In the Classroom
Use of Molecular Symmetry To Describe Pauli Principle Effects on the Vibration–Rotation Spectroscopy of CO2(g) M. L. Myrick,* P. E. Colavita, A. E. Greer, B. Long, and D. Andreatta Department of Chemistry and Biochemistry, University of South Carolina, Columbia, SC 29208; *
[email protected] A common experiment in the physical chemistry laboratory is the measurement of the infrared rotation–vibration spectrum of HCl(g), which allows students the opportunity to explore quantization of rotational states in diatomic molecules (1–7). In the HCl experiment, a heteronuclear diatomic molecule is studied in the gas phase. This simple vibration–rotation experiment affords an excellent opportunity for students to become familiar with the analysis of rotational constants and bond lengths in such molecules. Homonuclear diatomic molecules exhibit alternating spectral intensities in their rotational transitions owing to the equivalence of the atoms, with relative intensities that depend on the nuclear spins of the atoms. This is a consequence of the Pauli principle, one of the fundamental concepts in atomic physics and chemistry. While these molecules are simpler in a molecular sense than HCl, they are more difficult to study in the undergraduate laboratory because their infrared vibrational transitions are forbidden. Raman spectroscopy is required to perform rotational studies of these molecules. The opportunity to spectroscopically demonstrate the Pauli principle is thus generally lost to the undergraduate laboratory because of the more complex apparatus required. In a recent experiment reported in this Journal (8), the infrared-allowed transitions of carbon dioxide, a linear, symmetric triatomic molecule were presented. The presence of equivalent oxygen atoms in CO2 results in fine structure of the infrared transitions that is governed by the Pauli principle. We have found the CO2 vibration–rotation spectrum particularly useful as an addition to the HCl experiment mentioned above for several reasons. First, it uses the same apparatus as the HCl experiment and is even simpler to perform. Second, the familiarity gained with interpreting rotational fine structure in the HCl experiment is reinforced. Third, this experiment offers students the opportunity to observe a Q band in the infrared CO2 degenerate bending vibration, a phenomenon students generally hear about but rarely see. Fourth, it offers the opportunity to explain the infrared spectra observed for atmospheric gases, and how this can be used to make atmospheric measurements. Lastly, it gives students an opportunity to understand how the Pauli principle is properly interpreted in terms of the symmetry or antisymmetry of wavefunctions, allowing them to see further consequences of a rule that is the underlying cause of the Pauli exclusion principle they learned as freshmen: that no two electrons can possess the same set of quantum numbers in the same system at the same time. By using this experiment after the HCl experiment, students can focus more on the physical principle than on learning the basics of vibration–rotation spectroscopy. The argument used in our laboratories to derive the missing-level consequences of the Pauli principle on the rotational states of CO2 is presented, based in part on a related presenwww.JCE.DivCHED.org
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tation in ref 9. The effect of exchanging labels on the oxygen atoms in CO2 is derived by conceptually breaking down the exchange into rotation, vibration, electronic, and nuclear coordinate components. The sign of each of these suboperations is used to deduce the overall symmetry of each state with respect to label permutation. This analysis is used to deduce that all odd values of the rotational quantum number, J, are forbidden in the ground vibrational state of CO2, while all even J values are forbidden in the first vibrationally-excited state of the asymmetric stretching vibration. The same argument may be applied to the spectroscopy of other molecules of interest to undergraduates, such as atmospheric oxygen. The Pauli Principle and the Permutation Operator Exchange of the labels of identical particles in a wavefunction cannot change the probability distribution of the wavefunction. Since the distribution is given by the absolute square of the wavefunction, exchanging the labels of a pair of identical particles can change the sign of a total wavefunction without violating this rule. For example, if we write the total wavefunction of CO2 as a function of all the particles in the molecule (nuclei and electrons), the effect of changing the labels on, for instance, particles 1 and 2 can be represented in the form: ψtot(1, 2, 3, ...) = ±ψtot(2, 1, 3, ...) = ±P^12ψtot(1, 2, 3, ...) (1) In eq 1, the operator on the right is the permutation operator for particles 1 and 2, where particles 1 and 2 are identical particles, for example, electrons or some other particle. Identical in this case means that the particles are of the same type and, in the case of nuclei, are symmetrically equivalent to one another. In the case of CO2, exchange of the oxygen atom labels is illustrated as the