Research: Science and Education edited by Advanced Chemistry Classroom and Laboratory Joseph J. BelBruno Dartmouth College Hanover, NH 03755 Spectroscopic Selection Rules: The Role of Photon States Andrew M. Ellis Department of Chemistry, University of Leicester, University Road, Leicester LE1 7RH, UK; [email protected] Selection rules are vital in the interpretation of atomic and In quantum mechanics, all three Cartesian components of molecular spectra. The usual starting point for a derivation this vector cannot be known simultaneously (for a proof see of selection rules is the transition moment which, in intro- ref 2). Instead, there are only two simultaneously observable ductory spectroscopy courses, is normally taken on trust. quantities, the total angular momentum and any one of However, many students find the transition moment to be its Cartesian components. Which component is chosen is a somewhat obscure quantity that does little to reveal the arbitrary in the absence of some external constraint but it is underlying physics. Furthermore, the journey from the tran- conventional to select the z-component. As a result there sition moment to a particular selection rule is not always easy. are two quantum numbers that define the orbital angular This is especially troublesome when all one wishes to achieve momentum of an electron, the total quantum number l and is a justification of certain selection rules, rather than to give its corresponding projection onto the z-axis, quantum a full-blown account. number ml . An easier way of justifying selection rules is to apply con- Figure 1 summarizes the findings in a vector diagram. servation arguments to individual photon–atom (or photon– Similar diagrams are commonplace in textbooks dealing with molecule) interactions. One quantity that must be conserved the quantum theory of angular momentum. The total angular is angular momentum. To invoke this argument, the angular momentum, represented by the length of the vector, has the momentum of the photon must first be known. A transition magnitude that satisfies angular momentum conservation must cause a ll +1 h change in angular momentum state of the absorbing or emit- ting atom (or molecule) that compensates for the loss or gain The quantum number l may have any integer value includ- of photon angular momentum. ing zero. Although only one is indicated in the figure, there It is commonly assumed that a photon possesses an are 2l + 1 possible values of m ranging from {l to + l in unit ⁄ ⁄ l angular momentum of ± h where h = h/2π. In other words, steps. The values of the projection of the orbital angular mo- ⁄ it has a unit quantum of angular momentum and therefore mentum on the z-axis are ml h. unit changes in the angular momentum quantum number Strictly speaking, l and ml are not the only important are expected for an absorbing or emitting atom or molecule. angular momentum quantum numbers. Angular momentum Foss presented this argument in an article in this Journal some states also possess parity. Parity refers to the effect of an in- years ago (1) to rationalize selection rules such as ∆l = ±1 version of all spatial coordinates on the sign of the angular and ∆J = ±1, and it has also appeared in several textbooks momentum wave function. If there is no change in sign the (e.g., see refs 2–5). While it is a useful starting point, this parity is said to be even; a change in sign corresponds to a model fails to account properly for selection rules when sub- state of odd parity. For orbital angular momentum the parity is jected to closer examination. In this article a more compre- ({1)l . This is easily seen by considering the well-known shapes hensive, but nevertheless straightforward, justification of an- of s, p, d, and f orbitals. An inversion of coordinates about the gular momentum selection rules is presented. I begin with a atomic nucleus does not change the sign of s and d orbitals, very brief review of some key results from the quantum and so they possess even parity; but it does change the sign theory of angular momentum. I then show some of the flaws of p and f orbitals, so they must have odd parity. Parity places in the model used by Foss and others. Finally, I explain how restrictions on spectroscopic transitions, as will be seen later. these flaws disappear when a more complete quantum elec- trodynamic model of a photon is employed. Coupling of Two Angular Momenta The coupling of two angular momenta, l1 and l2, yields Review of the Basic Elements of the Quantum Theory a resultant L, which is the vector sum of the two individual of Angular Momentum Most undergraduate courses on quantum mechanics Figure 1. Vector model of quantized include a description of the angular momentum of a single angular momentum. The magnitude rotating or orbiting body. Many also