Analysis of the Infrared Spectra of Diatomic Molecules

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Analysis of the Infrared Spectra of Diatomic Molecules
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Analysis of the Infrared Spectra of Diatomic Molecules
W
Richard W. Schwenz
Department of Chemistry and Biochemistry, University of Northern Colorado, Greeley, CO 80639
William F. Polik*
Department of Chemistry, Hope College, Holland, MI 49423; *[email protected]
One of the most common physical chemistry laboratory
exercises involves the infrared spectrum of a heteronuclear
diatomic molecule, usually HCl, from which the internuclear
separation, the vibrational force constant, and higher-order
molecular constants can be determined (1–5). The reasons
for frequent use of this exercise include the relevance to the
harmonic oscillator and rigid rotor models taught in quantum
mechanics, the high accuracy of the spectroscopic measurements
(achieving essentially literature values for the line positions),
the existence of several isotopic variants of HCl allowing tests
of the Born–Oppenheimer approximation, and the opportunity for students to perform a rigorous data reduction and
error analysis to determine molecular parameters from spectroscopic data. It is this last feature with which this paper is
concerned. In particular, several methods of data reduction
are examined and an improved method using the multiple
linear regression feature available on most microcomputer
spreadsheet programs is presented that eliminates some of the
problems inherent in other methods.
Theory
displacement of the bond as vibrational energy increases.
While some of the spectroscopic literature includes more
molecular constants, these five are sufficient to describe the
infrared spectra recorded for most diatomic molecules in the
undergraduate physical chemistry laboratory. Thus, a
complete expression for the rovibrational energy levels of a
diatomic molecule is given by
∼
E (v, J ) = ωe(v + 1⁄ 2 ) +Be J( J + 1) – ωexe(v + 1⁄ 2 )2 –
De J 2( J + 1)2 – αe(v + 1⁄ 2 ) J( J + 1)
(3)
where all the constants have wavenumber units.
The lines in an absorption or emission spectrum arise
from transitions between two energy levels. The wavenumber ν∼ of a transition can be obtained by subtracting the wavenumber energy of the lower state from that of the upper state
∼
∼
ν∼(v′, J′,v′′, J′′) = E (v′, J ′) – E (v′′, J ′′)
∼
(4a)
∼
E (v′, J ′) – E (v′′, J ′′) =
ωe[(v′ + 1⁄ 2) – (v′′ + 1⁄ 2 )] + Be[ J′( J′ + 1) – J ′′( J ′′ + 1)] –
All textbooks, laboratory manuals, and original spectroscopy papers treat the infrared spectra of diatomic molecules
similarly. The molecules are first treated in the harmonic
oscillator–rigid rotor approximation with energy levels given by
E(v,J ) = [(h/2π)(k/µ)1/2](v + 1⁄ 2 ) + [h 2/8π2Ie ] J( J + 1) (1)
where v is the vibrational quantum number, J is the rotational
quantum number, h is Planck’s constant, k is the vibrational
force constant, µ is the reduced mass, and Ie = µre2 is the
moment of inertia with re being the internuclear separation.
Dividing through by hc yields the molecular energy and
parameters in wavenumber units (cm{1), which is indicated by
the use of a tilde,
∼
E(v,J )/hc = E (v, J ) = ωe(v + 1⁄ 2 ) + Be J( J + 1)
(2a)
ωe = (k/µ)1/2/2 πc
(2b)
Be = h/8 π 2cI e
(2c)
ωexe[(v′ + 1⁄ 2)2 – (v′′ + 1⁄ 2)2] – De[ J ′2(J′ + 1)2 – J ′′2(J′′ + 1)2] –
αe[(v′ + 1⁄ 2 )J′( J′ + 1) – (v′′ + 1⁄ 2)J ′′( J ′′ + 1)]
(4b)
where the single prime indicates the upper state and the
double prime indicates the lower state. In order to apply
eq 4b, appropriate quantum numbers (v ≥ 0, J ≥ 0) and
selection rules (∆J = ±1) must be used.
