JBCS

INTRODUCTION

The topic of atomic spectroscopic terms is a cornerstone in the curriculum of undergraduate chemistry courses due to its relevance to properly understanding the electronic structure of the atom. A thorough discussion of this topic reinforces important concepts such as spin-orbit coupling interaction, the Zeeman effect, and the fundamental role that different sources of angular momentum play in the atomic energy levels or terms. It also provides an opportunity to discuss the limitations of overreliance on electronic configurations to understand atomic properties.

Unfortunately, an emphasis on the procedures necessary to obtain the spectroscopic terms can easily degrade into an uninteresting game of producing meaningless symbols. A remedy to this issue involves properly introducing the students to some of the theoretical underpinnings of atomic states, and the strong connection between atomic terms and spectra. In this work, we present a comparative analysis of the two coupling schemes (LS coupling and jj coupling) used to obtain the term symbols. This comparison facilitates the discussion of their meaning and their limitations in the classroom. In this analysis we use experimental data to build several plots of the energy levels, and we also use the Landé interval rule as a diagnostic tool to indicate the most appropriate coupling scheme for an element.

The assignment of atomic energy levels is usually made considering the LS spin-orbit coupling scheme, also called Russell-Saunders coupling.^1-4 This scheme is presented in many undergraduate physical chemistry textbooks^5-7 and in inorganic chemistry textbooks,^8-13 because of its importance to understand the electronic spectra of coordination compounds, and in more specialized books, usually in the context of the electronic spectra of atoms, simple molecules, and coordination compounds.^14-23 Different approaches to obtain the LS terms for each electronic configuration can be found in the literature. One of the more convenient methods is presented by Orchin and Jaffé,¹⁸ Douglas et al.,⁹ and Hyde.²⁴ This method is straightforward, as it does not require the removal of terms that are not allowed by the Pauli principle.

The Russell-Saunders spin-orbit coupling scheme provides the LS terms that are best suited for describing the energy levels of the lighter elements. This is because the spin-orbit coupling is small compared to the electron-electron interaction energy for these elements. However, as we move down in the periodic table, multiplet structures begin to deviate from the Landé interval rule. This rule states that the energy difference between two neighboring energy levels in the same multiplet is proportional to the largest J value, as we will discuss below in more details.^17,21 This deviation is more pronounced for heavier elements, and the energy difference between neighboring energy levels in a multiplet does not follows the Landé interval rule. For instance, when descending in the periodic table, a set of states for an electronic configuration formed by one triplet and two singlets may become like one singlet and two doublets, providing clear evidence that the LS coupling scheme is not appropriate to describe the energy levels' structure of elements in the last periods of the periodic table. For these heavier elements, other spin-orbit coupling schemes, such as the jj coupling, are a better option. Procedures for obtaining the jj terms can be found in a number of books and articles, and will not be discussed here.^{14,20,23,25-31}

To our best knowledge, there are few works discussing the use of jj coupling to describe the atomic energy levels that are worth to mention. Rubio and Perez²⁷ demonstrated the unfolding of energy levels using the jj scheme for s¹p¹, p², and p³ configurations. Gauerke and Campbell²⁵ presented a systematic analysis in considering the jj scheme to general configurations, but it is restricted to certain groups of the periodic table and to neutral states. Haigh²⁸ applied the jj coupling for neutral states and excited configurations and indicated that in many cases an intermediate coupling scheme between LS and jj is more appropriate. Schemes for obtaining the spectroscopic terms for both couplings using boxes and arrows diagrams³⁰ or microstate tables³¹ have also been devised. The proper identification of which spin-orbit coupling scheme should be considered has been shown to be critical to explain the trends observed in the periodic table, such as bond energies³² and electronic entropy.³³

THEORETICAL CONSIDERATIONS OF THE COUPLING SCHEMES AND THE LANDÉ INTERVAL RULE

The rigorous derivation of a quantum-mechanical operator expression representing the spin-orbit coupling effect requires a relativistic treatment, which is beyond the scope of this work. Nevertheless, there is an approximate approach to obtaining the operator that ultimately yields the correct expression for the case of hydrogen-like atoms. Herein, we will provide a brief introduction to the origin of spin-orbit coupling. For a more detailed treatment, we recommend readers to consult the references cited in this section.

