Research: Science and Education
Minimizing the Black Box Effect: Using Normal Mode Analysis to Integrate Computational Methods into the Physical Chemistry Course Delphia F. Harris*† Department of Chemistry and Physics, University of St. Thomas, 3800 Montrose Blvd., Houston, TX 77006; *
[email protected] Julio F. Caballero Argentine Consultants in Education, 2306 Branard, Houston, TX 77098
Computational methods have become an integral part of chemistry and are included in the education of future chemists. Some authors have effectively linked computational exercises with spectroscopic experiments (1–3). In particular, many molecular modeling packages are sophisticated and user friendly enough that it is possible to begin using them with little knowledge of either the methods or the theoretical basis on which they are founded. This ease of use may be an advantage when the software is used primarily for visualization. In physical chemistry, however, a program that requires minimal user input may obscure the process through which the calculated results are obtained and the theoretical basis upon which they rest. To minimize this “black box” effect (4), we have chosen to implement the area of normal mode analysis as an introduction to computational methods for students in physical chemistry. A critical component of the experience is the use of current research literature to obtain the necessary input values for the program package available from the Quantum Chemistry Program Exchange (QCPE) (5). An important feature of this package is that it requires significant ongoing interaction by the user. Additional normal coordinate analysis programs adapted to operate on personal computers are also available (6 ). The selection of small molecules of current interest in the literature makes this experience accessible as well as interesting and stimulating. Students explore the effects of symmetry and isotopic substitution on the infrared spectrum of one of these molecules and conduct a vibrational analysis during a period of several weeks as a case study for their course work (7, 8) and an introductory research experience. Selection of the Chemical Species Two criteria were established for the species to be studied in this project. First, the molecules must be small, owing to the increased complexity of the analysis for larger molecules. Second, they must be of current research interest. Why repeat the analysis of molecules such as H2O or CO2, for which the vibrational analysis was resolved many years ago? Students can experience an element of research in this project by analyzing a molecule that is the subject of current research. A wealth of small halogen-containing species appear in † Current address: LeMoyne-Owen College, 807 Walker Av., Memphis, TN 38126.
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the literature as possible contributors to the multiple pathways of stratospheric ozone depletion. It is now fairly well established that halogen atoms and reactive free radicals (Cl, ClO, Br, BrO, I, and IO) are involved in the chemistry of ozone depletion. Through chemical reactions these species are sources of triatomic and tetraatomic intermediates (XOO, OXO, XOXO, OXXO, and XOOX, with X = Cl, Br, I) that by photodissociation or chemical reaction participate in the ozone depletion scheme (9). The best-characterized series of intermediates is the chlorine oxides (10–15). Bromine oxides have been less studied (16–20), and little work has been published on the iodine oxides (21, 22). Students select molecules from the above series that include different symmetries and facilitate comparisons between halogenated species. The amount of information in the literature and the unresolved issues vary from species to species. Once the literature values for the vibrational frequencies and structural parameters have been obtained, not only can a normal mode analysis be conducted, but specific issues associated with the status of research on that species can be explored. In this way students are able to experience chemical research by determining new molecular information on the chemical species. Computational Method At the beginning of the semester students are assigned to do a literature search for vibrational frequencies and structural parameters (bond lengths and bond angles) for one of the species in the series above. Two months later they submit the results of the literature search and their calculated atomic coordinates before conducting the analysis using the General Vibrational Analysis System program package for personal computers (5). A general use license upgrade with permission to reproduce the documentation was purchased. The programs were loaded on all the departmental personal computers and were made available for students to load onto their own computers. The software package is designed in modules that break the problem into its components, which require user input at each stage. This makes the analysis conceptually more accessible to the students than if it were all done in a single program based on initial input. Figure 1 illustrates the flow of the analysis using this package. On the basis of the information from the literature search, they calculate the Cartesian coordinates for the atoms in the
Journal of Chemical Education • Vol. 76 No. 9 September 1999 • JChemEd.chem.wisc.edu
Research: Science and Education
