Laporte s rule

The first two parts of the expression vanish exactly because of Laporte s rule, while the last two survive both parity and orbital selection rules to the extent that the mixing coefficients c and c are non-zero in noncentric complexes. 

Experimentally, spin-allowed d-d bands (we use the quotation marks again) are observed with intensities perhaps 100 times larger than spin-forbidden ones but still a few orders of magnitude (say, two) less intense than fully allowed transitions. This weakness of the d-d bands, alluded to in Chapter 2, is a most important pointer to the character of the d orbitals in transition-metal complexes. It directly implies that the admixture between d and p metal functions is small. Now a ligand function can be expressed as a sum of metal-centred orbitals also (see Box 4-1). The weakness of the d-d bands also implies that that portion of any ligand function which looks like a p orbital when expanded onto the metal is small also. Overall, therefore, the great extent to which d-d bands do satisfy Laporte s rule entirely supports our proposition in Chapter 2 that the d orbitals in Werner-type complexes are relatively well isolated (or decoupled or unmixed) from the valence shell of s and/or p functions. 

I like to recall his [M. von Lane s] question as to which results derived in the present volume I considered most important. My answer was that the explanation of Laporte s rule (the concept of parity) and the quantum theory of the vector addition model appeared to me most significant. Since that time, I have come to agree with his answer that the recognition that almost all rules of spectroscopy follow from the symmetry of the problem is the most remarkable result. 

A well-known selection rule concerning centrosymmetric systems (those with a center of inversion) is the Laporte s rule. For such systems, states are either g (even) or u (odd). Laporte s rule states that only transitions between g and u states are allowed i.e., transitions between two g states and those between two u states are forbidden. With the foregoing discussion, this rule can now be easily proved. For centrosymmetric molecules, the three components of the dipole moment vector are all u. For g g transitions, the overall symmetry... 

Note that all three transitions are g ++ u, in accordance with Laporte s rule. 

Crystal field, or d-d, transitions are defined as transitions from levels that are exclusively perturbed d orbitals to levels of the same type. In other words, the electron is originally localized at the central metal ion and remains so in the excited state. When the system has ( symmetry, Laporte s rule says that an electric-dipole allowed transition must be between a g state and an u state, i.e., u - g. Since all the crystal field electronic states are gerade ( g ), no electric-dipole allowed transitions are possible. In short, all d-d transitions are symmetry forbidden and hence have low intensities. The fact that the d-d transitions are observed at all is due to the interaction between the electronic motion and the molecular vibration. We will discuss this (vibronic) interaction later (Section 8.10). 

All of these electronic transitions are technically forbidden by Laporte s rule, which states that electronic transitions can only occur if the orbital angular momentum changes by 1 during the transition. Since this does not occur for transitions from one d state to another d state, or from one f state to another f state, no absorption should occur for these ions. Fortunately, Laporte s rule is relaxed in solids due to the lack of perfect spherical symmetry, which results from the presence of a limited number of point sources, so that electronic transitions can occur with a low probability between 3d or 4f levels which are split by the fields of the neighboring ligands. The low probability of these transitions, however, does reduce the intensity of the absorption. As a result, ligand field induced transitions are much weaker than the charge transfer effects which occur in the ultraviolet. 

Let us note that such is the case for most of the transitions in the ions considered because most of them take place within the same configuration, either 4f or 3d. By Laporte s Rule only magnetic dipole transitions are allowed in first order. Then electric dipole transitions shall depend on the host whereas magnetic dipole ones are host independent. 

Optical transitions between 4f" levels used for optical pumping and stimulated emission are predominantly of electric-dipole nature. Although f-f transitions are forbidden by Laporte s rule, if the rare earth is located in a non-centrosym-metric site, odd-order terms in the expansion of the static (or dynamic) crystal-field admix states of higher, opposite-parity configurations, such as 4f" 5d, into 4f" and transitions become allowed. The oscillator strengths of transitions between J states are small, 10 . - While ab initio calculations of the probabilities for electric-dipole transitions are not possible, spectral intensities can be treated using the Judd-Ofelt approach discussed below. 

In general, excitation of a centrosymmetric system is forbidden if the initial and final orbitals both have the same symmetry for inversion (either both g or both u). This principle is known as Laporte s rule. 

According to Laporte s rule for IPA the transition between states having the same parity is forbidden. In the case of the non-centrosymmetric molecules both IPA and 2PA trarrsitions to the final state are allowed. On the other hand, the ground arrd final states of the centrosymmetric molecule can be both symmetric in respect to. u. u. ... 

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