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IGAN ST NIVERSITY UBRARI Willi \lllfllmwmuwlull 1293 00895 8724 ‘ This is to certify that the dissertation entitled The Preresonance Raman Interference Effect in Linear Polyenes presented by Isaac W. Sztainbuch has been accepted towards fulfillment of the requirements for Ph.D . degree in Chemical Physics d Majomssor Date September 26, 1990 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 LIBRARY Mlchigan State UnlversIty PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE |h———— —_ MSU lc An Afflrmdive Action/Equal Opportuntty Institution cmnmS-o.‘ The Preresonance Raman Interference Efl'ect in Linear Polyenes BY Isaac I. Betainbuch A DISSERTATION Submitted to Michigan state Univernity in partial fulfillnent of the requirements for the degree at DOCTOR OF PHILOSOPHY Chemical Physics Program 1990 ThePn The electron principl profiles through between Electrc ground Kramer tO-one ZlAg exPer: deCap. REP o fluor repre expel excit hidde ABSTRACT The Preresonance Raman Interference Efl'ect in Linear Polyenes BY Isaac I. sztainbuch The ‘presence of dipole-forbidden ("hidden") excited electronic states in centrosymmetric chromOphores can in principle be inferred from preresonance Raman excitation profiles (REPS). As the lexcitation radiation is tuned through the appropriate energy range, vibronic coupling between the state of interest and nearby "allowed" electronic states will produce interference effects in the ground state scattering intensity. We have used the Kramers-Heisenberg dispersion relation to establish a one— to-one correspondence between the vibrational modes of the 21Ag excited state and the interference features in the experimental preresonance REP of all-trans diphenyl- decapentaene (DPDP). The parameters utilized to predict the REP of DPDP were obtained without adjustment from absorption and fluorescence excitation spectra. The satisfactory representation and interpretation of the structure in the experimental spectrum establishes preresonance Raman excitation as a viable technique for characterizing such hidden states in non-fluorescing molecules. San-Ram Well and B Hindi, Po Michael, ' (Poland Naxis. ' Yitzh 1987), DEDICATED TO THE MEMORY OF Sara-Rance Sztainbuch - -- - -- - --(great-grandmother) Wolf and Bella Sztainbuch (grandparents) Hinds, Pole, Elke, and Esther Sztainbuch .............................. (aunts) Michael, Rafael, and Phillip Sztainbuch. _ - (uncles) Along with thousands of innocent men, women, and children from Deblin' (Poland) on 6 May 1942 they were deported to Sobibor, and then murdered by the Nazis. ' Yitzhak Arad, W (Indiana University Press, Bloomington, 1987), P. 390. ii adxiso Parke variox Sztai gtea ACKNOWLEDGMENTS I would like to thank Professor George Leroi for his guidance as my research advisor. Iwould also like to thank Professors R. Schwendeman, J. Kovacs, and P. Parker for serving on my guidance committee. Dr. Tom Atkinson assistance with various computer programs is much appreciated. My deepest gratitude is extended to my parents Mordechai and Elka Sztainbuch who were always available when needed. Their love and support has greatly facilitated the completion of this work. Special thanks goes to Hinda and Barry Feinstein for their encouragement, in spite Of their bewilderment as to why I am pursuing all this. Mike Gaisner deserve my appreciation for his friendship which goes back to our undergraduate years in physics at Hunter College. Surprisingly, his pessimistic view on life has always had the opposite effect on me. I am most grateful to have had the privilege of knowing Professor Joachim Weyl at Hunter College. Even during his last days he was cheerful, full of life, and totally devoted to his students and to the teaching of mathematics. His faith in me as a potential scientist has been an inspiration. Dean Weyl died on 22 July 1977 after a long struggle with cancer. This dissertation would have been dedicated to his memory had it not been for members of our family who were victims of the holocaust in Poland during the war, for whom it is dedicated instead. iii TABLE OF CONTENTS Page LIST or TABLES...........................................vi LIST or PIGURBS..........................................vii CHAPTER I INTRODUCTION................................1 CHAPTER II EXPERIMENTAL................................lO CHAPTER III THEORY......................................3l CHAPTER IV RESULTS.....................................45 4.1 Vibronic Features of the Preresonanace Raman Excitation Profile..............45 4.2 The 2119 Electronic Origin............52 CHAPTER V DISCUSSION..................................59 5.1 The Raman Interference Effect.........59 5.2 solvent Pclarisability Bffect.........65 5.3 The Semi-Analytical Transform from Absorption to Raman Bxcitation........68 5.3.1 Pranck-Condcn Pactors.................68 5.3.2 Bandwidth.............................87 5.3.3 Combinations and Overtcnes............88 5.3.4 Multimode Formulation.................94 5.4 Resonance Region......................96 CHAPTER VI CONCLUSIONSOOOOOOOOOOOO0.00.0000...0.0.00.0.102 iv Mmr—u -—.4 ’ IRYEKDI‘ APPENE REFER] Page APPENDIX IOOOOOCOCOCOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOO0.0.0.105 PART A. O O I O C O I C O C O C O C C O O C O O O C C O O O C O O O O I O O O O .107 PART B. O C O O O O O O O O O O O O O O O O O O O O O O O O C C O O O O O O O O .121 APPENDIX II0.000000000000000000000000000000000000000..00.1‘4 REFERENCES...O...OOOOOOOOOOOOOOOOO0.00.00.00.00.0I.0.000.150 TABLE Th3“ TAB] TX Ta TABLE TABLE TABLE TABLE Table Table I. II. LIST OE TABLES Page Displacement parem ers, A1 :0.005, in units of (emu) A, obtained by fitting the absorption spectra to equation (ll)..................................7o The llnu excited state vibrational modes (cm'l) and their corresponding Raman Franck-Condom integrals for the c=c symmetric mode used in the Raman excitation model...............................72 III. The 21A,; excited state vibrational IV. A. modes (cm'l) and their corresponding Raman Pranck-Condon integrals for the c=c symmetric mode used in the Raman excitation model...............................74 A flow diagram of the program .................120 Pranck-Condon Integrals used for the Raman Excitation P ofile model. (1-16 utilised or the l Bu state: and 1-6 for the 2 Ag state only.)......................146 Pranck-Condon Integrals used for the absorption spectium. model. (1-16 utilised or the l Bu state: and 1-6 for the 2 Ag state only.)......................14a vi - .‘4 g —_ . '—-—-— ... Figure Yigurl Piqu Pi Figure 1. Figure 2. Figure 3. Figure 4. LIST OF FIGURES Electronic energy level ordering for a typical linear polyene. A one- photon ( ) and two-photon (- -) dipole allowed transition are shown by their r spectiv arrows. The transition 1 .. 2 indicated by (-X-) represen s a one-photon Page forbidden transition............................3 Three Raman scattering processes as a function of incident radiation: 1. Ordinary stokes Raman scattering 00 << up 2. Preresonance Raman scattering 0. < a, 3. Resonance Raman scattering no ~ 01 .........8 The Raman spectrum of diphenyldeca- pentaene (DPDP) in cyclohexane/ chloroform. solution. at room temperature obtained with excitation frequency 21720 cm'l. The solvent lines of cyclohexane and chloroform are identified by t and +, respectively. Th! DPDP lines at 1160 cm and 1565 cm Absorption spectrmm of diphenyldeca- pentaene (DPDP) of ~lxlo' It in a 0.5 on cell at room temperature. The 0-0 band is at 419 nm (~2ssoo cm'l). The 0-1 band at 394 nm is not resolved to its vibronic components as it is at are unmarked.................12 77‘inrigur.18......0000000000000000.0.0.0.0001‘ vii Figure 6 Figure 7‘ Figure Piqurr Figure 5 . Figure 6. Figure 7. Figure 8. Figure 9. Page The power output o several dyes, pumped by a cw Ar laser, as a function of the dye laggr output wavelength (nm) range. .......................17 A schematic diagram of the apparatus utilised in obtaining the Raman spectrum........19 A typical Raman spectrum of DPDP in cyclohexane/chloroform solution at In excitation frequency of 22200 of" . The more intense band at 1565 cm" is the symmetric c=c stretch and the cyclohexane peak at 1445 cm"1 is used as an internal standard. The struc- ture of DPDP is also shown......................23 A typical Raman spectrum of DPDP in cyclohexane/chloroform. solution at various efcitation freguencies. ht ~2000 cm" below the 1 an stqfe, the cyclohexane peak at 1445 cm' is of greiter intensity (bottom). ht 21650 cm“ both peaks are of approximately equal intensity (center) . The DPDP 1565 cm- band is most intense for the excitation frequency of 22200 “-1 (top)00.0.00...OOOOOOOOOOOOOOOOOO00.00....25 .A sample of the preresonance REP segments after adjustment were made to compensate for differences in concentration. Each segment other than the central segment was multiplied by an appropriate value. Points marked x, which were not reproducible in all overlapping segments, were not included in the the final composite spectrum....................27 viii aqua Figure Figr Pi; Page Figure 10 . The experimental ([]-[]) prereso- nance Raman excitation profile of the symmetric c=c band of DPDP. The vertical bars correspond to the deviation error associated with each paint 0: th. ”POOOOOOOOOOOQOOOOOOOOOO0.00.0.0003° Figure 11. A two electronic state system with a displacement parameter shift, A. . The electronic transition momen s associated with the Raman process and the harmonic oscillator wave functions representing the ground and fundamental modes are illustrated. {Some authors express A2 by multiplying it by (ac/a)” this results in a unitless expression with a numerical value greater by a factor of 10 than tha given in (amu) A }.‘3"5 .IOOOOOOOOOOOOOOOOOOOOOO‘O Figure 12. Experimental preresonanace REF ([1- [Jl and the calculated preresonanace REP (———) (fourth v. mode is included)..........47 Figure 13. High resolution luorescence exci- tation of the 2 Ag state. The digitised data points were obtained from reference 9. The effect of including ( -— ) and excluding (- - -) the v. mode from the multimode formulation in the best fit model is shown. Although this effect is easilly seen here, it is not as apparent in the preresonance REP shown in Figure 12 and Figure 14................51 Figure 14 . The calculated preresonance Raman excitation profile of FOP (———) and an REP for which the 2 Ag oscillator strength is chosen to be vanishingly small f———)- The effect of excluding the v. mode from the multimode formulation for the REF (- - -) is ‘180 .honOOOOOOOOOOOOOOOOOOO0.0.0.....0.000....5‘ ix Figure Figure Figure Figure Figure Figure 15. 16. Expanded view in the region of the electronic origin. The lower REP curve (- - -) correpsond to the experimental data points and its corresponding error. The REP model Page b——-) is situated immediately above.............57 The experimental preresonance Raman excitation profile, and predicted excitation profiles for symmetric 2 modes (———) and for asymmetric modes (- - -) 17. A plot of the position of the first absorption (upper) and emission (lower) peaks of DPO as a function of the effective polarisability, (n2-1)/(n2+2), where is the solvent OCOOOOOOOOOOCOOOOOOOOOOO0..O0..0006‘ index of refraction.1 ..........................67 18. Visible-UV absorption spectrum of DPDP in EPA at 77 x ([J-[]) and the calculated absorption cross-section (——) . The effect of deleting the combination. band 3v1+v2 (---) from the calculated absorption cross- section is also shown...........................77 19a. The three fundamental Franck-Condon integrals v1, ya, a". Only v1 has negative values. The F-c origin peaks first for a given displacement paramenter A1. The amplitude of all except for v, approaches sero for A1>00‘...0..OCCOOOOCOOOOCOOOOOOOOOOOO00....0.080 19b. The Franck-Condon integral v1 and its overtones 2y1, and 3,1, The relative shift between the F-0 integrals is most apparent. It is clear that as A1 increases (at A1 > 0.3) these overtone amplitudes increases, but the fundamental v1 d.°r.‘..‘.OOOOOOOOOOOOOOIOOOOOOCOO....00.0.00.0082 figure 1 figure Figure Figu Pig Pit Figure Figure Figure Figure Figure Figure 19c. The Franck-Condon integral amplitude as a function of A1 for several combinations of v1, v2, Page and v.00.000000000000IOOOOOOIOOOOO...OO0.......084 19d. The Franck-Condon integral amplitude as a function of A1 for combinations of ml with u,..................86 The predicted resonance Raman exci- tation profile (———) and the REP with the combination band 3.3+»2 deleted (---). The deletion of this band affects the entire REP spectrum, in sharp contrast to the effect on the calculated absorption spectrum of Figur. 18.00.000.00000000000000000000.0.0...0.0.90 21. Expanded view of the 20000-22500 cm" region. for the REF shown in Pigur. zoOOOOOOOOOOOOOOOOOOOOOOO...0.0000.00....92 (a) The REP of DPDP with unadjusted displacement parameters (——). (b) The REP with the adjusted parameters (- -’eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeegg 23. (a) The effect. of adjusting the displacement parameters (- -) on the calculated room temperature absorption spectrum of DPDP. (b) The spectrum with the unadjusted parameters (-—)is consistent with that expected from Figure 4....................101 xi CHAPTER I INTRODUCTION For centrosymmetric molecules, such as all-trans linear polyenes, RfCH=CH+nR, in the absence of vibronic coupling approximately half of the electronic excited state manifold would be inaccessible from the ground state via conventional absorption spectroscopy because of the alternate parity dipole selection rule. As illustrated in Figure 1, group theory predicts selection rules for one photon absorption which require g»u transitions, while forbidding gag or uau transitions.1 Since the ground state of a centrosymmetric linear molecule is generally of g parity,2 then those excited electronic states which are also 9 states are "forbidden", although the associated vibrational levels of u symmetry are allowed. Thus a transition may be electronically forbidden but vibronically allowed. For two- photon absorption spectroscopy the opposite is true: g»g and u»u transitions are allowed,3 as shown by the dotted arrow in Figure 1. Such "hidden" states have been studied by two- photon induced fluorescence spectroscopy“!5 in which, for example, the polyene llAg » 21Ag transition is allowed. High resolution fluorescence excitation also has been utilized extensively in the characterization of the 21A,; Figure 1. Electronic energy level ordering for a typical linear polyene. A one-photon ( ) and two-photon 0- -fi dipole allowed 'transition. are shown by their respective arrows. The transition llAg 4 21Ag indicated by (-X-) represents a one- photon forbidden transition. Figure 1. state, investi‘ A polye either equilib: parity contin< molecu E ditnae State reso beer iJicj ecu: tile ilic den as; tin ‘32 [DE lit 31. fc: state, via experiments where the molecule under investigation is placed in an n-alkane Shpolskii matrix.6"10 A polyene placed in such a mixed crystal environment may either statically or dynamically deform from its C2h equilibrium symmetry, which leads to relaxation of the 11' Both ‘techniques, however, are parity’ selection rule. contingent upon an adequate fluorescence yield from the molecule of interest. Evidence that the ZlAg parity-forbidden state in diphenyloctatetraene (DPO) is the lowest singlet excited state was first provided by Hudson and Kohler12 via high resolution fluorescence excitation. This level ordering has been confirmed for several other all-trans polyenes, including the longest unsubstituted member studied to date, 6 This approach is, however, hampered by octadecaoctaene. the precipitous drop in fluorescence quantum yield with increasing chain length.7'l3 (Highly efficient radiationless decay pathways also exist for shorter chain polyenes, such as butadiene and hexatriene.5'11'14) Thus many linear polyenes of importance, particularly the longer chain members such as the carotenoids, have extremely low fluorescence quantum yields15 (compared to the UPC paradigm, for example), so that other methods must be utilized to locate the gerade excited states. Preresonance Raman excitation may provide a good alternative for the characterization of such parity forbidden excited states in many molecules. Although this technique is laborious, determining the location of the 21Ag state in carotenoids can be of great importance in understanding energy transfer between these antenna pigments and chlorophyll in the photosynthetic system.15'18 In previous studies the preresonance Raman excitation profiles of fi-carotene, lycopene and other long chain polyenes have been obtained.2°'22 However, the spectra were of uncertain reliability, and the assignment of the excitation profile features was not clearly established. Thus, the impetus for this project was the need to interpret the preresonanace Raman excitation profile of fi-carotene, for which reproducible spectroscopic features have been recently obtained.24 In the preresonance Raman process, as with conventional Raman scattering, an incident photon is destroyed, a scattered photon is created, and the molecular system undergoes a transition from an initial vibronic state |G> to a final vibronic state |F>. If wo<< up, that is the excitation frequency, (00, is far from an "allowed" electronic excited state (e.g. 113“), the process is known as ordinary Raman scattering. In this case the denominators in equation (2) of Chapter III can be taken as roughly a constant and (wo- wcp)‘ is the primary contributor to the Raman cross-section. For resonance Raman scattering the excitation frequency is nearly coincident with an electronic excited state 1 (i.e., wozwz), and the cross-section for certain modes is significantly enhanced26 (This scattering —._. _.