# Greener approaches for chemical synthesis: ball mill and microwave assisted synthesis of fluoxetine and duloxetine and enantioselective catalysed addition of organometallic reagents to aldehydes

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A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 313/761-4700 800/521-0600 Order Number 1339059 Hyperfine structure in the microwave spectra of the iodine fluorides IF7 and IF5 and of the weakly bound complex HCL ...NNO Shea, James Christopher, M.S. The University of Arizona, 1989 UMI 300 N.Zccb Rd. Ann Arbor, MI 48106 HYPERFINE STRUCTURE IN THE MICROWAVE SPECTRA OF THE IODINE FLUORIDES IF? AND IFg AND OF THE WEAKLY BOUND COMPLEX HC1...NN0 by James Christopher Shea A Thesis Submitted to the Faculty of the DEPARTMENT OF CHEMISTRY In Partial Fulfillment of the Requirements For the Degree of MASTER OF SCIENCE In the Graduate College THE UNIVERSITY OF ARIZONA 19 8 9 STATEMENT BY AUTHOR This thesis has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this thesis are allowable without special permission, provided that accurate acknow ledgement of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgement the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED: APPROVAL BY THESIS DIRECTOR n the date shown below: I -2 STEPHEN G. KUKOLICH Professor of Chemistry Date 3 TABLE OF CONTENTS LIST OF ILLUSTRATIONS Page 4 LIST OF TABLES 5 ABSTRACT 6 I.INTRODUCTION II.EXPERIMENTAL 1.Sample Handling 2.Instrumental 3.Data Treatment III.RESULTS AND DISCUSSION 1.Iodine Heptafluoride 2.Iodine Pentaflouride 3.HC1...NNO 7 13 13 16 22 27 27 33 37 APPENDIX: FITTING PROGRAM 40 REFERENCES 55 LIST OF ILLUSTRATIONS Figure I.Gas Manifold II.Diagram of Spectrometer III.Timing Diagram of Pulse Sequences IV.Example FID Signal V.Fourier Transform of Example Signal VI.Distorted Structure of IF7 Page 14 17 21 23 25 28 VII.FID Signal From IF7 Spectrum 30 VIII.Fourier Transform of IF7 FID 31 IX.Structure of IF5 34 LIST OF TABLES Table I. Instrumental Operating Parameters Page 18 II. Measured and Calculated Transition Frequencies for IF7 • * • . • 32 III. Comparison of IF5 Constants With Previous Studies 35 IV. Measured and Calculated Transition Frequencies for IF5 36 V. Transition Frequencies for HC1...NN0 38 VI. Transition Frequencies for Unknown Trimer Species 38 ABSTRACT A pulsed-beam fourier transform microwave spectrometer was used to measure the rotational spectra of iodine heptafluoride, iodine pentafluoride, and the weakly bound complex, HC1...NNO. For IF7, only five rotational .transitions were observed, and no resolvable hyperfine structure was detected. Based on this data, the A, B, and c rotational constants were determined to be 1746(3) MHz, 1732.0(8) MHZ, and 1553.0(2) MHZ, respectively. The existence of a pure rotational spectrum confirms that this molecule undergoes polar distortions. Twenty-two hyperfine components of the IF5 spectrum were recorded. The B rotational constant for this molecule was determined to be 2727.421(3) MHz, and the quadrupole coupling constant was found to be 1069.35(13) MHz. Though the work is still in progress on HC1...NN0, nine transitions have been recorded. In addition, five transitions have been recorded for an apparent trimer species composed of HC1, NNO, and an as yet unidentified third species. CHAPTER I: INTRODUCTION Microwave spectroscopy is used to determine the rota tional energy of molecules. The rotation referred to is the end over end motion of a molecule, and not internal bond rotation. Such a rotating molecule, if it possesses a dipole moment, can couple to an oscillating electric field. The rotation is quantized, and the energy differences be tween rotational states of the molecule correspond to fre quencies in the microwave range of the electromagnetic spectrum. Thus microwave spectroscopy can be used to deter mine the energy of the rotational levels for certain species. The magnitudes.of these energies are determined by rotational constants, and it is these constants which are of interest. A molecule has three rotational constants (A,B, and C) corresponding to the three molecular axes (a,b, and c, respectively). The constants are all defined analogously: A = h / 8*2Ia (1) Where Ia is the moment of inertia about the molecule's "a" axis. If a molecule's two larger rotational constants (A&B) are equal, it is termed an "oblate symmetric top", or simply an oblate top. The molecule IF5 is an oblate top, and IF7 is very nearly so. The energy for a rotational level of an oblate top is given by: E = BJ(J+l) + (C-B)K2 - Dj[J(J+l)]2 (2) - DjkJ(J+1)K2 - DkK4 Where the quantum number J is the total rotational angular momentum of the molecule, and the quantum number K is the r> • projection of the angular momentum on the molecule's sym metry axis. above. B and C are rotational constants as described Lastly, the constants Dj, DJK, and DK represent centrifugal distortions of the molecule due to its rotation. When an atom is present which possesses a nuclear spin (I) of greater than 1/2, the resultant quadrupole moment must be considered. The nuclear spin (I) can couple with the molecule's rotational angular momentum (J), giving a new total angular momentum (F). F is given by the formula: F = J+I, J+I-l, J+I—2 f , |J-I| (3) So for a molecule such as IF5, containing iodine (1=5/2), in the J=1 rotational state, the allowable values of F are: F = 1 + 5/2 = 1 + 5/2 - 1 = 7/2 = 5/2 = 1 + 5/2 - 2 = |1 - 5/2| = 3/2 Of course these splittings of J states become more numerous at higher values of J, as can be seen in table III. The exact energy of each of these states is dependent upon the strength of the quadrupole coupling. The quad- rupole coupling constant, eqQ, is a product of the fundamen tal charge, e; the potential field gradient, q, due to a non-symmetrical