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UBC Theses and Dissertations

The ultraviolet absorption spectrum of thionylimide and the rotational energy formulae for linear polyatomic… Allegretti, John Michael 1971

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AND i THE ROTATIONAL ENERGY FORMULAE FOR LINEAR POLYATOMIC MOLECULES IN 2 I AND 3 Z ELECTRONIC STATES IN WHICH ONE OR MORE QUANTA OF A DEGENERATE BENDING VIBRATION IS EXCITED b y JOHN MICHAEL ALLEGRETTI A.B. (Honors) Univers i ty of Ca l i f o r n i a , Berkeley, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Chemistry We accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1971 In present ing th i s thes i s in pa r t i a l f u l f i lmen t of the requirements fo r an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ib ra ry sha l l make i t f r ee l y ava i l ab le for reference and study. I fu r ther agree that permission for extens ive copying of th i s thes i s for s cho la r l y purposes may be granted by the Head of my Department or by h i s representat ives . It is understood that copying or pub l i ca t i on o f th i s thes i s f o r f i nanc i a l gain sha l l not be allowed without my wr i t ten permiss ion. Department of The Un iver s i t y o f B r i t i s h Columbia Vancouver 8, Canada Thionylimide, HNSO, was prepared and i t s u l t r a v i o l e t spectrum recorded for the f i r s t time. Two t rans i t ions are seen; a long ser ies of bands in the 2700$ region and a continuum ending at o approximately 3440A. Franck-Condon overlap ca lculat ions were carr ied out to ascertain how large a shape change was occurring in the 2700$ system. Reasons for the d i f fu se nature of the bands in the 2700A* system and the continuous nature of the absorption in the 3440$ region are given. In the second part of th i s thes i s , formulae for the rotat iona l 2 3 energies of l i nea r polyatomic molecules in z and z e lect ron ic states in which one or more quanta of a degenerate bending v ib rat ion o are excited are derived. It i s found in z e lect ron ic states that the spin doubling and the i-type doubling are independent provided the rotat iona l constant B i s much larger than y» the spin rotat ion constant. In z e lect ron ic states departures from the normal t r i p l e t spin pattern occur at low N values, when the v ibrat iona angular momentum i s s t i l l mainly coupled to the axis of the molecule. The e f fec t i s enhanced i f x , the spin-spin in teract ion constant, is large compared to B. At high N values the normal t r i p l e t pattern is approached as the v ib rat iona l angular momentum i s uncoupled from the axis of the molecule. PAGE Abstract i L i s t of Tables and Figures i i i Acknowledgment v Introduction 1 CHAPTER I THE ULTRAVIOLET ABSORPTION SPECTRUM OF THIONYLIMIDE 3 1-1 Experimental 3 a. Preparation 3 b. Apparatus 5 c . I dent i f i ca t i on of product 5 d. Preparation notes 6 e. Spectroscopic experiments 6 1-2 Results 13 1- 3 Discussion 23 a. Ground state normal coordinates 23 b. Franck-Condon ca lcu la t ion 24 c. E lectron ic configuration and 4 0 Electron ic states of HNSO CHAPTER II ROTATIONAL ENERGY FORMULAE FOR LINEAR POLYATOMIC 4 7 MOLECULES IN 2 E AND h ELECTRONIC STATES IN WHICH ONE OR MORE QUANTA OF A DEGENERATE BENDING VIBRATION IS EXCITED 2- 1 Derivation of the HamiTtonian and i t s matrix elements 47 2-2 Results and Discussion 5 5 References 78 Appendix 1 81 Appendix 2 84 - n i -L i s t of Tables and Figures Table i Page 1 Observed Bands of HNSO and DNSO 21 2 Force Constants and Calculated Frequencies 33 3 Results of Franck-Condon Overlap Calculat ions ' 37 (AR S 0=AR N S = .158, A c t N S 0 = 15°) 4 Character Tables for C and Point Groups and 45 Correlat ion of the I rreducible Representations of the C s and C^y Point Groups 5 Elements o f f diagonal in the v ib rat iona l quantum 64 3 numbers for the = 0 level in z e lect ron ic states 6 E f fec t i ve Operators Ar i s ing from Second-Order 65 Transformation of the Hamiltonian 7 Matrix Elements of Terms in Table 6 66 2 8 Matrix fo r = 1 f o r z E lectron ic State Correct 67 to Second Order 9 Case (a) and Case (b) Matrices for v 2 = 2 in a z 68 Electronic State 3 10 Case (a) and Case (b) Matrices fo r v^ = 1 in a z 70 ' E lectron ic State 3 11 Case (a) and Case (b) Matrices fo r = 2 in a I 72 Electronic State Figure . Page 1 Thionylimide Preparation System 10 2 Experimental Set up for the 1.5 meter Bausch and Lomb Spectrograph ' 11 3 Experimental Set Up for the 21 f t Eagle Spectrograph 11 4 4 Meter White Cel l and the Order Separator 12 5 DNSO U.V. Spectrum 16 6 HNSO U.V. Spectrum (high pressure) 17 7 HNSO U.V. Spectrum (low pressure) 18 8 HNSO 2700H System 19 9 HNSO 3440^ Continuum 20 10 Infra-Red Spectrum of Matrix Isolated HNSO at 4°K 22 11 Calculated Trans i t ion Envelope for A R ^ s = 0.158 38 and Aa^Q = 15° 12 Correlat ion of A R n s and A a ^ g for Certain System Or i g in - 39 Absorption Maximum Separations 13 Possible Potential Energy Curves fo r Thionylimide 46 3 14 Calculated T r i p l e t S p l i t t i n g Patterns fo r Vibronic z, 74,75 , 7 6 a,b,c o -5 J n and J A levels f o r HCCN, C 2 N 2 and NCN 15 Departure of the a-type Doubling Pattern from the Formulae 77 Av = qN(N+l) for the \ Vibronic Level of HCCN ACKNOWLEDGMENT The author wishes to thank Dr. A . J . Merer for his advice and encouragement in carrying out the research which th i s thes is i s based on and during the wr i t ing of th i s thes i s . The author also wishes to thank Peter Tchir and Dr. R.D. Spratley for providing matrix i s o l a t i on in f ra - red spectra of thionlyimide and the i r help with the v ib rat iona l analys is programs, and l a s t but not l e a s t , E.J.F. who gave quite a hand in putting th i s thesis together. This thes i s consists of two parts which are bas i ca l l y unrelated. The f i r s t part deals with the molecule thionyl imide (HNSO or DNSO) and i t s u l t r a v i o l e t spectrum. This study i s prompted by the fact that the f i r s t strong u l t r a v i o l e t absorption system of the i so -e lec t ron i c molecule S0 2 has yet to be analyzed, despite many attempts (1,2). I t i s reasoned that , thionyl imide being i soe lect ron ic with S0 2 and having a s im i l a r shape, an understanding of the u l t r a v i o l e t spectrum of th ionyl imide would help decipher the 29008 system of S0 2- In the end, though, due to the lack of information ava i lab le from the th iony l imide spectrum resu l t ing from i t s d i f fuseness, the opposite occurred and prel iminary resu l t s from S0 2 were used to help expla in the u l t r a v i o l e t absorption spectrum of th ionyl imide. In t h i s thes i s there i s described the preparation of th ionyl imide and the f i r s t observation of i t s u l t r a v i o l e t spectrum. Two absorptions have been seen: a long series of d i f fuse bands in the 2700-2100$ region and a continuum at longer wavelengths. Franck-Condon overlap ca lcu lat ions were carr ied out for the 2700$ system to see how large a shape change i s necessary to explain the observed spectrum, and whether th i s shape change correlates with the shape change occurring in the 2900$ system of S0 2 . Reasons fo r the di f fuseness of the bands in the 2700$ system and the continuous nature of the long wavelength system are given based on the e lec t ron i c structure of thionyl imide. The second part of th i s thesis deals with the der ivat ion and discuss ion of rotat iona l energy formulae for l i nea r polyatomic molecules in doublet and t r i p l e t z e lect ron ic states i n which one or more quanta of a degenerate bending v ib rat ion are exc ited. As ye t , no theoret ica l treatment ex i s t s for th i s top i c (3 ) , though with the advent of f l a sh photolysis techniques, several examples of such molecules are now known. In pa r t i c u l a r , a system of HCCN at 3200-3400A\ which i s probably a ^z~ - ^z~ t r a n s i t i o n , shows v ib ra t iona l "hot" bands in the H-C-C degenerate bending v ib rat ion which have so fa r defied rotat iona l analysis according to the conventional energy level formulae for z states (4). The hamiltonian has been wr i t ten in " e f f e c t i v e " form ( i . e . wr i t ing the electron spin interact ions as parameters to be determined experimentally) and i t s matrix elements given in a Case (a) representation correct to second order. The resu l t s have been given i n Case (b) notation fo r some cases of i n te res t . The results are discussed i n re lat ionsh ip to what would be seen in actual molecules. CHAPTER I THE ULTRAVIOLET SPECTRUM OF THIONYLIMIDE 1-1 Experimental a. Preparation of HNSO and DNSO The commonest methods of making thionyl imide are the hydrolys i s of NSF and the d i rec t gas phase reaction of NH3 and SOClp ( 5 , 6 ) . In a l l experiments reported here, t h i ony l -imide was produced by the l a t t e r method. This was because the s t a r t i ng material was read i ly ava i lab le and resu l t s in a purer product than by the hydrolysis of NSF. Since SO2 i s always formed as a by-product during any preparation of thionyl imide, one must be extremely careful to choose conditions which minimize the amount of SO2 formed, s ince SO2 absorbs throughout the u l t r a v i o l e t . Kirchhoff (7), during his microwave studies of th iony l imide, discovered that th ionyl imide could be prepared reasonably free of SO2 by using low pressures of NH3 and S 0 C 1 2 in the exact stoichiometr ic proportions f o r the react ion 3 N H 3 + S 0 C 1 2 + 2NH 4C1 + HNSO ( 1 ) Higher pressures of reactant gases resulted in a higher percentage of SO2 produced. Accordingly, K i rchhof f ' s condit ions, 12 t o r r NHo and 4 t o r r S 0 C 1 ? , were followed for a l l experiments. I t was found that the amount of S0 2 impurity produced was very sens i t i ve to traces of water present in the apparatus or reagents (presumably from the d i r ec t hydrolysis of S0C1 2)- Thus the glass apparatus had to be wel l flamed under vacuum pr io r to the preparation, and the ammonia had to be ca re fu l l y dr ied over sodium metal. S0C1,, was pu r i f i e d by trap to trap d i s t i l l a t i o n to remove SC^, HC1, and dissolved a i r . I t was never possible to prevent the formation of S 0 2 e n t i r e l y , and an average sample of thionyl imide contained 5% S0 2 as judged from the u l t r a v i o l e t spectrum. This i s presumably because react ion (1) i s not the only reaction occurring (5,6). A major d i f f i c u l t y with preparation of HNSO i s that whenever the gas i s trapped in dry ice or l i q u i d nitrogen, i t immediately polymerizes. I t i s thus not possible to pur i fy HNSO by f rac t i ona l d i s t i l l a t i o n (see below). DNSO was prepared in an analogous way to HNSO except that NDg was used. The heavy ammonia was both prepared in the lab and obtained from a cy l inder. The NDg prepared in the lab was synthesized using Mg 3N 2 and D20 in the fol lowing react ion (8): 6 D20 + Mg 3N 2 — > 2 ND^ + 3 Mg(0D)2 (2) In the preparation an excess of Mg 3N 2 i s used to produce a very dry sample of ND^. The D20 used was from Stohler Isotopes (99.8% D). The Mg 3N 2 was from Alpha Inorganics and had to be heated in vacuo p r io r to use to remove " l i g h t " water hydroxides. The bottled ND^ was from Merck, Sharpe, and Dohme and was 99% D. The samples of DNSO prepared were ascertained to be at least 70% D using matrix in f ra - red techniques. j b. Apparatus and preparative method The thionyl imide was prepared in a standard a l l glass vacuum l i ne incorporating e i ther a s ix or two twenty-two l i t e r reaction vessels and a U tube trap (see f i g . 