Why the Lower-Energy Term of Singlet Dioxygen Has a Doubly Occupied ␲* Orbital

Research: Science and Education

Terry S. Carlton Department of Chemistry, Oberlin College, Oberlin, OH 44074; [email protected]

Introductory treatments of molecular orbitals draw spe- 1 ؉ cial attention to the ground state of dioxygen, which is un- Σg usual in being a triplet state and hence paramagnetic. Those 1 treatments explain these properties in terms of the numbers ∆g and spins of electrons in two degenerate orbitals. Dioxygen’s ؊ 3 first two excited states, known collectively as singlet oxygen, Σg also deserve special pedagogical attention (though probably not until an inorganic or physical chemistry course). The relative energies of these two spin-paired singlet terms are sur- Figure 1. Occupancy of π* orbitals in the three lowest-energy terms prising: the term with a pair of singly occupied degenerate of O2. orbitals is higher in energy than the term in which one of those orbitals is doubly occupied and the other is vacant. Ground states (both atomic and molecular) behave quite differently: electrons occupy singly as many degenerate or- orbitals (referred to subsequently as π* orbitals) and in the bitals as possible. One might erroneously conclude that a pair relative spins of the two electrons in this pair of orbitals (Fig- of electrons always have lower energy in separate degenerate ure 1). The electronic energies of the 1Σ term and the 1∆ term orbitals than when they are in the same orbital. Brief consid- are 158 and 95 kJ͞mol higher respectively than that of the eration of the three lowest terms of dioxygen can show stu- 3Σ ground-state term (3). dents that the actual cause of the ground-state pattern is the Because oxygen is a linear molecule, the component of energy advantage of groups of electrons with the same spin. orbital angular momentum about the molecular axis (the z This article will present a simple physical explanation axis) is quantized. The quantum number Λ is the absolute ⁄ for the relative energies of the lowest two singlet terms of O2. value of this z component in units of h. In the three terms The explanation invokes π+ and π− molecular orbitals, which under consideration, orbitals other than π* make a net con- are eigenfunctions of orbital angular momentum about the tribution of zero to Λ. The two π* orbitals are π−* and π+*, molecular axis. where the subscript indicates whether each electron in that orbital has an orbital angular momentum of − h⁄ or + h⁄ . Thus 1 1 the 1∆ term, for which Λ is 2 (hence the ∆), has both elec- πππ+ =+()xyi ; πππ− = ()xy− i 2 2 trons in the same π* orbital (either π−* or π+*), whereas Λ is zero in the 1Σ term (and in the 3Σ term) because the orbital Though πx and πy orbitals are much more commonly used angular momentum of the π−* electron cancels that of the than π+ and π− orbitals in discussions of diatomic molecules, π+* electron. the former are inappropriate for situations (such as the present The probability densities of the π−* and π+* orbitals are one) in which orbital angular momentum is under consider- figures of revolution about the z axis (4–6). These two prob- ation. Fortunately, we may ignore the fact that π+ and π− ability–density functions are identical, whereas πx* and πy* wave functions involve complex numbers, since the present orbitals are rotated from each other by 90Њ about the z axis. explanation will depend only on their probability densities, (The sign of angular momentum is unspecified for πx* and which are real. πy* orbitals, making them unsuitable for describing singlet or triplet oxygen.) Background Explanation Singlet oxygen, which can be formed chemically or by photosensitization, is highly reactive. It causes desirable and Two π* electrons of O2 do not avoid each other any bet- undesirable biological damage (1) and has applications in syn- ter merely by being in different π* orbitals instead of the same thesis and photodynamic therapy (2). It is a mixture of two one. Because the π−* and π+* orbitals have identical one-elec- electronic states, the 1Σ and 1∆ terms. The latter term is re- tron probability-density distributions in space, the energy of garded as the more important reactant because of its longer electrostatic repulsion between one electron’s charge distri- lifetime. bution and that of the other is unaffected by whether the Dioxygen’s first two excited electronic states and its two electrons are in the same π* orbital or different ones. ground state all have the electronic configuration Yet the 1∆ term, in which both π* electrons occupy the 2 4 2 1 [Be2]3σg 1πu 1πg . The electronic structures of these three same orbital, is more stable than the Σ term, in which one states differ in the occupancies of the two antibonding 1πg electron occupies the π−* orbital and the other occupies the