distribution of the electrons and other charges about the nucleus of interest; and the nuclear quad rupole moment for the given atom, Q. Thus the splitting for IF5 and IF7 would be expected to differ because of differing molecular symmetries (C4v and C2, respectively) and differ ing electronic charge distibutions. When quadrupole coupling is considered, the energy level expression (equation 2) becomes: E = BJ(J+l) + (C-B)K2 - Dj[J(J+l)]2 (4) - DjkJ(J+1)K2 - DkK4 + {eqQ [3K2/J(J+l) - 1] Y(I,J,F)} Where Y(I,J,F) is Casimir's function: Y(I,F,J) = [3C(C+1) - I(1+1)J(J+l)] / (5) [81(21-1)(2J-1)(2J+3)] In equation 5, the symbol, C, is not the c-axis rotational constant, but a quantity known as Casimir's coefficient, given by: C = F(F+l) - 1(1+1) - J(J+1) (6) The above energy expression actually represents only a first order correction for the quadrupole coupling, though the fitting program used in this research (see appendix) calcu lated the energy to second order. The calculation of second order corrections entails a considerably longer equation, so for the sake of brevity, I leave its format to a reference text1. The energy of a transition between two rotational levels is given (in terms of J, K, I, and F of the lower state) by: 2B(J+l) - 4Dj(J+l)3 - 2DJK(J+1)K2 AE = (7) + eqQ {[3K2 /(J+l)(J+2) - 1] Y(I,F',J+l) - [3K2 /J(J+l) - 1] Y(I,F,J)} (Again, this shows only first order correction even though the calculations were done to second order.) So, by deter mining several transition frequencies, and assigning them the correct quantum numbers, one can fit the rotational and quadrupole coupling constants of a molecule to satisfy the above equation. Microwave spectroscopy is commonly used to obtain struc tural information about molecules. As seen in equation 1, measurement of the rotational constants, in essence, deter mines the moments of inertia. A molecule's moments of iner tia are functions of the masses of the atoms (which are known precisely), and the distribution of these atoms (bond lengths and angles) with respect to the center of mass. Thus three precise rotational constants may be sufficient data to solve for the variables of a molecule's structure. Often, though, there are more bond lengths and angles than can be solved for with three data points, and it is neces sary to use isotopic substitution. Since changing only the mass of one atom will result in new moments of inertia this gives an extra set of data points for each substitution. This additional data is usually enough to solve for all the variables (bond lengths and angles) in a molecule's struc ture. In the case of the iodine fluorides, isotopic sub stitution was not possible, hence the study was only taken as far as rotational and quadrupole coupling constants. In addition, as a true symmetric top, IF5 has a dipole along only one axis, which means only one rotational constant value can be found. Another application of microwave systems like the one used here is the study of weakly bound complexes (WBC's). Weakly bound complexes are species such as van der Waal's complexes. The binding force in these cases is much, much weaker than the force of the normal chemical bond. This means that the typical dissociation energy of a WBC is low enough that they have negligible lifetimes at room tempera ture. However, due to the rotational cooling that results in the pulsed-beam, weakly bound complexes will form with enough longevity to be observed. The high sensitivity of the instrument also aids in the study of such species. The complex HC1...NN0 is a WBC, the study of which is still in progress, due to complications which are discussed below. CHAPTER IX: EXPERIMENTAL Sample Handling The gas manifold is shown in figure I. stainless steel. All parts are of The manifold is pumped out by a rough pump through the lowermost port, and the remaining four ports are used for gas input. These four ports all have swagelock fittings to accommodate lecture bottles or the carrier gas line. One port is commonly used for a line which connects, by means of a simple "T", to either an Argon or a Neon cylinder. For work where weakly bound complexes are studied, a lecture bottle of each constituent is attached to an available port. The system contains both a one liter side tank and a half liter tank in line to the pulsed valve (not shown). The two tanks are wrapped in heating tape to aid in working with less volatile species, and when pumping out the system. A pressure gauge is used to guide the making of gas mix tures. For the iodine fluorides, a mixture of slightly less than one percent in carrier gas (Argon was used) was found to give a better signal strength than a more concentrated mixture. For complexes, a one to two percent mixture of each component in the carrier gas is common. The pressure in the manifold is usually raised to one atmosphere or •TD PULSED VALVE 1 UTER TANK CARRIER GAS <Ar OR Ne) GAS A GAS 8 slightly higher with the carrier gas. However, the mixture is introduced into the system until the manifold pressure drops to approximately four hundred torr. Furthermore, the mixture is usually diluted with Ar back up to the starting pressure two or three times. No loss of signal strength is commonly observed with these dilutions. When the signal finally falls off, a new mixture is made. Between studying different gases, the system is pumped down and heated at least several hours, if not overnight. The pulsed valve sends the gas pulses into the resonant microwave cavity, inside a vacuum chamber. The vacuum cham ber is evacuated to the microtorr range by a ten inch diffu sion pump with a mechanical backing pump. After passing through the chamber gases are exhausted through the building's fume hood system. There is one other aspect of the gas handling system, which becomes important