Teaching the Spin Selection Rule: An Inductive Approach - Journal of

Article pubs.acs.org/jchemeduc

Teaching the Spin Selection Rule: An Inductive Approach Judith A. Halstead* Department of Chemistry, Skidmore College, Saratoga Springs, New York 12866, United States S Supporting Information *

ABSTRACT: In the group exercise described, students are guided through an inductive justification for the spin conservation selection rule (ΔS = 0). Although the exercise only explicitly involves various states of helium, the conclusion is one of the most widely applicable selection rules for the interaction of light with matter, applying, in various ways, to atoms and molecules of all sizes. Connections are made among several concepts routinely covered in physical chemistry courses including the Pauli Principle, orthonormal wave functions, overlap integrals, atomic term symbols, multiplicity, radiative lifetimes, fluorescence, and phosphorescence. Detailed student directions are included in the Supporting Information.

KEYWORDS: Upper-Division Undergraduate, Physical Chemistry, Inquiry-Based/Discovery Learning, Atomic Spectroscopy, Fluorescence Spectroscopy, Quantum Chemistry different atomic species”1 or “two different kinds of helium”.2 The validity, for very light atoms, of the assumptions used in this exercise is demonstrated by the very unusually long radiative lifetime of the helium 23S1 state, 8000 s.3,4 States with accessible, fully allowed radiative transitions in helium have lifetimes in the nanosecond range.5 For larger atoms and molecules, the assumption that the spatial component and the spin component of each system act independently is not as good as it is for helium. For the case of larger atoms and molecules, the term “spin-forbidden” needs to be interpreted in terms of relative probability. Forbidden transitions are less likely for heavy atoms and molecules than otherwise similar spin-allowed transitions. For example, in dyes (organic molecules with extensive pi systems) phosphorescence, a radiative spin-forbidden triplet-to-singlet transition is “less allowed” than fluorescence, a radiative singlet-to-singlet transitions, but phosphorescence is still observed in many molecules as a weak transition. The lifetimes of excited states of organic molecules undergoing fluorescence are generally in the nanosecond to microsecond range, whereas the lifetimes of those undergoing phosphorescence are generally much longer, in the microsecond to second range.6,7 Chi, Im, and Wegner8 studied a series of oligofluorenes at 77 K and found the fluorescence radiative lifetimes for singlet states emitting to the ground state to be hundreds of picoseconds. Phosphorescence radiative lifetimes, emissions from the triplet state for the same series of molecules under the same conditions, were a few milliseconds.8 Another example of the relative nature of the term “spin forbidden” for heavier atoms and molecules is the relative natural radiative lifetimes of the Hg 63P1 and 61P0

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he spin selection rule is essential in discussing the difference between fluorescence and phosphorescence and explains the existence of two forms of helium, ortho-He and para-He. It is one of the most widely applicable spectroscopic selection rules for the interaction of light with matter, applying, in various ways, to large atoms, small atoms, and molecules of all sizes. Prior to discussing spectroscopic selection rules, physical chemistry courses and textbooks distinguish between symmetric and antisymmetric wave functions and state the Pauli principle: wave functions for describing a many-electron system must be antisymmetric with respect to the exchange of any two electrons. Generally, a connection is not made between the spin selection rule and the concepts of symmetric and antisymmetric wave functions. In fact, after presenting the Pauli principle, physical chemistry courses and texts often do not return to or use the Pauli antisymmetry principle again. In the exercise described here, teams of students are guided through an inductive argument using concepts and techniques that they learned when studying the Pauli antisymmetry principle. They conclude that spin is conserved during transitions between electronic states. This group activity reinforces and makes connections among several concepts routinely covered in physical chemistry courses (e.g., the Pauli Principle, orthonormal wave functions, overlap integrals, atomic term symbols, multiplicity, radiative lifetimes, fluorescence, and phosphorescence). This guided-discovery exercise can be used in the classroom or as an assignment to be completed outside of class in groups. Although it is most appropriate for physical chemistry classes, it may also be useful for some other upperlevel courses. Transitions between para-helium and ortho-helium are so rigorously forbidden that they were once considered “two © XXXX American Chemical Society and Division of Chemical Education, Inc.