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orbitals when the corresponding electrons have paired spins), thereby lowering both the electron–nuclear energy and the total energy (7). This stabilization due to unpaired spins is the physical reason for Hund’s spin rule: for a given configuration consisting of closed shells plus degenerate electrons 2 π؊* π؉* (e.g., np ), the lowest-energy term generally has the largest total spin S (8). Hund stated this rule for atoms, but it also Figure 2. Directions of electronic circulation about the z axis in the applies to molecules. π* orbitals. Thus the ground state of dioxygen is the 3Σ term not because of but in spite of the fact that its two π* electrons occupy different orbitals. Energy lowering due to high spin more than offsets the energy cost of having the π* electrons π+* orbital. The reason is illustrated by Figure 2. Because two in orbitals with opposite orbital angular momenta. In con- electrons in different π* orbitals move in opposite directions trast, because the π* electrons of singlet terms are paired, they about the z axis, they must arrive from time to time at the are not restricted to different orbitals, nor does antisymme- same angle about that axis despite the tendency to keep apart try prevent their close approach. Only relative orbital angu- owing to electron repulsion. Because two electrons in the lar momentum causes the 1∆ term to be lower in energy than same π* orbital move in the same direction about the axis, the 1Σ. they could avoid each other better than if they were in different π* orbitals. Electrostatic repulsion ensures that they do Pedagogical Considerations avoid each other instead of moving together in close prox- imity. Thus electron correlation (specifically, angular corre- I suggest beginning with a quick review of the molecu- lation) in a doubly occupied π* orbital causes the 1∆ energy lar orbitals that are occupied in the ground state of dioxygen. to be lower than the 1Σ. As the last step of this review, write on the board two hori- The assertions in the preceding paragraph are valid not zontal lines representing the degenerate 2pπ* orbitals, and just classically but quantum mechanically even though pre- ask the class to specify their occupancy and the spin of each cise knowledge of an electron’s angular momentum about an electron. After obtaining the correct answer, write the remain- axis precludes knowledge of its angular position about that ing two diagrams (unlabeled) from Figure 1, but beside, not axis. On the other hand, one should not interpret Figure 2 above, the first one. Then label each diagram with the word rigidly by assuming that the two π* electrons in the 1∆ term singlet or triplet, possibly with advice from the class.1 follow a definite path or remain diametrically opposite, or State that the two singlet states are the first two excited that their periods of circulation about the z axis are equal or states of dioxygen. Ask the students to predict which of the even constant. Because an electron’s distance from this axis two is lower in energy. I suspect that they will be surprised varies, constant angular momentum does not guarantee con- by the correct answer and that this surprise will enhance their stant angular frequency. Nevertheless, even if one of two π* interest in your explanation based on angular momentum and electrons circulating in the same direction sometimes over- Figure 2. You can then explain the term symbols (omitting took the other electron, this could not occur as often as an +, –, and g) in a diagram like Figure 1. encounter at the same angle by two π* electrons with oppo- Point out that for atomic and molecular configurations, site angular momenta. the energy effect of spin is typically greater than that of orbital angular momentum. This is why the ground term has Contrast with Ground States the largest spin compatible with the ground configuration, even though other terms may have larger angular momenta. In the aufbau of atomic or molecular ground states, Statement of Hund’s spin rule follows naturally, as would an double occupancy of an orbital in a degenerate set occurs only explanation of its basis in the Pauli principle. when unavoidable. Why does single occupancy have prior- Kauzmann (9) and Atkins and Friedman (10) have used ity? Not because electrons in different orbitals must occupy an angular-momentum rationale like the present one to jus- different regions of space. Degenerate pairs of atomic orbit- tify Hund’s angular-momentum rule for atoms. This rule als with opposite values of lz have identical probability-den- states that for a given configuration consisting of closed shells sity distributions, like the π− and π+ orbitals of dioxygen. plus degenerate electrons, the lowest-energy term from among Instead, only by occupying different orbitals can two those with the largest spin is the one with the largest angu- electrons have the same spin, and electrons with the same lar-momentum quantum number L (8). spin are never at the same position and rarely in close prox- Nevertheless, it is not feasible to extend to atoms the imity (on the atomic scale). This strong avoidance is a con- classroom presentation of the angular-momentum rationale. sequence of the Pauli principle, which guarantees that under Many atomic orbital diagrams do not represent eigenfunctions exchange of spatial coordinates for any pair of electrons with of L2 and hence may not be labeled with a value of L. Any the same spin, the wave function is antisymmetric and hence atomic configuration for which Hund’s angular-momentum tends toward zero as the electrons approach each other. This rule is relevant has so few orbital diagrams with well-defined strong avoidance by electrons with unpaired spins enables L that one cannot use them persuasively to justify the angu- orbitals to shift toward the nucleus (compared to the same lar-momentum rationale.