when complexes are being studied. Since the gas mixture for a typical complex contains con stituent A, constituent B, and the carrier gas, any combina tion of these three species may complex and add to the ob served transitions. Considering that species have been ob served containing four constituents (though this is statis tically unlikely), the possible complications can be great. To help alleviate this problem, each recorded signal is gas tested. This process involves putting in mixtures of A and B with a different carrier gas (Ne is usually used for this, while Ar is used for scanning); and also putting in mixtures of only one constituent at a time with the carrier gas. If the signal is still present in the former case, it provides evidence (though not conclusive) that the signal belongs to the complex of interest. If the signal is still present in the latter case, it generally proves that the transition belongs to a different species. Instrumental The instrument used for this work was a pulsed-beam, fourier transform, microwave spectrometer. Its basic set up is shown in figure II; instrumental operating parameters are listed in table I. The operating range of the instrument is four to eighteen gigahertz. This range is divided into three bands: four to eight GHz (C-band); eight to twelve GHz (X-band); and twelve to .eighteen GHz (Ku-band). The oscil lators and waveguide used in the system are band specific, so to change between bands requires switching these com ponents. To facilitate this procedure, the necessary com ponents are mounted on sections of plywood, one for each band. Changing bands can be accomplished in about fifteen minutes. Operation of the spectrometer is as follows. The frequency of interest is generated by the stimulat ing oscillator ("stimul. osc" in illustration). This is a YIG tuned Gunn diode oscillator for X- and Ku-bands, and a YIG tuned transistor oscillator for C-band. The stimulating oscillator is phase locked twenty megahertz above the har- STIMUL. DSC. PIN1 IBM PC SCDPE PIN2 MOLECULAR FREQUENCY LOCAL OSC. 20MHz +/FID AMP FID (0-1 MHz) 20 MHz REFERENCE AMP TWR Table I INSTRUMENTAL OPERATING PARAMETERS Operating Range 4 - 1 8 GHz Resolution 10 - 30 kHz Precision +/- 5 kHz Bandwidth 1 MHz Optimal Scan Rate 3 0 MHz/hr. Sensitivity(per rotational state) 1 x 1014 molecules Gas Pulse Rate 0.5 - 5 HZ Rotational Temp in Beam <20 K Digitizing Rate 20 MHz monic of a stable frequency which is generated by a program mable, crystal source frequency synthesizer (not shown). The frequency from the stimulating oscillator is sent to three locations: a frequency counter (not shown), to monitor it precisely; the resonant cavity (discussed later); and a second phase lock loop with another oscillator of the same type ("Local osc"). Part of the signal from this local os cillator is mixed with the stimulating frequency to produce a twenty megahertz reference signal which will be important later. The part of the stimulating frequency that was sent to the sample cavity goes through a diode switch ("PINl"), and then a circulator which sends it to an antenna inside a Fabry-Perot cavity. The cavity is tuned by moving the mir ror opposite the antenna. Before the signal coming out of the cavity gets to the main detection system, part of it is split off and sent to a monitoring oscilloscope (not shown). Using the signal on this scope, it is possible to tune the cavity to produce the least reflection of signal. Once the cavity is tuned, the actual experiment begins. The instrument is controlled by an adjustable pulse genera tor. The pulse time sequence is shown in figure III. The first of four pulses goes to the gas valve that lets the gas into the chamber from the manifold. The backing pressure on this nozzle is approximately one atmosphere. The chamber is evacuated to the microtorr range by a ten inch diffusion pump. pulse. This results in a supersonic expansion of the gas The resultant cooling achieves a rotational tempera ture of less than twenty Kelvin. After the gas pulse has had time to reach the central region of the cavity (seventy-five microseconds to three milliseconds), the second control pulse is sent to the diode switch, PIN1. This lets in a pulse of microwaves from the stimulating oscillator lasting from one tenth to ten microseconds. A delay of from one tenth to fifteen microseconds exists before the next microwave pulse, so that the initial stimulating pulse can die down. The third of the four pulses goes to the diode switch, PIN2. This lets the emission signal from the cavity (less ten percent that goes to the oscilloscope used for tuning) through to the detection system. If the stimulating fre quency was within a few hundred kilohertz of a transition frequency for the sample, the outgoing signal will be the original stimulating frequency plus or minus the offset from the exact transition frequency. If no transition was near the stimulating frequency, the signal out will be the same as the signal in. After passing through the PIN2 switch, the signal is mixed with the local oscillator frequency, which results in a twenty megahertz signal plus or minus any offset from a possible transition. is more easily amplified. This new lower frequency After amplification, it is mixed once again, this time with the twenty megahertz reference 21 LJLJ Ld CxD 1 CD LDQ_ 00 z:i £L Figure III: Timing Diagram of Pulse Sequences signal generated earlier. The result is that only the off set frequency (if any) remains. This is amplified and detected, then sent to a transient waveform recorder("TWR"). The fourth and final control pulse triggers the TWR to receive the signal. The TWR sends the analog signal to an oscilloscope for viewing, and also digitizes the signal to be sent to an IBM PC for signal averaging and