For the HCl molecule, the infrared absorption spectrum
consists of about 20 pairs of closely spaced lines for both the
fundamental (v′′ = 0 → v′ = 1) and first harmonic (v′′ = 0 →
v′ = 2) bands for each of the H and D isotopic variants. The
where
and
are the harmonic vibrational frequency and rotational constant, respectively.
Three correction terms are then usually introduced to
this energy level expression. The first of these, the anharmonicity constant ωexe, arises from the departure of the vibrational energy levels from the harmonic approximation, in part
to allow for bond dissociation. The second, the centrifugal
distortion constant De, arises from the stretching of the
molecule as it rotates faster. The third, the vibration–rotation
interaction constant α e, arises from the increase in mean
1302
Figure 1. Infrared absorption spectrum of the fundamental band
for HCl.
Journal of Chemical Education • Vol. 76 No. 9 September 1999 • JChemEd.chem.wisc.edu
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spacing within a pair arises from the slight difference of the
molecular constants for the 35Cl and 37Cl isotopic variants.
Figure 1 presents the fundamental band of HCl. The overall
appearance of the spectrum is due to the ∆J = ±1 rigid rotor
selection rules, which result in a P branch (∆J = {1) and an
R branch (∆J = +1) of almost equally spaced pairs of lines
surrounding the band origin. The spacing between each pair of
lines is approximately 2Be. The relative intensity of each pair
of lines arises from the 2J + 1 degeneracy of the rigid rotor
energy levels, the Boltzmann distribution of molecules in the
ground state rotational levels, and the rotational absorption
intensity factors (squared dipole transition moment) (6 ). The
relative intensity of the lines within each pair is due to the
relative isotopic abundance of 35Cl and 37Cl.
Analysis of the Spectrum
Average Spacing
The simplest method for analyzing the infrared spectrum
of a diatomic molecule relies on the harmonic oscillator–rigid
rotor energy level expression of eq 2. In this approximation,
the spacing between all the lines is averaged, this average is
set equal to 2Be, and the moment of inertia Ie and internuclear
separation re are calculated. The force constant k for the bond is
simply calculated from the wavenumber of the band center.
This method has the virtue of simplicity. It also has the disadvantage of being vastly oversimplified, as it neglects the
additional terms present in eq 3 that do not occur in eq 2.
This method is suitable for homework problems in textbooks,
but not for the quantitative analysis of an actual spectrum.
Successive Differences
It is readily apparent from the spectrum of HCl that the
lines are not equally spaced, but that the spacing between
lines in the P branch expand and lines in the R branch contract
as J increases. The method of successive differences found in
laboratory manuals (2, 5, 6 ) accounts for this observation by
recognizing that the transition wavenumbers in the spectrum
can be approximated as a parabolic function of an index variable m where m = J ′′ + 1 for the R branch and m = {J ′′ for
the P branch. Neglecting the effect of De (since it contributes significantly only at high J ) and using this substitution,
the transition wavenumbers in the fundamental band are
found to be given by
ν∼(m) = ωe – 2ωexe + (2Be – 2αe)m – αem2
(5)
Typically, a successive difference plot of the wavenumber difference between adjacent transitions versus m is constructed
allowing the determination of Be and αe from a linear fit to
∆ν∼(m)
= ν∼(m
+ 1)
– ν∼(m)
= (2Be – 3αe) – 2αem
(6)
The fundamental band origin ω10 = ωe – 2ωexe may be calculated by rearranging eq 5 to isolate ωe – 2ωexe and substituting
the values of Be and αe along with the observed transition
wavenumbers for several values of m near the band center.