1). Pumping was by means of a rotary pump and an o i l d i f fu s i on pump. Pressures were measured with a thermocouple gauge and a manometer f i l l e d with Dow Corning 707 s i l i c one f l u i d (12.8 mm o i l = 1 mm rig). The actual preparation of thionyl imide involved f i r s t f i l l i n g the reaction vessel with dry ammonia to the correct pressure and then trapping the ammonia out i n the U tube trap with l i q u i d nitrogen. The reaction vessel was then f i l l e d with SOC^- The ammonia was then evaporated back into the reaction bulb where i t reacted immediately with the S0C1 2» forming a yellow-white coating mostly of NH^Cl on the glass wal l s . c. I dent i f i ca t i on and analysis of reactant products Pos i t i ve i d e n t i f i c a t i o n of the gas produced as thionyl imide was accomplished by in f ra - red and mass spectrometric techniques. The in f ra - red spectrum was taken of a matrix i so lated sample at 4°K. The thionyl imide was d i lu ted with argon in the r a t i o of 1:800. Samples were deposited at 120 microns pressure on a Csl window. [See re f . (9) for a complete descr ipt ion of the matrix i s o l a t i on technique.] Spectra were taken on a Perkin-Elmer 225 in f ra - red spectrophotometer. The matrix i so lated spectrum was i dent i ca l with the gas phase spectrum except for small s h i f t s caused presumably by in teract ions with the matrix^and the absence of rotat iona l s t ructure (see f i g . 10). The mass spectrum of thionyl imide was taken on a MS 9 mass spectrometer. The mass spectrum showed a weak parent ion peak at m/e = 63 (HNS0+) and another peak at m/e = 15 (NH) +. There was a strong peak at m/e = 48 (S0) + , but the S0 + may have come from S0 2 f o r there was also a strong peak at m/e = 64 (S0 2) . d. Notes on other attempts to prepare pure thionyl imide Attempts to produce large pressures of thionyl imide by condensing successive batches i n a trap and reconst i tut ing the monomer from the polymer by heating the trap to 70°C as reported by Schenk (10) only y ie lded large amounts of S0 2 and very l i t t l e th iony l imide. I t had been hoped that HNSO free of S0 2 could be produced by th i s method, but i t turned out that Schenk 1s resu l t s could not be reproduced at a l l . e. Spectroscopic experiments Prel iminary u l t r a v i o l e t spectra were taken on a Bausch and Lomb 1.5 meter Eagle spectrograph, the experimental arrangements having been as in f i g . (2). The lamp shown in f i g . (2) was a P.E.K. 75 watt high pressure xenon lamp. The one lens shown focused an image of the lamp on the s l i t . . The c e l l shown i s a 75 cm a l l quartz c e l l with Suprasi l windows. With a 60 y s l i t and c e l l pressures ranging from 4 to ^ 0.2 t o r r , exposure times were of the order of 4 seconds using Kodak SA-I f i l m . For c a l i b r a t i o n a 25 m.a. Westinghouse Iron hollow cathode lamp was used. Higher reso lut ion spectra (approximately 150,000 resolving power) were taken in the second order of a 21 f t . Eagle spectrograph as shown in f i g . (3) with the previously mentioned l i g h t source and c e l l . To lower the exposure times, a 22.5 cm focal length c y l i n d r i c a l lens was used to focus the image of the arc on the s l i t as a l i n e . A Corning 9-54 f i l t e r placed between the lamp and the c e l l was used to prevent photolysis of th iony l imide. With a 20 y s l i t and c e l l pressures ranging from 4 t o r r to less than 0.1 t o r r , and using Kodak SA-I p la tes , exposures were of the order of two minutes. Ca l ibrat ion was with a 120 m.a. i ron hollow cathode lamp of the l ab ' s own design. In these experiments a moderately intense, but d i f f u se , band system l y i ng in the region 2700-2100$ (to be discussed below) was discovered. Temperature studies of the bands of th i s system were carr ied out using the same experimental arrangement as above except that the c e l l was wrapped with heating tape. The c e l l temperature was cont ro l led by regulating the applied voltage to the heating tape. The c e l l was heated in 25° steps to a maximum of 100°C. The c e l l temperature was monitored using a Chromel-Alumel thermocouple in conjunction with a Leeds and Northrup m i l l i v o l t potentiometer. The next experiment was a search fo r weaker t rans i t ions of th ionyl imide at longer wavelengths in the 3600 $ region. To do th i s a 4 meter White type (11) mul t ip le re f lex ion c e l l was b u i l t and used i n conjunction with a reaction vessel consist ing of two twenty-two l i t e r globes. The c e l l was constructed of 85 mm pyrex tub ing, to the ends of which 6 inch sections of 3 inch diameter pyrex pipe end-pieces with 0-ring grooves were blown on. Normal pyrex pipe flanges were used to attach the end plates which sealed the c e l l (see f i g . 4). One c e l l end-plate carr ied the Suprasi l entrance and e x i t windows, while the other had the external contro ls fo r the D mirrors. Inserted in the c e l l v/as a"boat" to the f ron t of which was attached the "shouldered" mirror. The number of t raversa l s of the c e l l was contro l led by manipulation of the D mi r ror s . The c e l l was set up as in f i g . (4) using the previously mentioned lamp. Care had to be taken so that the incoming cone of l i g h t matched the aperture of the White c e l l so as not to produce stray l i g h t r e f l ec ted o f f the ins ide walls of the c e l l (cone too large) or to u n d e r f i l l the D mirror (cone too smal l ) . This c r i t i c a l adjustment was done by moving the 35 cm lens shown in f i g . (4) r e l a t i v e to the l i g h t source. After the l i g h t had passed through the White c e l l , a 55 cm c y l i n d r i c a l lens focused the l i g h t on the s l i t of the spectrograph. Corning 0-56 and 7-54 f i l t e r s were used to stop photolys is of the sample by l i g h t with a wavelength less than 3000A, which was found to produce decomposition products on the mirrors unless i t was el iminated. With the White c e l l set for 24 t r a ve r sa l s , or an e f f ec t i ve path length of 96 meters, and a 80 y main s l i t , exposures were of the order of four minutes using Kodak IIa-0 plates and c e l l pressure ranging from 2.6 t o r r to 0.23 t o r r . The 120 m.a. i ron hollow cathode lamp was again used fo r c a l i b r a t i o n . The White c e l l was also used to look at the weak " t a i l " of the 2700$ system of thionyl imide. The c e l l was set up as previously described except that the Corning f i l t e r s were replaced with an order separator in f ront of the main s l i t , cons i st ing of a f o r e s l i t , concave mi r ror , and quartz prism (see f i g . 4). The order separation was necessary because the f i l t e r s could not completely el iminate the intense overlapping f i r s t order rad iat ion (\ ^ 5200$) from the xenon arc. Using Kodak I-D f i l m , an 80 y main s l i t , a 2 mm f o r e s l i t , gas c e l l pressures of the order of 0.2 t o r r , and 4 t raver sa l s , exposures were of the order of 5 to 10 minutes. Ca l ibrat ion was as previously described. None of the White c e l l experiments were repeated with DNS0, and for DNS0 only the 2700$ system was photographed. Measurements were made from contact pr ints of the plates. These are probably good to ± 25 cm~^. REACTION VESSEL 0 STOP COCK TO CELL O 0 0 U-TUBE TRAP INLET PORTS OIL MANOMETER () O O O A A /V A o I " © ~ > TO PUMPS FIG. 1 THIONYLIMIDE PREPARATION SYSTEM 22.4 cm CYLINDRICAL LENS 4-35 cm LENS 0 0 SLIT 75 cm QUARTZ GAS CELL XENON LAMP 21' EAGLE SPECTROGRAPH FIG. 3 35 cm LENS GAS INLET C) "SHOULDERED MIRROR" D MIRRORS 55 cm CYLINDRICAL LENS ORDER SEPARATOR SPECTROGRAPH FIG.(4)4 METER WHITE CELL AND 2T EAGLE SPECTROGRAPH MAIN CONCAVE MIRROR FORE SLI & SLIT PRISM Low reso lut ion absorption spectra of thionyl imide taken on a Cary Model 14 spectrophotometer show moderately strong absorption in the 2700-1900$ range (see f i g . 5,6,7). The absorption maximum comes at 2170$; the system obviously continues past 1900$ but could not be observed for experimental reasons. On the long wavelength side of the absorption maximum the system loses i n tens i t y rap id ly and i s no longer d i scern ib le at 2700$ where i t i s l o s t under the t a i l of the strong 2900$ system o'f S0 2 (see f i g . 6). The main features of the Cary spectra are a long ser ies of absorption bands at the long wavelength end of the system, which can be followed from 2689$ to 2387$ (see f i g . 5,6,7). The bands are most d i s t i n c t at lower energies and, at the absorption maximum, they are no longer d i s ce rn ib le . At high reso lut ion the bands noted above were found to be en t i r e l y d i f f u s e , not even showing any p a r t i a l l y resolved rotat ional s t ructure (see f i g . 8) . The band posit ions are given in Table (1). Upon examination of Table (1), one w i l l notice that the bands are not regu lar ly spaced and cannot be f i t t e d into a s ingle progression. The 2700$ system of DNS0 showed several d i s t i n c t di f ferences from that of HNSO. The bands which stood out well in the spectra of HNSO no longer do so in DNS0 (compare f i g . 