topmost operation in Figure 1. A nonequilibrium position in the asymmetric stretching vibration has been illustrated (with appropriate exaggeration) in the figure, and the nuclei of the oxygen atoms have been labeled as particles 1 and 2. As the figure shows, exchange of the labels affects nothing else about the molecule; even the nuclear spins (α and β in Figure 1) remain fixed. All total wavefunctions must be either symmetric (+) or antisymmetric (−) with respect to the permutation of identical particles. “Bosons” is the name given to particles that always give the symmetric result. Other particles, called “Fermions”, always give the antisymmetric result. The particle spin is the factor that determines which rule will be obeyed by the particle. Particles with 1/2-integral spins (e.g., electrons, some nuclei, etc.) always obey the antisymmetric rule. The exchange of two electrons, for example, always inverts the sign of the wavefunction: they are Fermions because
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their spins do not have an integer value. Electrons follow the rule known to most undergraduates as the Pauli exclusion principle because if two electrons could have identical quantum numbers, their labels could be exchanged with no change in sign. But the wavefunction must change sign when electron labels are exchanged because electrons are Fermions. Consequently, no two electrons can possess the same quantum numbers. Some common nuclei and their nuclear spins (I ) are: 12C, 14C, 16O, 18O, 32S
I = 0:
2α
180° rotation (whole molecule) 2β
1α
1β nuclear spin exchange
2β
180° rotation (electrons)
1α 180° rotation (vibration)
2β
2H, 14N
I = 1: I = 3/2:
11B, 33S, 35Cl, 37Cl
I = 5/2:
17O 10B
The reader will note that nuclei with odd numbers of nucleons are always Fermions, while those with even numbers of nucleons are invariably Bosons. The major isotope of oxygen is a Boson with I = 0. Implications of the Pauli Principle for Rotational Levels In the case of a homonuclear diatomic molecule like 1H2, the two hydrogen nuclei are indistinguishable. Their spins are 1/2 and the exchange of the nuclear labels must invert the sign of the total H2 wavefunction. Nuclei in polyatomic molecules are also subject to the restrictions of the Pauli principle whenever they are identical and are in symmetrically equivalent positions. ψtot, the total molecular wavefunction described in eq 1, can be separated (at least formally) into subcomponents that help to evaluate the consequences of the Pauli principle. These subcomponents are the electronic wavefunction (a function of the coordinates of the electrons), the vibrational wavefunction (a function of internal vibrational coordinates), the rotational wavefunction (a function of external coordinates) and the nuclear spin wavefunction (a function of the spin coordinates of the nuclei). The decomposition of the total wavefunction is illustrated as: (2)
ψtot = ψelψvibψrotψnuc
When we apply the permutation operator to exchange the labels of a pair of identical 16O nuclei, the effect of the operation on all these terms, taken together, must be symmetric. The effect of permuting the labels of the two oxygen nuclei in CO2 can be analyzed by imagining a path between the initial state and the permuted state in Figure 1 by choosing steps that can each be analyzed separately. Symmetry can be used to help us in this analysis. We can propose a set of symmetry operations that accomplish the task we desire by operating on the electronic, vibrational, rotational, and nuclear wavefunctions separately. Such a series of operations is illustrated as the bottom pathway in Figure 1. An arrow extending from the center of the molecule has been added to serve as a placeholder for the electronic coordinates of the molecule, so that we can see the effects of our operations on 380
2β
1α
1H, 3H, 13C, 13N, 15N, 15O, 31P, 19F
I = 1/2:
I = 3:
P12
1α
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Figure 1. Permutation of the labels of the two oxygen atoms in CO2. Top path: effect of the permutation operator. Bottom path: a set of symmetry operations giving the effect of the permutation operator. The symmetry of the overall operation can be deduced from the symmetries of the rotational states (1st step), electronic state (2nd step), vibrational state (3rd step) and nuclear spin state (4th step). CO2 is illustrated in a nonequilibrium position in the asymmetric stretching vibration to track behavior of the vibrational coordinates. An arrow serves to keep track of the electronic coordinates.
the electronic coordinates. Also, the asymmetric stretching vibration has been shown in a nonequilibrium position so that the effect of operations on it can be analyzed. The justifications in the following paragraphs are provided for students who are unfamiliar with group theory and symmetry labels. If laboratory students are familiar with group theory, the symmetries of the electronic and vibrational states provide a direct means for obtaining the results summarized in Table 1. The group theory-based approach should be preferred over the following, more qualitative statements if it is feasible for a particular group of students.