The intercept (m = 0) of a plot of ν∼(m) – (2Be – 2αe)m + αem2
versus m yields the best estimate of the fundamental band
origin ω10. Analysis of the first overtone band is similar, with
the transition wavenumbers given by
ν∼(m) = 2ωe – 6ωexe + (2Be – 3αe)m – 2αem2
(7)
and the difference between adjacent transitions by
∆ν∼(m) = ν∼(m + 1) – ν∼(m) = (2Be – 5αe) – 4αem
(8)
The fundamental band origin ω10 may be combined with knowledge or measurement of the first overtone ω20 = 2ωe – 6ωexe
to determine the values of ωe and ωexe.
The successive differences method readily introduces the
concept that vibration–rotation interaction is responsible for
the unequal spacing of spectroscopic transitions in the HCl
spectrum. However, this method results in an inaccurate value
of Be due to neglect of the centrifugal distortion term De,
often yields inconsistent values of B e and αe from the independent analyses of the fundamental and overtone bands,
and does not correctly calculate the uncertainties in Be and
α e due to statistical correlation between the coefficients of m
and m2. It should be noted that the successive difference
method can be extended to determine De by retaining it in
the energy level expression to yield an equation for ν∼(m) that
is cubic in m (6, 7 ). This equation can be analyzed by least
squares fitting to cubic polynomial or by taking the second
successive difference ∆2ν∼(m); however, this method is not used
in any physical chemistry laboratory manual owing to its
complexity.
Combination Differences
An analysis technique used by spectroscopists prior to
modern computational abilities was the combination differences method (6, 8–10). In this method, it is recognized that
certain pairs of spectral lines share a common upper or lower
state, allowing an energy difference between states to be
calculated. For example, the R( J ′′ = 0) and P( J ′′ = 2) transitions both have J ′ = 1 in the upper state; thus, the difference
in transition energies for the R( J ′′ = 0) and P( J ′′ = 2) lines is a
direct measurement of the energy separation between J ′′ = 0
and J ′′ = 2 in the lower state. Formulas involving the rotational constants of the upper and lower vibrational states and
the centrifugal distortion constant can be derived by taking
the expression for spectroscopic transitions, substituting upper
state quantum numbers using the rigid rotor selection rules,
identifying appropriate pairs of combination difference states,
and eliminating the upper or lower state energy (1, 3, 4, 6 ).
The two simplest resulting formulas are
ν∼R( J ′′) – ν∼P( J ′′) = 2(2J ′′ + 1)Bv′ – 4[( J ′′ + 1)3 + J ′′3]De (9)
and
ν∼R( J ′′) – ν∼P( J ′′ + 2) =
2(2J ′′ + 3)B0 – 4[( J ′′ + 1)3 + ( J ′′ + 2)3]De
(10)
where Bv ′ = Be – αe(v′ + 1⁄ 2 ) is the rotational constant of the
upper vibrational state and B0 is the rotational constant of the
lower vibrational state. A formula involving the band origin
can be similarly derived:
ν∼R( J ′′) + ν∼P( J ′′ + 1) =
2[v ′ωe – v ′(v ′ + 1)ωexe] + 2( J ′′ + 1)2(Bv ′ – B0)
(11)
To determine the values of B1 and De from the fundamental
band, for example, eq 9 is linearized by dividing through by
2(2J ′′ + 1) and a plot of [ν∼R( J ′′) – ν∼P( J ′′)]/[2(2J ′′ + 1)] versus
{4[(J ′′ + 1)3+ J ′′3]}/[2(2J ′′ + 1)] is constructed. The intercept
equals B1 and the slope is {De. Similar linearized plots of eqs
JChemEd.chem.wisc.edu • Vol. 76 No. 9 September 1999 • Journal of Chemical Education
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10 and 11 yield B0, another determination of De, ω10, and
the difference between upper and lower state rotational constants
B1 – B0. Analysis of the first overtone band yields B2, two
additional values of De, another value of B0, ω20, and B2 – B0.
