Quim. Nova, Vol. 48, No. 1, e-20250006, 1-7, 2025 http://dx.doi.org/10.21577/0100-4042.

20250006

THE ROLE OF jj COUPLING ON THE ENERGY LEVELS OF HEAVY ATOMS

Education
Lucas A. L. Diasa, Thiago M. Cardozoa and Roberto B. Fariaa,*,
a
Instituto de Química, Universidade Federal do Rio de Janeiro, 21941-909 Rio de Janeiro – RJ, Brasil

Received: 10/07/2023; accepted: 01/31/2024; published online: 05/03/2024

The study of atomic spectroscopy has a profound relationship with quantum mechanics and its comprehension. Since Bohr’s success
with a theory for the atom, it has been established that spectrum lines originate from transitions between different states. Moreover,
the analysis of complex spectra reveals that there are groups of spectral lines that compose what is called a multiplet. To rationalize
the multiplet structure, we need models to characterize and label the quantum states. The characterization of the atomic states is a
common subject in classes of physical chemistry and inorganic chemistry, and there are basically two approaches to infer and label
the energy states: (i) the LS coupling scheme and (ii) the jj coupling scheme. Usually, only the LS coupling is presented, and this
ends up omitting its underlying assumptions. Here, we present a survey of both approaches, highlighting their premises and their
adequacy towards different groups of the periodic table, ions, and excited states configurations. We show that by using benchmark
data together with the Landé interval rule, the appropriate use of the jj coupling to understand the atomic spectra of heavier atoms
is easily conveyed.

Keywords: spin-orbit coupling; LS coupling; jj coupling; spectroscopic terms; atomic energy levels.

INTRODUCTION The Russell-Saunders spin-orbit coupling scheme provides

the LS terms that are best suited for describing the energy levels
The topic of atomic spectroscopic terms is a cornerstone in the of the lighter elements. This is because the spin-orbit coupling
curriculum of undergraduate chemistry courses due to its relevance is small compared to the electron-electron interaction energy for
to properly understanding the electronic structure of the atom. A these elements. However, as we move down in the periodic table,
thorough discussion of this topic reinforces important concepts multiplet structures begin to deviate from the Landé interval rule.
such as spin-orbit coupling interaction, the Zeeman effect, and the This rule states that the energy difference between two neighboring
fundamental role that different sources of angular momentum play energy levels in the same multiplet is proportional to the largest J
in the atomic energy levels or terms. It also provides an opportunity value, as we will discuss below in more details.17,21 This deviation
to discuss the limitations of overreliance on electronic configurations is more pronounced for heavier elements, and the energy difference
to understand atomic properties. between neighboring energy levels in a multiplet does not follows
Unfortunately, an emphasis on the procedures necessary to obtain the Landé interval rule. For instance, when descending in the
the spectroscopic terms can easily degrade into an uninteresting periodic table, a set of states for an electronic configuration formed
game of producing meaningless symbols. A remedy to this issue by one triplet and two singlets may become like one singlet and two
involves properly introducing the students to some of the theoretical doublets, providing clear evidence that the LS coupling scheme is
underpinnings of atomic states, and the strong connection between not appropriate to describe the energy levels’ structure of elements
atomic terms and spectra. In this work, we present a comparative in the last periods of the periodic table. For these heavier elements,
analysis of the two coupling schemes (LS coupling and jj coupling) other spin-orbit coupling schemes, such as the jj coupling, are a
used to obtain the term symbols. This comparison facilitates the better option. Procedures for obtaining the jj terms can be found
discussion of their meaning and their limitations in the classroom. in a number of books and articles, and will not be discussed
In this analysis we use experimental data to build several plots of the here.14,20-23,25-31
energy levels, and we also use the Landé interval rule as a diagnostic To our best knowledge, there are few works discussing the use
tool to indicate the most appropriate coupling scheme for an element. of jj coupling to describe the atomic energy levels that are worth
The assignment of atomic energy levels is usually made to mention. Rubio and Perez27 demonstrated the unfolding of
considering the LS spin-orbit coupling scheme, also called Russell- energy levels using the jj scheme for s1p1, p2, and p3 configurations.
Saunders coupling.1-4 This scheme is presented in many undergraduate Gauerke and Campbell25 presented a systematic analysis in considering
physical chemistry textbooks 5-7 and in inorganic chemistry the jj scheme to general configurations, but it is restricted to certain
textbooks,8-13 because of its importance to understand the electronic groups of the periodic table and to neutral states. Haigh28 applied the
spectra of coordination compounds, and in more specialized books, jj coupling for neutral states and excited configurations and indicated
usually in the context of the electronic spectra of atoms, simple that in many cases an intermediate coupling scheme between LS and
molecules, and coordination compounds.14-23 Different approaches to jj is more appropriate. Schemes for obtaining the spectroscopic terms
obtain the LS terms for each electronic configuration can be found for both couplings using boxes and arrows diagrams30 or microstate
in the literature. One of the more convenient methods is presented tables31 have also been devised. The proper identification of which
by Orchin and Jaffé,18 Douglas et al.,9 and Hyde.24 This method is spin-orbit coupling scheme should be considered has been shown to
straightforward, as it does not require the removal of terms that are be critical to explain the trends observed in the periodic table, such
not allowed by the Pauli principle. as bond energies32 and electronic entropy.33

