Article Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX
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The Pauli Principle: Effects on the Wave Function Seen through the Lens of Orbital Overlap David R. McMillin*
J. Chem. Educ. Downloaded from pubs.acs.org by MOUNT ROYAL UNIV on 08/10/18. For personal use only.
Department of Chemistry, Purdue University, West Lafayette, Indiana 47907, United States ABSTRACT: The Pauli exclusion principle is a key postulate of the quantum theory and informs much of what we know about matter. In terms of electronic structure, the lone, deceptively simple mathematical requirement is that the total wave function be antisymmetric with respect to the exchange of any two electrons. However, visualizing the effect on the electron distribution presents a formidable challenge to students because of the complexity and number of variables involved. Approaches to gaining insight developed here include a quantitative analysis of two-dimensional configuration spaces as well as a qualitative assessment of interference patterns. For 2px12py1 states as representative examples, the analysis reveals that the Pauli principle directs electrons with opposing spins to codistribute along axes passing through the nucleus in the plane of the occupied orbitals. On the other hand, electrons with the same spin are never simultaneously in the same place or on the same axis; instead, they preferentially distribute in perpendicular directions. KEYWORDS: Upper-Division Undergraduate, Graduate Education/Research, Physical Chemistry, Inorganic Chemistry, Quantum Chemistry, Theoretical Chemistry ost students find the Pauli exclusion principle to be a useful but abstract concept. To promote a better understanding of the phenomenon and its consequences, this contribution describes methods for visualizing key aspects of the wave functions involved. Historically, Pauli formulated the principle against the backdrop of Bohr theory by anticipating a fourth degree of freedom for the electron and positing that no two electrons can have the same set of four quantum numbers.1,2 The fourth degree of freedom later turned out to be the intrinsic spin of the electron. By excluding the possibility of electrons sharing the same quantum numbers, Pauli inferred that each atomic orbital can house a maximum of two electrons and thereby rationalized the basic contours of the periodic table. A more encompassing statement is that no two electrons can ever be in the same quantum state at the same time.3 As modern quantum theory developed, the concept broadened, and the mathematical statement of what one might more generally call the Pauli principle became that the overall wave function must be antisymmetric with respect to the interchange of any two electrons.2,4−7 However, it is not intuitively obvious what that entails. Because this remarkable principle also accounts for Hund’s rule as well as the possibility of high spin ground states in transition metal complexes,8,9 students of electronic structure ought to have at least a qualitative understanding of how the Pauli principle shapes wave functions. The difficulty is that developing conceptual understanding typically entails a process of visualization,10,11 but depicting the electronic wave function is immensely challenging because of the number of degrees of freedom involved. Thus, even when
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the system has only two electrons, a graph of the wave function involves as many as seven dimensions, more specifically, three for locating each electron and another for the value (amplitude) of the wave function. Seeking to facilitate student understanding, educators have found ways to reduce the dimensionality. For example, in exploring the triplet-state wave function of the H2 molecule, Dunbar froze one electron in place to show the effect it has on the other.12 Similarly, using density functional theory, Harbola and Sahni fixed the location of a particular electron of argon in order to find the distribution of the “Fermi hole” that occurs in the density associated with electrons of the same spin.13 Linnett was able to take a different approach in studying the states associated with the 1s12s1 configuration of the helium atom because the spatial wave functions depend only on the two electronic radii.14 Both electrons therefore distribute freely in the resulting contour plots, which show that antisymmetrization insures that electrons enjoy a favorable radial correlation when the spins are parallel. The discussion to follow shows it can be useful to combine aspects of both approaches and explores 2px12py1 states as a representative application.15 Here one finds that there is significant angular as opposed to radial correlation. Finally, the analysis shows that an examination of interference patterns alone can provide qualitative insight into the divergent electron distributions of singlet and triplet states. The Received: May 31, 2018 Revised: July 28, 2018
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DOI: 10.1021/acs.jchemed.8b00407 J. Chem. Educ. XXXX, XXX, XXX−XXX
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SPATIAL DISTRIBUTIONS IN CONFIGURATION SPACE To obtain some insight into the nature of ψ, it is useful to focus on a two-dimensional (2D) cut of the multidimensional configuration space. Suppose, for example, the two electrons roam along some combination of the axes t and u in Figure 1A.
qualitative approach readily extends to other electron configurations and provides concrete insight into difficult-toportray electron distributions.