The molecular constants Be and αe are then determined from
the intercept and slope of a plot of Bv ′ versus v ′ + 1⁄ 2 , and De
is taken to be its average determined value. The constants ωe
and ωexe are determined from the values of ω10 and ω20.
The difficulties encountered by students using this
method are legendary. Problems include understanding the
derivation of the combination difference formulas, the immense
amount of data that must be handled to produce the three
graphs per vibrational band per isotope required to determine
the molecular constants, and resolution of inconsistent values
of molecular constants determined from independent fits. In
addition, spectroscopists recognize that the combination
differences method introduces errors when missing or blended
lines occur, alters the weights of lines of differing J values
because of the form of the linearized equation, and completely
neglects the statistical correlation between the molecular
constants in the upper and lower states (11).
A New Method of Analysis
We suggest here that a different analysis method be used
by physical chemistry students, which takes advantage of the
relative simplicity of the energy expression in eq 3 and the
ability of a microcomputer spreadsheet program to perform
multiple simultaneous regression on a large amount of data.
The preceding data analysis methods can be frustrating to
students because of the repetitive calculations, the graph
production, and the indirect method of obtaining molecular
constants. In addition, the methods themselves suffer from
inherent problems in determining molecular constants when
the energy expression coefficients are highly correlated. The new
method dramatically simplifies the effort of data reduction,
provides a more direct connection between experimental data
and molecular constants, and yields results that are more
accurate and precise than the traditional methods.
Fundamental and Overtone
In our method, the wavenumber and the associated
quantum numbers of each transition are entered into a spreadsheet row. The data occupy five columns of the spreadsheet,
one for the transition wavenumber, one each for the upper state
vibrational and rotational quantum numbers, and one each for
the lower state vibrational and rotational quantum numbers.
Thus, five pieces of data are entered for each observed transition: ν∼, v′, J ′, v′′, J ′′. The same information is entered into
a new row for each transition observed in the spectrum.
Five columns are subsequently computed that allow the
determination of the molecular constants directly via multiple linear regression. These five columns are computed from
the five terms in the wavenumber transition expression, eq
4b. Each term in eq 4b consists of a molecular constant multiplied by a combination of quantum numbers for the upper
state minus the same combination of quantum numbers for
the lower state. For example, Be is multiplied by J ′( J ′ + 1) –
J ′′( J ′′ + 1). The five computed columns consist of the quantum number expressions that multiply the five molecular constants that are being determined. Thus, these five columns
will contain formulas for (v′ + 1⁄ 2) – (v′′ + 1⁄ 2 ), J ′(J ′ + 1) –
1304
J ′′(J ′′ + 1), (v′ + 1⁄ 2 )2 – (v′′ + 1⁄ 2 )2, J ′2(J ′ + 1)2 – J ′′2(J ′′ + 1)2,
(v′ + 1⁄ 2 )J ′(J ′ + 1) – (v′′ + 1⁄ 2 )J ′′(J ′′ + 1), in which each formula
is calculated from the quantum numbers in the first five
columns of the spreadsheet. The most straightforward procedure is to create these formulas in the first row and then
copy them into the rest of the rows.
The remaining step is to initialize and perform the
multiple linear regression. The regression variables must be
defined in the spreadsheet. The y (or dependent) variable is
the wavenumber of the transitions. The x (or independent)
variables consist of the five computed columns defined in the
previous paragraph, which contain the quantum number
expression associated with each molecular constant. Since the
form of the energy expression does not contain a constant
term, the regression should have a defined zero value for the
intercept. Once the regression is performed, the output will
include five x coefficients with their associated errors and the
standard error of the regression analysis. The five regression
coefficients are ωe, Be, { ωexe, {De, and { αe, which correspond
directly to the energy expression coefficients of eq 3. The
coefficient errors are the errors in each molecular constant.
The standard error is a measure of how well the observed
transitions fit eq 4b. The equilibrium internuclear separation,
vibrational force constant, and their associated uncertainties
are easily calculated from the regression output using eqs 2b
and 2c.