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dx.doi.org/10.1021/ed200795t | J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

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states, both of which have well-known transitions to the ground 1 S0 state. The Hg 63P1 state has a natural radiative lifetime of 122 ns9 compared to the Hg 61P0 state that has a lifetime of approximately 1 ns.10 While the traditional, deductive, and expository-based methods of teaching are still common in chemical education today, there has been considerable discussion of inductive teaching methods in recent decades.11−15 The activity described here can be considered a discovery approach based on the categories suggested by Domin:16 this exercise takes an inductive approach, the outcome is predetermined and the procedure is given to the students. Activities or laboratory experiments with these characteristics have also been labeled “guided-inquiry” activities.6,2 The exercise described here is not a laboratory experiment. However, rubrics designed for laboratory experiments are referred to because, to date, rubrics designed to categorize undergraduate laboratory experiments16,17 rather than in-class or homework exercises dominate the literature. Buck and co-workers17 describe a rubric for “characterizing the level of inquiry in the undergraduate laboratory”. By their rubric,17 an experiment or activity, like the one presented here (in which the problem, theory, and procedure are given by the instructor but the results and conclusions are arrived at by the students) is categorized as a “level 1” inquiry activity or “guided inquiry”. Kirschner et al.18 provide evidence that teaching using “strong instructional guidance” is more effective for novice to “intermediate learners” than “instruction using minimal guidance”.

Physical chemistry students have some experience integrating spatial wave functions that have continuous variables. Although spin wave functions do not really depend on continuous variables, it is convenient in this exercise to treat them as if they can be integrated over spin variables, s1 for electron 1 and s2 for electron 2. The key point is that we define the one-electron spin functions αi and βi as orthonormal functions. Students may already know that the one-electron spin functions αi and βi for electron i form an orthonormal set. In the portion of Part I where students are guided through an analysis of spin wave functions, the following notation is introduced:

∫

(2a)

all − si

∫

β*(si)β (si)dsi = ⟨βi ,βi ⟩ = 1 (2b)

all − si

∫

α*(si)β (si)dsi = ⟨αi ,βi ⟩ = 0 (2c)

all − si

∫

β*(si)α (si)dsi = ⟨βi ,αi⟩ = 0. (2d)

all − si

The student instructions also note that elementary 2-electon spin function inner products can be factorized into the product of two one-electron spin function inner products. For example:

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THE EXERCISE One appropriate time to use this exercise in physical chemistry courses is after the introduction of atomic term symbols and before discussion of spectroscopic selection rules. Students are divided into groups of three for an exercise investigating the effect of spin on the likelihood that a spectroscopic transition will occur. There are three parts to the exercise and instructors can choose to use some or all of the parts.

⟨α1β2 , α1β2⟩ =

∬

α(s1)*β(s2)*α(s1)β(s2) ds1 ds1

all − s1, all − s 2

=

∫

α(s1)*α(s1) ds1

all − s1

∫

β(s2)*β(s2) ds2

all − s 2

= ⟨α1 ,α1⟩⟨β2 ,β2⟩ = 1 × 1 = 1

(3a)

and

Part I: Determination of the Spin−Orbital Wave Functions for the Ground State and First Two Excited States of Helium

⟨α1α2 , β1β2⟩ =

∬

α(s1)*α(s2)*β(s1)β(s2) ds1 ds1

all − s1, all − s 2

In the first part of the exercise, students are led through a series of questions helping them discover the appropriate spin− orbital wave functions corresponding to the 1s2 and 1s12s1 electronic configurations of He. The exercise does not require the students to have prior familiarity with spectroscopic term symbols. In the beginning of Part I, the concepts of symmetric and antisymmetric wave functions, the Pauli principle, and orthonormal sets are reviewed as students answer and discuss the questions in the exercise. The second section of Part I deals with the ground state of He arising from the 1s2 electronic configuration. Chemists often approximate the wave function for each possible state of an atom with a spin−orbital wave function, χi. These spin− orbital wave functions are a product of a function of the electronic spatial coordinates only (Ψi) and a spin function (ϕi), which depends only on electron spin coordinates. That is, χi = ψϕ i i