recording. Data Treatment When a transition occurs in the sample, a superposition of rotational states has been created by the stimulating radiation. The excited molecules relax, and the emission signal decays with time. The signal is referred to as a free induction decay (FID). An example of one of the FID signals found for IF5 is- shown in figure IV. This signal was digitized by the TWR and sent to the IBM PC, where it was averaged. The actual signal was averaged for about two minutes to achieve the signal to noise ratio shown. The gas valve is opened only on every other pulse sequence, and the computer subtracts the alternate signals to improve the sig nal to noise ratio. This signal represents fifty gas pulses, therefore it is approximately seven times stronger than the original emission. This signal (figure IV) shows the interesting feature of a second signal superimposed on the first. If one tran sition were present, the signal would be an exponentially damped sinusoidal curve. Close examination of the illustra- *T1 <>q n a £ I —1 t fD bd i—i tn w H' <ra 3 fa 0 8 16 X JJ_ 24 32 40 TIME IN MICROSECONDS 48 ro tri tion will show a second sinusoidal curve of lower intensity superimposed upon the main FID. This indicates a second transition frequency lies slightly further away from the stimulating frequency, but still within the bandwidth of the instrument. After a reasonable signal to noise ratio is achieved, the signal is recorded on a floppy disk. Once a number of signals have been compiled, they are, in turn, run through a fast fourier transform (FFT) program. This converts the signal from the time domain, as it is recorded, to the fre quency domain. figure V. The FFT of the example signal is shown in It does indeed show the two transitions indicated by the FID. The two transitions were assigned the quantum numbers shown in accord with prior predictions (see chapter III, section 2). It can be seen that the frequency scale on the horizon tal axis of figure V is rather low. This is because the frequencies shown are the offset frequencies with respect to the stimulating oscillator rather than the absolute fre quencies of the transitions. This is due to the fact that the FID signal is independent of whether it is above or below the stimulating frequency. To alleviate the uncer tainty, once a transition is found, the stimulating oscil lator is stepped up or down a few hundred kilohertz. By comparing a few FFT's from nearby frequencies, it is easy to decide upon the location of the transition. Figure V: Fourier Transform of Example Signal The next step in the analysis is to assign quantum num bers to the transitions, based on spectral predictions. These assignments are used as data points for a least squares fitting program (see appendix). The new values for rotational constants output from the fitting program are used to refine the spectral predictions. This is an itera tive process, which becomes increasingly refined as more transitions are located and recorded. CHAPTER III: RESULTS AND DISCUSSION Iodine Heptafluoride The initial vibrational investigation^ of iodine hep tafluoride concluded that the molecule possessed D5tl sym metry. For a seven-coordinated molecule such as this, a pentagonal bipyrimidal structure would indeed be expected. «•> However, results of an electron diffraction study suggested that the equatorial fluorines were not exactly coplanar, giving a distorted Dg^ symmetry. This latter determination was refuted by a later electron diffraction study4 which supported the D5h symmetry model. Another vibrational study5 was done which also concluded Dg^ symmetry. This was not the end of the controversy, as yet another electron dif fraction paper6 returned to the distortion theory, and reported angles of distortion for both the axial and equatorial fluorines (see figure VI). Furthermore, a subse quent vibrational study7 also agreed with the distortion model, based on observed violations of selection rules for vibrational combination bands. In addition to these structural studies, a study was done which confirmed the existence of slight polar distortions®. been obtained. To date, though, no rotational spectrum had Any dipole existing as a result of the above 28 Figure VI: Distorted Structure of IF^ mentioned distortions would be small, and the added com plications of hyperfine splitting might well have resulted in signals too weak to be found. As is shown by the FID signal in figure VII, a weak signal can be observed. This FID signal was averaged for a much longer time than the IFg example shown in the previous chapter. Whereas the IF5 sig nal is shown approximately seven-fold enhanced, figure VII shows an IF7 signal that has been enhanced twenty-fold. corresponding fourier transform shown in figure The VIII is a similarly weak but discernable signal. The A, B, and C rotational constants for IFy were found to be 1746(3) MHz, 1732.0(8) MHz, and 1553.0(2) MHz, respec tively. The standard deviation for the fit was 0.459 MHz. Table II gives the observed and calculated frequencies for IF7. Two comments need to be made regarding table II. The first is in regard to the number of transitions observed. A great deal of time was spent searching for additional tran sitions, but the search was eventually abandoned. The dif ficulty in finding more than five signals was attributed to the sample being used up. IF7 is not currently commercially available, and the special equipment necessary for fluorine synthesis was unavailable to this author. The second comment regards the two different measured frequencies listed for the third transition. an impurity in the IF7 sample used. IFg exists as A transition frequency belonging to iodine pentafluoride lies very close to one Figure VII: FID Signal From IF^ Spectrum a r»CM CD rv. CD