*
Department of Chemistry, Vanderbilt University
Nashville, TN 37235
* J. Mol. Spectrosc . 226 , 137 (2004).

Personal Retrospective:
(1) TSF yields surprisingly precise information about population distributions from “poor”
ESD spectra of OH [Mendenhall, et al ., Nucl. Instrum. and Meth .
B33 , 834 (1988)].
(2) Impressive performance found for TSF applied to ESDproduced CN spectra [Xu, et al .,
J. Chem. Phys. 93 , 5281 (1990)].
In (a), fits yield a 2T Boltzmann rotational distribution, with T s of
91(3) (19%) and 665(7) K (800 spectral points).
384
OH A

X (0–0)
CN B

X (0–0)
(nm) 389

(3) Later work on CN spectra of higher quality yields even more impressive results. [For similar, see Xu, et al ., Phys. Rev.
B . 48 , 8222 (1993).]
10
8
6
4
CN B  X from ESD off NaCl
Experimental
CN B

X Fit Model
Quadratic Background: b
0 b
1 b
2
= 0.056(5)
= 2.0(5)

10
–3
= -1.2(5)

10
–4
Population Distribution: 3T Boltzmann
T
1
T
2
T
3
= 161(8) K; c
1
= 519(13) K; c
2
= 19 K (fixed); c
3
= 0.68(4)
= 1.16(4)
= 0.056(5)
Line Shape: Gauss-Lorentz (~ Voigt)
D 
1/2
= 0.335(3) Å; f
G
= 0.82(6)
2
0
3840 3850
Calculated
3860

(Å)
3870 3880 3890
1-1 Band: I (1–1)/ I (0–0) = 0.061(6)
Calibration:
D  ( Å) =
A sin [ k (
 – 
0
)] + B
A = 0.133(8);

(2nd Harmonic!):
0
= –2.6(2) Å
D 
2
= A
2
A
2
= –0.030(7);
B = –1.010(5) sin [2 k (
 – 
2
)]

2
= 0.23(38)
Reduced c
2 = 1.13

(4) OH A

X spectra recorded at high resolution on a CCD array require inclusion of OD (natural abundance ~ 0.014%) in fit model [ J. Chem. Phys . 114 , 3465 (2001)].
300 0
200 0
100 0
0
75 80 85 90

(Å) - 3000 Å
95 100
10

0
-1 0
OH A

X emission , from
Tesla discharge spectrum of water vapor in Ar, as recorded at ~0.03Å resolution on a CCD array
(800 channels). Normalized
LS residuals below.
4
2
0
8
6
70
71 72 73 74

(Å) - 300 0 Å
75 76 77
5
0

-5
Blowup of 6-Å region of spectrum, showing residuals with and without OD in fit model. OD abundance fit parameter = 2.1(2)

10
–4
.

All examples so far have involved prior knowledge of the spectroscopic constants. Similar methods have worked on congested spectra requiring estimation of the constants.
Calibration
I
2
D'

A'
22-0 band
Experimental
32521
Calculated
32523 32525
Wavenumber (cm
–1
)
32527 32529
Experimental and fitted spectra for D'

A' transition in I
2
, as excited in a free jet expansion
[Zheng, et al ., J. Chem. Phys.
96 , 4877 (1992)]. Adjustable fit parameters include B' , B'' , and

0
, yielding apparent errors of 0.0008 cm
–1
, 1.3

10
–5 cm
–1
, and 1.4

10
–5 cm
–1
. The latter two represent relative errors of
0.05%.

• Just how good are these derived parameters?
• In the case of the fitted spectroscopic parameters, how do these results compare with those obtained by the traditional Measure, Assign, Fit (MAF) approach?
• First question is relevant, because these are nonlinear LS fits, so there is no guarantee on the
V -based statistical errors. However, the relative errors are much less than 10%, so these parameters would appear to satisfy the “10% rule of thumb” [ J. Phys. Chem. A 104 , 2834 (2000)].

