Letter Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX
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On Teaching Molecular Term Symbols: From the Atomic Term Symbols in a Two-Dimensional World to the Molecular Term Symbols in a Three-Dimensional World Yingbin Ge*
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Department of Chemistry, Central Washington University, Ellensburg, Washington 98926, United States ABSTRACT: In the article “A Systematic Method for Determining Molecular Term Symbols for Diatomic Molecules Using Uncoupled-States Orbital Diagrams” in this Journal, Campbell introduced a systematic method to determine the molecular term symbols for diatomic molecules. This method is introduced in various undergraduate Physical Chemistry textbooks because of its practicality. However, there remains one question that may baffle some students: Why is |ML| = Λ used when determining the molecular term symbols whereas it is |ML| ≤ L for atomic term symbols? This letter aims to provide a more comprehensive yet clear answer to this question, which, I hope, will help students better understand molecular term symbols and polish their creative problem-solving skills. KEYWORDS: Upper-Division Undergraduate, Physical Chemistry, Analogies/Transfer, Problem Solving/Decision Making, MO Theory, Theoretical Chemistry term symbols, whereas ML can adopt 2L + 1 values (|ML| ≤ L) for atomic term symbols. This letter aims to provide a more comprehensive explanation: When the internuclear axis of O2 is chosen to be the z axis (Figure 1a), the rotation of the πy* orbital about the x or y axis separates this orbital from the nuclei and thus is unphysical (Figure 1b−c), whereas its rotation about the z axis is allowed (Figure 1d) and, in fact, barrier-less, as the πx* and πy*orbitals rotate concertedly.
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systematic method for determining molecular term symbols for diatomic molecules was proposed by Campbell in 1996.1 This method is introduced in various Physical Chemistry textbooks.2−6 The O2 (or B2) molecule is often used as an example to illustrate the determination of the molecular term symbols for its π2 electron configuration: 3Σ−g , 1Δg, and 1Σ+g .1−7 What is missing, or not fully explained, in these papers and textbooks is the reason why ML can only be equal to ±Λ for molecular
Figure 1. (a) The antibonding πy* orbital of O2. (b−d) The πy* orbital after a 2D rotation about the x or y or z axis, respectively. Received: May 6, 2018 Revised: July 24, 2018 © XXXX American Chemical Society and Division of Chemical Education, Inc.
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DOI: 10.1021/acs.jchemed.8b00334 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Letter
Overall, Figure 1 shows that the rotation of the πy* orbital is strictly a two-dimensional (2D) rotation about the z axis. Similarly, the πx* ± iπy*orbitals cannot rotate about the x or y axis but can rotate freely about the z axisjust imagine a cylindrical electron density distribution. Therefore, the proof for ml = ±l and ML = ±L for the rotation of atomic orbitals in a 2D atom8 can be extended to 3D diatomic molecules such as O2. In short, the wave function for a 2D atomic orbital is ψ =
better understand the molecular term symbols and to develop creative problem-solving skills.
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Corresponding Author
*E-mail:
[email protected]. ORCID
Yingbin Ge: 0000-0001-5315-9312
R(r)Φ(ϕ).8 R is the radial function of r (r = x 2 + y 2 ) and Φ(ϕ) = eimlϕ, where ml = ±l and l is the magnitude of the total angular momentum of the 2D rotation.8 Similarly, the wave function of the πx* ± iπy* orbital of the 3D O2 is ψ = Z(z) R(r)Φ(ϕ), where the Φ function is exactly the same as that in a 2D atom. Because the atomic and molecular terms are determined by the angular function Φ (neither Z nor R is relevant here), the molecular terms of the 3D O2 with a π2 electron configuration are essentially the same as the atomic terms of a 2D atom with a p2 electron configuration. Table 1
Notes
The author declares no competing financial interest.
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ml
ML = 1
↓↑
∑ ml ,i
MS =
i
∑ ms ,i i
L (or Λ), S
2D atomic term (or 3D molecular term) D (or 1Δ)
−2 2
0 0
2,0
1
↓↑ ↑ ↑ ↓
↑ ↓ ↓
0 0 0
1 0 −1
0,1
3
↓
↑
0
0
0,0
1
REFERENCES
(1) Campbell, M. L. A Systematic Method for Determining Molecular Term Symbols for Diatomic Molecules Using Uncoupled-States Orbital Diagrams. J. Chem. Educ. 1996, 73 (8), 749−751. (2) McQuarrie, D. A.; Simon, J. D. Physical Chemistry: A Molecular Approach; University Science Books: Sausalito, CA, 1997. (3) Cooksy, A. Physical Chemistry: Quantum Chemistry and Molecular Interactions; Pearson: Boston, 2014. (4) Atkins, P.; de Paula, J. Elements of Physical Chemistry, 6th ed.; Oxford University Press: Oxford, 2013. (5) Atkins, P.; de Paula, J.; Keeler, J. Physical Chemistry, 11th ed.; Oxford University Press: Oxford, 2017. (6) Engel, T.; Reid, P. Physical Chemistry, 3rd ed.; Pearson: Boston, 2012. (7) Carlton, T. S. Why the Lower-Energy Term of Singlet Dioxygen Has a Doubly Occupied Π* Orbital. J. Chem. Educ. 2006, 83 (3), 477−479. (8) Ge, Y. Let Students Derive, by Themselves, Two-Dimensional Atomic and Molecular Quantum Chemistry from Scratch. J. Chem. Educ. 2016, 93 (12), 2033−2039.
Table 1. Six Microstates of the p2 Electron Configuration of a 2D Atom (or the π2 Electron Configuration of a 3D O2 Molecule)
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AUTHOR INFORMATION
S (or 3Σ)
S (or 1Σ)
illustrates that the p2 electron configuration of a 2D atom results in six microstates, which are grouped into the 1D, 3S, and 1S atomic terms.8 Also in Table 1, but enclosed in the parentheses, the π2 electron configuration of a 3D O2 molecule results in the same six microstates, which are grouped into the 1 Δ, 3Σ, and 1Σ molecular terms. The only difference between the molecular term symbols of a 3D O2 molecule (π2) and the atomic term symbols of a 2D “carbon” atom (p2) is the use of Greek vs English letters. Similarly, one can determine the molecular terms for more complicated electron configurations of any diatomic molecule. For example, the molecule term symbols for a δ2 electron configuration of a diatomic molecule are 1Γ, 3Σ, and 1Σ given the 1G, 3S, and 1S atomic term symbols for the d2 electron configuration of a 2D atom.8 In conclusion, I would like to advocate using the above explanation in the classroom when teaching molecular term symbols, and in Physical Chemistry textbooks, because determining the molecular term symbols via deductive reasoning would be more effective than memorizing the rules. It is nonideal to have students memorize that |ML| ≤ L should be used for atomic term symbols but |ML| = Λ for molecular term symbols. Whereas by visualizing the two unphysical rotations and the one allowed rotation of the πy* orbital of O2 (Figure 1), students should be able to justify the extension from the derivation of the 2D atomic terms8 to the determination of the 3D molecular terms. This extension will enable students to B
DOI: 10.1021/acs.jchemed.8b00334 J. Chem. Educ. XXXX, XXX, XXX−XXX