LU ZD CJ LU ce Lu ui CNJ no D cn Figure VIII: Fourier Transform of IFy FID 32 Table II MEASURED AND CALCULATED TRANSITIONS FOR IF? Calc. fMHzl M-CCMHz} J 9504.338(11) 9503.748 0.590 9504.372(5) 9504.507 10951.517(3) 10951.568(3) 10951.545 Measured(MHz! Kn J' K ' K ' 2 E 1 2 -0.135 2 0 Tl -0.028 0.023 2 12610.003(3) 12610.170 -0.167 12610.033(3) 12610.197 -0.164 Kn 3 E 1 3 2 3 0 3 0 2 3 2 1 3 1 3 4 1 4 3 0 3 4 0 4 belonging to iodine heptafluoride. observed and resolved. Both transitions were Unfortunately, my predictions were not precise enough to assign either to one or the other molecule. This transition was left out of the IF5 fit, but was due to lack of data, no fit was done for IF7. There are conclusions which can be drawn from this study, however. It is further confirmation that IF7 does distort to a slightly polar configuration. Also, consider ing the agreement between the data and the predictions based on the work of Adams et al.6, it gives additional support to their structure. Finally, note should be made that these predictions included no quadrupole coupling. Again, con sidering the agreement with the data, it is possible that the high symmetry of the molecule produces an extremely small potential field gradient, q. This would result in an analogously small quadrupole coupling constant, eqQ, which could produce splitting too small to be observed. Iodine Penatfluoride As mentioned above, iodine pentafluoride existed as an impurity in the IF^. Unlike IF7, however, IFg has a strong dipole moment (see figure IX), and has been the object of two previous microwave studies9'10. As summarized in table III, these two studies showed some disagreement, as well as limited resolution in determining the quadrupole coupling constant. The results of this study agree more closely with the results of the later of the two previous studies, though in o\ o Figure IX: Structure of IFg Table III COMPARISON OF IF5 CONSTANTS WITH PREVIOUS STUDIES CONSTANT: Bp(MHz) eaO fMHz) Bradley et al. 2727.551(46) 1056.6(4.9) Truchetet et al. 2727.4217(10) 1067(10) Present Work 2727.421(3) 1069.35(13) 36 TABLE IV MEASURED AND CALCULATED TRANSITIONS FOR MEASURED fMHz) CALCULATED f MHz ^ M-C(MHz) IF 5 F F' K (J: 0-1 transitions) 5402.600 .042806 5627.491 .000179 5/2 5/2 7/2 5/2 0 0 10887.256(2) 10868.576(4) 10845.688(3) 10846.056(3) 10730.954(5) 10817.854(33) 10923 .636(8) 10960.881(8) 10983 .124(4) 11034.903(4) 11042.716(4) 11062.809(1) (J: 1-2 transitions) 10887.208 .047903 10868.543 .032462 10845.734 -.046507 10846.008 .047952 10731.007 -.053618 10817.895 -.041179 10923.703 -.067212 10960.938 -.057061 10983.140 -.016164 11034.918 -.015770 11042.786 -.070807 11062.771 .037552 7/2 5/2 3/2 7/2 3/2 5/2 7/2 5/2 3/2 5/2 7/2 5/2 9/2 7/2 1/2 9/2 1/2 5/2 5/2 3/2 3/2 5/2 5/2 7/2 0 0 0 1 1 0 1 1 0 1 0 1 16293.898(1) 16332.344(5) 16336.401(6) 16341.737(8) 16343.116(6) 16352.022(3) 16378.802(7) 16406.845(10) (J: 2-3 transitions) 16293.928 -.030318 16332.366 -.022708 16336.350 .050043 16341.703 .033183 16343.153 -.037983 16351.969 .052429 16378.756 .045795 16406.814 .030151 5/2 5/2 5/2 5/2 9/2 11/2 7/2 9/2 7/2 7/2 9/2 11/2 7/2 9/2 5/2 7/2 0 1 1 0 0 0 1 1 5402.643(6) 5627.492(5) with much better resolution of eqQ. The observed and calcu lated transition frequencies are listed in table IV. The overall standard deviation of the fit is .046 MHz. HC1...NNO The final species studied in this thesis is the weakly bound complex of hydrogen chloride with nitrous oxide. As explained in chapter I, WBC's are transitory species at room temperature, but are fairly long-lived in our vacuum chamber where the molecular beam is expanding. The work on this complex is part of a collaborative effort of the Kukolich research group at the University of Arizona. The transi tions which have been observed so far are listed in table V. No definite assignments have been made for these transitions as of yet. The difficulty arises from two factors, both due to the chlorine atom. The first is that the isotope fairly abundant with respect to 35 C1 37 C1 is (approximately a 1:3 ratio). This produces a second set of transitions for the ") —t H CI...NNO complex which are difficult to distinguish from close-lying H35C1...NN0 transitions. is the quadrupole moment of the The second difficulty chlorine atom. This produces splittings which make the assignment of quantum numbers to the observed transitions extremely difficult. An interesting side note to this study is the discovery of what appears to be a trimer species. When the gas test procedure described in chapter II was applied to the transi tions listed in table VI, the initial results indicated they Table V OBSERVED TRANSITIONS IN THE SPECTRUM THE COMPLEX HC1...NNO (MHz^ 11426.421(5) 11771.343(10) 11773.655(3) 11776.468(25) 11777.858(2) 11784.271(16) 11785.373 (3) 11800.000(17) 11801.793(8) Table VI OBSERVED TRANSITION FREQUENCIES OF UNKNOWN TRIMER fMHz) 7695.748(4) 7695.896(3) 7700.700(10) 11471.613(6) 15312.304(10) 15375.462(6) recquired HC1, NNO, and Ar to be present. Later tests with high purity argon did not show a transition, yet the signal returned when the initial argon cylinder was used. Multiple attempts with HC1, NNO, and high purity argon revealed no signal. Yet the signal was present when a small amount of air from the lab was allowed into the mixture. This result is reasonable, considering that Ar is distilled from liquid air, and could likely contain other atmospheric components. The transitions listed in table VI seem to definitely belong to a trimer of HC1, NNO, and a third constituent. APPENDIX: FITTING PROGRAM This program is adapted from the fitting program that the our group uses for asymmetric tops. There are only minor changes in the main program. The important dif ference is in the subroutines which provide the calculated