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SYMMETRIC AND ANTISYMMETRIC FUNCTIONS Step one of the process is to develop wave functions that change sign when two electrons exchange places: ψ (1, 2) = −ψ (2, 1)
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(1)
where 1 and 2 schematically denote the coordinates of electrons 1 and 2, respectively. The case with two electrons and two orbitals is convenient because it is possible to write the wave functions as products of spatial and spin components: ψ = (spatial function) × (spin function) such that one can separately assess the exchange symmetry of each factor. In turn, one normally approximates the spatial wave function as a product of one-electron wave functions. Consider the following function as a starting point ψ (1, 2) = x(1)y(2)
(2)
where for convenience x(1) denotes electron 1 housed in a 2px orbital and y(2) denotes electron 2 in a 2py atomic orbital. Because the individual 2p orbitals involved in eq 2 form part of the basis for a representation of the full rotation group, product functions like eq 2 are also potential basis elements. Beyond that, theory shows that it is possible to partition the product functions into two distinct subsets by forming combinations as follows:16 1 [x(1)y(2) + x(2)y(1)] 2
(3)
1 [x(1)y(2) − x(2)y(1)] 2
(4)
where eq 3 represents a symmetric direct product and eq 4 represents an antisymmetric direct product. Equation 4 is antisymmetric in that it changes sign when electrons 1 and 2 trade places. Equations 3 and 4 also have different symmetries because no symmetry operation transforms one combination into the other. The two combinations also have different symmetries with respect to the exchange of identical particles, and they exhibit quite distinct electron distributions, vide infra. In addition to the spatial functions there are four associated spin functions to consider, because there are two electrons and two spin states for each. It is not necessary to dwell on them in detail, but for completeness, three of them correspond to the MS = 1, 0, and −1 spin states associated with the S = 1 triplet state, viz., the α(1)α(2) = α(2)α(1) state associated with MS = 1. Those three combinations are all symmetric with respect to electron exchange. The lone antisymmetric combination belongs to the S = 0 state: 1 [α(1)β(2) − α(2)β(1)] 2
Figure 1. (A) Contour plots of the 2px and 2py atomic orbitals. Axes t and u bisect opposing overlap zones (speckled). Overlap is net positive (negative) in the gray (pink) zones. (B) A two-dimensional configuration space. The constraint is that both electrons must reside on the t axis from part A. The displacements of electrons 1 and 2 along that axis define a point in configuration space with coordinates (t1, t2), both in italic font to distinguish from axes highlighted in bold.
Case 1 occurs when both electrons reside on the same axis, namely t, and Figure 1B reveals how a 2D array of coordinates (t1, t2), arises.17 That 2D subspace is an abstract construct, but each quadrant embodies physical content. Thus, quadrant I in Figure 1B reports all possible ways the two electrons can simultaneously coexist on the positive side of the t axis.
(5)
In that the combined product wave function must be antisymmetric with respect to the exchange of electrons, eq 3 is the relevant spatial function for the S = 0 singlet state, and eq 4 belongs to the S = 1 triplet level. B
DOI: 10.1021/acs.jchemed.8b00407 J. Chem. Educ. XXXX, XXX, XXX−XXX
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Figure 2. Plots of the spatial components of singlet and triplet wave functions for the 2p2 system described in Figure 1. (A) Plot of the singlet function (eq 3) with both electrons constrained to reside on the t axis. (B) Plot of the triplet function (eq 4) corresponding to panel A. (C) Plot of the singlet function with one electron constrained to reside on the t axis while the other electron is on the u axis. (D) Plot of the triplet function corresponding to panel C. In panels A and D, the floor of the plot contains 2D contours of the function. All distances in units of Bohr radii expressed in angstroms.
