In the Classroom
Introducing Relativity into Quantum Chemistry Wai-Kee Li* Department of Chemistry, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong *
[email protected] S. M. Blinder* Department of Chemistry, The University of Michigan, Ann Arbor, Michigan 48109-1055, United States and Wolfram Research, Inc., Champaign, Illinois 61820-7237, United States *
[email protected];
[email protected] In the short time span between 1926 and 1928, English physicist Paul Dirac (1902-1984) played a pivotal role in two important developments in the history of physics: transformation theory (a unification of Heisenberg's and Schrödinger's formulations of quantum mechanics) and incorporating relativity into quantum mechanics. In this article, an attempt is made to introduce the important and chemically significant results of relativistic quantum mechanics to chemistry students and teachers. The presentation is essentially nonmathematical. Indeed, not a single equation is included, so this article is mainly qualitative. Electron Spin When chemistry students first encounter quantum mechanics in their physical chemistry course, the first system they treat is often a lone electron trapped in an infinitely deep potential well, the so-called particle-in-a-box problem (1 2,). When the box is one-dimensional, the solutions (energy levels and wavefunctions) are characterized by a single quantum number. When the box is two-dimensional, that is, when it is a square or a rectangle, the solutions require two quantum numbers. When the box becomes three-dimensional, as in the case of a cube, the solutions will have three quantum numbers. So the moral of the story is: in quantum mechanics, the solutions of a k-dimensional Schrödinger equation will have k quantum numbers. Indeed, the solutions of the Schrödinger equation for the hydrogen atom (with three variables r, θ, and j) have three quantum numbers, n, l , and ml . The electron possesses, in additional to its orbital motion, an internal degree of freedom called the spin. Spin is somewhat analogous to the daily rotation of the earth, which complements its annual revolution around the sun. An electron is said to have “spin 1/2”, which, according to the arcane rules of quantum mechanics, provides a fourth quantum number ms, with the possible values þ1/2 or -1/2. Spin was first postulated by G. E. Uhlenbeck and S. Goudsmit in 1925 to explain some anomalies in atomic spectra. Wolfgang Pauli exploited this additional degree of freedom, even before the advent of quantum mechanics, to formulate his exclusion principle, which states that no two electrons in an atom can exist with the same set of four quantum numbers. The concepts of spin and the exclusion principle made it possible to understand the periodic structure of the elements. Although electron spin was already there before Dirac extended quantum mechanics to take account of relativity, he gave electron spin a rigorous mathematical justification. Dirac believed that the two great advances in 20th century physics;Einstein's theory of relativity and quantum mechanics; should be part of a unified theory. (Today the most challenging
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unsolved problem in physics involves the unification of gravity with quantum mechanics.) Specifically, Dirac recognized the need to put special relativity into the Schrödinger equation. The resulting modification of the Schrödinger equation, which describes the relativistic behavior of the electron, gave rise to the Dirac equation. Miraculously, the existence of electron spin automatically emerged, without the need for any additional postulate. Also accounted for was the electron-spin magnetic moment, a byproduct of a charged particle with angular momentum, with the correct g-factor of 2. (By contrast, the orbital motion of an electron has g = 1.) To be historically accurate, it should also be pointed out that Erwin Schrödinger actually wrote down the first relativistic wave equation (in 1925, and in his notebook!) as he looked for an equation to describe the de Broglie waves for electrons. When he applied this equation to the hydrogen atom, he failed to obtain the correct fine structure of the hydrogen spectrum because he did not take electron spin into consideration. So he abandoned this equation and settled for the nonrelativistic equation that now bears his name. Two years later, Oskar Klein and Walter Gordon resurrected this equation and found that it correctly described the behavior of spinless particles (such as pions). So now it is known as the Klein-Gordon equation. Even the standard procedure for quantization of a classical equation of motion shows the pervasive influence of relativity. The fundamental variables of a system can be formulated as a “four vector” (x1, x2, x3, x4), with x1, x2, x3 = √x, y, z, respectively, and x4 = ict (where c is the speed of light and i = -1). The energy-momentum four vector is correspondingly defined by (p1, p2, p3, p4), with p1, p2, p3 = px, py, pz, respectively, and p4 = iE/c (where E is energy). The fundamental quantization prescription is then given by the operator replacement pj f -i(h/2π)∂/∂xj, j = 1-4, where h is Planck's constant. (In modern usage, the fourth components of the four vectors are replaced by the real variables x0 = ct and p0 = E/c.) Nodes of an Electronic Wavefunction In the radial probability distribution function of a (hydrogenic) 2s electron, r2|R2,0|2, there is a node at r = 2 au (1, 2). In other words, there is zero probability of finding this electron between r and r þ dr, with r = 2 au, although there is finite probability of finding the electron in the regions of r < 2 au (region I) and r > 2 au (region II). So how can the electron go from region I to region II (or the other way around) without going through the node? Or, in an analogy with baseball, how can the electron go from first to third without touching second base? Many approaches to this question have been
