded in digital form on magnetic tape under the control of computer software. Then under the control of the operator the computer is used to adjust the baseline and remove any anomalies in the data. The data arealso smoothed digitally by the method of least squares. The above steps prepare the data for deconvolution, which enhances the spectral resolution, presently by as much as 2.5 over the original data. The method employed is a modification of one developed and used for resolution enhancements of a much lesser degree by Van Cittert19 and Burger and Van Cittertgo'21 The spectral lines in the deconvoluted data can then be measured using a combination of the data manip- ulating "power" of the computer and the knowledge and judg'f '2 of the scientist. A computer program has been written to display simultaneously,the lines,along with a mirror reflection and their difference. Based on the symmetry of the above information the scientist chooses the line centers. The remainder of this chapter will deal in more detail with the theory and application of these oper- ations. The following chapter will deal with the method 34 of deconvolution. If the spectra are to be deconvoluted, they must meet some basic criteria, i.e. a) the signal to noise ratio should be greater than 200:1 after RC and digital smoothing, b) the 100% absorption and 0% absorption levels must be fixed to known values, c) the full width at half height, (FWHH), of single lines must each contain a sufficient number of data points, e.g. approximately 30 points for a resolution enhancement of 3. If the data is not to be deconvoluted, less stringent restrictions can be set. Acquisition of Data Since the spectrum is obtained nearly linearly in time, it is simplest and sufficient to sample the data, via the analog-to-digital converter, linearly in time. The time between samples is chosen to yield the number of samples—per-FWHH required. However, were we able to do so, we would sample at exactly equal frequency inter- vals. We depend on the frequency of the spectrometer to be changing linearly in time over a range much greater than the FWHH of a single line. The voltage signal to be sampled is zero offset and 35 electronically amplified such that its range is within that required by the analog-to—digital converter, i.e. -l to +1 volt. Before recording the spectrum, the operator should record the "zero" signal, i.e. 100% absorption, and again whenever a change in that zero is made. This information is required for adjusting the baseline. Noise or Error in the Data Noise or errors in either coordinate of a data point affects smoothing, deconvolution, etc. The sources of noise or error are different for the two coordinates. The method of driving the monochromator in our spectro- meter is the greatest source of error in the abscissa of the data points, is only partly random in character, and tends to become worse at the slowest drive rate. Therefore, it ultimately determines what the longest observation time may be. Noise in the ordinate of the data points is elec- tronic in nature and mostly stems from the detector circuitry and the associated electronics. The latter noise is generally "white" in character and can be de- creased by increasing the period of observation. Noise whose frequency is much higher than that of the signal can be averaged out, whereas noise whose frequency is 36 similar to that of the signal, cannot be distinguished from it. A third source of noise is regular in character and comes from 60 Hertz power and 90 and 450 Hertz chop- pers found in and around the spectrometer. Noise of this character becomes a problem when the data is sampled at regular intervals, for it can then be folded into the data, becoming a false signal of frequency less than F/2 where F is the sampling frequency. This signal is commonly said to come from aliasing of Nyquist folding.22 Smoothing of Data To average out the high frequency noise, we use a combination of RC filtering and digital smoothing. Averaging of repeated runs has not been possible because our monochromator is not driven sufficiently linearly and reproducibly. Stewart23 has given a good introduction to the effects of RC smoothing on individual lines, as well as references to other work on the subject. The principal effects include loss of height, increase in FWHH, drag of the peak, and asymmetry in the shape of the line. Edwards and Strome24 have made a careful study of these effects for both Gaussian and Lorentzian single lines, as well as the effects on doublets. They conclude that the optimum 37 value of RC is near 0.1 FWHH of single lines, independent of whether or not the data is to be digitized and that if the data is to be digitized the optimum RC value is independent of the number of samples per FWHH. Digital smoothing is easily done by convoluting a "smoothing function" with the data. The "smoothing func- tion" chosen may drastically alter the result,e.g. one may obtain any derivative of the data by selecting the appropriate function (see Appendix IV). To maintain the symmetry of the original data and a point for point cor- respondence between the original data and the convolved result, the "smoothing function" must be symmetrical about a single point in it. We have chosen to use a smoothing function which gives us the same result as obtained by fitting the data by the the method of least squares to a running cubic polynomial, i.e. for the ordinate of each data point,y, is substituted b0, whose value is determined by fitting the equation _ 2 3 y — bO + blx + bzx + b3x to the spectral region immediately surronding that data point. This operation, as done by convolution, requires that the data should be linear in the abscissas over the range which the fitting is done(see Figure V). In a study of the effects of digital least squares 38 Fig. V. Smoothing Function For a 13 Point Smooth. X OX OX X OX X OX X OX —6 6 c_i = c1 c0 = 0.174825 cl = 0.167832 c2 = 0.146853 c3 = 0.111888 c4 = 0.062937 c5 = 0 c6 =-0.076923 The numbers, Ci’ (referred to as convolutes) are convoluted with the data to be smoothed; the result will be a least squares fit to a cubic polynomial about each point, provided the points are equally spaced in the values of the abscissas. 39 smoothing of calculated Gaussian and Lorentzian single lines, calculated doublets, and observed lines, we25 have shown that the most important parameter is the smoothing range compared to the FWHH of the lines and that the optimum value of the smoothing range is about 0.7 FWHH. In addi- tion we have shown that multiple smoothing by successive least squares fitting to a shorter range cubic polynomial is no longer a least squares fit to a cubic polynomial, and is less effective in improving the signal to noise ratio. than a single least squares fit to a polynomial of the proper range. To reduce the amplitude of aliased frequencies we have chosen to multiple sample, i.e. average 2n rapidly but equally spaced samples into one digitzed data point, which is then stored on the magnetic tape. The methods of smoothing just discussed will reduce high frequency noise. For low frequency noise, such as drift due to detector response, we use a method of ad- justing the baseline. Base Line Adjusting_ Our data is absorption spectra and therefore the baseline corresponds to zero absorption or 100% trans— mission. This means two things. First of all the 100% absorption level must be known, since it can not be obtained directly from the data, and this level will remain unchanged throughout the experiment as long as 40 the amplifier "zero" or offset is not changed. Secondly, changes in the baseline, which occur due to changes in detector response, source intensity, and other sundry causes, affect the "gain" of the signal, and so, must be compensated for by changing the gain in the digitized data. Once the data is in digital form "zero" and 100% absorption must be defined in terms of numbers. For convenience of calculation as well as need for precision, we have chosen "zero" to correspond to 0000 (010) and 8 100% absorption to correspond to 20008 (1024 There- 10)' fore absorption lines actually appear as peaks in the data and henceforth will be treated as peaks. After the data is recorded, the difference between the 100% absorption octal value recorded at the beginning of the run and the octal number 20008 must be added to the data. And then the "gain" of the data must be changed in a continuous way so that the base line of the spectra is set to zero. The computer program which can do this operation is written so that a variable, t, whose'value can be changed continuously by the Operator by turning a knob while viewing the data, alters the data values by the formula y' = 2000 - (2000 8 + t)(20008 - y). 