comparison of different vmi fitting formulas/procedures

Comparison of different VMI fitting

formulas/proceduresHRH – 29/7/14

• 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+b2P2(cosq)

• 2) I=(1+b2fP2(cos ))q (1+b2phP2(cosq)+b4phP4(cosq)), b2f=-0,621

• 3) I =(1+b2fP2(cos ))q (1+b2phP2(cosq)+b4phP4(cosq)), b2ph=2

• 4) I=(1+b2fP2(cosq)+b4P4f(cosq)) (1+b2phP2(cosq)+b4P4ph(cosq)), b2f=-0,621

• 5) I=(1+b2fP2(cosq)+b4P4f(cosq)) (1+b2phP2(cosq)+b4P4ph(cosq)), b2ph=2

Peak D – fit 1

I=1+b2P2(cosq)

Peak G – fit 1

I=1+b2P2(cosq)

• 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.

J‘ b2 (Peak D) Db2 b2 (Peak G) Db2

1 1,736 0,0438 1,4957 0,06212 1,7794 0,0686 1,3848 0,0443 1,285 0,121 0,95699 0,09854 1,3288 0,0596 0,75156 0,0835 1,3399 0,163 0,96959 0,07296 1,3663 0,108 0,86906 0,1047 1,1234 0,114 0,82279 0,1258 1,264 0,115 0,93651 0,1039 0,94178 0,128 0,82183 0,104

Peak D – fit 2

I=(1+b2fP2(cos ))q (1+b2phP2(cosq)+b4phP4(cosq)), b2f=-0,621

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

Peak G – fit 2

I=(1+b2fP2(cos ))q (1+b2phP2(cosq)+b4phP4(cosq)), b2f=-0,621

• 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 b2

ph parameter, had to be held constant at 2, so the fits may potentially be better if the b2

f parameter would be a little higher, e.g. -0,5.• Therefore, a second fitting procedure with the same formula is

performed, only the b2ph parameter is held constant at 2, while the b2

f parameter is fitted for in order to asses uncertainties in the b2

f parameter.

J‘ b2ph(D) Db2

ph b2ph(G) Db2

ph b4ph(D) Db4

ph b4ph(G) Db4

ph

1 2 0,4 2 0,3 0,66733 0,0921 0,80455 0,09572 2 0,5 2 0,2 0,85066 0,104 0,76978 0,07383 2 0,1 1,8209 0,0466 0,38389 0,0782 0,19826 0,05714 2 0,1 1,6595 0,0778 0,65814 0,05 0,31288 0,09425 2 0,2 1,8662 0,0564 0,61404 0,133 0,48108 0,06656 2 0,2 1,7319 0,0777 0,69863 0,107 0,20313 0,09657 1,9693 0,09 1,6239 0,0601 0,65144 0,106 -0,07881 0,07988 2 0,1 1,7294 0,0575 1,0183 0,105 0,093789 0,0759 1,8054 0,114 1,7059 0,112 0,78424 0,143 0,40065 0,135

Peak D – fit 3

I =(1+b2fP2(cos ))q (1+b2phP2(cosq)+b4phP4(cosq)), b2ph=2

Peak G – fit 3

I =(1+b2fP2(cos ))q (1+b2phP2(cosq)+b4phP4(cosq)), b2ph=2

• 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 b2

ph parameter is held constant at 2. Also, this supports the theory that b2

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.

J‘ b2f (D) Db2

f b2f(G) Db2

f b4ph(D) Db4

ph b4ph(G) Db4

ph

1 -0,28962 0,0329 -0,38481 0,0483 0,27967 0,0558 0,50603 0,09692 -0,27523 0,0495 -0,46083 0,0333 0,39473 0,0866 0,55499 0,07393 -0,51094 0,0481 -0,67907 0,0263 0,17107 0,121 0,35664 0,0794 -0,53672 0,0228 -0,67928 0,0429 0,52906 0,0559 0,51031 0,1455 -0,51832 0,0782 -0,55389 0,0904 0,42582 0,201 0,55389 0,09046 -0,48471 0,0634 -0,68386 0,044 0,50461 0,145 0,39142 0,1337 -0,63412 0,0436 -0,78454 0,0323 0,68301 0,124 0,32558 0,09638 -0,5415 0,046 -0,71418 0,0364 0,8816 0,135 0,33409 0,09969 -0,68071 0,0485 -0,61516 0,0681 0,94286 0,151 0,46706 0,205

• The average of the fitted values for b2f is -0.50±0.14(standard

deviation). The previously calculated REMPI value of b2f is -0.621, so it

falls just inside the standard deviation of the fitted values. • Using the value of b2

f is -0.621 is therefore totally justifiable for the next fitting procedures where the b4

f parameter is added to increase the quality of the fits themselves.

