Article pubs.acs.org/jchemeduc
Using Atomic Orbitals and Kinesthetic Learning To Authentically Derive Molecular Stretching Vibrations Adam J. Bridgeman,*,† Timothy W. Schmidt,† and Nigel A. Young‡ †
School of Chemistry, The University of Sydney, NSW 2006, Australia Department of Chemistry, The University of Hull, HU6 7RX, United Kingdom
‡
S Supporting Information *
ABSTRACT: The stretching modes of MLx complexes have the same symmetry as the atomic orbitals on M that are used to form its σ bonds. In the exercise suggested here, the atomic orbitals are used to derive the form of the stretching modes without the need for formal group theory. The analogy allows students to help understand many conceptually difficult aspects of the vibrational spectroscopy of polyatomic molecules, including IR activity, degeneracy, and orthogonality. The approach intentionally requires active participation and is nonmathematical and “low tech”. It is suitable for large classes of chemistry majors and nonmajors. KEYWORDS: Second-Year Undergraduate, Inorganic Chemistry, Physical Chemistry, Hands-On Learning/Manipulatives, Inquiry-Based/Discovery Learning, Misconceptions/Discrepant Events, Group Theory/Symmetry, IR Spectroscopy, Spectroscopy
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spectroscopy for which the shapes of atomic orbitals are assumed knowledge. In our implementation, the approach is developed through active learning worksheets and peer discussion groups in large lectures held in tiered seating settings. The worksheets follow the methods developed by the POGIL project5 that has shown the effectiveness of student centered and active learning methods in chemistry. Two worksheets are included in the Supporting Information. Alongside discussions and a requirement for students to commit their ideas to paper, the classroom work also lends itself to encouraging students, and sometimes the lecturer, to become physically active as they “act out” the vibrations. Alongside providing an opportunity to maintain or prolong student attention in class,6 these activities greatly assist in planting the learning experience in the long term memory. The use of physical movement to enhance learning (kinesthetic learning) is well established in secondary education,7 yet it is also a practice that can be used in higher education.8 Recent studies of the use of movement in statistics9 and computer sciences teaching,10,11 for example, suggests that, when opportunities arise, such techniques are worth exploring in the sciences too.
eing able to visualize the motions of the atoms involved in a molecular vibration is key to understanding and demystifying the basics of vibrational spectroscopy, including the application of selection rules and the factors affecting vibrational frequencies. Group theory is a very powerful tool for generating vibrational modes. Unfortunately, it is also, when first encountered at least, extremely abstract. It requires a dedicated course or series of workshops on the mathematical tools involved before the more satisfying applications such as derivation of the number, spectroscopic activity, and form of the molecular vibrations are within a student’s grasp. Its highly conceptual nature also means that, if it is taught at all, it may be included toward the end of the undergraduate degree or in an elective. In spectroscopy courses that precede a formal course in group theory, it is therefore common to simply provide illustrative examples only such as the symmetric and asymmetric modes of linear and bent triatomics. In lectures and online, these are often portrayed using the excellent visualization tools which are available.1−4 These are very powerful but they do not readily allow students to construct their own understanding and mental images. Displaying an animation may actually be a didactic experience in a lecture and may provide only surface learning that is not readily transferrable. When students discover or work out the form of vibrations themselves, such as in a group theory course, they are more actively engaged and the learning becomes more authentic. In this paper, we discuss one way we use to enable students to develop an understanding of the form of molecular vibrations and to construct simple symmetry modes without formally being taught group theory. It uses an analogy between the well-known form of the atomic orbitals and the phase of the atomic motions. It is thus suited to an introductory course on © XXXX American Chemical Society and Division of Chemical Education, Inc.
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RELATIONSHIP BETWEEN ORBITALS AND VIBRATIONS The correspondence between the ligand group orbitals and the normal vibrational modes of complexes or molecules with the general formula MLx is well-known.12 As shown in Figure 1 for a trigonal planar ML3 molecule, the in-plane (δ||), out-of-plane (δ⊥) and stretching (νs) motions of each L can be obtained by appropriately aligning and combining the local x, y, and z
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number and shape of the atomic orbitals are offered usually with minimal mathematical justification. However, students rapidly become accustomed to these properties as well as the meaning of nodes and degeneracy in the context of building the periodic table. The least well understood property of atomic orbitals for most students at this stage is probably the meaning of the phase of the orbital, which is commonly presented using shaded or colored lobes (or in older texts + and -), or is glossed over completely.