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In the Classroom
evoked, from those citing the uncertainty principle to arguments that in the quantum world an electron does not move in a classical fashion. But an attractive rationalization is suggested by the Dirac equation. It turns out that the Dirac electronic wavefunction ψ is a fourcomponent spinor;there is a twofold contribution representing the two spin states and another duality determining whether the particle is an electron or a positron. (The four-component wave function is sometimes erroneously attributed to the four-dimensional structure of spacetime in relativity, but the relativistic Klein-Gordon equation also describes a spinless particle.) The four components of ψ, designated, ψ1, ψ2, ψ3, and ψ4, give a probability density function ψ†ψ of the form |ψ1|2 þ |ψ2|2 þ |ψ3|2 þ |ψ4|2. For the 2s wavefunction, at r = 2 au, among these four terms, the first two are zero, whereas the sum of the last two are minutely positive. In other words, the “node” of the 2s wavefunction actually has a very small finite value at r = 2 au when relativity is taken into account. So now the 2s electron can go through the “node” at r = 2 au without any conceptual difficulty. The Prediction of the Positron In addition to accounting for electron spin, the Dirac equation also unexpectedly implied that an electron can have either positive or negative energy. To avoid this apparent dilemma, Dirac assumed that, in a given atom, all of the infinite number of negative energy levels are filled--this being the inherent nature of what we perceive as the “vacuum”, whereas the positive energy levels are filled by the electrons of that atom from the ground up. When a member of the negative energy “sea” is excited to a positive energy state, the resulting hole in the negative energy “sea” will appear to be a particle with a positive charge. In 1928, only two subatomic particles were known: electron and proton. Hence, Dirac at first thought this particle was a proton. However, it was later shown that the hole had to have the same mass as an electron. It cannot be a proton, so Dirac proposed there should be a new particle with electronic mass as well as a “positive electronic” charge, and this particle should also have a spin of 1/2. In 1932, this prediction became a reality as Carl Anderson observed this particle's existence in cosmic rays. He called this new particle a “positron”. Dirac originally got his idea about a filled “sea” of negativeenergy electrons from X-ray spectroscopy on atoms. An X-ray photon can promote an inner-shell electron to an excited level, thus, leaving behind a hole in the inner shell. This may be viewed as “pair production” of a positive-energy electron plus a negative-energy positively charged hole. When the photon falls back, an inverse process of pair annihilation results. From a more modern viewpoint, the heavy philosophical baggage of an infinite Dirac “sea” can be avoided by the development of quantum field theory. The key idea is that wavefunctions are promoted from simple functions to particle creation and annihilation operators. Known as second quantization, this procedure enables electrons and positrons to be created, from a more rational vacuum. Lastly, the positron may also be called an antielectron; it is the antimatter counterpart of the electron. Annihilation occurs when a positron and an electron collide, and two or more photons (or other particles) are produced. Thus, the Dirac equation first predicted the existence of antimatter and thereby opened up a whole new aspect of particle physics. Particle in a One-Dimensional Box Now let us return to the particle-in-a-box problem, but this time with relativity considered. The mathematical treatment (3) 72
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of this problem is considerably beyond the scope of this article. Still, some interesting qualitative results can be extracted. When the box is one-dimensional, the wavefunction ψ has just two components, ψ1 and ψ2, and the probability density function ψ†ψ is |ψ1|2 þ |ψ2|2. Nonrelativistically, the wavefunction (and hence the probability density function) of the first excited state has a node at the midpoint of the box. As we have seen, when relativity is turned on, this node acquires a small nonzero component. Another crucial question is: when does relativity become an issue for this physical system? Specifically, at what box length should we take relativity into account? As it turns out, the critical length L0 is h/2πmc, where is m the electronic mass, and h and c have been defined earlier. After putting in the numbers, L0 is calculated to be