*e-mail: faria@iq.ufrj.br
2 Dias et al. Quim. Nova

THEORETICAL CONSIDERATIONS OF THE COUPLING where:

SCHEMES AND THE LANDÉ INTERVAL RULE
(7)
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 represents the central field Hamiltonian. is the electron-
operator that ultimately yields the correct expression for the case of nucleus potential in the central-field approximation, where
hydrogen-like atoms. Herein, we will provide a brief introduction to each electron moves independently of the others. The term
the origin of spin-orbit coupling. For a more detailed treatment, we accommodates electron correlation and is defined as:
recommend readers to consult the references cited in this section.
To gain some physical intuition about the nature of the interaction, (8)
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 A good approximation for in the many-electrons case is given
to the nuclear Coulomb potential and the motion of the electron. The by:5,34,37,38
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 (9)

(1)
where:

In the hydrogen atom, the electron is under the influence of a (10)

Coulomb potential, and the electric field can be expressed by minus
the gradient of this potential. Since it is a central potential:
The generalization to many electron atoms is not as straightforward
(2) 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
Substituting Equation 2 in 1 and remembering the definition of interaction, and also dipolar interactions with the spin magnetic
angular momentum (l = r × p) we have: moments. Details can be found in literature.39-45 For our proposes,
Equation 9 will be sufficient.
(3) 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
Since the interaction of a magnetic dipole with a magnetic field is independent particle model (IPM), which is often the Hartree-Fock
given by –µ × B, and the magnetic dipole associated with the electron method. This approach maintains the system’s complete symmetry
spin angular momentum is µ = –(e/m)s, we have: and degeneracies. When are considered, the problem is less
symmetric, and some degeneracies are broken. If , then is
(4) treated as a perturbation and the electron correlation dominates. Since
commutes with the orbital angular momentum operator,

The spin-orbit interaction energy associated with Equation 4 , spin angular momentum operator, , and the total
is often referred as the Γ factor, which can be shown to satisfy the
following expression:36 angular momentum operator, , the atom energy levels can be
labelled as eigenvalues of , , , and . This is the famous
(5) 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
where γ is a constant proportional to the square of the fine structure . Here, some degeneracies are lost for the states with nonzero
constant (≈ 1/137.036). The constants l, s and j correspond to angular orbital angular momentum. In this case, the unperturbed Hamiltonian
momentum quantum numbers. Here, l represents the orbital angular commutes with , and , which makes it convenient to
momentum, assuming natural numbers 1, 2, 3, ..., s is the spin angular label the eigenstates of each electron by the quantum numbers (n, l, j,
momentum and is equal to +½, and j is the total angular momentum, mj) rather than (n, l, ml, ms). This is the so called jj coupling scheme,
which assumes values such as l + s, 1 + l – 1, …, |l – s|. which is more appropriate when treating heavy atoms, where
In general, atoms possess rotational invariance that reflects its makes the spin-orbit interaction dominant. In both coupling schemes,
spherical symmetry and leads to angular momentum conservation. J is associated with a (2J + 1)-fold degeneracy. When the two terms,
This symmetry gives rise to electronic configurations with degenerate and , are considered, only the total angular momentum J is a
energy levels. The energy levels of an atom with Z electrons are good observable.
obtained from the Hamiltonian:37 Next, we will follow closely the enlightening treatment presented
by White36 to illustrate the different regimes of LS and jj coupling.
(6) To exemplify the procedure, we choose a dp configuration, such as
Vol. 48, No. 1 The role of jj coupling on the energy levels of heavy atoms 3