8 41 In this fashion, slow changes in the baseline of the spec- tra are easily removed. Since at times it is oftenyvery diffisult: for a physicist to tell where the baseline of a spectrum is, it is impossible to have the computer automatically determine it. In practice, first the zero level, then the gain should be corrected before Smoothing digitally is done, so that any "staircasing" due to roundoff errors and other minor errors which creep in while correcting spu- rious data values will be smoothed out. After these operations have been completed accurately, the spectrum is ready for deconvolution. Deconvolution The radiation signal, as it passes through the spec— trometer, is changed by the spectrometer. In effect one can represent the spectrometer with its slits and optical components by a function which is convolved with the spectrum. The electronic components in the system likewise alter the voltage from the detector and can be represented by a function convolved with the spectrum. The end result is that the true spectrum is observed convolved with numerous other functions representing the components of the system. The convolution of these "numerous other functions" together, quickly takes on a Gaussian character 42 and is called the instrument function.26 Originally our purpose in deconvolution of the spec- trum was to remove this instrument function from the spectral data and thereby enhance the resolution of the 27 have found that one data. More recently we and others can actually enhance the resolution of the spectrum by using a function which is greater than just the instrument function and includes doppler broadening, etc. Numerous methods of deconvolution have been proposed and used with moderate success. Resolution enhancement of a factor of approximately 1.4 is apparently obtainable by most of the methods. Since Jansson and the Florida State University group28 obtain resolution enhancement of approximately 2, we have adopted and improved their method for our minicomputer and can now obtain a resolu- tion enhancement of approximately 2.5. The method is described in detail in the next chapter. Measuring Line Centers After the data is deconvoluted the centers of the spectral lines are measured and transformed into wavenumbers for fitting to a Hamiltonian. When the lines are well resolved, numerous methods of measuring the lines work. For partially resolved lines, it has generallly been found not possible to give the computer sufficient 43 criteria to automatically measure the line centers. The program developed and written by Hurlock and Hanratty29 to measure the lines, displays the spectral region under consideration and that same region reversed about the center of the screen. The operator decides where the line centers exist using the symmetry of the display as well as difference between the line and its reverse, which can also be displayed, as criteria. The measurement is made in terms of the nearest half separa- tion of points (see Figure VI). The Fabr y—Perot fringes are measured automatically by the computer program simply from using an a-priori- separation and minimizing the difference between the line and its reversal. If the fringes are properly calibrated, the program transforms the measurements of the infrared lines directly into wavenumbers. 44 Fig.VI. Measuring Line Centers by Line Reversal. Upper Traces: The region of the spectrum selected by the operator and that same region reversed about the cursor. Lower Trace: The difference between the region and that region reversed. CHAPTER IV DECONVOLUTION OF SPECTRA As described in the previous chapter, the observed spectral lines consist of the "real" spectral lines convolved with a number of other functions. Calling W the "real" spectrum, S the observed spectrum, and A the convolution of these "other" functions, we can write (I) S(y) = f W(x) A(y-x) dx = fw A (IV-l) We desire to solve Eq.(IV-l) for W, that is to deconvolute S. Theoretaically this can be done most easily by dividing the Fourier transform of S by the transform of A and transforming the result back. In general, the results of such an operation on experimental spectra have been unreliable. In the recent years a number of people, Ref. 28, 30,31,32,33, and 34, have tried with success different methods of pseudo-deconvolution. In these methods the solution to Eq.(IV-l) is obtained indirectly by an iterative method without transforming the functions. Of these methods, the one used by Jansson, Hunt and Plyler28 seemed to be most successful, yielding the 45 46 greatest improvement in resolution. Van Cittert Method Jansson et.a1. adopted a method developed by van 19 which is based on the Fredholm solution to Cittert integral equations of the first kind. Jansson altered van Cittert's method by adding a variable over-relaxation parameter which,in effect, added boundary conditions to the solution, i.e. he constrained all ordinate data values to remain within a given range,viz. 0% to 100% absorption. We have studied Janssonds method in detail,. especial- ly in relation to Gaussian shaped lines, and have found ways of improving and extending the method. In van Cittert's method the first approximation of W in the solution of Eq.(IV-l) is Wl = S. The second is w2 = wl + (s - fA WI) The nth_is wn = wn_l + (s - fA wn_l) (IV-2) which can be written as = — - I coo _. Wn nS n(n l)Sl/2’ + + Sn-l (IV 3) where S1 = fA Wl Sn = [A Sn-l = fS An—l A = A , A = fA A 47 Eq.(IV-3) can also be written as Wn = fW Xn (IV-4) where Xn = nA - n(n-1)Al/2! + n(n-l)(n-2)A2/3! + --- + An-l It is proved by Burger and van Cittert21 that if |1 - JAI < 1 (IV-5) then Wn + W as n + m where JA is the Fourier transform of A. This in turn implies that Xn behaves as a delta function as n + w. The condition, Eq.(IV-S), is satisfied by Gaussian shaped lines as well as by our spectral lines. Hence, the method should theoretibally' work for deconvoluting our spectra. In Figure VII we give an example of the function A = exp(-x2) = X1 and its related functions X2 and X5. Note the negative lobes which appear in the latter func- tions. When using this method to deconvolute,similar lobes appear in the early iterations for the approx- imation of W, e.g. in W5. When deconvoluting a Gaussian line, S, using a Gaussian function for A, we have found that if the half width of A, A is less than approx- A! imately .7 AS, the lobes do eventually disappear. But as the half width of A approaches that of S, then even in numerous iterations, the lobes do not disappear (see Figure VIII), thereby limiting the resolution enhancement 48 X and X for X = A = exp(-x2). Fig.VII. The Functions X 2, 5 1 ll ‘J\ “r 'WM‘ 5 Fig.VIII. Results of Deconvolution by van Cittert's Method. 49 possible by this method. 35 Jansson ’added to Eq.(IV-2) a variable over-relaxa- tion factor, here denoted by a, so that Eq.(IV-2) becomes W n wn_1 + a(s - fA wn_l) (IV—6) In numeric notation this becomes J=m Wi(n) = Wi(n-l) + n(yi)[Si - ji-m A_jWi+j(n-l)] (IV-7) where yi = Wi(n-l). Jansson: chose a(yi) = b(l.0 - 2.o|yi - o.5|) (IV-8) where "b" is a selected constant. The triangular shaped function, which is plotted in Figure IX, forced the data ordinate values to stay within the real experimental boundaries, i.e. 0 S yi S 1. Adding the boundary conditions enabled Jansson et. al.28 to obtain resolution enhancements near a factor of two without the negative lobes showing up. In fact, they increased the resolution of their data beyond the opticalf’,71 resolution limit of their spectrometer.27 We also observed that we could enhance the resolution of our spectra beyond the limit for our spectrometer.36 The next question is what is the maximum resolution enhancement possible in practiee and can it be success- fully obtained for real spectra? 50 Houomm Hepomm COHHMKMHmm mcHHHm> coHumeHom mcHHHm> mch A HHHMHDSGMHHB o n HHHMUHUMHGMSO o Hmuuowmm Hk , .mGOHuocsm COHummemm HmHsmGMHHE can 0Huwupmso we .xH.mHm 51 Maximum Resolution Enhancement It is quite easily shown that a Gaussian line convoluted with another Gaussian line is a Gaussian. The relationship between the line widths at half heights for Eq.(IV-l) when S, A, and W are Gaussian lines is easily found to be A /A = [1 - (A /A )21'3 (IV-9) s w A s where AS, A and AW are the FWHH for S, A, and W respec- A, tively. We shall define the resolution enhancement to be AS/AW. From the plot of Eq.(IV-9) in Figure X, one can readily see that for large values of AA/A the maximum 8' resolution enhancement wiD.be limited by the precision of the ordinate values of the data. That is, when AA/As is approximately 0.95 a 2% change in that value can make a 20% change in the resolution enhancement. It should be noted that the largest value AA/As can be is one. If AA/AS should exceed one,then for lines in the spectrum which are singlets, i.e. separated from all other lines, the deconvolution process will diverge, and for lines which are very close to one another, i.e. so that there combined half width approaches the:ha1f Width of A, the deconvolution process will yield a narrow single line. IA 2% change in AA/AS is not unexpected in our data and in fact for a run which covers several 100 cm-1, changes 52 TATE}: - 2 u zimq 3.6:: 3.3163 Ho :5: u < .coHuocnu mcHunHo>cooov mo mmZE u m .mcHH uo>uomno no mmzh H 2.5 n m. <HC.SH an coHusHo>coowo msomcmHHDEHm ucHom .AOHI>HC.6H an coHusHo>coomt m>Hmmmoosm HGHOQ mo uHsmmH can Am xv HmsHmHHO Ho HHDmmH can Am xv HmcHoHHO / j .mumHmHHB mo coHusHo>noomQ msomcwuHDEHm ucflom .m> m>Hmmmoosm ucHom .Hx.mHm 56 as the area of the whole multiplet was shifted to the left. The point simultaneous method conserves peak cen- troids throughout the many iterations, but does not con- serve the area under the peaks. That is, for large values of b in Eq.