Selection rule

In physics and chemistry, a selection rule, or transition rule, formally constrains the possible transitions of a system from one quantum state to another. Selection rules have been derived for electromagnetic transitions in molecules, in atoms, in atomic nuclei, and so on. The selection rules may differ according to the technique used to observe the transition. The selection rule also plays a role in chemical reactions, where some are formally spin-forbidden reactions, that is, reactions where the spin state changes at least once from reactants to products. In the following, mainly atomic and molecular transitions are considered.

In physics and chemistry, a selection rule, or transition rule, formally constrains the possible transitions of a system from one quantum state to another. Selection rules have been derived for electromagnetic transitions in molecules, in atoms, in atomic nuclei, and so on. The selection rules may differ according to the technique used to observe the transition. The selection rule also plays a role in chemical reactions, where some are formally spin-forbidden reactions, that is, reactions where the spin state changes at least once from reactants to products. In the following, mainly atomic and molecular transitions are considered. m_{1,2} = \int \psi_1^* \, \mu \, \psi_2 \, \mathrm{d}\tau, where \psi_1 and \psi_2 are the wave functions of the two states, "state 1" and "state 2", involved in the transition, and is the transition moment operator. This integral represents the propagator (and thus the probability) of the transition between states 1 and 2; if the value of this integral is zero then the transition is "forbidden". In practice, to determine a selection rule the integral itself does not need to be calculated: It is sufficient to determine the symmetry of the transition moment function \psi_1^* \, \mu \, \psi_2. If the transition moment function is symmetric over all of the totally symmetric representation of the point group to which the atom or molecule belongs, then the integral's value is (in general) not zero and the transition is allowed. Otherwise, the transition is "forbidden". The transition moment integral is zero if the transition moment function, \psi_1^* \, \mu \, \psi_2, is anti-symmetric or odd, i.e. y(x) = -y(-x) holds. The symmetry of the transition moment function is the direct product of the parities of its three components. The symmetry characteristics of each component can be obtained from standard character tables. Rules for obtaining the symmetries of a direct product can be found in texts on character tables. In vibrational spectroscopy, transitions are observed between different vibrational states. In a fundamental vibration, the molecule is excited from its ground state (v = 0) to the first excited state (v = 1). The symmetry of the ground-state wave function is the same as that of the molecule. It is, therefore, a basis for the totally symmetric representation in the point group of the molecule. It follows that, for a vibrational transition to be allowed, the symmetry of the excited state wave function must be the same as the symmetry of the transition moment operator. In infrared spectroscopy, the transition moment operator transforms as either x and/or y and/or z. The excited state wave function must also transform as at least one of these vectors. In Raman spectroscopy, the operator transforms as one of the second-order terms in the right-most column of the character table, below. The molecule methane, CH 4 , may be used as an example to illustrate the application of these principles. The molecule is tetrahedral and has T d symmetry. The vibrations of methane span the representations A 1 + E + 2T 2 . Examination of the character table shows that all four vibrations are Raman-active, but only the T 2 vibrations can be seen in the infrared spectrum. In the harmonic approximation, it can be shown that overtones are forbidden in both infrared and Raman spectra. However, when anharmonicity is taken into account, the transitions are weakly allowed. In Raman and infrared spectroscopy, the selection rules predict certain vibrational modes to have zero intensities in the Raman and/or the IR. Displacements from the ideal structure can result in relaxation of the selection rules and appearance of these unexpected phonon modes in the spectra. Therefore, the appearance of new modes in the spectra can be a useful indicator of symmetry breakdown. The selection rule for rotational transitions, derived from the symmetries of the rotational wave functions in a rigid rotor, is ΔJ = ±1, where J is a rotational quantum number. The emitted particle carries away angular momentum, with quantum number , which for the photon must be at least 1, since it is a vector particle (i.e., it has = 1 − ). Thus, there is no radiation from E0 (electric monopoles) or M0 (magnetic monopoles, which do not seem to exist). Since the total angular momentum has to be conserved during the transition, we have that \mathbf J_\text{i} = \mathbf{J}_\text{f} + \boldsymbol{\lambda}, where \|\boldsymbol{\lambda}\| = \sqrt{\lambda(\lambda + 1)} \, \hbar, and its projection is given by \lambda_z = \mu \hbar; and where \mathbf J_\text{i} and \mathbf J_\text{f} are, respectively, the initial and final angular momenta of the atom. The corresponding quantum numbers and ( -axis angular momentum) must satisfy |J_\text{i} - J_\text{f}| \le \lambda \le J_\text{i} + J_\text{f} and \mu = M_\text{i} - M_\text{f}. Parity is also preserved. For electric multipole transitions \pi(\mathrm{E}\lambda) = \pi_\text{i} \pi_\text{f} = (-1)^{\lambda}, while for magnetic multipoles \pi(\mathrm{M}\lambda) = \pi_\text{i} \pi_\text{f} = (-1)^{\lambda+1}. Thus, parity does not change for E-even or M-odd multipoles, while it changes for E-odd or M-even multipoles. These considerations generate different sets of transitions rules depending on the multipole order and type. The expression forbidden transitions is often used, but this does not mean that these transitions cannot occur, only that they are electric-dipole-forbidden. These transitions are perfectly possible; they merely occur at a lower rate. If the rate for an E1 transition is non-zero, the transition is said to be permitted; if it is zero, then M1, E2, etc. transitions can still produce radiation, albeit with much lower transitions rates. The transition rate decreases by a factor of about 1000 from one multipole to the next one, so the lowest multipole transitions are most likely to occur. Semi-forbidden transitions (resulting in so-called intercombination lines) are electric dipole (E1) transitions for which the selection rule that the spin does not change is violated. This is a result of the failure of LS coupling. J = L + S is the total angular momentum, L is the azimuthal quantum number, S is the spin quantum number, and M_J is the secondary total angular momentum quantum number. Which transitions are allowed is based on the hydrogen-like atom. The symbol \not\leftrightarrow is used to indicate a forbidden transition. In hyperfine structure, the total angular momentum of the atom is F = I + J, where I is the nuclear spin angular momentum and J is the total angular momentum of the electron(s). Since F = I + J has a similar mathematical form as J = L + S, it obeys a selection rule table similar to the table above.

Text derived from https://en.wikipedia.org/w/index.php?oldid=1338553539 under CC BY-SA 4.0; modifications have been made.