the 3d 4p excited configuration of Ti2+ ion. Starting from energy of electronic configuration with the same L, the one with highest spin
the configuration, the average energy for the spherically symmetrized multiplicity presents the lowest energy. For the terms with the same
atom, the addition of Coulomb repulsion (Γ1), exchange effects (Γ2), spin multiplicity, the one with larger L value is the more stable.
and spin-orbit interactions give rise to the different terms and levels. Equation 12 contains an essential rule for a multiplet structure.
In this particular case of two electrons, there are four individual First, we note that Γ3 and Γ4 comprise the spin-orbit interaction energy.
angular momentum quantum numbers, l1, l2, s1 and s2, corresponding For a given L and S, the difference between the fine structure levels
to the orbital angular momentum and the spin angular momentum is proportional to the level with the larger J value. From Equation 12,
of electron 1 and electron 2, respectively. These four quantities can we can infer the difference between levels in a term with adjacent J
form four spin-orbit terms: l1 with s1 (Γ3), l2 with s2 (Γ4), l1 with s2 (Γ5) values, explicitly showing that the intervals are proportional to the
and l2 with l1 (Γ6). For each coupling there is an energetic relation: larger J values:

Γ3 = γ3(l1 × s1) Γ5 = γ5(l1 × s2) Γ4 = γ4(l2 × s2) Γ6 = γ6(l2 × s1) (11) (13)

in total analogy to Equation 5.

Depending on the considered atomic system, some of these Equation 13 formulates what is known as the Landé interval
interactions will be dominant, while others will be considered rule. For example, for the 4s 4p excited configuration of calcium,
negligible. In the LS coupling, the energy terms Γ1 and Γ2 predominate the triplet term energy levels46 are equal to 3P0o (15157.901 cm-1),
over Γ3 and Γ4, while Γ5 and Γ6 are negligible. In the jj coupling, Γ3 and P1 (15210.063 cm-1), and 3P2o (15315.943 cm-1), which gives a ratio
3 o

Γ4 predominate over Γ1 and Γ2, while Γ5 and Γ6 are again negligible. between the energy levels equal to 2.03, very close to the theoretical
Considering an ideal case of the LS scheme, we only must deal value of 2 based on the Landé interval rule. The agreement between
with Γ1, Γ2, Γ3, and Γ4. We know that in the LS coupling l1 and l2 the spectral data and these theoretical previsions can be used to
add up to generate L, while s1 and s2 give rise to S. In a classical validate the choice of the Russel-Saunders coupling. The adequacy
model, we can imagine the spin-orbit interaction energy as being of the LS coupling can be related to: (i) the large gaps compared to
a consequence of the precession of L and S around the resultant J. the fine structure of the terms (e.g., singlet-triplet gaps compared to
Additional interaction energy is due to the coupling of the vectors l1 triplet fine structure) and (ii) the intervals between the levels in a term
with s1, and l2 with s2, or Γ3 with Γ4. Adding Γ3 and Γ4, we obtain the following the Landé interval rule.36,47
following interaction energy: Now, let’s consider the idealized case of the jj coupling between
two valence electrons. In this scheme Γ3 and Γ4 predominates over
(12) 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
In Figure 1 we can see the cumulative addition of gamma of its spin and orbit values:
interactions in the unfolding of energy levels. In the LS coupling, Γ1
correspond to Coulomb correlations neglecting Pauli’s principle and J = j1 + j2 (14)
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 Adding up Γ1 and Γ2 we get:
trends that are observed in the spectra can be understood considering
Hund’s rules. In LS coupling, among the terms arising from the same (15)

Equation 15 is in total analogy with Equation 12, although we

cannot obtain a rule like the Landé interval rule because the constant
Ajj 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.
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
Figure 1. Schematic representation of the effect of inclusion of successive the most appropriate choice.5,37 However, these schemes correspond
Γ factors for a dp configuration in the LS coupling. Note that the diagram to extreme cases, and many elements on the periodic table have
is not to scale. Figure 1S (Supplementary Material) presents the Ti2+energy electronic structures that do not fit into any of them. As a result,
values for these terms intermediate coupling schemes have been proposed, such as the
4 Dias et al. Quim. Nova