about 0.004 Å. When the box length is about L0 or smaller, the electron is relativistic. So we now know we are justified to apply the nonrelativistic free-electron model (1, 2) to treat the π electrons of a conjugated polyene such as butadiene because the length of this box is nearly 6 Å, or about 1,500L0. Properties of Heavy Elements In the special theory of relativity, when an electron is traveling with a velocity v that is an appreciable fraction of the speed of light, its effective mass m behaves like m0[1 - (v/c)2]-1/2, where m0 is the rest mass of the electron. In atomic units, speed of light c is approximately 137. Thus, the average orbital velocity Ævæ of a 1s electron in an atom is approximately Z au, where Z is the atomic number. Hence, for an electron in Au, with Z = 79, we have Ævæ/c ≈ 79/137, or 0.58. So this 1s electron travels at a significant fraction of the speed of light. Moving at this speed, mass m of the electron increases to 1.23m0, which has a considerably effect on the radial distribution of the electron. Recall from Bohr's atomic theory that the radius of the nth orbit is proportional to n2/m. Thus, the ratio between the relativistic 1s radius to its nonrelativistic counterpart is approximately (1.23m0)-1/m0-1, or 0.81. This implies that relativistic effects have contracted the 1s orbital in Au by about 20%. Similar contractions in a heavy atom also occur for the higher s orbitals because these become orthogonal to one another. The result is a lowering of the energies of all s orbitals. This can also be seen from the Bohr atomic model, where energies are proportional to -m. When relativity is turned on, the heavier electron makes the ground-state energy lower, or more negative. In any event, this is known as the direct relativistic effect and accounts for orbital contraction and stabilization of s orbitals and, to a lesser extent, p orbitals. In contrast, the valence d and f orbitals in Au are expanded and destabilized by relativistic effects. This is because the contraction of the s orbitals increases their shielding magnitude, which gives rise to a smaller effective nuclear charge for the d and f electrons. This is known as the indirect relativistic orbital expansion and destabilization. In addition, if a filled d or f subshell lies just inside a valence orbital, that orbital will experience a larger effective nuclear charge that will lead to orbital contraction and stabilization. This is because the d and f orbitals expand when their shielding from the nucleus is reduced. This is also believed to make a contribution to the lanthanide contraction. The concomitant relativistic stabilization of the 6s orbital and the destabilization of the 5d orbital in Au narrow the energy gap between these two orbitals. As a result, the [...5d106s1]2S1/2 f [...5d96s2]2D3/2 transition in Au is observed in the visible region; this absorption of blue and violet region accounts for the yellow
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In the Classroom
luster of gold. In contrast, the corresponding [...4d105s1]2S1/2f [...4d95s2]2D3/2 transition in Ag (Z = 47) lies in the UV region. In addition, the stabilization of the 6s orbital gives rise to a large ionization energy for Au, which in turn leads to gold's greater stability: gold does not tarnish in air, whereas silver does. By a similar argument, the relativistic contraction of the 6s2 subshell in Hg (Z = 80) makes this element relatively unreactive: the so-called inert pair effect described in many inorganic chemistry texts. Indeed, the tightly bound low-energy pair of 6s electrons in Hg makes it comparable to He, nearly a noble gas. Thus, there is reduced interatomic attraction in Hg, which is responsible for it remaining a liquid at room temperature. The relativistic effect on the property of heavy metals is now a topic covered in books and journal articles (1, 2, 4, 5). This short section gives only the highlights. Conclusion This article attempts to present to chemistry students and teachers some important and chemically interesting results of relativistic quantum mechanics. These results include a mathematical justification for electron spin, a possible rationalization
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of the mystery surrounding the nodes of wavefunctions, the prediction of antimatter, the relativistic treatment of the onedimensional particle-in-a-box problem, and the anomalous properties of some heavy metals. It is hoped that this nonmathematical presentation will be accessible to chemists. A few additional references (6-8) are cited for those who wish to delve more deeply into this topic. Literature Cited 1. See, for example , Blinder, S. M. Introduction to Quantum Mechanics; Elsevier/Academic Press: Amsterdam, 2004. 2. See, for example , Li, W.-K.; Zhou, G.-D.; Mak, T. C. W. Advanced Structural Inorganic Chemistry; Oxford University Press: Oxford, U.K., 2008. 3. Alberto, P.; Fiolhais, C.; Gil, V. M. S. Eur. J. Phys. 1996, 17, 19–24. 4. Norrby, L. J. J. Chem. Educ. 1991, 68, 110–113. 5. Guerrero, A. H.; Fasoli, H. J.; Costa, J. L. J. Chem. Educ. 1999, 76, 200. 6. Pitzer, K. Acc. Chem. Res. 1979, 12, 271–276. 7. Pyykkö, P.; Desclaux, J. Acc. Chem. Res. 1979, 12, 276–281. 8. Pyykkö, P. Chem. Rev. 1988, 88, 563–594.
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