(IV-8), the area under the curve Wn can be drastically different from that of the original curve, S, and the area may be set into wild oscillations in magni- tude from one iteration to another. To prevent such oscillations from beginning, small values of b must be used, and as a result more iterations are required to obtain convergence. we have found additional ways of changing the method, such that the oscillations are reduced. Quadratic Relaxation The discontinuity in the triangular over-relaxation factor given by Eq.(IV-B) tended to increase the proba- bility of oscillations. By changing the shape of this factor from triangular to quadratic, i.e. by using a(yi) = 4b(yi - yi) (IV—ll) and thereby eliminating the discontinuity and flattening the "top" of a, larger values of b could be used, resulting in faster convergence. Figure IX,page 50, demonstrates the shape of this function. 57 The value of a which is used to determine the value of the data point Wn from Eq.(IV-7),corresponding to the point marked x,on the Figure is as shown. The value of y in Eq.(IV-ll) is the ordinate value at the point to be calculated taken from the result of the previous iter- ation. Hence the curve of the spectral line shown in the Figure must be that of the function Wn-l for the nth iteration. Normalization Let us now consider Eq.(IV—7) in more detail. From Figure XII, a plot of S, W, and fAWl,where A and S are Gaussian lines, one can readily see that on the first iteration the direction of change in the ordinate values, i.e. (S - fAWl), will be wrong between the points p, where curves S and [AW cross,and q, where curves W and S 1 cross. Eq.(IV-U) will then yield W which passes through 2 point p rather then point q. Since point p is farther out into the wing of the line than point q is,and because the value of a increases as one approaches the peak;. of the line, the area under W2 will be greater than the area under S. If one knows point q, then the curve [AW1 could be scaled before subtraction to pass through point q, and point q would become an invariant. Since point q should be in the result of deconvolution, the result of the first iteration should have a line shape closer to that 58 Fig.XII. A Plot of S, W, and [A Wl for Gaussian Lines. The curve W2 = Wl + (S w [A Wl), which is not shown, will pass through point p, whereas, it is desirable for W2 to pass through point q, since q is a point in the expected deconvoluted result, i.e. W. 59 of the desired result, i.e. W, and convergence should proceed at a faster rate. For Gaussian lines the necessary scaling factor can be calculated in terms of the resolu- tion enhancement,for determining W2. But for subsequent iterations the scaling factor is very difficult to deter- mine. An alternative to using a scaling factor for the integral fAW1,J, which has a similar effect, is to scale W2, or more generally,Wn, to the area of S. The scaling can be done by the computer at the time of calculating the next pass and is not difficult to do. In this way, ize: using Eq.(IV—7) and scaling the results of each pass before making the next pass, the peak centroids and the area can be conserved. Smoothing During the Time of Deconvolution We have programmed the deconvolution process on a PDP-lZ computer which has a twelve bit word length. The original data and the results of each iteration are stored as a fixed point number whose magnitude is generally between 0 and 1048 (these values correspond to 0% absorp- 10 tion and 100% absorption respectively). Hence each data point is truncated before it is stored and truncation errors show up in the results. These errors appear as high frequency noise, which do not grow in magnitude unless large values of b are used. This "noise" does not 6O seem to remain "fixed" relative to the spectrum, but rather changes with each pass. However if the result of each pass is smoothed or even the result of every few passes is smoothed, then this high frequency "noise" is smoothed to lower frequency ripples close to the fre- quency of the deconvoluted lines. The ripples are enhan- ced in later iterations or passes and becomes a permanent part of the result. Therefore we do not smooth the results between passes except once or twice when close to convergence. Conclusion Our goal has been to obtain the maximum resolution enhancement with dependable results. To do this we need to use an "instrument function",A, whose half-width is a few percent smaller than the half—width of the observed lines, S. Since our spectrometer does not give us a symmetrical line, we prefer to use the narrowest smooth singlet which appears in the observed lines and has the characteristic line shape. This line can be made narrower by deconvoluting it with a narrow Gaussian function. The spectrum is deconvoluted using point simultaneous, EQ-(IV-7), quadratic over-relaxation, Eq.(IV-ll), method with normalization of the results between each iteration and smoothing of the results at completion of approximately 61 the sixth iteration and the twelfth iteration when con- vergence is nearly reached. Small values of b are used to keep the results from"blowing up: i.e. 3,1,l,l,6,l,l etc. Resolution enhancements of approximately 2.5 have been obtained with reasonable results. CHAPTER V DATA ANALYSIS The spectra of HDS and D28 were recently run in the 1 to 3930 cm‘l, which includes the region from 3520 cm- (1,0,1) and the (0,2,0) bands of HDS and the (2,0,0) and the (1,0,1) bands of D23, as well as some weaker bands which have not yet been identified. These spectra have been digitized and recorded on magnetic tape, digi- tally smoothed, baseline adjusted, and deconvoluted by the methods outlined in Chapters III and IV. The lines are now in various stages of measurement and analysis. Figure XIII is a picture of the Q-branch of the (1,0,1) band of HDS mentioned above as it appears before and after deconvolution. It is typical of the resolution enhancement we obtain by deconvolution on the minicomputer. The line width of the original spectrum is approximately 0.05 cm-1 and that of the result after deconvolution is approximately 0.02 cm-l. The maximum resolution for the grating in this region is 0.024 cm-l; but the measured maximum rESOlution for the grating is only 70% of the 37 theoretixmfli resolution. Hence the deconvolution has 62 63 —p. use 2%» .. an: 1%. .pmpsHo>coomo mom Ho pcmm m >+ H -.3 3 > man we Hocmnmno was .HHHx.wHH 64 narrowed the line width beyond that theoretically pos- sible for the grating. This implies that we have also removed by deconvolution part of the Doppler and collision (pressure) broadening effects. The computer programs ICDFIT, ISPECFIT, and SPECFIT2 have been used in the analysis of infrared spectra which were run before we could digitally record spectra. We include in this work three ofthose bands, viz. the (2,0,0) (Te-H stretch) band of the prolate molecule HDTe and the Coriolis coupled bands (1,0,0) and (0,0,1) of the oblate molecule H28. The HDTe band was run by N. K. Moncur and the H28 bands were run by the author. The running conditions for all of the these bands are listed in Table V. The assignments of lines for these bands was done by Prof. T. H. Edwards and so only the results will be included here.39’4O The (2,0,0) Band of HDTe The HDTe band contains absorption lines of the five different isotopes of Te, viz. 130Te, 128Te 126Te 125T 124T I I er and e. Over 900 transitions were identified and used to determine the molecular constants both from combination differences and from line fits. The data from each isotope was first fit separately and then the data for all of Table V. Grating Calibration Detector Date of Run Source Pressures(torr) Path Length Chart Resolution Calibration Fit Temp. of Gas 65 Experimental Conditions. H23 (370 - 32.50) v and v 1 3 300 groves/mm 1§E_Order Single Pass N 0 (1,2,0) H 1 (1-0) PbS-type P @770K #1 Chart IR-8/21/67 #2 ChartIIR-8/23/67 Carbon Rod 27700K #1 - 50,60 #2 - 54 6.4 m Multiple Traverse Cell 20.09 cm‘1 0.002 250C HDTe (51O - 460) 2V1 300 groves/mm 2nd Order Sifigle Pass HCl (2-0) 3rd Order CO (2-0) an_0rder PbS-type O #1 Chart 2-3/18/73 #2 Chart 5 300 watt Zr Arc 3.5 mixture 2.6 for HDTe 6.6 m White Cell 20.03 cm-1 -500C 66 the isotopes were fit simultaneously. We have obtained a very good fit of the ground state combination differences for molecular isotope HD13OTe for each of the three Hamiltonians given in Chapter I. The prolate form of the planar Hamiltonian, Eq.(I-3) was used; the KFC Hamiltonian, Eq.(I-S), with T6 = 0 and H6 = 0 was used; and the Watson Hamiltonian, Eq.