HRH 29714 This is a comparison between different fitting procedures of anisotopy parameters for peaks D amp G in the E0 state of HBr 1 I1 b 2 P 2 cos q 2 I 1 b 2f ID: 690691
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Slide1
Comparison of different VMI fitting formulas/procedures
HRH – 29/7/14Slide2
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+
b
2
P
2
(cos
q
)
2)
I=(
1+
b
2f
P
2
(cos
q
))
(
1+
b
2ph
P
2
(cos
q
)+
b
4ph
P
4
(cos
q
)),
b
2f
=-0,621
3
) I =(1+
b
2f
P
2
(cos
q))
(1+
b
2ph
P
2
(cos
q
)+
b
4ph
P
4
(cos
q
)),
b
2ph
=2
4) I=(1+
b
2f
P
2
(cos
q
)+
b
4
P
4f
(cos
q
)) (1+
b
2ph
P
2
(cos
q
)+
b
4
P
4ph
(cos
q
)),
b
2f
=-0,621
5)
I=(
1+
b
2f
P
2
(cos
q
)+
b
4
P
4f
(cos
q
)) (
1+
b
2ph
P
2
(cos
q
)+
b
4
P
4ph
(cos
q
)),
b
2ph
=2Slide3
Peak D – fit 1
I=1+
b
2
P
2
(cos
q
)Slide4Slide5
Peak G – fit 1
I=1+
b
2
P
2
(cos
q
)Slide6Slide7
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.Slide8
J‘
b
2
(Peak D)
Db
2
b
2
(Peak G)
Db
2
1
1,736
0,0438
1,4957
0,0621
2
1,7794
0,0686
1,3848
0,044
3
1,285
0,121
0,95699
0,0985
4
1,3288
0,0596
0,75156
0,083
5
1,3399
0,163
0,96959
0,0729
6
1,3663
0,108
0,86906
0,104
7
1,1234
0,114
0,82279
0,125
8
1,264
0,115
0,93651
0,103
9
0,94178
0,128
0,82183
0,104Slide9
Peak D – fit 2
I=(1+
b
2f
P
2
(cos
q))
(1+b2ph
P2(cosq)+b
4phP4(cosq)),
b2f=-0,621Slide10Slide11
Nota bene: The „f“ & „ph“ labellings were accidentally switched in the figures.Slide12
Peak G
– fit 2
I=(1+
b
2f
P
2
(cos
q))(1+
b2phP2(cosq)+
b4phP4(cosq
)), b2f=-0,621Slide13Slide14
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
b
2
ph
parameter, had to be held constant at 2, so the fits may potentially be better if the
b
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 b2ph parameter is held constant at 2, while the
b2f parameter
is fitted for in order to asses uncertainties in the b2f parameter.Slide15
J‘
b
2
ph
(D)
Db
2
ph
b
2
ph(G)
Db
2ph
b4ph(D)
Db
4
ph
b
4
ph
(G)
Db
4
ph
1
2
0,4
2
0,3
0,66733
0,0921
0,80455
0,0957
2
2
0,5
2
0,2
0,85066
0,104
0,76978
0,0738
3
2
0,1
1,8209
0,0466
0,38389
0,0782
0,19826
0,0571
4
2
0,1
1,6595
0,0778
0,65814
0,05
0,31288
0,0942
5
2
0,2
1,8662
0,0564
0,61404
0,133
0,48108
0,0665
6
2
0,2
1,7319
0,0777
0,69863
0,107
0,20313
0,0965
7
1,9693
0,09
1,6239
0,0601
0,65144
0,106
-0,07881
0,0798
8
2
0,1
1,7294
0,0575
1,0183
0,105
0,093789
0,075
9
1,8054
0,114
1,7059
0,112
0,78424
0,143
0,40065
0,135Slide16
Peak D – fit 3
I =(1+
b
2f
P
2
(cos
q))
(1+b2ph
P2(cosq)+b
4phP4(cosq)),
b2ph=2Slide17Slide18
Peak G
– fit 3
I =(1+
b
2f
P
2
(cos
q))(1+
b2phP2(cosq)+
b4phP4(cosq
)), b2ph=2Slide19Slide20
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
b
2
ph