include an elementary of the angular momentum, repre- account of the coupling of angular momenta. Some key facts sented by the length of the vector, is from both topics are summarized below without proof. ll +1 h Single Body The vector precesses (not shown) about the z axis, producing inde- Angular momentum is a vector quantity. An example is the terminate components in the x and orbital angular momentum of an individual electron in an atom y directions but a well-defined com- (the possibility of coupling is dealt with in the next section). ponent along the z axis. JChemEd.chem.wisc.edu • Vol. 76 No. 9 September 1999 • Journal of Chemical Education 1291 Research: Science and Education angular momentum vectors, Photon energy. This is familiar from elementary physics— it depends linearly on the frequency ν; that is, E = hν. L = l + l (1) 1 2 Angular momentum. The quantum mechanics of photon In quantum mechanics, the total orbital angular momentum angular momentum has much in common with any other is quantized and has a quantum number L, which is restricted type of angular momentum. A photon can be ascribed a total to the values angular momentum quantum number, j, and a projection ≥ ≥ |l1 + l2| L |l1 – l2| (2) quantum number mj. The magnitude of this momentum is This important result is central to the arguments about con- given by servation of angular momentum presented later. jj +1 h ⁄ Transitions between Angular Momentum States: The and the projection along the z axis is mj h. Photons may Failure of the Simple Model possess any positive integer j, but j = 0 is impossible; that is, photons cannot have zero angular momentum. Photons possess quantized angular momentum. As dis- Parity. An angular momentum state of a photon must have cussed by Foss (1), experiments have shown that all photons have ± ⁄ parity. For a given j both odd and even parities are possible. an angular momentum of h along the axis of propagation. Proof of the above statements are rather involved and This is known as the helicity of a photon. The two possible can be found in books on quantum electrodynamics (see, for signs refer to the two possible states of circularly polarized example, refs 6, 7). However, the take-home message is very light, left and right circular polarization. Linearly polarized simple. The different possible angular momentum states and light can be thought of as an equal mixture of left and right parities can be identified with different types of photons. Notice circularly polarized light, while any other state of polarization that, in contrast to the electron orbital angular momentum, (e.g., elliptical polarization) will contain unequal proportions both even and odd parities are possible for each value of the of photons with opposite helicities. photon angular momentum quantum number j. Taking j =1 Now consider a transition in which an electron moves and j = 2 as examples, the following types of photon occur: from one atomic orbital to another by photon absorption or j = 1, parity = odd electric dipole photon emission. According to the Laporte selection rule, only tran- sitions for which ∆l = ±1 are allowed. This selection rule can j = 1, parity = even magnetic dipole photon be justified on the grounds of parity conservation, as will be j = 2, parity = even electric quadrupole photon shown later. But it should also be possible to show that it is j = 2, parity = odd magnetic quadrupole photon consistent with angular momentum conservation. The argument In a classical electromagnetic field, the electric and mag- that is normally employed, as mentioned earlier, is that since netic parts contain various multipole components: dipole, a photon possesses unit angular momentum, ± h⁄ , only a quadrupole, and onwards to higher multipoles. In quantum compensating unit change in the electron orbital angular electrodynamics, we can think of these parts as arising from momentum is compatible with the conservation of overall different types of photon, and the names of the photon states angular momentum. given above reflects this. However, a trivial example will show that this explana- It can be shown (6, 7) that all photons, regardless of j tion is flawed. Consider the transition l = 1 ↔ l = 0. The or parity, must possess an angular momentum along their conservation of angular momentum requires that the vector direction of propagation of ± h⁄ , thus concurring with experi- sum of all angular momenta is zero. For l = 0 the orbital ment (1, 8); this is the helicity referred to earlier. However, angular momentum is zero and so in a transition to l = 1 the it is important to recognize that the direction of propagation magnitude of the angular momentum gained by the electron is not the same as the axis (z) along which the projection must exactly equal that of the photon that is being absorbed.