Subject Chemistry Paper No and Title Module No and Title Module Tag 8 and Physical Spectroscopy 5 and Transition probabilities and transition dipole moment, Overview of selection rules CHE_P8_M5
TABLE OF CONTENTS 1. Learning Outcomes 2. Introduction 3. Transition Moment Integral 4. Overview of Selection Rules 5. Summary
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 M 3 ij 8π! 2 Bij = = M 2 2 ij 6ε " 3h 4πε 0 ( ) 0 where the radiation density is expressed in units of Hz. The quantity M! is known as the transition moment integral, having the same unit as dipole ij 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 0 cos(ωt) that interacts with a transition dipole. The dipole moment vector is! µ = e r!, 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 ii + * = ψ ˆ µψ dτ i i and
! M jj + * = ψ ˆ µψ dτ j j respectively. The symbol! M ij + * = ψ ˆ µψ dτ signifies the transition dipole moment for a transition from the i j i 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 3 3 5 7 = x 3 ; f ( x) = 2x + 3x + 5x f ( x) + x 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) = x 2 and f(x) = 2x 4 + x 6. On the other hand, functions such as f(x) = 2x 4 + x 5 are neither even nor odd, and possess no symmetry properties. The figure on the right is a plot of an even function (y = x 2 ). 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
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 = x 3. 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:
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 N 2 or F 2 possesses no permanent dipole moment and is hence microwave inactive. Molecules having a spherical shape, such as
the tetrahedral molecule SiH 4, 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 molecules in food. Water is a polar molecule and can absorb microwave radiation, making the water molecules rotate faster and transfer heat to neighbouring molecules. Radar systems also use microwave radiation. very little loss of signal occurs in transmission because most of the molecules in the air: nitrogen, oxygen and carbon dioxide are microwave inactive. Vibrational spectroscopy In this case, the gross selection rule is that there must be a change in the dipole moment of the molecule during a vibration for the vibration to be infrared active. Since homonuclear molecules do not possess a permanent dipole moment and there is no change in the dipole moment when the molecule vibrates, they are also infrared inactive. Heteronuclear diatomic molecules like HCl are IR active. However, some of the vibrations of molecules like carbon dioxide, which possess no dipole moment are infrared active. Carbon dioxide has three modes of vibration (Fig. 1).
Figure 1 Vibrational modes of carbon dioxide In the first mode, the two CO bonds compress or expand in phase, so that the net dipole moment remains zero. In the second, bending, mode, the molecule becomes bent like water and acquires a dipole moment. This motion is hence infrared active. In the third mode, asymmetric stretch, one CO bond compresses while the other stretches, leading to an asymmetric distribution of charge and a net dipole moment. This vibration is also infrared active. Raman spectroscopy Vibrations are also observed using Raman spectroscopy. The mechanism of interaction with the electromagnetic radiation is different in this case and is based on the scattering of radiation. The electric field induces a dipole moment in the molecule. The molecular property that must change for a rotation or vibration to be Raman active is the polarizability, which measures the extent to which the electron distribution of a molecule can be disturbed by an electric field. Thus it is not necessary that a molecule possess a dipole moment for it to be Raman active. Homonuclear diatomic molecule like hydrogen,
nitrogen, etc. also display rotational Raman spectra, because they have anisotropic polarizabilities it is easier to induce a dipole moment along their bond axis than in other directions. In other words, they are more polarizable along the bond axis. When these molecules rotate, their polarizability changes in shape. However, spherically symmetrical molecules like methane are rotational Raman inactive. Homonuclear diatomic molecules also show vibrational Raman spectroscopy. The symmetric stretch vibration of carbon dioxide (Fig. 1) is not infrared active, but it is Raman active, as the polarizability increases when the molecule is stretched and decreases when it is compressed. Electronic spectroscopy For homonuclear diatomic molecules, there are certain selection rules that can be easily deduced. The first is the Laporte selection rule derived above, according to which the parity of the wave function must change during a transition. If the Hamiltonian of a system is spin-independent, the system cannot change its spin angular momentum during a transition. We thus have the ΔS = 0 selection rule. It must also be emphasized that these rules apply to electric dipole transitions only. The radiation also has a magnetic field, and magnetic dipole transitions may also be induced. However, they are much weaker than electric dipole transitions, and are only of importance when the selection rules do not permit electric dipole transitions. The reason is that the magnetic field (B) is much weaker than the electric field (E = cb) since the value of the speed of light (c) is very large. Higher order (electric quadrupole) moments may also contribute to the transition. Certain forbidden transitions (such as n π*) are also observed, although with very low intensities. Some other mechanisms (vibronic coupling. i.e. coupling of electronic and vibrational wave functions) allow these transitions to take place. Similarly, the spin selection rule is also frequently violated, 5. Summary The probability of a transition is proportional to the square of the transition dipole moment. The transition dipole moment is the transient dipole moment induced in the molecule during a transition. It involves an integral of a triple product of the excited state wave function, the dipole moment operator and the ground state wave function.
Though a difficult quantity to evaluate, often it is enough to know the symmetry properties of the three quantities in the triple product to predict the selection rules. Transitions between levels of same symmetry are laporte forbidden