-_ _ process inciden level 0 the gro between Raman s. in resc process. hidden plot 0' functic Plot 1: A featur and 1 under] benzer Shower of di‘ SoUrc 0th8r and J liteI prerc SYnmh Cans process differs from resonance fluorescence where the incident photon is absorbed, followed by a decay to a lower level of that electronic excited state and then emission to the ground state.) Scattering from excitation in a range between normal and resonance is defined as preresonance Raman scattering. The excitation frequency may, however, be in resonance with a dipole-forbidden state. These three processes are depicted in Figure 2. The presence of such hidden states is manifested as an interference pattern in a plot of the intensity of a Raman vibrational band as a function of the energy of the incident radiation. Such a plot is known as an excitation profile As early as 1971 Rimai et al.17 reported interference features in the Raman excitation profiles (REPS) of retinal and retinol, which were interpreted as arising from underlying triplet states. With the 182“ excited state of benzene used as an example, Korenowski et al.19 subsequently showed that the scattering of totally symmetric fundamentals of dipole-forbidden electronic transitions can be a possible source of measurable Raman cross-sections. A number of other experimental and theoretical studies of preresonance and resonance interference effects have been reported the literature.21'23'25'27'28 In this work we present a relatively high resolution preresonance Raman excitation profile for the strongest symmetric stretching mode of all-trans diphenyldecapentaene, €5H5(CH=CH)5C5H5 (structure shown in Figure 7), in solution. Figure 2. Three Raman scattering processes as a function of incident radiation: 1. Ordinary Stokes Raman scattering wo << w! 2. Preresonance Raman scattering wo < w! 3. Resonance Raman scattering wo ~ wp -7'f- - |F> 1III III |G> Raman Preresonance Resonance scattering Ramen Raman scattering scattering .Figure 2. The diphenyldecapentaene (DPDP) molecule was selected for this study primarily because data for the 21A,; and 118u excited states are available from high resolution absorption, as well as from two-photon and conventional fluorescence excitation spectra.9'29 We demonstrate a one- to-one correspondence between the predictions of the theoretical excitation profile and the features of the experimental REP, through the utilization of parameters obtained from the one-photon spectra. This enables us to propose an interpretation for the preresonanace Raman excitation features in terms of the underlying hidden electronic state. ||' CHAPTER II EXPERIMENTAL Pure samples of all-trans diphenyldecapentaene, prepared according to procedures refined by Spangler,3°'31 were kindly supplied by Professors R. R. Birge and C. W. Spangler. Samples were further purified to remove undissolved residue by filtering DPDP crystal flakes in hot cyclohexane solution, first through a paper filter and then through a syringe filter (0.25 pm pores) until all undissolved particles were removed. The solvent employed for the Raman measurements was a mixture of cyclohexane and chloroform at a volume ratio of 10:1, where the sole purpose of the chloroform was to keep the ~10'4M DPDP in solution. Equally important, chloroform has no Raman spectral features in the region of interest. The Raman spectrum of the DPDP solution is shown in Figure 3. UV-visible absorption spectra of the samples, such as that shown in Figure 4, were obtained with a Perkin-Elmer lambda 5 scanning spectrophotometer; no changes in the edectronic absorption were discerned from spectra obtained prior and subsequent to each Raman excitation profile measurement. 10 Figure 3. 11 The Raman spectrum of diphenyldecapentaene (DPDP) in cyclohexane/chloroform solution at room temperature obtained with excitation frequency 21720 cm'l. The solvent lines of cyclohexane and chloroform are identified by * and +, respectively. The DPDP lines at 1160 cm'1 and 1565 cm.1 are unmarked. Figure 4. 13 Absorption spectrum of diphenyldecapentaene (DPDP) of ~1x10’4M in a 0.5 cm cell at room temperature. The 0-0 band is at 419 nm (~23800 cm'l) The 0-1 band at 394 nm is not resolved to its vibronic components as it is at 77K in Figure 18. 14 *3 .v sunbum E5 cameo—c.6- HE” E in a... 1885...... eouoqaosqv 15 A Coherent model 100 Argon ion laser, operating in the multiline UV (MLUV) mode (essentially 363.8 nm and 351.1 nm), was used to pump a Coherent model CR-599 cw dye laser. Dyes used were stilbene 420 and coumarin 480, to cover the excitation range: 20000-22700 cm'l. The dye laser output power was kept constant at 45 mW and recorded to be 20 mW at the sample site. A concentration of 1.5 x 10'3M of stilbene 420 (also known as stilbene 3) dissolved in ethylene glycol had minimum lasing power output of 25 mW within the range 21400- 24080 cm'1 when pumped by 3.5 Watts MLUV from the Ar+ laser. The maximum power output was measured at 22700 cm'l, where the dye laser power exceeded 200 mW. A significant drop in laser power output ocurred within approximately 50 hours, indicating a chemical decomposition of the dye. Coumarin 480 (also known as coumarin 102) does not absorb well at 364 nm or 351 nm. However, an energy transfer from stilbene 420 to coumarin 480 can be employed, where the emission of stilbene 420 will be absorbed by coumarin 480 whose fluorescence is utilized to initiate the lasing process. To a liter of stilbene 420 solution 0.75 grams of coumarin 480 dissolved in 75 m1 of benzyl alcohol is added (Final solution concentration of ~2.9 x 10‘3M).32 Pumped by ~4 Watts of MLUV the available range with a minimum power output of 45 mW began at 19960 cm'l; the 1 maximum power peaked at 21400 cm- , (The power output of various dyes generally has a Gaussian like shape as shown in Il' 16 Figure 5. The power output of several dyes, pumped by a cw Ar+ laser, as a function of the dye laser output wavelength (nm) range.32 17 . n ennui.— .Ecv Eggt; ‘fiifl..z- .' ="i-ufihgl 4 xEras-ESiIZ-Esd . 176.322.: 1.: i .. m 2.. I‘...r‘4.-!I‘. 14. / i... m \ or»: I ........._.. a... t \\ A!“ ‘ . ti... 5.. \ Ml \iflal... Figure 6. 18 A schematic diagram of the apparatus utilized in obtaining the Raman spectrum. ._.—.___—— ._.‘ 19 Monochromator computer control ‘I Illuminator & Sample Chamber Double Monochromator ._l Spex 1401 a I « . RCA PHT c31034a | l I r 4 I , I x-r ' Recorder __ DYE Coherent Laser CR-599 A I l I coherent Ar+ Innova 100 Laser Figure 6. 20 Figure 5. Thus it follows that the total available range from coumarin 480 under our experimental conditions is from ~2oooo - 22800 cm'l). The Raman scattered radiation from room-temperature samples was collected at 90° to the incident laser beam, passed through a computer-controlled Spex 1401 double monochromator , and detected by an RCA C3 1 0 3 4A photomultiplier tube cooled to -20°C. A schematic diagram of the apparatus utilize to obtain the Raman spectra is shown in Figure 6. Each Raman excitation profile data point, which is an average of several measurements, was determined by obtaining the ratio of the area of the C=C symmetric stretch of DPDP at 1565 cm"1 to the area of the 1445 cm"1 band of cyclohexane. This can be done since the 0-0 absorption band for cyclohexane is at ~40000 cm'l. Thus being far from resonance for cylclohexane or chloroform the preresonance REP range of 20000-22700 cm"1 corresponds to ordinary Raman scattering. Therefore by taking the ratio of the two peaks the [(u-ugp)‘ /(u-ugrp')‘] ~1, canceling the v‘ effect does cancel. Consequently the Raman intensity of the REP is entirely due to DPDP. The ratio is obtained by first drawing, a base line connecting both peaks, this is followed by enlarging the peaks, (using a Xerox copier' with no distortion), to minimize the associated error in the cutting and weighing. Each individual peak is then cut out and weighed with an analytical balance. Since the paper density III" 21 is essentially constant, the weight ratio will be identical to the ratio of the corresponding peak areas. Figure 7 and Figure 8 shows such a typical Raman spectrum. The points in regions where the REP showed intensity ratios departing from the expected monotonic preresonance increase were repeated with several different samples on different occasions; each time these interference features were confirmed. The Raman excitation profile of DPDP over the range 20000-22700 cm'1 was obtained in several overlapping segments. The DPDP concentration at the lower end, the 20000-21000 cm'l region, was adjusted to be slightly higher than its concentration at the upper end; however for all segments it was maintained in the ~lo'4M range. Concentration adjustment was necessary for otherwise the height of the Raman-active ground state C=C symmetric stretch would be either too intense or too weak relative to the nearby cyclohexane reference band, which was used as an internal standard. All REP segments were then scaled to the concentration of a central segment in the 21200-21700 cm"1 range. As an example the segment identified by A-A in Figure 9 is multiplied by a factor of 0.1825 and the segment identified as []-[] is scaled by a factor of 1.28. The central segment o-o is left unchanged. Data points of an REP segment which were not reproducible in the REP segments, such as the ones labeled by X, were not utilized in the final composite preresonance REP. Once normalized, an Figure 7. 22 A typical Raman spectrum of DPDP in cyclohexane /chloroform solution at an excitation frequency of 22200 cm’l. The more intense band at 1565 cm"1 is the symmetric C=C stretch and the cyclohexane peak at 1445 cm'1 is used as an internal standard. The structure of DPDP is also shown. 1445 cm'1 23 Figure 7 . Figure 8. 24 A typical Raman spectrum of DPDP in cyclohexane /chloroform solution at various excitation frequencies. At ~2000 cm'1 below the 118“ state, the cyclohexane peak at 1445 cm"1 is of greater intensity (bottom). At 21650 cm'1 both peaks are of approximately equal intensity (center). The DPDP 1565 cm'1 band is most intense for the excitation frequency of 22200 cm'1 (top). Figure 9. 26 A sample of the preresonance REP segments after adjustment were made to compensate for differences in concentration. Each segment other than the central segment was multiplied by an appropriate value. Points marked X, which were not reproducible in all overlapping segments, were not included in the the final composite spectrum. 27 . a 9:5: CIEov «rearrange: OOVNN 8m — N GO? pN 88w DOVON b F b L b b P b b b h b h f b b b L P i b i- . OO OIICIIICIIICICIThil \I lm.O .. 1.. \I a ~ 4 a. es re .1. r. - . p .. fie h. \K ‘\ s ‘ I X The. —. s s s .. ION . . e 1 . 1 e. .. A 3.. x V 0a . A 8.. x v 010 Ind m A no.6 .. V 4.. P P In P ”Ii h h I? b b h b F Iii P P i it IIPII.I|IPIIII.L (suun mmmv) flaw-“nu: “some averag profil Obtains compos excita C analyt' develor prograr These disper: (equat. The re harmon formu1 Vibrat StateI the e Pr0fii 28 average relative scattering intensity for each excitation profile point and its corresponding error distribution were obtained from the repeated measurements. The final composite REP, the intensity ratio Ic=C/Iref vs. dye laser excitation energy, is shown in Figure 10. Computer programs for determining, graphically and analytically, the REP and the absorption cross-section, were developed for an IBM AT personal computer. (The detailed program listing is given in part B of the appendix I.) These programs are based on the Kramers-Heisenberg dispersion relation and the absorption cross-section (equations (4) and (11), respectively, in the next Chapter.) The requisite Franck-Condom integrals were calculated using harmonic oscillator eigenfunctions in the multimode formulation (three or four modes). A total of 23 vibrational states, 7 for the 21Ag state and 16 for the llBu state, were found necessary for an adequate description of the experimental spectra, both for the Raman excitation profile of DPDP and its absorption cross-section. 29 Figure 10. The experimental ([]-[]) preresonance Raman excitation profile of the symmetric C=C band of DPDP. The vertical bars correspond to the deviation error associated with each point of the REP. 3O . ou ensur— CIEov mcobEsco>cz coeum com .N DON . N cowoN oooow bPI—LIFPFLr-lbbib bquiPLlllrLerLbLibl 8.o eh elsefle 4 .TQOIIII...III..O.I... 4 a... e V. 1.1.43... t. 1...... s \ ' F? LlrkiLI:.;.-p-I-Ll||—Iluplllrllrl. PILLinrir . L. r .— --..I psi.-- pinhllr (suun Mommv) Kuwaiti: Mamas T transi excite molecu where scatte fine compor 1 Krame] Pa ' vibror incide OPQrat 31 CHAPTER III THEORY The Raman scattering cross-section for the Stokes transition from an initial vibronic state | G) to a final excited state |F>, for a system of randomly oriented molecules, is given by33 25 3 2 2 4 2 aGF = " [—]<£lpalG> <£|pp|G> + ”o + 1F 9 II 3| H w (2) pa wg- wo + if hat") 2 1 2 ‘where if) is the complex damping factor associated with the ‘vibronic state |1> of energy fiwl, hwo is the energy of the incident radiation, pp(a) is the total dipole moment operator (nuclear and electronic) in the p(a) direction, and dil‘a momen vibra excit compo the t: final More0\ an all (2) d SBCOI‘) aPPrc initj fina: of t] where respe mOmen 32 <2|p0|G> is the ath component of the transition dipole moment associated ‘with the transition from the initial vibrational level of the ground electronic state |G> to the excited vibronic State |£>. Similarly is the pth component of the transition dipole moment associated with the transition from the |2> excited vibronic state to the final vibrational level of the ground electronic state, |F>. Moreover, in the near preresonance and resonance region of an allowed electronic transition the first term in equation (2) dominates and we may neglect the contribution from the second term, the antiresonance term. We next apply the adiabatic Born-Oppenheimer approximation. The wavefunctions corresponding to the initial ground state, G, the intermediate state, 2, and the final excited state, F, may then be expressed as the product of the nuclear and electronic wavefunctions. wg(r,R) = og(r)eo(R) .4 |G> = |g>|0> t,(r,R) = ¢e(r)9v(R) «4 |2> = |e>|v> WF(T,R) = og(r)en(R) .4 |F> = |g>|n> where r, R are the electronic and nuclear coordinates respectively. We now write the total transition dipole moment as : fl = #nuc + #e 33 The 'transition moment in the .numerator of equation (2) becomes: (Flpp|£> = + (2a) = }|e> + }|e> or = <0|pnuc|u> + }|v> (2b) Since the electronic eigenfunctions are orthonormal, that is J Q*g(r)ée(r) d3]: = Sge where age is the Kronecker delta, the first term in equation (2b) will vanishe for gee and the above reduces to (Flpp|2> = }|u> Similarily for <2|pa|G>. Thus the first term of the polarizability tensor, apa, in expression (2) becomes X (11' [flpJge |v> and [pa]eg a are components of the pure electronic transition moment, and where |0> and |n> are the initial and final electronic ground state 34 vibrational levels, respectively. For a totally symmetric vibration of a centrosymmetric linear polyene, only one diagonal element of the polarizability tensor, app, need be evaluated.36 Since the electronic transition moment is a slowly varying function of the internuclear distances, it may be expressed as a rapidly converging power series expanded about ‘the equilibrium. position in. the normal coordinate 90-27 = [pgemo +2; [g%g§]Q§Qn-Qo) + (3a) The numerator of equation (3), , then becomes = x } or d [ + z [5%gfi Q0 + ...] a X [ + g [5%53 Q:UI(QK-Q0)IO> + ...] } 35 The first term in each factor simplifies, giving ={ [ [I‘geJQo (“I”) 4'; [%fg]qo + g [ggf9]0:u|(Q‘-Qo)|o> + ...] } After expansion, the product 2 = Iflgel Q0 (3b) + [#geJQo g [g%§]Q:an.-Qonv> d + [4.919. g [55—59]Qol0>} where the high order terms has been neglected. For a single normal coordinate, Q“, the product 2 = lpge|Q0 du _ + [pge1Qo [555:]Qo 3 + (eagle. [Tag]0.} (3C) 36 For a given excited electronic state e, equation (3) becomes w-wo _ 1 2 - .{ lugelqo 2 . .r Ue U ‘0 an + [#geJQo [fi‘flQ‘ ]Q0 §e ‘0”:in + 1F (nlv> + [#ngQ [60. ]QOX .e ”u‘ ”0 +£1F. } (3d) The first term in the series (known also as the A-term) will dominate provided that the magnitude |(apge/6Q)0|<< |(pge)ol. This will be valid for a centrosymmetric linear polyene in the preresonance or resonance region for a totally symmetric vibrational mode.37 In addition, for the case of a dipole forbidden state, the need for the second term (the B-term) in the expansion can be avoided by introducing an empirical value for the forbidden transition moment.19 Under these conditions, the polarizability expression for such a system with two excited electronic states, e and 5, reduces to _ 1 Ee< e|0> 2 <-s|5> app " ; Iflgel2 u v + ll‘gsi 2 u v (4) w-wOU-I-ll‘ w-w+lI‘ v vsu 0 v 37 where the coefficients |pge|2, |pgs|2 are proportional to the corresponding oscillator strengths for the transition from the ground state to the electronic states e and s, respectively. Namely, feg =[(8th)/(6fie2)]ulpeg|2, where u is the wavenumber of the transition and Ipl is the electronic transition moment.38 In the limit of no Duschinsky mixing (normal coordinate rotation), the multidimensional Franck-Condon integral can be factored into products of one-dimensional integrals39'41 m = IT (5) i=1 where m is the number of normal modes. In particular, for Raman scattering arising from the ath, totally symmetric fundamental mode, the multimode expression becomes (Ig|;e><;e|ag > = <1aglvae> lfla(6) For example, in the case where the ath mode corresponds to V1 , in a multimode formulation involving only three modes: 1, V2, v3 and for excitation in resonance with the vibrational fundamental V2 in the excited state, equation (6) reduces to <;e| 6g> = [<0e| 09>]V1 [<°g| le><1e|0g>1y 2 (7) x []y3 38 The Raman Franck-Condon integrals utilized in the REP model are shown in Appendix II, Table A. We may describe the Franck-Condom factors by the harmonic oscillator eigenfunction basis. These overlap integrals are a function of the displacement of the excited state potential curve for each normal coordinate from the corresponding ground state position, Aa . (Qag'Qae)or and of the corresponding vibrational frequencies in the ground (vg)a_ and excited, (Ve)a electronic states.39'41'43’ The relationship of ground and excited electronic potential curves to one another is exemplified in Figure 11. The one-dimensional overlap integrals may be evaluated by using the Manneback recurrence relationszzé'44l45 <0|0> = Mxex - :2 __E." "g A2 ] (8) He + vg P f; ye + ”9 eg < I +1) = m “M < | -1) + v “2 V V )%< -1Im > ”emg mye+ugvem9 mye+ug ”e 9 V V ‘/ 1 ' 41cc ' ”2 ug' (v +1 > - - ” Zig_:_£fi < -1 > + m %2(" V )%< |m -1> e lmg - u+I Ve + ug ”e lmg u+I we + 119 ”e 9 Y V y 1 ' 43c 'ug Ve' - [v+1] [ fi ] we + ygAeg (10) 39 Figure 11. A two electronic state system with a displacement parameter shift, Aeg. The electronic transition moments associated with the Raman process and the harmonic oscillator wave functions representing the ground and fundamental modes are illustrated. {Some authors express A2 by multiplying it by («C/h)v; this results in a unitless expression with a numerical value greater by a factor of 10 than that given in (amu)%A }.43r45 40 Ekuwgy figure 11. 