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Figure 1. Coordinate system for: (A) the local displacements and (B) the local bonding of the ligands in a trigonal planar ML3 molecule.
STRETCHING MODES Given the relationship noted above between the stretching modes and the ligand group orbitals, it is thus possible to derive the stretching modes by matching with the atomic orbitals. Students are presented with two ideas: 1. The number of stretching modes is equal to the number of bonds in the complex 2. The form of the stretching modes can be constructed by analogy with the atomic orbitals on the central metal used to make σ bonds. If the students have knowledge of hybrids, the second point can be clarified by stating that the atomic orbitals needed are just those used to construct hybrid orbitals along the bond directions. Thus in CH4, the student will choose to use the carbon 2s, 2px, 2py and 2pz orbitals. In BH3, the student will choose to use the boron 2s, 2px and 2py as 2pz is not used if sp2 hybridization is assumed. However, this step is not actually necessary. If an inappropriate atomic orbital is chosen, such as the boron 2pz orbital in BH3, the student will find that no stretching mode can be constructed and will begin to construct a bending motion instead. Similarly, if a student uses an orbital of a higher l quantum number than is necessary, they will simply construct a mode that is the same as one of those already derived or will construct a rotation. A few examples will make the utility of this analogy clear. The topic of normal modes is commonly introduced using the simplest case: a triatomic molecule such as CO2 or H2O. For these molecules, there are only two possible stretching modes: the symmetric and asymmetric combinations of the individual bond stretches. Given this result, students are then shown the analogy between the shape of the atomic orbitals of C and these motions. This is illustrated in Figure 2a and 2b, taken from worksheet 1 in the Supporting Information, for the symmetric and asymmetric stretches, respectively. The direction of the arrow represents the relative motion of the atoms and is chosen by directly analogy with the phase of the s and pz atomic orbitals on C, respectively. When the lobe of the atomic orbital
coordinates. The z-axis of each ligand is directed toward the central M atom. The σ-component of each M−L bond points along the bond. The z-coordinates, and hence the M−L stretches, transform identically to the M−L σ-bonds: a1′ + e″. The π-components of the bond lie in two orthogonal planes that are perpendicular to the bond vector, just as the bending motions do. The x-coordinates, and hence the M−L in-plane bends, transform identically to the in-plane M−L π||-bonds. The y-coordinates, and hence the M−L in-plane bends, transform identically to the out-of-plane M−L π⊥-bonds. Similarly, for a tetrahedral molecule such as CH4, the σ bonds and the C−H stretching modes both transform as a1 + t2. The symmetry and, by extension, the form of the vibrations can thus be derived if the ligand group orbitals are already known. As we sought a method suitable for a course that precedes any formal group theory, such an approach is limited, as it requires derivation of the ligand group orbitals. Less satisfactorily, the ligand group orbitals can be presented to the students. It is our experience that the abstract nature of molecular orbitals means that many students find the visualization of molecular orbitals and their construction to be difficult. Vibrations are a much more tangible concept. They can be animated, either by a person or on a computer, and a knowledge of them leads to the satisfaction of being able to assign vibrational spectra at a fairly high level. Being able to understand the relationship between the molecular shape and the number of peaks in the vibrational spectrum is rewarding. Such understanding can be used, for example, to immediately explain the difference between the structural information provided by vibrational and NMR spectra. In both, the number of peaks is used to help detect any symmetry in the molecular structure. In NMR, the number of peaks is simply related to the equivalent nuclei. In vibrational spectroscopy, however, it also provides evidence on the molecular shape. To extract such information, the student must be able to appreciate how the change in geometry during a vibration affects the dipole moment and polarizability. It can be counterproductive to introduce ligand group orbitals before normal modes. The meaning of phase, nodes, eigenvalues, normalization, and degeneracy are all, in our view, better explained in the context of vibrations than through molecular orbitals and molecular orbital diagrams. Derivation of normal modes using the method outlined below allows students to use their existing “symmetry instincts” and to develop their intuition of how group theory works when applied to chemical problems. When group theory cannot be assumed knowledge, the construction of ligand group orbitals is often presented by matching with the atomic orbitals of the central atom. Atomic orbitals are presented in introductory courses, in the first year of an undergraduate degree or perhaps even at high school. The
Figure 2. Excerpt from a worksheet on stretching vibrations showing the analogy between the atomic orbitals and the stretching vibrations in CO2. B
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Figure 3. Excerpt from a worksheet on stretching vibrations showing the analogy between the atomic orbitals and the stretching vibrations in CH4 and H2O.