5 and 6). The bands i n DNS0 appear not to have been sh i f ted to any great extent, though new bands at 2543$ and 2530$ have appeared and the band that was at approximately 2535$ in HNSO has disappeared. The band that was 2517$ in HNSO has been sh i f ted to 2515$ in DNSO. The other bands in the spectrum of DNSO do not appear to have changed pos i t ion r e l a t i v e to the bands in the spectrum of HNSO. The high reso lut ion spectra of the 2700$ system of HNSO showed three very weak bands between the bands of the main progression. Due to the lack of contrast in the DNSO spectrum, the corresponding bands could not be seen. Upon heating the c e l l , these weak bands of HNSO appeared to gain in tens i ty r e l a t i v e to the other bands as the c e l l temperature rose, thus showing them to be "hot" bands a r i s ing from excited v ibrat iona l levels in the ground s tate . Since the bands of the main progression did not change in i n ten s i t y , they are " c o l d " bands. The long path experiments in the 3400$ region did not d isc lose any sharp or banded absorption of HNSO but only a continuum beginning at approximately 3440$ (see f i g . 9). Since S0C1 2 also has continuous absorption near th i s reg ion , i t was necessary to show that the two continua were not i d e n t i c a l . Comparison of the new continuous absorption and spectra of S0C1 2 taken under s im i l a r conditions showed that the two continua were not the same. There are no other species involved in the preparative reaction that have a continuous absorption in the u l t r a v i o l e t . The long path experiments in the 2685$ region did not reveal any new bands in the t a i l of the 2700$ system of HNSO but did v e r i f y the existence of a band at 2686A\ From 2686$ and to longer wavelengths, the 2900A* system of S 0 2 r i ses in i n ten s i t y and i t becomes impossible to separate the weak and d i f fu se HNSO bands from the strong, sharp S 0 ? absorption. ' - - - - - \ J ! t 0-9^  * t 1 i -0.8- — — — Fig. 5 DNSO 3.7 to r r Taken on a Cary Model 14 Spectrophotometer — 1 j- I -i 1 — 1 I _ _ l —{ - • - . ! , - J * 1 "I ' \ u-1 ! 1 -0.6 M — i ' ( 1 1 i t "1—~ E S i i. 4 -1 -0.5-— L 1 J V i — i - — i — : : L 1 1 t \ i r i — • • -0.4 . . . oybiem or 5U 9 — -0.3- , - •• — — \ — 1 A -0.2-[ — $ — -0.1-— — 4 — 2500 St 3 0 0 . 0 A . _ L t | —*~l .1 ; — 1 1 — i L _ —1—1 1 1 -0.9i i ! ... j . . . —q * f- — i '4 0 8- 1 l - i — 1 " ig. 6 HNSO 2.6 to r r "aken on a Cary Model 14 Spectrophotometer -=y== — F - =h=d 1 — f l — — H — ^ _n 7_ 4 \j •/ —i- j si 1 —It ? (/I -D A 1 ! .« 1 i il J | ! -JH — --0.5i 1 — — i 1—-1 _ i , i i 1 -0.4-\ \ J • S j j \!\ >—4 , i -* -7— 1 — i — rii'.o » i — i — - f - d —v — i — — r — — ^ . s I 2900 X System of S0 ? —. — \ — h — 1— 1 -0.2-... ..j. . ... . > f 1 J 1 V\j \ — T % t — v - V ' — ' — J 1 ! 2500 ?\ 3000 $ 0.8 = N H 3 : ^0.7 Fig. 7 HNSO 0.8 t o r r Taken on a Cary Model 14 Spectrophotometer 0.6 A 3 ? 0.5 U J C J CO cc o OO CO 0.4 0.3 0.2 2900 X System of S0 2 V 0.1 2500 3000 X 1,5 Cel l empty 2 2.6 to r r 3 0.7 t o r r 4 0.23 t o r r 3441 A V FIG. 9 The 3440 A Continuum of HNSO taken on a 21 1 Eagle Spectrograph in conjunction with 4 meter White Cel l set for 24 t raversa l s TABLE 1 ! A. HNSO Bands of the 2700A system X Obs. AX Vac. X Vac. v Vac 2689.23 A 0.80 I 2690.03 Si 37174 2671.82 0.79 2672.61 37417 2652.14 0.79 2652.93 37694 2636.25 0.79 2637.04 37921 2620.51 0.78 2623.29 38120 2612.59 0.78 2613.37 38265 2603.92 0.78 2604.70 38392 2593.46 0.78 2594.24 38544 2583.80 0.77 2584.57 38691 2572.86 0.77 2573.63 38856 2555.23 0.77 2556.00 39124 2535.53 0.76 2536.29 39428 2517.03 0.76 2517.79 39717 2496.80 0.75 2497.55 40039 2479.70 0.75 2480.45 40315 2458.55 0.74 2459.29 40662 2443.71 0.74 2444.45 40909 2423.28 0.74 2424.02 41254 2407.81 0.73 2408.54 41519 2386.76 0.73 2387.49 41885 6. DNSO BANDS 2635.31 0.79 2636.10 37935 2612.23 0.78 2613.01 38270 2606.78 0.78 2607.56 38550 2573.36 0.77 2574.13 38848 2556.70 0.77 2557.47 39127 2543.14 0.76 2543.90 39310 2530.08 0.76 2530.84 39513 2515.10 0.76 2515.86 39748 HOT BANDS FIG. 10 Infra-Red Spectrum of Matrix Isolated HNSO taken on a Perkin-Elmer 225 Spectrophotometer at 4°K 1-3 Discussion and Interpretat ion of Results As stated in the int roduct ion, the reason for in teres t in th ionyl imide i s i t s s i m i l a r i t y with S0 2. In the fol lowing d i scuss ion, arguments w i l l be made to further th i s point and resu l t s from S 0 2 w i l l be applied to thionyl imide to help explain the observed resu l t s . a. Ground state normal coordinates o To understand the 2700 A system of thionyl imide and the v ibrat ions act ive i n the system, one must f i r s t determine the normal modes of v ib ra t ion in the ground state of the molecule. Richert (12), i n his normal coordinate ana lys i s , treated thionyl imide as a t r ia tomic molecule with-the N-H group as a s ing le atom. He used the set of diagonal force constants given in Table (2). For th i s work, a f u l l normal coordinate analysis using a f u l l set of diagonal force constants (that i s neglecting a l l i n te rac t ion force constants) was carr ied out. The normal coordinate analys is was carr ied out using the computer programs wr i t ten by Schachtschneider (13). The force constants were varied u n t i l the calculated frequencies of HNSO, DNSO, HN 3 4S0, and DN3 4S0 agreed to wi th in approximately 5 cm"^ of the observed matrix i s o l a t i o n frequencies except for the v ibrat ions involving a hydrogen atom. No attempt was made to ref ine the force constants to achieve a better f i t of the calculated frequencies due to the problem with anharmonicity and matrix ef fects (see below). The f i n a l force constants are shown in Table(2),along with the observed and calculated frequencies. The large errors in the ca lcu la HN stretch ing and HNS bending v ibrat iona l frequencies can be . a t t r ibu ted to anharmonic ef fects which are always large in molecules of th i s type. The anharmonic ef fects ar i se from the f a c t that the H and D "sample" d i f f e ren t parts of the v ib rat iona l potent ia l s and thus the anharmonicity a f fects them quite d i f f e r e n t l y . No attempt was made to correct the observed frequencies for anharmonicity to get a better f i t . Tab!e( 2) includes the re su l t ing matrices and L -" ', where i i s the matrix that transforms the vector of normal coordinates Q_ to internal coordinates S_, b y the equations S = LC: (1) o r , i n reverse, £ = L _ 1 S (2) See Appendix 1 for a discussion of the formalism of the FG matrix methods of Wilson, Decius, and Cross (14). b. Franck-Condon ca lcu la t ion for the bands of the 2700$ region Now that the form of the normal coordinates in the ground state i s known, one can use the Franck-Condon p r i nc ip le to determine what v ibrat ions are act ive in the t r an s i t i on and how large a shape-change i s necessary to explain the observed t r an s i t i on . The basis of the Franck-Condon p r i nc ip l e i s that the e lect ron ic t r an s i t i on in a molecule i s assumed to take place so rap id ly compared to the v ib ra t i ona l motion that the internuclear distance can be regarded as f i xed during the t ran s i t i on (15). Translated into quantum mechanical language, the strongest v ibrat iona l t rans i t ions according to the Franck-Condon p r i nc i p l e are those with the largest values of the where and ^ " are the v ib rat iona l wave functions in the upper and lower e lec t ron ic states respect ive ly. The re l a t i ve in tens i t y of a given band (M,N) in the e lect ron ic t r an s i t i on i s equal to the square of th i s i n t e g r a l . Thus one i s cor re lat ing an "overlap" between the upper and lower e lec t ron ic state v ibrat iona l wave functions to the in tens i t y of a given band. As noted prev ious ly, the 270o8 system of thionyl imide consists of a long ser ies of bands that did not undergo any great change upon deuteration, though the bands did lose much of t he i r i n tens i ty r e l a t i ve to the background. From the fac t that a long series of bands i s observed, one knows from the Franck-Condon p r i n c i p l e that a large change of shape i s occurring in the t r an s i t i on . From the fact that the observed series of bands i s r e l a t i v e l y insens i t i ve to deuterat ion, one can say that the v ibrat ions act ive in the t r a n s i t i o n , and therefore the shape change, do not involve the hydrogen atom and are centered in the S-0 end of the molecule. See Table (2) fo r a compar-ison of the ground state HNSO and DNSO frequencies. To get some idea of which v ibrat ions might be strongly act ive in the e lec t ron ic t r an s i t i on and how much of a change of shape would be necessary to Franck-Condon overlap integral R^, defined as (3) produce the observed absorption maximum, several rough Franck-Condon overlap ca lcu lat ions were done. Integrals of the type shown in eq. (3) were f i r s t evaluated by Hutchisson(16) and l a t e r by Wagner (17), and Ansbacher (18). In general, such integra ls are not easy to evaluate because Q' ? Q " ; i n cer ta in cases, however, and by using various approximations, overlap in tegra l s of the type shown in eq. (3) can be quite easy to ca l cu la te . Smith and Warsop (19) have developed a method fo r ca l cu la t i ng in tegra l s of the type f o r large changes in shape. The integra l shown in eq. (4) represents the overlap between the zero v ib rat iona l leve l of the lower e lec t ron ic state with the v ib rat iona l l e v e l , M, in the upper e lect ron ic s tate. To evaluate integra l s of the type shown in eq. (4), one begins by r e l a t i ng the upper state normal coordinate Q' to the lower state normal coordinate Q" by the fol lowing re la t ionsh ip : where A i s a square matrix which transforms the vector of ground state normal coordinates into the vector of upper state normal coordinates, and d_ i s the change in the o r i g i n of the normal coordinates. Such a re lat ionsh ip i s necessary so that OJ and Q" are functions of the same var iable so that the integra l over dU,1 w i l l have some meaning. (4) Q/ = AOJ' + d (5) To s impl i f y the re lat ionsh ip between Q1 and Q", A i s set to the un i t matrix. Setting A equal to the unit matrix i s r a t i ona l i z ed by Coon, et . al_. (20), who show that , for band progressions that or ig inate in the v = 0 level of the ground e l ec t ron i c s ta te , the v ib rat iona l wave function in the ground state i s l a r ge l y concentrated around the o r i g in of the normal coordinate and thus I £ , L = 0 (6) for a large percentage of the time. This can eas i l y be seen i f one graphs the harmonic o s c i l l a t o r wave functions fo r the v = 0 l e v e l . From th i s one can see that OJ w i l l be very in sens i t i ve to A (since Q" i s zero most of the time) but w i l l be sens i t ive to d. With A set to the unit matr ix, eq. (5) becomes OJ = OJ' + d (7) and thus eq. (4) becomes RJVJQ =JV(Q ' ) * " (Q' - d) dQ' (8) Smith (21) has shown that fo r R^Q under the above conditions eq.