Russell-Saunders terms

It is convenient to classify an atomic state in terms of total orbital angular momentum L and total spin S (capital letters always are used for systems of electrons small letters are reserved for individual electrons). This Russell-Saunders LSMzMs scheme will now be described in detail. 

Let us take the lithium atom as an illustrative example. The atomic number (the number of protons or electrons in the neutral atom) of lithium is 3- Therefore the orbital electronic configuration of the ground state is (lj ) (2r)k The ground-state LSMlMs term is found as follows  

Admittedly the lithium atom is a very simple case. To find the term designations of the ground state and excited states for more complicated electronic structures, it helps to construct a chart of the possible Ml and Ms values. This more general procedure may be illustrated with the carbon atom. The carbon atom has six electrons. Thus the orbital configuration of the ground state must be (ls)X2.s)X2py. It remains for us to find the correct ground-state term. 

First a chart is drawn as shown in Fig. l-8a, placing the possible values of Ml in the left-hand column and the possible values of Ms in the top row. We need consider only the electrons in incompletely filled subshells. Filled shells or subshells may be ignored in constructing such a chart since they always give a contribution Ml = 0(L = 0) and Ms = 0(T = 0). (Convince yourself of this brfore proceeding.) For carbon the configuration (2 ) is important. Each of the two p electrons has 1=1 and can therefore have mi = - -1, 0, or —1. Thus the values possible for Ml range from -)-2 to —2. 

The next step is to write down all the allowable combinations (called microstates ) of mi and m values for the two p electrons and to place these microstates in their proper Ml, Ms boxes. The general form for these microstates is 

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In tier (1) of the diagram (for the electronic structure of iron(III)), only the total energy of the five metal valence electrons in the potential of the nucleus is considered. Electron-electron repulsion in tier (2) yields the free-ion terms (Russel-Saunders terms) that are usually labeled by term ° symbols (The numbers given in brackets at the energy states indicate the spin- and orbital-multiplicities of these states.)... 

Subsequently, LFMM was extended with an explicit spin-pairing energy (18). This is an empirical term with the same functional form as the AOM parameters (Eq. 7) and is significantly easier to implement than to express the ligand field stabilization energy as a function of the many-electron Russell-Saunders terms (60). Preliminary indications (61) for Fe(II) species suggest the approach is satisfactory and we expect to be able to use LFMM to study phenomena like spin crossover. 

It has been pointed out above that two electrons in the Is orbital must have their spins opposed, and hence give rise to the singlet state So, with no spin or orbital angular momentum, and hence with no magnetic moment. Similarly it is found that a completed subshell of electrons, such as six electrons occupying the three 2p orbitals, must have S — 0 and L = 0, corresponding to the Russell-Saunders term symbol lS0 such a completed subshell has spherical symmetry and zero magnetic moment. The application of the Pauli exclusion prin-... 

We then applied this formula to various types of single-electron wave functions, for example s,p,d, /, g, and to wave functions for various Russell-Saunders terms characterized by integral values of the quantum number L. 

Example 7.3-1 (a) Into which states does the Russell-Saunders term d2 3F split in an intermediate field of Oh symmetry (b) Small departures from cubic symmetry often occur as a result of crystal defects, substituent ligands, and various other static and dynamic perturbations. If some of the IRs of O do not occur in the group of lower symmetry, then additional splittings of degenerate levels belonging to such IRs must occur. Consider the effect of a trigonal distortion of D3 symmetry on the states derived in (a) above. 

Example 8.2-1 Examine the effect of spin-orbit coupling on the states that result from an intermediate field of O symmetry on the Russell-Saunders term 4F. Correlate these states with those produced by the effect of a weak crystal field of the same symmetry on the components produced by spin-orbit coupling on the 4F multiplet. 

Exercise 8.2-1 Verify the DPs necessary to determine the spin-orbit splitting of the intermediate-field states derived from the Russell-Saunders term 4F. 

We are now in position to derive the electronic states arising from a given electronic configuration. These states have many names spectroscopic terms (or states), term symbols, and Russell-Saunders terms, in honor of spectroscopists H. N. Russell and F. A. Saunders. Hence, the scheme we use to derive these states is called Russell-Saunders coupling. It is also simply referred to as L-S coupling. 