energies. These, too, are modifications to existing software. I am unsure as to the origins of the various pieces of programs used. I have adapted and configured them to do the job of fitting the hyperfine structure of a symmetric top molecule with only one nuclear spin. The actual code is fairly well commented. $DEBUG $LARGE $NOFLOATCALLS C THE FOLLOWING PROGRAM WILL FIT THE HYPERFINE STRUCTURE OF A C SYMMETRIC TOP (ONE NUCLEAR SPIN); IT WAS CREATED BY MODIFYING C THE PROGRAM "FITSPEC" (USED FOR ASYMMETRIC TOPS) IMPLICIT REAL*8(A-H,0-Z) REAL*8 JACOB CHARACTER*79 TITLE1,TITLE2,TITLE3 CHARACTER*15 PNAMES,VNAMES CHARACTER*1 HOW,IFPROP,IHOW DIMENSION ALPHA(21,1),DELTAA(21,1),Y(150),Y0(150),YP(150), 1JAC0B(150,21),TJAC0B(21,150),W(150),XINV(21,21),PNAM ES(21), 2AINC(21),HOW(21),EVECT(150,1),TEMP(21,150),V(21,21),ASAVE(21), 3 DSAVE(21),IQNUMS(150,5),C0RRM(21,21),ALLVAR(21),VNAM ES(21), 4IFFIT(21) C THIS VERSION ALLOWS 21 PARAMETERS AND 150 DATA PT. MAX. C PROGRAM TO PERFORM GENERALIZED NON-LINEAR LEAST SQUARES FIT BY THE JACOBIAN C METHOD. SEE HYUNYONG KIM, J. CHEM. ED., VOL.47, pp. 120122 (1970). C ALLVAR IS THE MATRIX OF VARIABLES SENT TO THE SUBROUTINE FSUB FOR C CALCULATION OF THE Y VALUES. ALPHA IS THE MATRIX OF PARAMETERS OUT OF C ALLVAR WHICH ARE ACTUALLY BEING FIT, DELTAA IS THE MATRIX OF CORRECTIONS C TO THE PARAMETERS, Y IS THE MATRIX CONTAINING THE DATA AS READ IN, YO C IS THE MATRIX CONTAINING THE CALCULATED VALUES COR RESPONDING TO Y, WITH C THE CURRENT VALUES OF ALPHA, YP IS THE MATRIX CONTAINING CALCULATED Y'S C FOR USE IN CALCULATING THE DERIVATIVES, JACOB IS THE JAC0BIAN MATRIX C CONTAINING THE DERIVATIVE OF EACH Y WITH RESPECT TO EACH PARAMETER, C TJACOB IS JACOB'S TRANSPOSE, W IS THE MATRIX CONTAINING THE WEIGHTS OF C EACH Y, XINV IS A MATRIX USED FOR STORING THE INVERSE OF • JACOB*TJACOB C PNAMES IS A MATRIX CONTAINING THE NAMES OF THE PARAMETERS IN ALPHA, AINC C IS THE MATRIX CONTAINING THE INCREMENTS TO USE ON ALPHA WHEN CALCULATING C THE DERIVATIVES, HOW IS A MATRIX TO INDICATE HOW TO CAL CULATE AINC, C EVECT IS THE ERROR VECTOR ORC Y-YO, TEMP IS A TEMPORARY MATRIX NEEDED IN C THE CALCULATIONS, V IS THE VARIANCE-COVARIANCE MATRIX AND CORRM IS THE C MATRIX OF CORRELATION COEFFICIENTS. C LAST MODIFIED 4/21/88 TO GET WEIGHTING CORRECR FOT CAL CULATION OF STD. C DEVIATIONS OF PARAMETERS. LAST MODIFIED 7/29/89 TO OPERATE WITH C HYPERFINE STRUCTURE FOR A SYMMETRIC TOP. C C SET MAX. NUMBER OF PARAMETERS AND DATA PTS. POSSIBLE W/ ABOVE DIMENSIONS MAXPS=21 MAXDPS=150 OPEN(5,FILE='SYMFIT.IN',STATUS='OLD') OPEN(6,FILE='SYMFIT.OUT',STATUS='NEW') C INITIALIZATION C SSOLD=l.0D16 KOUNT=0 Z=O.ODO DO 30 1=1,MAXPS DSAVE(I)=Z ASAVE(I)=Z 10 20 30 ALLVAR(I)=Z IFFIT(I)=0 ALPHA(I,1)=Z DELTAA(1,1)=Z AINC(I)=Z DO 10 J=1,MAXPS CORRM(I,J )=Z V(I,J)=Z XINV(I,J)=Z DO 20 J=1,MAXDPS TEMP(I,J)=Z JACOB(J,I)=Z TJACOB(I,J)=Z CONTINUE DO 40 J=1,MAXDPS W(J)=Z EVECT(J,1)=Z Y(J)—Z Y0(J)=Z YP(J)=Z DO 40 1=1,10 IQNUMS(J,I)=0 40 C C INPUT SECTION C DATA FILE SHOULD CONSIST OF THE FOLLOWING: C LINES 1-3 TITLE LINES .79 CHARS OR LESS EACH C LINE 4: THE NUCLEAR SPIN IN UNITS OF 1/2 C LINE 5: THE NUMBER OF CYCLES THE FIT SHOULD RUN - INTEGER FREE FORMAT C LINE 6: THE NUMBER OF VARIABLES TO BE FIT,"NVARS" - IN TEGER FREE FORMAT C NOTE- THIS INCLUDES BOTH THE FIXED AND FIT VARI ABLES. C C NVARS LINES C EACH LINE CONTAINING FIVE ITEMS; C 1) THE PARAMETER NAME (15 CHARS OR LESS), C 2) THE INITIAL VALUE OF THE PARAMETER - REAL FREE FOR MAT; C 3) HOW TO INCREMENT THE PARAMETER - IF AN P IS HERE, THE PARAMETER WILL C BE INCREMENTED A PROPORTIONAL AMOUNT FOR TAKING THE DERVIVATIVE. IF AN F C IS HERE, THE PARAMETER WILL BE INCREMENTED BY A FIXED AMOUNT. 1 CHARACTER. C 4) THE FRACTION OR FIXED AMMOUNT TO INCREMENT THE PARAMETER WHEN C CALCULATING DERIVATIVES. REAL - FREE FORMAT. C 5) A 1 TO INDICATE THAT THIS VARIABLE WILL BE FIT. OTHERWISE, VARIABLE C WILL BE HELD FIXED TO THE VALUE INPUT - INTEGER FREE FOR MAT C C THE NUMBER OF DATA PTS. "NDPTS" AND THE NUMBER OF QUANTUM NUMBERS "NQNUMS" C FOR EACH DATA POINT - 2 INTEGERS FREE FORMAT (THIS VER SION OF THE PROGRAM C IS DESIGNED FOR FIVE QUANTUM NUMBERS: J,J',F,F',K C C NDPTS LINES C EACH LINE CONTAINS FOUR ITEMS; C 1) THE DATA POINT - REAL FREE FORMAT C 2) NQNUMS MANY QUANTUM NUMBERS (F IN UNITS OF 1/2)- IN TEGERS FREE FORMAT C 3) A 'Y' IF PROPORTIONAL FITTING IS DESIRED - 1 CHAR. FREE FORMAT C 4) A WEIGHT OF THE DATA PT. - REAL FREE FORMAT C READ( 5,*)TITLE1 READ(5,*)TITLE2 READ( 5,*)TITLE3 WRITE(6,1)TITLE1,TITLE2,TITLE3 READ(5,*)ISPIN WRITE(6,*)'THE NUCLEAR SPIN INVOLVED IS: ',ISPIN,'/2' WRITE(6,*) WRITE( 6 , * ) ' THE FOLLOWING VARIABLES WERE FIXED DURING THE FIT 7 WRITE(6,51) READ(5, *)KMAX READ(5,*)NVARS NPARMS=0 C READ IN VARIABLES AND ASSIGN FIT ONES TO ALPHA DO 50 1=1,NVARS READ(5,*) VNAMES(I),ALLVAR(I),IHOW,AINCR,IFFIT(I) IF( IFFIT(I).EQ.1) THEN NPARMS=NPARMS+1 PNAMES(NPARMS)=VNAMES(I) ALPHA(NPARMS,1)=ALLVAR(I) HOW(NPARMS)=IHOW AINC(NPARMS)=AINCR ELSE WRITE(6,61) VNAMES(I),ALLVAR(I) ENDIF 50 CONTINUE WRITE(6,*) READ(5,*) NDPTS,NQNUMS DO 60 1=1,NDPTS READ(5,*) Y(I),(IQNUMS(I,J),J=1,NQNUMS),IFPROP,W(I) 60 IF(IFPROP .EQ. 'Y') W(I)=W(I)/Y(I) C CALCULATE YO 65 CALL ASSIGN(MAXPS,ALLVAR,ALPHA,VNAMES,PNAMES, NVARS,NPARMS, 1IFFIT) CALL SYMFTSUB(MAXPS,MAXDPS,NDPTS,NPARMS,ALLVAR,YO,IQNUMS,ISPIN) C CALCULATE JACOBIAN MATRIX "JACOB" AND ITS TRANSPOSE DO 80 J=l,NPARMS IF(HOW(J).EQ.'F') THEN DELA=AINC(J) ELSE DELA=AINC(J)*ALPHA(J,1) ENDIF IF(DELA.EQ.O.ODO) DELA=1.0D-6 ALPHA(J,I)=ALPHA(J,1)+DELA , CALL ASSIGN(MAXPS,ALLVAR,ALPHA,VNAMES,PNAMES,NVARS,NPARMS, 1 IFFIT) CALL SYMFTSUB(MAXPS,MAXDPS,NDPTS,NPARMS,ALLVAR,YP,IQNU MS,ISPIN) ALPHA(J,1)=ALPHA(J,1)-DELA DO 70 1=1,NDPTS JACOB(I,J)=(YP(I)—Y 0(1))/DELA 70 TJACOB(J,I)=JACOB(I,J)*W(I) 80 CONTINUE C CALCULATE ERROR VECTOR DO 90 1=1,NDPTS 90 EVECT(I,1)=Y(I)-YO(I) C C DO PRESCRIBED MATRIX OPERATIONS C123456789123456789212345678931234567894123456789512345678 CALL MMULT(MAXPS,MAXDPS,MAXDPS,MAXPS,MAXPS,MAXPS,NPARMS,N .DPTS, 1NPARMS,TJACOB,JACOB,XINV) CALL MTRXIN(MAXPS,XINV,NPARMS) CALL MMULT(MAXPS,MAXPS,MAXPS,MAXDPS,MAXPS,MAXDPS,NPARMS,NPARMS, 1NDPTS,XINV,TJACOB,TEMP) CALL MMULT(MAXPS,MAXDPS,MAXDPS,1,MAXPS,1,NPARMS,NDPTS,1,TE MP, 1EVECT,DELTAA) C CALCULATE VARIANCE OF OVERALL FIT SS=0.0D0 WNDPTS=0.0D0 DO 100 1=1,NDPTS WNDPTS=WNDPTS+W(I) 100 SS=SS+(Y(I)-YO(I))*(Y(I)-YO(I))*W(I) SS=SS/(DBLE(NDPTS)-DBLE(NPARMS)) SDEV=DSQRT(SS) C CALCULATE THE VARIANCE-COVARIANCE MATRIX. 