(8)takes the form R M Q = ( 2 " M q M ! ) 1 / 2 exp(l/4 y 2 P ) ) M l ^ L j ^ M° %=o (M - 2t)l tl u 2 ( v ' v " ) 1 / 2 v "v ' / l / 2 . where q = N / p = — y = a ' d v" v'"-+ v" a _ 4TT 2 VC X = v " - v ' (9) h v" + v ' Thus, to carry out the ca l cu l a t i on , a l l one needs to know are the lower and upper state frequencies and d^, the s h i f t in the o r i g i n of the normal coordinate being considered. The vector of normal coordinates, i s related to that of the in te rna l symmetry coordinates by eq. (2). The vector of the s h i f t s i n the or ig ins of the normal coordinates, d_, is related to A S ^ the vector of changes i n the internal symmetry coordinates, i n the same manner. d = L ' 1 AS ( 1 0 ) One should remember that in th iony l imide, with the change in an interna l symmetry coordinate being equal to the change in the corresponding interna l coordinate, AS_ represents d i r e c t l y the changes in bond angles and bond lengths between the upper and lower s tates. To ca lcu la te d_ one must know the shape change occurring in the t r a n s i t i o n . In th iony l imide, due to the diffuseness of the observed bands, no ro ta t iona l analys is could be done and thus i t was impossible to determine the structure of thionyl imide in the excited s tate. Thus, to ca lcu late d_, an appropriate excited state structure fo r t h i o n y l -imide had to be assumed. The change of shape occurring in the 290oH system of S0 2 was assumed to be the same as the shape change occurring in the 270o8 system of th ionyl imide. This i s j u s t i f i e d by the facts that the change of shape occurring in thionyl imide i s occurring on the S O end of the molecule, that both t rans i t ions have s im i l a r v ib ra t i ona l s t ructures , and that both t rans i t ions are the f i r s t strong t rans i t ions seen in each molecule. In a prel iminary band contour analysis of the (OIO)-(OOO) band [Metropol i s ' s v ibrat iona l numbering (2)] of the 290oK system of S0 2 , Dixon (22) has found that i n the excited s t a te , R S Q =1.58 H o !° and ctQrjg = 105 . Since in the ground state R Q N = 1.43 'A and ancn = 11 XS0 '0S0 (22), AR S 0 = 0.15 8 and /&nC;n = " 1 5 ° f o r the t r an s i t i o n . Using these l0S0 changes in bond lengths and bond angles and set t ing A R S0 " A RNS (11) one can ca lcu late d_. Since the shape change has been reasoned to be taking place around the S atom, only the overlap integrals f o r v 2 , V g , and [see table (2) fo r frequency assignments] need be considered since only these v ibrat ions w i l l have appreciable d^'s. Using eqs. (10), and (11), d = L" AR AR AR NH NS SO Aa HNS AaNSO AyHNSO = L " L 0 0.15A 0.15A 0 0.26R 0 (12) Thus the ind iv idua l d^'s of i n te res t become d 2 = (1.936 - 1.995)(0.15) + (0.028)(0.26) .= 0.052 d 3 = (1.923 + 2.596)(0.15) - (0.142)(0.26) = 0.674 d 5 = (1.067 + 0.874)(0.15) + (3.352)(0.26) = 1.163 From the above values of fo r v 2, v 3, and vg, one can see that v 3 and w i l l provide most of the i n ten s i t y . One would therefore expect a long series of band progressions in both v 3 and V5. The reason d 2 i s small even though the v ibrat ion i s centered on the S-0 end of the molecule, i s that i t i s mainly an asymmetric stretching v i b r a t i on , thus the change in i s counteracted by that in R S Q. Since we cannot be sure of the v ibrat iona l analysis of the 2700$ system of th iony l imide, the excited state frequencies cannot be determined unambiguously. The NSO bending v i b r a t i on , v^, was assumed to be 285 cm"^ (the predominant v ib ra t i ona l i n te rva l seen in the t r a n s i t i o n ) , by analogy with the 2900$ system of S0 2 , where the upper state bending v ibrat ion (v 2' = 318 cm~^) i s s i m i l a r l y prominent. This assignment i s supported by the three previously mentioned "hot" bands which are a l l separated from bands known to be temperature-insensit ive by the ground state frequency v 5 = 453 cm~^ to within ± 25 cm~^ ( i . e . fo r these d i f fu se bands with in the accuracy of measurement). The other low-ly ing frequency, the tors ional v ibrat ion Vg", at 759 cm~^, cannot give r i s e to "hot" bands of th i s type because, since i t i s a non-t o t a l l y symmetric v i b r a t i on , i t must obey the se lect ion rule A Vg = 0, ± 2 , then, "hot" bands a r i s ing from the leve l Vg" = 1 are most l i k e l y to be sequence bands (A V=0). However, the assignment of these "hot" bands as tors ional sequence bands cannot be e n t i r e l y ruled out, though i t i s less l i k e l y than t he i r assignment as the f i r s t members of ground state progressions in v^" , s ince these are favoured by the Boltzmann d i s t r i bu t i on law. was estimated using the ru le given by Birge (24 ) f o r diatomic molecules: \ 2 I" ' r to = constant ' ( 14 ) where r = internuclear distance and co = v ibrat ion frequency. Thus: r , 2 c o ' = r " 2 co " <•>' = (p-) 2 oo" = 859 cm" 1 ( 15 ) A computer program wr i t ten by Malm and Merer was used to evaluate eq. (9 ) using the previously determined d^.'s and frequencies. The results of these ca lcu lat ions are given in Table ( 3 ) and are graphed in F ig. ( 1 1 ) . Upon examination of the spectrum of thionyl imide taken on the Cary 14 spectrophotometer ( f i g . 6 ) , one can see that the bands are not d i scern ib le along the whole t r a n s i t i o n , and one must consider the p o s s i b i l i t y that the absorption maximum and the banded structure may not -correspond to the same excited s ta te . The absorption maximum comes -at approximately 2170A (46083 cm - 1 ) and the f i r s t d i scern ib le band i s at 2690A* (37175 cm" 1 ) . I f the observed absorption maximum and the banded structure are taken as a r i s i ng from the same t r a n s i t i o n , the Franck-Condon maximum l i e s at least 8900 cm" 1 from the system o r i g i n . The value shown,in f i g . (11) on the basis of Dixon's resu lts for the 290oH system of SO2 i s approximately 6000 cm" 1 . Thus the ca lcu lated t r an s i t i on i s too "short" in re l a t i on to the observed t r an s i t i on provided the banded structure and the observed absorption maximum belong to the same t r an s i t i on . Further ca lcu lat ions were done to corre late changes in ARJ^. ( = AR^Q) and A a ^ Q i n the t r an s i t i on to the " length" of the t r a n s i t i o n (or the pos i t ion of the Franck-Condon maximum); thus i t was possible to see how large a shape change was necessary fo r the Franck-Condon maximum to l i e 8900 cm""' from the system o r i g i n . The resu l t s are shown in f i g . (12). From f i g . (12) one can see that to explain the value 8900 cm~^ in th iony l imide, one needs A R ^ s = 0.18$ and A a N S 0 = 0.28 Radian or other combinations from the graph. Thus one must accept approximately R^ = 1.69$, R^Q =1.63 and = 1 1 4 ° in the exc i ted state of thionyl imide to explain the 2700$ system of th ionyl imide as a s ing le t r a n s i t i o n . Such large changes in shape are approaching the l i m i t for -n -> T T * t rans i t ions i f one assumes one i s seeing such a t r an s i t i on in thionyl imide. The largest change of shape known for a IT -* T T * t r an s i t i on i s in 0 2 where there i s a 33% change i n the internuclear d istance; thus in th ionyl imide where the IT molecular o rb i t a l i s spread over two bonds,one would expect that the maximum change in the NS or SO bond would be 16.5% which is 0.25$ in the NS bond and 0.24$ in the SO bond. These changes are not much larger than those l i s t e d above which are needed to explain the 2700$ system of thionly imide as a s ing le t r a n s i t i o n . Table 2 Comparison of Reported Gas Phase Frequencies and Matrix Frequencies Meaning •1 v l v 2 v 3 v 4 v 5 (HNSO Q branches) 1 Matrix (HNSO) 3345 cm 1261 1090 911 453 755 3309 cm 1249 1083 900 447 754 'NH v 0 S N asym. v0SN aNH aNS0 YHNS0 sym. NS SO NH f a NSO R icher t ' s Force Constants 8.2 mdyne A - 1 8.6 " 0.6 " f a HNS fY HNSO .Author's Force Constants 7.48 mdyne A" 1 8.4 6.1 0.598 0.51 0.22 Table 2 (continued) Frequency f i t s fo r various isotopes using the force constants of t h i s work HNS320 HNS340 OBS. FREQ. CALC. FREQ. DIFFERENCE PERCENT ERROR (cm" 1) (cm" 1) (cm" 1) i j 1 3309.0 3321.6 -12.6 -0.381 2 1249.0 1253.0 -4.0 -0.318 3 1083.0 1067.9 15.1 1.393 4 900.0 928.3 -28.3 -3.140 5 754.0 754.9 -0.9 -0.117 6 447.0 437.8 9.2 2.061 OBS. FREQ. CALC. FREQ. DIFFERENCE PERCENT ERROR (cn f 1 ) (cm - 1 ) (cm" 1) 1 2 3 4 5 6 0.0 1234.0 0.0 0.0 0.0 0.0 3321.6 1239.3 1064.4 923.9 753.9 434.8 0.0 •5.3 0.0 0.0 0.0 0.0 0.0 -0.429 0.0 0.0 0.0 0.0 OBS. FREQ. CALC. FREQ. DIFFERENCE PERCENT ERROR (cm" 1) (cm - 1 ) (cm - 1 ) 1 2450.0 2432.4 17.6 0.718 2 1245.0 1244.4 0.6 0.050 3 1048.0 1043.0 5.0 ' 0.477 4 752.0 724.0 28.0 3.717 5 592.0 590.9 1.1 0.192 6 400.0 412.9 -12.9 -3.227 OBS. FREQ. CALC. FREQ. DIFFERENCE PERCENT ERROR (cm" 1) (cm" 1) (cm- 1) 1 0.0 2432.4 0.0 0.0 2 1230.0 1230.0 -0.2 -0.018 3 0.0 1037.6 0.0 0.0 4 0.0 721.2 0.0 0.0 5 0.0 589.6 0.0 0.0 6 0.0 410.6 0.0 0.0 Eigenvector and Inverse Eigenvector Matrices for HNS 0 Based on the Force Constants of th i s Work ,EIGENVECTORS FREQUENCY = 3321.9 cm" 1 \ 1.0312 -0.0335 0.0005 -0.0429 0.0429 0.0 FREQUENCY = 1259.4 cm" 1 0.0082 0.2417 -0.2219 -0.4022 0.0202 0.0 FREQUENCY = 1075.3 cm" 1 0.0062 0.1750 0.2104 -0.3769 -0.0738 0.0 FREQUENCY = 932.7 cm" 1 0.0086 0.1102 0.0079 0.8863 -0.1230 0.0 FREQUENCY = 746.6 cm" 1 0.0 0.0 0.0 0.0 0.0 1.2215 FREQUENCY = 433.7 cm" 1 -0.0017 0.0158 0.0115 0.0707 0.2834 0.0 EIGENVECTOR INVERSE FREQUENCY = 3321.9 cm" 1 0.9680 -0.0386 0.0006 -0.0034 0.0086 0.0 FREQUENCY = 1259.4 cm" 1 0.0535 1.9358 -1.9954 -0.2196 0.0284 0.0 FREQUENCY = 1075.3 cm" 1 0.0555 1.9230 2.5959 -0.2824 -0.1420 0.0 FREQUENCY = 932.7 cm ' 1 0.1021 1.6093 0.1288 0.8824 -0.3144 0.0 Table 2 (continued) FREQUENCY = 746.6 cm" 1 0.0 0.0 0.0 0.0 0.0 0.8187 FREQUENCY = 433.7 cm" 1 -0.0916 1.0666 0.8740 0.3255 3.3521 0.0 Table 3 i Calculated values of R^ for v 3 and with R<-Q =0.15$ and Act „ =0.26 Radian NSO UPPER STATE FREQUENCY = 859.00 cm" 1 UPPER STATE FREQUENCY = 285.00 cm LOWER STATE FREQUENCY = 1050.00 cm" 1 LOWER STATE FREQUENCY = 450.00cm V' v" = 0 v' v" = 0 0 0.055549 0 0.036467 1 0.140021 1 0.103645 2 0.245643 2 0.202506 3 0.346050 3 0.313296 4 0.414857 4 0.405846 5 .0.436694 5 0.452940 6 0.411496 - — 6 0.442376 7 0.351593 7 0.381074 8 0.274826 8 0.290026 9 0.197751 9 0.194110 10 0.131543 10 0.112692 F ig . 