α*(si)α (si)dsi = ⟨αi ,αi⟩ = 1

=

∫ all − s1

α(s1)*β(s1) ds1

∫

α(s2)*β(s2) ds2

all − s 2

= ⟨α1 ,β1⟩⟨α2 ,β2⟩ = 0 × 0 = 0

(3b)

The notation introduced in 2a through 2d is standard mathematical inner product notation. Students may or may not have already encountered inner product notation in mathematics classes. However, because the notation used is entirely defined within the exercise, no prior experience with inner product notation is necessary. The inner product notation is optional. Instructors who believe their students may be uncomfortable with it can use the integral notation for the entire exercise. However, the exercise responses are quicker and easier for the students to write, and easier for the instructor and students to read, when written in inner product notation rather than integral notation. After completing their work on the possible spatial components and the possible spin components separately, students are asked to combine the appropriate spatial wave function and spin wave function to determine the spin−orbital wave function for the He ground state. The exercise reminds

(1)

In assuming that the wave function can be separated in this way, we are considering the interaction between the spatial component and the spin component of the wave function to be “vanishingly small”.19 B



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the students that the spin−orbital wave functions have the following properties: • they can be written as a product of spatial and spin components, • they are consistent with the Pauli principle, that is, because electrons are indistinguishable fermions, the 2electron wave functions must change sign (be antisymmetric) when the labels on any two electrons are exchanged. For the He ground state (1S0, a singlet, with paired electrons, both in the 1s orbital), there is no way to write the spatial component of the wave function as antisymmetric, so the spin component must be antisymmetric. Guided by the questions in the exercise, the students arrive at a complete normalized spin− orbital wave function for the He ground state, 1S0: χgs (r1, r2, s1 , s2) = (1s(r1)1s(r2))

χ1, +1 (r1, r2, s1 , s2) =

[α(s1)α(s2)] χ1, −1 (r1, r2, s1 , s2) =

(4)

Part II is the heart of the entire exercise and, unlike in Parts I and III, the students are arriving at answers they most likely have not seen before. Unless students are very well prepared for this type of experience, Part II may be best done in the presence of the instructor. The instructor can monitor the progress of various student groups and provide helpful hints as appropriate. For helium, a specific case of a two-electron system with Hamiltonian Ĥ o (in the absence of light), the corresponding orthonormal spin−orbital eigenfunctions may be more generally written χj (r1, s1, r2, s1) where j labels the state. The probability, P (i → j), that the system undergoes a spectroscopic transition from the state χi to the state χj depends, among other things, on the relationship between the final and initial states and the nature of the perturbation. Specifically, for a system initially in state i and under the influence of Ĥ p, a perturbing Hamiltonian, the dependence of P (i → j) on χi and χj is given by eq 9:20,21 P(i → j) ∝ |

χi* (r1, s1 , r2, s2)Ĥ pχj 2

(r1, s1 , r2, s2) ds2 d3r2 ds1 d3r1|

(9)

where the total Hamiltonian, Ĥ = Ĥ o + Ĥ p. In electronic spectroscopy, when a system undergoes a transition from one electronic state to another under the influence of light, the perturbing Hamiltonian is due to an electromagnetic field. A spectroscopic transition is forbidden if P (i → j) = 0 and allowed if P(i → j) ≠ 0. The term “forbidden” is used as a relative term in spectroscopy, as is discussed more in Part III. The electric field component of the electromagnetic radiation (EMR) does not act on the spin component of the wave function.22 In general, magnetic fields do interact with electron spin, but we will consider the magnetic field component of EMR to be sufficiently small that the perturbation Ĥ p does not act on the spin component of the wave functions. Hence, eq 9 can be rewritten as eq 10:

(5)

