1. Comparison of different VMI fitting formulas/procedures HRH – 29/7/14

2. This is a comparison between different fitting procedures of anisotopy parameters for peaks D & G in the E(0) state of HBr. 1) I=1+  2 P 2 (cos  ) 2) I=(1+  2f P 2 (cos  (1+  2ph P 2 (cos  )+  4ph P 4 (cos  )),  2f =-0,621 3) I =(1+  2f P 2 (cos  (1+  2ph P 2 (cos  )+  4ph P 4 (cos  )),  2ph =2 4) I=(1+  2f P 2 (cos  )+  4 P 4f (cos  )) (1+  2ph P 2 (cos  )+  4 P 4ph (cos  )),  2f =-0,621 5) I=(1+  2f P 2 (cos  )+  4 P 4f (cos  )) (1+  2ph P 2 (cos  )+  4 P 4ph (cos  )),  2ph =2

3. Peak D – fit 1 I=1+  2 P 2 (cos  )

5. Peak G – fit 1 I=1+  2 P 2 (cos  )

7. As we can see from the first fitting formula, the fit is okay for the low J‘s, namely J‘=1, 2. However, the fit gets progressively worse with increasing J‘s. We can assume that the fitting formula may therefore be alright for transitions that are solely parallel in character, but with perpendicular increments in the nature of the transition, the fit gets worse.

8. J‘  2 (Peak D)  2  2 (Peak G)  2 11,7360,04381,49570,0621 21,77940,06861,38480,044 31,2850,1210,956990,0985 41,32880,05960,751560,083 51,33990,1630,969590,0729 61,36630,1080,869060,104 71,12340,1140,822790,125 81,2640,1150,936510,103 90,941780,1280,821830,104

9. Peak D – fit 2 I=(1+  2f P 2 (cos  (1+  2ph P 2 (cos  )+  4ph P 4 (cos  )),  2f =-0,621

11. Nota bene: The „f“ & „ph“ labellings were accidentally switched in the figures.

12. Peak G – fit 2 I=(1+  2f P 2 (cos  (1+  2ph P 2 (cos  )+  4ph P 4 (cos  )),  2f =-0,621

14. As opposed to the first fit. This fit crumbles a bit for J‘=1,2. However, for the the higher J‘s, the fit becomes progressively better, again, as opposed to the first fit. However, in almost all cases for the D peak, the  2 ph parameter, had to be held constant at 2, so the fits may potentially be better if the  2 f parameter would be a little higher, e.g. -0,5. Therefore, a second fitting procedure with the same formula is performed, only the  2 ph parameter is held constant at 2, while the  2 f parameter is fitted for in order to asses uncertainties in the  2 f parameter.

15. J‘  2 ph (D)  2 ph  2 ph (G)  2 ph  4 ph (D)  4 ph  4 ph (G)  4 ph 120,420,30,667330,09210,804550,0957 220,520,20,850660,1040,769780,0738 320,11,82090,04660,383890,07820,198260,0571 420,11,65950,07780,658140,050,312880,0942 520,21,86620,05640,614040,1330,481080,0665 620,21,73190,07770,698630,1070,203130,0965 71,96930,091,62390,06010,651440,106-0,078810,0798 820,11,72940,05751,01830,1050,0937890,075 91,80540,1141,70590,1120,784240,1430,400650,135

16. Peak D – fit 3 I =(1+  2f P 2 (cos  (1+  2ph P 2 (cos  )+  4ph P 4 (cos  )),  2ph =2

18. Peak G – fit 3 I =(1+  2f P 2 (cos  (1+  2ph P 2 (cos  )+  4ph P 4 (cos  )),  2ph =2

20. The fits for the D peak give rather promising results. It may be indicative of a „true“ parallel nature of the D peak, where the  2 ph parameter is held constant at 2. Also, this supports the theory that  2 f may be a bit higher than -0.621. The fits for the G peak, are good for the low J‘s (where the G peak exhibits the greatest paralell nature), but they get worse for higher J‘s, which stands to reason because the G peak exhibits a greater blend of a parallel and perpendicular transition with increasing J‘s, which has already been established.