11 Calculated t r an s i t i on envelope, for AR N S = AR S 0 = 0.15 A and A a N S Q = 15° •18 .20 . .22 .24 .26 .28 .30 .32 A aNS0 (Rad.) F ig. 12 Corre lat ion of A R n s and toNS0 with d i f f e ren t system or ig in-absorpt ion maximum separations C. E lect ron ic States of Thionylimide To achieve an understanding of the absorption spectrum of HNSO, i t i s necessary to estab l i sh the possible e lect ron ic states of HNSO. There are two possible ways to proceed: e i ther the e lect ron ic states of HNSO can be correlated with the states of the i soe lect ron ic i molecule, S0£, or the e lect ron ic states of HNSO can be derived from a reasonable set of d i s soc iat ion products. These two methods w i l l each be considered. States of HNSO derived from d i s soc ia t ion products The most probable d i s soc ia t ion products w i l l r e su l t from the rupture of the weakest bond in the molecule. Some relevant d i s soc ia t i on energies are given below (23). S 0 2 ( g ) * S 0 ( g ) + °(g) A H = 5' 9 e' v' N S ( g ) * N (g ) + s ( g ) A H = 5 * 6 1 e * v : H N 0 ( g ) - H ( g ) + N 0 ( g ) AH = 2.11 e.v. N H 3 (g ) * H (g) + N H 2(g ) A H = 4 ' 3 e - v « Although the bond energy of the NS radica l i s ce r ta i n l y greater than that in HNSO, i t i s reasonable to expect the N-H bond to be the weakest in the molecule. Thus the most l i k e l y d i s soc iat ion process i s H N S 0 ( j r H ( g ) • N S 0 ( g ) The ground state of HNSO w i l l be formed from the d i s soc iat ion products i n t h e i r ground states. The H atom w i l l be in the I s 1 2 S atomic ground s tate . The nature of the ground state of NSO i s more d i f f i c u l t to determine. NSO i s i soe lect ron ic with N0 2 and w i l l probably have corresponding e lectron configurations and states. I t i s noted that N0 2 belongs to the C 2 y point group, whereas NSO belongs to the C g point group. (See Table(4)for C 2 v and C g character tables and the cor re la t ion of the i r reduc ib le represent-ations of the C 2 v and C $ point groups.) N0 2 i n i t s ground state has the fol lowing electron conf i g -uration (3): ( 5 a i ) 2 ( l a 2 ) 2 ( 4 b 2 ) 2 (6a,) . . . % ( C 2 v ) , 2 A ' (C g ) Here the numbers (5,1,4,6) refer to the number of times a certa in representation has appeared in the molecular o rb i t a l conf igurat ion. Thus 5a-j refers to the f i f t h a, molecular o r b i t a l . One can see that the NSO rad ica l in i t s ground state w i l l probably be in 2 a A' s t a te . Thus the ground state of HNSO w i l l be formed from 2 2 a ( S) H atom and a ( A ' ) NSO r a d i c a l . Using the bui ld ing up 1 3 p r i nc ip l e s (23) one obtains ' A' states of HNSO. It i s seen that , f o r molecules with even numbers of e lectrons, no p a r t i a l l y f i l l e d degenerate o r b i t a l s , or any near l y ing u n f i l l e d o r b i t a l s , Hund's Rule (23) does not hold and the s ing le t has the lower energy. Thus, according to th i s approach the ground state of HNSO w i l l be the s ing let and the t r i p l e t w i l l be the f i r s t excited s ta te . The f i r s t excited state of N0 2 has the fol lowing electron conf igurat ion (23): ( 5a , ) 2 ( l a 2 ) 2 ( 4b 2 ) 2 ( 2b ] ) . . . 2 B , ( C 2 y ) , 2 A " (C g ) Thus the f i r s t exc ited state of NSO w i l l be a A" s tate. From a ( 2S) H atom and a ( 2 A") NSO radica l one obtains ( 1 , 3 A " ) HNSO. As we sha l l see, the t r i p l e t w i l l have the lower energy, and the s i ng le t w i l l have the higher energy. A potential energy diag based on these resu lts i s shown in f i g . (13). Corre lat ion of the E lectronic States of S0 2 and HNSO S 0 2 has the fol lowing ground state and low-ly ing excited state e lectron configurations (23). . . . ( l a 2 ) 2 ( 4b 2 ) 2 ( 6a - j ) 2 , ] A, (ground state) . . . ( l a 2 ) 2 ( 4 b 2 ) 2 ( 6 a 1 ) 1 ( 2 b 1 ) 1 , 1 ' 3 B 1 ' . . . ( l a 2 ) 2 ( 4 b 2 ) 1 ( 6 a 1 ) 2 ( 2 b 1 ) 1 , ] ' 3 A 2 . . . ( l a 2 ) 1 ( 4 b 2 ) 2 ( 6 a 1 ) 2 ( 2 b 1 ) 1 , 1 The f i r s t excited state of S0 2 , g iv ing r i s e to the t r an s i t i on at 3900A\ has been i d e n t i f i e d as a 3 B, state (25). Evidence obtained by Hochstrasser and Marchetti (26) from matrix i s o l a t i o n studies shows another t r i p l e t state ly ing close to the 3B-j s t a te , which i s probably the 3 A 2 s tate. The 290oH system of S O 2 , though much stronger than the 390oH system of S0 2 , i s not as strong as would be expected for an allowed s ing le t t r an s i t i o n . Dixon (22) has suggested that i t may be a ^A 2 - ^A, v ib ron ic -allowed t r an s i t i on where the ^A 2 state i s interact ing v ia the b 2 antisymmetric stretching v ibrat ion with a higher B, e lect ron ic s tate to give ^B-, - ^A, v ibronic bands. On the basis of these resu l t s the fol lowing energy level diagram fo r S0 2 i s drawn. 29622 cm" 1 ] A * 27000 cm" 1 3A, 25767 cm ' 1 3 B 0 cm ' 1 ] A Using the co r re la t i on table for C 2 v and C $ point groups, the lowest four states in HNSO are predicted to be 1 A ' , 3 A " , 3 A " , and 1 A " by co r r e l a t i on with the known states of SO,,. This method gives the same resu l t s as the previous method except that the f i r s t t r i p l e t i s 3 A " instead of 3 A ' . •• - •• This d iscuss ion shows that the ground state of thionyl imide i s most l i k e l y to be a e lec t ron i c s ta te . The f i r s t two excited e l e c t r on i c states are given as t r i p l e t s by both methods. As noted in the experimental sect ion, approximately 0.40 meter-atm. were necessary to observe the weak continuum in the 3440$ region. Such weak t r an s i t i on s are usual ly associated with spin forbidden t r an s i t i on s . The second t r i p l e t which i s not observed may l i e close to the f i r s t t r i p l e t or under the 2900$ system of S0 2 , i n e i t he r case being undetectable. The 2700$ system of thionyl imide i s probably the f i r s t s i ng le t t r an s i t i o n . Di f fuse bands are usual ly associated with pred i s soc iat ion. Pred i s soc iat ion occurs when an e lect ron ic state with d iscrete energy levels has the same energy as an unbound or repuls ive e lect ron ic s tate. Thus A1 shown in f i g . (13) may be predissocfat ing the ^A" upper state of the 2700$ system. One should note that, as predissoc iat ion gets stronger, the d i f fuse v ibrat iona l bands themselves may no longer be d i scern ib le above the background. It i s therefore possible that the observed Franck-Condon Maximum and the banded structure can be from the same excited e lect ron ic state given the acceptance of the large shape change proposed in Section B. Table 4 Character Tables for the Point Groups C $ and C E a v ( y z ) C2v E C 2 a v (xy ) a v ( yz ) 1 A 1 1 y.z : R x A l 1 1 1 1 z A" 1 -1 X R y ,R z A 2 1 1 -1 -1 B l 1 -1 1 -1 X R y B 2 1 -1 -1 1 y Corre lat ion of the I r reducible Representations of the C g and C Point Groups u 2v C s A r B 2 A' A 2 > B l A" N-H F ig . ( 1 3 ) Poss ible Potent ia l Energy Curves for Thionylimide CHAPTER II 2 3 E and z E lectronic States of Linear Molecules  i n which One or More Quanta of a Degenerate Bending  V ibrat ion i s Excited 2-1 Der ivat ion of the Hamilton!an and i t s Matrix Elements For a l i n e a r polyatomic molecule in a z e lect ron ic s tate, one normally assumes that the molecule i s better described by Hund's Case (b) coupling where the spin i s almost " f r e e " , being only coupled to the rotat ion of the molecule, due to the absence of any i n te rac t i on coupling the spin to the axis of the molecule such as spin o rb i t coupling. In a z e lect ron ic s tate, a Hund's Case (a) representat ion, where the spin and v ibrat iona l angular momenta have well defined components along the axis of the molecule, i s a good approximation as long as the molecule i s not rotat ing or i n low rotat iona l quantum l e ve l s . At high rotat iona l quantum numbers, a Hund's Case (b) representation i s a better de sc r i p t i on , and the resu l t s obtained using a Case (a) hamiltonian w i l l be transformed into Case (b) notation as needed. The hamiltonian for a l i nea r polyatomic molecule in a mu l t i p le t £ e lec t ron ic state in which one or more quanta of the degenerate bending v ibrat ion i s excited i s given below, fol lowing Van Vleck (27) and Watson (28). H = B ( r ) . LVW2 + (WV 2 ] +2x(Sz2"k2) + ( Th e f i r s t term i s the ordinary rotat ional energy and involves only the x and y components since the rotat ional angular momentum vector i s perpendicular to the molecular ax i s , which i s taken the components of the t o t a l , the sp in, and the v ibrat iona l angular momentum. The second term i s the d ipo le-d ipo le spin-spin i n t e r -act ion , where x is the spin-spin in teract ion constant. The th i r d term i s the sp in- rotat ion i n te rac t i on . The l a s t term i s a quasi sp in-rotat ion in teract ion where the rotat ional motion ar ises from a degenerate bending v ib ra t i on . This has been discussed by Chang and Chiu (29), but has not yet been seen experimentally. Eq. (1) can be rewritten by f i r s t expanding and subst i tut ing with the fol lowing re la t ionsh ips : as the z ax i s . In eq. (1), J ( X j V ) > s ( x , y ) , and G (x,y) are j u s t (2) Thus eq. (1) becomes H = B ( r ) [ J 2 - J z 2 + S 2 - S z 2 ] + S z 2 (2x+ Y ) -S 2 ( Y +|x) + Y ' G z S z + (2-r-B)(J +S_+J_S +) -B(J +G_+J_G +) + B(S+G_+S_G+) + B(G x 2 +G y 2 ) ( 3 ) We now have the hamiltonian in a form in which we can discuss i t s matrix elements. The matrix elements of J and S are well known [see Van Vleck (27) and Hougen (30)] and are given below: J 2|J,P> = J(J+1) ti 2|J,P> S2|S,Z> = S(S+1) Ti 2jS,z> J |J,P> = Ph[J,P> Sz|S,Z> = ETi|S,E> J _ | J ,P> = [ J ( J +1 ) - P ( P±1 ) ] 1 / 2 Ti|J,P±l> S+|S,z> = [ S ( S + l ) - z ( E ± l ) ] 1 / 2 Ti|S,Z±l> (4) Here P i s the eigenvalue of the operator J z > P i s equal to U+z), where i i s the v ibrat iona l angular momentum quantum number, i . e . the eigenvalue of G z , and z i s the eigenvalue of S z . S has the normal sign of i in i t s