Rotation of the Whole Molecule The first step on the path to accomplish the exchange of labels is rotation of the whole molecule by 180⬚. This motion takes place during free rotation of the molecule and is governed by the rules for rotational states. The derivation of the rigid rotor wavefunctions and quantized energy levels is identical to the derivation for the spatial component of atomic wavefunctions, meaning that the lowest ( J = 0) rotational state has symmetry properties similar to an s orbital. The J =
Table 1. Summary of Pauli Principle Analyses Based on Figure 1 Pauli Principle Contribution
Coordinates
States
Rotational
J = Odd
−
Rotational
J = Even
+
Electronic
G.S.
+
Vibrational
v=0
+
Vibrational
v=1
−
Nuclear
1
+
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Figure 2. One cycle of the symmetric stretching vibration. All states of this vibration are symmetric with respect to rotation of the vibrational coordinates, meaning that the ground vibrational state is also symmetric.
Figure 3. One cycle of the asymmetric stretching vibration. Rotation of the vibrational coordinates changes the phase of this vibration by 180 degrees. Like any oscillatory function, this inverts the sign of the oscillation. The first vibrational state of the asymmetric stretch is thus antisymmetric with respect to rotation of the rotational coordinates.
0 state has symmetry properties similar to a p orbital, and so forth. Like s, d, and other atomic orbitals with even values of the quantum number l, rotational states with even values of J are symmetric (+) with respect to 180⬚ rotation. Like p, f, and other atomic orbitals with odd values of l, rotational states with odd values of J are antisymmetric (−) with respect to 180⬚ rotation.
(180⬚) earlier. Like a sine function, it changes sign when this happens. In other words, the rotation of the vibrational coordinates converts the first excited state of the asymmetric stretching vibration into its negative. In summary, the lowest vibrational level (v = 0) of the asymmetric stretching vibration is symmetric with respect to inversion of the vibrational coordinates. However, the v = 1 level of this vibration is antisymmetric with respect to inversion.
Rotation of the Electronic Wavefunction. The second step along the path in Figure 1 is a 180⬚ rotation of the electronic wavefunction, leaving the nuclei and spins and vibrational wavefunctions unchanged. Most molecules in their ground electronic states (although there are exceptions such as NO and O2) are in totally symmetric electronic configurations. CO2 is not an exception to this rule-of-thumb, meaning that the molecule is symmetric (+) with respect to the rotation of electronic coordinates.
Rotation of the Vibrational Wavefunction. The third step along the path in Figure 1 is the rotation involving only the vibrational coordinates. Symmetry with respect to this operation requires a bit of explanation. Consider Figure 2, which shows the symmetric stretching vibration of CO2 and how the bond lengths change during the stretch for one full cycle (exaggerated). At any point in this vibration, if you rotate the vibrational coordinates (e.g., exchange the two C⫽O bondlengths), the molecule would be unchanged. All the vibrational levels of this vibrational mode, including the lowest, are thus symmetric with respect to this type of symmetry operation. Since the lowest vibrational level of all of CO2’s vibrations (including the asymmetric stretch) are exactly the same (i.e., there is only a single lowest vibration level, and it has zero quanta of every vibration), this lowest vibrational level is symmetric with respect to the rotation depicted in Figure 1. The asymmetric stretching vibration, however, is a bit different. This vibration is illustrated in Figure 3. As illustrated in the figure, rotation of the vibrational coordinates converts the molecule into what it looked like half a cycle www.JCE.DivCHED.org
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Nuclear Spin Exchange The fourth and final step along the bottom path in Figure 1 is that of nuclear spin exchange. Nuclear spins of identical nuclei couple together to form symmetric and antisymmetric combinations, just as two electrons can couple to form collective spin states. An excellent discussion of this can be found in refs 10 and 11. In the case of electrons and other particles with spins of 1/2, this coupling can produce spin states with a net spin of zero, or spin states with a net spin of one. There is a single S = 0 state that can be formed from two electrons, while there are three S = 1 states that can be created. The former of these collective states is called a singlet spin state (for obvious reasons), and it is antisymmetric with respect to exchange of the spins. The latter is called a triplet spin state, and it is symmetric with respect to exchange of the spins. The number of symmetric and antisymmetric spin states that can be created from two particles whose spins are coupled depends on the spins of the individual particles. A general equation for the number of symmetric and antisymmetric spin states is easily derived as, total symm = (I + 1)(2I + 1)
(3)
total antisymm = (I )(2I + 1)
(4)
where I is the spin quantum number. In the case of identical oxygen nuclei and other pairs of nuclei with I = 0, the total number of antisymmetric nuclear spin functions is zero. There is a single symmetric nuclear spin function.