Fundamental Only
The preceding treatment assumes that measurements
have been made on both the fundamental and first overtone
bands of HCl. Should measurements be made only on the
fundamental band, then information is not present to determine both the harmonic frequency ωe and the anharmonicity
ωexe, and the matrix inversion in the regression routine will
fail. In this case, the known vibrational quantum numbers,
v′ = 1 and v′′ = 0, should be incorporated into the expression for transition wavenumbers, eq 4b, resulting in
ν∼(v′=1, J ′,v′′=0, J ′′) =
ωe – 2ωexe + Be[ J ′( J ′ + 1) – J ′′( J ′′ + 1)] –
De[ J ′2( J ′+ 1)2 – J ′′ 2( J ′′+ 1)2] –
αe[ 3⁄2 J ′( J ′+ 1) – 1⁄2 J ′′( J ′′ + 1)]
(12)
Three data columns are needed for the values of ν∼,
J ′, and J ′′. Three formula columns are used to compute
J ′( J ′ + 1) – J ′′( J ′′ + 1), J ′ 2( J ′ + 1)2 – J ′′ 2( J ′′ + 1)2, and
3⁄ J ′( J ′ + 1) – 1⁄ J′′( J ′′ + 1). Since eq 12 contains a constant
2
2
term, that is, a term not multiplied by a quantum number
expression, the regression is performed with three independent variables that correspond to the three formula columns
and the intercept must be allowed to vary. The three regression coefficients will correspond to Be, {De, and { αe, and
the
intercept will
correspond to the vibrational origin, ω10 =
∼
∼
E(v′=1, J ′=0) – E (v′′=0, J ′′=0) = ωe – 2ωexe. The coefficient
errors are the errors in each molecular constant, and the
intercept error is the error in the vibrational origin. An
alternate method of fitting the constant term in eq 12 is to
include a column of ones for it in the spreadsheet and perform
the regression with four independent variables while defining
the intercept to be zero. The vibrational origin will then
correspond to the coefficient for the column of ones. This
Journal of Chemical Education • Vol. 76 No. 9 September 1999 • JChemEd.chem.wisc.edu
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alternate method is particularly useful if the spreadsheet
program does not output the error for the intercept.1
Calculation of Residuals
Regardless of whether data from the overtone band are
used, it is critical to use two more columns in the spreadsheet to calculate the predicted transition wavenumber
from the determined constants and to calculate the residual
difference between observed and predicted wavenumber for
each transition. This not only illustrates how to use molecular
constants to calculate spectroscopic transitions, but also serves
as an essential check for data entry or calculational errors.
The regression method is numerical rather than graphical,
and data entry errors are not easily caught before performing
the regression analysis. Since line positions are determined
with high precision (typically 0.2 cm{1 or better) with modern
FTIR instrumentation and eq 3 fits the observed energy levels
extremely well over the observed quantum number range, the
calculated residuals should be on the order of the precision
of the spectrometer. It is very easy to pick out a transition
with a data entry error, as its residual will be much larger
than the standard error of the regression. Also, errors in the
first-order spectroscopic constants (ωe and Be) may easily be as
small as 1 part in 105 with data taken on a FTIR instrument,
so the existence of a data entry or calculational error is flagged
by an unreasonably large error for a regression constant.