(I-6), through second order was used. Each of them fit the data 1 to approximately 0.004 cm_ , but the planar Hamiltonian converged to the fit in the second iteration through the fitting process (see Chapter II, p.26), while the other Hamiltonians required more iterations. The constants obtained from the planar fit and the KFC fit are listed in Table VI. The transitions and the ground state combination differences for all of the isotopes were fit simultaneously by the method described on pp.27-28 to the Watson Hamilton— ian to second order constants. We found that the iso- tOpes could be simultaneously fit by allowing a quadratic dependence of v on mass , a linear dependence of the 0 rotational constants on mass, and treating the coefficients of the fourth—power angular momentum terms to be indepen- dent of mass. The coefficients of the sixth-power angular momentum terms were not found statistically significant. A very good fit of the 917 identified transitions was 67 Table VI. Ground State Constants of HDl3oTe. Planar . _1 KFC Hamiltonian -1 Hamiltonian cm T6: 0 cm A 6.16896 1 0.0002* 2 6.16952 1 0.0003 B 3.11061 1 0.0003 2 3.11072 1 0.0003 0 2.04217 1 0.0002 2 2.04168 1 0.0002 Taaaa —0.000902 1 0.00001 11 -0.0000585 1 0.000005 Tbbbb -0.000227 1 0.00002 12 -0.000013171 0.0000005 Taabb 0.000029 1 0.00004 13 -0.0002248 1 0.000003 Tabab -0.001124 1 0.00003 14 —0.000101 1 0.00001 65 -O.0004964 1 0.000009 K =-0.482 Total # of Points Fit =249 Standard Deviation of Fits=0.004 cm"1 *95% Simultaneous Confidence Interval equals four times the standard errors of the coefficients. Table VII. 68 Molecular Constants for 2v 1 of HDTe. 130 All isotopes were fit simultaneously relative to HD Te. Watson Ground -1 Upper Hamiltonian State cm State cm ~* Z 6.169355 1 0.0005 5.838694 1 0.0003 7* X 3.110844 1 0.00024 3.110497 1 0.00045 ”‘k Y 2.041369 1 0.00034 2.005928 1 0.00024 01 —O.00003478 1 0.0000024 -0.00003438 1 0.000002 Dz —0.0004951 1 0.000009 -0.0004984 1 0.000008 D3 0.0003055 1 0.000008 0.0003104 1 0.000006 D4 -0.00002185 1 0.0000025 -0.00002178 1 0.000002 D5 —0.0003081 1 0.00001 -0.0003034 1 0.00001 62 0.000433 1 0.00008 0.000399 1 0.00006 x E 0.000347 1 0.00014 0.000333 1 0.00014 5y 0.000201 1 0.00007 0.000196 1 0.00007 v 4063.971 1 0.0026 0 0 gm 0.1100 1 0.002 gmm 0.0009 1 0.0004 Total # Lines Fit = 917 1 Standard Deviation of Fit 0.0035 cm- 69 obtained, viz. =0.0035cm-l. The assigned transitions are listed in Appendix V and the constants obtained for the fit are listed in Table VII. The mass dependence, Am, as required for Eq.(II-lO) is given by Am = 130 - atomic weight of the Te isotope.. The conStantEX is the variation in X*, whereas Em is the linear mass dependence of the band center. The (1,0,0) and (0,0,1) Bands of HAS In contrast to the HDTe spectra, only lines due to H2328 were analyzed in the Coriolis coupled (1,0,0)- (0,0,1) bands? Since H28 is an oblate molecule,the IIIr representation was used. The planar Hamiltonian was evaluated in this representation and used to fit the coupled states. Attempts were made to fit the ground state combination differences to the Watson Hamiltonian and to the KFC Hamiltonian with T6 = 0, but they were not successful, apparently implying that the condition given in Table II, i.e. X # Y, is not met to the order of magnitude necessary (see comment on p.9), Hence the reason for using the planar Hamiltonian. Table VIII lists the constants obtained for the two bands and Appendix VI lists the assigned transitions. The ground state constants given by Snyders. were used in fitting the transitions. +See Figure XIV. Table VIII. Molecular Constants for v and 03 of H S. Planar v —1 -l Hamiltonian 1 cm 3 cm * v0 2614.41361 i 0.006 2628.5473 i 0.06 A 10.20408 1 0.005 10.145 i 0.009 B 8.88882 i 0.005 8.926 t 0.004 C . 4.66161 i 0.002 4.670 t 0.003 T -0.0079 i 0.0001 -0.0063 i 0.0007 aaaa 28 Tbbbb -0.00476 1 0.00007 -0.0043 i 0.0004 Taabb 0.004578 i 0.00007 0.005 1 0.001 Tabab -0.001405 1 0.00006 -0.0023 i 0.0006 HK «5.5x10"8 i 0.4x10'8 -1.5x1o' i 0.6x1o‘7 # Lines 194 60 Coriolis Constants: G2 0.05 i 0.04 cm— G -0.269 f 0.003 cm" XY Total # Lines Fit: 254 Standard Deviation of Fit= 0.006 cm—1 *95% Simultaneous Confidence Interval equals six times the standard errors of the coefficients. 71 .-...o 85 OOON 896E no u._ :2 0m "a N m H m m mo 9 can 9 mo muuommm >m>udm .>Hx.mHm CHAPTER VI CONCLUSION This work described the vibration-rotation Hamiltonians recently developed for the asymmetric-top molecule and outlines how they can be used in fitting observed mole- cular infrared spectra for molecules of any symmetry what- soever. Computer programs were written which put this theory into practice for observed data. The programs were then used to successfully fit three different Ham- iltonians to the ground state energy levelrdifferences of the prolate XYZ molecule, HDTe. The Watson Hamil- tonian was used to fit simultaneously all five of the molecular isotopes which were observed in the (2,0,0) band of HDTe. The planar Hamiltonian was used to fit the Coriolis coupled (1,0,0) - (0,0,1) bands of the oblate XYX molecule, H28, With the number of Hamiltonians now available to the infrared spectroscopist the "best" Hamiltonian for differ- ent molecules may have to be found. We have not found advantages in using the "reduced"Hamiltonians,as compared with using the planar Hamiltonian, for triatomic mole- cular spectra. In fact the planar Hamiltonian has the 72 73 advantage of not having the restrictive conditions on the rotational constants and also fits the observed data as well, if not better than do the "reduced" Hamil- tonians and uses fewer constants to make the fit. Using several Hamiltonians to fit the same data may be help- ful in finding lines which are incorrectly assigned. This work also outlined in detail a process of digitally smoothing and deconvoluting spectra, which was programmed on a PDP-12 minicomputer. A number of inter- esting improvements were made in the method of deconvolu- tion among which are"quadratic over-relaxation," desira- bility of point simultaneous deconvolution, and normal- ization in deconvolution. Infrared bands of HDS and D28 were processed by the minicomputer using the programs developed, and enhanced the resolution of the spectra of those bands by as much as 2.5 over that of the original data. All of the computer programs for the minicomputer were written in assembly level language so that the full power of the computer could be utilized. The programs are capable of processing large volumes of points in a reasonable length of time, e.g. approximately 45 minutes per pass for 225,000 points in the deconvolution process. The data can be monitored continuously by the operator during all operations. He can take full control over 74 the value of every data point in every step of an operation, or he can allow the program to automatically control the operation. These programs should facilitate the finding of new methods and the best procedure in the deconvolution pro- cess. Currently ten to thirteen iterations are necessary for convergence in deconvolution; with the right procedure this number of iterations may be reduced. The minicom- puter facility,if properly used, could be developed into an even more useful tool. REFERENCES l. K. T. Chung and P. M. Parker, J. Chem. Phys. 38, 8 (1963). 2. J. M. Dowling, J. Mol. Secstroscopy 6, 550 (1961). 3. T. Oka and Y. Morino, J. Phys. Soc. Japan 16, 1235 (1961). 4. N. K. Moncur, Thesis, Michigan State University (1967). 5. L. E. Snyder, Thesis, Michigan State University (1967). 6. W. B. Olson and H. C. Allen Jr., J. Res. Natl. Bur. Std. (U.S.) A67, 359, (1963). 7. K. T. Chung and P. M. Parker, J. Chem. Phys. 43, 3865 (1965). 8. F. X. Kneizys, J. N. Freedman and S. A. Clough, J. Chem. Phys. 44, 2552 (1966). 9. K. T. Chung and P. M. Parker, J. Chem. Phys. 43, 3869 (1965). 10. K. K. Yallabandi and P. M. Parker, J. Chem. Phys. 49, 410 (1968). 11. J. K. G. Watson, J. Chem. Phys. 46, 1935 (1967). 12. J. K. G. Watson, J. Chem. Phys. 48, 4517 (1968). 13. L. H. Ford, K. K. Yallabandi, and P. M. Parker, J. Mol. Spectroscopy 30, 241 (1969). 14. G. W. King, R. M. Hainer and P. C. Cross, J. Chem. Phys. 11, 27 (1943). 15. G. Steenbeckeliers,Ann.Soc. Sci. Bruxelles I (Belgium) 85,163 (1971). 16. M. A. Efroymson, Mathematical Methods for Digital Computers, Ralston and Wilf (Eds.), New York, 1960, pp. 191-203. 75 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 76 K. N. Rao, C. J. Humphreys, and D. H. Rank, Wavelength Standards in the Infrared, Academic Press, New York, 1966, pp. 160-2. T. L. Barnett, Thesis, Michigan State University (1966). P. H. van Cittert, Z. Physik. 9, 298 (1931). H. C. Burger and P. H. van Cittert, Z. Physik. _9, 722 (1932). H. C. Burger and P. H. van Cittert, Z. Physik. 89, 428 (1933). E. J. Gauss, (J. A. Blackburn, Ed.), Spectral Analysis: Methods and Techniques, Marcel Dekker, Inc., New York, 1970, pp. 30-31. J. E. Stewart, Infrared Spectroscopy: Experimental ;Methods and Techniques, Marcel Dekker, Inc., New York, 1970, pp. 439-464. T. H. Edwards and D. R. Strome, Paper AAl, 1973 Sympo- sium on MoleCular Spectroscopy and Structure, Columbus, Ohio. T. H. Edwards and P. D. willson, Paper B1, 1972 Symposium on Molecular Spectroscopy and Structure, Columbus, Ohio. Same as Ref. 23, except pp. 231-251. P. A. Jansson,private communication. P. A. Jansson, R. H. Hunt and E. K. Plyler, J. Opt. Soc. Am. 69, 596, (1970). S; C. Hurlock and J. R. Hanratty, Paper AA4, 1973 Symposium on Molecular Spectroscopy and Structure, Co- lumbus, Ohio. A. L. Khidir and J. C. Decius, Spectrochimica Acta 18, 1629 (1962). A. F. Bondarev, Optics and Spectroscopy 12, 282 (1962). L. C. Allen, H. M. Gladney and Glarum, J. Chem. Phys. 40, 3135 (1964), R. H. Jones, R. Venkateraghavan, and J. W. Hopkins, Spectrochimica Acta 23A, 925 (1967). 34. 35. 36. 37. 38. 39. 40. 