For the weak field case, we have the situation where the crystal field interaction is much weaker than the electronic repulsion. In this approximation, the Russell-Saunders terms 3F, 3P, 1G, lD, and 5 for the d2 configuration are good basis functions. When the crystal field is turned on, these terms split according to the results given in Table 8.4.2 ... 

Russell-Saunders terms (L, S) refer to energy states arising out of an electronic configuration when electron repulsions are included. The terms are denoted as L, the orbital angular momentum and S, the spin angular momentum. We will now summarize the various attributes in one-electron and many electron systems. ... 

It is well-known that the electron repulsion perturbation gives rise to LS terms or multiplets (also known as Russell-Saunders terms) which in turn are split into LSJ spin-orbital levels by spin-orbit interaction. These spin-orbital levels are further split into what are known as Stark levels by the crystalline field. The energies of the terms, the spin-orbital levels and the crystalline field levels can be calculated by one of two methods, (1) the Slater determinantal method [310-313], (2) the Racah tensor operator method [314-316]. 

Figure 7.1 Microstates and Russell-Saunders terms for a d transition metal ion...

Table 7.2 Russell-Saunders terms for d-block ions ...

Figure 7.6 An energy level diagram for the ion Pr showing the effect of spin-orbit coupling on the Russell-Saunders terms...

Atomic states characterized by S and L are often called free-ion terms (sometimes Russell-Saunders terms) because they describe individual atoms or ions, free of ligands. 

Nickel(II) is a 3d ion. From this configuration the Russell-Saunders terms G,... 

Atomic states characterized by S and L are often called free-ion terms (sometimes Russell-Saunders terms) because they describe individual atoms or ions, free of ligands. Their labels are often called term symbols. Term symbols are composed of a letter relating to the value of L and a left superscript for the spin multiplicity. For example, the term symbol corresponds to a state in which L = 2 and the spin multiplicity (25 + 1) is 3 marks a state in which L = 3 and 25 -I- 1 = 5. 

The value of performing intermultiplet spectroscopy has been demonstrated by optical results on ionic systems. Well defined atomic spectra from intra-4f transitions have been measured up to 6 eV in all the trivalent lanthanides (except, of course, promethium) [Dieke (1968), Morrison and Leavitt (1982) see fig. 1 based on Carnall et al. (1989)]. Each level is characterised by the quantum numbers L, S, J, F), where L and 5 are the combined orbital and spin angular momenta of the 4f electrons participating in the many-electron wavefunctions, and J is the vector sum of L and 5. The quantum number F represents the other labels needed to specify the level fully. It is usually the label of an irreducible representation of the crystal field and we shall omit it. The Coulomb potential is responsible for separating the 4f states into Russell-Saunders terms of specific L and S, while the spin-orbit interaction is diagonal in J and so splits these terms into either 25-1-1 or 2L -I-1 levels with 7 = L - 5 to L -f 5. Provided the spin-orbit interaction is weaker than the Coulomb interaction, as is the case in the lanthanides, the resulting levels consist of relatively pure L, 5, J), or in spectroscopic notation states. These 27-1-1 manifolds are then weakly... 

Fig. 1. Energy levels of trivaient lanthanides below 43000 cm (5.3 eV) arranged according to the number q of 4f electrons. Excited levels known frequently to luminesce are indicated by a black triangle. The excited levels corresponding to hypersensitive transitions from the ground state are marked with a square. For each lanthanide, J is given to the right (in the notation of atomic spectroscopy, ] is added to the Russell-Saunders terms as lower-right subscripts). When the quantum numbers S and L are reasonably well-defined, the terms are indicated to the left. It may be noted that the assignments and F< in thulium(lll) previously were inverted these two levels with 7 = 4 actually have above 60% of H and F character, respectively. Calculated 7-levels are shown as dotted lines. They are taken from Carnall et al. (1968) who also contributed decisively to the identification of numerous observed levels, mainly by using the Judd-Ofelt parametrization of band intensities.

The crystal-field splitting of the and Russell-Saunders terms can be unravelled from high-resolution emission spectra of both srid Di trarrsitions and excitation spectra of the (/ = 0-2) F (/ = 0 and 1) transi-... 

TABLE II Russell-Saunders Terms for Free lons ... 

FIGURE 6 Russell-Saunders terms of a c/ free ion with and without spin-orbital coupling. 

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