41 Thus the FTanck-Condon integrals which constitute the zlAg vibrational states utilized in the model (Plus the overtone, shown here for completeness only) are: y __ 2(v v )‘ _ «c ll v 2 (0'0) _ W exp[ ? fl A eg] (8) From equation (9) <0 |1 > = 2['° Z V (V )% A (9a) e9 T337337” From equation (10) <1 |o > = -2 12 Z Yin—1131;.A (10a) From equation (9) “WW-3)Z 4‘3 (”gVe)3/2 2 <1g|19> = ——————_— - —_— 2 A99 Ve + Vg h (Ve + Ug) (9b) From equation (10 1 V9 ' Ve 41C Ve(Vg)2 2 <0 |2 > = __ ________ + Ae g e 2 9 I2 Ve + Vg fl (He + Vg) (10b) [Zuc]% % h [Vel’g] 41": (V9) 3/2 (Ve) 4/2 2 ‘<1g|2e> = Aeg (Ve + Hg) h Ve + Vg - Z - - Z 2<0 0> [Va] [”9 w.) 4["e] [Vg] Aeg I (10c) 42 In the specific case of the linear polyenes, for the electronic transition from the llAg ground state to the llBu excited state we may take ”e 9! ug, where ye, :19 are the vibrational frequencies of a given mode for the corresponding electronic states. [Of course, this approximation does not apply to the 11AgazlAg (hidden) transition, because of large frequency differences, as much as 250 cm"1,6 associated with vibrational modes of linear polyenes in the ZlAg excited state.) This leads to considerable simplification in the above recurrence relations, equations (8)-(10) «c v <0|o> = exp[-fi —— 5 (Aeg) 2] (3') U % = [5:1] [4;C / uZ +[-—1J%—§— Aeg (9’) K m " [71:ng l—‘éfi Aeg <10” 43 Thus the Franck-Condon integrals utilized for the REP model are: (0'0) = exp[- Egg (Aeg) 2] (8’) From 9' x Z (03' lg) =2 [ £2] :5 Aeg (0'0) (9a,) From 10’ Z a c <1e|og> = '2[1i ] 15 Aeg (lOa’) 2 <1g|1e> = [1 - g5 u (Aeg) ]<0|0> (9b') = [1% ] u (neg)2 (lOb’) 21c K x «c 2 , <19|2e> = [h ] V :7] (Aeg) "' 1 (Aeg) <0|0> (10C ) 3/2 -1 _ «c 2 -1 «Cu 2 <1g|3e> - 1 - I2 { 57:](Aeg) - 1} T5 h ](Aeg) <0|0> (loe’) 44 1/2 -1 <09l4e> = :- [Kill] (Aeg) <3e|Og> (lOf') 1/2 1 1 c , <1g|4e> = §[<4el°9> - 5 [*fi”] (Aeg) <3e|1g> (lOg ) The absorption cross-section39'4o'46'47 is given by “abs 3 g (11) P” is the homogeneous linewidth of the vibronic state |n> having energy hwy, and hwo is the energy of the incident radiation. The Franck-Condon integrals utilized for the model absorption spectrum are collected in Appendix II, Table. B (The IRaman Franck-Condon integrals differs from ordinary Franck-Condom integrals in being a product of two Franck-Condon integrals rather than a single integral). As with the Raman cross-section, the sum is over the vibronic subspace associated with the electronic state e. CHAPTER IV RESULTS 4.1 vibronic Features of tho Prorosonanaco Raman Excitation Profile The. experimental REP for the symmetric» C=C stretch (hereafter designated ul) of DPDP in cyclohexane at room temperature in the preresonance region from 20000-22700 cm-1 is shown in Figure 10 on page 22. A vertical line designates the reproducibility of each data point. As the incident radiation is tuned towards the llBu dipole-allowed excited state, for which the 0-0 absorption maximum lies at 23,750 cm'1 in cyclohexane, it is expected that the Raman intensity would continue to increase monotonically,27 if the llBu state was the lowest excited singlet state. (This is simulated in Figure 12 by a calculated excitation profile for which the 21Ag oscillator strength is chosen to be vanishingly small.) Instead, we observe spectroscopic features which result from constructive and destructive interference between an underlying, weakly-allowed state and the nearby llBu excited state. The departure from the smooth intensity increase in the preresonance REP of DPDP shown in Figure 12 manifests itself 45 46 Figure 12. Experimental preresonanace REP ( []-[] ) and the calculated preresonanace REP ( ——— ) (fourth v mode is included). 8 47 .NA auscuu ClEov €0on533 §NN COG pN CON pN 860m OOOON D b h b P D .P P tr b P D b i. r? b p P b b b TL r P b O I. .xuo II !.|I......... ...-3...... ... . 1 100A. T e mow.— Too; r IOQN I. 1 184.. L D P L [D b F D b b b # b b b L b b P b r P P I (suun Momqav) msuawl Mamba 48 1 initially as a steep ascent which begins at ~21100 cm" and continues to ~21300 cm"1 , where it culminates in a plateau on which there are several undulations. The lower energy features, at approximately 21350 cm’l, and 21425 cm'l, as well as the higher energy components at approximately 21700 cm.1 and 21900 cm'l, dominate the preresonanace region. A relatively sharp drop in intensity is measured near 21975 cm'l, before the excitation profile resumes its monotonic increase in intensity as the strongly allowed llBu state is approached. From the experimental data, the depth of this fairly sharp dip appears to be a function of solvent polarizability (i.e. concentration of chloroform in cyclohexane ). Moreover, it seems to be part of the vibronic pattern associated with the feature at ~21900 cm'l. Unlike peaks in fluorescence or absorption spectra, due to the overlapping interference effects of several vibrations the structure in the Raman excitation profile is not easily directly associated with the 21Ag vibrational modes. Consequently the interpretation of the preresonance Raman excitation region requires additional analysis, which is provided by a theoretical approach. In general, for linear polyenes the C-C and C=C vibrational modes dominate in fluorescence, fluorescence excitation, and absorption, for both the 21Ag and the 118“ excited states.11'48 For DPDP, the most intense band seen in high resolution fluorescence excitation of the 21Ag state is assigned by Horwitz et al.9 to the symmetric C=C stretch, 49 at a 1750 cm"1 shift from the electronic origin (The fluorescence excitation spectrum from reference 9 was manually fit and digitized; it was then simulated by a theoretical model as shown in Figure 13. The absorption cross-section expression of equation (11) is employed for this purpose. A quick look at the fluorescence excitation spectrum for the llBu state of reference 9 and the 77K absorption spectrum in Figure 18 will convince the reader of this.) It seems reasonable to assign by direct comparison the strongest feature in the preresonance Raman excitation profile, at approximately 21900 cm'l, to the same C=C vibration. A similar comparison for the remaining spectroscopic features is not as straightforward, primarily due to the presence of the interference effects. It is here that the Raman excitation model is of paramount importance. The interference features generated by the model on the basis of very few parameters, obtained without adjustment from other sources, replicate the experimental Raman excitation spectral data as illustrated in Figure 14 .. Although the simplified model does not account for all the details, it does satisfy our primary objective by providing an interpretation for the salient aspects of the excitation profile. It enables us to establish a one-to-one correspondence between the prominent Raman interference features, belonging to the hidden 21159 excited state, and the vibrational frequencies previously determined by various Figure 13. 50 High resolution fluorescence excitation of the 21Ag state. The digitized data points were obtained from reference 9. The effect of including ( — ) and excluding (- - -) the us mode from the multimode formulation in the best fit. model is shown. .Although this effect is easilly seen here, it is not as apparent in the preresonance REP shown in Figure 12 and Figure 14. 51 COtNN L (p . bl . nu encode AplEov «tooncgo; COW—N CON—N b p . oomow SOON 1P bib IPL * L b L L800 10¢.o 18.6 wow.— 100.— (suun mum) KIEWOIUI sp« id 5P in vi 52 spectroscopic techniques. Thus the vibrational mode identified as the C-C stretch in the fluorescence excitation 9 spectrum of Figure 13 is manifested in the REP as an interference feature on the plateau at approximately 21425 cm'l, as seen in Figure 14. This corresponds to a vibrational frequency of 1250 cm'l. (A number of fundamental vibrational frequencies are observed in the 1000-1400 cm':l region for a typical linear polyene349'501 such modes may account for some of the additional structure observed in the vicinity of 21425 cm'l.) The remaining prominent interference features can be interpreted in terms of combination bands of only three polyene fundamental modes, which we will call V1 (symmetric C=C stretch), V2 (symmetric C-C stretch) and v3 (symmetric bend). [A fourth mode, assigned as a phenyl vibration (us) in the fluorescence excitation,9 is included for completeness. Although its presence is clearly observed in the fluorescence excitation model, it has very little effect on the preresonance REP.] 4.2 The Zing Electronic Origin The electronic origin for the 21Ag excited state, being of gerade vibronic symmetry, is inaccessible by a one-photon process from the ground state. However, such origins have been detected by high resolution fluorescence excitation when the guest molecule is situated in a symmetry-perturbing 53 Figure 14. The calculated preresonance Raman excitation profile of DPDP (—) and an REP for which the 21Ag oscillator strength is chosen to ibe vanishingly small (-——- ———). The effect of excluding the u, mode from the multimode formulation for the REP (- - -) is also shown. 54 .en euauuh ClEov Eooncgoz COGNN Dawn—N GOO—N canON OOOON b P L L-r * b ir - L p b b P b p b b h b r b e OOAV v \ a \\\\ \\\\ I \\ 19.0 \ \\. s \ s \\ I \ roan. \ \ \ s x l ‘ r O \ ON . x .\ e t e x \ \ ‘s TOO a \ F Lillrir P L 'P L b b ¥ P P b IIIPII r P b b (mun 00mm) WWW-ll 55 environment, such as a Shpolskii matrix.7'10 The electronic origin of the ZlAg state of DPDP has been observed at 20130 cm-l via absorption and fluorescence excitation measurements on the molecule isolated in an n-decane host at 4.2 K.9 (In hosts where polyenes retain their center of symmetry the origin is absent, yet strong spectroscopic progressions built on a low-frequency promoting mode are observed.51) In the preresonance Raman excitation of DPDP, two weak 1 are observed in an otherwise features separated by ~200 cm" featureless region in the vicinity of the reported origin, one at 20175 cm"1 and the other at 20375 cm-l. An expanded view of this region of the REP is shown in Figure 15 . Two possible interpretations are: (1) Two low-frequency vibrational modes, and (2) The 21Ag electronic origin and a nearby low-frequency vibrational mode. Low energy vibrational modes are commonly observed for 1 6,49,51 the polyenes in the 50-300 cm' range; first interpretation would thus place the 21Ag origin below 20175 cm'l. However ‘we favor' the second. possiblityu .A low- frequency vibrational mode at 210 cm'l, assigned to a symmetric bending motion, is seen in fluorescence excitation of DPDP9 and it may correspond to the second feature (with the higher intensity) in the REP. The first feature would then correspond to the 21Ag electronic origin; this assignment cannot be ruled out, because even if it is assumed that the electronic transition moment pge(Qo)=0 at 56 Figure 15. Expanded view in the region of the electronic origin. The lower REP curve (- - -) correpsond to the experimental data points and its corresponding error. The REP model (———) is situated immediately above. 57 8mg L ¥ . an ensur— CIEoV 90on53: r F 8mg T + L L ammou + w t L i wood Feud (suun Momqsv) Kassuazm “£10198 58 equilibrium, vibrationally-induced intensity will occur because of the fluctuation in pge due to zero—point vibrations of the nuclei.52 The extreme sensitivity of this forbidden transition to small perturbations is demonstrated by polyenes for which the origin is absent in one-photon spectra in a particular, matched n-alkane host, but is observed in other n-alkane hosts differing by only one or 11 Thus small static distortions give two carbon atoms. visibility to the zlAg origin; so might small dynamic distortions due to asymmetric vibrations and collisions in solution. We therefore assign the weak peak at 20,175 cm'1 to the origin of the 21Ag state of DPDP in cyclohexane solution at room temperature. DISCUSSION 5.1 The Raman Interference Effect The contribution of weakly allowed states to the Raman excitation profile intensity may be understood by considering the Raman scattering intensity, which is proportional to |apa|2, where a is the polarizability pa tensor. When a is expressed by the Kramers-Heisenberg pa dispersion relation of equation (4), it is seen that "mixing" cross terms arise, which may reduce or enhance the Raman intensity. This is due, in part, to the Franck-Condon factors, for which the sign is a function of the displacement parameter, A1. In sharp contrast to Raman excitation, each term of the absorption cross-section, given by equation (11), contributes solely to the intensity of a single state, since 'the absolute square of ‘the overlap integral leads only to the enhancement of its intensity: hence, there is no "mixing" of states. As an illustration, consider IapPI2 obtained from expression (4) which is given by53 59 60 A12 2 In I = 2 pp i=1 6wi’ + F12 j>pge. For a given vibrational mode j of the weakly allowed electronic state e, Aj will be very small. Its contribution to the first term of equation (12) may therefore be neglected. The Raman cross terms will then be the only measurable contribution to the Raman intensity, which otherwise may not be detectable within the signal-to- noise limits of the experiment. The interferences in the preresonace REP of DPDP, shown in Figures 7 and 9, demonstrate this effect. The proximity of the two electronic excited states is a crucial factor in the determination of the intensity level for the interference features in the REP. This is easily seen from the cross term expression of equation (12). If the incident radiation wo is in resonance with state j then 61 8wj=0 and the maximum contribution of the cross terms will occur when the separation between the two coupled electronic excited states is at a minimum, since then Swi will also be smallest. Symmetry-forbidden transitions generally have oscillator strengths in the 10"3 to 10"2 range,10 in comparison to ~10”6 for a singlet-triplet transition.23 Strong dipole-allowed states, on the other hand, have oscillator strengths of 1.0-1.5.10'38 For example, the 11Ag .. 118“ oscillator strength for UFO in EPA at 77K is 1.5;2'10"12 for the llAg .. 21119 transition it is 0.05.10 The ratio of the ZlAg transition oscillator strength to that of the llBu state is thus 0.03. The value which best fits our' experimental data is 0.04 (This ‘would approximately correspond to the ratio of lpge|2 and |p98|2 in equation 4 of the polarizability tensor, app); this ratio is the only parameter which was allowed to vary in our treatment of the DPDP preresonance REP. The value is obtained by taking the ratio of the multiplicative constant 425 and 10225 in program lines 24020 and 24060, respectively of Appendix I part B. From ‘the argument already introduced. that g»g transitions are parity forbidden it would seem natural to assume that the preresonance Raman excitation profile structure would be of ungerade character. Then the overlap integrals for the Raman scattering arising from the ground state v1 , in a multimode formulation involving four modes: 62 V51, val, uaz, ”as and for excitation in resonance with the vibrational fundamental V2 in the excited state would be given by: = [<1g|oe>]$,,1[<1e|09>]ay2[]av3 (13) where the superscripts s and a are the symmetric and asymmetric vibronic modes, respectively. This results in a preresonance Raman excitation profile whose features, as shown in Figure 16, are weaker than those in Figure 10, where it was assumed that the associated vibrational level in the 21Ag state are totally symmetric. Moreover, for the best fit value the oscillator strength ratio is about 0.08, a significant deviation from DPO. The arguments employed to account for the observation of the 21Ag origin in the REP apply as well to the intensity of symmetric fundamental modes in the hidden electronic state. Consequently the vibrational modes which are associated with the electronic 21Ag state, which give rise to the preresonance REP features are assumed to be of totally symmetric character. 63 Figure 16. The experimental preresonance Raman excitation profile, and predicted excitation profiles for symmetric 21Ag modes (———) and for asymmetric modes (- - -). 