Adoption of the antisymmetric direct product in eq 4 as the wave function prevents the two electrons from being in the same place at the same time in the triplet state.2,5 Interference between the two components of the superposition plays an essential role. The interference pattern associated with eqs 3 and 4 is different, and ultimately that is why the singlet and triplet wave functions exhibit very different electron distributions. Interference occurs when both product functions have nonvanishing values, most importantly in the overlap domains identified by the speckled patterns in Figure 1A.18 It is because axes t and u both pass through strong overlap areas that they provide nice windows into the resulting distributions. To complete the picture, consider case 2, the subspace that arises when electron 1 resides along the t axis and electron 2 falls on the u axis. Under this constraint x(1)y(2) = −x(2)y(1), and in this subspace the singlet wave function vanishes (Figure 2C). On the other hand, the product functions in eq 4 end up adding constructively and yield the triplet-state wave function depicted in Figure 2D. The plots in Figure 2 reveal that the angular relationship between the electrons is an important consideration, and systematic examination of interference patterns reveals trends. Figure 1A reveals that the local overlap zones define a cloverleaf pattern. Analysis of the algebraic signs associated with the participating 2px and 2py orbitals shows that the interference in eq 3 is constructive, enhancing the singlet state, if electrons 1 and 2 are both in the same zone or in diagonally opposite zones. The same electron arrangements are, however, less favorable for the triplet state because the interference is destructive (eq 4). Destructive interference occurs in the
Quadrant II consists of all ways electron 1 can be on the negative side of axis t while electron 2 is on the positive side, and so forth. A companion subspace, case 2, arises when electron 1 distributes along axis t while electron 2 roams along the perpendicular axis u. Here axes t and u are high-symmetry choices in that they bisect the axes associated with the two p orbitals. These axes are convenient for the present purposes, but one can choose others in the same plane, vide infra. To complete the figure, the contour plots in Figure 1A represent the atomic orbitals that appear in eqs 3 and 4. For simplicity the calculations that follow employ 2p orbitals of the hydrogen atom. Figure 2A is a contour plot of the singlet wave function (eq 3) in the case 1 subspace, when both electrons reside on the t axis. For calculating the amplitude of ψ for each constellation of electrons, the framework in Figure 1A is useful, whereas configuration space is a construct useful for compiling the amplitude as a function of all possible coordinates. Note that the amplitude goes to zero at each axis in Figure 2A because for an electron to fall on the t1 or t2 axis, the other electron has to be at the origin (Figure 1B). Each product function in eq 3 therefore vanishes because 2p orbitals have a node at the nucleus. Within each quadrant of Figure 2A, however, analysis shows that the two product functions have nonzero values and add together constructively in eq 3. Thus, along t x(1) = y(1) and x(2) = y(2), so that x(1)y(2) = x(2)y(1) (see Figure 1A). In contrast, the product functions combine destructively in the triplet state because of the minus sign in eq 4; indeed, the corresponding wave function vanishes everywhere in the same subspace (Figure 2B). C
DOI: 10.1021/acs.jchemed.8b00407 J. Chem. Educ. XXXX, XXX, XXX−XXX
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from first principles. It is, however, easier to derive them from well-known terms of the O2 molecule, because its ground state also entails a π2 configuration. In order of increasing energy, the three lowest-energy terms of the dioxygen molecule are 3 + 1 Σg , Δg, and 1Σg+.22 The transition from O2 to NH entails a descent from D∞h to C∞v symmetry, and Figure 4 shows the
singlet state when the electrons occupy adjacent zones, which turns out to be a favored arrangement for the triplet state. Thus, at any point in time, the electrons are more likely to reside at right angles to each other in the triplet state, while in the singlet state they are prone to fall on one axis or another. It is also possible to assess the distributions more quantitatively, e.g., that of the triplet wave function. Consider any set of two axes in the xy plane, and assume they occur at angles θ1 and θ2 relative to the y-axis as in Figure 3. If the
Figure 4. Correlations between term states of O2 and NH. Energies relative to the ground-state triplet.