To gain some physical intuition about the nature of the interaction, we can interpret the spin-orbit coupling effect classically as arising from the interaction of the spin magnetic moment with the magnetic field generated by the motion of the nucleus relative to the electron. This magnetic field can be expressed in terms of the electric field due to the nuclear Coulomb potential and the motion of the electron. The magnetic field (B) experimented by a charged particle moving with momentum (p) in the presence of an electric field (E) is given by:^34,35

In the hydrogen atom, the electron is under the influence of a Coulomb potential, and the electric field can be expressed by minus the gradient of this potential. Since it is a central potential:

Substituting Equation 2 in 1 and remembering the definition of angular momentum (l = r × p) we have:

Since the interaction of a magnetic dipole with a magnetic field is given by -µ × B, and the magnetic dipole associated with the electron spin angular momentum is µ = -(e/m)s, we have:

The spin-orbit interaction energy associated with Equation 4 is often referred as the Γ factor, which can be shown to satisfy the following expression:³⁶

where γ is a constant proportional to the square of the fine structure constant (≈ 1/137.036). The constants l, s and j correspond to angular momentum quantum numbers. Here, l represents the orbital angular momentum, assuming natural numbers 1, 2, 3, ..., s is the spin angular momentum and is equal to +½, and j is the total angular momentum, which assumes values such as l + s, 1 + l - 1, ., |l - s|.

In general, atoms possess rotational invariance that reflects its spherical symmetry and leads to angular momentum conservation. This symmetry gives rise to electronic configurations with degenerate energy levels. The energy levels of an atom with Z electrons are obtained from the Hamiltonian:³⁷

where:

represents the central field Hamiltonian. is the electron-nucleus potential in the central-field approximation, where each electron moves independently of the others. The term accommodates electron correlation and is defined as:

A good approximation for in the many-electrons case is given by:^5,34,37,38

where:

The generalization to many electron atoms is not as straightforward as presented above, where we consider only one-body magnetic spin-orbit interaction. A more rigorous approach would also include two-body magnetic interactions, the coupling of the spin-other-orbit interaction, and also dipolar interactions with the spin magnetic moments. Details can be found in literature.^39-45 For our proposes, Equation 9 will be sufficient.

However, it is important to evaluate the magnitude of the energetic contributions from and . As a first approximation, we consider only , where each electron moves independently, following an independent particle model (IPM), which is often the Hartree-Fock method. This approach maintains the system's complete symmetry and degeneracies. When are considered, the problem is less symmetric, and some degeneracies are broken. If , then is treated as a perturbation and the electron correlation dominates. Since commutes with the orbital angular momentum operator, , spin angular momentum operator, , spin angular momentum operator, and the total angular momentum operator, , the atom energy levels can be labelled as eigenvalues of , , , and . This is the famous Russell-Saunders or LS coupling, and it is the most applied approach, which is more appropriate to light atoms. If , then can be seen as a perturbation, where the unperturbed Hamiltonian is . Here, some degeneracies are lost for the states with nonzero orbital angular momentum. In this case, the unperturbed Hamiltonian commutes with , and , which makes it convenient to label the eigenstates of each electron by the quantum numbers (n, l, j, m_j) rather than (n, l, m_l, m_s). This is the so called jj coupling scheme, which is more appropriate when treating heavy atoms, where makes the spin-orbit interaction dominant. In both coupling schemes, J is associated with a (2J + 1)-fold degeneracy. When the two terms, and , are considered, only the total angular momentum J is a good observable.

Next, we will follow closely the enlightening treatment presented by White³⁶ to illustrate the different regimes of LS and jj coupling. To exemplify the procedure, we choose a dp configuration, such as the 3d 4p excited configuration of Ti²⁺ ion. Starting from energy of the configuration, the average energy for the spherically symmetrized atom, the addition of Coulomb repulsion (Γ₁), exchange effects (Γ₂), and spin-orbit interactions give rise to the different terms and levels. In this particular case of two electrons, there are four individual angular momentum quantum numbers, l₁, l₂, s₁ and s₂, corresponding to the orbital angular momentum and the spin angular momentum of electron 1 and electron 2, respectively. These four quantities can form four spin-orbit terms: l₁ with s₁ (Γ₃), l₂ with s₂ (Γ₄), l₁ with s₂ (Γ₅) and l₂ with l₁ (Γ₆). For each coupling there is an energetic relation:

in total analogy to Equation 5.

