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PHYSICAL SPECTROSCOPY) MODULE No. : 5 (TRANSITION PROBABILITIES and TRANSITION DIPOLE MOMENT. OVERVIEW of SELECTION RULES

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PHYSICAL SPECTROSCOPY) MODULE No. : 5 (TRANSITION PROBABILITIES and TRANSITION DIPOLE MOMENT. OVERVIEW of SELECTION RULES ____________________________________________________________________________________________________ Subject Chemistry Paper No and Title 8 and Physical Spectroscopy Module No and Title 5 and Transition probabilities and transition dipole moment, Overview of selection rules Module Tag CHE_P8_M5 CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 5 (TRANSITION PROBABILITIES AND TRANSITION DIPOLE MOMENT. OVERVIEW OF SELECTION RULES) ____________________________________________________________________________________________________ TABLE OF CONTENTS 1. Learning Outcomes 2. Introduction 3. Transition Moment Integral 4. Overview of Selection Rules 5. Summary CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 5 (TRANSITION PROBABILITIES AND TRANSITION DIPOLE MOMENT. OVERVIEW OF SELECTION RULES) ____________________________________________________________________________________________________ 1. Learning Outcomes After studying this module, • you shall be able to understand the basis of selection rules in spectroscopy • Get an idea about how they are deduced. 2. Introduction The intensity of a transition is proportional to the difference in the populations of the initial and final levels, the transition probabilities given by Einstein’s coefficients of induced absorption and emission and to the energy density of the incident radiation. We now examine the Einstein’s coefficients in some detail. 3. Transition Dipole Moment Detailed algebra involving the time-dependent perturbation theory allows us to derive a theoretical expression for the Einstein coefficient of induced absorption ! 2 3 M ij 8π ! 2 B M ij = 2 = 2 ij 6ε 0" 3h (4πε 0 ) where the radiation density is expressed in units of Hz. ! The quantity M ij is known as the transition moment integral, having the same unit as dipole moment, i.e. C m. Apparently, if this quantity is zero for a particular transition, the transition probability will be zero, or, in other words, the transition is forbidden. This forms the basis for the selection rules for transitions. Selection rules tell us the possible transitions among quantum levels due to absorption or emission of electromagnetic radiation. Incident electromagnetic radiation presents an oscillating electric field E cos(ωt) that interacts with a transition dipole. The dipole moment vector ! ! 0 ! is µ = er , where r is a vector pointing in a direction of space. By definition, the dipole moments of two given states i and j are ! +∞ M * ˆ d ii = ∫ψ i µψ i τ −∞ and CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 5 (TRANSITION PROBABILITIES AND TRANSITION DIPOLE MOMENT. OVERVIEW OF SELECTION RULES) ____________________________________________________________________________________________________ ! +∞ M * ˆ d jj = ∫ψ j µψ j τ −∞ respectively. ! +∞ The symbol M * ˆ d signifies the transition dipole moment for a transition from the i ij = ∫ψ j µψ i τ −∞ state to the j state, and signifies a transient dipolar polarization created by an interaction of electromagnetic radiation with a molecule. 4. Overview of Selection Rules ! M ij is a triple integral involving the complex conjugate of the excited state wave function, the dipole moment operator and the wave function of the ground state. Though the quantum mechanical solution of wave functions for the ground and excited states is complicated, it is often possible to deduce the selection rules from the symmetry properties of the wave functions alone. From well-known fact of mathematics, that a function f(x), which reverses sign on replacing x by –x is an odd function, and a function which does not change sign under a similar operation is an even function. Some examples of odd functions are f (x) = x + 3x3 ; f (x) = 2x3 + 3x5 + 5x7 You may note that all powers of x in such expressions are odd, so that the function changes sign on reversing the sign of x; in other words, f(-x) = f(x). Similarly, functions having all even powers of x are even functions, and for these f(-x) = f(x). Examples are f(x) = x2 and f(x) = 2x4 + x6. On the other hand, functions such as f(x) = 2x4 + x5 are neither even nor odd, and possess no symmetry properties. The figure on the right is a plot of an even function (y = x2). It is also a well-known fact of mathematics that +a the integral ∫ ydx represents the area under the curve −a of y against x between the two limits. It is apparent CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 5 (TRANSITION PROBABILITIES AND TRANSITION DIPOLE MOMENT. OVERVIEW OF SELECTION RULES) ____________________________________________________________________________________________________ that this is just twice the integral between the limits 0 and a, and is non-zero. Now, let us examine a similar graph for an odd function, y = x3. The figure on the left shows that the area is positive for positive values of x and negative for negative values of x, so that the integral between –a and a vanishes because of the cancellation of the positive and negative areas. Exactly the same reasoning applies for deducing the selection rules from the transition moment integral. If the triple product is an odd function, it will vanish on integration and the corresponding transition will be forbidden. Similarly, if the triple product is even, the transition may be allowed. Wave functions that do not change sign on reflection (r → -r) are said to be of even parity and those that change sign are of odd parity. The dipole moment operator transforms as r, since for a single electron, the dipole moment operator is er, where e is the electronic charge. Thus it is of odd parity. For the transition moment integral to survive, therefore, the product of the two wave functions should also be of odd parity, since odd × odd = even. This is only possible if one is odd and the other even, since odd × even = odd. We immediately have the Laporte selection rule, which states that the wave function should change its parity during a transition. This rule is also stated as follows: g ↔ u, but g ↮ g and u ↮ u. This notation originates from the German gerade for wave functions of even parity, denoted as g, and ungerade for wave function of odd parity (denoted by u), in the case of centrosymmetric molecules. An immediate application of this rule is in atomic spectroscopy. Consider the atomic orbitals shown below: CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 5 (TRANSITION PROBABILITIES AND TRANSITION DIPOLE MOMENT. OVERVIEW OF SELECTION RULES) ____________________________________________________________________________________________________ It is apparent that the s and d orbitals are of even parity and hence transitions cannot take place between them. However, the p orbitals are of odd parity and hence transitions between the s and p orbitals may be possible. Though these are allowed transitions by electric dipole selection rules, the argument does not explain why s to f transitions are forbidden, though these transitions involve a change of parity. It is important to remember that the symmetry selection rules only tell us which transitions are forbidden by symmetry, but it does not follow that the remaining transitions are allowed. They may be forbidden for reasons other than symmetry. In the present case, conservation of angular momentum requires that the angular momenta of the atom and photon should remain constant. Since the photon has an intrinsic angular momentum of one, it can either add one unit or subtract one unit of angular momentum from the atom. In other words, the l quantum number can either decrease or increase by one unit, or Δl = ±1. Hence, an s electron can only be promoted to a p orbital. Every kind of spectroscopy has two parts to the selection rules: a gross selection rule and a specific selection rule. The gross selection rule states the requirements for an atom or molecule to display a particular spectrum and the specific selection rule states the changes in quantum numbers accompanying the transitions. The gross selection rule is often easy to predict based on the requirements for effective interaction with the electromagnetic field. Rotational spectroscopy A molecule must have a transition dipole moment that is in resonance with the electromagnetic field for rotational spectroscopy to be used. Polar molecules have a dipole moment and are thus microwave active. The term means that they will exhibit rotational spectra, which is usually observed in the microwave region. As the molecule rotates, the dipole moment vector also oscillates. If the oscillation is in resonance with the electric field, absorption or emission of radiation is induced. In contrast, a homonuclear diatomic molecule like N2 or F2 possesses no permanent dipole moment and is hence microwave inactive. Molecules having a spherical shape, such as CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 5 (TRANSITION PROBABILITIES AND TRANSITION DIPOLE MOMENT. OVERVIEW OF SELECTION RULES) ____________________________________________________________________________________________________ the tetrahedral molecule SiH4, are also microwave inactive for the same reason. However, when the molecule undergoes fast rotation, a transient dipole moment may be induced and a rotational spectrum observed. The intensity of the rotational spectrum is proportional to the square of the dipole moment, and so highly polar molecules such as HCl display strong absorptions in the microwave. Microwave spectroscopy finds several uses. Microwave ovens operate on the rotational excitation of water
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