Table 1. Transition Wavenumbers
and Assignments for Fundamental
and First Overtone Bands of H 35 Cl
v ′′
v′
J ′′
J′
∼
ν/cm{1
0
1
0
1
2905.995
0
1
1
2
2925.581
0
1
2
3
2944.577
0
1
3
4
2962.955
0
1
4
5
2980.689
0
1
5
6
2997.788
0
1
6
7
3014.202
0
1
7
8
3029.941
0
1
8
9
3044.965
0
1
9
10
3059.234
0
1
10
11
3072.771
0
1
11
12
3085.600
0
1
12
13
3097.550
0
1
13
14
3108.914
0
1
14
15
3119.418
0
1
15
16
3129.099
0
1
1
0
2864.834
0
1
2
1
2843.315
0
1
3
2
2821.249
0
1
4
3
2798.641
0
1
5
4
2775.499
Comparison of Methods
0
1
6
5
2751.817
0
1
7
6
2727.624
The regression method of data analysis is computationally superior to both the successive-difference and combination-difference methods. To demonstrate this, an H35Cl data
set recorded with a Nicolet 730 FTIR spectrometer is presented in Table 1. This data set has been analyzed using each
of the methods described above. A spreadsheet with these
analyses is available on the World Wide Web (12, 13). Implementation of the regression method using Mathcad is also
possible (14, 15). The results of these analyses are compared
to accepted literature values (16, 17) in Table 2. The regression
analysis results are both more precise and more accurate than
the successive-difference and combination-difference methods.
The successive-difference method neglects the centrifugal
distortion constant De, and its effect is therefore incorporated
into the determination of the other constants. This is apparent
in the successive difference plot of Figure 2, in which a
systematic curvature is observed resulting in a significant
underestimate of Be when the slightly curved data set is fit to
a straight line. The combination-difference method results
from a linear least squares analysis with equal weights are also
given in Table 2. The numerical difficulties of this method
are apparent from the deviation from linearity near the
intercept in the plot of Figure 3 (the result of fitting eq 10)
and from the inconsistent De values among plots despite the
fact that all values should be the same at this level of theory.
Owing to division by 2(2J ′′ + 3) when linearizing eq 10, for
example, the y uncertainties for low J ′′ data points are greater
than high J ′′ data points, resulting in larger deviation of the
data from the best fit near the intercept in Figure 3. The
determination of De by averaging its value from four separate
analyses causes an unrealistically large error for De in Table 2.
The regression method simultaneously determines all the
molecular constants without neglecting any terms. In addition,
0
1
8
7
2702.907
0
1
9
8
2677.697
0
1
10
9
2651.932
0
1
11
10
2625.689
0
1
12
11
2598.979
0
1
13
12
2571.861
0
1
14
13
2544.220
0
1
15
14
2516.141
0
2
0
1
5687.494
0
2
1
2
5705.926
0
2
2
3
5723.158
0
2
3
4
5739.109
0
2
4
5
5753.793
0
2
5
6
5767.262
0
2
6
7
5779.441
0
2
7
8
5790.312
0
2
8
9
5799.833
0
2
9
10
5808.141
0
2
1
0
5646.969
0
2
2
1
5624.896
0
2
3
2
5601.612
0
2
4
3
5577.185
0
2
5
4
5551.571
0
2
6
5
5524.865
0
2
7
6
5496.971
0
2
8
7
5467.968
0
2
9
8
5437.895
JChemEd.chem.wisc.edu • Vol. 76 No. 9 September 1999 • Journal of Chemical Education
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Table 2. H35Cl Molecular Constants Determined by Various Methods
Value (SD) /cm{1
Quantity Average
Spacing
ωe
2860(37)
Successive
Difference
2989.215(23)
2989.206(55)
2989.281(34)
2989.74
2990.97424
Be
ωexe
10.41(12)
10.58982(85)
10.58919(89)
10.5909
10.593404
9.8(31)
Combination
Difference
Multiple
Regression
Ref 16
Ref 17
—
51.763(8)
51.765(18)
51.796(12)
52.05
52.84579
De
αe
—
—
0.0005041(181)
0.0005206(29)
0.00053
0.000532019
—
0.30421(118)
0.30318(57)
0.30167(10)
0.3019
0.307139
ω10
—
2885.689(8)
2885.676(18)
2885.690(41)
—
2885.9775
ω20
—
5667.852(5)
5667.823(7)
5667.789(97)
—
5667.9841
Deviation a
49
1.431
0.172
0.046
—
—
aRoot-mean-square
residual.
of the energy level expression, eq 3, and the concept that
spectroscopic transitions arise from transitions between energy
levels, eq 4, over the details of a data analysis method that is
specific to the analysis of diatomic rovibrational spectra.