64 . on euauuh CIEUV «tobEscgoz ocean coo — N OON pm oomow oooow IF Lr Ll + L P b L-rL .P by L III P b 'r * I? L LL L b b b \r b 80° ...e 9 _ . . MW known-.-.. 1 ’4‘“. a. \ $ \. 18o \ TON; f 4 men.— s Tofu 7 a road fllhntc pliP--lbl£Lil.P P p k p p h-L1€.-: L . .2, .7 p- b 3.. .,,,- .. p- IL (suun mmmv) msuowl “nosey 65 5.2 Solvent Polarizability Effect As the solvent polarizability increases, the absorption band for the strongly-allowed polyene 113“ state shifts to lower energy.1°'12'54'55 The wavenumber shift is given bylo'll 5V = k feg[(n2-1)/(n’+2)], where su is the shift in cm"1 , n is the refractive index of the solvent, fag is the oscillator strength for the gee transition, and k is a molecular constant. This effect is illustrated by the DPDP molecule, where the llBu electronic origin is shifted from 23050 cm'1 in EPA at 77K29 to 23750 cm“1 in cyclohexane at 300K. The ZlAg state, however, is in general little affected by solvent polarizability. In fact, a plot of the emission band of UPC as a function of the effective polarizability, (n2-1)/(n’+2), reproduced in Figure 17 from reference 10 reveals that the slope is nearly horizontal (only 0.1 times that of a similar plot for the strong llAg » llBu absorption).10 The llsu state shift from 23068 cm'l, reported by Horowitz et al.,9 to 23750 cm'1 for DPDP in cyclohexane at room temperature. Thus the corresponding predicted shift for the 21Ag state origin will be (0.10)(682) ~ 50 cm'l. Therefore, to account for the differences in solvent and temperature, the 77K fluorescence excitation spectrum reported in reference 9 was shifted 50 cm'1 higher before it was employed in the interpretation of the preresonance REP of DPDP in cyclohexane. 66 Figure 17. A plot of the position of the first absorption (upper) and emission (lower) peaks of UPC as a function of the effective polarizability, (n2-1)/(n2+2), where n is the solvent index of refraction.10 excmmou ENERGY (cmT'I 10000 67 28 f— f T r 27- ABSORPTION 26 r— 25- 24L- 2 ‘mguou 22 w L L l O J .2 .3 (oz-0N0: 0 2) riguro 17. 68 5 . 3 The Semi-Analytical Transform from Absorption to Raman Excitation 5.3.1 Pranck-Condon rectors The theoretical REP is obtained from magnitude square of the diagonal of the polarizability tensor, which are written in terms of the Kramers-Heisenberg dispersion relation of equation (4). The Franck-Condon integrals are expressed in terms of harmonic oscillator eigenfunctions, which require experimental values of the following parameters: 1191, and vei, the ground and excited state vibrational frequencies for mode i, and IAilr the shift in the potential minimum between the ground and the excited state for that vibrational mode. The vibrational modes and displacement parameters utilized for the zlAg and llBu excited states of DPDP are listed in Tables I-III. The values of vi required to calculate the Franck- Condon parameters for the llBu excited state were obtained directly from the absorption and fluorescence excitation spectra of DPDP shown in Figures 3 and 15.9,29 The mode displacement parameters listed in Table I for the llBu excited. state *were (obtained. by "fitting" the absorption cross-section of equation (11) to the experimental absorption spectrum of DPDP in EPA glass at 77K,29 as shown 69 TABLE 15 Displacement parameters, Ai 10.005, in units of (amu)%A, obtained by fitting the absorption spectra to equation (11). 70 21Ag ilsu Parameter 4-mode 3-mode 3-mode 2-mode A],1 0.36 0.40 0.17 0.11 sz 0.30 0.32 0.15 0.24 AV 0.37 0.35 0.10 - 71 TABLE II. The llBu excited state vibrational modes (cm-1) and their corresponding' Raman Franck-Condon integrals for the C=C symmetric mode used in the Raman excitation model. 72 Assignment u Au F-C integral x 10‘"3 900 23750 - 274 us 23950 200 8.11 V2 24950 1200 109 u? + V3 25150 1400 3.24 V1 25300 1550 -92.3 V, + 93 25500 1750 -2.73 2v2 26150 2400 43.7 2V2 + V3 26350 2600 1.30 V1 + 92 26500 2750 -36.9 2111 26850 3100 -171 2v1 + V3 27050 3300 -5.06 2V2 + ”1 27700 3950 -14.7 2u1 + 92 28050 4300 -68.3 3v1 28400 4650 -77.8 291 + 2V2 29250 5500 -27.3 391 + V2 29600 5850 -31.1 73 TABLE III. The 21Ag excited state vibrational modes (cm-1) and their corresponding Raman Franck- Condon integrals for the C=C symmetric mode used in the Raman excitation model. 74 Assignment 9 Au F-C integral x 10'3 V00 20175 - 7.70 V3 20375 200 3.12 v2 21425 1250 12.3 v, + us 21625 1450 4.98 V: 21725 1550 2.14 ”1 21925 1750 15.6 :11 + V3 22125 1950 6.31 75 in Figure 18. A two-mode model inadequately reproduces the structure at higher energies (see Figure 18); addition of the low-frequency bending mode provided satisfactory simulation. Each A1 was varied until the following best-fit values were obtained: Ac=c=0.17 (amu)%A, Ac-c =0.15 (amu)%A, and A”: =0.10 (amuf‘A. These mode displacements compare closely to those of octatetraene, for which Ac=c = 0.14 (amu)%A and Ac-c = 0.13 (amu)%A, respectively,56 for the only two modes reported. Similarly, for the 21Ag excited state we use the relative peak intensities from the experimental DPDP 9 reconstructed in Figure fluorescence excitation spectrum, 13, to determine the A1 given in Table I. Both three-mode and four-mode models were employed. The electronic origin, "00! and the low frequency vibrational mode, 113, are the least intense among the prominent vibrational states. To simulate this condition for the 21Ag "absorption" model, the polyene mode displacements must have values greater than those of the llBu excited state, since for the latter the origin is the most intense peak. These larger values of the 21Ag displacement parameters can be understood on the basis of a plot of A1 as a function of the Franck-Condon amplitude, which shows that as the value of the displacement parameter increases, the intensity of the electronic origin decreases from its peak value, while that of fundamental 76 Figure 18. Visible-UV absorption spectrum of DPDP in EPA at 77 K ([]-[]) and the calculated absorption cross— section (———). The effect of deleting the combination band 3u1+u2 (---) from the calculated absorption cross-section is also shown. 77 r5111.u~t-.i|u illriiuhui .L- . on 0.25: CIEUV 9055:5663 u illbittlibit 80mm b L 'br b r b --P.!-«|Pliiub!l-lr ... P a r8.N— (suun Mommv) 441303101 78 vibrational modes increases. In Figure 19a, where the overlap integrals for a 3-mode model are plotted, the V00 formulation, namely [<1g|03>]y1[]u2 x []ys, peaks at 0.15 (amu)%A and then begins a steep decline; that for V1 that is, [<1e|°g>]ul x[<0e|09>]V2[]ys peaks at 0.30 (amu)%A, for v2 namely [<1g|0e>]y1[<1elog>]y2 X[]y3 peaks at 0.24 (amu)%A, and for V3 the product peaks at 0.55 (amu)%A. Since the overtones 2v1 and 3v1 are shifted relative to 91 as illustrated in Figure 19b, a large A1 will result for these modes in acquiring significant intensity. For’ completeness Franck—Condon integrals of various combination modes are illustrated in Figure 19c and Figure 19d. The values selected for DPDP are supported by results for diphenylhexatriene, for which the displacement parameters AVI and A”: are significantly smaller in the llBu state than in the 21Ag excited state.55 Moreover, for the dipole-forbidden 132“ electronic state of benzene, the A-value found19 for the symmetric ring breathing mode (992 cm'l) is 0.33 (amu)%A, which agrees well with the values listed for DPDP in Table I. The overlap integrals are not as sensitive to V1 as they are to A1. A plot of the overlap integrals as a function of Vi will show a curve that is nearly a constant. Therefore the experimental frequency differences for the llAg and 118,4 states of DPDP, about 50 cm.1 for both the 7 symmetric C-C and the symmetric C=C stretches,5 may be 79 Figure 19a. The three fundamental Franck-Condon integrals VI, 92, us. Only V1 has negative values. The F-C origin peaks first for a given displacement paramenter Ai- The amplitude of all except for v3 approaches zero for A1 > 0.4 80 .oou canonh emmectoa EmEmucEmfi 000.0 005.0 000.0 00¢.0 00nd 00 p .0 000.0 PhthbeDPbbebbbbD-DbbD—bbbbbbPPPbPPPhbbebbbbbbbbebthbbhbbbbebbb-bb-bbbhnPbbbe-Pbbbhhbbthbe 00> -..- T . .> I. H ~> ..n r n> -.. 4T T’fiT I 00ml 1 VI 4 100Pl T "\ bbbbebebbbPhPbPPbPbbbththprthbbbbDDhDbbbePPbD-nPPbDPPPbePbWbPPFPbPDPbbPbDP—bhhbb-b-b-Dbbb epmudwv 81 Figure 19b. The Franck-Condon integral v1 and its overtones 2v1, and 391. The relative shift between the F-C integrals is most apparent. It is clear that as A1 increases (at A1 > 0.3) these overtone amplitudes increases, but the fundamental v1 decreases. 82 ion Insof— tomeotoQ EmEouoEmfi 005.0 000.0 00¢.0 000.0 009.0 000.0 DPPb-Pbebbbbbbehb-b:PbbebPDPPD_PDDDPDDDDDbPhD-Ph.DDbbbbbbbbh-Pthbbhbbhhhbb DON ,EPZL CEEbbEEEEEttpLC-.C r. 9: . . Z — . . . ...PpEbLL} .> .>~ \ Sn -... I epmudwv 83 Figure 19c. The Franck-Condon integral amplitude as a function of A1 for several combinations of v V and V3. 1' 2’ 84 .00« Ghana» 030E800 EoEmocEmfi 005.0 000.0 00¢.0 000.0 on —.0 000.0 DDPPbPDDbbPDPhPDDPb—PbbtthDb.bbP-bbbPthPPPbfibDbPEPbbPDPbbb8bbbDPDPPDLPPbbbP 0 I n>+N> .. I n>+ —> ll —>+N>N --.. N>+ p) l PPPFDPDbPDPPDPbE—bbbPPPFDDPPDPPhbDPEPPPPDPDimbbDPbEPbbbbbbb-bbPPbbbeDD: 00! T Yogi T io¢l ' .lONII epmudwv 85 Figure 19d. The Franck-Condon integral amplitude as a function of Ai for combinations of VI with V2. 86 .oo« ensues touoEotoa acmEoooEmfi 005.0 000.0 00¢.0 000.0 00.0 000.0 bbb’bPDPDDbDDbPDDbD-bPD...-DP.PDPP-PbbbbbbbbDbbPP-DbbeDPPPPhbbb-DbPDbDDDDFPP N>+ —>n II .. N>N+ —>N ..-- x / i N>+p>N II. \ DI-P-DbbbPD.PPPbDbPPP-EDPDDPDDlib-PDDPPPbDDbDth—bEbfipbbthP—tPPPDbP Dubbh 00I Iowa... ..IOVII I ..IONII 9901“de 87 neglected for the purpose of determining the Franck-Condon integrals. As a result, the recurrence relations of equations (8)-(10) simplify considerably, as shown by equations (8')-(10’). However, for the transition to the 21Ag excited state a similar approximation would result in significant error, because of the expected large change in the C=C symmetric stretching frequency. For DPDP this mode is observed at 1750 cm'l,9 200 cm'1 higher than the ground state value. Such a large increase for the C=C stretching mode in the 21Ag excited state is typical for linear 6,7,9,58 polyenes; as the chain length increases so does the frequency of this vibration. Theoretical results initially predicted a slight decrease in frequency.59 Among possible explanations,6'7'5°'61 the unexpected increase has been attributed to vibronic coupling between the 11Ag ground state and the 21Ag excited state. 5.3.2 Bandwidth A linewidth value (HWHM) of Pi = 500 cm"1 for the vibrational modes in the llBu state was obtained by fitting the model absorption cross-section of equation (11) to the room temperature experimental absorption spectrum of DPDP in cyclohexane (Figure 4). Confirmation was provided by fitting the upper end of the experimental preresonance profile to the tail of the calculated REP; a lower limit of 88 500 cm”:L for the linewidth was determined. This seems reasonable, since a P1 value of 650 cm'1 has been measured 62 for the REP of fi-carotene in cyclohexane. A narrower linewidth (250 cm'l) was reported by the same authors for B— carotene in isopentane.62 For each vibrational band of the 21Ag excited state a half bandwidth value of 50 cm'l, typical of vibrational linewidths in solution, was selected. For the purpose of mathematical simplicity the damping parameters, Pi, were all assigned the same numerical value within a given electronic excited state. 5.3.3 Combinations and overtones By including combinations and overtone bands in the model for the 113“ excited state, not only did the absorption spectrum of the model improve, but the calculated preresonance Raman excitation profile also better fit the experimental spectrum of Figure 10. The omission of the three highest energy absorption bands (see Figure 20), comprised of overtones and combinations of the 1lBu excited state fundamentals, would cause the low-energy preresonance REP tail in the 21119 region, 20000-21000 cm'l, to be significantly misaligned, and, would predict a curvature ‘which does not coincide with the experimental data. 89 Figure 20. The predicted resonance Raman excitation profile (———) and the REP with the combination band 391w? deleted (---). The deletion of this band affects the entire REP spectrum, in sharp contrast to the effect on the calculated absorption spectrum of Figure 18. 90 D b b .on unseat 3153 9.035236; 0000N P ir 000§N b t 1|} I. .1 -l i. L OOOON 100A. TOQd— wOQdN (suun Monng) K}!SUO}U| 91 Figure 21. Expanded view of the 20000-22500 cm-1 region. for the REP shown in Figure 20. 92 .m« ousudh ClEov ntonEscgoz OO¢NN coo—N OONpN OOQON OOOON b p r by. * ’ Ly b P ir L ‘7 b k p D L h b lb! hr h F L b coo (Pl Pull»...4i--.¥lll?rLilLl|if.|L.il.PlibsLnibxllbl;bi..-.. In . -;_ 1 p? i- : Li .. --Ihlll (swan mmcw) met-law: 9400198 93 This is interesting, since these 11B“ states modes are at least 4000 cm“1 from the nearest prominent 21Ag vibronic feature. Their effect on the preresonance region of the calculated REP given in Figure 20 appears to be negligible. However, on the expanded scale shown in Figure 21 it is clear that the model does not coincide as well with the experimental data when even the single highest energy combination band (3v1+u2) considered in our analysis is deleted from the calculated REP. The preresonance intensity calculated. with the 'truncated ‘model is higher' than the prediction shown in Figure 12, which is already higher than the experimental observations. The overlap integrals for the 113“ C=C stretch, its overtones and combinations are negative for the corresponding A i value (Tables I-II, and clearly seen in Figures 16a-16d ), which has the effect of lowering the model excitation profile curve. The antiresonance term in the Kramers-Heisenberg dispersion relation of equation (2) has a similar effect; however it was sufficiently small in this case to be neglected. From the tabulated parameters and the Kramers- Heisenberg expression the "best fit" calculated REP was obtained. In principle we could have solved the absorption cross-section equation (11) for A1 (taking v1 and Pi directly from the absorption spectrum), and then inserted these expressions into equation (4) to obtain the Raman polarizability. In practice, however, obtaining the A i graphically by fitting the absorption equation to the 94 experimental absorption spectrum at 77K (Figure 18) was found to be more appropriate. 5.3.4 Hultinoda Formulation The multidimensional Franck-Condon integrals of equation (5) involve all of the 3N-6 vibrational modes. However, the absorption spectrum of DPDP, like those of other linear polyenes, is dominated by two symmetric carbon- carbon stretching modes, 91 (C=C) and 92 (C-C) , and their progressions, where the remaining fundamental vibrations are much weaker. A two-mode model, <1102|v1u2>, may therefore be expected to offer a reasonable description of such spectra. In fact, for the llBu absorption two fundamental modes describe the 0-0 and 0-1 band systems adequately, but not the higher energy transitions. For the two-mode model as well as well as for the three-mode model, only the 0-0 and the 0-1 bands of Figure 18 are utilized to determine the "best" fit and hence A i (The fit for the higher energy bands are incidental). The inadequacy of this two-mode approximation is also apparent in the values for the displacement modes obtained: Au1 = 0.11 (amu)%A, and A112 = 0.24 (amu)"'A, which deviate significantly from the ‘values, Av1 = 0.14 (amu)%A and 692 = 0.13 (amu)%A, obtained for octatetraene.56 95 However, when a three-mode model, <110203|u19293>x (111112113'010203), is employed, the description of the absorption and Raman excitation spectra improves for both excited states. Moreover, enhanced intensity in the 21Ag REP near 20350 cm'l, 21600 cm'1 and 22100 cm'1 can be accounted for by the low-frequency node v, and combination bands, 92-», and u1+u,, respectively. Horwitz et al. interpreted a band in the fluorescence excitation profile of DPDP, shifted from the 21Ag origin by 1550 cm’l, as the benzene u, mode, belonging to the phenyl end groups.9 Inclusion of this mode in the calculated preresonance REP provided only slight improvement to the experimental fit, as noted in Figure 14, although for the model fluorescence excitation spectrum of Figure 13 the improvement is clearly noticeable. All calculated model spectra for the 11Bu excited state were determined by using three modes in the multimode formulation: a four-mode model was utilized for the zlAg state. The mode displacements, A1! in the three-mode model were consistent with values presented in the literature, as already noted. Interestingly, increasing the multimode formulation to include four vibrational modes, namely C-C, C=C, and two low-frequency vibrations or one low frequency mode and another higher frequency fundamental, did not improve significantly the excitation profile for the preresonance region nor did it considerably affect the calculated resonance REP. It follows that for further 96 significant improvement in the theoretical description of the DPDP spectra, all 3N-6 normal modes should be included in the multimode formulation. 