commutation re lat ionsh ips but J does not. This anomalous sign of i comes from the fact that , although J has the normal sign of i in i t s commutation re lat ionsh ips when J i s referred to a space f i xed axis system, when J i s referred to the molecule f ixed axis system the anomalous sign of i appears i n the commutation re lat ionsh ips fo r J . The sign change comes from the d i rec t ion cosines which are used to transform the space f ixed axes to the molecule f ixed axes. The matrix elements of G + are more complex. To obtain them, one s tar t s with the de f i n i t i o n of the Cartesian component G^, (a=x,y,z) G a = £ r L l (5) which re lates G to the vector of the normal coordinates Q and the i r conjugate momenta P_. c01 i s the Co r i o l i s coupling coe f f i c i en t matrix and is defined such that (^. a) u v = ( i H a A t r ) u V (u and v are v ibrat ions) (6) where i_ i s the transformation matrix that transforms the mass weighted Cartesian displacement coordinates, cj_, to normal coordinates, Q. Q = i SL • <7) M a i s one of Meal and Polo ' s cross product matrices [see Meal and Polo (31)]. One should note that £ a i s an antisymmetric matrix and therefore c a u v = - ^ a ^ t Most of the s coe f f i c i en t s are zero for l i nea r molecules. The coe f f i c i en t C u v i s only non-zero when the d i r ec t product of the i r reduc ib le representations of the v ibrat ions u and v transforms as a rotat ion in the point group to which the molecule belongs. Thus in a symmetric t r iatomic l i nea r molecule, only the a,. + and ir v ibrat ions w i l l have non-zero Co r i o l i s coupling u u r a coe f f i c i en t s between them. Therefore only t, ^  U32) wi 11 be non-zero because [v 0 x hT, = n„ x o,,+ = TT 2 3 u u g where iTg i s the species of the rotat ions Rx and Ry in the D ^ point group. For an unsymmetrical t r iatomic molecule only C a 2 1 , ? a , 2 » ? U32> a n < ^ ? t*23 W 1 " ' 1 b e n o n _ z e r o ' F o r large molecules, the expression for G^ w i l l become unwieldy. This thes is w i l l therefore be r e s t r i c ted to unsymmetrical t r iatomic molecules, but the fonnulae derived can be generalized fo r larger molecules. Eq. (5) fo r unsymmetrical t r iatomic molecules becomes G a " kll.3 <Wk + W 2 > <8> Now defining G as '; V G x t i G y <9> eq. (5) gives G± - Jl,3 <* 1 W 2 ± * ' W k ^ = ( x ' y ) ( 1 0 ) and f i n a l l y G ± = t 'zkW « k P Z ± T 1 Pk'z±3 .<"> The matrix elements of Qk and for a harmonic o s c i l l a t o r are given by Wilson, Decius, and Cross (32) and are <vk + l|P k|v k> = i A t i v k ( v k + l ) <v k|P k|v k + 1> =- iATiv k (v k +l) < v k + 1 l Q k l V = / £ 2 1 ( v . + i ) OTT V k K ^ v k l Q k l v k + 1 > = / & 2 1 (v.+l) (12) The matrix elements of i P 2 + +" Q2+. f o r the doubly degenerate harmonic o s c i l l a t o r are given by M o f f i t t and L iehr (33), but w i l l be given here in a s l i g h t l y more useful form. <v2 ± l , £ + l | P + l v 2 i > = ± i [ % h v 2 ] 2 [ ( v 2 + l ) ± ( £ + l ) ] 2 <V2 ± 1 , £ - 1 | P J V 2 A > = ± i C 3 2 h v 2 ] J 2 [ ( v 2 + l ) + ( £ - l ) : S s h , • , <v2 ± l , n + l | Q + | v 2 £ > = L 8 ^ ] 2 [ ( v 2 + l ) ± U + l ) ] 2 <v2 ± l , £ - l | Q j v 2 i > = [ | ^ - f t ( v 2 + l ) + ( i i - l ) ] J s ( 1 3 ) Now i t c a n be s e e n t h a t f o r G_ a n d G + t h e r e w i l l be f o u r t y p e s o f n o n - z e r o m a t r i x e l e m e n t s f o r . e a c h . <v2 + 1,211, v R + l | G ± | v 2 £ v k > = <v2 - 1 , £ ± 1 , v k + l | G ± | v 2 ) i v k > = <v2 + l , a ± 1 , v k - l | G ± j v 2 ^ v k > = <v2 - 1 , £ ± 1 , v k - l | G ± | v 2 ^ v k > = + n ? 2 k * k [ ( v k + i ) ( v 2 ± £ + 2 ) ] ± t f ? 2 k n k [ ( v k + 1 ) ( v 2 n . ) ] 3 5 + - h ? 2 k ^ k [ v k ( v 2 ± i i + 2 ) ] ^ ( 1 4 ) w h e r e k = 1,3 a n d , f o l l o w i n g M i l l s ( 3 4 ) , w i t h t h e d e f i n i t i o n s / V 2 + A ( 1 5 ) ,R . v2 B e f o r e b e g i n n i n g t o e v a l u a t e t h e e n e r g y l e v e l s o f t h e h a m i l t o n i a n g i v e n i n e q . ( 3 ) , a s u i t a b l e n o t a t i o n f o r t h e b a s i s f u n c t i o n s m u s t be s p e c i f i e d . S i n c e t h e h a m i l t o n i a n i s s e t up i n a C a s e ( a ) r e p r e s e n t a t i o n , w h e r e , a s n o t e d p r e v i o u s l y , t h e s p i n a n d v i b r a t i o n a l a n g u l a r momenta h a v e w e l l d e f i n e d c o m p o n e n t s a l o n g t h e m o l e c u l a r a x i s , a b a s i s d e n o t e d b y < v 2 , £ , v k , J , P , E | w i l l be u s e d , T h e s y m b o l s a r e as p r e v i o u s l y d e f i n e d . In eq. (3) the f i r s t four terms are diagonal in a l l quantum 2 2 numbers. The l a s t term G x + Gy > a s shown by Hougen (35), only contributes small amounts to the anharmonicity constants x^> x 2 3 ' and w i l l be neglected. The f i f t h term which represents the sp in -rotat ion i n te r ac t i on , i s off-diagonal in z but "diagonal in the 1 v ib ra t iona l quantum numbers. The s i x th term involves G + and i s 1 therefore off-diagonal in the v ibrat iona l quantum numbers. It i s th i s term which i s responsible for the Co r i o l i s in teract ion which resolves the degeneracy of states d i f f e r i n g only in the sign of 1 and brings about the f am i l i a r Ji-type doubling. The seventh term i s also off-diagonal i n the v ibrat iona l quantum numbers and represents a "gyroscopic" sp in -v ib ra t ion i n te rac t i on . Since we are interested in the rotat iona l energy levels in a given v ib ra t iona l l e v e l , the elements off-diagonal in the v ib ra t i ona l quantum numbers must be taken into account co r rec t l y . This has been accomplished by the use of the Van Vleck transformation [Kemble (36)] <vt'|X|v'i->- .1 <Vi.'lHlv^<vi"|H|V£"> ( 1 6 ) For the case V2=v k =£=0 in a t r i p l e t s tate, the appl icat ion of the Van Vleck transformation can be eas i l y demonstrated. The v 2 = v k = £ = 0 l e ve l w i l l only i n te rac t with the v 2 = \ = l > ^ = ± 1 leve l s v ia the s i x th and seventh terms. Shown in Table(5)are the matrix elements of these two terms. Upon appl icat ion of the Van Vleck transformation, one obtains the second order correct ion to the matrix fo r the v 0=v,=a=0 v ib rat iona l l e v e l . The resu l t i s shown below. <0 0 0 J - l - 11 <0 0 0 J O 0[ <0 0 0 J 1 1| <0 0 0 J - l - 1I B 2[4J(J+1)] . -4B 2 /2J(J+1) 0 (17) <0 0 0 J 0 0 <0 0 0 J 1 11 symmetric B 2[4J(J+l)+8] . j -4B 2 /2J(J+1) B 2[4J(J+1)] One can see that carrying, out the Van Vleck transformation f o r higher v ib ra t iona l levels w i l l be very tedious. To avoid t h i s procedure the resu lts of carrying out the Van Vleck transformation in the general case were invest igated, and the general forms of the resu l tant terms are tabulated in Table(6). The matrix elements of these operators can be evaluated using eqs. (4,14) and are given in Table(7) in terms of q, the normal 5,-type doubling parameter fo r s i n g l e t s ta te s ; a , the Co r i o l i s contr ibut ion to the e f fec t i ve B value f o r a given v ibrat iona l l e v e l ; and g, the Co r i o l i s contr ibut ion to g22> the anharmonicity constant from the standard v ibrat iona l energy formula The matrix elements of the f i r s t three terms of the hamiltonian shown i n eq. (3) are diagonal, and using eq. (4) are simply E (v 9 ,£ ) = co 0 (v 9 + 1) + g 0 9 J i 2 (18) < V v k J P z l H d i a g J V v k J P z > = B[J(J+1)-U+E) 2 + S ( S + 1 ) - Z ] 2 + E 2 ( 2 X + Y ) - S ( S + l ) ( Y + f x ) + Y ' j 6 E The f i f t h term in eq. ( 3 ) , the sp in- rotat ion i n te rac t i on , i s off-diagonal in z, but is diagonal in v 2 , £, and v^. <v 2iiv ! KJP,z±l |H J > s |v 2 z v k JPz ) = i (JsY-BJEJCJ+D-C^+zJCii+ztDl^sCs+D-ECzil)]^ ( 2 0 ) I I t i s seen from eqs. (19,20) and Tab le(7 ) that i t i s possible to incorporate B and a y into an e f f ec t i ve B ( B y e ^ ) for a given v ib ra t iona l level since the quantum number dependence of B and a. a y are the same. 2-2 Results and Discussion 2 3 The resu l t s deal with both z and . z_electronic states i n which v 2=l ,2 and v k=0, where v 2 corresponds to a degenerate bending v ib rat ion and v^ corresponds to a stretching v ib ra t i on . These cases are the ones of prime in te res t because they are the cases most l i k e l y to be observed experimentally. U t i l i z i n g eqs. (19,20) and Tab le(7 ) , the matrix correct to second order can be immediately wr i t ten down in the Case (a) representation. Since a l l the resu l t ing matrices are doubly symmetric, they can be fac to r i zed into smaller sub-matrices using the Wang transformation ( 3 7 ) [see Appendix 2 ] . These sub-matrices correspond to e i the r the Kronig (+) or (-) rotat iona l l e ve l s . The resu l t ing sub-matrices can be diagonal ized. The results can be transformed to a Case (b) representation as needed. E E lec t ron ic States For the leve l V2=l, a=±l, v ^O , in a E e lect ron ic state one obtains the matrix shown in Table (8)using eqs. (19,20) and Table(7). Use of the Wang transformation y ie ld s the fol lowing pa i r of matr ices. |*,s> h\1,h>±\-1,-h>) !{| l,-a5>±|-l,3s>} ft ft •= ( | l , J s > ± | - l , - % > > ft - {| l . - % > ± | - l , J s > > ft symmetric - ( B r 3 s Y ) / J ( J+ l ) - f ±*2q -j /[J(J+D+%][J(J+1)-|3 B-,[J(J+I ) + ^ : - ^ Y These two matrices ( d i f f e r i n g in the sign of the q terms) can ea s i l y be diagonalized to give four energy l e ve l s . The general energy equation i s given below. E = B 1 [ ( J + J 2 ) 2 - l ] - ^ q ( J + % ) ± (21) >2n2 2 i2 r { [%Y , -B 1 ±%q(J+Js )^^+[J 5 Y-B 1 ±J 2 q(J+%)^]^[ (J+^) c - l ] } ' 5 (22) One should notice that in generating the four energy l e ve l s , one must be consistent in the choice of sign of the q terms, but th i s choice i s independent of the sign of the square root. Since the matrix can be diagonalized exact ly , one ju s t has to subst i tute (N=J+*j) and (N=J-%) to obtain the resu lts in Case (b) notat ion. Thus eq. (22) becomes The above equations correspond to the formulae that H i l l and 2 Van Vleck (27) have derived for n e lect ron ic states i f the terms in y and q are omitted, and Y ' i s substituted for the sp in -orb i t 2 coupling constant, A . Thus one can see by analogy to n e lect ron ic states tha t , unless Y ' i s la rge, the s p l i t t i n g i t w i l l cause w i l l be quenched by the spin uncoupling, and w i l l merge into the sp in- rotat ion i n te rac t i on s p l i t t i n g as the molecule rotates fas ter and fa s te r ; that i s , the e f fec t of Y ' G z $ z i n coupling the spin to the molecular axis w i l l be overcome by the term - ( B - ? S Y ) (J +S_ + J_S + ) . I f Y 1 i s neg l i g i b l e and B i s large compared with Y> the terms under the square root sign in eqs. (23,24) can be s imp l i f i ed , and the equations 2 2 reduce to the equations used fo r n v ibronic levels of E e lect ron ic states given by Johns (38), where the spin doubling and £-type doubling are separate e n t i t i e s . These equations are given here. (23) [¥r-B n ±^q(N+l)][N 2 +2N]} (24) F,(N)=B[N(N+1)-1] + hrW ± JsqN(N+l) (26) Thus i f y' i s neg l i g ib le and B,±^q(J+32)>>y, the spin s p l i t t i n g and £-type doubling are indeed separate. When two quanta of a degenerate bending v ibrat ion are exc i ted, E U=0) and A (£=±2) v ibronic states are formed. Using eqs. (19,20) and Table(7),the corresponding matrix can be wr i t ten down. Upon appl icat ion of the Wang transformation one produces the two 3 x 3 matrices given in Table(9). To obtain the Case (b) representation, where the rotat iona l energy i s diagonal, i t i s necessary to f i nd the matrix such that To obtain the transformation matrix fo r the A sub matrices one proceeds to set up the problem as fo l lows. Written out i n f u l l eq. (27) becomes (27) 2 B[(J+%) 2-2] -B/(J+%) 2-4 B[(J+J5 ) 2 -6] S -B/(J+i?)