45 DO 110 1=1,NPARMS DO 110 11=1/NPARMS 110 V(I,II)=SS*XINV(I,II) C CHECK FOR DIVERGENCE IF(SS.GT.SSOLD) GO TO 1000 C RESET NO DIVERGENCE SWITCH IDIV=0 C THIS IS CONVERGING BUT NOT YET DONE SSOLD—SS CALL OUTPUT(Y,YO,ALPHA,KOUNT,SDEV,V,MAXDPS,MAXPS,NDPTS,NPARMS, 1PNAMES,IQNUMS,NQNUMS,W) C CORRECT ALPHAS BUT SAVE ALPHA'S AND CORRECTIONS IN CASE A DAMPENED FIT C IS DESIRED ON NEXT CYCLE DO 120 1=1,NPARMS ASAVE(I)=ALPHA(I,1) DSAVE(I)=DELTAA(1,1)*0.1D0 120 ALPHA(1,1)=ALPHA(1,1)+DELTAA(1,1) C CHECK FOR TOO MANY ITERATIONS IF(KOUNT.EQ.KMAX) GO TO 1200 KOUNT=KOUNT+1 C RE-ITERATE FOR NEW ALPHAS GO TO 65 C C SECTION FOR DIVERGENCE - WILL ATTEMPT DAMPENED FIT C CHECK TO SEE IF DAMPENED FIT HAS BEEN ATTEMPTED ON PREVI OUS CYCLE 1000 IF(IDIV.NE.0) GO TO 1300 IDIV=1 WRITE(6,*) WRITE(6,*) 7 ***** CAUTION ATTEMPTING DAMPENED FIT! ***** ' DO 1010 1=1,NPARMS 1010 ALPHA(1,1)=ASAVE(I)+DSAVE(I) GO TO 65 C C TOO MANY ITERATIONS SECTION C 1200 WRITE(6,31) KMAX GO TO 1500 C C DIVERGENT EVEN AFTER DAMPENED FIT C 1300 WRITE(6,*) ' FIT STOPPED DUE TO DIVERGENCE EVEN THOUGH r 1'DAMPENED FIT HAS BEEN TRIED.' WRITE(6,*) C C ALL DONE SECTION C C CALCULATE AND OUTPUT CORRELATION MATRIX 1500 DO 1505 1=1,NPARMS DO 1505 J=1,NPARMS CORRMfl,J)=0.0D0 1505 CORRMfl,J)=DSQRT((V(I,J)*V(I,J))/(V(I,I)*V(J,J))) WRITE(6,*) WRITE(6,*) ' CORRELATION MATRIX ' WRITE(6,*) DO 1507 1=1,NPARMS 1507 WRITE(6,11)PNAMES(I),(CORRM(I,J),J=1,NPARMS) WRITE(6,*) WRITE(6,41) DO 1510 1=1,NPARMS DO 1510 J=l,NDPTS 1510 WRITE(6,21)Y(J),ALPHA(I,1),JACOB(J,I) 1 FORMAT(/A79,/A79,/A79/) 11 FORMAT(IX,A15,15F8.4) 21 FORMAT(IX,F20.10,IX,G20.10,IX,F15.5) 31 FORMAT(IX,/' FIT STOPPED BECAUSE',13,' ITERATIONS HAVE BEEN', 1' COMPLETED. FIT MAY NOT',/' YET HAVE CONVERGED TO THE BEST', 2' POSSIBLE VALUES.') 41 FORMAT(8X,'DATA PT.',12X,'PARAMETER',12X,'DERIVATIVE') 51 FORMAT(//' VARIABLE',15X,' FIXED AT ') 61 FORMAT(IX,A15,3X,G20.10) END SUBROUTINE OUTPUT(Y,YO,ALPHA,KOUNT,SDEV,V,MAXDPS,MAXPS,NDPTS, 1NPARMS,PNAMES,IQNUMS,NQNUMS,W) IMPLICIT REAL*8(A-H,0-Z) CHARACTER*15 PNAMES DIMENSION Y(MAXDPS),Y0(MAXDPS),ALPHA(MAXPS,1),V(MAXPS,MAXPS) DIMENSION PNAMES(MAXPS),IQNUMS(MAXDPS,10),W(MAXDPS) WRITE(6,1) KOUNT DO 10 1=1,NPARMS SDA=DSQRT(V(I,I)) 10 WRITE(6,11)1,PNAMES(I),ALPHA(1,1),SDA WRITE(6,21)SDEV WRITE(6,31) DO 20 1=1,NDPTS RESID=Y(I)-YO(I) 20 WRITE(6,41)Y(I),W(I),Y0(I),RESID,(IQNUMS(I,J),J=1,NQNUMS) 1 FORMAT(IX,'***********************************************— **' f ******************************' 2//' ITERATIVE CYCLE # ',12//' 3 'STD. DEV.') # PNAME',20X,'VALUE',15X, 11 FORMAT(IX,I2,2X,A15,2X,G20.10,2X,G20.10) 21 FORMAT(//IX,' WEIGHTED STD. DEV. OF OVERALL FIT= ',G30.5//) 31 FORMAT(//6X,'DATA POINT',IX,' WT. ',7X,'CALC,15X,'RESID', 16X,'QUANT. NOS.') 41 FORMAT(IX,F15.6,IX,F8.3,IX,F15.6,IX,F15.6,10I3) RETURN END C C SUBROUTINE ASSIGN(MAXPS,ALLVAR,ALPHA,VNAMES,PNAMES,NVARS,NPARMS 1IFFIT) C SUBROUTINE TO ASSIGN THE FIT VARIABLES "ALPHA" TO THE VARIABLES USED C IN THE FUNCTION (OR CALLED BY ) FSUB. IMPLICIT REAL*8(A-H,0-Z) CHARACTER*15 VNAMES,PNAMES DIMENSION VNAMES(MAXPS),PNAMES(MAXPS),ALLVAR(MAXPS) 1ALPHA(MAXPS,1),IFFIT(MAXPS) NUMP=0 DO 10 1=1,NVARS IF(IFFIT(I).EQ.1) THEN NUMP=NUMP+1 ALLVAR(I)=ALPHA(NUMP,1) ENDIF 10 CONTINUE RETURN 48 This part of the program is the subroutine that outputs the transition frequencies according to the assignments of the input data points (measured transitions). $LARGE $NOFLOATCALLS SUBROUTINE FSUB(MAXPS,MAXDPS,NDPTS,NPARMS,ALLVAR,Y,IQNUMS) IMPLICIT REAL*8(A-H,0-Z) INTEGER EINDEX,OFFSET DIMENSION ALLVAR(MAXPS),Y(MAXDPS),IQNUMS(MAXDPS, 10) DIMENSION E(452) AEFF=ALLVAR(1) BEFF=ALLVAR(2) CEFF=ALLVAR(3) DELTJ=ALLVAR(4) DELTJK=ALLVAR(5) DELTK=ALLVAR(6) DELJ=ALLVAR(7) DELK=ALLVAR(8) HJ=ALLVAR(9) HJK=ALLVAR(10) HKJ=ALLVAR(11) HK=ALLVAR(12) SHJ=ALLVAR(13) SHJK=ALLVAR(14) SHK=ALLVAR(15) JMIN=20 JMAX=0 DO 10 1=1,NDPTS IF(JMIN.GT.IQNUMS(1,1)) JMIN=IQNUMS(1,1) IF(JMIN.GT.IQNUMS(1,4)) JMIN=IQNUMS(1,4) IF(JMAX.LT.IQNUMS(1,1)) JMAX=IQNUMS(1,1) 10 IF(JMAX.LT.IQNUMS(I,4)) JMAX=IQNUMS(1,4) IF(JMIN.LT.1) JMIN=1 OFFSET=(JMIN)*(JMIN)-1 WRITE(*,*) ' CALLING DASYME ' CALL DASYME(JMIN # JMAX,AEFF,BEFF,CEFF,DELTJ,DELTJK,DELTK,DELJ, 1DELK,HJ,HJK,HKJ,HK,SHJ,SHJK,SHK,E,ME) DO 20 1=1,NDPTS J=IQNUMS(1,1) KP=IQNUMS(1,2) K0=IQNUMS(1,3) EINDEX=(J+1)**2-J+KP—KO-OFFSET EG=E(EINDEX) J=IQNUMS(I,4) KP=IQNUMS(1,5) K0=IQNUMS(1,6) EINDEX=(J+1)**2-J+KP—KO-OFFSET EU=E(EINDEX) Y(I)=DABS(EU—EG) RETURN END This last part of the program is the subroutine which calculates the energies of the specified levels. it is a modified version of the program, "symtop.for", which can be found in the Kukolich lab. Again, I do not know how many modifications to the original code that "symtop.for" repre sents. I have deleted all extraneous I/O statements, and completely eliminated the section that calculated transi tion frequencies. It is more efficient to have the preceeding subroutine calculate only the specified transi tions, as opposed to all within a given range. $DEBUG $STORAGE:2 $NOFLOATCALLS $LARGE SUBROUTINE FTSYMNRG(B,DELTJ,DELTJK,DELTK,EQQ,DXZ,DXX,SA,SB,IS PIN, 1FMAX,KMIN,KMAX) C PROGRAM TO CALCULATE ENERGIES FOR A SYMMETRIC TOP MOLECULE WITH 1 NUCLEUS C HAVING A SPIN BY EXACT DIAGONALIZATION OF THE HAMILTONIAN MATRIX. C