77 B. D. Saksena, K, c, Agarwal, D. R. Pahwa and M. M. Pradhan, Spectrochimica Acta 24A, 1981 (1968). P. A. Jansson, J. Opt. Soc. Am.,§0,184, (1970). P. D. Willson, J. R. Hanratty and T. H. Edwards, 1972 Symposium on Molecular Spectroscopy and structure, Columbus, Ohio. Test results for the grating as furnished by the manufacturer, Bausch and Laumb, when the grating was new. E. Whittaker and G. Robinson, An Introduction to Numerical Analysis, Dover Publications,Inc., New York,l967, pp.29l-5. T.H. Edwards, P. D. Willson and N. K. Moncur, 1972 Symposium on Molecular Spectroscopy and Structure, Columbus, Ohio. T. H. Edwards, P. D. Willson and W. H. Degler, Paper WJ21, 1969 Spring Meeting of the Optical Society of America, San Diego, Calif. APPENDICES APPENDIX I NONVANISHING ANGULAR MOMENTUM MATRIX ELEMENTS Listed below are the nonvanishing matrix elements for the angular momentum operators of the various Hamiltonians evaluated in the symmetric top basis w(J,K). 11/2 [f-K(Kil)]l5 X = 1/2 [f-K(Kil)];5 f = J(J+l) j = f—K2 gt: ((J;K—1)(J;K)(J:K+1)(J:K+2)]l5 mi: [um-2)(J¢K+3)(J;K-3)(J:K+4)];‘gi The diagonal terms, where x1 is equal to: P2 = K p; = p: = gj P2 = f P: = K2 (pip: + ngj) = (pip: + Pipi) = jK4 78 79 4 _ 4 _ .2_ . 2 Py _ PX — (33 23+3K )/8 P4 = f2 P4 = K4 2 P2P2 = sz 2 (P2P 2 + P2P 2) = (P2P 2 + P2P 2) = sz x z z x y z z y 2 2 2 2 _ .2 . 2 (PXP y + PyP X) — (j +23-3K )/4 P2(P2 — P2) = o y P; = P: = (Sj3-10j2+25jK2+8j-20K2)/16 P6 = f3 P6 = K6 Z P4P2 = f2K2 2 P2P4 = fK4 Z 2P 4 4P 2 _ 2P 4 4P 2 _ 2 .2_ . 2 (P2P y + PyP z) — (pzp x + PXP z) — K (33 23+3K )/4 2P 4 4P 2 _ 2P 4 4P 2 _ .3 .2_ . 2_ . 2 (PXP y + PyP x) — (PyP X + PXP y) — (3 +63 193K 83+20K )/8 2P 2P 2 2P 2P _ 2 2 (PXP zP y + PYPP z i) - %j 2K 2-[ 9+ + (K- 2)2 g_ 21/8 The off-diagonal terms, where x1 is equal to: (PP +PP) x y y x ll H- H' \ N LQ .... 80 p; =-p: = 4(3—212K)gi (P5P p: +pipj) = “(Pip Z + P2 zPi) = %(K2i2K+2)gi P2(P: - Pi) =—;5fgi [P:(P: - Pi) + (P: - P§)P:] =-(K2¢2K+2)gi P: =-p: = [1532;603K—703+105K(K¢1) :125K+136]gi/64 (pip P; + P3P pi) = -(p§p p: + pip§)= (3 23:43K+63- -41K2;102K- -72)g_ /32 (pip p: + PfiP Pi) = —(P:P : + pip:)= P[K4+(K:2)4]gi (pip p; + pgp pi) = —(P:P: + P:P:)= 15(K212K+2)[f—(K2:2K+2)]gi P4(P: - Pi) = ~15f2gi P2[P:(Pi - Pi) + (pi-p p§)p:] =-(K2:2K+2)fgi [P:(P: - Pi) + (Pi-P P;)P:1 =-[K4+(K:2>41gP/2 The off-diagonal terms, where x1 is equal to: 4 4- _ PY — PX — mi/16 2P 2 2 2 _ (PXP Y + P yPX) —-mi/8 p6 = p6 =(3' $12K-20)m /32 y PX J i 2P 4 4P2 _ 2P4 4P 2 _ 2 (P2P y + PyP z) — (PZP X + PXP 2)— (K i4K+8)mi/8 2P 2 2 2 2P 2 (PXP zPy + PszP X)=-(K+2) 2m i/8 4 P2 2 4 4 2 . = + = _ (p2 XPY + P; PX) (PyPX PXPy) ( 3+4K+4)mi/16 81 The off-diagonal terms, where x1 is equal to: Py =-PX = [(J-PK-S) (J-T-K—4) (JiK+5) (JiK+6)]%m+/64 2 4 4 2 __ 2 4 4 2 __ _ _ _ (PXPY + PyPX) — (PyPX + PXPy) — HJ$K_5)(J+K 4)(JiK+5) (JiK+6)]2m+/32 APPENDIX II JACOBI METHOD OF DIAGONALIZATION Consider a Hermitian matrix A, which we desire to diagonalize all a12 a13 "° alN a21 a22 ... A = a3l o o o aNl If for i # j, there exists an Iaijlmuch greater than any other off diagonal element, we may neglect all but ai. and diagonalize the two by two matrix aii aij cl b a.. a.. b* c 31 JJ 2 The eigenvalues of a Hermitian matrix are given by Determinate (a - A Y ) = 0 (A-II-l) mn r mn The eigenvectors are given by n2 - ArYn£)S£r = 0 82 83 1r 2r mm where is the eigenvector. U) 000 Nr It then follows that the unitary transformation matrix is S11 s12 s13 "' SlN s21 S22 "' S = 531 no. SN1 which diagonalizes A, i.e. S+A S = D where D is the matrix of eignvalues, which we will denote by A. For our two by two submatrix Eq.(A-II—l) becomes _ .. - 'k = (Cl 1) (c2 1) bb 0 which has the solutions 1 = (c +c )/2 + [(c -c )2/4 + bb*1g5 (A-II-2) i 1 2 ‘ l 2 If U is the unitary transformation matrix which diagonalizes the small submatrix then We get the four equations which follow from diagonal- ization 84 (c1 - Aim:i + b8i = 0 (c2 - 1i)8i + b*ai= o (A-II-3) From normalization, i.e. U+U = l, we get 2 2 _ a+ + 8+ — 1 a3 + a_ = l * a+a_ + B+B_ — 0 (A-II-4) These equations may be solved for d+, a_, 8+, and 8_ except for an arbitrary phase factor. We will choose that factor so that a+ and B_ are real. Let us define -c2)2/4 + bb*] (A-II-S) R = (cl-1+) = (cl-c2)/2 - [(cl then it follows that (CZ-1_) = -R and from Eqs.(A-II-4) and (A-II-S) we get m + II -Ra+/b R8_/b* (A-II-6) Q ll Using Eq.(A-II-6) and Eq.(A-II-4) we get Q II + s_ = (1 + RZ/bb*)% * e = -a_ = -(1 + Rz/bb*)%R/b (A-II-7) 85 In summary: c b A 0 a -8 c b a 8 U1. 1 U: + = * i 'k b c2 0 A_ B a b c2 -8 a where 1+ = cl - R 1_ = c2 + R a = (1 + Rz/bb*)—% B = aR/b R = (c -c )/2 - [(c -c )2/4 + bb*]% 1 2 1 2 If the condition a.. is much greater than any 13 other off diagonal element does not hold, one may use the above method iteratively to find a series of trans- formations, each of which will cause the largest remaining off diagonal element to go to zero. The iteration can be carried until the precision desired is obtained. This is the Jacobi method of diagonalization. Each submatrix in the diagonalization affects other off diagonal elements as well as the elements in the sub- matrix. Let us now derive that effect. 8 Consider the 4 x 4 matrix all alz al3 al4 A = azl a22 a23 a24 a3l a32 a33 a34 a4l a42 a43 a44 Let us choose a rotation involving the a l 0 0 0 0 a 0 B _ $1 = I 811: 0 0 l 0 0 -8* 0 a so that * all (aalz 8 a14) (aa -Ba ) (a -R) A'=S+AS = 21 41 22 l l a (ma -B*a ) 31 32 34 * (B a21+aa4l) 0 6 24 al3 (aa23‘8a43) a33 (Ba23+aa43) term. Then (8a +da 12 0 l4) (Ba32+aa34) (a44 +R) Initially the transformation matrix is the identity matrix. So that initially _ f then + + l _ A — SISOAsosl , and iterating + . _ + 2A 82 - 823 H-+ .1. SOAS S S A = S o 1 2 so that finally _ + AT — s A s where s = Soslsz"'SN We see, therefore, S11 S12 S21 S22 50 = S31 S32 s41 S42 then Sll S _ _ 21 s — 5051 S 31 S41 for the case at hand, S13 S14 S23 S24 ’ S33 s34 S43 S44 (aalz’a*al4) (a522'B*324) (a532’B*a34’ 87 * “"542"8 S44) S13 S23 S33 S43 Generalizing from the example we for a rotation involving the a (“512'8 Sik) s! = 13 S.. 13 { (Bsi£+asik) 2k term if if if j j j if (8512+asl4) (8522+a524’ (8332+a334) (8542+aa44) get the relations (1 < k). II 2° II x 88 ( o if i = 2,k; 3 = 2,k (aPP-R) if i = j = 2 (akk+R) if i = j = k aij = [ (aa P-s*aik) if 3 = 2; i ¢ £,k (BaiP+aaik) if j = k; i # £,k a3: if i = 2,k; 3 y £,k L aij if i g 2,k; 3 ¢ 2,k If computerized, the diagonalization cannot be im- proved once 1+ and 1_ do not change, that is when 4 R — c2 where C1 > c2 and N is the number of bits which store the '2 mantissa of c2 in the computer. Eq.(A-II-S) can be written as R =[(cl-c2)/2][l - (1 + 4bb*/(cl-c2)2)a] Since R + 0 as bb* + 0, for small R N_ * - R — bb /(c1 c2) Therefore, the most natural place to stop diagonalization is when * bb < c2(cl-C2) for all b's. l) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 89 The necessary calculations are BAB bb* CDF = (cl-c2)/2 R = cup - [(cpp)2 + BApla (l + Iva/BABY!5 mode real real real real complex real real ALFA = BETA = (R)(ALFA)/b cl-R c2+R ALFA * S(I,L)-S(I,K) * CONJG(BETA) BETA * S(I,L)+S(I,K) * ALFA BETA * A(I,L)+A(I,K) * ALFA ALFA * A(I,L)-A(I,K) * CONJG(BETA) complex complex complex complex APPENDIX III LISTING OF PROGRAM ISPECFIT )Rlo34 90 05/03/73 PRDSRAN ISPECFIT 130 CONNDN/BLKp/P(1934O25’1PI(2025)*NP(3’0PH(4925) COWfloN/3LKAP/AP(1505OT) CON4DN /PMAK/HAMPVISZIVIBXYIMSPMILPS CONNDN /BL5/AODATA DBSNO.AVEWHTP NOVARS LNOUTPSIGDIF .NFXT. 1XOUT.!'30EN,IFSTEP.IFPRen,SIGMAY1,DELSIG DIN: NSIDN IHEAD(10) C0440N A(25.25) 5(23. 25),AS(600).JIN(2:2503)PINDEX.NVAR(24.2)pIBN CDMADN/BLKl/PAR 19. 4, .v 3,.NPAR 24 4 EQJIVA- ENCE (NPAR(1).NPA§1)C(NP4&(49);NPAR2) EQJIVA. ENCE (PAR(1)pPARl)o(PAR(20).PGDl) (PAR(39). PARz),(PAR(58), 1PGDZ) DIN: NSIQN DA7A(36)030NST(30) CON4DN/3L3/NT(1 09),DELPAR(35) TYP: (INTEGER HAM C CONSTANIS 44ICH ARE DEP: NDENT ON THE MAXIMUM ARRAY IN COMMON BLKP NODATAso NH84325 JMIVBO 5 JMAX=49 C 35 PARAMETERS MAY BE VARIED END’HLE 50 END’ULE 51 END’HLE 52 D0 4'IS1P76 4 PARTI)!O. 115 CONTTNJE SIGNPIB. NFITii : C MSPM MUST BE GE 1 T0 HRITE oUT MATR‘X ELEMENTS IN PMAK5,175 VALUE Is THE L; C NUMBER PMA.v3(3),(PAR(1,4),1E4,19’ FORRPT (SE15. /6E1g 0/1 [8 I, CONTINJE NVAQKZUoi)=NVAR(23.?)=NvAR(21.1)=NVAR(21 2):: D0 1? I: 1119 DO 12 J: 20402 IJIJ/z NvARKI IJ)= IF (PARII.JII NVARII. IJ)81 CONTINJE IF (IFNOVAR.NE. ) GDTO 616 C INDICATEIPARAHETERS T0 VARY 5 5 606; REA):515.(INPAR(1,J),1:1,24),J=1,4) FORRRT (2412) CONTINJE NGS=0 P NOVARsl DO 1P J31. 44 6 1:9.2 IF((FU- L. E0. )GO TO - IF((!.LY.1U).OR.