Peak D – fit 4

I=(1+b2fP2(cosq)+b4P4f(cosq)) (1+b2phP2(cosq)+b4P4ph(cosq)), b2f=-0,621

Peak G – fit 4

I=(1+b2fP2(cosq)+b4P4f(cosq)) (1+b2phP2(cosq)+b4P4ph(cosq)), b2f=-0,621

• As expected, upon addition of the b4f 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

b2f parameter, a last fitting procedure is performed, where the b2

ph

parameter is held constant at 2, in order to assess the b2f parameter,

and compare with the results from fitting procedure #3.

J‘ b2ph(D) Db2

ph b2ph(G) Db2

ph b4ph (D) Db4

ph b4ph(G) Db4

ph b4f (D) Db4

f b4f (G) Db4

f

1 2 0,3 2,0033 0,0574 0,31722 0,0944 0,3315 0,0975 0,2231 0,0431 0,2423 0,0379

2 2 0,4 1,9815 0,0465 0,42925 0,115 0,40623 0,078 0,2107 0,0434 0,18096 0,0291

3 2,0581 0,0821 1,7373 0,0559 0,22512 0,132 0,004815 0,0978 0,071208 0,0404 0,086301 0,0384

4 2 0,1 1,3865 0,0814 0,52362 0,0612 -0,32186 0,162 0,066419 0,021 0,29685 0,0819

5 2 0,1 1,7627 0,0661 0,44456 0,193 0,24716 0,112 0,070767 0,0576 0,08921 0,0376

6 2,0174 0,109 1,4352 0,0859 0,4414 0,179 -0,50398 0,169 0,13451 0,0639 0,35508 0,094

7 1,9663 0,109 1,5345 0,0765 0,64474 0,173 -0,29621 0,144 0,002815 0,0555 0,12284 0,0732

8 1,9921 0,112 1,5614 0,0583 0,81411 0,174 -0,30951 0,111 0,076525 0,0458 0,23343 0,0582

9 1,7519 0,142 1,2646 0,0888 0,6697 0,234 -0,66099 0,186 0,052254 0,0822 0,56668 0,127

Peak D – fit 5

I=(1+b2fP2(cosq)+b4P4f(cosq)) (1+b2phP2(cosq)+b4P4ph(cosq)), b2ph=2

Peak G – fit 5

I=(1+b2fP2(cosq)+b4P4f(cosq)) (1+b2phP2(cosq)+b4P4ph(cosq)), b2ph=2

• Again, the fits are very good although the assumption that b2ph =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 b2

f parameter, solely from the D peak, as we did with the results from the 3rd fitting procedure.

J‘ b2f (D) Db2

f b2f (G) Db2

f b4f (D) Db4

f b4f(G) Db4

f b4ph (D) Db4

ph b4ph(G) Db4

ph

1 -0,21128 0,00491 -0,5899 0,0533 -0,10205 0,0461 0,22174 0,0464 0,3507 0,0624 0,33117 0,0763

2 -0,14125 0,0786 -0,6153 0,0421 -0,1568 0,0648 0,16884 0,0383 0,53006 0,0968 0,42822 0,061

3 -0,57352 0,0822 -0,81928 0,0391 0,056482 0,066 0,15737 0,0392 0,15732 0,117 0,28638 0,0611

4 -0,52464 0,0451 -1,0146 0,0387 -0,0134 0,0423 0,33497 0,0374 0,53768 0,0628 0,35472 0,0686

5 -0,50259 0,159 -0,79558 0,047 -0,01483 0,132 0,15652 0,0432 0,43238 0,218 0,46425 0,0741

6 -0,60582 0,1 -0,93997 0,051 0,12935 0,0918 0,28921 0,052 0,4244 0,144 0,24457 0,0849

7 -0,65523 0,0924 -0,93937 0,0442 0,02394 0,0933 0,21515 0,0538 0,66815 0,138 0,23746 0,0722

8 -0,61602 0,0954 -0,92887 0,0345 0,070867 0,0814 0,28525 0,0401 0,82272 0,146 0,21492 0,0559

9 -0,83022 0,0903 -1 0,1 0,17213 0,0967 0,38531 0,0264 0,82612 0,153 0,25587 0,09

• The averaged values of the fitted b2f 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 b2

f =-0.621 is justifiable in the fits including b4f for improved fitting

curves.