Depending on the considered atomic system, some of these interactions will be dominant, while others will be considered negligible. In the LS coupling, the energy terms Γ₁ and Γ₂ predominate over Γ₃ and Γ₄, while Γ₅ and Γ₆ are negligible. In the jj coupling, Γ₃ and Γ₄ predominate over Γ₁ and Γ₂, while Γ₅ and Γ₆ are again negligible.

Considering an ideal case of the LS scheme, we only must deal with Γ₁, Γ₂, Γ₃, and Γ₄. We know that in the LS coupling l₁ and l₂ add up to generate L, while s₁ and s₂ give rise to S. In a classical model, we can imagine the spin-orbit interaction energy as being a consequence of the precession of L and S around the resultant J. Additional interaction energy is due to the coupling of the vectors l₁ with s₁, and l₂ with s₂, or Γ₃ with Γ₄. Adding Γ₃ and Γ₄, we obtain the following interaction energy:

In Figure 1 we can see the cumulative addition of gamma interactions in the unfolding of energy levels. In the LS coupling, Γ₁ correspond to Coulomb correlations neglecting Pauli's principle and gives rise to the S, P, D, F, G,. terms. Γ₂ split terms with different multiplicities and separates, such as singlets from triplets.²² The trends that are observed in the spectra can be understood considering Hund's rules. In LS coupling, among the terms arising from the same electronic configuration with the same L, the one with highest spin multiplicity presents the lowest energy. For the terms with the same spin multiplicity, the one with larger L value is the more stable.

Figure 1. Schematic representation of the effect of inclusion of successive Γ factors for a dp configuration in the LS coupling. Note that the diagram is not to scale. Figure 1S (Supplementary Material) presents the Ti²⁺energy values for these terms

Equation 12 contains an essential rule for a multiplet structure. First, we note that Γ₃ and Γ₄ comprise the spin-orbit interaction energy. For a given L and S, the difference between the fine structure levels is proportional to the level with the larger J value. From Equation 12, we can infer the difference between levels in a term with adjacent J values, explicitly showing that the intervals are proportional to the larger J values:

Equation 13 formulates what is known as the Landé interval rule. For example, for the 4s 4p excited configuration of calcium, the triplet term energy levels⁴⁶ are equal to ³P₀^o (15157.901 cm^-1), ³P₁^o (15210.063 cm^-1), and ³P₂^o (15315.943 cm^-1), which gives a ratio between the energy levels equal to 2.03, very close to the theoretical value of 2 based on the Landé interval rule. The agreement between the spectral data and these theoretical previsions can be used to validate the choice of the Russel-Saunders coupling. The adequacy of the LS coupling can be related to: (i) the large gaps compared to the fine structure of the terms (e.g., singlet-triplet gaps compared to triplet fine structure) and (ii) the intervals between the levels in a term following the Landé interval rule.^36,47

Now, let's consider the idealized case of the jj coupling between two valence electrons. In this scheme Γ₃ and Γ₄ predominates over Γ₁ and Γ₂. The total angular momentum is now given by the sum of the total angular momentum of each electron, which is only the sum of its spin and orbit values:

Adding up Γ₁ and Γ₂ we get:

Equation 15 is in total analogy with Equation 12, although we cannot obtain a rule like the Landé interval rule because the constant A_jj is not the same for a given multiplet. Figure 2 shows a schematic representation of the interaction energies for two electrons in a dp configuration. We should note that the primary effect that separates the energy levels in this electronic configuration is the spin-orbit coupling associated with the factors Γ₃ and Γ₄. Further separation of the terms occurs when j₁ and j₂ couples to give the total angular momentum J. This interaction is associated with the sum of Γ₁ and Γ₂.