1 1 (1s(r1)2s(r2) − 1s(r2)2s(r1)) 2 2 [α(s1)β(s2) + α(s2)β(s1)]

∫ ∫ ∫ ∫ all − r1 all − s1 all − r2 all − s 2

where ψ = 2−1/2(1s(r1)2s(r2) + 1s(r2)2s(r1)) is the symmetric spatial component and ϕ = 2−1/2[α(s1)β(s2) − α(s2)β(s1)] is the antisymmetric spin component. The other appropriately paired spin−orbital wave functions corresponding to the 1s12s1 electronic configuration correspond to the 23S1 triplet and may be represented as: χ1,0 (r1, r2, s1 , s2) =

(8)

Part II: Discovering the Spin Selection Rule

1 1 (1s(r1)2s(r2) + 1s(r2)2s(r1)) 2 2 [α(s1)β(s2) − α(s2)β(s1)]

1 (1s(r1)2s(r2) − 1s(r2)2s(r1)) 2

In the absence of a magnetic field, these three states (eqs 6, 7, and 8) are degenerate and correspond to the first excited triplet energy level, 23S1. Students are also told that the overall electron spin for each of these three states is S = 1. The exercise instructions tell the student to check their answers with the instructor before proceeding to Part II. It may be useful to only hand out Part II after students had successfully completed Part I. Alternatively, Part I can be given as homework before Part II is done in class or during a laboratory session.

where Ψ = 1s(r1)1s(r2) is the symmetric spatial component, ϕ = 2−1/2[α(s1)β(s2) − α(s2)β(s1)] is the antisymmetric spin component, the vectors r1 and r2 refer to the positions of electrons 1 and 2, respectively, s1 and s2 are spin coordinates for electrons 1 and 2, respectively, α refers to one spin orientation (e.g., ms = 1/2 or up), β refers to the other spin orientation (e.g., ms = −1/2 or down) and 1s(ri) refers to a 1s hydrogen-like spatial orbital occupied by the ith electron. Students are reminded that the overall electron spin for this state is S = 0. As written, the exercise has the students normalize the spatial components of the wave functions, the spin components of the wave functions and the overall spin−orbital wave functions for each state. (Normalization can be time-consuming for some students and is not essential for Part II. Instructors may decide to not have their students use normalized wave functions to shorten the exercise.) The third section of Part I deals similarly with the excited states of helium corresponding to 1s12s1 electronic configuration. Working as a group, students determine which of the spin wave functions can be paired with each of the appropriate spatial wave functions. They then can write an approximate spin−orbital wave function with both spatial and spin components consistent with the Pauli principle guided by the exercise questions. They conclude that the first excited singlet state (21S0) of He may be represented as: χ0,0 (r1, r2, s1 , s2) =

(7)

[β(s1)β(s2)]

1 [α(s1)β(s2) 2

− α(s2)β(s1)]

1 (1s(r1)2s(r2) − 1s(r2)2s(r1)) 2

(6) C



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the basis of these four cases only, they propose a general spin selection rule for electronic transitions. Although a few students needed a bit of prompting, most students, with varying degrees of self-confidence, conclude that transitions with ΔS equal to zero are spin-allowed and transitions with nonzero ΔS values are spin-forbidden. If students need additional prompting, the instructor might start with helping the students summarize their results so far. For example, ask “For how many transitions have you evaluated Pspin?”, “For each of these transitions is ΔS zero or nonzero?” and “Can you make a reasonable guess for the relationship between Pspin and ΔS based only on these four cases?” The exercise instructions tell the student to check their answers with the instructor before proceeding to Part III. Instructors may choose to hand out Part III only after students have successfully completed Part II. Alternatively, Part III can be given as homework.

