Table 2. C ~Character V Table Showing the Irreducible Representationfor NH3
Table 1. C ~Character V Table
1
1
12
0
1
1
-1
1
12
0
-6
4
24
A1
3
NH3 A2
NH3
E
no. of unshifted atoms times the character
E
C3
nu(EJ
nu(C3)
m(od
h(R)
f(EJ
4C3)
40")
4R)
rnntrihnntinn
1
xm
2
X(C3)
a4 1
VA t
I R-
aE 1
v4
2 VE
Y
2-?,v
0
X.
0
Rx,Ryxz, yz
E
C3
4
1
2
ndR)
3
0
1
f(R)
12
0
2
XdR)
1
2
3
d(R)
12
0
6
XdR)
0"
no. of unshifted atoms times the character contribu-
3
Xi(0")
4R)
times the number of operations of the class
XdR)
A?
aA I
RZ
0
tion
times the character contribution
-
-1
2
NH3
6
total spectral terms less rotational and translational vibrational terms
I R-
Raman
Raman VA, + VE IR-active; VA,+ VE Raman-active v , inactive vibration bands
Those terms that remain, 24, + 2E, will be the IR-active, Raman-active, and the inactive vibrational terms. The IRactive terms, 2AI + 2E, are designated by x , y, andz in the second column from the right of the character table. The Raman-active terms, 2A1 + 2E, are designated by any listing in the far-right column of the character table. The inactive vibrational terms are any terms present in the reduced representation t h a t a r e neither IR- nor Raman-active. Ammonia has no inactive bands. Table 2 summarizes all these observations. This process predicts four IR-active and four Raman-active bands for ammonia. This compares favorably with the four lines found in the IR spectra and the four lines in the Raman spectra for ammonia as reported by Herzberg (6). Molecular Geometry Prediction
With the use of these spreadsheet templates, molecular vibration calculations are easy enough to allow routine prediction of the most likely geometry for a molecule. One can eliminate unlikely geometries for a molecule by comparing the number of predicted lines in the IR and Raman spectra with those actually found. Sulfur Tetrafluoride
Sulfur tetrafluoride is a molecule predicted to have a seesaw structure by the Valence Shell Electron Pair Repulsion Theory (VSEPR). Now let us use molecular vibration calculations and known IR data to predict a possible struc-
I R-
A2
E
3
1
4
1
1
2
2
0
2
UY
total spectral terms less rotational and translational vibrational terms
I R-
Raman Raman 4 IR-active 0 inactive vibration bands
4 Raman-active
ture for this molecule, ignoring the principles of the VSEPR Theory. Jolly ( 4 )suggests three possible shapes for the SF4molecule. One of the shapes, involving a central sulfur atom surrounded by four fluorine atoms, could conceivably have a tetrahedral shape and thus be represented by the Td point group. Spreadsheet calculations using the automatic approach for the Td point group are represented in Table 3. These calculations would predict that SF4would have two IR-active bands and four Raman-active bands. A sewnd possible structure for SF4would have the central sulfur atom surrounded by four fluorine atoms with a seesaw shape and thus be represented by the C2, point group. Similar spreadsheet calculations for the C2" point group appear in Table 4. These calculations would predict SF4to have eight IR-active bands and nine Raman-active bands. The third structure suggested by Jolly ( 4 )involves a central sulfur with four surroundinp fluorine atoms-me at a greater distance than the otheFthree. This suggests C3" symmetry and seems unlikely. (Why would one S-F bond be different from the other three?) So it will not be further discussed in this paper. Because the IR spectra for SF4 (as observed by Dodd, Woodward, and Roberts (7)) shows at least five bands, the Tdsymmetry is eliminated. The C2,.symmetry calculations predict more than five IR hands and thus may indicate the geometry of the SF4 molecule. Although thk is negative Volume 71 Number 6 June 1994
487
Table 3. Molecular Vibration Calculations for SF4 with TdSymmetry
Table 4. Molecular Vibration Calculations for SF4 with Cz~Symmetry
x
x
5
2
1
1
3
0
-1
-1
1
15
0
-1
-1
3
1
0
3
6
6
15
0
-3
4
18
3
no. of unshifted atoms timesthe character contribution
x
5
1
3
3
3
-1
1
1
15
-1
3
3
1
1
1
A2
1
SF4
0
E 1
-
Tl
X{R)
S F4
T2
; 1
3
times the number of operations of class
times the number of operations of class -
A1
no. of unshilted atoms times the character contribution
total spectral terms LSS rotatid and translational
4
2
less rotational and translational vibrational terms
8 IR-active bands
Rarnan-adive bands
Raman
Raman
IR-
Raman
I
2 IR-active bands
Rarnan-active bands 0 inactive vibration bands 4
evidence-because some untested symmetry might fit the experimental data as well or better--Cotton, George, and Waugh (8)and Tolles and Gwinn (9) have confvmed this structure using nuclear magnetic resonance and microwave spectroscopy. Conclusion In the past only simple molecules have been used in molecular vibration calculations when these principles are taught to students. Using a spreadsheet, the drudgery of all the calculations has beeneliminated, and the principles can be easilv illustrated. Possible structures for molecules can be elimhated by comparing calculated values for the number of IR- and Raman-active bands with ex~erimentally determined spectra. This will present the kudents with a ~racticala ~ ~ l i c a t i oofne r o u ~ theorv to which thev can relate.
..
-
A
. . These calculations were made on an IBM PSI2 Model 50 using a Quattro Pro 1.0 spreadsheet program. However,
488
0 inactive vibration bands
Journal of Chemical Education
the calculations are very simple and can be made on any computer using any spreadsheet program. Copies of MSI DOS Quattro templates for several character tables and a typical homework assignment can be obtained by sending a blank disk (5.25 in. or 3.5 in.) and $2.00 for postage and handling to the author. Acknowledgment I wish to thank my inorganic chemistry class for inspiring me to use spreadsheets in this manner. I wish to acknowledge the encouragement to write this article given to me by David Jeter of Rhodes College and Dale Johnson of the University of Arkansas. Liierature Cited 1. Butler, I. S.:Harrod, J. F Inorg.nle
Chemiafry. Prineiph and Applimtiom; BenjaminlCurnmhgs: Redwood C i a CA, 1989. 2. G o t h . F A .Chamlml Applications of Cmup Theory, 2nd ed.; W k - h t e r s i e n c e : N P ~ Y - k 1911 3. Carter, R. L.J Cham. Educ. 1981,68,373374. 4. Jallx W L. The Synthesis and Chometwizotion afInwganic Compaunds: PrentieHall: Enelewod Cliifs.NJ. 1970:oo 300-306. 5. JoUx W. L.The Synthesis and Chvmeferizotion ofInorganC Compounds: Prenfiee Hall: Englovod Cliffs,NJ, 1970;p 301. 6. Hersberg, G. Infmrpd and Raman Spectetro afPolpfomicMolmules; Van Noatrand: Rineetm. NJ. 1945:p 295. 7. Dodd, R.E.: Woodward, L.A.;Roberts,H. L. l h m . Fomdoy Soc. 1956.52,1052. 8. Cotton. E A,:. Gwree. . J. W.: Waueh. .. J. S.J. Chem Phvs 1958.28.994, 9. Tolles, W.M.;Gwhn, W.D. J. Chom. Phys. 1982.36.1119. ~
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