C atom at the center and the H atoms on four of the corners. The derivation of the stretching modes is illustrated in Figure 3a,b, taken from worksheet 1 in the Supporting Information. Alongside the simple prediction of IR activity that follows from knowledge of these modes, the degeneracy of the three t2 modes is readily apparent to the students as they construct and act out the vibrations. Indeed, some students simply draw one of these and state that the other two are equivalent and simply related by rotation. The orthogonal nature of the modes is also apparent. In small and advanced group tutorials, it may be reasonable to extend this exercise by asking students to use a different definition of the axes, with one axis passing along a C−H bond. The stretching modes of water were derived above by descent in symmetry from a linear to a bent triatomic. As sp3 hybrids are often used to form the σ-bonds and lone pairs of water, it is also possible to generate the vibrations using the same set of atomic orbitals used above for CH4, as illustrated in Figure 3c. As can be seen, the four modes of CH4 become two pairs of identical vibrations, which correspond to stretching and compression parts of the symmetric and asymmetric modes of H2O. The stretching modes of other complexes can be derived in the same way. For the square planar, octahedral, and trigonal bipyramidal cases, d-orbitals on the central atom must be included. Figure 4 shows the construction of the b1g vibration of a square planar complex and the eg vibrations of an octahedral complex (see worksheet 1 in the Supporting Information). These illustrate a number of important points that often confuse students when vibrations are simply presented. As the z-axis is contained in nodal planes of the dx2−y2 orbital, there is no motion of the two M−L bonds of the octahedral complex which lie along this axis in the matching stretching mode. In the dz2 orbital, the lobes are larger along the z-axis and so the amplitude of the stretch of the bonds in this direction is also larger. Although the motions of the six bonds
is shaded black, the corresponding motion is a stretch of the bond. When the lobe of the atomic orbital is shaded white, the corresponding motion is a compression of the bond. Figure 2c shows that a student choosing to use a dz2 orbital on the central atom will generate an identical vibration to that found by analogy to an s-orbital in Figure 2a. Figure 2d shows that use of a dxz or dyz orbital leads to a molecular rotation. Clearly, only one-half of the vibration is depicted. At the end of this half of the vibration, the relative motions are reversed to produce the familiar vibrational oscillation. To underline this, half of the class can be given instructions in which the correspondence between the shade of the lobe and the motion is reversed. This can lead to some useful discussion of the meaning of the phase of the atomic orbital, based on the more tangible concept of the vibration. The corresponding motions of water can be established without further work using descent in symmetry: that is, just by considering the same vibrations of a bent molecule. At this point, the IR activity of the modes can be discussed and contextualized by reference to atmospheric chemistry.13 Students are asked to consider both the size of the dipole moment and the direction in which it points at all stages of the vibration. If either of these factors changes, the vibration is IR active. This is also a good point to address the difference between IR and NMR: the bonds are equivalent in CO2 and in H2O but only one stretching mode is IR-active in the former while both are active in the latter. Without knowledge of the form of the molecular vibrations, it is easy for students to confuse the different information contained within the number of peaks in an IR and in a NMR spectrum. Using this analogy, the students can then be set the task of deriving the form of the stretching modes and the number of IR bands expected for CH4. As noted above, the analogy requires the use of the carbon 2s, 2px, 2py and 2pz orbitals and these are most conveniently directed toward the faces of a cube with the C
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Figure 6. Possible choreography to demonstrate the stretching modes of a square planar MX4 complex or molecule.
standing that is constructed is deep and is carried over into subsequent years and into the laboratory. Indeed, it is gratifying to watch students acting out vibrational modes when discussing spectral assignments. If a student elects to take the material further in a formal group theory course, the basis of the analogy and the nature of the results obtained become apparent. For these students, the exercise is very useful as a way of developing their symmetry intuition. For example, the meaning of terms such as “breathing modes” and “totally symmetric” become more evident when discussed in the context of the vibrations illustrated in Figures 3−5. Similarly, identification of the irreducible representations of the modes is a good exercise in showing the meaning of the symbols in character tables.