molecule, assign the point group for the molecule, define the internal coordinates, and predict the number and symmetry of IR and Raman active bands using the methods learned in class (7, 8). They then estimate the initial internal force constants by comparison with similar compounds or from published values, if available. This constitutes the input for the UMAT program. UMAT calculates the symmetry coordinates and symmetrizes the initial force constants. The students compare these results with the symmetry species of the vibrations for their molecules, which they determined by hand using character tables (7, 8). Through the construction of the initial force constant matrix, it becomes evident how many parameters will be adjusted to fit the vibrational frequencies, in wavenumbers, and students realize why isotopic substitution is needed in order to have enough constraints to determine the force constants. The symmetrized force constants can be optimized to fit the experimental frequencies; however, the students note that force constants in the literature are typically reported for internal coordinates (10, 14). They use the Cartesian coordinates they calculated previously as input for the BMAT program, which can be used to produce the B matrix for use in optimizing the internal force constants and the Bd matrix for use in the determination of root-mean-square amplitudes. The ATOM2 program has a variety of functions, depending on the option code selected. The B matrix, vibrational frequencies, and initial force constants are used as input for this program. Students manually adjust the diagonal force constants first and then both the magnitude and sign of the off-diagonal interaction constants in order to improve the agreement of the calculated frequencies with the observed frequencies. Through this process they learn about the interaction between vibrational modes and the impact of the change of various force constants on the calculated frequencies. The FFIT program is then used for refinement of the force constants. Preliminary manual adjustment of the force constants is necessary not only from an educational perspective, but also because FFIT is merely a mathematical minimization algorithm, which does not take into account the symmetry of the normal coordinate generated (5). The students select the convergence criteria and the maximum number of iterations and have the opportunity to explore the impact of these values on the resultant force constants. They learned to ask themselves, with the aid of the instructor, whether the optimized force constants are physically reasonable or whether the routine should be repeated with different criteria. The optimized force constants are used as input into ATOM2. The final results include not only the optimized force constants but also the symmetry coordinates previously obtained, the potential energy distribution, the atom displacement matrix, and the root-mean-square amplitudes. Students draw diagrams representing the atom displacements for each normal mode. The potential energy distribution matrix indicates the percentage contribution of each internal coordinate to any given normal coordinate.
Owing to symmetry, there are only 4 unique force constants. UMAT clearly demonstrates the separation of the modes into a1 and b1 symmetry species (assuming that the plane of the molecule is the xz plane), as discussed in the lecture course. The four force constants for OClO and the diagonal force constants for OBrO optimized via the QCPE program package were within the range of values reported in the literature (10, 19). The results for these two molecules provided the necessary validation to proceed with other species for which force constants have not been reported. The force constants for OIO obtained in this project are presented in Table 1. For OXO, the force constants decrease as follows in the series X = Cl, Br, I: fr = 6.72, 5.38, and 5.10 mdyn A{1 and fα = 1.70, 1.06, and 0.48 mdyn A rad{2. Care has been taken to report each value with the proper use of significant figures (23) based on an assessment of the number of digits and uncertainty in the frequencies used for the analyses. These differ from compound to compound. As is typical, the program’s output values for force constants and average deviations have more digits than can be reported legitimately.
ClOO, BrOO, IOO Each of these species has Cs symmetry. There are 6 unique force constants for species in this series. All force constants for ClOO were optimized using the QCPE program package. Again, these results agreed within a few percent with published values (10). Since only three frequencies are available for BrOO and IOO, only the diagonal force constants could be optimized. A comparison of force constants for BrOO and IOO is given in Table 1. The only frequencies available in the literature for IOO are estimated (22) and include a value of 275 cm{1 for the IOO angle bending mode and 150 cm{1 for the I–O stretching mode. As indicated in Table 2, this results in a larger force constant for the IO stretch than for the IOO bend, in contrast with the results for ClOO and BrOO. Table 2 includes the potential energy distribution matrix for IOO and shows that using the published frequencies (22) leads to almost complete mixing of the IO stretching and angle bending modes. This mixing also contrasts with the results for ClOO and BrOO and provided an opportunity for students to explore changes in
Literature Search Cartesian coords. Internal coords. Initial force constants
UMAT Program
Cartesian coords. Internal coords. Frequencies Initial force constants
BMAT Program Bdmatrix B matrix
Symmetry coordinates Symmetry force constants
ATOM2 Program
FFIT Program
Results
OClO, OBrO, OIO Each of these species has C2v symmetry. For these species, the symmetry coordinates differ from the internal ones.