corresponding term labels for the NH system, as inferred from functions found in the character tables.23 In each term symbol the superscript on the left designates the spin multiplicity (2S + 1) quantum number, while the Greek letter denotes the symmetry type. O2 has a center of symmetry, and its terms have gerade (g) symmetry. The superscripts on the right reflect the symmetry with respect to vertical planes passing through the molecular axis.24 For further insight, note that the vertical planes are all in the same class because rotation operations interconvert them. Therefore, one can choose a conveniently oriented plane to deduce the character. As the plane that bisects the x- and y-axes interchanges the 2px and 2py orbitals, one sees that eq 3 is symmetric (+) and that eq 4 is antisymmetric (−) with respect to reflection through a vertical plane. In accordance with Hund’s rule, the triplet state falls at the lowest energy. The corresponding 1Σ states fall at higher energies, and the separation is much greater in the case of the NH molecule because the electrons reside on only one atom and have less space in which to distribute. More specifically, the top-to-bottom energy differences are 13,000 cm−1 in O2 and 22,100 cm−1 in NH.21,22 A more advanced application relates to the n-π* excited states of carbonyl compounds, where the singlet−triplet splitting is much smaller. In these systems, the nonbonding (n) orbital is effectively a 2p orbital of the oxygen atom. The π* orbital is in turn an admixture of 2p orbitals residing on the carbon and oxygen atoms, though richer in carbon character. The only significant overlap between the n and π* orbitals therefore occurs between two perpendicular 2p orbitals of the oxygen atom. Because of the reduced amplitude of the 2pπ orbital of oxygen, there is less overlap, and the singlet−triplet splitting in formaldehyde amounts to only about 3,500 cm−1.25 In closing this section, it is worth discussing the relative energy of the 1Δ states. It seems at first counterintuitive that a singlet state that has electrons paired in the same 2p orbital would be more stable than one in which the electrons reside in separate atomic orbitals. However, there are two factors to
Figure 3. Axes t and u are at a right angle to each other. Axis u makes an angle of θ2 with respect to the positive y direction. Sweeping through values of θ2 from 0 to −π/2 defines all choices of axes that are 90° apart, leaving axes in adjacent quadrants.
displacements along each axis are t1 and u2, respectively, then within that subspace one can write the triplet function in the following form: T : t1u 2 f (t1 , u 2)sin(θ1 − θ2)
(6)
where f(t1, u2) entails a product of various constants and an exponential. From an angles standpoint, the function achieves π maximum amplitude when θ1 − θ2 = 2 . Inspection of Figure 3 shows this result means the axes must be in adjacent quadrants. Therefore, all else being the same, the optimum distribution of the triplet state occurs within the plane when the electrons reside on axes that are 90° apart. Such regimentation prevents the electrons from screening the nucleus as efficiently as they otherwise might, which is important in determining why the triplet state occurs at lower energy in accordance with Hund’s rule.19,20
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EXAMPLE SYSTEM This presentation focuses on understanding how the Pauli principle affects electron distribution, but relating the results to actual systems may enhance perspective. A case in point is the NH molecule, which has a 1σ22σ23σ21π2 electronic configuration.21 In fact, the orbitals with π symmetry are essentially pure 2p atomic orbitals of the nitrogen atom. The ground state therefore effectively involves a 2p2 configuration that gives rise to singlet and triplet states, each involving singly occupied 2p orbitals, as discussed above. Double occupation of each orbital is, however, also possible so long as the electrons have opposite spins. In addition, there are two orbitals to choose from; hence, the third state that arises is orbitally degenerate. One can derive symmetry labels for all three energy states, or terms, D
DOI: 10.1021/acs.jchemed.8b00407 J. Chem. Educ. XXXX, XXX, XXX−XXX