Figure 2. Schematic representation of the effects of different Γ factors for a dp configuration in the jj coupling. Term levels are not in order of energy

The treatment above of spin-orbit interaction for two electrons furnishes the grounds for the general problem of many-electrons atoms. Of course, as the number of electrons grows, more gamma terms emerge, and we need to realize which of them are important and which are negligible. The expressions presented above for these ideal coupling schemes most used in the literature (LS and jj) can be used to verify their validity for each set of experimental data. In systems where the electronic correlation is not negligible, the spin-orbit coupling can be treated as a perturbation, and the LS coupling is the most suitable. Otherwise, in cases where spin-orbit coupling predominates over the effect of electronic correlation, jj coupling is the most appropriate choice.^5,37 However, these schemes correspond to extreme cases, and many elements on the periodic table have electronic structures that do not fit into any of them. As a result, intermediate coupling schemes have been proposed, such as the jK coupling and LK coupling, but these are out of the scope of the present work.²²

COMPARISON BETWEEN LS AND jj COUPLING SCHEMES

Table 1 presents the LS and jj terms for equivalent electrons in the configurations s¹, s², and p¹ to p⁶, and for nonequivalent electrons in configurations s¹p¹, s¹p³, and p¹d¹. The notation used for the jj terms follows that used by Haigh.²⁸ In this case, the jj terms are given by the term symbol (j₁, j₂, j₃,...)_J, where the values inside the parenthesis are the total angular momentum quantum numbers of each electron, j, and the subscript at the right side of the parenthesis is the total angular momentum quantum number, J, of the atom.

In the case of LS terms, Hund's rules provides that the term with highest spin multiplicity is the ground term.^5,19,21,22 When there is more than one term with the highest multiplicity, the term with the highest L value will be the lowest in energy. In addition, in the multiplets with more than one possible J value, the one with lowest J will be the lowest energy level for electronic configurations less than half filled. For electronic configuration more than half filled (p⁴ configuration, for example), the highest J will be the lowest energy level. The parity symbol (^o) is used when the sum of all azimuthal quantum numbers is odd (a p³ electronic configuration, for example, in which all electrons have l = 1). In the case of configurations containing non-equivalent electrons, such as excited states, the Hund's rules do not apply.

In the case of jj terms, Hund's rules indicate that the term with the lowest j values (inside the parenthesis) will be the lowest energy term.²⁹ In the case of multiplets (more than one J value as a subscript in the right side of the parenthesis), the highest J will be the lowest energy level. Unlike LS coupling notation, there is no indication of the spin multiplicity of each term in the jj terms. Even though a spin multiplicity is no longer defined when the spin-orbit coupling is strong, we can use the same terminology to refer to the structure of a multiplet described in the jj coupling. The multiplicity of a multiplet structure can be obtained from the number of J values. For example, the term (3/2,3/2, 1/2)_5/2,3/2,1/2^o for the p³ electronic configuration behaves like a triplet considering that there are three J values 5/2, 3/2, and 1/2. The parity symbol (^o) is used similarly as in LS terms. Similarly, Hund's rules do not apply to configurations containing non-equivalent electrons, just like in the case of LS coupling.

Comparison between LS and jj coupling schemes shows that the number of terms with a given J value is exactly the same in both. For example, for a p⁴ electronic configuration, as indicated by the LS terms ³P_2,1,0, ¹D₂, and ¹S₀, there are two levels with J = 2, two with J = 0, and one with J = 1. In the jj coupling, as indicated by the terms (3/2,3/2,3/2,3/2)₀, (3/2,3/2,3/2,1/2)_2,1, and (3/2,3/2,1/2,1/2)_2,0, we have the same J values, but the multiplet structure resembles to two doublets and one singlet. Moreover, by the Hunds' rules, in the LS coupling the lowest energy state is a triplet and in the jj coupling it is a doublet like.

For the lighter elements, the LS coupling is more appropriate, but the heavier elements depart from this kind of coupling and get closer to the jj coupling, as shown in Figure 3 for the p² electronic configuration. Similar figures are given by other authors,^14,19 but these are qualitative and do not show a step-by-step progression for all elements in the group.