^-4 (N=J-^) (N=J+%) (28) B[N(N+l)-4] 0 0 B[N(N+l)-4] The Case (b) matrix i s in terms of N and not J . To wr ite the Case (b) matrix in terms of J , one substitutes N=J+% and N=J-5$ respect ive ly , f o r the two N values. The Case (b) matrix becomes \ r ,2 17 B 0 0 J 2 + 2J -1 4 Now, solving for the matrix _S , one obtains fo r the lower 2 17 eigenvalue, B(J — t h e fo l lowing simultaneous equations ( J+%) 2 -2 - J 2 +^ -/(J+Js)Z-4 ( J + % ) 2 - 6 - J 2 ^ Y 1 ' H 2 _ _ Therefore: .-1 '22 2(J+h) 2(J+3s) For the upper eigenvalue, B(J +2J—^) one s i m i l a r l y obtains '21 --1 >11 2(J+35) 2(J+^) L2'(0-^)J 2(J+%)J *3 Since the hamiltonian was set up in a molecule-f ixed axis system, care must be taken with the phases of the resu l t s when applying them to a space f ixed axis system due to the previously mentioned anomalous sign of i in the commutation re lat ionsh ips of J . This would be important in doing in tens i t y ca lcu lat ions for the branches of a band involving such a state i^see Hougen (39)]. Transforming the ent i re Case (a) matrix resu l t s in the Case (b) matrix shown in Table(9). The y ' terms have been omitted from the Case (b) matrix because they are l i k e l y to be neg l i g i b l e . There are terms o f f diagonal in y in the Case (b) matrix that do not normally appear in the Case (b) formalism. These terms have been retained because the sp in- rotat ion in teract ion in the Case (b) hamiltonian i s usual ly wr i t ten H S-R (31) This formalism i s only good for z v ibronic s tates , where N z i s zero, and in the above case N z i s d e f i n i t e l y non-zero. This i s discussed by Freed (40) and Carrington, et.. a l . (41). E E lectron ic States The formulae fo r E e lect ron ic states are of in teres t because 3 data are ava i lab le fo r the zero point levels of molecules in z states (NCN, HCCN, C 2 N 2 ) , but as yet no bands involving degenerate v ib rat iona l levels have been analyzed. One can ca lcu late energy leve l patterns fo r v 2=l,2 and determine what form the predicted energy leve l pattern w i l l take. In t r i p l e t states therL are ef fects from A, the spin-spin i n te rac t i on , which are not present i n the doublet s tates. As in doublet s tates , the v ibrat iona l angular momentum w i l l upset the normal spin s p l i t t i n g pattern un t i l the rotat ion of the molecule quenches i t by uncoupling i t from the ax i s . For the v 2=l leve l (vi<=0) in a E e lec t ron ic s ta te , one obtains from the Wang transformation two 3 x 3 matrices. These are shown in Table( lO). To determine the corresponding Case (b) matrices one proceeds in the same manner as fo r the v2=2 level i n a E e lect ron ic s tate. The re su l t i s given in Table(lO) below the Case (a) matrices. For v2=2 the Wang transformation y ie ld s a 4 x 4 and a 5 x 5 matr ix, shown in T a b l e ( l l ) . The (-) signs in the matrix correspond to the 4 x 4 matrix and the (+) signs to the 5 x 5 matr ix. The 4 x 4 matrix i s comprised of the 3 3 three A states and the E-j s tate. The very complicated Case (b) matrix i s given below the Case (a) matr ix. A computer program wr i t ten by Merer was used to diagonalize the Case (a) matrices for v,=1,2.for the fol lowing cases of i n te re s t : the a 3 z u + state of C 2 N 2 (42), the X3z~ state of NCN (43), and the 3z states involved in the 3200$ t r an s i t i on of HCCN (4). There are published values for B, x, and y fo r the v=0 levels of C 2 N 2 and NCN. There i s no information for g, but q can be ca lcu la ted. Data fo r HCCN are l im i t ed . The value Ixl = 0.43 cm - 1 for the ground state of HCCN i s obtainable from the ESR spectrum (44). The B value i s approximately 0.36 c m - 1 . If the sp in- rotat ion constant i s small (0.001),x in the upper state i s estimated as being 1.7 cm - 1 or 0.8 c m - 1 , depending on the sign of x in the ground s t a te , from the observed head-head separations. Figures (14 a,b,c) show the resu l t s of these ca lcu lat ions although no attempt i s made to show a-type doubling e f f ec t s . The term B[N(N+l)-£ ] has been substracted from the energy. Examination of f i g s . (14 a,b,c) reveals that the n and 3 3 A v ibronic l eve l s do not have the same form as the z v ibronic l e ve l s . I t i s also seen that , depending on the r e l a t i ve sizes 3 3 3 of B, x, and y» there must be a convergence of n, A, and z 3 v ibronic states at high N values. However, the normal z energy leve l s are not conformed to at low N values. In C 2 N 2 where X and B are of the same order of magnitude and Y i s large, 3 3 3 the patterns f o r A, n, and z converge very quickly and only 3 for low values of N w i l l any deviat ion from the normal z pattern be seen. The opposite i s observed in HCCN where there i s 3 3 3 quite a large dev iat ion between n, A , and z v ibronic l e ve l s . As mentioned previous ly, the £-type doubling i s not shown in f i g s . (14 a ,b ,c ) , although i t was ca lcu lated for the v,=l level of HCCN. At high N values the normal i - t ype doubling pattern for s i ng le t s i s followed given by the equation Av = qN(N+l) (32) but at low N values the £-type doubling pattern i s e r r a t i c . With q set at 0.003 cnf^, the theoret ica l £-type doubling was ca lcu lated and plotted in f i g . (15) with the deviat ion from eq. (32) as a funct ion of N. In the analys is of actual bands (yet to be done), it w i l l be 3 found that the normal E combination relations,based on the 3 formulae f o r E leve l s given by M i l l e r and Townes (45), used for evaluating the parameters A and y w i l l break down as the rotat iona l branches are followed back to the band o r i g i n . This can be i l l u s t r a t e d by the fol lowing re la t i on sh ip , F 1 (N- l)-F 2 (N-l)+F 3 (N+l)-F 2 (N+l)=-(2x- Y ) (33) where,as N decreases,the l e f t hand side of the equation w i l l become 3 smaller than the r i gh t hand s ide. For N=10 in the A v ibronic leve l of the given state of HCCN,the d i f ference w i l l be nearly 0.4 cm~^. Also,the i n ten s i t i e s w i l l probably not fo l low the normal 3 E pattern and i t may be necessary to examine the form of the i n t e n s i t i e s to analyze the bands. Elements o f f diagonal in the v ibrat iona l quantum 3 numbers for the v 2 = 0 level in z states < 1,-1,0,0,-2,-1 | < 1 , - l , 0 , J , - l ,0| < 1 ,-1,0,J,0,1 | < 1,1,0,J,0,-1| < 1,1,0,J,1,0| < 1,1,0,J,2,1| |000J-1-1: |000J00= |000J11: •Btf ^2j(J+l)-4 2B?$ -Be$ /2T(J+TT 2Bc$ -Bs$ /2J(J+1) Bc<& /2J(J+1) -2B C$ Be* /2T{J+l7 -2Bs$ Bc$ /2JCJ+D-4 Table 6 Ef fect ive Operators Ar i s ing from Second-Order Transformation of the Hamiltonian Selection Rules Term A£ AP A E Meaning B 2 ( J + J_G_G + + J_J +G +G_)/Av 0 0 - Diagonal a and g 2 2 term B 2 (S + S G G + + S S +G +G_)/Av 0 - 0 Diagonal a term resu l t ing from spin -B 2(J_S +G +G_ + J +S_G_S +)/Av 0 ±1 ±1 V ibrat ional Y term -B 2 ( J + S + G_ 2 + J_S_G + 2)/Av ±2 ±1 TI AP = ± 1 A-type doubling a l l m u l t i p l i c i t i e s B 2 ( J + 2 G 2 + J 2 G + 2 ) / A v ±2 ±2 , - \ Normal £-type doubling for s ing lets . B 2 ( S + 2 G 2 + S 2 G + 2 ) / A v ±2 - +-Z ' "Diagonal"£ -type doubling for t r i p l e t s Av = (w0 ± ai.-) i CJl i Table 7 Matrix Elements of the Ef fect ive Operators Given in Table 6» in Harmonic Approximation <v 2J« E | H | v 2 J , = g* 2- %[J(J + l ) -U +E ) 2 +S(S + l ) -x 2 ] ~ 1 , _ <v 2 J£ z IHI v 2J. I ± 2 Z > = ^q/v 2T£)(v 2±£+2)/J(J+l )-(fc+E) (£+E±l )/J.(J+l )-(£+Z±l)(£+Z±2) <v 2J£ Z ± 2|H| v 2 J£ ± 2 Z > = lq/ev 2T£)(v 2±£+2)/S(S+l)-Z(Z±l)/S(S+l)-(Z±l)(Z±2) <V2J£ Z ± 1|H| V 2J£ ± 2 Z > = -35q/(v2+£)(v2±£+2)/J(J+l )-( £+E±l) (£+Z±2)/S(S+l )-Z(Z±l ) <V2J£ Z ± 1|H| V 2J£ Z > = c^/J(J+l)-(£+Z)(£+Z±l)/S(S+l )-Z(Z± l ) where / 2 2 2 2\ / 2 2 \ q = -4B 2 I (hk "k i 2 k _ ^ k _ \ _ 2B 2 f l + 4 I ?2k_^2_ \ , i . e . such that £-type doubling e k/2 ' I ' — s - I k^2 V V k w 2 + ( \ / "2 \ u 2 - u 2 k £ of the v 2 = 1 level of a ^z e lectron ic state is given by Av = qJ(J+l) , / 2 2 2 2 \ = " 2 B ! £ / ^ 2 k " k V2 " C 2 k *k ^ v 2 + 2 ^ \ i . e . such that the e f fec t i ve B-value fo r the level v 2 i s given by ( B f i - £ v ) , and g = 2B 2 ^ /iz^a* •+ ? 2 k 2 * k 2 \ * T h e q u a n t i t y ^ h a s b e e n s u P P r e s - s e d h e r e -2 w2 "2 "k Matrix fo r v 2 = 1 for V E lectronic States Correct to Second Order \l,h> » ' 2 5 |1.¥> \1,-h> |-1.