(I.3+.19))Go To 14 IJ:(J*1’/2 IF((J.EQ.1)GC T0 22 K:J.2 NPARMI.J,=NPAR(I;K, IF(VPAQ(IaJ).NE.U)NVAR(I.IJ)=1 )R1.3A 92 05/03/73 GO ID 16 22 K=J02 NPAIK1.JI=NPAR(1;KI+NPAR ?1.3A 36 38 94 05/03/73 JAG\.IN-1 CAL .BYHDIG (A IER) CAL IFJRMEPl (IFPAV, NMAx,JEO PAR1,P. NP CAPP) JEOS4 ISA4:1vDEX ; CAL.IF3?HEP1 (-1;NMAx.JEO.PAR1.P.NP.CAPPI IF (ISAV.EQ.INDEX) 30 To 38 INDEK:ISAV S IER=IPRT IF (JASAIN.EC.U) CA-L PMAKE (J.JE0.NHAX.PAR1,P.PI.NPI N=0 CAL tHweT (AINIJEO JAGAIN NHAX PARI. PH PIN?) CAL .IsanIG A IER CAL .IFJRMEPI IIFPAv,NMAx.JE0.PAR1.P.NP.CAPPI J=Jb 60 T 19 C GROUND STATE ENERGIES ARE DONE AND STORED IN A3 50 C COVTINJE INDEFIVDEX COVTINJE LPS=LPSZ INDEksIVDE JBAVDa34 UPPER IBV=2 C CALCULATE T45 UPPER STATE ENERGIES 33 CAP°FCA°PAIPAR2.JBAVDaHAM:IHEAD NPARZ .PGDZ.ISOI IF (NPfi?l22.4).OR.V I IIPRINT197 V? I ).NPARI22:4) IF (NpAaczs 4).0R. v IIIIPRINT 108. v: III.NPARI23.4I IF (NPAR(24 4,.09. v (3))PRINT 109 VS‘S).NPAR(24 4, IF (SI3DIF. GE. DELSIS) GO TO 100 IF INFIT. GT. AFITS) 30 To 10 IF (NFITIE0.AFITSI IFCOEN=1 IF (NFIT.EQ.AFITSI IFPRED=0 J=J1HN IF (J.LT.U) i=5 COVTINJE N=0 JAGAIN=D JEo:I s NPI1I=J S N°I3)=J/2+1 ISAISIIDFX CAL-|F3RMEP1 I-1;NMAx.JEo.PAR2.P.NP.CAPPI IF IISAV.EQ.INDEX) so To 40 INosk-ISAV s IER=IPRT CAL-l =IAKE (J.JEO.VMAX,PAR1.P,PI.NP) CAL-IHAVGT CAL.IFJRMEP1 IIFPAV,NMAx.JEo,PAR2.P,NP.CAPP) JAGKJNsl JED:A ISAJhIVDEX CAL.IF3?MEP1 ('1.NMAX.JEO.PAR2.P,NP.CAPP) IF IIsAv.Eo.INDEx) so To 44 INDEKsISAV S IER=IPRT IF‘(JASAIN.EC.C) CALL PMAKE IJ.JEQ,NHAx,pAR1,p,p1,Np, N=0 CAL_'NAVGT IA.N.JEo.JAGAININHAX,PAR2.PH,P.N=> CAL.IsvaIC(A.IER, CAL.IF3?MEP1 IIFPAV.NMAx.JED,PAR2.P.NP.CAPP) J=Jo IF I5.SI.JMAX )GO T3 9a GO ID 33 UPPER STAES ARE CALCULATED ONCE covrINJi FORIHI (toNUFBER 0F LEVELS CALCULATED a .163 PRIIW 10L1.IADEX RENIND 51 SUHIHT=0.o [=0 NTCI C FIND THEIG.5.C.D. AND P_ACE THEM ON TAPE 60 61 C CALCULATE TRANSITION FREQUENCIES AND COMPARE THEM TO OBSERVED FREQUENC;L 1:191 REA33(51) JU.KNU}KPJ.JL,KNL,KPL,OBSFRE.HT2.ISYOP,IDAINDUIINDL IF (50:951)67,61 CONTINJE ISTRHISD-ISTCP ILsJINIIgINDL) s IU=JIN(2oINDU) FREDBAS‘IU)~AS(IL)*V0(1)*VU(2)*ISTP¢VC(3)'ISTP'ISYP Do 52 IJ=1.2 IULFdL IF ‘lJOEQ|2) IUL=IU ' DO 52 11:1.19 62 NsuvARIII.IJ) S IF (N.Eo.0) GO TO 62 CONSN(v)=AP(AnlUL) 1231J*2 FREJBFREQ+ISTPwPAR(II.12)*CON$T(N) COVYINJE R1.3A 96 05/03/73 DATAINJVAR>=CBSFRE ~FREQ IF (WTCI)) 985,985.982 982 DIF’EaABS (DATA(NOVAR)) IF (DI’FE.LE.BDLN) 30 To 986 984 NTIVfl+1 IF (NT.3T.1) GO TO 983 PRIIN 228 983 PRIIN 236.JUaKNU}KPJ.JL,KNL.KPL:OBSFRE.FREO,DATAINOVAR).HT2.I NT‘I).OOD 985 OBSqDaDBSND-i. 986 NUfizfi SUMJHT =SUMHFT +NT(I) Do 54 IJ=1.4 ISFtl IF ((IJ.EQ.2).OR.(IJ.EO.4)) ISF=ISTP DO 54 11:1.19 N=V=HRIII.IJI $ IF (N.Eo.0) GO To 64 IN: (1J61)/2 $ NN=NVAR 5 IF (V.E0..)Go To 69 DATK.(q) = ISF 69 ISF;IS’*ISTP HRITE (SC) JL.KNU.K=U.JL.KNL;KPL.OBSFRE .FREO: 1 (D‘IA‘LD'L=11NOVAR)0IOHTZI[STOPOID GO to so 0 FIT THE DAIA 67 COHIINJE OBSVD x NODATA COBSVO AVEJHTISUMNHT/OBSNO RENIND 5t REHIND 5: DO 108 1:1.35 108 DEL=HRII>=0. CA ’[RESRESS NFIf’s VFIT+1 Do 70 1:1;4 DO 7D (31.19 N=V3HRIKII) 3 IF (N.EQ. I Go TO 70 PARIK;I1 a PARIK;I,+DELPAR(N,APARFRAC 7o covrINJ: Do 76 I:a,3,2 NaV’HRIgn.I) 3 IF IV.E0,.) GO TO 74 PARI1.I) : pARI1;I) 6 DE PARtn)*PARFRAC/2.0 PAR(2,I) , PAR(2;I) t 05 PAR(N)*PARFRAC/2.C 74 N:V=AR<21.I) 5 IF (V.E0,u) GO TO 76 PAR(1.I) = PAR<1;II . DELPAR(N)*PARFRAC/2.U PAR(2.I) = pARI2;I) . DELPARIN)*PARFRAC/2.C R1.3A 76 230 97 05/03/73 CONTINJE DO 30 1:22.24 N=N=HRII;4) 5 IF (N.EQ. I Go To an K=I~21 Vfl¢<§:VDIK) . DELPAQIN,,PARFRAC COVY‘NJE 818 8 ASS((SIGMA-SISMAY1)/SIGMAV1) SIGfifiaSICMAYl IF (NGsoEG. ) GO TO 2 GO TD 1' CONTINJE GO TO 115 FORIRY {/BXt VC 3 *F1”.3a6X.!2.dUXtBAND CENTER.) FORQRT(*~*29X.512.5o13) FoaflfiT (9X'DEVC = r EEX,E 5.5.13) FORAHTtt1 UPPER*5 . ONE t7XtCA C CD*3X*DBSERVED CDaBXcNEICHYw) FOR4AT (//i THE EIGHTS 0F *HE FOLLOWING LINES HILL NON BE 5 1ET ID ZERO*/* UPPER J K- K+ LCHER J K- K. w7XtOBSERvED FREQUENCY 1. CIICALC FREQUENCY67X*DIFFERENLEtlzxthIGHT*) ECEIAI (2(6X.313).7X.E16.9 .9X.F16.9 .7x.510o3:4on1o.2.110 ) N R1.3A 98 05/03/73 SUBROUTINE PPAKE (J JED NMAX PAR.P PtoNP) COMAON IPMAK/HAM.VI32.VIBxy.MSPM.LPS THIS PROGRAM CALCULATES THE MATRIX ELEMENTS OF THE OPERATORS IN J K SP C THE DIM: ~N$ION OF P MUST BE 9*4aIJM XI 2+1) DIMENSION P(19.4}NMAX)I%I(2.NMAX).NP(3) DIMENSION PARI3) TY’E INTEGER HAM C ZERO TAE 3 VATRIX M=1 MMINMAX019t4 DO 5t!:M’.MM 6 P(I)=o. o MMCZFNMAX DO 3:181,MM 8 PIIIIIJQC N=0 IFIJ El C)GO TO 386 THE K :NEN ELEMENTS ARE CALCULTED +ND STORED S= :PATRATE FROM THE ODD GO TO (L 2 4. 4) JEO K IS ElEN KEO:(J/2)t2 GO TO 5 K Is 03D KEO$£(J*1)/2)*2-1 PPF:U¢(J*1) K 8 ME) +2 NKI so To 37 C OBLATE PLAVAR FCRM R I PARI2)/(PAR(1)+°ARIZI I RIRtR S I PA?(1)/(FAR(1)+=AR(2)I 3:808 360 NK ==N<-5 KsK-E PKa(hK KN 81-4 IF (KNoLflul) KNSI IF (KN.3T.4) KN :4 GO ID (362)3641366038C, KN 362 RPM = aPF-PK C CALCULATE THE DIAGONAL TERMS IF (NK.LE.O) GO TO 383 GO UIb'O NO 0 P<131;V‘I = o5*PPK P<2.1.NI S a 5.3 371 NKaMKp1 KSKIZ PKI = PPF*°K*PK P11 IINK) = PK'DKPPK PI! 1. VK,=K1VIBZ IF ( (50 E0. ) 30 To 38; IF (KEJ' KI 2660260 264 PPKCPP7-PK 266 G=(JpK-1)*w(K+2.)w(K+a. )v(K¢2.))a{~1.) P11 2 VK)= .5*VIBXY*SQRT(~(J-K}t(JaKt1)*(J¢K*2) PPG 8 SQRYCG) 20 PM 8:CJ-K-3)*(J-K-2)* Go To 38o IF (K53 .K-4 ) 24.1 PPK:PP’-PK G ' (J-KP1)*(J-K)*(JoK¢1)t(J¢K+2) PPG a SQRT(G) PM ==(J- K 3).¢J- -K- 2>¢(J¢K+3)¢(J+K+4) PPM I 9PG*SQRT(PM) PN t (J K-STttJ- K- -4)O(J+K*S)t(J+K+6) co TD (75 71’ LP COVI’TNJE‘0 PPM : aPMSGRTIPN) NKaVK¢3 P(13.4 VK) a - .5-PPVt.03125 PT11.4.VK) : -P(10 4. NK) P(14,4.NK) 8 .03125tPPN NKIVK;3 COVT1NJE‘. IF (K.ST. -2) 60 T0 10 c qur FORILAST TAPE OPERATION To COMPLETE 380 26 3o CONTTNJET _ _ NP(138J S NP(2)!JEO S NP(3)=N COVYVNJB IF ("SPV LT.1) RETURN IF (UNIT H59") 26 3 BUF’ER OUY (PSPM.1) (P(10101)0NP(3)’ RETJRN END 3R1.3A nannnnn CO 104 suaaOuTINE REGRESS COMVON/SLKAP/VECTOR(37.37).DATA(36)g AVE(36) SIGMA(36).C0EN(36)3 1S!GTCOT36) IKDExcaé), AP(15. 600) C0qqp~ xehsxnouATA. .oasNo.AVENKT NOVAR.&N0UT.SIGOIT .NOPROB. ixour 1:00 N,IF$TEP. IPPREDoSIGHAY1.DELSG COHTDN/3L3/HY(1 on).DELPAR(35I EQU!VA ENCE-= VECTQR (IgJI . IVECToRII.NoVPLI w VECToR (JaNOVPLI -/ IEgTOR (NOVPL} NOVPL)) DO 590 1 B 1. NOV‘R AVEIII - VECTORII.NOVPLI / VECTORINOVPLJNOVPL) IF (ITAVEI 9o..735.71o PRINT 50 . PRINT 20, II.AVE(II.!-1.NOVHI) PRINT . g 25, AVEINOVAR) IF (IFTEBD) 9 0.780.740 PRINT 55_ RRtVN' 35,(‘proVFCYORVITJ)0J!1INOVV1’I PRIVR 4U, (I‘VECTOR(19N0VAR)¢I91ONOVH1 PRINT . 45, VECTORINoVARaNOVAR) NOSYEP gtfii ASSION ‘132 T0 NUMBER DEPQIa QRVECTORINOVPL.NOVPL) Do son I,-, .NOvAR IFIVECYORIIIVI) 792,794,810 PR1VR 7931 I GO TO 915 RRiVW 7950 I SIGTIQII a 1.0 so To 300‘ . _ SIGTIIIIJQDSCRT (VECTOR (1.1)) VECTORIIIIIT’1.O no 386 I-- 1.NOVNI 1P1 _I o 1 6 no 330 Jsa 1P1. NOVAR w. - VECTORII;J)-q vEcToRII.JI /( SIOKAIII. stanI VECTDRIJII) a VECTORIloJI IP (IPSOEN) 9?0:951 .67: PRINT 60 NONNE s NOVTI - 1 no sea I a 1; NOVMZ [P R 1R. 1 . PR NW 35,(‘oJOVECTOR‘10J)0J31P11N0VM1’ PRINT 40,(I.VECTOR(I.NOVAR).I=1.NOVMI, COVYINJE GO To NEXT.<10331531) NOSfEP QINOSTEP d If (VESTOR(NOVAR;N%VAR))1002010 201915 NST’Hi . NOSTEP . 1 I I 3 32 3333 II 33331! 33 I 0R1.3A 1015 1016 1017 1010 020 030 1035 1040 1041 1042 1060 1080 1090 1100 1120 1150 140 ‘ 150 904 1190 7a 1180 1190 1110 1210 1220 1050 1230 903 1240 1260 1245 153‘; ‘1250 1270 290 280 1300 1310 1311 131% ,1314 1315 106 05/03/73 PRINT 1004. KSTPN1 GO T01915 DEFRIIOEFR-1.- IF! WpEVT.NE u) DEFR= DEFRIZ. 0 IF (DE’R ) 1 17,1017. 1010 PRINT 1019 .NOSTEP GO TO 015 SIGT a SIGMAINOVAR) IDSQRT VMIVII 0.0 VMAT‘ g 0.0 NOIVII 0 1 DO 1053 I = 1 NOVHI IF (VESTOR (I I), 1 42010500106 PRINT 1044. I. NoST: P GO TO 915 IFIIECTORII. I) - TO ) 165 .1090.1080 VAR I VECTORII NOVA?) * VECTORINOVAR I) / VECTORII I) IFIIIRI1100.1 5 .1110 NOINI, NOIN . 1 INDEKINOIN) I I COEVKNDIN) ' VECTORII.NOVAR) t SIGMAINOVARI / SIGMA (I) SIGTCOIVDIN) a ISIGY I sIGMAII)) *DSQRT IVE; TORI! III IF (VMIVI 116 011700934 COVTINJE PRIVT 906 GO TO 915 IFIIIR . VMIKII 50.1050,11W VHIVII VAR NOVIN I I. GO TO 1050 . 1 IT (N12 . VMAXI1050.105 .1210 VHAKI! V‘R . NOVKX I I COVIINJE IF (NOIVI 9 3.1240.1245 COVTINJET PRINT 907 GO TO 915 PRIVT, SIGTAY‘ISIGY GO TO 1356 IF( 1?: VS?) 9 0.1246.1250 CNST I 0. 0 00 TD 1300 , CNsT'g AVEINCVAR) D0 1233 I = 10N0IN J I INDEX‘I) CNST'I CNST - IcoENIII 0 AVEIJII F IFSTEP, 9 0.132 .1310 INOEVT) 1311.1311.1313 (VECTORINOVAR.NOVAR)/ DEFR) 65.SIGY PRIVW 91,NOSTEP,K GO To 3 4 PRINT 1 1 92 NOSTEP.K PRINT 70 FLEVEL, SIGY, CNST. 1(IVJEXIJ1000EN(J)0 SIGMCOIJI.J=1 NOIN) GO TO VJMBER: (13209158 ’ TY'Z777??? 7 1777777? 7777??? ‘7 "Z L.SA 1320 1330 340 345 1350 360 361 1370 1390 1391 392 393 '1400 1420 1430 1450 1460 1440 1410 1470 1490 500 48g 1510 1530 540 520 1550 1560 1380 38 1570 1571 1530 949 1581 583 K a NOEVF I 0 00 TO FLEJEL s VMAX . DEF? / (VECTORINOVAR.NOVAR)~- IF (EFIV n FLEVEL) .1391 IF (EFIVI 138 .1380.137: K , Noflkx NOEVN I K IF¢<0-1392.1392.14o CovrINJE Patvn 1395. 00375? GO TO 915 00 1010 I-s 1,N0VAR IF (1-<) 1430.1410.1430 Do 1943 J =_1. NOVAR IF 4.4) 1460.144o.1460 ,VECI 6(K01C’ COVTTNJE covrINJE“ 00 198) x = 1. NOVAR IF (I.() 15 0,1480.1530 VECTOR (I.K) CovIINJE DO 1523 J = 10 NOVA? IF (J-(I 1540.1520.1540 VECTOR1‘1J) : COVTINJE GO TO 1000 PRIVU 75. 107 FLEIEL 3 VMIN * DEF? / VECTOR (HOVAR,NOVAR) IFIEFOJT t FLEVEL) 1350, 1350; NDTIN I344 1370.136101SRO 05/03/73 VHAX) 0011;0> a VECTORIToJ) . (VECTOR(I.K) . VECTOR (K.J> / VECTOR , - VECTOR (1.x) / VECTOR («.K) VECTOR (KgJ) / VECTOR (KaK) 1.0 / VECT0R(K0KI NOSTEP 1r (IrerP) 9 0.157 . 500 ASSIGN 00 to 90:00 IF (TFCDEN.NE.0) GO TO 1531 158 1310 IFCDEN11 ASSIGN 1581 T0 NEXT GO TO 565 IF( TF’QED) 2 COVTTNJE PRTTN To NUqB R . _ 1536.(L;vefiroa(L.L).L=1.Novnz) DELSIGI‘BS(SIGMAY1vSIGY3/SIGMAY1 00 949 1:1.NCIN K . INDEXII) DELanR¢<>=005N.15709.10 NRJN GO r01590 3 COVVINJE REleD 50 c ENAVDARD DEVIATI3N FROM RESIDUALS VARJEV 8 (VARDEVtSNDATA)/(SNHT* 9 F15. 6: 6H x1I2. 76) vs chg HI 9 F 5. 6. 2 . 6H x<12. 7H) vs x112. H) 6 F15. 40 F0916? (1H 76 x 12.12HI vs Y =F15. 6. 1 6H x112. 12H) vs Y =F15. 6, 2 5“ X(12. 2H) VS Y 3F 45 FOR 6? (1H 21H 9 VS V =a'156 6I 50 F0916? 11H063H AVERAGE VALUE OF . VARIABLES/I I 55 F0916?