Potential energy distribution Atom displacement matrix Root mean square amplitudes
Figure I. General vibrational analysis package.
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the normal mode analysis as the frequency for the 275 cm{1 band was varied over a range of 275 to 400 cm{1. Through this exercise the students were able to see how changes in a single variable, one frequency, influence the results of the analysis. The potential energy distribution matrix calculated using 375 cm{1 for the angle bending mode is also shown in Table 2. Students were able to see that a higher frequency for the angle bending mode leads to force constants and potential energy distributions for IOO that are more consistent with the other halogens, but are left with the question of whether IOO will actually reflect this trend when experimental frequencies are published.
Tetraatomic Species The inclusion of at least one tetraatomic molecule is beneficial in expanding the concepts already explored. The calculation of the atomic Cartesian coordinates is significantly more challenging. Assigning the point group and the use of symmetry considerations in determining the number of unique force constants for these species reinforces and strengthens what students learn from the triatomic species. Tetraatomic species, including ClOOCl and ClOClO, have also been studied in this work. The peroxide structure for ClOOCl with C2 symmetry has 17 experimental frequencies available (14, 24). One vibrational mode has not been observed experimentally. An ab initio value for this frequency (15) is used. Jacobs et al. (14) have reported 13 force constants. The force constants in the current work compare well, within a few percent, with the published values, except for the fτ. Seven experimental frequencies, including isotopic substitution, have been reported for ClOClO (14 ). The two lowest-frequency modes have not been reported. The structure and the low-frequency modes were obtained from ab initio values (15). Continued work is in progress for this molecule and others in the tetraatomic series. Thus far, combination and overtone bands have been discussed but not included in the normal mode analysis. Use of these frequencies and the inclusion of anharmonicity effects could be incorporated as a more advanced aspect of the project. Learning Outcomes and Conclusion Junior and senior students in physical chemistry are introduced to current experimental and theoretical research on small halogen oxides through their reading of the current research literature. Each student works with a different triatomic species or a pair of students work together on a tetraatomic species. For some species they find conflicting assignments for infrared bands in early papers, which are later resolved in the literature. They find differing amounts of data available for the various species and adapt their analysis to the information available. This experience provides the opportunity for students to perform a computational vibrational analysis on a molecule as a case study for material learned in the course. One laboratory period and three class periods were set aside for students to learn to use the program and to work on their analysis. During these sessions they talked to each other about the analysis and had opportunities to ask the instructor questions. A handout explaining how to build data files and showing the conceptual flow chart in Figure 1 were provided. A list of the printed outputs to be included and
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Table 1. Force Constants and Structural Parameters for Some Triatomic Compounds of Stratospheric Interest Variablea
OIOb {1
BrOOc
IOO (275 cm{1)d
fR /mdyn A
—
10.37
10.6
fr /mdyn A{1
5.1
0.343
0.4e
fα /mdyn A rad {2
0.48
0.911
0.3e
{1
frr /mdyn A
0.52
—
—
Frequencies (no.)