5.4 Resonance Region On the basis of the previous analysis the resonance REP of DPDP, which has not been obtained experimentally to date, can be predicted. If one employs the As and us given for the llBu state in Tables I and II, the resonance REP shown in Figure 20 is obtained. The vibronic structure in the REP in resonance with the 118.4 state for polyenes in general is also governed by Franck-Condon progressions formed by two dominating totally symmetric modes: the more prominent being the C=C stretch at voo+1550 cm-l, and a slightly less intense C-C stretch at 11004-1200 cm'1.11'48 For DPDP both fundamental modes fall within the 0-1 band. The electronic origin at 23750 cm"1 is the most intense band. The overtones and combinations which comprise the remaining bands are listed in Table II. The inclusion of the us mode :not only gives structure to the expected REP at high tenergies, but as noted earlier it also improves the fit of the experimental data in the preresonance region. The calculated resonance REP intensity of the 0-1 band of DPDP is slightly weaker than the 0-2 band, which is contrary to the observed REPs of the longer chain 97 polyenes,63'65 such as fi-carotene. This discrepancy may be due to the unadjusted displacement parameters, which were not optimized to the experimental DPDP REP solvent environment; it may also be due to the CH3 bending fundamental of fi-carotene, which is not present in DPDP. A 10% reduction in the A1 values will remove this discrepancy, as is seen clearly in Figure 22, and slightly improve the fit at the tail of the experimental preresonance REP of DPDP. On the other hand, using lower A1 values than those obtained from the absorption spectrum of DPDP at 77K will result in an inaccurate prediction of the room temperature absorption spectrum, as illustrated in Figure 23. Measurement of the resonance REP of DPDP would resolve this question. 98 Figure 22. (a) The REP of DPDP with unadjusted displacement parameters (—) . (b) The REP with the adjusted parameters(- -). 99 fir f 22500 T 40.00-.-—-._ 20000 (1005 8, 8 (1004 10.004 2 (Shun 440411901) KMSWWI wovenumbers (cm- 1) Figure 22. 100 Figure 23. (a) The effect of adjusting the displacement parameters (- -) on the calculated room temperature absorption spectrum of DPDP. (b) The spectrum with the unadjusted parameters (——) i 5 consistent with that expected from Figure 4. 101 . nu ensue.— CIEOV 909E236; (shun bommv) Kuwaiti: TOOJ. (b\ b 1P) [8 hr {Pi \[Pi b, \iP lb L, (P h \L '0!!! O ... I O 000 0000 lo 0 00 I I. e o e. o e o e 00 e 00 9. 00 e I e 00 0 e O o 00 O . .. o e .I e e a e o e o e o e a e O o a e e e o o o O o o o .. .. e i . o 0 o e l O \\ O O O 0 a o 000‘ 00 o .. .. Q. o e o u o e o o o e o o o o o c o a o o o u . u o 1| 0 o o 0 o o o o a o o o o o o o o o o o . .. o o o e o u . o e o o 1 a e o c o e . lb h (iii blxillbl by \P II?) P‘ (b) til) P, b b) )P. h b CHAPTER VI conclusions Evidence of the low—lying 21119 excited state of the DPDP molecule has been observed in the preresonance Raman excitation profile. Moreover, a one-to-one correspondence between interference features in an experimental Raman excitation profile and the vibrational modes of a dipole- forbidden state has been established for the first time. Preresonance Raman excitation thus promises to be suitable for the detection of weakly allowed states in molecules having low fluorescence quantum yield. Raman excitation profiles might be exploited to help resolve the excited state level ordering of the 21Ag and 55 and diphenyl- 118u states for diphenylhexatriene butadiene.66 The locations of these states have differing dependence on the solvent polarizability, which can account for the inconclusiveness of their relative positions for the diphenyl polyenes and for their unsubstituted 57'68 The shapes and intensities of counterparts . interferences in the REP due to the presence of a dipole- forbidden state depend strongly on the proximity of the coupled allowed state, and a solvent mixture might be "tuned" to highlight the presence of the hidden state. 102 For p-carotene and other biological polyenes, firm establishment of the 21Ag excited state energy may help answer crucial questions regarding energy transfer processes 15:13:59: As noted earlier,a in photosynthetic systems. detailed report of the preresonance Raman scattering from 8- carotene is to be published shortly.24 The manuscripts which describes this research project has had an excellent reception. A. C. Albrecht of Cornell stated that my "A-state Raman studies are already an impressive contribution." A similar response came from Z. G. Soos at Princeton, R. M. Hochstrasser at Pennsylvania, and others (including B. E. Kohler of Riverside who refereed the paper). At present, obtaining a high resolution Raman excitation profile point by point is both laborious and time consuming. We have established, the preresonance REP as a viable technique for the detection of dipole forbidden states: however, for it to be regularly and practicaly useful, computerization of the REP apparatus and the data analysis will be required. 103 104 APPENDICES 105 APPENDIX I 106 A complete listing of the program written in Basic for an IBM compatible computer, which was utilized to calculate the Franck-Condon factors, REPs, and the Absorption Cross- Section, is given in Appendix I part B. A flow diagram of the program is also included. In order that an interested reader may better follow the rationale under which the REP portion of the program was developed, the discussion in Appendix I part A is offered. APPENDIX I PART A Program lines 25295-27400 (labeled as the first equation) determine the Raman excitation profile, based on the Kramers-Heisenberg dispersion relation (see Theory Chapter III). Initially the program was written for six energy levels (cl-15). It became apparent that six terms are inadequate to represent the excitation profile of the 21Ag and llBu excited states: consequently it was necessary to increase the number of terms from six to a yet undetermined number, n. This was done not by reworking the REP model for n terms, but by developing an algorithm which made it possible to build on the existing expression. The mathematical expressions utilized are shown in this section. Past the sixth term, one term at a time was added until the model REP, consisting of 23 terms, gave an adequate description of the experimental data. Although an algorithm consisting of a loop program would have been more efficient than this approach, the calculation time and memory capacity was not a problem. Thus there was little incentive for a major mid-course correction (minimizing the risk of errors was also a factor). Nevertheless, had a large number of terms for the REP model, been anticipated a more efficient program would have been written. 107 108 To extend the number of terms from m to k we may use the following approach (the use of brute force is lengthy and prone to errors). Let m m e [mum] 3 i we wish to evaluate m m e * sk = { Z cj + ck }{ Z 03 + ck } (2) J J [ Z cj ][ X C; J + [ 2 Cj ]c; + ck[ Z c; ] + (Ck)(C;) J i j i Sm+ [ § Cj J01: + Ck[ )j: C;] + (Ck)(Cl:) (3) where Sm has aready been determined. As an illustration, for 7 terms (k=7) and the center expressions for K=m+1 becomes. 6 t 6 * * S7 3 35+ [ X Cj ]C7 + C7[ g: Cj ] + (C7) (C7) (4) 3 109 or i * * * * (c1 + c2 + c3 + c, + cs + C. )c., = c‘c, + czc, + cac7 + c‘c7 * * + csc, + csc, similarly * 'k * * * * i * 'k * c.,(c1 + c.2 + c3 + c, + c15 + c,’ ) = c.,c1 + c.,c2 + c.,c3 + c.,c‘ * * +0., (35 + C7 C6 A5 Let Cj = (ch + iFj) * [ j j ] [ 1 ] ejc, (ch + 1F ) (Ac, - 1P7) * * * we combine ejc7 and C7Cj and if Aj = Aj using the result from equation (9) the expression for 7 terms becomes 6 (ch Ar, + Pj P7) (A7)2 81.7 = s + 22A°A + (5) 6 ' j 7 2 2 2 2 2 2 j (ch + Pj )(Ae7 + P, ) AC7 + P7 Thus to the existing S. we add the above expressions to obtain 8,. 110 To determine 86 we begin with S8 3 . A) j ch + in Evaluating |R123|2 is straightforward thus 3 A 3 A* 33 = 2 j . I [X j 3 ch + 1rj j ch - irj Taking the product and if Aj is real we have 1 1 (6) (7) (Ac2 + 1P2)(Ac1 - 1P1) (A:1 + 1P1)(Ac2 - in) J (3) 1 1 + A2A3 + (A:3 + 1P3)(Ac2 - in) (Ac2 + 1P2)(A£3 - 1P3) J 1 1 + ASA1 + (A:3 + iPS)(A£1 - 1P1) (A:1 + 1F1)(Ac8 + [ (A1)2 (A...)2 + (A:3 + ir3)(Ac1 - irl) (A:1 + in)“:3 — 1P3) (A3)” (A:3 + 1P3)(Ac1 - iP‘) + - ...] 111 Each of the above pairs can be further simplified if we note that 1 1 + (Ark + iPk)(Atj - in) (Aej + in)(Atk - iFk) (Aej + in)(Aek - iPk) +(Ack + iPk)(Aej - in) (Ask + iPk)(Ack - iPk)(A¢j - iFj)(ch + iFj) (chAzk-iA:ij+iAckPj+PkPj)+(chAek-1chrk +iA¢krj +PkPj) (Ask + iPk)(Ack - irk)(ch - iFj)(Asj + iFj) (ZAtjAtk +2PkPj+(-1chrk +iAeij)+ (iAckrj -iAeij) [(Aek)’ + (rk>’1[<4ej)’ + (rj)’1 2(chAck +PkPj) [(Aek)’ + (Pk)’][(ch)’ + (Pj)’] 112 Therefore 2A3 A1(Ac1 Ar, + P1 P3) |R123|2 = 2 2 2 2 (10) (A83 + P3 ) A21 + P1 + 2A1 [A2(Ac2 A81 + F2 P1)] (A812 + r12) 6:13 + r,” + 2A2 [A,(A¢, Ar, + r3 r2)] (A822 + ir22) Ar,” + it,“ 3 (As)2 + X > 5:1 A‘s: + F82 Renaming the indices 456 instead 123, |R456| is similarly defined (to be used below). 113 Thus to increase |R123|2 to six terms we may write A4 A5 A6 I6 = R123 + + + (11) Ac‘ + iP‘ Ass + 1P5 Ar. + iI‘o * * * 2 2 A4 A5 A6 '_ |R123| '+ |R456| '+ R123 + + Ac‘ - iF‘ A85 - ifs AC6 - 1P6 * A‘ A5 A6 + R123 + + (12) At‘ + 1F‘ AC5 + 1P5 A:6 + if, The first two terms already have been evaluated (For |R456| just exchange 123 for 456 in the final expression of |R123|)° Expanding the remaining two terms, namely 114 A4 A6 Ac x + + (13) Ae‘ - iP‘ Ass - 1P5 Ac. - 1P. (14) [ A. As Ac x + + At‘ + if, A85 + 1P5 At. + 1F. For clarity consider only one of the possible nine terms of the products above A1 Aj Ak Ar + + (15) A81 + iI‘i Atj + iI‘j Atk + irk Air " irr and its complement (equation 14 ) * s * A1 Aj Ak Ar + + (16) Ari - iFi Atj - in Ask - 1Pk Aer + iPr For each two corresponding pair of terms in equation 15 and 16 we obtain the following expressions * * A1 Ar A1 Ar + (17) Ati + iI‘i Atr " irr A81 "' iI‘i Atr + 11‘:- 115 which reduces to * 1: (Ai)( Ar ) ] [ “‘1 )(Ar ) + (18) (Ari + iPi)(Acr - irr) (Ari - 1P1)(Acr + irr) The numerator above can be factored out provided A1 and Ar are real 1 1 = (A1)(Ar)[ + <19) (Ari +113)“:r —irr) (Ari -iF1)(Acr +irr) or (as done earlier) 2(Aci Aer + F1 Pr) = (Ai)(Ar)[ (2°) (Ari2 + iI‘i’)(Acr2 + irr’) 116 Repeating the same for each pair, therefore * * * ' A: A2 A: A4 + + r A:1 - lI‘1 Ac, - if: Ac. - 1Pa Ae‘ + 1P, J ' * A1 A2 A3 A5 + 4 + + A81 + 1P1 A82 + 1P2 A6. + 1?, Ass - 1P5 * * * A1 32 A: A5 + + Ac1 - 1Pl A62 - 1P2 As. - 1Pa Ass + ifs . * A1 A2 A3 A6 + 4 + + Acl + 1P1 A8, + 1P2 Ar, + 1r3 Ac, - 11‘6 117 From each of the expressions in the large braces we obtain P [ ZAIA‘UH:1 Ac‘ + P1 P‘) J [ ZAZA‘(A¢2 Ac‘ + P2 P‘) + (At-22 + 1‘2”)(Ae22 + r22) (11:,2 + r,’)(Ac22 + r22) b ‘ 2A3A4(A£3 A8‘ + P3 F‘) + L (22) (Ar,2 + r3”)(Ae22 + P2”) or ZA‘ 3 As(A£s AC‘ + [‘3 1“) = )3 (23) (A822 + F22) s 1 A632 + P32 The same is done with the remaining two terms in the large braces on the previous page. Thus the final expression is 2A4 3 As(A£s At‘ + rs P‘) (Ac‘2 + P22) s=1 Ass2 + P32 L (24) 2A5 3 As(A£s A85 + rs 1‘5) 2 (A852 + r52) s=1 A852 + rs2 2A6 3 As‘Ats A‘C + F8 F6) 2 2 2 2 2 2 + IR...I + IR...I (AC. + P0 ) s=1 Ats + rs J To expand to n>6 we proceed as prescribed on the second page. 118 APPENDIX I PART B 119 A flow diagram of the program 120 As—agv '50-.— nx .n geese 55.8.? a $8.250 «as... 8.5.5 3.3. .— 33 g _ _ A 3.8: .83 8 «fine 8: 2888 seal-ac e 58.88 _ 100 110 120 130 200 205 210 215 220 225 230 235 240 245 250 255 260 265 121 APPENDIX I PART 3“ REM * ----- * lines 110-140 set up the screen * ------- * CLS SCREEN 9 : REM ==== EGA -—-—- PALETTE 0,3 :REM ==== Blue background ===== CLS:LOCATE 6,10 PRINT "---- PLOTTING AND DATA IANIPULATION PROGRAM ---- BY ISAAC SZTAINEUCH " LOCATE 10,10 PRINT "Program to create and modify data points (x,y). Lines 2635 and 2655 are for performing calculation on x and y. Press [enter] for a default step. Respond to questions in ( )." PRINT "Equations of lines 24000- or 35000- or 50000 will be plotted and its points will be stored in a file (If out of memory delete equations 50000-, temporarily)" PRINT " ------ PRESS * TO START OR STOP -------- " B$=INKEY$ IF B$="*" THEN GOTO 245 GOTO 230 CLS:LOCATE 10,10 PRINT " (1) Create a NEW FILE (2) MODIFY an existing file (3) Create and Plot a New File by Equations [lines 24K, 35K, 50K]" PRINT " (4) Same as 3 but graph points are CONNECTED (5) Plot a Data File (6) Plot TWO Files simultaneously (x,y scale of file 1) ' PRINT " (7) Same as 6 but points NOT CONNECTED (8) Plot TWO files but NOT AUTOSCALE (+ for a 3rd file, * to end) (9) Plot THREE Files (otherwise it is same as 6)" INPUT " (10) Perform calculation on equations [line 40000-1 (11) Perform calculation on equations [line 60000-1 ”:Y 270 295 300 310 320 330 340 350 360 370 380 400 410 420 425 430 440 450 460 470 475 480 485 490 495 499 500 510 515 520 530 532 535 540 545 550 555 560 570 580 590 600 1140 270 1150 1160 1180 1195 1200 1250 122 GOTO 1140 REM * ------- * Lines 300-380 sets-up a table * ------- * LOCATE 6,11 LINE (35,90)-(530,65),12,B LINE (75,65)-(75,170),5 LINE (178,65)-(178,170),10 LINE (285,65)-(285,170),5 LINE (287,65)-(287,170),5 LINE (410,65)-(410,170),10 LOCATE 6,7:PRINT "N":LOCATE 6,15:PRINT "X":LOCATE 6,31:PRINT "Y":LOCATE 6,43:PRINT “X'”:LOCATE 6,59:PRINT "Y’" RETURN REM -- 400-450 determines max.,min of x,y in the file -- IF N=1 THEN LET Q=U IF ABS(U)>M THEN LET x-U IF ABS(T)>K1 THEN LET K1=T IF T0,error otherwise **** RETURN REM ** ( to line 4005 ) ** REM --- 545-560 places a numerical value at tick mark -- LOCATE 25,2 :PRINT Q LOCATE 25,38:PRINT ((M-Q)/2)+Q LOCATE 25,71:PRINT USING "##.###0000":M LOCATE 3,4:PRINT USING "##.###AA**":K1 REM --- 530-550 generates tick marks on the x,y-axis --- T1=10 : REM ---- This is the number of tick marks ----- FOR N=0 T0 T1 F826+N*((619-26)/T1) F2=26+N*((324-26)/T1) LINE (F,330)-(F,320),15 LINE (20,F2)-(30,F2),15 NEXT N IF K230 THEN GOTO 600 LINE (25,25)-(620,176),4,B : REM *** DRAW CENTER LINE IF K2<0 *** LOCATE 23,5:PRINT USING "##.##AAAA";K2 RETURN : REM *---* see line 9175 *---* IF 3=Y OR 4=Y OR 1=Y THEN GOTO 2700 :REM see line IF 10=Y THEN GOTO 40000 IF 11=Y THEN GOTO 60000 IF 5=Y OR 6=Y OR 7=Y OR 8=Y OR 9=Y THEN GOTO 4000 REM ---- 1200-2600 for modifing existing file, Z$ ---- CLS LOCATE 8,8 : INPUT ”Existing file name";Z$ LOCATE 10,8 INPUT "New modified file name":A$ 1400 1410 1440 1445 1450 1500 1550 1565 1575 1585 1600 1650 1675 1680 1690 1692 1700 1725 1740 1750 1785 1800 1850 1900 1950 2000 2050 2100 2150 2250 2295 2300 2350 2360 2400 2450 2500 2510 2520 2530 2540 2550 2560 2570 2580 2590 2595 123 LOCATE 12,8 : INPUT "Enter New Text":T$ LOCATE 13,8 : PRINT TS LOCATE 15,8 : INPUT ”Enter value to screen ":8 : REM *---* See 1690,92 *---* GOSUB 2600 OPEN "I",#1,Z$ FOR N=0 TO 10000 IF N80 THEN GOTO 2300 IF "n”=F$ THEN GOTO 1675 LOCATE 3,7 : INPUT "Modify manually (n)”;F$ : CLS IF "n"=F$ THEN GOTO 1675 LOCATE 6,8 : INPUT "Insert Line (i)”;H$ IF ”i"=H$ THEN GOTO 1850 INPUT #1,W ON ERROR GOTO 2530 REM *-* Numerical values in the original TITLE must be deleted (e.g. date) *-* Bz=88 : REM * ------ * If more than one value is to be deleted enter a value for 82(next step) * ----- * IF W=0 OR W=B OR W=82 THEN GOTO 1675 INPUT #1,S IF ”n“=F$ THEN GOTO 1785 LOCATE 8,32:PRINT N,W,S IF "n"=F$ THEN GOTO 2150 LOCATE 8,8 : INPUT "Delete Line (d)":G$ :IF ”d"=G$ THEN GOTO 2520 LOCATE 10,25 : INPUT "New X Value"3P LOCATE 10,50 : INPUT ”New Y Value":Q1 : CLS IF P=0 THEN GOTO 2050 IF P<>0 THEN LET W=P IF Q1=0 THEN GOTO 2150 IF Ql<>0 THEN LET S=Ql GOSUB 2635 IF N<>0 THEN GOTO 2360 REM *=*=*=* Modified file is stored in As *=*=*=*=* OPEN ”O",#2,A$ PRINT #2,T$:GOTO 2520 GOSUB 300 LOCATE 8,6: PRINT N :LOCATE 8,13: PRINT WzLOCATE 8,25:PRINT USING "##.###“‘A";S:LOCATE 8,39:PRINT USING "##.###‘A“”;U:LOCATE 8,54:PRINT USING ”##.###“AA";T GOSUB 400 PRINT #2,U,T IF EOF(1) THEN GOTO 2530 NEXT N LOCATE 15,28:INPUT ”Plot this file ( y or n )";C$ IF ”y"<>C$ THEN GOTO 11000 REM - lines 2560-2590 prepares the file to be plotted - CLOSE #2 CLS:OPEN "I", #2,A$ V=2 GOTO 9150 END 2600 2605 2610 2615 2620 2625 2630 2635 2640 2645 2650 2655 2660 2670 2680 2698 2700 2750 2800 2825 2860 2900 2950 3000 3050 3100 3125 3150 3200 3995 4000 4005 4010 4025 4030 4040 4045 4050 4100 4125 4150 4200 4250 4300 4320 4350 4400 4450 4475 124 REM ---- 2600-2670 For modification of data points ---- CLS LOCATE 8,8 : INPUT ” Enter new slope ”:A2 LOCATE 10,8: INPUT " Enter new vertical shift ( intercept )"3A3 LOCATE 12,8: INPUT " Change nm to cm-1”:Q$ CLS RETURN : REM ----- TO 1445 ------- A1=1E+07 : U=(1/W)*A1 :REM -*- Change nm to cm-l -*- IF ”y" = 05 THEN GOTO 2650 A181 : U=W*A1 :REM -*- ENTER X VALUE CHANGES IN A1 -*- IF A2=0 THEN GOTO 2660 T-(S*A2)+A3 IF A2<>0 THEN GOTO 2680: REM*2640 & this leaves T unchanged by default** T=S+A3 RETURN : REM ----- To 2150 ------ REM*** STEPS 2700-3200 ARE FOR CREATING A NEW FILE *** CLS : LOCATE 10,10 : INPUT ”File Name":Z$ LOCATE 12,10 : INPUT "Text":T$ OPEN "O",#1,Z$ PRINT #1,T$ : LOCATE 13,10 : PRINT T$ IF 3=Y OR 4=Y THEN GOTO 5000 FOR N=0 TO 10000 LOCATE 16,15 : INPUT "X value";W LOCATE 16,35 : INPUT "Y value";S LOCATE 18,15 : PRINT N,W,S PRINT #1,W,S REM IF W80 AND S=0 THEN GOTO 12000 : REM -- To terminate a new file -- NEXT N GOTO 9000 REM -*-*-*-*-*-* 4000-4475 plots file 1 -*-*-*-*-*-*- CLS: LOCATE 4,8: INPUT "File 1 to be plotted";Z$ IF 8=Y THEN GOSUB 470 IF Y=5 THEN GOTO 4050 LOCATE 12,8 :INPUT ”File 2 to be plotted":P$ IF Y<>9 AND Y<>8 THEN GOTO 4050 LOCATE 20,8 :INPUT "File 3 to be plotted";SZ$ IF 8=Y THEN GOTO 9000 CLS:OPEN "I",#1,Z$ INPUT "Enter value to screen" :8 REM IF 8=Y THEN GOTO 4370 FOR N=1 TO 100000! INPUT #1,U IF U=0 OR U=89 OR U=88 OR U=87 OR U=90 OR U=B THEN GOTO 4200 : REM *-* Number (U) in the title that needs to be screened -*-*- INPUT #1,T IF EOF(1) THEN GOTO 9000 GOSUB 400 PRINT U,T NEXT N END 4495 4500 4550 4600 from 4650 4700 4750 4780 4790 4800 4825 4850 125 REM -*-*- 4500-4750 file 2 to be plotted -*-*- CLOSE #1: V=1 OPEN 'I',#1,P$ J=1 : H=4 :REM *-* J81 prevents first & second graph being connected ( H sets the graph color to brown ) *-* Kz=9 :REM ---- To get it to plot file 3 after plotting file 2 ---- GOTO 9300 END REM * ------- * 4800-4950 file 3 to be plotted * ----- * IF Y<>9 THEN GOTO 11200 CLOSE #1: V81 OPEN ”I”,#1,SZ$ J-l : Hal :REM * ------ * J81 prevents first & second graph from being connected ( H sets the graph color to blue 4870 4900 4950 4995 5000 ) *-———* Kz=99 :REM -- To get to end after plotting file 3 --- GOTO 9300 END REM -------- 5000-6000 Plots the equation ------- LOCATE 14,10 : INPUT ”Plot Equation (0), (1), or (2) [ENTER for 0] ”3 5025 5050 5075 5100 5125 5150 5175 5200 5225 5250 5300 5400 5600 5800 6000 9000 9100 9125 9150 9175 9200 9225 9250 9265 9275 9285 9300 9325 LOCATE 16,10 INPUT "Enter min range for x in f(x) ":0 LOCATE 18,10 INPUT "Enter max range for x in f(x) ":M LOCATE 20,10 INPUT "Enter incremental steps for x";L FOR W=Q TO M STEP L IF HH=O THEN GOSUB 24000 IF HH=1 THEN GOSUB 35000 IF HH=2 THEN GOSUB 50000 REM ---- 5225 determines max y value for auto plot ---- IF ABS(S)> K1 THEN LET K1=S IF S< K2 THEN LET K2=S PRINT w,s PRINT #1,w,s NEXT w GOTO 9000 END CLOSE #1 Val CLS: OPEN "I”,#1,Z$ REM LINE (25,25)-(620,325),7,BF GOSUB 500 H=10:REM - This determines the color of graph (green) - C=Z:W0=Wl IF KZ<>9 THEN GOTO 9275:REM ** kz presence of file 3 (Also in line 4650)** IF EOF(1) THEN GOTO 4790:REM *** file 2 completed start plotting 3 *** IF Kz<>99 THEN GOTO 9300:REM**(kz) presence of file 3 (Also in line 4870) IF EOF(1) THEN GOTO 11200:REM - file 3 plot completed - LOCATE 1,1:INPUT #V,W B$=INKEY$ 9350 9375 9400 9450 9500 9550 9600 9650 9695 9700 9750 9790 9800 9845 9850 9900 9950 9975 10000 10010 10025 10050 10100 10120 10150 10175 10200 10300 10400 10420 10500 10550 10600 10625 10650 10700 10725 10750 10800 10850 10900 11000 11100 11200 126 IF B$="*" THEN GOTO 11300 : further plotting --- ON ERROR GOTO 10025 IF W80 OR W=89 OR W=90 OR W=BZ THEN GOTO 9300 IF W5 THEN GOTO 10900 : REM - To display W at another location - LOCATE 1,1 :PRINT USING "##.