Spectroscopic state

In a manner similar to that by which the atomic states were designated as s, p, d, or /, the letters S, P, D, and F correspond to the values of 0, 1, 2, and 3, respectively, for the angular momentum vector, L. After the values of the vectors L, S, and / have been determined, the overall angular momentum is described by a symbol known as a term symbol or spectroscopic state. This symbol is constructed as Ps+1)Lj where the appropriate letter is used for the L value as listed earlier, and the quantity (2S + 1) is known as the multiplicity. For one unpaired electron, (2S + 1) = 2, and a multiplicity of 2 gives rise to a doublet. For two unpaired electrons, the multiplicity is 3, and the state is called a triplet state. 

or 1 units in length. Therefore, the resultant, R, for these combinations can be written as 1 + l21, 

In L-S coupling, we need to determine the following sums in order to deduce the spectroscopic state of an atom  

Note that if all of the electrons are paired, the sum of spins is 0, so a singlet state results. Also, if all of the orbitals in a set are filled, for each electron with a positive value of m there is also one having 

In the simple example just given, only one spectroscopic state is possible. In many cases, more than one spectroscopic state can result from a given electron configuration because the electrons can be arranged in different ways. For example, the electron configuration np2 could be arranged as 

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There are numerous methods for solving the time dependent Scln-ddinger equation (A3.13.43). and some of them were reviewed by Kosloff [118] (see also [119. 120]). Wlienever projections of the evolving wave fiinction on the spectroscopic states are usellil for the detailed analysis of the quanPun dynamics (and this is certainly the case for tlie detailed analysis of IVR), it is convenient to express the Hamiltonian based on spectroscopic states I... 

Here, = q ([) ) are the wave fiinctions of the spectroscopic states and the coefficients are detennined from the initial conditions... 

At this stage we may distinguish between excitation involving different electronic states and excitation occurring within the same electronic (ground) state. Wlien the spectroscopic states are located in different electronic states, say the ground (g) and excited (e) states, one frequently assumes the Franck-Condon approximation to be applicable ... 

The advantages of INDO over CNDO involve situations where the spin state and other aspects of electron spin are particularly important. For example, in the diatomic molecule NH, the last two electrons go into a degenerate p-orbital centered solely on the Nitrogen. Two well-defined spectroscopic states, S" and D, result. Since the p-orbital is strictly one-center, CNDO results in these two states having exactly the same energy. The INDO method correctly makes the triplet state lower in energy in association with the exchange interaction included in INDO. 

The remarkable thing is that the HF model is so reliable for the calculation of very many molecular properties, as 1 will discuss in Chapters 16 and 17. But for many simple applications, a more advanced treatment of electron correlation is essential and in any case there are very many examples of spectroscopic states that caimot be represented as a single Slater determinant (and so cannot be treated using the standard HF model). In addition, the HF model can only treat the lowest-energy state of any given symmetry. 

The two following lines present the results obtained later by Rerat et al. (17) the method consists in adding one more term in the expression of i) given by Eq.l4. He keeps the dipolar factor from the summation on the spectroscopic states l n)) he retains only the first one of the symmetry of interest, thus there is no extrapolation procedure on the other hand, he adds the Slater determinants l m) which contribute to the perturbation of the ground state by the operators... 

On the other hand, NMR spectra appear in general as the average of the spectra of the two spin states [36, 153]. This observation determines an upper limit for the spin-state lifetime shorter than the nuclear spin relaxation time Tl = l/ktH < lO s. In general, therefore, either the superposition or the average of the particular spectroscopic properties of the two spin states is observed, subject to the relative magnitude of lifetime of the excited spectroscopic state and the rate of spin-state conversion. The rate /clh is thus estimated... 

For this arrangement, the sum of spins is 3/2 and the L value is 3. These values give rise to the / values of 3 + 3/21, 3 + 3/2 — 1, . .., 3 - 3/21, which are 9/2, 7/2, 5/2, and 3/2. Because the set of orbitals is less than half filled, the lowest / corresponds to the lowest energy, and the spectroscopic ground state for Cr3+ is 4F3/2- The spectroscopic states can be worked out for various electron configurations using the procedures described above. Table 2.6 shows a summary of the spectroscopic states that arise from various electron configurations. 

Although we have not given a complete coverage to the topic of spectroscopic states, the discussion here is adequate for the purposes described in this book. In Chapter 17 it will be necessary to describe what happens to the spectroscopic states of transition metal ions when these ions are surrounded by other groups when coordination compounds form. 

Table 2.6 Spectroscopic States Arising for Equivalent Electrons. ...

Electron Configuration Spectroscopic States Electron Configuration Spectroscopic States... 

The spectroscopic ground state for a certain first row transition metal is 6S5/2. (a) Which metal is it (b) What would be the ground state spectroscopic state of the +2 ion of the metal described in (a) (c) What would be the spectroscopic ground state for the +3 ion of the metal described in (a) ... 

As in the case of atomic orbitals and spectroscopic states (see Chapter 2), we use lowercase letters to denote orbitals or configurations and uppercase letters to indicate states. It should also be pointed out that the a1 and b1 orbitals are a bonding orbitals, but the b2 molecular orbital is a nonbonding 7r orbital. 

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... 

As we have seen, an understanding of spin-orbit coupling is necessary to determine the spectroscopic states that exist for various electron configurations, dn (see Section 2.6). Because they will be needed frequently in this chapter, the spectroscopic states that result from spin-orbit coupling in dn ions that have degenerate d orbitals are summarized in Table 18.1. 

Table 18.1 Spectroscopic States for Gaseous Ions Having dn Electron Configurations0. Ion Spectroscopic states...

Table 18.2 Splitting of Spectroscopic States in a Ligand Fielc a...

Gaseous Ion spectroscopic state Components in an octahedral field Total degeneracy... 

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