Figure 3. Connection between LS and jj terms for the p² electronic configuration of group 14 elements. Energy levels from NIST⁴⁶

As can be seen in Figure 3, the lowest energy state for carbon is the triplet ³P_2,1,0 whose terms are very close from each other. Going down in the periodic table, germanium starts to show a greater separation between the terms and for lead it is not possible consider that the lowest state is a triplet anymore. It looks clearly like a singlet, which is better described by the jj term (1/2,1/2)₀, which has the same J value as the lowest term of the triplet ³P_2,1,0. In addition, the LS terms ¹S₀ and ¹D₂ become closer to each other and can be assigned to the doublet like structure (3/2,3/2)_2,0. It is worth to note that the assignment of the terms for Pb in the NIST site⁴⁶ are given in jj notation, following the work by Wood and Andrew.⁴⁸

Elements of group 15 present a similar progression of the energy levels, as shown in Figure 4. As can be seen, the lowest energy term for all elements is a singlet with J = 3/2. However, the lowest LS energy term (J = 1/2) from the doublet state ²P_1/2,3/2^o gets close to the terms from the doublet ²D_3/2,5/2^o when going from N to Bi and these three energy levels can be considered as a triplet like (3/2,3/2,1/2)_5/2,3/2,1/2^o in the case of Bi. For this element the highest energy term (J = 3/2) from the doublet ²P_1/2,3/2^o gets apart from the others and is better described as a singlet like (3/2,3/2,3/2)_3/2^o following the jj coupling.

Figure 4. Connection between LS and jj terms for the p³ electronic configuration of the group 15 elements. Energy levels from NIST⁴⁶

The rearrangement of the terms from the LS coupling scheme to the jj coupling is even more interesting in the case of group 16. As can be seen in Figure 5, the term with J = 1 of the triplet ³P_2,1,0, which is between J = 0 and J = 2 for oxygen, sulfur and selenium, crosses the term with J = 0 in Te and gets closer to the term ¹D₂ for the Po. This rearrangement allows one to consider that the lowest energy term for Po is the doublet-like (3/2,3/2,1/2,1/2)_2,0, followed by another doublet-like (3/2,3/2,3/2,1/2)_2,1 at higher energy. It is worth to mention that the reordering of the lowest energy J values from 2-1-0 in the oxygen to 2-0-1 in the Po is a consequence of the change in the coupling scheme from LS to jj.

Figure 5. Connection between LS and jj terms for the p⁴ electronic configuration of the group 16 elements. The (;) which appears in some jj term symbols indicates that after this symbol it is shown the j value of the s electron. Energy levels from NIST⁴⁶

Another fact that can be appreciated in Figure 5 is the behavior of the lowest energy terms for the excited electronic configuration np³(n+1)s¹. These terms can be indicated as the singlets ⁵S₂^o and ³S₁^o (they have the symbols of quintet and triplet, respectively, but they have only one J value and because of this they can be called singlets). As we move from oxygen to Po, these terms keep close to each other and can be better considered like a doublet (3/2,1/2,1/2;1/2)_2,1. This behavior indicates that the jj coupling is more appropriate to describe the lowest terms of this excited electronic configuration containing nonequivalent electrons, even for the lighter elements.²²

The behavior of the terms for positive ions is like that observed for neutral atoms with the same electronic configuration. As can be seen in Figure 6, for the ions +1 of group 15 with a p² configuration, the lowest energy terms for N⁺ and P⁺ belong to the triplet ³P_2,1,0. For the heavier elements this LS term split to form a jj singlet like (1/2,1/2)₀ and a doublet like (3/2,1/2)_2,1 in the same fashion as it was observed in Figure 3 for the neutral elements of the group 14 (note the very different energy scale in Figures 3 and 6).