###A“‘“":W GOTO 10050 LOCATE 15,27 :PRINT DONE IF Y=6 OR Y=7 OR Y=8 OR Y=9 THEN GOTO 4500 : ********* This begins second plot ********* IF Y<>8 THEN GOTO 11300 REM --- To stop from REM 11210 11215 11220 11225 11230 11299 11300 11305 11310 11320 11330 11340 11350 11400 11500 11900 12000 24000 24005 24010 24015 24020 24030 24035 24040 24045 24050 24060 24070 24075 24090 24095 127 B$=INKEY$ IF B$="/” THEN GOTO 11300 IF B$="+" THEN GOTO 4800 LOCATE 1,1 :PRINT "PRESS / or +" GOTO 11210 REM Lines 11300-11350 places a name label for each plot LOCATE 3,47 : PRINT ZS LOCATE 3,47 : PRINT A$ LOCATE 3,58 : PRINT P$ LOCATE 3,68 : PRINT 82$ LINE(368,25)-(440,25),10,B LINE(450,25)-(525,25),4,B LINE(535,25)-(620,25),9,B IF B$-"*" THEN GOTO 11900 :REM -prevents being trapped in a loop- HI=5 : GOTO 10050 : REM ==== see line 10550 ==== LOCATE 1,1 : REM -- This is to place cursor at origin END REM --- Enter function in program line 24000-34999 --- REM ----- Raman Excitation Profile Parameters ---- REM ** t11=> VEl delta, t22=> VE2, t33=> VE3 delta, K= mult const for Ag REM ** t1=> V1 delta, t2-> V2 , t3=> V3 delta, C= mult const for Bu state REM Ag ----- Tll=.36 :T22=.3 :T33=.37 :T44=.11 : K8425 VE1=1750 :VE2=1250 :VE3=200 :VE4=1550 :VG1=1550 :VGZ=1200 :VG3=200 :VG4=1550 E1=20175 :E2=E1+(VE3) :E3=E1+(VE2) :E4=E1+(VE1) :E5=E1+(VE1)+(VE3) :E6=E1+(VE2)+(VE3) :E23*=Ei+(VE4) G1=50 :G2=G1 :G3=Gl :G4=G1 :GSaGl :G6=Gl :G23*=Gl REM ----- REM =============m Bi; -- T1=.17 :T2=.15 :T3=.1 : C=10225 :Vl=1550 :V2=1200 :V3=200 :E7 -23750 :E8 =E7+V1 :E9 =E7+V2 :E10=E7+V3 :E11=E7+2*(Vl) :E12=E7+2*(V2) :E13=E7+(V1+V2) :El4=E7+(V2+V3) :E15=E7+(V1+V3) E16=E7+2*(V1)+V2 :El7=E7+2*(V2)+V1 :E18=E7+3*(V1) :El9=E7+(2*(V1)+2*(V2)) :E20=E7+(2*(V1)+V3) :E21=E7+(2*(V2)+V3) :E22=E7+3*(V1)+V2 G7=500 :G8=G7 :G9=G7 :GlO=G7 :G11=G7 :G12=G7 :Gl3=G7 :Gl4=G7 :GlS=G7 :Gl6=G7 :Gl7=G7 :Gl8=G7 :619=G7 :G20=G7 :621=G7 :622=G7 REM ------- 128 24098 REM ---- Franck-Condom integrals ( Ag state ) --- 24100 F01=2*(.1217)*(((VE1)*(vc1)A.5)/((VE1)+(VC1)))*(T11) *(4*((VE1)*(VG1))/(((VE1)+(VG1))*2))*EXP(2*(.0148) *((T11)‘2)*(((VEI)*(VGI))/((VEl)+(V61)))) 24200 F02=4*(((VE2)*(VGZ))/(((VE2)+(VGZ))*2))*EXP(2 *(-0148)*((T22)‘2)*(((VEZ)*(VGZ))/((VE2)+(V62)))) 24300 F03=4*(((VE3)*(VG3))/(((VE3)+(VG3))02))*EXP(2 *(-0148)*((T33)“2)*(((VE3)*(VG3))/((VE3)+(V63)))) 24350 F04=4*(((VE4)*(VG4))/(((VE4)+(VG4))“2))*EXP(2 *(-0148)*((T44i‘2)*(((VE4)*(VG4))/((VE4)+(VG4)))) 24400 P01=2*(.1217)*(((VGI)*(VE1)A.5)/((VE1)+(VG1))) *(T11)*(4*((VE1)*(VGI))/(((VE1)+(VGI))‘2))*EXP(2 *(-0148)*((T11)‘2)*(((VEI)*(VGI))/((VEl)+(VGl)))) 24450 P02=4*(.Ol48)*((((VE1)*(VG1))Al.5)/(((VE1)+(VG1))02)) *((T11)‘2)-2*(((VE1)*(VGI))‘-5)/((V31)+(V61)) 24500 F05=(P01)*(P02) 24600 F06=4*(.0148)*((((VGZ)*((VE2)A.5))/((VE2)+(VGZ)))02) *((T22)‘2)*4*(((VE2)*(VGZ))/(((VE2)+(VGZ))“2)) *EXP(-2 *(.0148) *((T22)‘2)*(((VE2)*(VGZ))/((VE2)+(VGZ) ))) 24700 F07=4*(.0148)*((((VG3)*((VE3)‘.5))/((VE3)+(VGB)))02) *((T33)‘2)*4*(((VE3)*(VG3))/(((VE3)+(VG3))‘2)) *EXP(-2*(.Ol48)*((T33)*2)*(((VE3)*(VG3))/((VE3)+(VG3)))) 24750 F08=4*(.0148)*((((VG4)*((VE4)*.5))/((VE4)+(VG4)))*2) *((T44i‘2)*4*(((VE4)*(VG4))/(((VE4)+(VG4))‘2)) *EXP(2 *(.0148) *((T44)‘2)*(((VE4)*(VG4))/((VE4)+(VG4)))) 24895 REM -------------------------------------------------- 24900 REM (v1) (V2) (V3) (v4) F01=<1|0><0|0> F02=<0|0><0|0> F03 =<0|0><0|0> F04=<0|0><0 0> F05=<1 1><1 0> P06=<0 1><1 0> F07 =<0 1><1 0> F08=<0 1><1 o> 24905 REM -------------------------------------------------- 24950 A1=K*(F01)*(F02)*(F03)*(F04) :A2-K*(F01)*(F02)*(F07)*(F04) :A3=K*(F01)*(F06)*(F03)*(F04) 24960 A4=K*(F05)*(F02)*(F03)*(F04) :A5=K*(F05)*(F02)*(F07)*(F04) :A6=K*(F01)*(F06)*(F07)*(F04) :A23=K*(F01)*(F02)*(F03)*(F08) 25000REM - 25110 REM ////// Franck-Condom integrals (Bu state) ////// 25120 REM F1=EXP(-(.0148*(((T4)02)*(V4)))) 25125 F2=EXP(-(.0148*(((T3)Az)*(v3)))) 25130 F3=EXP(-(.0148*(((T2)02)*(V2)))) 2 25135 F4=.122*((V1)0.5)*(Tl)*EXP(-.0148*(V1)*((T1)“2)) 25140 F5=.0148*(V2)*((T2)Az)*EXP(-.0148*(v2)*((T2)A2)) 25145 F55=.0148*(V3)*((T3)*2)*EXP(-.0148*(V3)*((T3)*2)) 25150 REM F01=.0148*(V4)*((T4)‘2)*EXP(-.0148*(V4)*((T4)*2)) 25155 F6=.122*((Vl)‘.5)*(Tl)*(.0148*(Vl)*((T1)*2)-1) *EXP(-.0148*(V1)*((T1)02)) 25160 F7=((.0148)*2)*((V2)*2)*((T2)A4) *EXP(-(.0148*(V2)*((T2)“2))) 25165 F77=((.0148)“2)*((V3)“2)*((T3)A4) *EXP(-(.Ol48*(V3)*((T3)‘2))) 129 25170 REM F02=((.0148)*2)*((V4)*2)*((T4)*4) *EXP(-(.0148*(V4)*((T4)‘2))) 25175 F9=(.00127)*((V1)“l.5)*(.0148*(V1) *((T1)*5)-2* ((T1)*3))*EXP(-.0148*(V1)*((T1)‘2)) 25180 F10=(8.88E-06)*((V1)“2.5)*((.010465)*(V1) *((Tl)*7)-(2.414) *((T1)*5))*EXP(-.0148*(V1)*((T1)02)) 25210 F11=((.3333)*(.0148)A3)*((V2)A3)*((T2)A6) *EXP(-(.0148*(V2)*((T2)02))) 25242 REM ------------------------------------------------ 25244 REM (V1) (V2) (V3) F4=<1 0><0 0> F3=<0 0><0 0> F2=<0|0><0|0> F6=<1 1><1 0> F5=<0 1><1 0> F55=<0 1><1 0> 25245 REM E9=<1 2><2 0> P7-<0 2><2 0> F77=<0 2><2 0> F10=<1 3><3 0> F11=<3 0> 25248 REM ------------------------------------------------ 25250 A7=C*(F4)*(F3)*(F2) :A8 =C*(F6)*(F3)*(F2) :A9=C*(F4)*(F5)*(F2) :A10=C*(F4)*(F3)*(F55) :A11=C*(F9)*(F3)*(F2) :A12=C*(F4)*(F7)*(F2) 25255 :A13=C*(F6)*(F5)*(F2) :Al4=C*(F4)*(F5)*(F55) :A15=C*(F6)*(F3)*(F55 :A16=C*(F9)*(F5)*(F2) 25265 A17=C*(F6)*(F7)*(F2) :A18=C*(F10)*(F3)*(F2) 25270 :A19=C*(F9)*(F7)*(F2) :A20=C*(F9)*(F3)*(F55) :A21=C*(F4)*(F7)*(F55 :A22=C*(F10)*(F5)*(F2) 25290 REM ------------------------------------------------ 25294REM 25295 REM \\\\\\\\ REP EQUATIONS BEGIN HERE \\\\\\\ 25300 11 -A1‘(((Bl-W)‘(E3-W)+01'63)/(((Bl-W)‘2) +(Gl)“2)) ‘(A3/(((E3-W)"2) + (63)?» 75400 12-A2‘(((E2-W)‘(B1-W)+GZ‘Gl)/(((E2-W)‘2)+ (02)?» '(AI/«(Bl-WYZ) + (007)) 25500 I3-A3’(((E3-W)‘(B2-W)+ 63‘62)/(((E3-W)‘2)+(G3)‘2)) ‘(AZ/(«EZ-WYZ) +(G2)‘2)) ”600 l4 -A4‘(((FA~W)‘(E6-W) + G4‘Gé)/(((E1-W)‘2) + (607)) ‘(A6/(((E6-W)"2) + (66)?» 25700 15 -A5'(((E$-W)‘(FA-W) + GS'G4)I(((F5-W)‘2) + (65)?» ‘(A4/(((B4-W)‘2) + (007)) 75800 16-A6‘(((B6-W)'(ES-W) + 66‘05)/(((E6-W)"2) + (66)“2» '(AS/(((E5-W)"2) + (GS)‘2)) 3900 R1 - (Al‘2)/(((El-W)‘2) + 61‘2) 26000 RZ-(m)/(((EZ-W)‘2)+GZ‘2) 26100 B-(m)/(((E3-W)‘2)+G3‘2) 26200 R4 . (A4‘2)/(((Bt-W)“2) + 64‘2) 26300 B-(ASY)/(((F5-W)‘2)+GS‘2) 26400 R6- (A6‘2)/(((E6-W)‘2) + 66‘2) 26500 X1 -A1'(((El-W)‘(Bt-W) +GI‘G4)/(((El-W)"2)+(Gl)‘2)) '(A4/(((FA-W)‘2) + (04)“2» 26600 XZ-A2‘(((E2-W)'(B¢-W)+GZ‘G4)/(((E2-W)‘2)+ (62)?» ‘(A4/(((E4-W)“2)+ (04)”2» 26700 X3-A3‘(((E3-W)'(B4-W)+ G3‘G4)/(((E3-W)‘2)+ (63)?» ‘(M/(«m-WYZ) + (64)?» 26800 Y1 -A1‘(((El-W)‘(E5-“0+61‘GS)/(((E1-W)‘2)+(Gl)‘2)) ‘(A5/(((P5-W)“2) + (65)“2» 26900 YZ-A2‘(((E2-W)‘(ES-W)+ GZ'GS)/(((E2-W)‘2) +(GZ)‘2)) ‘(AS/(((E5-W)‘2) +(GS)"2)) 27000 Y3-A3‘(((E3-W)‘(PS-W)+G3‘65)/(((E3-W)“2) +(G3)"2)) ‘(AS/«(ES-WW) + (65)?» T7100 21 -A1‘(((Bl-W)‘(E6-W)+Gl‘66)/(((E1-W)‘2) +(Gl)"2)) ‘(A6/(((E(rW)‘2)+ (607))27200 ZZ=A2'(((EZ- W)‘(B6-W)+GZ‘G6)/(((E2-W)"2)+(02)"2)) ‘(A6/(((FbW)‘2)+(G6)‘2)) 27300 Z3-A3‘(((B3-W)‘(Fb-W) + G3‘GG)/(((EB-W)‘2) +(G3)‘2)) ‘(AG/«(Eé-WYZ) + (06)?» 27400 Sl-(R1+R2+R3+R3+R4+R$+R6) +2‘(Il+12+13+l4+15+16+X1+x2+x3+Y1+Y2+Y3+21 +ZZ+Z3) 27410 REM U01-UO6 0: R7 are an extcntion for a 7th term 27420 001 -A1‘(((Bl-W)‘(B7-W) + GI‘GD/(«El-WYZ) + (01)?» ‘(A7/(((E7-W)“2) + (67)?» 130 27430 U02 - A2‘(((E2-W)'(E7-W) + GZ‘G7)/(((E2-W)“Z) + (62)“2» ‘(A7/(((B7-W)“2) + (607)) 27440 U03 =A3‘((CE3~W)‘(B7-W) + G3‘G7)/(((F3-W)‘2) + (63)“2» '(A7/(((E7-W)‘2)+ (0777)) 27450 004 =A4‘(((B4-W)’(E7-W) + G4'G7)/(((E4-W)‘2) + (607)) '(A7/(((E7-W)‘2) + (007)) 27460 005 =A5'(((E5-W)‘(E7-W) +GS‘G7)/(((1'5-W)‘2)+(05)‘2)) ‘(A7/(((E7-W)‘2)+(G7)"2)) 27470 006 - 6‘(((B6-W)‘(E7-W)+66‘G7)/(((E6-W)‘2)+(G6)‘2)) '(A7/(((E7-W)"2) +(G7)"2)) 27480 R7 - (A7“2)/(((B7-W)“2) + 672) 27490 82: SI +2‘(U01 +U02+UOB+U04+UOS+UO®+R7 27495REM --------- endot‘hhtermexpresions ---=------ 27500 REM D01-DO7 & R8 are an extention for an 8th term 27510 001 - A1'(((El-W)‘(BB-W) + 01 ‘G8)/(((El-W)‘2) + (007)) ‘(A8/(((E8-W)‘2) + (08)?» 77520 002*M‘(((EZ-W)'(E&W) + GZ’G8)/(((BZ-W)‘2) + (62)?» ‘(AB/(((EB-W)‘2)+(GS)‘2)) 27530 D03=4‘\3‘(((133-W)‘(138-W)+G3'(38)/(((133-W)"2) + (03)?» ‘(AB/(((EB-W)‘2)+(G8)‘2)) 27540 004 ' A4‘(((E4-W)'(EB-W) + 04‘68)/(((Bt-W)‘2) + (607)) ‘(AB/(«EB-WYZ) + (68)?» 27550 005 -As'«(mwr(mW) + Gram/(«3402) + (65)“2» '(As/«(mvvm «68)?» 77560 DO6-A6‘(((EGW)'(EB-W) +(36‘<38)/(((|?l>-W)"2) + (007)) ‘(A8/(((EB-W)‘2)+(G8)‘2)) 27570 007 ' A7'(((E7-W)'(EB-W) + G7‘GB)/(((E7-W)"2) + (67)?» ‘(AB/(«EB-WYZ) + (68)?» 27580 R8'(48’?)/(((Efi-W)“2)t68"?) 27590 S3— 82+2'(DOI+D02+D03+DO4+D0$+D06+DO7)+R8 27595 REM end of 8th term expressions 27600 REM 801-808 & R9 are an extention for an 9th term 27610 801 =Al‘(((El-W)‘(B9-W) + (31 ‘G9)/(((El-W)“2) + (Gl)“2)) ‘(A9/(((B9-W)“2) + (G9)“2)) 27620 W-AZ‘(((E2-W)‘(E9-W) + Gz‘(39)/(((132-\'V)‘2)+(C32)"2)) ’(A9/(((E9-W)‘2) + (69)?» 27630 W=A3‘(((E3-W)'(B9-W) + G3'G9)/(((E3-W)‘2) +(G3)"2)) '(A9/(((E9-W)"2) + (09)“2» 27640 304 =A4‘(((FA-W)‘(B9-W) + G4‘G9)/(((FA-W)‘2) + (007)) ‘(A9/(((E9-W)‘2) + (69)?» 27650 905 'A5'(((F5-W)'(E9-W) + 05‘G9)l(((E5-W)‘2) + (05)?» ‘(A9/(((B9-W)"2) + (69)?» 27660 306'A6‘(((E6-W)‘(E9-W) + G6'G9)/(((E6-W)"2) + (607)) ‘(A9/(((B9-W)‘2) + (09)?» 27670 907 'A7'(((B7-W)‘(E9-W) + G7‘G9)/(((E7-W)‘2) + (07)“2» ‘(A9/(((E9-W)‘2) + (09)?» 27680 1308==48‘(((13’8-W)‘(E9-\1V) + G8’G9)/(((133-W)“2)+(08)”2» ‘(A9/(((E9-W)“2) + (09)?» 27690 R9=(1“9‘2)/(«Bil-WT?) + 09‘2) 27695 54- S3+2'(BOI+BOZ+BO3+BO4+BO$+BO6+BO7+B(B)+R9 27697REM ...... endot'9thtermerpreesions ------ 27700 REM H01-H09 .9 R10 are an extention for an 10th term 27710 H01 - A1‘(((El—W)’(ElO-W) + Gl'GlO)/(((El-W)"2) + (Gl)"2)) ‘(AlO/(((B10-W)"2) + (GlO)"2)) 27720 H02 -A2‘(((E2-W)'(BlO-W) + GZ'GIO)/(((E?AV)‘2) + (GZ)“2)) '(AlO/(((BlO-W)"2) +(610)"2)) 27730 H03 - A3‘(((E3-W)‘(E10-W) + G3‘GlO)/(((E3-W)“2) + (G3)‘2)) ‘(AlO/(((E10-W)"2) + (GlO)“2)) 27740 H04 =A4'(((E4-W)'(BIO-W) +G4'GIO)/(((FA-W)‘2) +(G4)“2)) '(A10/(((810-W)‘2) + (010)?» 27750 H05 =A5‘(((E5-W)'(E10W) +Cfi‘610)/(((E5-W)‘2)+(05)‘2)) '(A10/(((E10-W)"2) +(610)‘2)) T7760 “06*A6'(((E&W)‘(EIGW)+06‘610)/(((136-W)"2) +(06)"2)) ‘(A10/(((BIO-W)"2) +(010)‘2)) 27770 H07 =A7'(((B7-W)‘(BlO-W) + G7‘GlO)/(((E7-W)“2) + (G7)“2)) ‘(AlO/(((E10-W)"2) + (GlO)"2)) 27780 Hm -A8'(((EB-W)'(EIO-W) + 08‘610)/(((E&W)‘2) + (08)?» ‘(A10/(((BIO-W)‘2) + (010)?» 27790 H09=A9‘(((B9-W)‘(E10-W) + G9‘GIO)/(((E9-W)‘2) + (09)?» ‘(A10/(((E10-W)‘2) + (010)?» 27795 R10 - ((A10)‘2)/(((ElO-W)"2) + (610)?) 27797 85- S4+2‘(HOI +H02+H03+H04+H0$+H06+H07+Hm+H09)+R10 27799 REM end of 10th term expmiom moo REM [DI-[010 0 R11 are an extention for an 11th term 27810 101 -A1‘(((El-W)‘(E11-W) + Gl‘Gll)/(((El-W)“2) + (Gl)“2)) ‘(All /(((B11-W)“2) + (61 1)‘2)) T7820 1m'=:‘\2‘(((E?-W)‘(131l-W) + 02'611)/(((132-W)"2) + (02)“2» '(A11/(((Ell-W)"2) + (011)?» 27830 W'M‘(((E3-W)'(E11-W) + G3‘Gll)/(((E3-W)"2) + (63)?» ‘(A11/(((E11-W)“2) + (011)“2» 77840 1104"“'\4‘(((E4-W)‘(E1 1-“0 + G4‘Gll)/(((FA-W)‘2) +(G4)"2)) ‘(AII/(«Ell-WYZ) + (0107)) 77850 1115 =A5‘(((E’5-W)‘(Bll-W) + GS‘Gll)/(((P5-W)‘2)+ (05)?» ‘(Ml/(«Ell-WYZ) + (611)?» 27860 W=A6‘(((E6-W)‘(Ell-W) + G6'Gll)/(((BG-W)‘2)+ (007)) ‘(A11/(((Ell-“0"2) + (611)?» 77870 I’07"A‘\‘7'(((B7-W)’(Ell-W) + 67’611)/(((E7-W)"2) +(G7)‘2)) ‘(A11/(((E11-W)‘2) + (011)?» 27880 1113 ‘A8'(((EB-W)‘(EI 1-W) + 08‘611)/(((138-W)“2) + (08)?» '(All/(«Ell-WYZ) + (011)“2» 27890 1109='M"(«PS-“0‘031 1-“0 + G9‘Gll)l(((B9-W)‘2)+(G9)"2)) ‘(All/«(Ell-WYZ) + (011)?» 131 27900 11110 :A10‘(((E10-W)‘(Ell-W) + 610‘611)/(((Elo -W)?) + (610)?))’(A11/(((El l-W)?) + (61 1)?» 27910 R11 - ((A11)?)/(((E11-W)?) + (611)?) 27920 S6=55+2-(un+w2+1m+m+w+ms+1m+1m+Im+w1m+ R11 27930REM --------- endofllthtemespresions ----------- 28000REM JOI-JOllaRIZuemextentionforanIZthter-m 28010 .101 -A1 ‘(((Bl-\V)‘(E12-W) + 61'612)/(((E1-W)?) + (61)?» ‘(AlZ/(((Blz-W)?) + (612)?» 28020 J02-:‘12‘(((l'i7v-W)’(Ell-W) + 62'612>/<((mvo*2)+(62)*2)) ‘(A12/(((1312-W)‘2) + (012)?» 28030 103-M'(((E¥-W)'(EIZ-W) +G3'612)/(((B-W)"2) + (63)?» ‘(AIZ/(((E12-W)"2) + (612)“2» 28040 J0‘'-A4'(((134-W)MED-W) +G4'612)/ «(m-WW) + (607)) ‘(A12/(((E1?rW)"2) + (612)?» 28050 105 =A5’(((B.W)‘(E12-W) + GS‘GIZ)/(((F5-W)‘2)+ (65)?» '(A12/(((E12-W)“2) + (012)?» 28060 ’06 =A6'(((E6-W)'(E12-W) + 06'612)/(((Eb-W)‘2) + (06)?» ‘(AIZ/(((E12-W)‘2)+(012)‘2)) 28070 107 =A7'(((B7-W)‘(E12-W) +G7‘612)/(((E7-W)‘2) + (607)) ‘(AIZ/(«EIZ-WYZ) +(612)‘2)) 28080 108-A8‘(((E8-W)‘(E12-W)+ 08‘012)/(((Efi-W)‘2)+ (68)?» ‘(AIZ/(((E12-W)"2) +(GIZ)‘2)) 28090 109 -A9‘(((E9-W)‘(E12-W) +G9‘612)/(((B9-W)‘2) + (09)?» ‘(A12/(((E12-W)‘2) + (012)?» 281K) 1010 -A10‘(((E10-W)‘(1312-W) + 610‘612)/(((E10-W)?) + (610)?»‘(A12/(((1312-W)?) + (612)?» 28110 1011 -A11'(((E11-W)‘(Blz-W) + 611‘612)/(((B11-\V)?) + (611)?))'(A12/(((Elz-W)?) + (612)?» 28120 R12- ((A12)?)/(((F.12-W)?) + (612)?) 28130 87- S6+2'(101 +Joz+103+104+105+ms+107+108 +JO9+1010+1011)+ R12 28140 REM end of 12th term expressions 2821) REM 001-0011 8: R13anean mention foran 13th term 28210 001 -= A1 ‘(((E1-W)'(E13-W) + 61 ‘613)/(((E1-W)?) + (61)?» ‘(A13/(((Bl3-W)?) + (613)?» 28220 002 =A2‘(((E2-W)‘(Bl3-W) + 62‘613)/(((E2»W)?) + (62)?» ‘(A13/(((E13-W)?) + (613)?» 28230 OCB -A3'(((E.3-W)'(El3-W) + 63‘613)/(((B3-W)?) + (63)?» ‘(A13/(((1313-W)?) + (613)?» mo 004 =A4‘(((E’A—W)‘(El3-W) + 04‘613)/(((E‘-W)"2) + (007)) ’(A13/(((Bl3-W)‘2) + (013)?» 287-50 005 ‘AS‘(((E$-W)‘(El3-W)+ 05‘013)/(((F5-W)‘2) + (0577)) ‘(A13/(((E13-W)‘2) +(Gl3)“2)) 28260 006-A6‘(((Bé-W)‘(E13-W) +06‘Gl3)/(((B(>W)"2) + (06)?» ‘(A13/(((E13-W)‘2)+ (613)?» 28270 007 -A7‘(((E7-W)‘(El3-W) + 67‘013)/(((E7-W)"2) + (07)?» ‘(Al3/(((E13-W)‘2)+(Gl3)‘2)) 28280 COB-M‘(((B8-W)‘(Bl3-W) + 08‘613)/(((EB-W)‘2) + (08)?» ‘(A13/(((Bl3-W)"2)+ (013)?» 28290 009-A9‘(((B9-W)‘(El3-W) + 09‘613)/(((E9-W)“2) + (69)?» ‘(A13/(((E13-W)‘2) + (01:02» 28310 0010 -A10' (((ElO-W)'(El3-W) + 610‘613)/(((ElO-W)?) + (610)?))'(A13/(((B13-W)?) + (613)?» 28310 0011 -A11 ‘(((E1 1-W)‘(El3-W) + 611 '613) /(((El 1-W)?) + (611)?»‘(A13/(((B13-W)?) + (613)?» 2833) 0012 . A12‘(((E12—W)‘(El3-W) + 612‘613)/(((E12-W)?) + (612)?))'(A13/(((El3-W)?) + (613)?» 28330 R13 . ((A13)?)/(((El3-W)?) + (613)?) 28340 SB - S7+2'(001+002+003+004+005+006+007+0(B+009 +0010+0011+0012)+R13 28350REM -------- endotl3thtermerpresions -------- 284m REM Ql-Ql3& R14ereanextention foran 14th term 28410 01 -A1 ‘(((E1-W)‘(E14-W) + 61‘614)/(((El-W)?) + (61)?» '(A14/(((Bl4-W)?) + (614)?» 28420 02 :- A2‘(((Ez-W)‘(E14-W) + 62'614)/(((Ez-W)?) + (62)?» ‘(A14/(((E14-W)?) + (614)?» 28430 03-M‘(((E3-W)'(E14-W) +G3‘Gl4)/(((E3-W)‘2) + (63)?» ‘(A14/(((E14-W)"2) + (614)?» 28440 04 -M‘(((FA-\V)‘(El4-W) +G4‘Gl4)/(((Et-W)"2) +(G4)“2)) ‘(A14/(((Bl4-W)‘2)+ (0107)) 2860 05-M‘(((F-5-W)‘(El4-W) +GS‘GI4)/(((F5-W)‘2) + (05)“2» ‘(A14/(((El4-W)“2) + (0107)) 28460 06-M’(((B6-W)‘(El4-W) +G6'Gl4)/(((B6’W)“2) + (66)?» ’(A14/(((E14-W)"2) +(Gl4)"2)) 28470 07 -A7‘(((B7-W)‘(El4-W) + G7‘Gl4)/(((B7-W)"2) + (607)) ‘(A14/(((El4-W)‘2) + (614)?» 28480 08 -A8‘(((E8—W)‘(El4-W) + 68'014)/ «(884107) + (08)?» ‘(A14/(((E14-W)"2) + (614)?» 28490 09 - A9‘(((B9-W)‘(El4-W) + 69‘614)/(((B9-W)?) + (69)?» ‘(A14/(((El4-W)?) + (614)?» 28500 010 - AlO‘(((ElO-W)‘(El4—W) + 610‘614)/(((ElO-W)?) + (610)?» ‘(A14/(((E14-W)?) + (614)?» 28510 011 -A11’(((El 1-W)‘(E14-W) + 611‘614)/(((E11-W)?) + (611)?))‘(A14/(((E14-W)?) + (614)?» 2851) 012 =A12‘(((ElZ-W)‘(El4-W) + 612‘614)/(((E12-W)?) + (612)?))‘(A14/(((E14-W)?) + (614)?» 28530 013 - Al3‘(((El3-\V)'(E14-W) + 613‘614)/(((B13-\V)?) + (613)?))‘(A14/(((El4-W)?) + (614)?» 28540 R14 3 ((A14)?)/(((E14-W)?) + (614)?) 28550 S9-88+2‘(01 +02+Q3+Q4+05+Q6+Q7+08+Q9+010+011 +012+Ql3)+R14 28560 REM “'” end of 14th term expressions “”“‘ 2871!) REM N1-N14 .1 R15 areanextention foran 15th term 23710 N1 'A1’(((Bl-W)‘(EIS-W) + 01'615)/(((El-W)‘2) + (6173)) ’(A15/(((315-W)“2)+(015)‘2)) 132 28720 NZ-A2‘(((E?rW)‘(E15-W)+ GZ‘GIS)/(((E2-W)‘2) + (02)?» '(AIS/(((BIS-W)‘2) + (615)?» 28730 N3-AB‘(((B-W)'(B15-W) + 63'615)/(((133-W)‘2)+(G3)‘2)) ‘(A15/(((E15-W)‘2) + (015)?» 28740 N4 -M‘(((B4-W)‘(EIS-W) + 64‘610/(((B4-W)‘2) + (04)?» ‘(AlS/«(ElS-WW) +(615)‘2)) 28750 N5 -A5‘(((ES-W)’(El$-W) + GS'GIS)/(((F5-“0‘2)+(GS)‘2)) ‘(AU/(((E15-W)‘2)+(615)"2)) 28760 N6-A6‘(((B6-W)‘(E15-W) + 06‘615)/(((86-W)‘2) + (66)?» ‘(A15/(((E15-W)‘2) + (615)?» 28770 N7 -A7‘(((E7-W)‘(B15-W) + G7'GIS)/(((B7-W)‘2) + (607)) '(A15/(((E15-W)“2)+(615)‘2)) 28780 M=A8‘(((E8-W)’(EIS-W) + GB‘GIS)/(((BB-W)“2) + (68)?» ‘(A15/(((E15-W)"2)+(015)‘2)) 28790 N9-A9‘(((E9-W)‘(BIS-W) + G9‘GIS)/(((B9-W)“2) + (09)?» '(A15/(((EIS-W)‘2) +(GIS)‘2)) 28800 N10 -A10‘(((E10-W)'(El$—W) + 610‘615)/(((Elo-W)?) + (610)?» ‘(A15/(((E15-W)?)