S. M. Condren
Christian Brothers University Memphis, TN 38104
Advanced Inorganic Chemistry, with a Physical Chemistry prerequisite, is a course that can serve to bring together many different theoretical concepts. In an attempt to make several concepts relevant, I have incorporated a discussion of symmetry with its implications to infrared (IR) and Raman spectroscopy as described by Butler and Harrod (1). This involves teaching some basic concepts of group theory math and symmetry point group character tables in making molecular vibration calculations as described by Cotton (2). However, making molecular vibration calculations using group theory can be a very laborious task, even using the tabular method described by Carter (3).
The many group theory calculations can be made quickly and easily with the use of a spreadsheet. The calculations will be done automatically once the experimenter has supplied information on the number of unshifted atoms for each of the symmetry operations in the point group. The student then check his/her results with the number of observed IR and Raman bands to verify that his/her prediction of the number of unshifted atoms is correct. If they have incorrectly predicted the number of unshifted atoms, the spreadsheet will predict the wrong number of IR and Raman bands. Negative and fractional values for the number of bands immediately indicates a problem with their predictions.
One particular application for molecular vibration calculations is to predict the most likely geometry of a molecule based upon the infrared and Raman spectra as described by Jolly (4). Once proficient in predicting the number of unshifted atoms, the student can select the most likely geometry based on group theory calculations and known IR and Raman spectra for a given compound. Since these calculations are so painless with the use of a spreadsheet, the student may be inclined to investigate several compounds.
Molecular Vibration Calculations
Before the molecular vibration calculations can be made, a spreadsheet template must be loaded or constructed. To construct the template the appropriate character table for the point group under investigation is typed into the spreadsheet, leaving a blank line between each row of the character table.
Below the character table, a section of the template will be devoted to the generation of the reducible representation for a particular molecule under investigation. Below this the irreducible representation is created and the rotational and translational spectrum terms are removed to produce the vibrational terms. Then the number of IR and Raman active bands are designated and the total is computed.
Construction of Spreadsheet Templates
Ammonia, a molecule in the C3v point group, will be used to illustrate the construction of a template for these calculations.
Before filling in the equations for the lines left blank in the character table, the reducible representation must be calculated. The reducible representation is required in making the calculations for the multiplier for irreducible representation that will appear in the blank lines of the character table. The first step in constructing the reducible representation is to determine the number of atoms that remain unshifted, nu(R), after each symmetry operation, R, is performed. Jolly provides a table of values for character contributions, f(R), for all symmetry operations (5). The character contribution for a symmetry operation, f(R), times the number of unshifted atoms following a symmetry operation, nu(R), is the part of the reducible representation, c (R), for that symmetry operation, see eq 1.
c (R) = nu(R)*f(R) (1)
The number of operations of the class of symmetry operation, d(R), from the character table, is multiplied times c (R) to produce C i(R), see eq 2.
C i(R) = d(R)*c (R) (2)
In the lines left blank in the character table, a molecular specific element, Mi(R), is created for each element of the character table. These elements are created by multiplying the character element, c (R), from the character table times the value obtained for each symmetry element, C i(R), see eq 3.
Mi(R) = c (R)*C i(R) (3)
The sum of each of these new rows is multiplied by the inverse of the order, h, of the symmetry elements, see eqs 4-5.
ai = 1/h*S RMi(R) (4)
ai = 1/h*S Rc (R)*C i(R) (5)
This value is the multiplier of the irreducible representations, ai, and is the number in the spreadsheet directly under the Mulliken label for that row of the character table. Table 1 shows the symbols for the variables that are placed in the blank cells of the modified character table. The reducible representation, G red, is a sum of the products of these multipliers and their irreducible representations.
For NH3 the reducible representation is 3A1 + A2 + 4E. To determine which terms are the molecular vibrations, the rotational and translational terms must be removed. The rotational terms are designated by Rx, Ry, and/or Rz in the second column from the right of the character table. The translational terms are designated by x, y, and/or z in the second column from the right of the character table. For NH3 the rotational terms are 1A2 and 1E, and the translational terms are 1A1 and 1E, see Table 2. Those terms that remain, 2A1 + 2E, will be the infrared active, Raman active, and/or the inactive vibrational terms. The infrared active terms, 2A1 + 2E, are designated by x, y, and/or z in the second column from the right of the character table. The Raman active terms, 2A1 + 2E, are designated by any listing in the right most column of the character table. The inactive vibrational terms are any terms which are present in the reduced representation but which are neither IR or Raman active. Ammonia does not have any inactive bands. Table 2 summarizes all these observations.