<160776 RESIDUAL SUNS 0F SDUA .RES AVD CRCSS PRODUCTS//) 60 F0916?11Ro696 PARTIAL CORRELATI .0N CO:FFICIENTS//) 65 F0916? (25H STANDARD ERROR 0F V = 512,5I 7o F0916? (11H F LEVEL 512.5/25H STANDARD ERROR or V g E12.5/12H SONSTANT E12. 5/55H VARIABLE GDEFFICIENT STD ERR 209 IF :0EF // (16H x- ~13.816. 9 3X.E16. 9II 75 F0916? ( 0H COMPLETZ— D 15.20H STFPS OF REGRESSIONI 80 FDRAAT (8H F5.D .2H F12.S .3H F12. 5. 2H F12. SI 85 FOR467(//4X. 5HUPPER.6X.5HLDNER.FXwPREDIcTED- 7x «OBSERVEDt 5x tCAL 1CULATED~ 4x~CBS- -CAL3¢3XRDEVIATIONtSXUHEIGHTBZXtATOM IDtZXtNRUNt) 9o F0916? I22H15TEPUIS: REGRESSIDN //12H PROBLEM N0 I // 3H N0 0F 1DATA . I5 //18H N0 3F VARIABLES E I10 //30H REIGHTE 0DEG EES 0F FR 2EED3M 3 Fig.2 //28H F LEVEL To ENTER VARIAB E 3 F12. 9 //29H F LEvE 3L 731R:' 1DvE VARIABLE a F13. 9 /// I 91 F0916? 19HUSTEP No.15 I19H VARIABLE REMOVED IBI 92 F0916? (9HQSTEP No.15 72 H VARIABLE ENTERING IBI _ 93 F0916? (5H RUN F6. .3H F10.5.3H F13.5.3H F10,5,3H F1”. 5 3H F10.5/(14H F1 .503“ F13!503H F10.5.3H F10, 5.311 F10. SII 654 F0916? (31H zERo NJMBER OF DATA so LONG.) 793 F034“? (31H ERROR stloUAL SQUARE VARIABLE ‘4' 31H Is NEGATIVE. PROB 1LEH TER1INATED I 795 F0916? (1Ho13 H VARIABLE 15.13H Is CONSTANT I 905 F0916? (42H ERROR IV CONTROL CARD. PROBLEM TERMINATEDI 906 FoRAAT (25H ERROR, VMIN P US $0 DNGI 907 F0916? (26H ERRDR.VDIN H NUS. S LDND I 918 F0916? (xxx/a THE wz' IGHTS 0F TREFDLLDHING LINES NILL Now BE SET T0 1ZER3lt/(2015//) I 1304 F0916? (1H0 37HY SDUARE NoN- PDSIIIVE TERMINATE STEP I SI 039 FOR16T I H029H ND MDRE DEGREES FREEDOM STEP I 5 I in 4 F0R16T (1Hu10H SQUARE X-I5.17H NEGATIVE. So 0N0 15.6H STEPSI 1395 F0916? 12H K- . STEP :6. 7H SDLDNGI 1586 F0916? 24H DIAGONAL ELEMENTS //2 H VAR.N0. VALUE/x 1119 I 7; F16 6II 654 FOR4‘T ‘***1U X,2Ho-) 1656 F0R46T (99910 X. 1H*) 1657 FORflkf 6 M91 1X.1H6) 1659 F0916? 2?2x.3I3I. 3 15.7.2F12. 7 F11. 2.2x.I4. AS. 151 670 F0916? 1212x.313I.3 3'2FTZ .4 F112.2x.!4.A5.ISI 790 F0946? (1 STE. DEV. SAFEU IEA ED IRDM ow HEIGHTED INES - *E12.5) 1000 F0916? (.aSINCE THE PR TED FREQUENEIES ARE NOT ALCULATED.N0 1HE13HTS RILL BE ALTERED THIS ITERATIDN.«I Mr’f HF: HPI H91 HF: HWF MD‘ NJ! MPI HP] r ,. in) Q 1') .J 1 \ ‘I f 3 \- H 1 MI M I H I M i M I M I H I M I M i M I H I LJ 591.3A 05/03 73 END 110 ’ 31.39 10 111 05/03/73 FUVSFIDV CAPPA (PAR.JBAND.HAH .IHEAD.NPAR.PGDEL.ISOI DIMENSION x121I.D(11.19I DIHENSIDN PAR<19I .IHEAncluI .PGDEL119I,NPA9<24.2I TYREIIITEDER HAM;D.x DATA.(TX(II.I=1.19I=BHUA , x ,.BH B . V =.8-1 c g_z =.8H0 . T1 =. 18H 2i=.86 T3 =.8H T4 =,8H TS =.8H ' T6 8.8H0 H1 =. 28H 43 :,8F H3 3,8H H4 =,8H . H5 =,8H H6 3,8H H7 :. 38H 4 3:8H H9 FISH H10 =08HSIX*Y’/2 08H (X'Y’lz ) DO 10 131011 DO 10 J:1.19 DII.JI-BR A 0(1.1)28HPX2 S 0(1,2)=BHPY2 $ DI1I3):BHP22 DI4.1):5HPLAAAR C FINDS CAPPA AND PRINTS 3UT THE STATE CONSTANTS 428 420 426 20 121 22 122 23 123 40 250 4 IF (PA? 1I.GE.PAR(2)IGO T0 428 PRIVT 2 1 API’RRIZI S PARI2)=°AR(1I S PAR11I=AP CDVTINJE API°IRIII BuPAfigz) C:PAR(3) x<1I=81oA = x = $X(2)=8H 8 = v = s x<3I=8H C = z = D15.1I-BHDBLATE IF 1B.3T.0I 00 To 426 0.3 s 3,6P s AP=RAR(3I x<1I=a1DB = x = s x<2I=aH C = Y = S x13I=81 A = z 9 D(5.1I:5HPROLATE ' IF IB.EE.API Go To 426 P9110221 . PARISII’ARI1) $PAR(1)=AP B=A3I$ AP=PAR(3) CA°?61(2.*B-AP-C)/(AP-CI. IF ‘JB‘VDDNEOBH GRDUND) GO TO ?0 PRIVN 217 PRIVT 218.116EADINI.N=1,BI 00 T0 121.22.231 HA1 PRIVW 121 F0916? (*UPLANAR-7 =ARAMETER HAMILTONIAN.HchUR-TRESIS .I DIS.1)33H GO I0 40 PRIVI 122 F0916? (toKFC-FORM.YALLABANDI AND PARKER.J.S.P.649.4101196BI6I 013.1IxSHKFc-N0N GO TO 10 PRIVN 123 F0916? 10 P-FORM.YALLABANDI AND PARKER.J.3.P..49.41o(1968ItI PRIVN 353.JBAND.ISO F0916? (1X.A89 STATEwlux *VARI (913*-ISOT0PE VARI.6/* CONSTANT VALUE19X*ABLE HA 1 1$SIEFFECT ABLE OPERATORQI GO TO (41:42.43) HA9 IF (CA’PA.LT. ) GO TO 51 C PLANAR 38 AVE 011.4 :3911/4I119 S D(2.4)=8HX4 . 926 s 013.4I=BRP24 6 Rt DI4.4)35H(PX2*P22 S 0(5.4)=8H * P229? 0 016.4Isaux2II Q1.39 112 05/03/73 0(1. 5)=5H(1/4)*(P S 0(2, 5)=8HY4 a 32* S DISI5I=8HPZQ D So D(4.5).8H(912.PZ2 0(166)=3H(1/4)*(2 DI4.6)83HPY2 9 FY $ 0(5 5I- 8H P22.P s s 012 6I= -BR«R« «SaPZA S S 0(5 D 6.5,:8HY2’) D 306,38H * (PXZ. .6)88H2*PX2) * $ D(6.6)8 8H58IPX28 D(7.6)83Hp22 + pz 5 018,6)38H219X2) 6 S 0(9.6I=8HR*IPY269 0(1I.6I.BH22.P22.P DI1.7II3H(1/4)*<2 D(4.7IIBHPx2) - 2 0(1.10)88HP6 0(1113I38HP26 GO TD 60 51 CONYINJE C PLANAR PRO.ATE DII.‘)85H(1/4)*(P D(4.4)85H(PZZ*PY2 0(115)83H(1/4)*(P 014.5)38H(PX2*PY2 D(1.6)83H(1/4)*(2 DI4.¢)=8HPX2 + PX DI7.6):8HpY2 6 pY D(1Jhb):8HY2.PY2tP 0(1. 7)85H(1/4)*(2 0(417):8HP22) - 2 DI1.10)88HP6 D(1.13I:8HP26 GO ID 50 42 CONYINJE C PHI-FOR“ D(1.4)I3HPX4 DI .5Ia8HPVA DI .7I=8H(PY21P22 0(168)88H(PXZ*P22 011.9I83HIPX21PY2 011.10I=BHPX6 0(1. 13I38H1PX21PY4 D(1.14)88H(PY2*PX4 DI1.15)88H(PY2*P24 DIl.16)88H(p22*pY4 D(1.17I38H(P22*Px4 D<1.17)8BH(PZZ*PX4 0(1.18)=8H(PX2*PZ4 D(1.19I88H(pX21p22 0(4019):8HX2) GO ID 60 43 COVTINJE C H‘FORM D(1.4):8HP4 DI117):3HP28(PX2- DI1.8):8H(PZZ*IBH6P2 6 56 s S 0(1011)88HP4tp22 S D( 06’38HPZ4 s 012.? 5 0‘2, 8’38H 9 PZZVP S S 0(2 9’88H t PYZSP S D(1.11)88HPY6 D(20l3)38H 0(2614)¢8H 0(2015)=3H D‘2.16’38H 0(2017)88H D‘2.17)38H PX‘flp D‘ZOle’IaH PZ4*P DI2.19)=8H*pY2 t p PYAtP 9x469 P248? PY4*P PXQwP vamefieunvumunw i-OO GFOTOO $ D(1.5)38HP2*P22 S 0(2,7)=8HPY2) S D(2,8)=8H2-PY2) t S 011.11IEBRRA.922 0(1614)=3Hp4*(Px2- 0(2.15)38H(px2'pY2 015.15IEBH2I 0(2016)=8H~PY2) t ‘vowuneuw )=8H 8 P229? S D! DI9.6I=BHR*(PX29P 013.7I=BHZ 6 Px24 D‘b.7’=8HPY2, S 011.12I88H921p24 I7I=BHYZI Dg.BI=BHXZ) DIB,9I=8HXZI I1012I88H926 013.13)38HX2) DI} 14I08HY2) 013 115)88HY2) DIS 16)88H22) 013.17I38H22) DI3617I88H22) 0(3'18)!8HX2) $69 UDQGHUHfiflflfi $ D(1.6I=BHPZ4 DI3,8I=8H (pxz-pY 12)38HP22PZ4 s 011. s 012.1.IaBHPsz s 013 S 013.16IsBH1Px2-P12 D‘3019)§8HY29PZZ*p 015)88H) * IPXZ -..3A 60 0 N 214 64 66 68 212 220 217 218 221 ’2 COVYTNJE DO 54 V81.19 05/03/73 IF (NPA?(N.1).OR.NPAR(N,2).ORngR(N) .0R.PGDEL(V)) 62.64 CONTINJE PRIQN 214.((X(N)}PA?(N),NPAR(N03)o(D(J:N)oJ81011))) roaan ( < A8’1X;E13.8 13 2x 15x IF (~212¢~.2I.0R.PGDEL1~I$ PAINT 216.PGDEL(V) FORN‘Y (*§*29XOE12.5013) COVTTNJE IF (NPA?<20,1I.Eo.bI GO To 66 PRINN 2 2.X(2 ) .NPAR(2 I ) IF (NPA (21:1).E0.D) GO T6 68 PRIVW 21?.x(21) ;NPAR(21.1I COVTTNJE F0241? ( A8,16X,I3) PRIVN_22§.CAPPA FOQIAT(/lt KAPPA = tF14.7) REYJRN F0?4IY(*19) FORflAT (1 A8) FOQ4KT (6 tittitt0N_Y TYPES I-R 3x luAB'A4) , CNPAR‘ZON) AND III-R WORK PROPERLY IN THIS PROuR 0621M -°-- THEREFORE THE VALUES X AND Y 0R X AND I HILL BE EXCHANG ED.) END R1.3A OCECMOC7CNOC3CTO<3C30M3C10 Q 40 FOR EAJH STATE JINCJAJIJ+4 IFPAV D 3 114 SUBIDUTINE FCRMEPl (IFPAVONMAXaIEO. PAR PONP CAPP) 05/03/73 cowqo~ A(25.25I.s<25.25I.AS(6ooI.JIN(2.250;I INDEXINVARI24IZIIIBN CowaoN/BIKAP/AP<15. quI 014: NSIDN P(19 4.NMAX) PAR(19).¢VEP(19I.NP(3) t 9 t i t a 9 t * w a t o t . t a w A t t t t w a t t g t g g t t a t . * THIS“ RJJTINE SORTS THE EIGENVALUES AND DETERMINES THE QUANTUM NUMB R THE DIAGONAL ENERGIES FROM A ARE STORED IN AG AT LOCATI THE AVERAGE VALUES ARE CALCULATED BY USING THE TRANSFORMAT'O MATIX AND APPLYING IT TO THE UNDIAGONALIZED OPERATOR VALUES OPTIONS ARE IFPAV =0.1 ENERGY BY RORE THAN 599, EXPECT21.OE-9 RT281.414213562 NT #0 HT!) J=V3K1I NL8VP(3) IF (NL.3T.0I GO TO 10 IF (IJ.VE.0).0R.(JE3.NE,1II RETURN INDIUIVIIBNIII IF (IND.EO. I RETURV INDEK:IVDEX+1 DO 40 131.15 APCIIIVDEXI=9.5 ASIINDEXIaU. JIVCIBVI1I=IADEX RETJRN C INITIA,UZE QUANTUM NUMBER IDENTIFICATION 10 0 FOR 21 22 23 24 C FOR 28 121 122 123 COVTINJE IF (CA’°,LT.LI GO T3 28 GO YD(21.22,23.24) JEO 3-R NOTATION USEIOBLATE KZI-Q S K1=J¢2 GO TO 30 K2!) 8 <1=J+1 GO TO 30 K23-1 S K1=J92 GO TD 30 K28~1 I K1=J91 GO TO 30 1-3 NOTATION USE ,PROLATE GO to (*21I15122 k23.1124) JEO g 'IJ/z 082925 23((J*1)/2)*2'J 2 T0 K18¢JISI9292 S K2= ((J 1I/2I92 J+1 GO to 30 K1:((J¢1)/2)*291 s (2=(J/2)*2'J"1 THE DIAGONAL AND AVERAGE ENERGIES ARE PRINTED ONLY IF EXPECTATIOV VALUE OF THE ENERGY IS DIFFERENT FROM THE THE DIAGONAL AND AVERAGE ENERGIES ARE PRINTED THE NORMALIZED EXPECTATION VALUES OF EACH OPERATOR IS PRIN E 9 t t p-. t t t t t t t t . t g t t t g t t t t t a t t r g t a t t t * THE DIAIO R1.3A 124 30 05/03/73 115 GO TO 30 K1=((J+1I/2)92+1 S <2=(J/2I92-J COVTINJE KP=<2 5 KN=K1 DO 55 NN=11NL C DERIVE THE OUATUp NUMBERS FOR THE STATE 50 KP: (P‘Z S KNgKN'Z INO=UwJ9J+KN91-KP ITARBJIVIIBNIINDI IF (ITA?.EO.UI Go T3 65 INDEKsIVDEX+1 IF (IF’AV.EO.-1) RETURN IF (INDEX.GT.6O I GO TO 242 JIV(IBV.IND)8INDEX D0 50 “Z: ’19 AVE’KN 2’86: C FORM THE IEX’ECTATION VA UES OF THE P OPERATORS IN H SPACE C THE AVERAG: OF PIN NI= SJM OvER I.J 0F SQIIINI9PIIILIASIL N) VSOSUN=0.0 DO 50 I‘laNL SIVVLSIIINN) DO 59 L'IONL MDI,bI+1 1F (”0.3T.4) GO TO 5U VSQ:SIVV95(L.NN) C FORM THEISDJARE OF THE DIAGQNAL ELEMENTS C THE 96 98 192 106 104 32 59 60 IF ("0.50.1I VSOSUM=VSOSUH9VSO IF (MD.VE.1) VSO=V5392.3 M=L 1F (JE3.FO.2I M=M+1 IF IIJED.EO.1).AND.(MD.EO.MI.AND,(MD.GT.1II VSO=RT29VSO DO 32 QI=1019 AVE°KLII=AVEP(LII9VSO*P(LIIMDIM) IF (M.ST. 4) Go To 32 FOL. DNIVG TRANSFORMS THE MATRIX ELEMENTS BY THE HANG TRANSFORM GO TO (94 98,1 2 1 4) JEO EOI. IFIIND.EO,1I.AND.