Figure 4. Excerpt from a worksheet on stretching vibrations showing the analogy between the b1g and eg atomic orbitals in square planar and octahedral complexes and the matching stretching vibrations.
are not the same in the two modes, overall each bond moves the same amount when the modes are considered as a pair. The comparison between the vibrations derived from the dx2−y2 functions is useful for discussing descent in symmetry and the transferability of the ideas being developed. In a trigonal bipyramidal complex, both the s orbital and the dz2 orbital of the same atom transform as a1′. In this case, the normal and symmetry modes are not equivalent, but the orbital analogy can still be used to derive approximate forms for the normal modes, as illustrated in Figure 5.
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EXTENSIONS If desired, the analogy can also be used to illustrate or, at least, introduce bending modes. However, as these may span irreducible representations for which no atomic orbitals exist, this may not generate a complete set. For most organic and inorganic chemists, it is the stretching modes that are of most interest. Most organic molecules do not possess the symmetry present in the complex illustrated in the examples above. Nevertheless, the exercise represents a good introduction to coupling between the vibrations of equivalent or nearly equivalent bonds, and this can be readily transferred to more complex systems. For example, once a student has a good understanding of the stretching modes of a trigonal pyramidal molecule such as NH3, the form of the stretching modes of a −CH3 group in a large organic molecule can be readily recognized. While the exercise presented here literally generates a “hand waving” treatment, it can also be used to introduce a more formal definition of normal modes. For example, if the asymmetric stretching mode of CO2, shown in Figure 2, is acted out, the small movement of the C atom that is required to conserve the center of mass arises naturally. Similarly, the use of atomic orbitals leads to orthogonality that a more formal treatment can build on. For students wishing to pursue these ideas further, an additional exercise can be constructed using a molecule such as ethylene in which there is no central atom. As shown in Figure 7, the four C−H stretching motions can be derived by using an s, px, py and dxy orbital on an imaginary atom at the center. This exercise is particularly suited to being followed up in a group theory course by asking students to identify the symmetry label of each mode. The exercise presented here has been used successfully in introductory spectroscopy courses at the second-year level in Australia and in the U.K. The approach is deliberately nonmathematical and is suitable to chemistry majors and nonmajors. We have found that it engages students of all ability levels as it is enjoyable yet readily allows for extension activities. Although it is intentionally hands-on and low tech, we also
Figure 5. Excerpt from a worksheet on stretching vibrations showing the approximate form of the two a1′ stretching vibrations in a trigonal bipyramidal ML5 complex.
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KINESTHETIC ACTIVITIES After constructing the vibrations in this way for modes that are analogous to the s, p, and d-orbitals of the central atom, the final task for students is to discover that only modes that are analogous to p-orbitals are IR-active. To ensure that these concepts are remembered, a kinesthetic activity can be used to complete the exercise. Figure 6 illustrates suggested choreography for the stretching modes of a square planar ML4 complex. Worksheet 2 in the Supporting Information shows how this can be used in a classroom setting. The learning outcomes for this exercise suit the needs and backgrounds of students taking an introductory course in spectroscopy and quantum mechanics. The active nature of the students’ learning means that, in our experience, the underD
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Figure 7. Excerpt from a worksheet on stretching vibrations showing the analogy between the atomic orbitals on an imaginary atom at the center of ethylene and the C−H stretching vibrations.
provide additional electronic resources, including animations of the modes discussed.
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ASSOCIATED CONTENT
S Supporting Information *
Worksheet 1, showing how atomic orbitals can be used to derive stretching modes, with CH4 and SF6 presented as exercises; extension activities on choosing which atomic orbitals to use and how to derive bending modes are also included; Worksheet 2, introducing students to the selection rules in Raman spectroscopy and presenting kinesthetic activities for the classroom. This material is available via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
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