Forbidden transitions, between states

The physical origins of the effect can be illustrated from Figure 2, which shows the energy level scheme for an I = 1 nucleus, such as coupled to an S = 1/2 electron spin. In the case considered here (i.e., the electron-nuclear interaction, the nuclear Zeeman interaction, and the nuclear quadrupole interaction, all of the same order), microwaves can induce both allowed and semi-forbidden transitions between states in the Mj = 1/2 manifold (a) and the Ms = - 1/2 manifold (/ ). Simultaneous excitation of both kinds of transitions by the echo generating microwave pulses gives rise to interference effects, which manifest themselves as variations in the echo amplitude and thus cause the modulation of the echo envelope. Where a number of nuclei are coupled to the same electron spin, the level scheme becomes more complicated, but it is possible to factor out contributions due to coupling with each nucleus in the overall modulation pattern. If v(U l2>" n) is the modulation function due due to coupling with n nuclei, then... 

Once the excited molecule reaches the S state it can decay by emitting fluorescence or it can undergo a fiirtlier radiationless transition to a triplet state. A radiationless transition between states of different multiplicity is called intersystem crossing. This is a spin-forbidden process. It is not as fast as internal conversion and often has a rate comparable to the radiative rate, so some S molecules fluoresce and otliers produce triplet states. There may also be fiirther internal conversion from to the ground state, though it is not easy to detemiine the extent to which that occurs. Photochemical reactions or energy transfer may also occur from S. ... 

Transitions of the d-d type are known as electric dipole transitions. The transition between states of different multiplicity is forbidden, but under certain circumstances it still may be seen, if only weakly. For example, Fe3+ has a 6S ground state, and all of the excited spectroscopic states have a different... 

Phosphorescence Photon emission. Phosphorescence involves a spin-forbidden radiative transition between states of different multiplicity, usually from the lowest vibrational level of the lowest excited triplet state, Tt. Ti(v = 0) - S0 + hv... 

Phosphorescence arises as the result of a radiative transition between states of different multiplicity, Ti —> So. Since the process is spin-forbidden, phosphorescence has a much smaller rate constant, kp, than that for fluorescence, kf ... 

Intersystem crossing is a spin-forbidden process between states of different multiplicity, so the magnitude of the spin-orbit coupling is important in controlling the rate of intersystem crossing. Transitions between... 

Selection rules. A transition between states ipi and ipj associated with light absorption or emission will be forbidden if ipi, er ipj) = 0. 

Both absorption and emission spectra have been recorded for a variety of octahedral chromium(III) complexes. For the systems of interest here, A/B 2. Inspection of Figure 2 leads to the expectation of three spin-allowed, parity-forbidden transitions between the iA2g and the other quartet states and two spin- and parity-forbidden transitions between the iA2g and the 2Eg and 2T2g states. Aqueous solutions of Cr(H20)s3+ display three bands with e 15 at 17,400, 24,500, and 38,000 cm-1, assigned respectively to the transitions iA2g- iT2g,... 

In addition, group theory can be used to assess when transition dipole moments must be zero. The product of the irreducible representations of the two wave functions and the dipole moment operator within the molecular point group symmetry must contain the totally symmetric representation for the matrix element to be non-zero (note that, if the molecule does not contain an inversion center, the operator r does not belong to any single irrep, except for the trivial case of Ci symmetry see Appendix B for more details). A consequence of this consideration is that, for instance, electronic transitions between states of the same symmetry are forbidden in molecules possessing inversion centers. 

The Laporte rule states that transitions between states of the same parity, u or g, are forbidden i.e. u - g and g - u but g +-> g and u +-> u. This rule follows from the symmetry of the environment and the invoking of the Bom-Oppenheimer approximation, But since, due to vibrations, the environment will not always be strictly symmetrical, these forbidden transitions will in fact occur, though rather weakly (oscillator strengths of the order of 10 4). All the states of a transition-metal ion in an octahedral environment are g states, so that it will be these weak symmetry forbidden transitions (called d-d transitions) that will be of most interest to us when we study the spectra of octahedral complexes. 

In the remainder of this section, we will consider only electric-dipole transitions. These are the strongest transitions, and account for most of the observed atomic and molecular spectroscopic transitions. (Magnetic-dipole transitions occur in magnetic-resonance spectroscopy.) When the integral d vanishes, we say that a transition between states n and m is forbidden. 

Hence Q — 0 unless i(fspin = i/spin. We have, then, the selection rule AS = 0. Transitions between states of different spin quantum number are forbidden. 

On the other hand, most chemists and many physicists leading with polyatomic organic molecules currently employ the mechanistic definitions advanced by G. N. Lewis and shown in Figure 1. Thus, fluorescence is defined as a radiative transition between states of like multiplicity, e.g., 5 x - So + hv. Phosphorescence is a radiative transition between states of different multiplicity. In organic molecules the process is usually associated with spin-forbidden transitions such as Ti - S0 + hv". 

In addition to the symmetry selection rule, there is also another selection rule on spin AS = 0, i.e., only the transitions between states with the same spin are allowed. The fact that we sometimes do observe spin-forbidden d-d transitions is mainly due to spin-orbit coupling, which will be discussed later in this section. 

A rule of quantum mechanics states that transitions between states of opposite symmetry are forbidden this is why the intensity of the outer lines falls to zero in the limit of An = 0. In between, in the strong coupling zone, the outer lines are diminished in intensity and this gives the leaning or house shape of the AB system. 

The d-d bands are usually relatively low in intensity compared to CT bands contrast the palish hues of familiar salts of Cr, Mn, Fe, Co, Ni, and Cu with the intense purple of Mn04. This can be explained within the CF model. The probability of a transition is governed by selection rules see Group Theory). In atomic spectra, transitions between states having the same / quantum number are forbidden if this rule were strictly obeyed, d-d transitions should not be observable. Moreover, if there is a center of symmetry, as in an octahedral or square coplanar complex, d-d transitions are forbidden although they can be observed, albeit relatively weakly. Molecular vibration can disturb the center of synunetry, and Vibronic Coupling lends intensity to d d absorption. A tetrahedral complex has no center of synunetry and the... 

Transitions between states of different spin multiplicities are forbidden. For example, transitions between 2 states are spin-allowed, but between... 

Emission or luminescence is referred to as fluorescence or phosphorescence, depending on whether it corresponds to a spin-allowed or a spin-forbidden transition, respectively. Similarly, radiationless transitions between states of the same multiplicity and of a different multiplicity are known as internal conversion (1C) and intersystem crossing (ISC), respectively. 

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