INCLUDES DISTORTION,QUADRUPOLE,DISTORTION OF THE QUADRUPOLE AND SPIN ROT. C H IS THE HAMILTONIAN MATRIX,EIGEN IS THE MATRIX OF EIGEN VECTORS, C E IS THE MATRIX WHERE THE ENERGIES ARE STORED IN THE E(J+1,2*F+1,K+1) C FORMAT ( I.E. THE ENERGY FOR J=0,F=5/2,K=0 IS AT E(l,6,l) ),T IS A MATRIX C CONTAINING THE TRANSITION FREQUENCIES AND IQN CONTAINS THE CORRESPONDING C QUANTUM NUMBERS IN THE ORDER J,J',F,F',K. ROGER BUMGARNER AUG.,85. C MODIFIED FOR K-LOOP 3-23-88 R.B. IMPLICIT REAL*8(A-E,G-H,0-Z) IMPLICIT INTEGER(F) DIMENSION H(12,12),EIGEN(12,12),E(40,40,6),IQN(200,5),T(200) ISW1=0 MDIM=12 LDIM=40 1002 1003 1005 1004 JDIM=200 DO 1002 I=1,MDIM DO 1002 J=1,MDIM EIGEN(I,J)=0.ODO H(I,J)=0.ODO DO 1003 1=1,LDIM DO 1003 J=l,LDIM DO 1003 K=l,6 E(I,J,K)=0.ODO DO 1004 I=1,JDIM T(I)=0.ODO DO 1005 J=l,5 IQN(I,J)=0 CONTINUE XI=DBLE(I)/2.ODO DO 2000 KI=KMIN,KMAX KIND=KI+1 XK=DBLE(KI) K=2*KI FMIN=1 ITEST=(-1)**I IF(ITEST.EQ.1) FMIN=0 DO 40 F=FMIN,FMAX,2 FIND=F+1 XF=DBLE(F)/2.ODO JMIN=ABS(F-I) IF(JMIN.LT.K)JMIN=K JMAX=F+I M=0 DO 20 J=JMIN,JMAX,2 XJ=DBLE(J)/2.ODO M=M+1 MP=M-1 DO 10 JP=J,JMAX,2 XJP=DBLE(JP)/2.ODO MP=MP+1 H(M, MP)=HAMIL( XI,XF,XJ,XJP,XK,B,DJ,DJK,DK,XA,DXZ,DXX, 1 SA,SB,ISW1) H(MP,M)=H(M,MP) 10 CONTINUE 20 CONTINUE IF(M.EQ.l) GO TO 25 CALL HDIAG(H,M,MDIM,0,0,EIGEN) M=0 25 DO 30 J=JMIN,JMAX,2 JIND=J/2+l M=M+1 E(JIND,FIND,KIND)=H(M,M) WRITE(6,31) J,F,KI,H(M,M) 30 CONTINUE 31 FORMAT(IX, ' J=',12,'/2',3X,' F=',12,'/2',3X,7 K=',I3,3X, 1 ' ENERGY(MHZ)= ',F15.5) DO 35 L=1,MDIM DO 35 LL=1,MDIM 35 H(L,LL)=0.0D0 40 CONTINUE 2000 CONTINUE M—0 C CALCULATE TRANSITIONS FOR DELTA J=+l, DELTA F=-1,0,+1 ,DELTA K=0 DO 3000 K=KMIN,KMAX KIND=K+1 DO 60 F=FMIN,FMAX,2 FIND=F+1 FLIND=FIND-2 FUIND=FIND+2 FU=F+2 FI>F-2 JMIN=IABS(F-I) IF(JMIN.LT.K)JMIN=K JMAX=F+I DO 50 J =JMIN,JMAX,2 IJ=J/2 IJP=IJ+1 JIND=J/2+l JPIND=JIND+1 C DELTA F=0 TRANSITIONS IF((E(JIND,FIND,KIND).EQ.0.0D0).OR. 1 (E(JPIND,FIND,KIND).EQ.0.0D0)) GOTO 45 M=M+1 T(M)=DABS(E(JPIND,FIND,KIND)-E(JIND,FIND,KIND)) IQN(M,1)=IJ IQN(M,2)=IJP IQN(M,3)=F IQN(M,4)=F IQN(M,5)=K C DELTA F-+1 TRANSITIONS 45 IF((E(JIND,FIND,KIND).EQ.0.0D0).OR. 1 (E(JPIND,FUIND,KIND).EQ.0.0D0)) GO TO 46 M=M+1 T(M)=DABS(E(JPIND,FUIND,KIND)E(JIND,FIND,KIND)) IQN(M,1)=IJ IQN(M,2)=IJP IQN(M, 3)=F IQN(M,4)=FU IQN(M,5)=K C DELTA'F-—1 TRANSITIONS 46 IF(FLIND.LT.l) GO TO 50 IF((E(JIND,FIND,KIND).EQ.O.0D0).OR. 1 (E(JPIND,FLIND,KIND).EQ.0.ODO)) GO TO 50 M=M+1 T(M)=DABS(E(JPIND,FLIND,KIND)E(JIND,FIND,KIND)) IQN(M,1)=IJ IQN(M,2)=1JP IQN(M, 3)=F IQN(M,4)=FL IQN(M,5)=K 50 CONTINUE 60 CONTINUE 3 000 CONTINUE CALL ORD(JDIM,5,T,IQN,M) END C COMMENT FUNCTION TO CALCULATE HAMILTONIAN MATRIX ELEMENT FUNCTION HAMIL(XI,XF,XJ,XJP,XK,B,DJ,DJK,DK,X,DXZ,DXX,SA,SB, 1 ISW1) C XI,XF,XJ,XJP,XK ARE THE REAL VALUES OF THE I,F,J,JP AND K QUANTUM NUMBERS. C B IS THE ROT. CONST, DJ IS THE DISTORTION CONST, X IS THE QUADRUPOLE C CONST, DXZ IS THE PARALLEL DISTORTION IN X, DXX IS THE PERP. DISTORTION C IN X AND A AND B ARE SPIN ROT CONSTS. ISW1 IS A SWITCH TO CALCULATE C ENERGIES TO FIRST ORDER ONLY. THIS IS JUST IN HERE AS A LEARNING TOOL IMPLICIT REAL*8(A-rH,0-Z) IF(XJ.LT.XK)GOTO 3 0 S=XJ*(XJ+1.ODO) C CALCULATE XA THE EFFECTIVE DISTORTED QUADRUPOLE STRENGTH XFACT=1.ODO IF((XJ.NE.0.0D0).AND.(XJ.EQ.XJP))XFACT=1.ODO3.0D0*XK*XK/S XA-XFACT*(X+DXX*(S-XK*XK)+DXZ*XK*XK) ER0T=0.ODO EQUAD=0.ODO ESPROT-O.ODO C CALCULATE DISTORTED ROTOR ENERGY (DIAG IN J) IF(XJ.EQ.XJP) EROT=B*S-DJ*S*S-DJK*S*XK*XKDK*XK*XK*XK*XK XJTEST=DABS(XJ-XJP) IF((XJTEST.NE.2.ODO).OR.(ISW1.EQ.1)) GO TO 10 C CALCULATE QUADRUPOLE ENERGY OFF DIAG BY 2 IN J Y=3.0D0*XA/(16.ODO*XI*(2.0D0*XI1.ODO)*(2.0D0*XJ+3.ODO)) Z — (XI+XJ+XF+3.ODO)*(XI+XJ+XF+2.ODO)*(XI+XJ-XF+2.ODO) Z=Z*(XI+XJ-XF+1.ODO)*(XJ+XF-XI+2.ODO)*(XJ+XFXI+1.ODO) Z=Z*(XI+XF-XJ)*(XI+XF-XJ- 1.ODO)/(2.0D0*XJ+1.ODO)/(2.0D0*XJ+5.ODO) XKSQ=XK*XK XJ1=(XJ+1.ODO)*(XJ+1.ODO) XJ2=(XJ+2.ODO)*(XJ+2.ODO) Z=Z*(1.ODO-XKSQ/XJ1)*(1.0D0-XKSQ/XJ2) EQUAD=EQUAD+Y*DSQRT(Z) 10 IF((XK.EQ.0.ODO).OR.(XJTEST.NE.1.ODO).OR.(ISW1.EQ.0)) GO TO 15 IF (XJ.EQ.O.ODO) GO TO 15 C CALCULATE QUADRUPOLE ENERGIES OFF DIAG IN J BY 1 FOR K'S .NE. 0 C (SEE TOWNES AND SCHAWLOW PAGE 157) EQUAD=3.ODO*XA*XK*(XF*(XF+1.ODO)-XI*(XI+1.ODO)XJ*(XJ+2.ODO)) EQUAD=EQUAD/(8.0D0*XI*(2.0D0*XI-1.ODO)*XJ*(XJ+2.ODO)) QSTUFF=(1.0D0(XK*XK/((XJ+1.ODO)*(XJ+1.ODO))))*(XI+XJ+XF+2.ODO) QSTUFF=QSTUFF*(XJ+XF-XI+1.ODO)*(XI+XF-XJ)*(XI+XJXF+l.ODO) QSTUFF=QSTUFF/( (2.0D0*XJ+1.ODO)*(2.0D0*XJ+3.ODO) ) QSTUFF=DSQRT(QSTUFF) EQUAD=EQUAD*QSTUFF C CALCULATE QUADRUPOLE AND SPIN ROTATION ENERGIES DIAG IN J 15 IF(XJ.NE.XJP)GO TO 20 S2=XI*(XI+1.ODO) S3=XF*(XF+1.ODO) C=S3—S2-S C (C IS CASIMR'S COEF.) S4=C*(C+1.0D0) Z=(3.0D0/4.ODO)*S4—S2*S Z=Z/(2.ODO*XI*(2.0D0*XI-1.ODO)*(2.ODO*XJ1.0D0)*(2.0D0*XJ+3.ODO)) EQUAD=-XA*Z XPROD=Z*XFACT WRITE(8,11)XJ,XF,XK,Z,XFACT,XPROD 11 FORMAT(IX,'J=',F3.1,' F=',F3.1/' K=',F3.1,/ COEFF=', 1F11.8,' QUAD FACT=/,Fll.8,' PROD'/F12.8) C LET CSEFF = THE EFFECTIVE SPIN ROT STRENGTH CSEFF=SA IF(S.NE.0.ODO) CSEFF=CSEFF+(SB-SA)*XK*XK/S ESPR0T=C*CSEFF/2.ODO 20 HAMIL=ER0T+EQUAD+ESPROT RETURN 3 0 HAMIL=0.ODO RETURN END REFERENCES 1. C.H.Townes and A.L.Schawlow, Microwave Spectroscopy (McGraw-Hill, New York, 1955). 2. R.C.Lord, M.A.Lynch, W.C.Schumb, and E.J.Slowinski, J. Am. Chem. Soc. 72., 522 (1950). 3. R.E.LaVilla and S.H.Bauer, J. Chem. Phys. 33., 182 (1960) 4. H.B.Thompson and L.S.Bartell, Trans. Am. Crystal Assoc. 2, 190 (1966). 5. H.H.Claassen, E.L.Gasner, and H.Selig, J. Chem. Phys. 49, 1803 (1968). 6. W.J.Adams, H.B.Thompson, and L.S.Bartell, J. Chem. Phys. 53, 4040 (1970). 7. H.H.Eysel and K.Seppelt, J. Chem. Phys. 56, 5081 (1972). 8. E.W.Kaiser, J.S.Muenter, and W.Klemperer, J. Chem. Phys. 53. 53 (1970). 9. R.H.Bradley, P.N.Brier, and M.J.Whittle, Chem. Phys. Lett. 11, 192 (1971). 10. F.Truchetet,R.Jurek, and J.Chanussot, Can. J. Phys. 56, 601 (1978).

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