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An Alternative Treatment of Angular Correlation Notes

Hückel (11), recapitulated by Kasha and Brabham (5), 1. The reader might object that the orbital diagram labeled 1 + also uses angular correlation about the z axis to explain rela- Σg in Figure 1 could belong to either a singlet or a triplet term tive energies of the 1Σ and 1∆ and even the 3Σ terms. Hückel (12). This is because it is compatible with either of these two compares for these terms the probability density’s explicit (unnormalized) wave functions: dependence on angles φ1 and φ2, where 1 and 2 refer to the π* electrons. Each term’s probability density includes the same ππ**+ ππ ** αβ− βα function of r1, r2, z1, and z2. After removing this common ()+ −−+ () factor, the normalized angular probability densities are

ππ**− ππ ** αβββα+ 1 ()+ −−+ () cos2 φφ− 1 + 2 ( 12) for Σg 2 π However, this consideration is unlikely to occur to students in the present context. This orbital diagram is the natural one for representing an oxygen molecule in which two electrons with paired spins 1 1 occupy singly both π* orbitals. Indeed, this is the only orbital dia- 2 for ∆g 1 + 4 π gram for the Σg term that I have found in published discussions of singlet oxygen.

1 2 Literature Cited 2 sin ()φφ12− 3 − 2 π for Σg 1. Singlet Oxygen, UV-A, and Ozone; Packer, L., Sies, H., Eds.; 1 The ∆ probability density is independent of φ1 and φ2. Methods in Enzymology, Vol. 319; Academic: San Diego, 2000. The 1Σ density is much higher than the 1∆ when the two 2. DeRosa, M. C.; Crutchley, R. J. Coord. Chem. Rev. 2002, 233, electrons are approximately 0Њ or 180Њ from one another, but 351–371. much lower than the 1∆ at approximately ±90Њ. Because the 3. Herzberg, G. Molecular Spectra and Molecular Structure. I. Spec- 2͞ interelectronic potential e r12 is highly nonlinear, densities tra of Diatomic Molecules, 2nd ed.; Van Nostrand: Princeton, at small angles dominate the repulsion energy. Therefore the NJ, 1950. energy of the 1Σ term is higher than that of the 1∆ term. How- 4. Ogryzlo, E. A.; Porter, G. B. J. Chem. Educ. 1963, 40, 256– 2 3 ever, the sin (φ1 − φ2) behavior of Σ strongly depletes the 261; Figure 16. 3 density for small values of φ1 − φ2, causing the Σ energy to 5. Kasha, M.; Brabham, D. E. In Singlet Oxygen; Wasserman, be lower than the 1∆. Indeed, Hückel proves that to first or- H. H., Murray, R. W., Eds.; Academic: New York, 1979; pp 2͞ der in energy (with e r12 as the energy perturbation), the 1–33. 1 1 3 ∆ energy is midway between the Σ and Σ energies. 6. Kasha, M. In Singlet O2; Frimer, A. A., Ed.; CRC Press: Boca Both the Hückel explanation and the present one based Raton, FL, 1985; Vol. 1, pp 1–11. on Figure 2 attribute the higher energy of the 1Σ term to the 7. Shenkuan, N. J. Chem. Educ. 1992, 69, 800–803. increased likelihood (compared to 1∆) that both electrons are 8. Hund, F. Linienspektren und Periodisches System der Elemente; near the same angle. The present explanation has two ad- Springer: Berlin, 1927; p 124. vantages. It can be presented quickly to students who have 9. Kauzmann, W. Quantum Chemistry: An Introduction; Aca- minimal acquaintance with wave functions. Secondly, the demic: New York, 1957; p 343. physical principal on which it is based—cancellation of two 10. Atkins, P. W.; Friedman, R. S. Molecular Quantum Mechan- electrons’ orbital angular momenta—applies just as well to ics, 3rd ed.; Oxford: New York, 1997; p 231. the exact wave function as to the orbital approximation on 11. Hückel, E. Z. Phys. 1930, 60, 423–456. which the Hückel explanation depends. 12. Campbell, M. L. J. Chem. Educ. 1996, 73, 749–751.

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