P(i → j) ∝

∫all−r1 ∫all−r2 ψi*(r1, r2)Ĥ pψ (r1, r2) d3r2 d3r1

2

2

∫ ∫

ϕi*(s1 , s2)ϕj(s1 , s2) ds2 ds1 (10)

all − s1 all − s 2

The impact of the spin component of the wave functions on the probability of a radiative transition between electronic states i and j, P(i → j), is then given by eq 11: 2

Pspin(i → j) =

∫ ∫

ϕi*(s1 , s2)ϕj(s1 , s2) ds2 ds1


(11)

The quantity inside the absolute value sign, a type of overlap integral, can be represented as follows, using inner product notation: ⟨ϕi ,ϕj⟩ =

∫ ∫

Part III: What Do Experimental Results Tell Us?

Part III could be a class discussion or a homework assignment following Part II. This part begins with a description of parahelium and ortho-helium. All of the singlet states collectively are known as para-helium and all of the triplet states collectively are known as ortho-helium. An energy level diagram for helium (which generally separates the para-helium and ortho-helium states) is helpful in discussing helium and is readily available.2,23 Students are asked to define fluorescence and phosphorescence and to explain the relative intensities of fluorescence and phosphorescence in terms of their results in the exercise. They are asked to consult the experimental literature and find experimental lifetimes associated with fluorescence and phosphorescence and to explain these lifetimes in terms of their results above.

ϕi*(s1 , s2)ϕj(s1 , s2 , ) ds2 ds1


(12)

or Pspin(i → j) = |⟨ϕi , ϕj⟩|2

(13)

If Pspin(i → j) is zero, the transition is spin-forbidden. If Pspin(i → j) is nonzero, the transition is spin-allowed. Of course, there is still a spatial component to the transition probability and the transition may be forbidden for other, non-spin-related, reasons. For spectroscopic transitions, various features of the spatial component of the transition probability, as well as the frequency of the transition, also affect the probability. The first set of questions in Part II guides students, working together, to evaluate Pspin(i → j) for the transition from 1 1S0 to 2 1S0. Several helpful hints are given in the instructions for the exercise. Students are reminded that the one-electron spin functions αi and βi form an orthonormal set and that 2-electon spin functions can be factorized into the product of two oneelectron spin functions. Students collect together the terms for electron #1 and collect together the terms for electron #2. Some students find the procedure straightforward as soon as they read the questions whereas others hesitate and need some extra prompting before proceeding smoothly. After evaluating Pspin(i → j) for the transition from 11S0 to 21S0, each group discusses together whether or not this a spin-allowed or spinforbidden transition. They are also asked to note whether ΔS for the transition is zero or nonzero. In the second section of Part II, students are asked to assign each of the three 23S1 states to individuals within their group. Students individually determine whether or not the transition from the 11S0 state to their assigned 23S1 state is spin-allowed or spin-forbidden. After all students have completed their work on the three transitions, they discuss within their group whether each transition is spin-allowed or spin-forbidden and also whether ΔS is zero or nonzero. Students may help each other if necessary. However, once students have successfully completed the evaluation of Pspin(i → j) for the transition from 11S0 to 21S0, they do not seem to find the evaluation of Pspin(i → j) for the other three transitions problematic. In the last section of Part II, the culmination of the exercise, students are asked to make a “great leap of inductive faith”. On

Concluding the Exercise Experience

In this exercise, we make two approximations: • the wave functions can be written as a product of a function of the electronic spatial coordinates only (Ψi) and a spin function (ϕi) which depends only on electron spin coordinates, and • the perturbing Hamiltonian, Ĥ p, due to the electromagnetic field does not interact significantly with the spin component of the atom. After working through a series of questions in teams, the students conclude that transitions between orthogonal spin states are forbidden and so ΔS = 0 for spin-allowed transitions. Spin also plays a role in the chemical kinetics of small molecules and atoms.24,25 Spin, although potentially an important consideration, is generally neglected in undergraduate classes on kinetics as most molecules of interest are singlets. The ground state of molecular oxygen is a good example of a common, important triplet state. Spin may be very important in reactions involving free radicals or unusual spin states, for example, in atmospheric chemistry, combustion, and photochemistry.