Figure 6. Connection between LS and jj terms for the p² electronic configuration of +1 ions of the elements of the group 15. Energy levels from NIST⁴⁶

One more example of the better description provided by the jj coupling in the case of the heavier elements is presented in Figure 7, which shows the energy of the terms for the excited electronic configuration ns²np¹(n+1)s¹ for the ions +1 of the elements of group 15. As can be seen in this figure, the lowest energy LS triplet ³P_2,1,0^o is very close to the singlet ¹P₁^o for the elements N and P. However, starting from As, these terms split into two jj doublet like, (1/2,1/2)_0,1^o and (3/2,1/2)_2,1^o. In addition, it is worth to note that for electronic configurations with non-equivalent electrons and excited states, Hund's rules do not apply, and the lowest energy term is the (1/2,1/2)₀^o, which has the lowest J value, and not the term (1/2,1/2)₁^o.

Figure 7. Connection between LS and jj terms for the ions +1 of the elements of the group 15 in the electronic configuration ns²np¹(n + 1)s¹. Energy levels from NIST⁴⁶

THE LANDÉ INTERVAL RULE AND THE SPIN-ORBIT COUPLING

As indicated above, if the LS spin-orbit coupling is followed, the Landé interval rule states that the energy difference between two energy levels in the same multiplet must be proportional to the largest J value, as shown in Figure 8.^17,21 For example, if the energy difference between the terms with J = 0 and J = 1 is equal λ, the difference between the terms with J = 1 and J = 2 must be equal to 2λ. This is another way to verify if the LS spin-orbit coupling is followed by an element. As can be seen from the percentages in the last column of Table 2, we can say that the Landé interval rule (and the LS coupling) is followed by the group 14 elements C, Si, and Ge, but for Sn and Pb the energy ratio (³P₂ - ³P₁)/(³P₁ - ³P₀) is far from the theoretical value equal to 2, indicating that the LS coupling is not a good choice for these elements. It is difficult to indicate a percentage error limit, above that the LS coupling scheme is not acceptable but a deviation higher than 30% is clearly a reasonable limit.

Figure 8. Separation between different energy levels (spectroscopic terms), for electronic configurations p², p⁴, and d², following the Landé interval rule and the LS coupling

In the case of group 16, as shown in Table 3, the energy ratio (³P₂ - ³P₁)/(³P₁ - ³P₀) is close to 2 for the elements O and S but departs significantly from this value for the elements Se, Te, and Po, indicating that the LS coupling is not a good model for these heavier elements. The negative values in the last column of Table 3 are a consequence of the inversion in the sequence of J values which can be observed in Figure 5.

Differently from main group elements, lanthanides and transition elements may present small, medium, or large deviations from the Landé interval rule, without a clear pattern as we move along a group or period in the periodic table. In this way, the discussion of the most suitable coupling schemes for these elements must be done with great caution and will not be addressed here.

IMPACT ON TEACHING

Teaching spectroscopic terms to undergraduate and graduate students is a hard work. It is a very abstract subject and the thinking based on the total energy of atoms requires to give up of the thinking based on electronic configurations and electrons in atomic orbitals. However, it was observed that presenting both, the LS and jj coupling schemes, improve the students understanding of the meaning of the spectroscopic terms. In addition, both, teachers and students, agree that a better and easier understanding is obtained by presenting a graphical comparison of the energy levels for the light and heavy elements in the same group in the periodic table. This makes clear the meaning and importance of the spectroscopic terms and each one of the LS and jj spin-orbit coupling schemes.

CONCLUSIONS

The proper understanding of the atomic energy levels is an important issue, which can be challenging to teach solely based on the LS coupling. Including the jj coupling scheme when teaching atomic energy levels can enhance students' understanding of spectroscopic terms and their significance, especially for the heavier elements of the periodic table. By illustrating the assignment of LS and jj terms graphically, it becomes clear that the jj coupling scheme is much more appropriate to describe the heavier atomic systems, while still recognizing that the LS coupling scheme provides the best description for lighter elements.

SUPPLEMENTARY MATERIAL

Complementary material for this work, energy values for the terms presented in Figure 1, is available at http://quimicanova.sbq.org.br/, as a PDF file, with free access.

ACKNOWLEDGMENTS

The authors thank the financial support from Ministério da Ciência, Tecnologia e Inovações (MCTI) CAPES (Grant 88887.690930/2022-00 (L. A. L. D.)), Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) (Grant E-26/201.542/2023 (L. A. L. D.)), and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) (Grant 305.737/2022-8 (R. B. F.)). This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - finance code 001.

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