(M.EO.2))AVEP(LII=AVEP(LI)QEOtVSO9PILII312I IFI(MD.EO,2).AND. (N.EO.3))AVEP(LI)IAVEP(LII9EO*VSO*PILII4I3I GO TO 32 EDI-1 GO TO 95 EDI-1 MDLtMDol IF((MD.LT. 4). AND. (M ED. MD)IAVEPI GO To 48 IF (0::F,LE.EXPECTI Go To 64 48 VSOSUHEVSOSUV-l IF (NT.EO.uI PRINT 226IEXPECT NTBVF+ PRINT 527,J..EO,NN.o pray. mr¢:—mcro>m<~ngw F“ N 128 083 3890.836 3899.890 3908.061 3913019U 3915.668 3918.212 3920.655 3925.705 3930.799 3935.931 3937.669 3942.747 3944.035 3947.871 3999.099 3909.172 3953.033 3959.182 3961.121 396%.333 396Q.683 3965.309 3969.517 3969.852 3970.050 3971.837 3971.837 3971.837 3972.203 3972.786 3974.997 3975.038 3977.183 3977.183 3977.183 3978.490 3979.107 3980.182 3981.635 3982.465 3982.465 3982.465 3982.586 3984.765 3985.386 3985.685 3987.673 3987.673 3987.714 OiS-CALC ~0.031 60601 ‘00691 DOUJZ 'U0071 0.002 -60095 -0062“ “60611 00605 ‘00026 -0001“ “00032 -0.00? -00D17 '90018 0.003 “U060? '00013 -UOUU“ -COUUZ -0.014 #0.002 “00601 UOUJ6 '60008 ‘00608 0.001 0.008 8.000 ‘00605 -Uobdg ‘00602 '00002 ’OOUUB -0.002 0.001 0.00“ “00601 00605 UOUJS ’00618 0.031 -0.005 0.000 '0.008 0.001 00601 0.LJ3 NT ISOTOPE 0.01 130 0.04 130 0.01 130 Jold 139 0.00 130 3010 139 0.01 130 0.91 13d JOOW 139 2.00 130 0002 150 0.10 130 0.01 130 2003 130 UQUQ 130 0.02 130 ZOJJ 13d ZOUJ 139 0.01 130 30UU 150 0.02 130 0.20 130 5.00 130 0.30 130 0.01 130 0.20 130 0.2J 134 0.20 133 floUl 130 0.03 130 0.02 130 0.02 130 003d 133 0.30 130 9030 130 0010 130 0.33 130 9.00 130 2.J0 130 0.10 130 0.10 130 0.01 130 0001 130 1.00 130 6.00 130 1.00 130 U016 13d hold 139 0.02 130 X I X | r.» 0&003mC15‘@(POJWCDdJDCD‘Jflhft‘mxfl H94 000Vtc~HNOHt®CDF*$(DU1®¢N\LF*QM(DRJOLDOmeoh‘OFUGMDO‘OFvUIN H H H ...-b ‘VUTNCPO‘W‘V5‘NCPUJ®\DC3®HP*F-wf\ K + CDF‘NP‘BJNF4CJvaF‘wCDRJWP*3‘:O‘RJWCNJ‘W(NR)WFV$‘W0‘O‘MOPQUIW‘VU3®(POJ0‘3 IJCVPV~C.HILL.Hr+CJPf‘hJNfVfiJPD‘L~R)NCNOdWP‘F’WfVC‘WrUF‘MCNI'NCN¥‘PTVCJWnLF‘NfUfiJF3‘F\flPOS‘GCMOJNCNP*$¢D¥‘WF‘UWOFVOURJWFUI‘WJ?00M L U) H 0\®\MU1W¢LU3N'V$‘P~DCDJTH(I‘Q$‘WCTP‘QJ@\DC:@CNBJW‘Q$‘NWNU1MCDOJC\DCJWFvP‘®\fl0‘m‘fli X I “\MRJCF*UlF-F$iL‘VCM4'PF*OJW£NOuWF‘C NP‘JTWP*00N(NRJQfVC9NCNfiJNCNOJHP‘C NTVF‘NIVC.H X 1. +.rJruwmxnh~:c~:*w-u£"tr»owuuvcu;~cc~01mcyouwso£‘ccanamxnr¢0nmr001wc~naHtmcuuwrxns~r 132 038 4072.807 4074.495 4074.572 4074.572 9075011“ 4075.499 4075.734 4075.790 4075.959 4076.607 4076.728 4076.772 4076.327 4077.762 4077.889 4077.986 4078.478 4079.058 4079.265 4079.497 4079.596 4080.338 4080.502 4080.608 4081.279 4081.319 4081.374 4081.764 4082.995 4083.149 4083.300 4083.300 4083.444 4083.619 4084.345 4084.503 4084.804 4085.085 4085.542 4085.573 4085.793 4086.254 4086.778 4087.460 4088.098 4088.676 4089.098 4089.731 4090.058 4090.228 OBS-CALC DOLJl 0.002 0.002 '00007 0.002 OQUUQ ‘0.004 -00001 0.005 -600Ub 0.004 0.002 0.003 ‘00001 0.005 ‘00004 0.007 U000? OOUU7 ‘00007 0.005 0.004 0.006 ’00003 00006 “00005 ‘00003 ’00001 0.007 ‘00007 ‘UOUUQ 0.003 00001 0.003 0.008 0.007 ’00002 ‘00010 0.001 0.005 0000“ ”00005 ‘00002 ’00006 -0000“ 0.005 GOUOB “00000 ’UoUUl ‘00005 HT ISOTODE 3.00 130 0.01 130 UolU 130 0.10 130 1.00 130 0.30 130 J010 130 0.10 130 0.10 130 1.00 130 0004 130 0.10 130 0.60 130 1.00 130 1.00 130 0.20 130 0010 130 0.04 130 0.10 130 0.01 130 1.00 130 1000 130 0.01 130 1000 130 0.10 130 0.01 130 0.10 130 0.30 130 0.10 130 JolJ 130 0.01 130 0.01 130 UOUQ 130 dolU 130 0.10 130 0.40 130 2.00 130 0.01 130 0090 130 0.01 130 2.00 13J U004 130 0.20 130 0.01 130 UOZJ 130 5000 130 0.10 130 0.40 130 0.20 130 0.04 130 L H H (FI‘:\OU1CCDI‘tC”:”VCJGCDOJQCP¢JVCNQHD‘4®\R‘JVCNP F‘H Hré HrA rAthtAhbp H F‘P‘H r4 H OFVRJHCNOJOCDU1H\flh‘erNJWKnc: X I (NRJHTVCJHCNOJFF0$W“F‘OCNF‘“TUCJHCN$‘FTVOUOF*RJMf‘OQNP‘R)®P*C;N(DP‘WCNP‘V‘VHJHCDHJN x ... rs» F‘H mrvn)©t¢h‘:CDF*G(NCDOrvh*VfVO‘®(Dk*O>m\flh‘mcnhhDCDDQI'N‘flh‘“ H F‘H ksH H hspra mr4r*m>wc~‘40rAcamc;nam L H F‘H H CD04:\EC)Q\fl0¢wCD$"®\DOJW‘M0006u‘00‘N‘4WxDO‘erdDN£D$‘@CPflJw H P‘Hf‘h‘H H F‘Pr‘ OP‘P*OIVRJ®‘Q$‘O.chppA NP‘RJWO‘C.RJNCNF‘WFUC>HFVCJfiJPF‘h'NCNOJHIVP‘C HPVR)NF‘C hia49P‘P'VnJNrUC @tTF‘UP‘h‘H X * p VCDC)®IMPUO‘UHV\DP‘©O*F*5‘“(NCLCEGCNP‘C eruao<~PoO‘H\nCJUHV'V~4$ N\mr*g.mtfic~o.®wfinam P rs» P‘H P‘H h‘H 133 OBS 4090.40b 4091.591 4092.323 4092.503 4093.262 4093.333 4093.782 4094.090 4094.255 9094.533 4094.030 4095.438 4U950704 4095.833 4095.905 4095.890 4097.131 4097.484 4098.081 4098.805 4098.844 4100.192 4100.192 4102.576 4102.576 4102.622 4103.433 4104.35b 4104.387 4105.649 4105.816 4105.816 4106.396 “1060687 4107.116 4107.954 4108.981 4108.986 4109.249 9109.299 4109.413 “1090568 4109.719 4111.228 4112.073 4112.073 4112.174 4112.352 4112.413 4112.813 OBS-CALC 0.002 UOUUD Oobdl UQUJS 0.001 Uobll -0.000 0.018 -UOLO3 “00002 'Uobd“ 0.001 0.001 ’Oobdl 0.001 ‘GOUJS 0.005 '00003 ‘UOUOZ 0.002 'UOUJ6 'GOUUB 0.010 0.000 0.010 'UOUUS -BOUJZ 'Uobdl ‘OQUJB DOLUU 0.032 UOCJS 00006 0.002 0.005 0.001 Oobfll 0.004 0.003 -OOUJZ GOUJ6 “0.001 “00002 0.001 ‘00033 -GOUUZ 0.000 -UODJZ 1-00007 00009 HT ISOTOPE 1.00 130 3.00 130 4.00 130 0010 13J 0.60 130 0.00 130 0.01 130 0.00 130 3.00 130 0023 13J UOQU 13J 0.01 150 2.00 130 loUU 130 0.10 130 0.01 130 loUU 130 0.01 130 2.00 130 1.00 130 001d 130 0.10 130 0.10 130 0004 13J 3.04 130 Uoul 13J 1.00 130 0.10 130 0.34 130 5.00 13d 100d 13d 1.00 133 0.40 130 0.20 130 Uodl 133 2.00 130 UOZJ 130 0020 13U 003“ 150 0.01 130 0.01 130 0.10 130 0.01 130 2.00 130 2.0J 130 2.00 13a 0060 130 0.10 150 JOUU 130 0.01 130 at 1'"! 1| ll ‘llll. II. 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' HHHUH bubOUMhJNWO\WO\fl#JHflHH‘30ID‘JOWOOVH\JOWIUPM‘JOMO\IJb&\OCVV€>OHm\flJuu(d\hO\flUI#‘AOHUNYVCBUib6dNM‘OP4 q x + cacuacaocaauuo rvnuvnunnvr~der-HW‘r-eruwm\nthJbathbJubyuucuuuucuouvnunuununnung“nurbpqdp-anh-Hw- 149 OBS 2764.792 2761.472 2608.588 2606.441 2602.834 2597.640 2591.155 2483.995 2576.865 2600.722 2597.003 2600.722 2594.018 2599.645 2597.003 2581.106 2592.547 2586.248 2589.682 2588.424 2586.661 2590.479 2583.988 2590.356 2560.424 2588.879 2585.477 2579.860 2578.747 2578.499 2578.203 2578.051 2575.379 2578.350 2572.881 2568.071 2568.029 2566.712 2565.884 2565.110 2563.168 2565.110 2557.397 2555.957 ?554.288 2552.905 2546.620 2619.759 2620.825 2622.862 2625.984 2629.869 2633.882 2637.416 2641.722 2626.793 26300435 26250170 2631.541 2624.092 2632.376 OBS-CALC 00019 .0.055 0.001 00.004 '0.001 0.006 0.003 06.231 «0.054 “00012 0.025 I0.004 0.003 $0.001 I0.009 03.995 0.004 00.006 00.044 «0.003 n0.000 0.002 900058 0.005 00.023 '0.002 0.004 90.005 00.007 0.017 0.020 0.011 0.002 '00024 ”00008 ”0.004 0.001 0.009 0.003 90.014 90.007 '0.016 0.019 0.033 00075 '0.024 0.021 «0.001 v0.001 I0.004 00.006 -n0.003 v0.016 $0.013 "0.152 0.001 ‘00002 “00001 "00002 00.009 00.019 WT 0 0 '9 0:08.. 0.70 0.00-u 0.01 0000-: 0.01 0.00., 0,01 0.01 0.00.. 0.02 0.00-9 0.00.- o,oo.. 0000'. 0.00.. 0 03 0.70 0.10 o,7o 0.06 o,25 o,on.. 0.90.. 0.00.. 0001 0025 1000 0000““ 0.00.. 0.06 BAND 100 100 zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo .zoo zoo {zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo zoo H1‘ g; O‘DCDiiaHQ¢DCD<0aWD‘J\H3‘Q‘40H3\IUNQ¢D‘J\IOHN\IJHD‘Q‘lOW’\IUI.<§(NOIOo~o\munm\runw\nauhAsaJLAuuouuc~0aucsutwnimrcnzmIanmruhzmr¢Hwa~AH+4H.+ H0‘ p. ‘OCDC3(“D‘QCDCD(HOCD‘J‘MW‘9‘40Wb\flUMNID‘J\IOHfi\flJbGHH‘JOW)\flULh x WCARJHWUCJWF‘k‘AFORJDCAQH‘F‘JMUNHD\flhWNOWNFUF‘&(ANJNr‘FMO\fl355(dRHVF‘HWfl\flJbbO(’0‘O\9UHJ\IU\IUMD1|..blbfiCflUCflOflU‘dOflNfUhJNIUG + W \hU1hwfi€dUL§¢dNHU\fl\rO‘Q\JOW)\JOWflJbOJ}\flULb(dULAIO\fl\JOWfl\anbHHChOthbthleubchflvi) Q hnvfidHWOCdOH‘F‘D.hh‘bTDCdFHUCANH‘C’F‘.FOOHUCDbJNF‘&I&CNK)NHflJL#(uOHUP‘QWHCflN)NW‘\flUIA(Afi0NH4lbOiMFUH”flI X (NouuuacunnvrunannacrvumsnvvaLans:-awunuouucunanuvxnpnmxnun.:saubasa.aonuuuowucunnvrunanuunowwfir9Hn-n.+ 153 OBS 2774.906 2615.077 2614.852 2615.936 2616.858 2601.175 2603.961 2602.580 2604.467 2605.488 2606.834 2592.551 2608.140 2593.084 2592.802 2592.550 2591.451 2592.550 2593.288 2582.545 2581.418 2581.191 2580.426 2579.603 2580.038 2577.728 2572.058 2570.617 2569.101 2568.899 2567.766 2640.727 2635.644 2630.010 2651.880 2648.354 2644.889 2651.587 2641.120 2660.454 2660.612 2659.009 2657.315 2656.048 2667.340 2667.468 2664.593 2665.038 2660.159 2675.940 2760.845 2696.636 2703.570 2714.512 2745.686 2750.247 2725.350 2741.113 2746.522 2757.143 2765.291 OBS-CALC .00099 90.038 5.951 -0.021 00.002 0.004 '0.002 00.025 0.021 0.021 0.033 n0.054 00.023 0.046 0.043 0.047 0.045 0.034 '0|016 00087 01069 00072 0.023 0.003 0.100 0.016 0.159 0.095 $0.025 90.039 u0.114 D0.053 0.004 00.046 n0.022 90.013 0.018 u0.006 90.002 0.005 0.001 0.017 .0.001 0.012 0.007 0.029 ~0.064 ~0.017 0.108 0.010 0.043 00.001 0.049 «0.017 0.245 0.006 .0'009 ’00010 09002 “09002 '00007 WT 0.00.- 0.00-. 0.00.. 0.01 0.00-- 0.00.. 0.06 0.02 0.01 0.05 0.00.. 0.00.. 0.02 0.01 0.04 0.00.9 0.00.. 0.00-. 0.01 0.01 0.01 0004'. 0000.. 0.00--- 0.00-- 0.00-- 0.00-- 0.00.. 0.00‘¢ 0.01 0.00.. 0.01 0,02 0.01 0.00.. 0.01 0.00.. 0.01 0.01 0.02 0.00.. 0.01 0.00.. 0.01 0.01 0.00.. 0.00.. 0.01 0.00“! 0.00-. 0.01 0.01 0.00.. 0.00.. 9.00.- 0.01 0.01 0.01 0.01 0.25 0.02 BAND 1 381 00! 001 661 001 001 001 00; 001 001 001 001 001 001 001 001 00; 00; 001 001 001 001 001 004 00; 001 001 001 001 001 001 001 001 001 001 001 00; 001 001 001 001 001 001 001 001 001 001 001 001 001 001 001 001 001 001 001 001 00? 001 uuooouoomwoomm g 74 nnunnouucuuHVULA¢souunol N + NMbbb-b‘dehuUUU ammonommuomwao Q N OMHHMHHOMNNPHO 79 f ”HUIUIUIUIUIVIAbA&-§b 154 OBS 2754.422 2755.040 2770.239 2771.983 2784.931 2799.293 2782.601 2782.708 7798.627 2533.575 2667.468 2782.708 2668.134 2527.641 OBS-CALC '0.004 0.009 -0.037 0.012 .0.007 ”0.143 0,010 0.020 0.040 0.066 0.029 0.020 -0.029 -0.023 BAND 001 001 001 001 001 001 001 001 001 001 001 001 001 00; MICHIGAN STATE UNIV. 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