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STUDENT EXPERIENCE AND RESPONSE Depending on the preparation of the class and the individual students, Part I takes the students anywhere from 45 to 90 min. To shorten the exercise substantially, the instructor might choose to have the students not normalize the wave functions. The essence of the exercise in Part II, the discovery of the spin selection rule, does not require normalized wave functions. D



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in physical chemistry which might otherwise be perceived by the student as unrelated.

Normalization, while potentially time-consuming, does provide practice in using important concepts discussed in lecture. Part I would make an appropriate homework assignment as it is largely review for many classes. If Part I is given as homework, the instructor should consider discussing eqs 3a and 3b and the factorization of 2-electron spin functions into the product of two one-electron spin functions, before assigning the homework. Similarly, depending on the preparation of the class and the individual students, Part II takes the students anywhere from 45 to 90 min. Because the students are arriving at answers they most likely have not seen before, the presence of the instructor while students work together on Part II is very helpful. Therefore, it is recommended that Part II is done in class or lab so the instructor can provide hints to any groups that get stuck. Part III could be a class discussion or a homework assignment following Part II. Student feedback, on an anonymous survey taken by the students after they had completed the exercise, indicates that the students believed this exercise was worthwhile. The instructor conducting the exercise was not the usual instructor for the class and did not know the majority of students. Eight of the eight students indicated that this exercise was “very helpful” or “somewhat helpful” in “understanding the difference between symmetric and antisymmetric wavefunctions”. Seven of the eight students indicated that this exercise was “very helpful” or “somewhat helpful” in “understanding and remembering the Pauli principle”. All of the eight students indicated that as a result of having completed the exercise, they will be “better able to answer questions on future exams related to symmetric and antisymmetric wavefunctions and the Pauli Principle”. All of the eight students indicated that they believed this exercise should be “included in second semester physical chemistry in future years”. One student added the comment that “everything seems a lot clearer now.”

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ASSOCIATED CONTENT

* Supporting Information S

A detailed student procedure and additional information for the instructor. 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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ACKNOWLEDGMENTS I gratefully acknowledge many helpful suggestions from, and fun conversations with, Christopher A. Chudzicki. REFERENCES

(1) Dushman, S. Elements of the quantum theory. IX. The helium atom. Part I. The perturbation method. J. Chem. Educ. 1936, 13, 132− 140. (2) Herzberg, G. Atomic Spectra and Atomic Structure (Spinks, J. translator); Dover Publications: New York, 1944. (3) Woodworth; Moos, H. W. Experimental Determination of the single photon transition rate between the 23S1 and 1 1S0 states of He I. Phys. Rev. A 1975, 12, 2455−2463. (4) McKinsey, D. N.; Brome, C. R.; Dzhosyuk, S. N.; Golub, R.; Habicht, K.; Huffman, P. R.; Korobkina, E.; Lamoreaux, S. K.; Mattoni, C. E. H.; Thompson, A. K.; Yang, L.; Doyle, J. M. Spectroscopy of isolated gas phase parahelium lifetime. Phys. Rev. A 2003, 67, 0627161−062716-9. (5) Heron, S.; McWhirter, R. W. P.; Rhoderick, E. H. Measurements of lifetimes of excited states of helium atoms. Proc. R. Soc. London, Ser. A 1956, 234, 565−583. (6) Berezin, M.; Achilefu, S. Fluorescence lifetime measurements and biological imaging. Chem. Rev. 2010, 110, 2641−2684. (7) McGown, L. B.; Nithipatikom, K. Molecular fluorescence and phosphorescence. Appl. Spectrosc. Rev. 2000, 35, 353−393. (8) Chi, C; Im, C; Wegner, G. Lifetime determination of fluorescence and phosphorescence of a series of oligofluorenes. J. Chem. Phys. 2006, 124, 024907-1−024907-9. (9) Halstead, J. A.; Reeves, R. R. Determination of the lifetime of the mercury 63Pl state. J. Quant. Spectrosc. Radiat. Transfer 1982, 28, 289− 296. (10) Smirnov, B. M. Physics of Atoms and Ions; Springer-Verlag: New York, 2003. (11) Briggs, M.; Long, G.; Owens, K. Qualitative assessment of inquiry-based teaching methods. J. Chem. Educ. 2011, 88, 1034−1040. (12) Prince, M. J.; Felder, R. M. Inductive teaching and learning methods: Definitions, comparisons, and research bases. J. Eng. Educ. 2006, 95, 123−138. (13) Spencer, J. New approaches to chemistry teaching. J. Chem. Educ. 2006, 83, 538−533. (14) Farrell, J. J.; Moog, R, S.; Spencer, J. N. A guided inquiry general chemistry course. J. Chem. Educ. 1999, 76, 570−574. (15) M., A.; Ricci, R. W. Discovery chemistry: Balancing creativity and structure. J. Chem. Educ. 1994, 71, 685−688. (16) Domin, D. A review of laboratory instruction styles. J. Chem. Educ. 1999, 76, 543−547. (17) Buck, L. B.; Bretz, S. L.; Towns, M. H. Characterizing the level of inquiry in the undergraduate laboratory. J. Coll. Sci. Teach. 2008, 38, 52−58. (18) Kirschner, P. A.; Sweller, J.; Clark, R. E. Why minimal guidance during instruction does not work: An analysis of the failure of