This process predicts 4 IR active and 4 Raman active bands for ammonia. This compares favorably with the 4 lines found in the IR spectra and the 4 lines in the Raman spectra for ammonia as reported by Herzberg (6).
Molecular Geometry Prediction
With the use of these spreadsheets templates, molecular vibration calculations are easy enough to make 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 to those actually found.
Sulfur Tetrafluoride
Sulfur tetrafluoride is a molecule that is predicted to have a "see-saw" structure by the Valence Shell Electron Pair Repulsion Theory (VSEPR). Now let us use molecular vibration calculations and known information about the IR spectrum to predict a possible structure for this molecule, ignoring the principles of the VSEPR Theory.
Jolly (4) suggests three possible shapes for the SF4 molecule. 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 SF4 would have 2 IR active bands and 4 Raman active bands.
A second possible structure for SF4 would have the central sulfur atom surrounded by four fluorine atoms with a "see-saw" shape and thus be represented by the C2v point group. Similar spreadsheet calculations for the C2v point group appear in Table 4. These calculations would predict that SF4 would have 8 IR active bands and 9 Raman active bands.
The third structure suggested by Jolly (4) involves a central sulfur with four surrounding fluorine atoms, one at a greater distance than the other three. This suggests C3v symmetry and appears to this author to be unlikely (why would one S-F bond be different from the other three) and will not be further discussed in this paper.
Since the IR spectra for SF4, as observed by Dodd, Woodward and Roberts (7), shows at least five bands. the Td symmetry is eliminated. The C2v symmetry calculations predict more than five IR bands and thus this symmetry may be the geometry of the SF4 molecule. Even though this is negative 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 confirmed this structure using nuclear magnetic resonance and microwave spectroscopy, respectively.
Conclusions
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 been eliminated, yet, the principles can be easily illustrated. Possible structures for molecules can be eliminated by comparing calculated values for the number of IR and Raman active bands which can be compared to experimentally determined spectra. This will present the student with a practical application of group theory with which they can relate.
Equipment
These calculations were made on an IBM PS/2 Model 50 using a Quattro Pro 1.0 spreadsheet program for the JCE article. However, the calculations are very simple and can be made on any computer using any spreadsheet program. Copies of templates for several character tables and a typical home work assignment can be obtained by clicking here.
Acknowledgements
I wish to thank my class in Inorganic Chemistry for inspiring me to use spreadsheets in this manner. I wish to acknowledge the encouragement in writing this article given to me by David Jeter of Rhodes College and Dale Johnson of the University of Arkansas.
Literature Cited
1. Butler,I.S.;Harrod,J.F. Inorganic Chemistry, Principles and Applications; Benjamin/Cummings; Redwood City, CA,1989.
2. Cotton,F.A. Chemical Applications of Group Theory, 2nd ed.;Wiley-Interscience;New York, 1971.
3. Carter,R.L. J.Chem.Educ.1991,68,535-536.
4. Jolly,W.L. The Synthesis and Characterization of Inorganic Compounds; Prentice-Hall; Englewood Cliffs, NJ, 1970; p 300-306.
5. Jolly,W.L. The Synthesis and Characterization of Inorganic Compounds; Prentice-Hall; Englewood Cliffs, NJ, 1970; p 301.
6. Herzberg,G. Infrared and Raman Spectra of Polyatomic Molecules; Van Nostrand, Princeton,NJ, 1945, p.295. .
7. Dodd,R.E.;Woodward,L.A.;Roberts,H.L. Trans.Faraday Soc.1956,52,1052.
8. Cotton,F.A.;George,J.W.;Waugh,J.S. J.Chem.Phys.1958,28,994.
9. Tolles,W.M.;Gwinn,W.D. J.Chem.Phys.1962,36,1119.
*"Group Theory Calculations of Molecular Vibrations Using Spreadsheets" by S. M. Condren J. Chem. Ed., Vol. 71, pg. 486, (1994).