18000
14000
B  ' = 0.0177
10000
0.01725
6000
0.016816
Spectra are synthesized for various B' and fixed
B'' (= 0.02804 cm
–1
) at
T rot
= 5 K. P and R lines are fully resolved at top, perfectly overlapped at bottom.
2000
-7 -6 -5 -4 -3
 – 
0
(cm
–1
)
-2 -1 0

•
Add random Gaussian error of known
 to independent variable in model. [Here proportional error is assumed, e.g.
, 5% of signal.]
•
Repeat N times to generate statistically equivalent spectra. Analyze each using weights w i
=
 i
–2
.
• Do sampling statistics in usual way, obtaining MC estimated values of each parameter b and its standard error
 bias) is b

. The MC precision of error (2 N ) b
/ N
–1/2
1/2
(= 2.2% for N b
(needed to evaluate
. The MC estimate of
= 1000).
 b has relative
* Recommended pedagogy — J. Chem. Educ. 82 , 157 (2005).

•
V -based standard errors for B in TSF approach are valid except where lines are doubled up; there “exact” predictions are conservative, while just off peak they are optimistic.
•
For B
 s, MAF outperforms TSF in doubled-up regions.
(First peak expanded)
0.001

B''
(cm
–1
) exact
TSF
MAF
0.0002
0.0001
0.0001
cm
–1
0.0000
TSF bias
0.0160
0.0170
0.0180
0.0190
0.0200
B ' (cm
–1
)
0.0165
0.0167
0.0169
0.0171
B ' (cm
–1
)

•
Similar results for

0
(points and curve at top), except that now TSF outperforms MAF everywhere in MC sampling.
0.0070
0.0050
0.0030

(cm
–1
)
0.0010
0.0008
(Linewidth parameter)
0.0160
0.0165
0.0170
0.0175
0.0180
0.0185
B ' (cm
–1
)

c 2
So why are the “exact” ( V -based) parameter
 s wrong when lines double up?
Exact too high
Examine c
2 surfaces for a clue.
Too low
10
8
6
0.016816
0.01650
0.01669
0.01700
1) Fix B'' = 0.02804 cm
–1
.
2) Compute spectra for indicated values of B' .
4
2
0
0.0270
0.0275
0.0280
0.0285
0.0290
B '' (cm
–1
)
3) Fit spectra with B'' frozen at values along the abscissa.
4) Results show c
2 surfaces with respect to B'' , for indicated B' .
Correct!

Histogrammed results for B'' from MC-generated spectra having various B' confirm this behavior.
160 120
0.0165
120
80
0.016816
0.01669
80
40
0
0.0276
0.0280
0.0284
0.0288
B '' (cm
–1
)
40
0
0.0279
0.0280
0.0281
0.0282
B '' (cm
–1
)

For MAF analysis of spectra, bias dominates statistical error except for regions where the R and P lines are either well resolved or perfectly coincident .
0.0004
0.0002
bias

0.0000
-0.0002
-0.0004
0.0160
0.0170
0.0180
B ' (cm
–1
)
0.0190
0.0200

Next step: Global analysis with B'' fixed. Now TSF method outperforms MAF everywhere, and exact and
MC error estimates agree.
( B' ) 1.0 10
-5

8.0 10
-6
6.0 10
-6
1 10
-6
5 10
-7
0
-5 10
-7
0.0160
0.0170
0.0180
0.0190
0.0200
B ' (cm
–1
)

Kicker: MAF results depend on peak-finding algorithm.
Histogram of MAF B'' values for B' = 0.01725 cm
–1
350
300
250
200
150
100
50
0
0.0272
0.5% noise
"
5% noise
Peak finder
(b)
0.0274
0.0276
B '' (cm
–1
)
0.0278
Peak finder
(a)
0.0280

Kicker 2: Good performance of TSF method is partly due to knowledge of the correct model.

Total spectrum fitting outperforms the traditional measureassign-fit approach to the analysis of congested diatomic spectra, except for the special case of perfectly coincident P and R lines, where MAF tacitly incorporates a special relation between the two B values.
A major advantage of TSF is lack of significant bias.
However, a major limitation of TSF is the requirement for a correct fit model. Model error can give bias comparable to the nominal statistical error.
A second significant limitation of TSF is unreliable parameter errors, from violation of the 10% rule of thumb. In the present case such violations are restricted to regions where the two B values are highly correlated, leading to very nonparabolic c
2 surfaces.