21. J‘  2 f (D)  2 f  2 f (G)  2 f  4 ph (D)  4 ph  4 ph (G)  4 ph 1-0,289620,0329-0,384810,04830,279670,05580,506030,0969 2-0,275230,0495-0,460830,03330,394730,08660,554990,0739 3-0,510940,0481-0,679070,02630,171070,1210,356640,079 4-0,536720,0228-0,679280,04290,529060,05590,510310,145 5-0,518320,0782-0,553890,09040,425820,2010,553890,0904 6-0,484710,0634-0,683860,0440,504610,1450,391420,133 7-0,634120,0436-0,784540,03230,683010,1240,325580,0963 8-0,54150,046-0,714180,03640,88160,1350,334090,0996 9-0,680710,0485-0,615160,06810,942860,1510,467060,205

22. The average of the fitted values for  2 f is -0.50±0.14(standard deviation). The previously calculated REMPI value of  2 f is -0.621, so it falls just inside the standard deviation of the fitted values. Using the value of  2 f is -0.621 is therefore totally justifiable for the next fitting procedures where the  4 f parameter is added to increase the quality of the fits themselves.

23. Peak D – fit 4 I=(1+  2f P 2 (cos  )+  4 P 4f (cos  )) (1+  2ph P 2 (cos  )+  4 P 4ph (cos  )),  2f =-0,621

25. Peak G – fit 4 I=(1+  2f P 2 (cos  )+  4 P 4f (cos  )) (1+  2ph P 2 (cos  )+  4 P 4ph (cos  )),  2f =-0,621

27. As expected, upon addition of the  4 f parameter, the fits have become exemplary. As before, the D peak exhibits a very pure parallel nature, while the G peak becomes more blended with increasing J‘s. To exemplify the justification of the the value of the REMPI calculated  2 f parameter, a last fitting procedure is performed, where the  2 ph parameter is held constant at 2, in order to assess the  2 f parameter, and compare with the results from fitting procedure #3.

28. J‘  2 ph (D)  2 ph  2 ph (G)  2 ph  4 ph (D)  4 ph  4 ph (G)  4 ph  4 f (D)  4 f  4 f (G)  4 f 120,32,00330,05740,317220,09440,33150,09750,22310,04310,24230,0379 220,41,98150,04650,429250,1150,406230,0780,21070,04340,180960,0291 32,05810,08211,73730,05590,225120,1320,0048150,09780,0712080,04040,0863010,0384 420,11,38650,08140,523620,0612-0,321860,1620,0664190,0210,296850,0819 520,11,76270,06610,444560,1930,247160,1120,0707670,05760,089210,0376 62,01740,1091,43520,08590,44140,179-0,503980,1690,134510,06390,355080,094 71,96630,1091,53450,07650,644740,173-0,296210,1440,0028150,05550,122840,0732 81,99210,1121,56140,05830,814110,174-0,309510,1110,0765250,04580,233430,0582 91,75190,1421,26460,08880,66970,234-0,660990,1860,0522540,08220,566680,127

29. Peak D – fit 5 I=(1+  2f P 2 (cos  )+  4 P 4f (cos  )) (1+  2ph P 2 (cos  )+  4 P 4ph (cos  )),  2ph =2

31. Peak G – fit 5 I=(1+  2f P 2 (cos  )+  4 P 4f (cos  )) (1+  2ph P 2 (cos  )+  4 P 4ph (cos  )),  2ph =2

33. Again, the fits are very good although the assumption that  2 ph =2, is not a very good assessment for all the J‘s in peak G. We will thusly calculate the average of the fitted values of the  2 f parameter, solely from the D peak, as we did with the results from the 3rd fitting procedure.

34. J‘  2 f (D)  2 f  2 f (G)  2 f  4 f (D)  4 f  4 f (G)  4 f  4 ph (D)  4 ph  4 ph (G)  4 ph 1-0,211280,00491-0,58990,0533-0,102050,04610,221740,04640,35070,06240,331170,0763 2-0,141250,0786-0,61530,0421-0,15680,06480,168840,03830,530060,09680,428220,061 3-0,573520,0822-0,819280,03910,0564820,0660,157370,03920,157320,1170,286380,0611 4-0,524640,0451-1,01460,0387-0,01340,04230,334970,03740,537680,06280,354720,0686 5-0,502590,159-0,795580,047-0,014830,1320,156520,04320,432380,2180,464250,0741 6-0,605820,1-0,939970,0510,129350,09180,289210,0520,42440,1440,244570,0849 7-0,655230,0924-0,939370,04420,023940,09330,215150,05380,668150,1380,237460,0722 8-0,616020,0954-0,928870,03450,0708670,08140,285250,04010,822720,1460,214920,0559 9-0,830220,09030,10,172130,09670,385310,02640,826120,1530,255870,09

35. The averaged values of the fitted  2 f values give -0,52±0.20, i.e. similar results as from the 3rd fitting, only with a larger standard deviation. We can therefore conclude that the use of the REMPI calculated value of  2 f =-0.621 is justifiable in the fits including  4 f for improved fitting curves.