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Application The symmetry of the total wavefunction in CO2 with respect to the exchange of the labels of the two oxygen nuclei must be (+) because oxygen nuclei are bosons. Since the nuclear spin and electronic wavefunctions in Table 1 are only symmetric (+), eq 2 tells us that the product of the vibrational and rotational wavefunctions must be symmetric as well. This only happens when both have the same symmetries. Thus, a given combination of vibrational and rotational wavefunctions is permitted only if both are (+) or both are (−) with respect to rotation by 180⬚. In other words, in the lowest vibrational level (v = 0) of the asymmetric stretching vibration, only rotational levels that are symmetric (even J ) are allowed to exist. In the first vibrational level (v = 1) of the asymmetric stretching vibration, only rotational levels that are antisymmetric (odd J ) are allowed to exist. Thus, in both the v = 0 and v = 1 vibrational levels, half the rotational states do not exist. The consequence of the Pauli principle on the vibration– rotation spectrum of CO2 is that, while P and R branch lines still occur in the spectrum, the missing states in the v = 0 level cause every other line in the spectrum to be absent. This effect would not be observed if the two oxygen atoms were not subject to the Pauli principle; indeed, the isotopic molecule 18OC16O has been observed to retain all its rotational lines (12). Observation of the CO2 spectrum provides arguably the simplest opportunity to observe direct effects of the Pauli principle, a component of quantum mechanics that most students (and instructors) find nonintuitive. Interpreting the spectrum without the use of this principle results in estimated C⫽O bond lengths that are smaller by a factor of √2 than for the same bond in carbon monoxide, for example. While this experiment is a simple extension from the common laboratory experimental measurement of the HCl(g) vibration– rotation spectrum, an even more apt experimental pairing is
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with the carbon monoxide diatomic infrared spectrum, since it provides a method for determining the C⫽O bond length in the absence of Pauli principle complications. Acknowledgments The authors gratefully thank R. Bruce Dunlap for the opportunity to work on new laboratory development in CHEM542L at University of South Carolina. PEC gratefully acknowledges support from the NSF graduate fellowship program. AEG, BL, and DA thank the Department of Chemistry and Biochemistry for support of this work. Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
Iannone, M. J. Chem. Educ. 1998, 75, 1188. Pu, Q. J. Chem. Educ. 1990, 67, 1034. Roberts, B. J. Chem. Educ. 1966, 43, 357. Stafford, F. E.; Holt, C. W.; Paulson, G. L. J. Chem. Educ. 1963, 40, 245. Rank, D. H.; Eastman, D. P.; Rao, B. S.; Wiggins, T. A. J. Optical Soc. Am. 1962, 52, 1–7. Rank, D. H.; Rao, B. S.; Wiggins, T. A. J. Mol. Spectrosc. 1965, 17, 122–130. Plyler, E. K.; Tidwell, E. D. Z. Electrochemie 1960, 64, 717– 720. Ogren, P. J. J. Chem. Educ. 2002, 79, 117–119. McHale, J. L. Molecular Spectroscopy; Prentice Hall: New Jersey, 1999; pp 223–225. Levine, I. N. Molecular Spectroscopy; John Wiley and Sons: New York, 1975; pp 178–183. Levine, I. N. Quantum Chemistry 4th ed.; Prentice-Hall: New Jersey, 1991; pp 261–267 Herzberg, G. Molecular Spectra and Molecular Structure: Infrared and Raman of Polyatomic Molecules, Vol. 2; Van Nostrand: New York, 1990; p 17.
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