%> B 1[J(J+1)-J ] -«SY , -%Y - ( B r J r r ) /J(J+1 )-J Symmetric B ^ J C J + l ) ^ ] - ^ -2 q/[J(J+l)+^][J(J+l)-f]p) - q / J(J+1) B^JtJ+ l ) -^ ] frl/ [J(J+D-^][J(J+1)-f] - (B r35Y ) / j ( J+ l ) - | -Table 9 Matrices fo r v2=2 in a  cl E lectronic State Case (a) representation 12 ' : 2 A 5| < 2 A 31 2 A 5/2 > '^3/2* 2 >-2a)2+4g22+B2[(J+Jsr-6]-35Y ( ^ - B ^ U ^ ) - 4 ] 2 V ^ K J + J j ) - l ] [ ( J % ) - 4 ] } 2 -2o)2+4g22+B2[(J+35)2-2] q { 2 [ ( J + % ) 2 - l ] } 1 / 2 { ± % ( j + % ) - l } symm 2u)2±(%y-B2) (J+3S)+B2(J+3S) Z-%Y The basis functions are the 'sum' functions of the Wang transformation fo r the upper signs in the th i r d column, and ' d i f fe rence ' functions fo r the lower signs; thus i f the plus signs are taken, i -the basis functions are of the type i 2 A 5 / 2 > = {\ i = 2, E = h>J> + U = - 2 , E = J>} For the z vibronic state th i s implies the F, (N=J-%) rotat iona l leve l s f o r the upper signs, and the F 2 (N=J+Jf) rotat ional levels for the lower signs. Case, (b) representation •A(F2)> •A(F1)> " A ( F o ) | -A(F-,)i 2o)2+4g22+[B2-Y/2(J+35)][(J+3s)(J+-|)-4] Y [ ( J - f ) ( J + f ) ^ / ( J + J s ) 2a)2+4g22+[B2+Y/2(J+%)] [(j-J2)(j+35)-4] symm Upper s ign: 0 3 Lower s ign: -ql(J-h) (J+35)(J+2) Upper s ign: q[(J-|)(J-3j)(J+%) ( J ^ Lower s ign: 0 2a)2+B2(j+%)(j+J5 ± l)±32Y(J+35+l) I CTi I Again fo r the £ s tate, the upper and lower signs correspond to F , and F 9 re spect i ve ly , so that the 2 2 2 2 non-vanishing off-diagonal elements occur only between A(F^)ahd z (F- j ) or between A ( F 2 ) and z ( F 2 ) . The terms in y'-have been omitted (see tex t ) . — " Case (a) representation | 3 V i 3v <3n2 | ±Jsqyj(J+l)[J(J+l)-2] symm B1 [J(J+1 )+l]-2Y±3sqJ(J+l) (SsY-B^q/ZJCJ+l) -Y , +B 1 [J (J+ l )+ l ]+2X- Y ±q 2 + 9 22 ~3~X ^ a s " 5 e e n S U D S t r a c t e c l f r o m t n e diagonal elements. As in Table 9, the basis functions are the Wang 'sum' or ' d i f f e rence ' functions for the upper and lower s igns, respect ive ly , fo r the q terms in the body of the table. Case (b) representation <=J+1(F3)> (N=J(F2): |N=J-1(F1)> cN=J+l <N=J! <N=J-1| -|x(J 2+3J-l)/(J+l)(2J+l) symm CB1±3sq-Y/(J+l)][J^+3J+l]±3sq -( YJ+2x)^/(J+ l ) v ^T+T -2A/(J-1 ) (J+2)/(2J+l) [B%)- Y/J(J+1 ) ] [ J +J-1] [Y(J+1 ) - 2X ]/JT/JV^J+T + J2q+|x[l-3/J(J+1)] [Bi^q+y/JJCJ -J- l ]±%q -|x (J 2 -J -3 )/ J(2J+1) o>2 + n a s been subtracted from the diagonal elements, and the terms involv ing Y 1 have been omitted. Table 11 Case (a) and Case (b) matrices for the v 2=2 level in E s ta te . (V,3A) < 3 A J | 3 A ? > < 3 A . < 3 A , | i3v 4g 2 2+B 2[J(J+l)-8]+2x- Y (J5Y-B2)/2J(J+1 )-\Z (q//?)/J(J+l)-2^(J+l)-6 4g 2 2+B 2[J(J+l)-2]-2 Y (35Y-B2V2J(J+1 )-4 -2qv/J(J+1 )-2 4g 2 2+B 2J( J+l )+2x-Y qv^di^JXJ+T)} B2J(J+1)+2X-Y -2q/2J(J+1 symm 2 ( ¥ r B 2 ) v O T d + T T i B 2 [J(J+l)+2]-2 Y ? In Table 9 the basis functions are Wang 'sum' functions for the upper 4 signs and ' d i f f e rence ' functions fo r the lower s i gns ; in the v 2 = 2 matrices, the s ingle | ZQ> function must be counted with the 'sum' functions 4 g iv ing a 5 x 5 matrix, whereas the 'd i f ference functions give a 4 x 4 matrix. The quantity -jX has been subtracted from a l l diagonal elements. =J+1> Case (b) MATRIX h |N=J> |N=J-1: 4x4:-5x5:' N=J> N=J+1> — — — — 4 X 4 ; - Q~ 2 ^ 4 4 g 2 2 + C B ^ J + 1 ) ] [ ( J + l ) 2JJ44X / j - 1 ) ( J + 3 ) 2 X _ /(J-l)(J-2)(J+2)(J+3) 5 x5 : —q>/J(J +1 ) (J + 2)(J +3) o / i o \ f i_i_c\ J+l / J(2J+1) 2J+1/ J(J+1) j N=J-1: <N=J+1 =N=J j <N=J-1| 4x4 : <N=J| 5x5: <N=J+1 <N=J-l'| ( J + 2 ) - 4 ] - | x ( J - 2 ) ( J + 5 ) 6 (J+1)(2J+1) symm 2co +4q A n - ] 2 i i 2Y (J+1) -4X/(J + 2) (J -2 ) 2 c o 2 + 4 g 2 2 ^ X L l j / ( J + 1 ) ( 2 J + i : +L B -jVry^ J( J + 1)-4] 4x4:--q/J(J+l)(J-l)(J+2) 5x5:- 0 oo 2 o ) 2 + 4 g 2 2 - | x ( ^ H i ) 1 u J J(2J+1) + (B+^f.)(J2-J-4) 4x4:- 0 5x5:- 0 q/T[jT /(J-2; 4x4:- 2W2+BJ(J+1)+|X-Y 5x5:- B(J+l)(J+2)+2co 2 2J+T -2X : Y ( j + 2 ) 2 x ( ± t2_ ) J 2J+1 The 5x5 The 4x4 matrix (Wang 'sum') i s Kronig ' - ' for J even matrix (Wang 'd i f ference) i s Kronig '+' fo r J even and vice versa fo r J odd /J(J+I) BJ(J-1 T J - f x . M + 2J+1 2to 0 ^Energy I CD ( c cu +-> + J c -0.2 h •0.4 -0.6 - -0.8 5 -1.0 I •1.2 •1.4 10 12 14 16 18 20 N 1 5 N 1 0 Fig. 15 Departure of the fc-type doubling of a hypothetical state of HCCN, with B = 0.36, X = 1.7, y = -0.001 and q = 0.003 c m ' 1 , from the case (b) formula Av = qN(N+l) for the v ibronic n level (v» = 1). 1. W.W. Watson and A.E. Parker, Phys. Rev. 37, 1484 (1930). J.H. Clements, Phys. Rev. 47, 224 (1935). R.K. Asundi and R. Samuel, Proc. Ind. Acad. S c i . 2A, 30 (1935). 2. N. Metropolis and H. Beut ler, Phys. Rev. 57, 1078 (1940). N. Metropol i s , Phys. Rev. 60, 283 (1941). ' 3. G. Herzberg, E lectron ic Spectra of Polyatomic Molecules, Van Nostrand, Princeton, New Jersey, U.S.A., pg. 75 (1966). 4. A . J . Merer and D.N. Trav i s , Canad. J . Phys. 44, 353 (1966). 5. P.W. Schenk, Ber. 65, 94 (1942). 6. M. Becke-Goehring, R. Schwarz, and W. Spress, Z. Anorg. A l l g . Chem. 293, 294 (1958). 7. W.H. K i rchhoff , J.A.C.S. 90. 2437 (1969). 8. G. Brauer, Preparative Methods of Inorganic Chemistry, Academic Press, New York, 1963. 9. A . J . Barnes and H.E. Hallam, Quarterly Rev. 2_3, 392 (1969). 10. P.W. Schenk, E. Krow, and H. Kartano-Soeratman, Monatsh 95, 710 (1964). 11. J . White, J . Opt. Soc. Am. 32, 285 (1942). 12. H. R ichert , Z. Anorg. A l l g . Chem. 309, 171 (1961). 13. J .H. Schachtschneider, Spectrochim, Acta.'l_9_, 117 (1963). 14. E.B. Wilson, J.C. Decius, and P.C. Cross, Molecular V ibrat ions, McGraw-Hil l, New York. 1955. 15. G.M. Barrow, Introduction to Molecular Spectroscopy, McGraw-Hil l, New York, 233 (1962). 16. E. Hutchisson, Phys. Rev. 36, 410 (1930). 17. M. Wagner, Z. Naturf. 14a, 81 (1959). 18. F. Ansbacher, Z. Naturf. 14a, 889 (1959). j 19. W.L. Smith and P.A. Warsop,-Trans. Far. Soc. 65, 1165 (1968). 20. J .B. Coon, R.E. De Wames, and, CM , Lloyd, J . Molec. Spectroscopy 8, 285 (1962). 21. W.L. Smith, J . Phys. B. Proc. Phys. S o c , 2, 1 (1969). 22. R.N. Dixon, pr ivate communication. 23. G. Herzberg, E lectron ic Spectra of Polyatomic Molecules, Van Nostrand, Pr inceton, New Jersey, U.S.A. (1966). 24. R.T. B i rge, Phys. Rev. 25, 240 (1925). 25. A . J . Merer, Disc. Far. Soc. 35_, 127 (1963). 26. R.M. Hochstrasser and A.P. Marchett i , J . Mol. Spec. 35, 335 (1970). 27. J .H. Van Vleck, Rev. Mod. Phys. 23, 213 (1951). 28. J.K.G. Watson, Canad. J . Phys. 46, 1637 (1968). 29. C T F . Chang and Y-N. Chiu, J . Chem. Phys. 53, 2186 (1970). 30. J.T. Hougen, The Calculat ion of Rotational Energy Levels and Line Intens i t ies in Diatomic Molecules. N.B.S. Monograph HS 115, Wash. D.C. U.S.A. 31. J .H. Meal and S.R. Polo, J . Chem. Phys. 24, 1119 (1956). 32. E.B. Wilson, J . C Decius, and P.C Cross, Molecular V ibrat ions, Appendix III McGraw-Hil l, New York, 1955. 33. W. M o f f i t t and A.D. L iehr , Phys. Rev. 106, 1195 (1957). 34. I.M. M i l l s , Pure and App. Chem. V\_, 325 (1965). 35. J.T. Hougen, J . Chem. PHys. 36, 519 (1962). 36. E.C. Kemble, The Fundamentals of Quantum Mechanics, Dover Pub l i cat ions , New York, U.S.A. (1958). j 37. S.C. Wang, Phys. Rev. 34, 243 (1929). ^ 38. J.W.C. Johns, Canad. J . Phys. 39, 1739 (1961). 39. J.T. Hougen, Canad. J . Phys. 42, 433 (1964). 40. K.F. Freed, J . Chem. Phys. 45, 4214 (1966). 41. A. Carrington, B.J. Howard, D.H. Levy and J.C. Robertson, Mol. Phys. 15, 187 (1968). 42. J .H. Callomon and A.B. Davey, Proc. Phys. Soc. 83, 335 (1963) 43. G. Herzberg and D.N. Trav i s , Canad. J . Phys. 43, 353 (1966)!. 44. R.A. Bernheim, R.J. Kempf, J.V. Gramas and P.S. S k e l l , J . Chem. Phys. 43, 196 (1965). 45. S.L. M i l l e r and C.H. Townes, Phys. Rev. 90, 537 (1953). Normal Coordinate Analysis To s o l v e t h e v i b r a t i o n a l p r o b l e m i n p o l y a t o m i c m o l e c u l e s , one b e g i n s by w r i t i n g down t h e h a m i l t o n i a n u s i n g c a r t e s i a n d i s p l a c e m e n t c o o r d i n a t e s f o r t h e i n d i v i d u a l atoms ( A X , , A y , , A z , , A X 2 , A y 2 , . . . A z n ) in matrix form H = T + V =h X + r M X + (1) Here M i s a 3N x 3N (N = number of atoms in the molecule) matrix with the fo l lowing diagonal form M Since e q . ( l ) i s .unwie ld ly because of the form of the potent ia l energy, one introduces normal coordinates, Q^, which one defines as f u l f i l l i n g the fol lowing re la t i onsh ip . = y i + r E Q + Q + r A Q >\j r\j % 'Xi 'Xi 'Xi In th i s equation i s the momentum associated with the normal coordinate, Q. , and A i s the diagonal matrix of the eigenvalues corresponding to the observed frequencies ( A ^ = 4TT c, in cm" 1 ) . Notice that k i s an index for the normal coordinates. Now we define the transformation matrix B that transforms the vector of the cartes ian displacement coordinates ( X ) into the vector of the internal displacement coordinates (S) (changes in bond angles and bond lengths). S = B X (4) Now we define the matrix G such that G = B M" 1 B + r (5) and the matrix L such that Now we define the matrix A such that 'X, The matrix A i s introduced only because B , not being square, lacks a genuine inverse. Now X = A S (8) and then therefore G G" 1 = B M" 1 B + r G " 1 = E (9) A = M" 1 B + r G" 1 ' (10) and X = M" 1 B + r G" 1 S (11) Now subst i tut ing eq. (11) into the f i r s t part of eq. (1) one gets 2T = S t r G ' 1 S (12) Now using eq. (6) one gets H = h Q + r L + r G" 1 L Q + h Qtr L t r F L Q (13) and therefore L + r G" 1 L = E and L ' + r F L = A 'Xj <\j <\, '\j 'Xj <\j <\j 'Xj or as usually written G F L = LA % f\, *\> *\j fXt In p r i nc ip l e the unknown F matrix could be calculated from the . G matrix and the observed frequencies (A); however, th i s i s computationally d i f f i c u l t . Instead a t r i a l force constant matrix i s subst i tuted, the calculated and observed frequencies are compared, and an appropriate adjustment i s made to the t r i a l force constant matrix. This process i s repeated u n t i l a sa t i s fac tory agreement between the calculated and observed frequencies i s obtained. The Wang Transformation i s a s i m i l a r i t y transformation that w i l l break down matrices that are symmetric about both diagonals (doubly symmetric) into smaller sub-matrices. I t i s equivalent to wr i t ing the matrix in terms of a new set of basis functions which are sums and differences of the old basis functions. That i s , the Wang Transformation i s such that S H' S" 1 = H ( 1 ) where H' i s doubly symmetric and H consists of two smaller matrices. One should note that the important thing for th i s thesis i s that the pre-transformed basis functions |£,E> are neither symmetric nor antisymmetric under the symmetry operation, a y , but the "Wanged" basis funct ions, which are now sums and differences of the old basis funct ions, have the required symmetry propert ies. The Wang Transformation i s shown below for both a 3 x 3 and 4 x matrix. 1 ft 1 0 1 0 ft 0 -1 0 1 A - D 0 A C D C B C D C A 0 0 B ft C 0 ft C A + D 1 ft [A - D] + 1 0 -1 0 ft 0 1 0 0 B ft Z ft C A + D 4 x 4 1 1 0 0 1 A C E F ft 0 1 1 0 C B D E 0 -1 1 0 E D B C -1 0 0 1 F E C A A - F C - E 0 0 C - E B - D 0 0 -0 0 B + D - C + E 0 0 C + E A + f 1 ft 1 0 0 -1 0 1 -1 0 0 1 1 0 1 0 0 1 A ± F C ± E C ± E B ± D 

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