Addition/Correction Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX
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Corrections to The Pauli Principle: Effects on the Wave Function Seen through the Lens of Orbital Overlap David R. McMillin* J. Chem. Educ. 2018, 95 (9), 1587−1591. DOI: 10.1021/acs.jchemed.8b00407 wo minor corrections are that the Σ+ terms at the top of Figure 4 are singlet, not triplet, states. Also, in the penultimate sentence before Figure 4 the ground term should be written 3Σ−. The other, more substantive concern relates to the spatial function discussed for the singlet state. The function (Ψ2s below) originally identified in eq 3 remains valid for the singlet excited state of formaldehyde, but not for the 1Σ+ state of the NH molecule. Because the two pπ orbitals of NH are degenerate, there are three distinct spatial functions that are symmetric with respect to exchange of the electrons. They appear as the top 3 entries in Table 1 which reveals that eq 3
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T
with each other in overlap zones formed by the participating orbitals.
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Table 1. Wave Functions and Term Symbols Ψ
σvaΨi
Spatial Function
Term
f(φ1, φ2)b
Ψ1
1 [x(1)x(2) + y(1)y(2)] 2
Ψ1
1 +
Σ
cos(φ1 − φ2)
Ψ2c
1 [x(1)x(2) − y(1)y(2)] 2
cos 4β Ψ2c + sin 4β Ψ2s
1
Δc
cos(φ1 + φ2)
Ψ2s
1 [x(1)y(2) + x(2)y(1)] 2
sin 4β Ψ2c − cos 4β Ψ2s
1
Δs
sin(φ1 + φ2)
Ψ3
1 [x(1)y(2) − x(2)y(1)] 2
−Ψ3
3 −
sin(φ2 − φ1)
Σ
REFERENCES
(1) Kearns, D. R. Selection Rules for Singlet-Oxygen Reactions. Concerted Addition Reactions. J. Am. Chem. Soc. 1969, 91, 6554− 6563. (2) Harding, L. B.; Goddard, W. A., III. The Mechanism of the Ene Reaction of Singlet Oxygen with Olefins. J. Am. Chem. Soc. 1980, 102, 439−449. (3) Salem, L. Electrons in Chemical Reactions: First Principles; John Wiley & Sons: New York, 1982; pp 67−72. (4) See also: Marmorino, M. G. Electron Correlation in the Singlet and Triplet States of the Atomic 2px12py1 Configuration. J. Chem. Educ. 2019, 96, 390−392.
Reflection through a vertical plane rotated φ = β from the positive xaxis. bAngular dependence on φ1 and φ2 with Ψi expressed in cylindrical polar coordinates (r, φ, z). As in Figure 1, the assumption is that the electrons are in the z = 0 plane. a
actually belongs to the 1Δ term. (For corresponding functions associated with the π*2 configuration of O2, see refs 1−3.) It is clear from column 3 in Table 1 that that Ψ2s and Ψ2c have the same energy due to the fact that a symmetry operation interconverts the two functions. Therefore, in terms of the notation used in eq 3, Ψ1 = x(1)x(2) + y(1)y(2) is the spatial wave function associated with the 1Σ+ state. In the context of cylindrical polar coordinates,4 the cos(φ1 − φ2) component in Table 1 describes how the wave function varies with the angle between electrons 1 and 2. The minus sign in the argument seems logical because in a Σ state the collective angular motions of the two electrons generate no net orbital angular momentum about the z-axis. The angular function is also helpful in interpreting the properties of Ψ1 vis-à-vis Figure 1A. In contrast to the 3Σ− ground state, the angular dependence of the 1Σ+ state maximizes when both electrons make the same angle with respect to the x-axis (φ1 = φ2) and therefore reside on a common axis, e.g., t or u in Figure 1A. Similarly, Ψ1 vanishes in the z = 0 plane when the two electrons reside on any set of perpendicular axes that intersect at the nucleus. Like Ψ2s, Ψ1 is a linear combination of two functions which interfere © XXXX American Chemical Society and Division of Chemical Education, Inc.
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DOI: 10.1021/acs.jchemed.9b00385 J. Chem. Educ. XXXX, XXX, XXX−XXX