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SUMMARY In this group activity, students discover for themselves a key concept in spectroscopy by using quantum mechanics principles previously discussed in class. The assumption that the spin component and spatial component of the wave function are independent is also an assumption we make when using Russell−Saunders coupling in the determination of atomic term symbols. The validity of this assumption for very light atoms is demonstrated by the very long radiative lifetime of triplet helium. For heavier atoms and molecules, where the spin angular momentum and orbital angular momentum cannot be reasonably considered as independent, the term “spin forbidden” needs to be interpreted in terms of relative probability. The conversation with students on spin states is extended to a discussion of the difference between fluorescence and phosphorescence. This exercise can be extended to include concepts as disparate as the role spin considerations play in the chemical kinetics of small molecules and atoms24,25 and the fluorescence of vegetables.26,27 Further extensions of this exercise might include a class discussion with students on why the triplet state is always lower in energy than the corresponding singlet state.28,29 The students should read the excellent article by Rioux28 before class if the relative energies are to be discussed. This exercise facilitates the students own use of theory to discover an important concept with practical consequences and makes connections between several concepts routinely covered E



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constructivist, discovery, problem-based, experiential and inquiry based teaching. Educ. Psychol. 2006, 41, 75−86. (19) Pilar, F. Elementary Quantum Chemistry; McGraw Hill; New York, 1968. (20) Scherrer, R. Quantum Mechanics: An Accessible Introduction; Benjamin Cummings: San Francisco, CA, 2006. (21) Griffiths, D. J. Introduction to Quantum Mechanics; Pearson Prentice Hall: Upper Saddle River, NJ, 2005. (22) Atkins, P. W. Molecular Quantum Mechanics; Oxford University Press: New York, 1983. (23) Hyperphysics, Department of Physics and Astronomy, Georgia State University. http://hyperphysics.phy-astr.gsu.edu/hbase/ quantum/helium.html (accessed Apr 2012) (24) Laidler, K. J. The Chemical Kinetics of Excited States; Oxford University Press: London, 1955. (25) Harvey, J. N. Understanding the kinetics of spin-forbidden chemical reactions. Phys. Chem. Chem. Phys. 2007, 9, 331−343. (26) MacCormac, A.; O’Brien, E.; O’Kennedy, R. Classroom activity connections: Lessons from fluorescence. J. Chem. Educ. 2010, 87, 685−686. (27) Muyskens, M.; Stewarts, M. Getting students of all ages excited about fluorescence. J. Chem. Educ. 2011, 88, 259. (28) Rioux, F. Hund’s multiplicity rule revisited. J. Chem. Educ. 2007, 84, 358−360. (29) Snow, R; Bills, J . The Pauli principle and electron repulsion in helium. J. Chem. Educ. 1974, 51, 585−586.

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Teaching the Spin Selection Rule: An Inductive Approach - Journal of

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