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(9) Natoli, S. N.; McMillin, D. R. Demonstrating Hund’s Rule in Action by Exploring the Magnetic Properties of Metal Complexes with 3dn and 4fn Configurations. J. Chem. Educ. 2018, 95, 126−130. (10) Garrett, A. B. Visualization: A Step to Understanding. J. Chem. Educ. 1948, 25, 544−547. (11) Jones, L. L.; Kelly, R. M. In Sputnik to Smartphones: A HalfCentury of Chemistry Education; Orna, M. V., Ed.; ACS Symp. Ser. 1208, American Chemical Society: Washington, DC, 2015; p 121− 140. (12) Dunbar, R. C. The Influence of Electrons on Each Other in a Molecule - Correlation of Electron Motions in H2. J. Chem. Educ. 1989, 66, 463−466. (13) Harbola, M. K.; Sahni, V. Theories of Electronic Structure in the Pauli-Correlated Approximation. J. Chem. Educ. 1993, 70, 920− 927. (14) Linnett, J. W. Wave Mechanics and Valency; Methuen: New York, 1960. (15) For an animation of a 2p2 contour assuming an electron in a series of fixed positions, see: Dill, D. Fermi holes and Fermi heaps. Notes on Quantum Mechanics.http://quantum.bu.edu/notes/ QuantumMechanics/FermniHolesAndHeaps.html (accessed Aug 2018), 2003. (16) Landau, L. D.; Lifshitz, E. M. Quantum Mechanics; AddisonWesley: Reading, MA, pp 329−336, 1958. (17) Schonland, D. S. Molecular Symmetry; D. Van Nostrand Company, Ltd.: New York, 1965. pp 49−50. (18) In overlap zones, electrons encounter each other or get “into each other’s way”. Michl, J.; Bonačić-Koutecký, V. Electronic Aspects of Organic Photochemistry; Wiley-Interscience: New York, 1990; p 38. (19) Rioux, F. Hund’s Multiplicity Rule Revisited. J. Chem. Educ. 2007, 84, 358−360. (20) Snow, R. L.; Bills, J. L. Textbook Error, 117 - Pauli Principle and Electronic Repulsion in Helium. J. Chem. Educ. 1974, 51, 585− 586. (21) Hay, P. J.; Dunning, T. H. Polarization CI Wavefunctions Valence States of NH Radical. J. Chem. Phys. 1976, 64, 5077−5087. (22) Salem, L. Electrons in Chemical Reactions: First Principles; John Wiley & Sons: New York, 1982; pp 67−72. (23) Strommen, D. P.; Lippincott, E. R. Infinite Point Groups. J. Chem. Educ. 1972, 49, 341−342. (24) Pilar, F. L. Plus and Minus States of Linear Molecules. J. Chem. Educ. 1981, 58, 758−760. (25) Turro, N. J.; Ramamurthy, V.; Scaiano, J. C. Principles of Molecular Photochemistry: An Introduction; University Science Books: Sausalito, CA, 2009; pp 63−66. (26) Kauzmann, W. Quantum Chemistry: An Introduction; Academic Press: New York, 1957.
keep in mind. One is that, as shown above, antisymmetrization directs the electrons of the latter singlet to bunch together. Second, the electrons in the 1Δ state combine to exhibit two units of orbital angular momentum about the molecular axis, in that the Lz quantum number is 2. As Kauzmann has described,26 all else being the same, a state achieves a lower energy whenever electron correlation produces net orbital angular momentum.
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PERSPECTIVE The Pauli principle is an inference from experiment and a postulate of the quantum theory. The lone, deceptively simple mathematical requirement is that the total wave function must be antisymmetric with respect to the exchange of any two electrons. In practice, fathoming how antisymmetrization affects the electron distribution is a challenge, even for a relatively simple two-electron system. Approaches described herein provide greater accessibility for students through zeroing in on two-dimensional configuration spaces or simply analyzing interference patterns. Within the plane defined by the occupied orbitals of a 2px12py1 state, the results reveal that the Pauli principle encourages electrons with opposing spins to codistribute along various axes, despite Coulomb repulsions. In sharp contrast, here, parallel spins never appear in the same place or on the same axis. Indeed, if the electrons have the same spin, the highest probability for the angular component of the wave function occurs when they distribute along axes that are perpendicular to each other.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
David R. McMillin: 0000-0002-6025-0189 Notes
The author declares no competing financial interest.
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ACKNOWLEDGMENTS The author thanks Adam Wasserman, Marcy Towns, and Timothy Zwier for useful conversations as well as Andreas C. Geiger and Hartmut Hedderich for skillful assistance with MATLAB plots. The author also thanks anonymous Reviewers for helpful suggestions and gratefully acknowledges Purdue University for resources provided.
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REFERENCES
(1) Pauli, W. On the connection of the arrangement of electron groups in atoms with the complex structure of spectra. Z. Phys. 1925, 31, 765−783. (2) Pauli, W. In The Dreams That Stuff Is Made Of; Hawking, S., Ed.; Running Press: Philadelphia, 2011; p 424−444. (3) Wilczek, F. The enigmatic electron. Nature 2013, 498, 31−32. (4) Urey, H. C. The structure of atoms with particular reference to valence. J. Chem. Educ. 1931, 8, 1114−1132. (5) Shadmi, Y. Teaching Exclusion Principle with Philosophical Flavor. Am. J. Phys. 1978, 46, 844−848. (6) McQuarrie, D. A.; Simon, J. D. Physical Chemistry: A Molecular Approach; University Science Books: Sausalto, CA, 1997. (7) Winn, J. S. Physical Chemistry; Harper Collins College Publishers: New York, 1995; pp 452−456. (8) Cann, P. Ionization Energies, Parallel Spins, and the Stability of Half-Filled Shells. J. Chem. Educ. 2000, 77, 1056−1061. E
DOI: 10.1021/acs.jchemed.8b00407 J. Chem. Educ. XXXX, XXX, XXX−XXX