+ (615)?» 28810 N11 -A11'(((E11-W)'(EIS-W) + 611'615)/(((E11-W)?) + (611)?))’(A15/(((E15-W)?) + (615)?» 2881) N12 -A12'(((E12-W)‘(E15-W) + 612‘615)/(((1312-W)?) + (612)?» ‘(A15/(((E15-W)?) + (615)?» 28830 N13 -A13‘(((El3-W)‘(El$-W) + 613'615)/(((Bl3-W)?) + (613)?» ‘(A15/(((E15-W)?) + (615)?» 2840 N14 -A14'(((El4-W)‘(B15—W) + 614‘615)/(((E14-W)?) + (614)?» “(AIS/((0315417)?) + (615)?» m R15-((A15)‘2)/(((l315-W)‘2)+(015)“2) 28860 S10-S9+2‘(N1+N2+N3+N4+N5+N6+N7+N8+N9+N10+N11 +N12+N13+N14)+R15 28870 REM '-‘-‘-'-‘- end of 15th term expressions -'-‘-'-‘- 29000REM M1-M15 a R16areanextentionforan 16th term 29010 M1 - A1 '(((E1-W)'(El6-W) + 61 '616)/(((El-W)?) + (61)?» '(A16/(((El6-W)?) + (616)?» 29020 MZ-A2‘(((EZ-W)’(El6-W) +GZ‘GI6)/(((E2-W)‘2) + (02)?» ‘(A16/(((316-W)“2) + (616)?)) 29030 M3-A3‘(((B3-W)‘(El6-W) + G3‘Glé)/(((E3-W)‘2) + (63)?» ‘(A16/(((E16-W)‘2)+(616)"2)) 29040 M4-A4'(((B4-W)‘(Bl6-W) + 64'016)/(((PA-W)‘2) + (64)“2» '(A16/(((E16W)‘2) + (616)?)) 29050 M5 -AS‘(((P-$-W)‘(Elé-W) + 65’616)/(((P5-W)"2) + (65)?» ‘(A16/(((E16-W)"2) + (016)?» 29060 M6-A6‘(((Bé-W)‘(BIC>“0 + 66‘616)/(((136-W)‘2) +(66)"2)) ‘(A16/(((E16-“0"2) + (616)?)) 29070 M7 - A7‘(((B7-W)‘(E16-W) + 67°616)/(((E7-W)?) + (67)?» ‘(A16/(((E16-W)?) + (616)?)) 29080 MB-M‘(((133-\‘/)‘(Elt5-W) + G8‘<316)/(((138-W)‘2) + (68)?» ‘(A16/(((B16-W)“2) + (016)?» 29090 M9-A9’(((E9-W)'(El6-W) + 09‘016)/(((E9-“0"2) + (G9)“2» ‘(416/(((E16-W)“2) + (616)?)) 29100 M10 -A10'(((Elo-W)'(El6-W) + 610‘616)/(((E10-W)?) + (610)?))‘(A16/(((E16-W)?) + (616)?)) 29110 M11 -A11‘(((E1 1-W)‘(El6-W) + 611‘616)/(((E11-W)?) + (611)?))'(A16/(((Bl6-W)?) + (616)?)) 29120 M12 -A12'(((E12-W)'(Bl6-W) + 612'616)/(((Bl2-W)?) + (612)?))‘(A16/(((E16-W)?) + (616)?)) 29130 M13 -A13‘(((E13-W)‘(El6-W) + 613'616)/(((El3-W)?) + (613)?» '(A16/(((El6-W)?) + (616)?)) 29140 M14 - A14‘(((Bl4-W)'(E16-W) + 614‘616)/(((E14-W)?) + (614)?))‘(A16/(((1316-W)?) + (616)?)) 29150 M15 -A15‘(((E15-W)‘(E16-W) + 615‘616)/(((E15-W)?) + (615)?))‘(A16/(((Bl6-W)?) + (616)?)) 29160 R16- ((A16)?)/(((El6-W)?) + (616)?) 29170 811- 810+2‘(M1+M2+M3+M4+M5+M6+M7+M8+M9+M10+M11 +M12+M13+M14+M15) + R16 29180 REM -‘-‘-‘-'- end of 16th term expressions -‘-‘-‘-‘- 29200 REM P1-P16 & R17 are an extention for an 17th term 29210 P1 -A1 '(((Bl-W)'(E17-W) + 61:617)/(((El-W)?) + (61)?» '(A17/(((E17-W)?) + (617)?» 29220 P2 - A2‘(((E2-W)‘(Bl7-W) + 62‘617)/(((m-W)?) + (62)?» '(A17/(((E17-W)?) + (617)?))29230 P3 = A3 ' (((E3- W)‘(E17-“0+G3‘Gl7)/(((B-W)‘2)+(63)“2» ‘(A17/(((Bl7-W)‘2)+(Gl7)"2)) 29240 P4-A4‘(((FA-W)‘(E17-W)+G4‘Gl7)/(((BA-W)"2)+(607)) ‘(A17/(((El7-W)“2) + (017)?» 29250 P5 -A5’(((F5-W)‘(Bl7-W)+GS‘GI7)/(((F5-W)‘2) +(GS)“2)) ‘(A17/(((El7-W)“2) + (017)?» 29260 P6-A6‘(((E6-W)‘(Bl7-W) +06‘Gl7)/(((E6-W)‘2) + (66)?» '(A17/(((Bl7-W)“2) + (017)“2» 29270 P7-A7'(((E7-W)‘(El7-W)+ 07‘610/(((E7.W)‘2) + (67)?» ‘(A17/(((El7-W)"2) + (017)“2» 29280 PB-A8‘(((EB-W)'(El7-W) + 68‘610/(«B-WYZ) +(G8)"2)) '(A17/(((E17-W)‘2) + (017)?» 29290 P9-A9‘(((B9-W)‘(El7-W) + 09‘610/(«B9-WYZ) + (69)?» ’(Al7/(((El7-W)‘2) + (617)?)) 29300 P10 - AIO' (((ElO-W)‘(E17-W) + 610‘617)/(((ElO-W)?) + (610)?» ‘(A17/(((Bl7-W)?) + (617)?)) 29310 P11 -A11’(((Ell-W)'(El7-W)+611‘617)/(((P.11-W)?) + (611)?))'(A17/(((E17-W)?)+(617)?)) 29320 P12-A12'(((E12-W)‘(El7-\V) +612'617)/(((812-W)?) + (612)?))'(A17/(((E17-W)?) + (617)?)) 29330 P13-A13‘(((E13-W)‘(El7-W) + 613‘617)/(((E13-W)?) + (613)?»‘(A17/(((El7-W)?) + (617)?» 29340 P14 -A14‘(((El4-W)‘(E17-W) + 614‘617)/(((Bl4—W)?) + (614)?))'(A17/(((E17-W)?) + (617)?)) 29150 P15 -A15‘(((EIS-W)‘(E17-W) + 615'010/(«EIS-WYZ) + (015)“2))‘(A17/(((E17-W)‘2) + (617)?)) 29360 P16 -A16'(((E16-W)‘(El7-W) + 616'617)/(((El6-W)?) + (616)?))‘(A17/(((Bl7-W)?) + (617)?)) 29370 R17-((A17)?)/(((El7-W)?)+ (617)?) 29380 812- $11 +2‘(P1 +P2+P3+P4+P5+P6+P7+PB+P9+P10+P11 +P12+P13+P14+P15+P16)+R17 133 29390 REM -‘-'-‘- end of 17th term expressions -'-‘-‘-‘- 294mm k01.k017& R18areanextention foran 18thterm 29410 K01 -A1 ‘(((E1-W)‘(E18-W) + 61‘618)/(((Bl-W)?) + (61)?» '(A18/(((E18-W)?) + (618)?» 29420 K02 'A2’(((E2-W)’(El8-W) + 02‘018)/(((E2-W)‘2) + (62)?» ‘(A18/(((Bl8-W)‘2) + (618)?» 29430 I"-03-43‘(((133-W)'(1313-W) + 03‘618)/(((E3-W)‘2) + (63)?» '(AIB/«(ElB-WYZ) + (018)?» 29440 KM-M‘(((B4-W)’(EIB~W) + 04'018)/(((Bt-W)“2) +(G4)“2)) '(A18/(((818-W)‘2)+(018)“2)) 29450 K05 -A5'(((F5-W)‘(E18-W)+ 05'618)/(((F5-“0“2) + (05)?» ‘(A18/(((E18-W)‘2)+(018)‘2)) 29460 K06=A6‘(((E6-W)’(BIB-W)+G6’018)/(((E6-W)‘2) + (06)?» ‘(A18/(((EIB-W)“2) + (018)?» 29470 K07-A7’ (((B7-W)‘(E18-W) + G7’Ci18)/(((E7-W)“2) + (07)“2» ‘(A18/(((E18-W)‘2) + (018)?» 29480 KOB'A8‘(((E8-W)‘(EIB-W) + G8‘GIS)/(((ES-W)‘2) + (08)?» ‘(A18/(((EIB-W)"2) + (618)?» 29490 K09 -A9'(((E9-W)‘(E18-W) + 09‘618)/(((E9-W)‘2) + (09)“2» ‘(A18/(((El&“0"2) + (618)?» 29500 [(010 -A10‘(((E10-W)‘(F.18-W) + 610‘618)/(((ElO-\V)?) + (610)?))‘(A18/(((E18-W)?) + (618)?» 29510 K011 -A11‘(((E11-W)'(E18-W) + 611'618)/(((Bll-W)?) + (611)?))‘(A18/(((E18-W)?) + (618)?» 29520 K012-A12‘(((BlZ-W)‘(EI&W) + 612°618)/(((Elz-W)?) + (612)?))‘(A18/(((E18-W)?) + (618)?» 2953) K013 -A13’(((El3-W)'(E18-W) + 613'618)/(((E13-W)?) + (613)?))‘(A18/(((E18-W)?) + (618)?» 29540 K014 :- A14‘(((E14-W)‘(P.18-W) + 614‘618)/(((E14-W)?) + (614)?))‘(A18/(((E18-W)?) + (618)?» 29550 [(015 -A15‘(((E15-W)‘(E18-W) + 615‘618)/(((F.15-W)?) + (615)?))‘(A18/(((818-W)?) + (618)?» 29560 [(016 - A16'(((El6-W)‘(E18-W) + 616'618)/(((El6-W)?) + (616)?))‘(A18/(((EIB-W)?) + (618)?» 29570 [(017 - A17'(((E17-W)‘(E18-W) + 617'618)/(((E17-W)?) + (617)?))‘(A18/(((1318-W)?) + (618)?» 29580 R18 - ((A18)?)/(((BlB-\V)?) + (618)?) 29590 813- S12 +2‘(K01 +K02+K03+K04+K05+K06+K07+K£B+K09+K010 +K011 +K012+K013+K014+K015+K016+K017)+R18 29595 REM -‘-‘-’-‘- end of 18th term expressions -‘-’-’-‘-‘ 30000 REM U1-U18 4 R19 are an extention for an 19th term 30010 U1 -A1‘(((El-W)‘(Bl9-W) + 61‘619)/(((B1-W)?) + (61)?» ‘(A19/(((EI9PW)?) + (619)?» 30020 U2 - A2‘(((EZ-W)‘(E19-W) + 62'619)/(((E2..W)?) + (62)?» ‘(A19/(((El9-W)?) + (619)?)) 110:!) U3 - A3’(((B-W)‘(E19—W) + 63‘619)/(((133-W)?) + (63)?» ‘(A19/(((Bl9-W)?) + (619)?)) 30040 04-A4‘(((FA-W)‘(El9-W) + 64'019)/(((E’4-W)‘2)+ (607)) ‘(A19/(((El9-W)“2) + (019m)) 30050 US -A5‘(((E5-W)‘(EI9BW) + 65’019)/(((P5-W)‘2)+ (5577)) ‘(419/(((EISLW)"2)+ (619)?)) 30060 06-M’(((86-W)'(Bl9-W) + 06‘019)/(((E6-W)"2)+ (06)?» '(A19/(((E19-W)“2) + (019)?» 30070 U7 -A7‘(((E7-W)°(El9-W) + 67‘619)/(((E7-W)?) + (67)?» ‘(A19/(((B19-W)?)+(619)?» 1180 U8 -A8‘(((EB-W)’(Bl9-W) + 68‘619)/(((EB-W)?) + (68)?» '(A19/(((Bl9-W)?) + (619)?)) 30090 U9-A9‘(((E9-W)'(Bl9-W) + 09’019)/(((E9-W)‘2)+ (69)?» ‘(A19/(((E19-W)“2) + (019)?» 30100 U10 -A10‘(((E10-W)‘(E19-W) + 610‘619)/(((ElO-W)?) + (610)?))‘(A19/(((El9-W)?) + (619)?)) 11110 U11 -A11‘(((Ell-W)‘(El9-W) + 611‘619)/(((Bll-W)?) + (611)?))’(A19/(((E19-W)?) + (619)?)) 30120 U12 -A12‘(((E12-W)‘(Bl9-W) + 612‘619)/(((ElZ-\V)?) + (612)?))’(A19/(((El9-W)?) + (619)?)) 1111) U13 -A13‘(((E13-W)‘(E19-W) + 613‘619)/(((Bl3-W)?) + (613)?))‘(A19/(((E19-W)?) + (619)?)) $140 U14 -A14‘(((Bl4-W)‘(Bl9-W) + 614‘619)/(((E14-W)?) + (614)?))‘(A19/(((E19-W)?)+ (619)?)) 30150 U15 =A15‘(((815-W)‘(E19-W) + 615‘619)/(((E15-W)?) + (615)?))‘(A19/(((El9-W)?) + (619)?)) 11160 U16- A16’(((E16-W)‘(El9-W) + 616‘619)/(((E16-W)?) + (616)?))‘(A19/(((E19-W)?) + (619)?)) 11170 U17-A17‘(((Bl7-W)‘(El9-W) + 617‘619)/(((E17-W)?) + (617)?))‘(A19/(((E19.W)?) + (619)?)) 30180 U18 =A18‘(((E18-W)'(Bl9-W) + 618‘619)/(((818-W)?) + (618)?))‘(A19/(((E195W)?) + (619)?)) 111% R19 - ((A19)?)/(((E19-W)?) + (619)?) 30195 Sl4-Sl3+2‘(U1 +UZ+U3+U4+US+U6+U7+UB+U9+U10+U11+012 +U13+U14+U15+Ul6+U17+ U18) + R19 1119‘] REM ‘-‘-‘-‘-'- end of 19th term expression -“-‘-'-‘-’ WREM D1-D19& Rmareanextentionforanmthterm 30210 DI -A1‘(((El-W)’(EZO-W) + 61‘020)/(((El-W)"2) + (01)?» ‘(AZO/(«Em-WYZ) + (020)?» 30220 DZ-AZ‘(((Ez-W)'(E20-W) + 02‘020)/(((E?rw)‘2) + (62)“2» ‘(A20/(((820-W)"2) + (020)?» 30230 D3-A?”(«133-W)11330-W) + 63‘620)/ «(133-Wm) + (03)“2» KAN/«(WWW + (020)?» 30240 D4 -M'(((E‘-W)‘(EZO-W) + 64‘620)/(((E4-W)‘2) + (007)) ‘(AZO/(((F10-W)"2)+ (620)?» 30350 I)5-M'(((E‘5-W)'(Em-W) + 05'020)/(((85-W)"2)+ (65)?» '(AZO/(«EZO-WYZ) + (020)?» 30260 D6*A6‘(((Pb-W)‘(mw) + 06‘020)/(((E6-W)‘2)+ (06)?» ‘(420/(((EZO-W)‘2) + (020)?» 30270 D7'A7‘(((E7-W)‘(EZO-W) + 67‘020)/(((B7-W)"2)+ (67)?» ‘(A20/(((820-W)‘2) + (020)?» 30230 DB -A8‘(((Efi-W)’(BZ0-W) + G8‘Gm)/(((BS-W)‘2) + (08)?» ’(A20/(((E20-W)‘2) + (020)?» 134 30290 D9-A9‘(((E9-W)‘(Ew-W) + G9‘Gw)/(((B9-W)“2) + (09)?» ‘(A20/(((E20-W)‘2) + (020)?» 30300 D10 . A10‘(((E10-W)‘(E%-W) + 610‘6%)/(((E10-W)?) + (610)?))‘(A%/(((E%-W)?) + (620)?» 11310 Dll -A11‘(((F.1 1-W)‘(E%-W) + 611‘6%)/(((E11-W)?) + (611)?»‘(A%/(((E%-W)?) + (6%)?» 303% D12 -A12‘(((E12-W)‘(E%-W) + 612'6%)/(((ElZ-W)?) + (612)?))'(A%/(((E%-W)?) + (6%)?» 30330 D13 =A13‘ (((Bl3-W)‘(E%-W) + 613‘6%)/(((E13-W)?) + (613)?))‘(A%/(((E%-W)?) + (6%)?» 1340 D14 -A14‘(((El4-W)‘(E%-W) + 614‘6%)/(((E14-W)?) + (614)?))'(A%/(((B%-W)?) + (6%)?» 30150 DIS -A15‘(((E15-W)‘(E?0-W) + 015‘020)/(((E15-W)“2) + (015)?»‘(AZO/(«E20-WW) + (020)?» %360 D16 -A16'(((l~316-W)‘(B%-W) + 616‘6%)/(((El6-W)?) + (616)?))‘(A%/(((E%-W)?) + (6%)?» %370 D17-A17‘(((E17-W)'(1m-W) + 617‘6%)/(((El7-W)?) + (617)?))‘(A%/(((E%—W)?) + (6%)?» 30380 D18 -A18‘(((E18-W)'(E%-W) + 618‘6%)/(((E18-W)?) + (618)?»‘(A%/(((E%-W)?) + (6%)?» 30390 D19 -A19‘(((El9-W)‘(E%-W) + 619‘6%)/(((E19-W)?) + (619)?))‘(A%/(((E%-W)?) + (6%)?» 30400 R20=((A20)‘2)/(((EZO-W)‘2)+(020)‘2) 30410 815- Sl4+2'(D1+D2+D3+D4+D5+D6+D7+D8+D9+D10+D11+D12 +D13+D14+ D15 +D16+ D17+D18+ Dl9)+ R% 504% REM -"-'-‘-'- end of 20th term expressions -’-'-‘-'-‘ 30500 REM 131-8% & R21 are an extention [Or an 21th term 30510 Bl - A1‘(((El-W)‘(E21oW) + 61'621)/(((El-W)?) + (61)?» ‘(A21/(((EZl-W)?) + (621)?» 30520 82 -A2‘(((EZ-W)‘(EZI-W) + 02°621)/ («124mm + (02)?» ‘(A21/(((EZI-W)‘2) + (621)?)) 30530 33 'A3‘(((E3-W)‘(E21-W) + G3‘GZI)/(((B-W)‘2) + (63)?» ‘(A21/(((E21-W)‘2)+(021)‘2)) 30540 84 'A4‘(((B4-W)‘(E21-W) + 04'621)/(((FA-W)"2) + (607)) ‘(A21/(((E21-W)‘2) + (021)?» 30550 85 'A5‘(((BS-W)‘(EZI-W) + 05‘621)/(((ES-W)‘2)+ (65)“2» ‘(A21/(((Ezl-W)‘2) + (621)?)) 30560 36 'A6‘(((E6-W)‘(ml-W) + 66‘021)/(((E6-W)“2) + (66)?» ‘(AZl/(«Ell-WYZ) + (021)?» 30570 B7 -A7'(((B‘7-W)‘(E21-W) + 67'021)/(((E7-W)“2) + (67)?» '(A21/(((E21-W)‘2) + (021)?» 30580 BB -A8‘(((EB-W)’(E21-W) + 08‘621)/(((E8-W)“2) + (08)?» '(AZI/(((EZI-W)‘2) + (021)?» 30590 B9 =A9‘(((E9-W)‘(E21-W) + 09‘621)/(((E9-W)‘2) + (09)?» ‘(A21/(((E21-W)"2) + (021)?» 30600 BIO =A10‘ (((ElO-W)‘(BZl-W) + 610‘ 621)/(((ElO-W)?) + (610)?))‘(A21/(((E21-W)?) + (621)?)) 30610 811 -A11‘(((El 1-W)‘(m1-W) + 611‘621)/(((E11-W)?) + (611)?))‘(A21/(((E21-W)?) + (621)?)) 306% 1312 - A12‘(((BlZ-W)'(EZl-W) + 612‘621)/(((E12-W)?) + (612)?))’(A21/(((E21-W)?) + (621)?)) 11630 813 - A13‘(((Bl3-W)‘(E21-W) + 613‘621)/(((El3-W)?) + (613)?»‘(A21 /(((E21-W)?) + (621)?)) 11640 814 - Al4‘(((El4-W)‘(E21-W) + 614‘621)/(((E14-W)?) + (614)?»‘(A21/(((F.21-W)?) + (621)?)) 30650 815 -A15'(((E15-W)‘(E21-W) + 015’021)/(((EIS-W)‘2) + (615)“2))'(A21/(((E21-W)"2) + (0207)) 11660 816 -A16‘ (((El6-W)'(m1-W) + 616‘621)/(((1316W)?) + (616)?))‘(A21/(((1321-W)?) + (621)?)) 11670 817 - A17' (((E17-W)‘(E21-W) + 617'621)/(((Bl7-W)?) + (617)?))‘(A21/(((E21-W)?) + (621)?)) 30680 818 -A18‘ (((ElB-W)‘(821-W) + 618‘621)/(((1318-W)?) + (618)?))‘(A21/(((EZl-W)?) + (621)?)) 11690 819 -A19‘ (((E19W)‘(E21-W) + 619‘ 621)/(((El9-\V)?) + (619)?))‘(A21/(((Em-W)?) + (621)?)) 30700 BZO-AZO‘(((E20-W)‘(B21-W) + 620‘621)/(((E20-W)‘2) + (620)?))‘(A21/(((E21-W)"2) + (021)?» 30710 R21 - ((M1)?)/(((EZl-W)?) + (621)?) 307% 816- 815+2‘(Bl+BZ+B3+B4+BS+B6+B7+BB+B9+BIO+BII +BlZ+Bl3+Bl4+BlS+Bl6+Bl7+1318+Bl9+B%)+ R21 30730 REM -'-‘-'-‘- end of 21th term expressions -‘-'-'-‘-‘ 310MREM H1-H21 a R22areanextentionforan22thterm 31010 H1 -A1 '(((E1-W)'(E22-W) + 61 '622)/(((E1o\V)?) + (61)?» '(A22/(((E22-W)?) + (622)?» 31020 H2 'A2‘(((E?rW)‘(EZ7rW) + 02‘022)/(((E2-W)‘2) + (62)?» '(422/(((E22-W)‘2) + (622)?» 31030 H3 -A3’(((E3-W)'(EZZ-W) + 03‘022)/(((F3-“0‘2) + (03)?» ‘(AZZ/(((E22-W)‘2) + (622)?» 31040 H4 -A4'(((B4-W)‘(En-W) + 04‘022)/(((Bt-W)‘2) + (607)) ‘(A22/(((Ez7v-W)‘2) + (022)?» 31050 HS -A5‘(((E5-W)'(EZZ-W) + 05‘622)/(((ES-W)“2) + (05)?» '(AZZ/«(EZZ-WYZ) + (622)?» 31060 Hé-M'(((E6-W)'(EZ2-W) + 66‘022)/(((Eé-W)‘2) + (66)?» ‘(A22/(((E22-W)‘2) + (022)?» 31070 H7-A7‘(((E7-W)‘(822-W) + G7'GZZ)/(((E7-W)‘2) + (67)“2» ‘(A22/(((E'22-W)?) + (622)?» 31080 "8 =A8‘(((EB-W)'(E22-W) + 08‘622)/(((E8-W)"2) + (08)?» ‘(A22/(((E2}W)"2) + (622m)) 31090 H9-A9‘(((B9-W)‘(E22-W) + 69‘022)/(((E9-W)‘2) + (09)?» '(m/(«EZZ-WYZ) + (022)?» 31100 H10 - A10‘(((BlO-W)'(Ezz-W) + 610‘622)/(((E10-W)?) + (610)?»‘(A22/(((E22-W)?) + (622)?» 31110 H11 -A11‘(((E11-W)‘(E22-W)+ 611'622)/(((E11-W)?) + (611)?))‘(A22/(((E22-W)?) + (622)?» 311% H12-A12‘(((ElZ—W)‘(E22—\V)+ 612‘622)/(((1312-W)?) + (612)?»‘(A22/(((822-W)?) + (622)?» 31130 H13-A13‘(((E13-W)'(E22-W) + 613'622)/(((Bl3-W)?) + (613)?»‘(A22/(((822-W)?) + (622)?» 135 31 140 H14 .. A14'(((El4-W)'(E22-W) + 614‘622)/(((E14-W)?) + (614)?))‘(A22/(((E22-W)?) + (622)?» 31150 H15 'A15'(((EIS-W)‘(E22-W) + G15‘(122)/(((1315-W)"2) +(015)‘2))‘(A22/(((EZZ-W)‘2) + (62273)) 31160 H16 -A16‘(((El6-W)‘(E22-W) + 616‘622)/(((El6-W)?) + (616)?))‘(A22/(((Ezz-W)?) + (622)?» 31170 H17-A17‘(((E17-W)'(E22-W) + 617‘622)/(((Bl7-W)?) +(617)?))‘(A22/(((E22-W)?) + (622)?» 31 180 H18 .. A18'(((E18-W)'(E22-W) + 618'622)/(((818-W)?) + (618)?))‘(A22/(((E22-W)?) + (622)?» 31190 H19 =A19'(((E19-W)‘(E22W) + 619’622)/(((El9-W)?) + (619)?))‘(A22/((CEzz-W)?) + (622)?» 31200 HID-Awflmm-WYWW) + G20‘622)/(((1330-W)“2) +(620)‘2))'(A22/(((EZ7»W)‘2) + (622)?» 31210 H21 -A21 '(((E21-W)'(Ezz-W) + 621'622)/(((m1-W)?) + (621)?))‘(A22/(((Ezz-W)?) + (622)?» 31220 R22- ((AZZ)‘2)/(((E22-W)‘2)+(622)‘2) 312% $17:- Sl6+2‘(Hl+H2+H3+H4+HS+H6+H7+HS+H9+ H10+H11+H12+H13 +H14+H15+H16+H17+H18+H19+H%+HZI)+R22 31240 REM '-‘-'-'-‘- end of 2%: term expressions -'-'-'-‘-‘ 31300 REM L1-L22 a R23areanextention foran 23th term 31310 L1 -A1 '(((El-W)'(Ez3-W) + 61 '623)/(((E1-\V)?) + (61)?» ‘(A23/(((E23-W)?) + (623)?» 31320 U-A2‘(((E2-W)‘(EB-“0 +GZ'GZ3)/(((E2-W)‘2) +(GZ)"2)) ‘(A23/(((EZ3-W)‘2) + (023)?» 31330 U-A3'(((E3-W)‘(E23—W) + G3‘GZ3)/(((E3-W)‘2) + (6377)) ‘(A23/(((EB-W)“2)+(GZ3)“2)) 31340 U=A4‘(((Bt-W)'(EB-W) + G4‘GZ3)/(((FA-W)“2) + (04)?» '(A23/(((E23-W)‘2) + (023)?» 31350 15 -A5‘(((F5-W)‘(E73-W) + 65‘623)/(((F5-W)‘2) +(05)“2)) '(AZi/«(Efl-WW) + (61377)) 31360 M-M'(((E6W)‘(E23-W) + G6‘GB)/(((Pb-W)‘2) + (66)?» ‘(A23/(((E23-W)‘2) + (623)?» 31370 L7-A7‘(((E7-W)‘(EZ3-W)+ G7‘GZ3)/(((E7-W)‘2) + (07)?» ‘(m/(((1323-W)‘2)+ (623)?» 31380 U-AB’(((E8-W)‘(E23-W) + 68‘623)/(((E8-W)“2) + (68)“2» ’(A23/(((E23-W)“2) +(GZ3)‘2)) 31390 U-A9‘(((E9-W)‘(BZ3-W)+ G9‘GZ3)/(((B9-W)"2) +(G9)"2)) '(A23/(((EZ3-W)‘2)+(GZ3)"2)) 31400 L10-A10‘(((810W)‘(E23-M + 610‘623)/(((810-W)?) + (610)?))'(A23/(((E23-W)?) + (623)?» 31410 L11 -A11‘(((E11-W)‘(Ez3-W) + 611 ‘623)/(((E11-W)?) + (611)?»‘(A23/(((E23-W)?) + (623)?» 314% L12 -A12'(((B12-W)'(Ez3-W) + 612‘623)/(((Elz-W)?) + (612)?))‘(A23/(((E23-W)?) + (623)?» 31430 L13 -A13'(((El3-W)‘(Ez3-W) + 613‘623)/(((Bl3-W)?) + (613)?»‘(A23/(((E23-W)?) + (623)?» 31440 L14 =A14'(((El4-W)'(Ez3-W) + 614'623)/(((El4-W)?) + (614)?))’(A23/(((Ez3-W)?) + (623)?» 31450 L15 =A15‘(((EIS-“0‘(EZ3-W) + GIS‘GZ3)/(((BIS-W)"2) + (615)?))‘(A23/(((EZ3-W)"2) + (623)?» 31460 L16 -A16'(((El(>W)‘(Ez3-W) + 616‘623)/(((El6-W)?) + (616)?))‘(A23/(((E23-W)?) + (623)?» 31470 L17 -A17‘(((E17-W)'(E23W) + 617‘623)/(((El7-W)?) + (617)?))‘(A23/(((1323»W)?) + (623)?» 31480 L18 =A18‘(((E18-W)‘(EB-W) + 618‘623)/(((E18-W)?) + (618)?))'(A23/(((1323-W)?) + (623)?» 31490 L19 -Al9‘(((E19-W)‘(EB-W) + 619‘623)/(((E19-\V)?) + (619)?))‘(A23/(((Ez3-W)?) + (623)?» 31500 L20-A20'(((E20-W)‘(Im-W) + 620‘623)/(((820-W)‘2) + (020)?))‘(An/(((E23W)‘2) + (033)?» 31510 1.21 -A21'(((EZl-W)’(Ez3-\V) + 621 ‘623)/(((E21-W)?) + (621 )?))‘(A23/(((E%-W)?) + (623)?» 31520 m-m'(((E22-W)‘(E23-W) + 622‘623)/(((E22-W)‘2) + (022)?))'(A23/(((E23-W)"2) + (613)?» 31530 R23-((A23)“2)/(((E23~W)‘2)+(023)‘2) 31540 818- Sl7+2'(L1+LZ+L3+L4+15+L6+L7+LB+L9+L10+L11+L12 +L13+L14+L15+L16+L17+L18+L19+120+121+L22)+R23 31545 GOTO 34850 31550 REM ‘-‘-‘-'-‘- end of 23th term expressions -‘-‘-‘-‘-‘ 31600 REM 11423 a R24areanextention foran 24th term 31610 Jl -A1 ‘(((E1-W)‘(1324-W) + 61 '624)/(((E1-W)?) + (61)?» '(A2A/(((E24-W)?) + (624)?» 31620 JZ-AZ‘(((E?«-W)‘(B24-W) + 02‘024)/(((EZ-W)"2) + (62)?» ‘(M/(((E24-W)‘2) + (624)“2» 31630 J3=A3‘(((E¥-W)‘(E24-W) + G3’Gu)/(((E3-W)‘2)+ (63)?» ‘(A24/(((EZ4-W)‘2) + (6207)) 31640 14 -A4‘(((PA-W)‘(El4-W) + 64’024)/(((E4-\V)‘2) + (607)) ‘(M/(((E24-W)‘2) + (024)?» 31650 15 =A5‘(((P5-W)‘(E24-W) + 66‘024)/(((E5-W)‘2) + (65)?» ‘(AI’A/«(E‘M-WW) + (674)?» 31660 J6-A6‘(((B6-W)‘(E24-W) + 66‘624)/(((E6-“0‘2) + (66)?» ‘(A24/(((E24-W)‘2) + (024)?» 31670 17 =A7‘(((E7-W)‘(E?A-W) + 07'624)/(((E7-W)‘2) + (67)?» ‘(A24/(((EZ4-W)“2) + (0207)) 31680 18 8A8‘(((Efi-W)‘(EI4—W) + 08‘024)/(((E8-W)“2) + (68)?» '(A24/(((E24-W)‘2) + (024)?» 31690 19 *A9‘(((E9-W)‘(E%W) + 69‘<324)/(((1‘59-W)"2)+ (09)?» '(A24/(((EZ4~W)‘2) + (6207)) 31700 JlO-A10‘(((Elo-W)‘(Em-W) + 610‘624)/(((E10-W)?) + (610)?))‘(A24/(((E24—W)?) + (624)?» 31710 111 - A11 ‘(((El 1-W)‘(E24-W) + 611 ‘62.4)/(((Bl l-W)?) + (611)?))‘(A24/(((EZ4—W)?) + (624)?» 317% 112 -A12‘(((E12-W)‘(1324-W) + 612‘62A)/(((1312-W)?) + (612)?))‘(A24/(((E24—W)?) + (624)?» 31730 .113 -A13‘(((E13-W)'(E24—W) + 613‘624)/(((El3-W)?) + (613)?))‘(A2A/(((E24-W)?) + (624)?» 136 31740 .114 - A14‘(((E14-W)‘(E24-W) + 614‘624)/(((1314-W)?) + (614)?))‘(A24/(((E24-W)?) + (624)?» 31750 115 =A15'(((E15-W)‘(E24-W) + G15’C324)/(((1315-W)“2) + (615)?))‘(A24/(((E24-W)‘2) + (6207)) 31760 116 =A16'(((El6—W)'(E24-W) + 616'624)/(((Bl6-W)?) + (616)?))‘(A24/(((mA-W)?) + (624)?» 31770 .117 aAl7'(((El7—W)‘(E24-W) + 617‘624)/(((El7-W)?) + (617)?))‘(A24/(((E24-W)?) + (624)?» 31780 .118 =A18‘(((E18-W)'(E24-W) + 618'624)/(((1318-W)?) + (618)?))