3
3
3
Av deviation/cm{1 f
—
0.1
—
R /pm
—
121
12 5
r/pm
180
229
240
α (deg)
120
116
120
= r (OO); r = r ( XO); α = ∠( XOO). bTwo experimental frequencies are from ref 21 . The additional frequency and the structural parameters are estimated in ref 22. cThree experimental frequencies are available from ref 18 based on 18 O for one mode. The remaining frequencies and structural parameters are ab initio values from ref 17. dEstimated frequencies and structural parameters from ref 22. eBased on the estimated frequencies f = .452 and f = .259. Only r α 1 significant figure is recorded, to reflect the uncertainty in the frequencies used. fAverage deviation of calculated frequencies from observed ones. For OIO and IOO the average deviation is much less than the uncertainty in the frequencies used. aR
Table 2. Force Constants and Potential Energy Distributions for IOO Force Constant Matrix O–O
∠ IOO I –O
O–O 10.56 ∠ IOO 0.26 I –O
0.45 O–O
∠IOO I–O
O–O 10.56 ∠ IOO 0.67 I –O
0.37
Potential Energy Distribution 1500 cm{1 275 cm{1 150 cm{1 .996
.000
.003
.003
.500
.498
.008
.500
.499
1500 cm{1 375 cm{1 150 cm{1 .996
.001
.004
.002
.145
.852
.002
.854
.144
the results to be explained in the project paper and oral presentation was also provided. Other faculty from the department were invited to listen to the project presentations. Those who attended remarked about the depth and sophistication of questions and discussions that followed the presentations. The similarity in species facilitated the students’ ability to engage each other in substantive discourse based on their own experience. The issue of the proximity of the estimated frequency for the IOO bend to that of the IO stretch (which greatly impacts the mixing of these two modes) is one example of the students’ experience of finding and exploring new problems that were not predicted at the start. This shows a clear case of a typical research pattern of finding “unexpected” new goals while trying to solve an initial problem. Initial force constants have been obtained for a number of species (BrOO, IOO, and
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Research: Science and Education
OIO) on the basis of frequencies and structural parameters currently available in the literature. The literature search indicates that no force constants have been reported for these species at this time. Acknowledgment This work was partially supported by Welch Departmental Research Grant No. AV-0024. Literature Cited 1. Mina-Camilde, N.; Manzanares, C. I; Caballero, J. F. J. Chem. Educ. 1996, 73, 804. 2. Sorkhabi, O.; Jackson, W. M.; Daizadeh, I. J. Chem. Educ. 1998, 75, 238. 3. Williams, D. L.; Minarik, P. R.; Nibler, J. W. J. Chem. Educ. 1996, 73, 608. 4. Bauer, S. H. J. Chem. Educ. 1990, 67, 692. 5. McIntosh, D. F.; Peterson, M. R. General Vibrational Analysis System; Quantum Chemistry Program Exchange (QCPE), Department of Chemistry, Indiana University: Bloomington, IN. The program is No. QCMP067 (1988) in the QCPE catalog. Additional information may be obtained by contacting the Quantum Chemistry Program Exchange, Creative Arts Building 181, Indiana University, Bloomington, IN 47405. Phone: 812/855-4784; Fax: 812/855-5539; Email:
[email protected].
6. Barlow, A.; Diem, M. J. Chem. Educ. 1991, 68, 35. 7. Barrow, G. M. Physical Chemistry, 6th ed.; McGraw-Hill: New York, 1996; pp 530–549. 8. Atkins, P. W. Physical Chemistry, 4th ed.; Freeman: New York, 1970; pp 426–459. 9. Wayne, R. P. Atmos. Environ. 1995, 29, 246. 10. Muller, H. S. P.; Willner, H. J. Phys. Chem. 1993, 97, 10589. 11. Peterson, K. A.; Werner, H.-J. J. Chem. Phys. 1992, 96, 8948. 12. Johnsson, K.; Engdahl, A.; Nelander, B. J. Phys. Chem. 1993, 97, 9603. 13. Byberg, J. R. J. Phys. Chem. 1995, 99, 13392. 14. Jacobs. J.; Kronberg, M.; Muller, H. S. P.; Willner, H. J. Am. Chem. Soc. 1994, 116, 1106. 15. Lee, T. J.; Rohlfing, C. M.; Rice, J. E. J. Chem. Phys. 1992, 97, 6593. 16. Tevault, D. E.; Walker, N.; Smardzewski, R. R.; Fox, W. B. J. Phys. Chem. 1978, 82, 2733. 17. Pacios, L. F.; Gomez, P. C. J. Phys. Chem. 1997, 101, 1767. 18. Tevault, D. E.; Smardzewski, R. R. J. Am. Chem. Soc. 1978, 100, 3955. 19. Muller, H. S. P.; Miller, C. E.; Cohen, E. A. Angew. Chem. Int. Ed. Engl. 1996, 35, 2129. 20. Chase, M. W. J. Phys. Chem. Ref. Data 1996, 25, 1069. 21. Gilles, M. K.; Polak, M. L.; Lineberger, W. C. J. Chem. Phys. 1992, 96, 8012. 22. Chase, M. W. J. Phys. Chem. Ref. Data 1996, 25, 1297. 23. Caballero, J. F.; Harris, D. F. J. Chem. Educ. 1998, 75, 996. 24. Stevens, P. S.; Anderson, J. G. J. Phys. Chem. 1992, 96, 1708.
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