Third, it gives students experience with the technique of
multiple linear regression. Physical chemistry students get
much experience fitting data to a straight line, y = mx + b
(linear regression), but almost no experience fitting data to
an equation of the form y = m1x1 + m2x2 + m3x3 + … + b
(multiple linear regression). This latter form is critical when
dealing with data that depend on more than one variable, as
is the case for most real data sets. Finally, it introduces students
to a method that roughly corresponds to the data analysis
method used by practicing spectroscopists, namely, multiple
nonlinear regression (nonlinear least squares fitting) to the
energy level expression (Hamiltonian matrix) of a molecule.
Conclusion
Analysis of the infrared spectra of diatomic molecules
by multiple linear regression in a spreadsheet dramatically
simplifies the reduction of spectroscopic data compared to
2 2 J ′′ + 3
νR J ′′ – νP J ′′ + 2
/ cm{1
no skewing of the weights is introduced from linearizing fitting
equations, correlation between the molecular constants in the
upper and lower state constants is accounted for, and each
observed transition is weighted equally. It should be noted
that equal weighting is appropriate for data acquired with
an FTIR instrument, as the wavenumber error in line positions is constant over the entire spectral range rather than
varying with wavelength, as occurs with grating or prism instruments. In all cases, care must be taken when comparing
fit values and literature values to make sure they use a consistent set of molecular constants and the same range of rotational and vibrational levels in the analyses. Inclusion of additional constants or more molecular states will affect the
values obtained in any analysis.
There are also pedagogical advantages to using the regression method to analyze the infrared spectrum of diatomic
molecules. First, it replaces the tedium of preparing and fitting
multiple plots with a much easier and far less time-consuming
use of computer technology. Students learn the concepts
underlying the experiment and gain experience analyzing data
using spreadsheets, rather than worrying about the details of
graph preparation. Second, the method emphasizes the form
3
{4 J ′′ + 1 + J ′′ + 2
2 2 J ′′ + 3
Figure 2. Successive difference plot, eq 6, for HCl fundamental
band with least squares linear fit. The systematic curvature of the data
points is caused by neglect of the centrifugal distortion constant
De. The lowered intercept of the least squares fit results in an underestimate for the value of the rotational constant Be.
1306
3
/ cm{1
Figure 3. Combination difference plot, eq 10, for HCl fundamental
band with least squares linear fit. Inclusion of the centrifugal distortion constant De results in a more linear plot than the successive difference plot in Figure 2. However, low J ′′ data points near the y-axis
have larger uncertainties and should be weighted less than high
J ′′ data points.
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the traditional successive difference and combination difference
methods. It also provides a more direct connection between
experimental data and the molecular constants. The resulting
molecular constants are shown to be both more accurate and
more precise than the results of traditional analysis methods.
Acknowledgments
This work was supported by National Science Foundation
grants CHE-8851651 (RWS) and CHE-9157713 (WFP).
Note
W
Supplementary materials for this article are available on JCE
Online at http://jchemed.chem.wisc.edu/Journal/issues/1999/Sep/
abs1302.html.
1. After submission of this paper, an alternative spreadsheet-based
analysis method was published which uses the Solver tool in Excel to
perform a nonlinear least squares optimization of a transition wavenumber expression to a data set (18). This method could be adapted to
optimize eq 4 or eq 12 and should give the same results as the multiple
linear regression method. The Solver method has the advantage of working for more complex, nonlinear functional forms. However, it is possible
for the method to converge at a local minimum giving an incorrect
result, and the determination of errors in fitting parameters is not
performed as easily as with the multiple linear regression method.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
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