Spin-forbidden radiative transition probabilities

Also in response theory the summation over excited states is effectively replaced by solving a system of linear equations. Spin-orbit matrix elements are obtained from linear response functions, whereas quadratic response functions can most elegantly be utilized to compute spin-forbidden radiative transition probabilities. We refrain from going into details here, because an excellent review on this subject has been published by Agren et al.118 While these authors focus on response theory and its application in the framework of Cl and multiconfiguration self-consistent field (MCSCF) procedures, an analogous scheme using coupled-cluster electronic structure methods was presented lately by Christiansen et al.124... 

A weakness of these methods lies in the limited number of zeroth-order states that are used for an expansion of the first-order perturbed wave function. In particular, it has been demonstrated that probabilities of spin-forbidden radiative transitions converge slowly with the length of the perturbation expansion.92... 

As stated in Chapter 1, transitions involving a change in multiplicity are spin forbidden. However, for reasons which we will consider later, such transitions do indeed occur although with very low transition probabilities in most cases. The intensity of an absorption corresponding to a transition from the ground state S0 to the lowest triplet state Tx is related to the triplet radiative lifetime t ° by the following equation[Pg.114]

Besides fine-structure splitting, the occurrence of spin-forbidden transitions is the most striking feature in which spin-orbit interaction manifests itself. Radiative spin-forbidden transitions in light molecules usually take place at the millisecond time scale, if the transition is dipole allowed. A dipole- and spin-forbidden transition is even weaker, with lifetimes of the order of seconds. Proceeding down the periodic table, spin-forbidden transitions become more and more allowed due to the increase of spin-orbit coupling. For molecules containing elements with principal quantum number 5 or higher (and the late first-row transition metals Ni and Cu), there is hardly any difference between transition probabilities of spin-allowed and spin-forbidden processes. 

This nonradiative process violates the Laporte rule, which states that only Franck-Condon transitions occurring between states having the same spin multiphcity are allowed, the others being forbidden. It is important to note that in quantum physics forbidden transitions are not impossible, but this means that their probability is extremely low, i.e., with a longer duration or timehfe of the transition. Therefore, the radiative transition occurring between a singlet with no vibrational excitation to a triplet state has a mean timelife (t ) of several microseconds (i.e., 1 ps = 10 s) to milliseconds (i.e., 1 ms = 10 s) ... 

Transitions with transition moments larger than a certain extent, are called the allowed transitions, and those with near-zero moments are called the forbidden transitions. For example, the transition between 0( P) 0( D) of O atoms is spin forbidden, and the light emission probability for 0( D) 0( P) is very small so that the radiative lifetime of 0( D) is long and the reaction of 0( D) with other molecule can be important in the atmosphere. 

Contrary to allowed transitions, for which the upper-state radiative lifetimes are in the nanosecond range, the lifetime of species whose transition to the ground state is spin forbidden lies in the range of microseconds or milliseconds to hours. One of the longest-lived (isolated) molecules is probably O2 in its a A state (5270 s) isovalent molecules with heavier atoms and reduced symmetry, such as SO, exist for only 0.45 s in this a A state (neglecting other processes). 

Finally, a B-spUne method has also allowed the evaluation of the radiative decay probabilities of the six vibrational levels of the metastable a X state of HeH . The transition a X+ X + is spin-forbidden, but acquires intensity through spin-orbit interaction with the singlet and triplet n states [111]. 

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