parameter is held constant at
2. Also, this supports the theory that
b
2f 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.Slide21
J‘
b
2
f
(D)
Db
2
f
b
2
f
(G)
Db2f
b4ph
(D)
Db
4
ph
b
4
ph
(G)
Db
4
ph
1
-0,28962
0,0329
-0,38481
0,0483
0,27967
0,0558
0,50603
0,0969
2
-0,27523
0,0495
-0,46083
0,0333
0,39473
0,0866
0,55499
0,0739
3
-0,51094
0,0481
-0,67907
0,0263
0,17107
0,121
0,35664
0,079
4
-0,53672
0,0228
-0,67928
0,0429
0,52906
0,0559
0,51031
0,145
5
-0,51832
0,0782
-0,55389
0,0904
0,42582
0,201
0,55389
0,0904
6
-0,48471
0,0634
-0,68386
0,044
0,50461
0,145
0,39142
0,133
7
-0,63412
0,0436
-0,78454
0,0323
0,68301
0,124
0,32558
0,0963
8
-0,5415
0,046
-0,71418
0,0364
0,8816
0,135
0,33409
0,0996
9
-0,68071
0,0485
-0,61516
0,0681
0,94286
0,151
0,46706
0,205Slide22
The average of the fitted values for
b
2
f
is -0.50±0.14(standard deviation). The previously calculated REMPI value of
b
2
f
is -0.621, so it falls just inside the standard deviation of the fitted values. Using the value of b
2f is -0.621 is therefore totally justifiable for the next fitting procedures where the
b4f parameter is added to increase the quality of the fits themselves. Slide23
Peak D – fit 4
I=(1+
b
2f
P
2
(cos
q
)+b4
P4f(cosq)) (1+b
2phP2(cosq)+
b4P4ph(cosq)),
b2f=-0,621Slide24Slide25
Peak G – fit 4
I=(1+
b
2f
P
2
(cos
q
)+b4
P4f(cosq)) (1+b
2phP2(cosq)+
b4P4ph(cosq)),
b2f=-0,621Slide26Slide27
As expected, upon addition of the
b
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
b
2
f parameter, a last fitting procedure is performed, where the b2ph
parameter is held constant at 2, in order to assess the b2f parameter, and compare with the results from fitting procedure #3.Slide28
J‘
b
2
ph
(D)
Db
2
ph
b
2
ph
(G)
Db2ph
b4
ph
(D)
Db
4
ph
b
4
ph
(G)
Db
4
ph
b
4
f
(D)
Db
4
f
b
4
f
(G)
Db
4
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
0,127Slide29
Peak D – fit 5
I=(1+
b
2f
P
2
(cos
q
)+b4
P4f(cosq)) (1+b
2phP2(cosq)+
b4P4ph(cosq)),
b2ph=2Slide30Slide31
Peak G – fit 5
I=(1+
b
2f
P
2
(cos
q
)+b4
P4f(cosq)) (1+b
2phP2(cosq)+
b4P4ph(cosq)),
b2ph=2Slide32Slide33
Again, the fits are very good although the assumption that
b
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
b
2
f
parameter, solely from the D peak, as we did with the results from the 3rd fitting procedure.Slide34
J‘
b
2
f
(D)
Db
2
f
b
2
f
(G)
Db
2
f
b
4
f
(D)
Db
4
f
b
4
f
(G)
Db
4
f
b
4
ph
(D)
Db
4
ph
b
4
ph
(G)
Db
4
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
0,09Slide35
The averaged values of the fitted
b
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
b
2
f
=-0.621 is justifiable in the fits including b4f
for improved fitting curves.