‘(A7A/(((Ez4-W)?) + (624)?» 31790 .119 -A19‘(((B19-W)'(E24-W) + 619‘624)/(((Bl9-W)?) + (619)?))‘(A24/(((E24-W)?) + (624)?» 31800 m-mamm-wrmvv) + 620‘624)/(((1320-W)‘2) + (020)?))‘(M/(((324-W)‘2) +(G2A)“2)) 31810 121 -A21 ’(((E21-W)'(EZ4-W) + 621 ‘624)/(((E21-W)?) + (621)?»‘(A24/(((E24-W)?) + (624)?» 31820 m-A22‘(((E22-W)'(EZ4-W) +022‘G2A)/(((EZZ-W)“2) +(022)‘2))‘(A24/(((E24-W)‘2)+6324)?» 31830 J23=-4031((1323-‘10‘(EM-W) + G23‘(124)/(((1315-W)"2) + (62311))‘(A24/ («WA-W?) + (024)?» 31840 R24 =((A24)‘2)/(((E24-W)‘2) +(G241‘2) 31850 819- 818+2‘(.11 +12+13+14+15 +.16+.17+.18+.19+.110+111 +112+Jl3+114+115+116+Jl7+118+119+1%+121 +122+123)+ R24 31860 REM ‘-'-'-'- end of 24th term expressions -‘-'-‘-‘ 32000 REM 01-024 & R3 are an extention foran 25th term 32010 01'A1‘(((B1-W)‘(E?5—W)+ GI'GZS)/(((El-“0“2) + (61)?» ’(Afi/(«E‘A’S-WYZH (025)?» 32020 02 =A2‘(((E2-W)‘(BZS-W)+ GZ‘GZS)/(((E2-W)‘2) + (62)“2» ‘(A25/(((E?5-W)‘2) + (025)?» 32030 03 -A3'(((ES-W)’(E?5-W) + G3‘GZS)/ «(133-“0?) + (03)?» '(Afi/(«EZS-WYZ) + (625)?» 32040 04-44‘(((E4-W)‘(Efi-W) + G4‘GZS)/(((B4-W)‘2)+ (0417)) ‘(A25/(((E3~W)‘2)+ (625)?» 32050 05 -A5'(((F5-W)‘(E3-W) + G$‘<13)/(((P-'>-W)"2)+(6517)) '(Afi/(«BZS-WYZ) + (03)?» 32060 06-M‘(((E6-W)‘(EZS-W) + 06‘025)/(((B6-W)‘2) + (6677)) ‘(Afi/«(El’S-WYZ) + (675)?» 32070 07 =A7‘(((B7-W)‘(E3-W) + 07‘6251/(((B7-W)“2)+ (67)?» '(Azs/(«mmm + (0%)?» 32080 08*M‘(((E8-W)‘(EZS-W) + G8'G?5)/(((E8-W)‘2) + (0817)) ‘(Afi/«(EZS-WYZ) + (0317)) 32090 09-49'(((E9-W)‘(E?5-W) + 09‘6251/(«E9-WYZ) + (69)?» ‘(A25/(((B?5—W)“2) + (025)?» 32100 010 =A10‘(((E10-W) ‘(Ezs-W) + 610‘675)/(((E10-W)?) + (610)?))’(A3/(((Efi-W)?) + (625)?» 32110 011 -A11‘(((El 1-W)‘(325-W) + 611‘6fi)/(((B1 1-W)?) + (611)?))‘(A25/(((E75-W)?) + (625)?» 321% 012=A12‘(((E12-W)‘(E%-W) + 612'675)/(((Elz-W)?) +(612)?))‘(A75/(((E3-W)?) +(675)?)) 32130 013 =A13' (((El3-W)'(Ezs-W) + 613'625)/(((B13-W)?) + (613)?»‘(A25/(((E?5-W)?) + (625)?» 32140 014 =- Al4‘(((E14-W)'(E75-W) + 614'6%)/(((E14.W)?) + (614)?))'(A25/(((825-W)?) + (625)?» 32150 015 =A15‘(((EIS-W)‘(E?5-W) + 615‘6251/(((EIS-W)"2) + (015)?»‘(A25/(«E25-WYZ) + (02517)) 32160 016 - A16‘ (((El6-W)'(E25-W) + 616'625)/(((E16-W)?) + (616)?))‘(A75/(((EZS-W)?) + (625)?» 32170 017 =A17‘(((E17-W)‘(E‘5-W) + 617‘625)/(((El7—W)?) + (617)?))'(A25/(((Ezs-W)?) + (625)?» 32180 018 - A18‘ (((Bl8-W)‘(825-W) + 618'675)/(((ElS-W)?) + (618)?»‘(A25/(((EZS-W)?) + (625)?» 32190 019 =A19‘ (((El9-W)‘(E5-W) + 619‘625)/(((El9-W)?) + (619)?))‘(A25/(((Efi-W)?) + (625)?» 32200 020-A20‘(((E20-W)‘(Efi-W) + 620‘675)/(((E20-W)“2) + (620)?)1‘(A25/(((E?5-W)“2) + (015)?» 32210 021 - A21'(((E21-W)‘(E%-W) + 621'675) /(((E21-W)?) + (621)?))‘(A25/(((1325-W)?) + (625)?» 32220 022=A22‘(((E22-W)‘(E15-W) + GZZ‘GZS)/(((EZZ-W)‘2) + (022)?))‘(A25/(((E3-W)‘2) + (025)?» 32230 03'A23‘(((EB-W)‘(Efi-W) + G23‘67-5)/ «(133-“07) +(GZ3)‘2))‘(A15/(((E75-W)"2) + (675)?» 32240 024 =A24‘(((E24-W)'(E25-W) +GZ4‘GZS)/(((E24-W)‘2) + (624)?))‘(A25/(((E75-W)“2) + (025)?» 32250 R75=((A25)‘2)/(((E25-W)‘2)+(Gfi)‘2) 32260 8%- Sl9+2‘(01 +02+O3+O4+OS+06+O7+08+O9+010+011 +012+013+014+015+016+017+ 018+019+ 0%+021+022+023+024)+R25 3270 REM '-'-‘-’- end of 25th term expressions -‘-‘-‘-‘ 34850 8:818 : REM USING ONLY 23 TERMS (see line 3 1545) 3 4 9 0 0 RETURN 34998 34999 35000 35050 35090 35095 35100 35125 35150 35175 35180 35195 35295 35300 35350 35400 35405 35990 35998 137 REM I “-- REM Lines 35000- represent the second Equation REM --- Enter function in program line 35000-38900 --- REM *** These equations represent Absorption as a function of cm-l ***** REM ** t11=> VEl delta, t22=> VE2, t33=> VE3 delta, K= mult const for Ag REM ** tl=> V1 delta, t2-> V2 C= mult const for Bu state , t3=> V3 delta, REM Ag ————— T11=.36 :T22=.3 :T33=.37 :T44=.11 :K=.65 VE1=1750 :VE231250 :VE33200 :VE4=1550 :VG1=1550 :VG231200 :VGB‘ZOO :VG4=1550 E1=20175 :EZ=EI+(VE3) :E3=E1+(VE2) :E4=El+(VEl) :E5=E1+(VE1)+(VE3) :E6=El+(VE2)+(VE3) :E23*=E1+(VE4) 61375 :GZ=GI 3G3=G1 :G4=Gl 3G5=G1 3G6=Gl :G23*=Gl REM ======== 3:388“: REM == Bu H ''''' T1=.17 3T2=.15 :T3=.1 : C=.4895 :V1=1550 3V2=1200 :V3=200 G7=170 :GS=G7 :GQ=G7 :GlO=G7 :Gll=G7 :G12=G7 :G13=G7 :GI4=G7 :G15=G7 :G16=G7 :Gl7=G7 2618=G7 :Gl9=G7 :GZO=G7 3G21=G7 3822=G7 :E7 =23060 :E8 =E7+V1 :E93E7+V2 :E10=E7+V3 :E11=E7+2*(Vl) :E12=E7+2*(V2) :E13=E7+(v1+v2) :El4=E7+(V2+V3) :E15=E7+(V1+V3) E16=E7+2*(V1)+V2 :El7=E7+2*(V2)+Vl :E18=E7+3*(V1) :El9=E7+(2*(V1)+2*(V2)) :E20=E7+(2*(V1)+V3) :E21=E7+(2*(V2)+V3) :E22=E7+3*(V1)+V2 REM === 31000-31800 are the Frank-Condon integrals REM \-\-\- Frank-Condon integrals ( Ag state ) \-\-\- 36000 F01=2*((((VE1)*(VG1))*.5)/((VE1)+(VG1)))*EXP(-(.0148) *((T11)‘2)*(((VEI)*(V61))/((VE1)+(V61)))) 36025 F02=2*((((VE2)*(v02))A.5)/((VE2)+(VG2)))*EXP(-(.0148) *((T22)“2)*(((VEZ)*(VGZ))/((VEZ)+(VGZ))1) 36050 F03=2*((((VE3)*(VG3))‘.5)/((VE3)+(VG3)))*EXP(-(.0148) *((T33)“2)*(((VE3)*(VG3))/((VE3)+(VG3)))) 36060 P04=2*((((VE4)*(VC4))A.5)/((VE4)+(VC4)))*EXP(-(.0148) *((T44)‘2)*(((VE4)*(VG4))/((VE4)+(VG4)))) 36061 36075 ) *(T11)*2*((((VEl)*(VGl))0-51/((VEl)+(VGI))) REM -------------------------------------------------- F05=2*((.0148)A.5)*(((vc1)*((VE1)A.5) /((VE1)+(VG1))) *EXP(-(.Ol48) *((T11)‘2)*(((VE1)*(VG1))/((VE1)+(VGI)7)) 138 36100 F06=2*((.0148)A.5)*(((VG2)*((VE2)A. *(T22)*2*((((VE2)*(v02))A.5)/((VE2)+(v02 *((T22)‘2) *(((VEZ)*(VGZ))/((VEZ)+(VGZ)) 36125 F07=2*((.0148)A.5)*(((v03)*((VE3)~. 5 / ) s g *(T33)*2*((((VE3)*(VGB))“.5)/((VE3)+(VG3) ) 5 ) ) ((VEZ)+(VG2))) EXP((.0148) /((VE3)+(VG3))) *EXP(-(.0148) *((T33)‘2)*(((VE3)*(VG3))/((VE3)+(VG3))) 36150 F08=2*((.Ol48)“.5)*(((VG4)*((VE4)“. *(T44)*2*((((VE4)*(VG4))A.5)/((VE4)+(vc4 )) )) ) )) )) ))/((VE4)+(VG4))) ))*EXP(-(.Ol48) *((T44)‘2)*(((VE4)*(VG4))/((VE4)+(VG4))) 36200 REM -------------------------------------------------- 36250 Al=K*(FOl)*(F02)*(F03)*(F04) :A2=K*(F01)*(F02)*(F07)*(F04) :A3=K*(F01)*(F06)*(FO3)*(F04) 36300 A4=K*(F05)*(F02)*(FO3)*(FO4) :AS=K*(F05)*(F02)*(F07)*(F04) :A6=K*(F01)*(F06)*(FO7)*(FO4) :A23=K*(F01)*(F02)*(F03)*(F08) 36495 REM ------------------------------------------------- 36500 REM (v1) (v2) (V3) (V4) F01= F02= F03=<0 0> FO4= F05=<0 1> F06=<0 1> F07=<0 1> F08= 36505 REM -------------------------------------------------- 36525 F1=EXP(-.Ol48*(((Tl)‘2)*(Vl/2))) 36550 36575 36585 36600 F2=EXP(-.Ol48*(((T2)*2)*(V2/2))) F3=EXP(-.0148*(((T3)‘2)*(V3/2))) REM F33=EXP(-.Ol48*(((T4)*2)*(V4/2))) F4=((.0148)A.5)*((V1)A.5)*(T1)*Exp(-.0148 *(V1/2)*((T1)‘2)) 36625 F5=((.0148)0.5)*((V2)A.5)*(T2)*Exp(-.0148 *(V2/2)*((T2)‘2)) 36650 E6=((.0148)A.5)*((V3)A.5)*(T3)*Exp(-.0148 *(V3/2)*((T3)‘2)) 36660 REM F66=((.0148)“.5)*((V4)“.5)*(T4)*EXP(-.0148 *(V4/2)*((T4)‘2)) 36675 F7=.0148*(V1)*((T1)‘2)*EXP(-.0148*(V1/2)*((T1)‘2)) 36700 F88.0148*(V2)*((T2)“2)*EXP(-.0148*(V2/2)*((T2)“2)) 36725 F9=.0148*(V3)*((T3)“2)*EXP(-.Ol48*(V3/2)*((T3)A2)) 36735 REM F99=.0148*(V4)*((T4)*2)*EXP(-.Ol48 *(V4/2)*((T4)“2)) 36750 FlO=((.5774)*(.Ol48)“1.5)*((Vl)“1.5)*((T1)A3)*EXP(- .0140 *(Vl/2)*((T1)*2)) 36890 REM ------------------------------------------------ 36900 A7 -C*(F1)*(F2)*(F3) :A8 =C*(F4)*(F2)*(F3) :A9 -C*(F1)*(F5)*(F3) :A10=C*(Fl)*(F2)*(F6) :A11=C*(F7)*(F2)*(F3) :A12=C*(Fl)*(F8)*(F3) 36905 :A13=C*(F4)*(F5)*(F3) :A14=C*(F1)*(F5)*(F6) :A15=C*(F4)*(F2)*(F6) :A16=C*(F7)*(F5)*(F3) 36915 :Al7=C*(F4)*(F8)*(F3) :A18=C*(F10)*(F2)*(F3) :A19=C*(F7)*(F8)*(F3) :A21=C*(F1)*(F8)*(F6) :A20=C*(F7)*(F2)*(F6) :A22=C*(F10)*(F5)*(F3) 36920 36925 F1, F7, 36998 37000 37010 37020 37030 37040 37050 37060 37070 37080 37090 37100 37110 37120 37130 37140 37150 37160 37170 37180 37190 37200 37210 37220 REM ------------------------------------------------- REM REM __ __Vl,V2,V3 F2,F3 <0 0> "“ F8,F9 <0 2> Rl=(W)*(Gl)*((Al*2 ) R2=(W)*(GZ)*((A2*2) R3=(W)*(GB)*((A3“2) R4=(W)*(G4)*((A4*2) R5=(W)*(GS)*((A5‘2) R6=(W)*(G6)*((A6‘2) R7=(W)*(G7)*((A7A2) R8=(W)*(GB)*((A8‘2) R9=(W)*(G9)*((A9‘2)/ R10=(W)*(G10)*((A10‘ R11=(W)*(Gll)*((All“ R12=(W)*(612)*((A12“ R13=(W)*(613)*((A13*2 R14=(W)*(Gl4)*((A14‘2 R15=(W)*(Gl§)*((AlS‘2 R16=(W)*(Gl6)*((A16“2 R17=(W)*(Gl7)*((Al7*2 R18=(W)*(Gl8)*((A18‘2 R19=(W)*(619)*((Al9“2 R20=(W)*(GZO)*((A20‘2 R21=(W)*(621)*((A21*2 /(( /(( /(( /(( /(( /(( /(( /(( (( 2) 2) 2) ) ) ) ) ) ) ) ) ) ( (E ( ( ( ( ( ( ( / / / / / / / / / / / / ( ( ( ( ( ( ( ( ( ( ( ( F4,F5,F6 F10 <0 <0 E1-W)‘2)+Gl‘2) ) 2-W)“2)+GZ‘2)) ) E3-W)*2)+G3‘2) E4-W)*2)+G4“2)) E5-W)A2)+6502)) E6-W)“2)+GGA2)) E7-W)A2)+G7‘2)) E8-W)“2)+GB*2)) E9-W)“2)+G9*2)) ((ElO-W)*2)+GlO‘2) (Ell-W)A2)+Gll“2) (E12-W)‘2)+612“2) (E13-W)*2)+613“2) (El4-W)‘2)+Gl4‘2) (E15-W)‘2)+Gl$‘2) ) ) ) ) ) ) ) (E17-W)“2)+Gl7*2)) (ElB-W)“2)+G18‘2)) (E19-W)‘2)+Gl9“2)) (EZO-W)“2)+620‘2)) (E21-W)*2)+621*2)) ( ( ( E ((ElG-W)*2)+Gl6‘2) ( ( ( ( ( R22=(W)*(622)*((A22“2)/(((E22-W)‘2)+622“2)) R23=(W)*(GZ3)*((A23*2)/(((E23-W)‘2)+623A2)) V1,V2,V3 1> 3> 39000 S=R1+R2+R3+R4+R5+R6+R7+R8+R9+R10+R11+R12+R13 +R14+R15+R16+R17+R18+R19+R20+R21+R22+R23 39500 RETURN 39995 REM 40000 CHAIN "cal",40000!:REM-*- Call on a program (otherwise out of memory) -*- 50000 REM 50001 REM 50002 REM *—*-* 50025 -*- Lines 50000- represent the Third Equation -*- *-*-* Enter function in program line 50000-51900 - REM *-*-*-*- Equations for calculation of the Franck-Condon integrals -*-*-*- t1=> V1 delta, t2=> V2 , t3=> V3 delta, C= mult const for Bu state T1=W :T2=.15 :T3=.1 : :V1=1550 :V2=1200 :V3=200 REM ------------------------------------------------- F2=EXP(-(.0148*(((T3)*2)*(V3)))) F3=EXP(-(.0148*(((T2)*2)*(V2)))) F4=.122*((Vl)‘.5)*(Tl)*EXP(-.0148*(Vl)*((T1)‘2)) 50050 REM 50100 C=1000 50195 50200 50250 50300 50350 50400 50450 50500 50550 50600 50625 50650 50655 50675 50695 50700 50710 50720 50725 50730 50750 50900 51000 59999 60000 ”gal" 40000 Equations Below -*—*-*- 40050 40095 40100 40195 40200 40250 40300 40350 40400 140 F58.0148*(V2)*((T2)*2)*EXP(-.0148*(V2)*((T2)*2)) P558.0148*(V3)*((T3)‘2)*EXP(-.0148*(V3)*((T3)*2)) F6-.122*((v1)A.5)*(T1)*(.0148*(v1)*((T1)02)-1) *EXP(-.0148*(Vl)*((T1)“2)) F7=((.0148)*2)*((V2)“2)*((T2)A4) *EXP(-(.0148*(V2)*((T2)*2))) F77=((.0148)‘2)*((V3)*2)*((T3)“4) *EXP(-(.0148*(V3)*((T3)*2))) F9=(.00127)*((Vl)*1.5)*(.0148*(V1)*((T1)*5)-2 * ((T1)*3))*EXP(-.0148*(V1)*((Tl)*2)) F10=(8.88E-06)*((V1)“2.5)*((.010465)*(Vl) *((Tl)‘7)-(2.414)*((T1)“5))*EXP(-.Ol48*(Vl)*((T1)‘2)) Fll=((.3333)*(.0148)*3)*((V2)*3)*((T2)A6) *EXP(-(.0148*(V2)*((T2)“2))) FF11=((.3333)*(.0148)A3)*((Vl)‘3)*((Tl)*6) *EXP(-(.0148*(V1)*((T1)“2))) F12=(.0148)*(Vl)*((Tl)“2)*(((FFll)/8)+((F10)/4)) REM A7=C*(F4)*(F3)*(F2) :A9=C*(F4)*(F5)*(F2) :A11=C*(F9)*(F3)*(F2) :A13=C*(F6)*(F5)*(F2) :A15=C*(F6)*(F3)*(F55) A17=C*(F6)*(F7)*(F2) :A19=C*(F9)*(F7)*(F2) :A21=C*(F10)*(F7)*(F2) 223=c*(r9)*(r3)*(rss) :A25=C*(F12)*(F3)*(F2) :A8 -C*(F6)*(F3)*(F2) :A10=C*(F4)*(F3)*(F55) :A12=C*(F4)*(F7)*(F2) :A14=C*(F4)*(F5)*(F55) :A16=C*(F9)*(F5)*(F2) :A18=C*(F10)*(F3)*(F2) :A20=C*(F9)*(F3)*(F77) :A22-C*(F10)*(F5)*(F2) :A24=C*(F4)*(F7)*(F55) REM --------- See 25244-5 for F# definition -------- S= A13 RETURN REM ----------------- CHAIN "cal",60000! REM -*-*-* 40000- is for Performing Calculation on ENTER Parameters in 40100 -*-*-*- REM *-*-* Equations for calculation of the Franck-Condom factors -*-*-*- REM t1=> V1 delta, t2=> V2, t3=> V3 delta, C= mult const for Bu state Tl=.17 :T2=.15 :T3=.l : C=1000 :V1c1550 :V2=1200 :V3=200 REM ----------------------------------------------------- F2=EXP(-(.0148*(((T3)*2)*(V3)))) F3=EXP(-(.0148*(((T2)“2)*(V2)))) F4=.122*((V1)“.5)*(T1)*EXP(-.0148*(V1)*((T1)“2)) F5=.0148*(V2)*((T2)*2)*EXP(-.0148*(V2)*((T2)*2)) F55=.0148*(V3)*((T3)“2)*EXP(-.0148*(V3)*((T3)“2)) 40450 141 F6=.122*((V1)“.5)*(Tl)*(.0148*(V1)*((T1)“2)-l) *EXP(-.0148*(V1)*((T1)‘2)) 40500 F7=((.0148)*2)*((V2)“2)*((T2)“4)*EXP(-(.0148 *(V2)*((T2)‘2))) 40550 F77=((.0148)“2)*((V3)“2)*((T3)“4)*EXP(-(.0148 *(V3)*((T3)“2))) 40600 F9=(.00127)*((Vl)*1.5)*(.0148*(V1)*((Tl)*5)-2*((Tl)A3)) *EXP(-.0148*(V1)*((T1)*2)) 40625 F10=(8.88E-06)*((Vl)‘2.5)*((.010465)*(Vl)*((T1)A7)-(2.414) *((T1)‘5))*EXP(-.0148*(V1)*((T1)‘2)) 40650 Fll=((.3333)*(.0148)“3)*((V2)*3)*((T2)*6)*EXP(-(.0148 *(V2)*((T2)‘2))) 40655 FF11=((.3333)*(.0148)A3)*((V1)“3)*((Tl)‘6)*EXP(-(.0148 *(V1)*((T1)‘2))) 40675 40695 40700 40710 40720 40725 40730 40750 40755 40760 40765 40800 40900 41450 41500 42000 42050 42100 42150 42200 42500 43000 49990 -338:====3=8=8-=8=====8==============g==: 49995 60000 8u====8================ 60010 60020 REM 8888' F12=(.0148)*(V1)*((Tl)‘2)*(((FF11)/8)+((FlO)/4)) REM ---------------------------------------------------- A7=C*(F4)*(F3)*(F2) :A8 =C*(F6)*(F3)*(F2) :A9=C*(F4)*(F5)*(F2) :A10=C*(F4)*(F3)*(F55) :All-C*(F9)*(F3)*(F2) 3A12=C*(F4)*(F7)*(F2) :A13=C*(F6)*(F5)*(F2) :A14=C*(F4)*(F5)*(F55) :A15=C*(F6)*(F3)*(F55) :A16=C*(F9)*(F5)*(F2) Al7=C*(F6)*(F7)*(F2) :A18=C*(F10)*(F3)*(F2) :A19=C*(F9)*(F7)*(F2) :A20=C*(F9)*(F3)*(F77) :A21=C*(F10)*(F7)*(F2) :A22=C*(F10)*(F5)*(F2) A23=C*(F9)*(F3)*(F55) :A24=C*(F4)*(F7)*(F55) :A25=C*(F12)*(F3)*(F2) REM ---------------------------------------------------- REM (V1) (V2) (V3) F4-<1 0><0 0> F3=<0 0><0 0> F2=<0|0><0|0> F6=<1 1><1 0> F5-<0 1><1 0> F55=<0 1><1 o> REM F9=<2 0> F7-<0 2><2 0> F77-<0 2><2 0> E10=<1|3><3|0> F11=<0 3><3 0> F12=<4 0> REM ---------------------------------------------------- CLS LOCATE 2,25 :PRINT 'Franck-Condon factors ” LOCATE 6,6 :PRINT A7,A8,A9,A10 LOCATE 9,6 :PRINT A11,A12,A13,Al4,A15,A16,A17, A18,A19,A20,A21,A22,A23,A24,A25 LOCATE 24,1 REM -*-*- Lines 42100-42200 prevents 42500 from responding instantly -*-*- B$=INKEY$ IF B$=”*” THEN GOTO 42500 GOTO 42100 CHAIN "V84” END REM REM *-*- Lines 50000- represent the Third Equation —*-* REM j-_-—--r _-----—--> REM -*- 60000- is for Performing Calculation on Equations Below -*-*-*-* ENTER Parameters in 60025-60030 *-*-*-* Ag —---—-—= 142 60025 T11=.36 :T22=.3 :T33=.37 :T44=.11 : K=1000 60030 VE1=1750 :VE281250 :VE38200 :VE4=1550 :V6131550 :VG281200 :VG3=200 :VG4=1550 60045 REM ‘3 - '- 60098 REM \-\-\-\- Franck-Condon factors ( Ag state ) -\-\-\-\ ééiéé'fiiéuiiéiéifi2Zééii£26éi$125522Miinééii55mm """ *(4*((VE1)*(VG1))/(((VE1)+(VGl))02))*EXP(-2*(.0148) *((T11)“2)*(((VE1)*(VGI))/((VE1)+(VGI)))) 60200 F02=4*(((VE2)*(V62))/(((VE2)+(V62))02))*EXP(-2*(.0148) *((T22)‘2)*(((VE2)*(VGZ))/((VE2)+(VGZ)))) 60300 F03=4*(((VE3)*(VG3))/(((VE3)+(VGB))*2))*EXP(-2*(.0148) *((T33)‘2)*(((VE3)*(VGB))/((VE3)+(VG3)))) 60350 F04=4*(((VE4)*(VG4))/(((VE4)+(VG4))*2))*EXP(-2*(.0148) *((T44)“2)*(((VE4)*(VG4))/((VE4)+(VG4)))) 60400 P01=2*(.1217)*(((VGl)*(VE1)A.5)/((VE1)+(VGl))) *(T11)*(4*((VE1)*(VGl))/(((VE1)+(VGl))02))*EXP(-2*(.0148) *((T11)“2)*(((VE1)*(VGI))/((VE1)+(VGI)))) 60450 P02=4*(.0148)*((((VE1)*(VGl))01.5)/(((VE1)+(V61))02)) *((T11)“2)-2*(((VE1)*(V61))‘-5)/((VE1)+(VGI)) 60500 F05=(P01)*(P02) 60600 F06=4*(.0148)*((((VGZ)*((VE2)A.5))/((VE2)+(V62)))02) *((T22)*2)*4*(((VE2)*(VG2))/(((VE2)+(V62))02))*EXP(-2*(.0148) *((T22)‘2)*(((VE2)*(VGZ))/((VE2)+(VG2)))) 60700 F07=4*(.0148)*((((VG3)*((VE3)A.5))/((VE3)+(VGB)))02) *((T33)*2)*4*(((VE3)*(VGB))/(((VE3)+(VG3))02))*EXP(-2*(.0148) *((T33)“2)*(((VE3)*(VG3))/((VE3)+(VGB)))) 60750 F08=4*(.0148)*((((VG4)*((VE4)*.5))/((VE4)+(VG4)))02) *((T44)02)*4*(((VE4)*(VG4))/(((VE4)+(VG4))02))*EXP(-2*(.0148) *((T44)“2)*(((VE4)*(VG4))/((VE4)+(VG4)))) 60895 REM .................................................... 60900 REM (v1) (V2) (V3) (v4) F01-<1 0><0 0> F02=<0 0><0 0> F03 =<0 0><0 0> F04=<0 0><0 0> F05=<1 1><1 0> F06=<0 1><1 0> F07 =<0 1><1 0> F08=<0 1><1 0> 60905 REM ...................................................... 60950 A1=K*(F01)*(F02)*(F03)*(F04) :A2=K*(F01)*(F02)*(F07)*(F04) :A3=K*(F01)*(F06)*(F03)*(F04) 60960 A4=K*(F05)*(F02)*(F03)*(F04) :A5=K*(F05)*(F02)*(F07)*(F04) :A6=K*(F01)*(F06)*(F07)*(F04) :A23=K*(F01)*(F02)*(F03)*(F08) 60970 REM == === 61000 CLS 61010 LOCATE 2,25 :PRINT ”Franck-Condon factors " 61025 LOCATE 6,6 :PRINT "origin ” 61050 LOCATE 6,26 :PRINT A1 61100 LOCATE 10,6 :PRINT "fundamentals" 61125 LOCATE 11,6 :PRINT ”(v3,v2,v1)” 61150 LOCATE 10,26 :PRINT A2,A3,A4 61160 LOCATE 14,6 :PRINT "v8 & combination" 61165 LOCATE 15,6 :PRINT "(v8,v1+v3, v2+v3)" 61170 LOCATE 14,26 :PRINT A23,A5,A6 62000 LOCATE 23,1 62050 REM -*-*- Lines 62100-62200 prevents 62500 from responding instantly -*-*- 62100 B$=INKEY$ : ----- 143 62150 IF B$=”*" THEN GOTO 42500 62200 GOTO 42100 62500 CHAIN ”ve4" 63000 END * For the REP and absorption model an additional term, namely E”, was added to the 21Ag state after the model was completed. For this reason the number is out of sequence. 1 * This program written in GW—Basic is easily converted to Quick-BASIC. This can be done by breaking down the REP equations into two expressions; and for earlier version of Quick-BASIC it is necessary to replace the "on error ...“ command with another appropriate command. Thereafter the program will run faster. 144 APPENDIX 11 145 Table A. Franck-Condon Integrals used for the Raman Excitation Profile model. (1-16 utilized for the 118u state; and 1-6 for the 21Ag state only.) 146 1. "oo [<19|0.><0.|09>],t[<09|0.><0.|09>]vz[<09|0.><0.|09>]; 2. ,3 [<1g|°.>‘°.|°q)]v‘[<09|10><1.l09>lvat<°9'10><10|og>1v3 3. ,3 [<19|°e>‘°e|°g>lv,[<°9|10"10|°9’]v,[‘oqlofi"°¢l09>]v, 4. u,+u, [<1g|0.><0.|0g>],‘[<09|1.><1.|09>1,2[<0g|1.><1,|09>]V3 5. v, . [<19'1.><1.|09>]y‘[<1.|09>]v2[<09I0.><0.|Og>lu3 6. v‘+v, [<19|1.><1.|09>],‘[<0g|0.><0.|0q>],2[<09|1.><1.|0g>]V3 7, 20, [<1g|0.><0.|09>],‘[lu3[‘°el09>]v3 8. 2y,+u, [<19|0.><0.|09>],1[<09|2.><2.|09>]v2[<09|1.><1.|09>]V3 9. v1+v2 [<19|1‘><1‘|°9>]'1[<09l1')<1‘|°9’]"2[<°9|°‘><°‘l09>]Vs 10. 29, [<19|2.><2.|09>j,‘(<0g|0.><0.|0g>1,3[<09|0.><0.|09>]y, 11. 2»,+y, (<19|2.><2.|09>],‘[<09|0.><0.|09>ly,[<09|1.><1.|09>]u, 12. 293+y‘ [<1ql1.><1.|09>]y‘[<09I2.>‘2.l°q’]v2(<°CI09>193 13. 2v,+v, [<19|2.><2.|09>],‘(<0q|1.><1.|09>],,[<09|0.><0.|09>]u, 14. 3y, [<1g|3.><3.|09>lu,t<°g|°o><°e|°q>lv,l<°9|°-"°¢|°9’1v. 15. zu,+zu, (<1q|2.><2.|09>1,‘[<09|2.><2.|09>],2(<0g|0.><0.|09>],3 16. 3v,+v, [<19|3.><3.|09>],‘[<09|1.><1.|09>],2[<09|0.><0.|09>]u, ‘Tllflll In 147 Table B. Franck-Condon Integrals used for the absorption spectrum model. 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