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
Figure 2. Schematic representation of the effects of different Γ factors for multiplicity is no longer defined when the spin-orbit coupling is
a dp configuration in the jj coupling. Term levels are not in order of energy 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
jK coupling and LK coupling, but these are out of the scope of the structure can be obtained from the number of J values. For example,
present work.22 the term (3/2,3/2, 1/2)5/2,3/2,1/2o for the p3 electronic configuration
behaves like a triplet considering that there are three J values 5/2,
COMPARISON BETWEEN LS AND jj COUPLING 3/2, and 1/2. The parity symbol (o) is used similarly as in LS terms.
SCHEMES Similarly, Hund’s rules do not apply to configurations containing
non-equivalent electrons, just like in the case of LS coupling.
Table 1 presents the LS and jj terms for equivalent electrons in the Comparison between LS and jj coupling schemes shows that the
configurations s1, s2, and p1 to p6, and for nonequivalent electrons in number of terms with a given J value is exactly the same in both.
configurations s1p1, s1p3, and p1d1. The notation used for the jj terms For example, for a p4 electronic configuration, as indicated by the
follows that used by Haigh.28 In this case, the jj terms are given by LS terms 3P2,1,0, 1D2, and 1S0, there are two levels with J = 2, two
the term symbol (j1, j2, j3,...)J, where the values inside the parenthesis 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,
Table 1. LS and jj terms for a few electronic configurations involving s and we have the same J values, but the multiplet structure resembles to
p orbitalsa 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
LS terms jj terms it is a doublet like.
s 1 2
S1/2 (1/2)1/2 For the lighter elements, the LS coupling is more appropriate,
s 2 1
S0 (1/2,1/2)0 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
p1 2
P3/2,1/2o (3/2)3/2o (1/2)1/2o
configuration. Similar figures are given by other authors,14,19 but
p2 3
P2,1,0 1D2 1S0 (3/2,3/2)2,0 (3/2,1/2)2,1 (1/2,1/2)0 these are qualitative and do not show a step-by-step progression for
(3/2,3/2,3/2)3/2o all elements in the group.
p3 4
S3/2o 2D5/2,3/2o 2P3/2,1/2o (3/2,3/2,1/2)5/2,3/2,1/2o As can be seen in Figure 3, the lowest energy state for carbon is the
(3/2, 1/2, 1/2)3/2o triplet 3P2,1,0 whose terms are very close from each other. Going down
(3/2,3/2,3/2,3/2)0 in the periodic table, germanium starts to show a greater separation
p4 3
P2,1,0 1D2 1S0 (3/2,3/2,3/2,1/2)2,1 between the terms and for lead it is not possible consider that the
(3/2,3/2,1/2,1/2)2,0 lowest state is a triplet anymore. It looks clearly like a singlet, which
(3/2,3/2,3/2,3/2,1/2)1/2o is better described by the jj term (1/2,1/2)0, which has the same J value
p5 2
P3/2,1/2o
(3/2,3/2,3/2,1/2,1/2)3/2o as the lowest term of the triplet 3P2,1,0. In addition, the LS terms 1S0 and
p6 1
S0 (3/2,3/2,3/2,3/2,1/2,1/2)0
1
D2 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
s1p1 1
P1o 3P2,1,0o (3/2,1/2)2,1o (1/2,1/2)1,0o
the terms for Pb in the NIST site46 are given in jj notation, following
(3/2,3/2,3/2;1/2)2,1o the work by Wood and Andrew.48
s1p3 5
S2o 3D3,2,1o 3P2,1,0o 3S1o 1D2o 1P1o (3/2,3/2,1/2;1/2)3,2(2),1(2),0o Elements of group 15 present a similar progression of the energy
(3/2,1/2,1/2;1/2)2,1o levels, as shown in Figure 4. As can be seen, the lowest energy term for
p1d1 3
F4,3,2o 3D3,2,1o 3P2,1,0o 1F3o 1D2o 1P1o
(5/2;3/2)4,3,2,1o (5/2;1/2)3,2o all elements is a singlet with J = 3/2. However, the lowest LS energy
(3/2;3/2)3,2,1,0o (3/2,1/2)2,1o term (J = 1/2) from the doublet state 2P1/2,3/2o gets close to the terms
a
For configurations containing only equivalent electrons, the lowest energy from the doublet 2D3/2,5/2o when going from N to Bi and these three
terms are indicated in bold. energy levels can be considered as a triplet like (3/2,3/2,1/2)5/2,3/2,1/2o
Vol. 48, No. 1 The role of jj coupling on the energy levels of heavy atoms 5

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

Figure 3. Connection between LS and jj terms for the p2 electronic configura-

tion of group 14 elements. Energy levels from NIST46

in the case of Bi. For this element the highest energy term (J = 3/2) Figure 5. Connection between LS and jj terms for the p4 electronic configura-
from the doublet 2P1/2,3/2o gets apart from the others and is better tion of the group 16 elements. The (;) which appears in some jj term symbols
described as a singlet like (3/2,3/2,3/2)3/2o following the jj coupling. indicates that after this symbol it is shown the j value of the s electron. Energy
levels from NIST46

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 3P2,1,0.
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 4. Connection between LS and jj terms for the p3 electronic configura-

tion 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 3P2,1,0,
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 Figure 6. Connection between LS and jj terms for the p2 electronic configura-
mention that the reordering of the lowest energy J values from 2-1-0 tion of +1 ions of the elements of the group 15. Energy levels from NIST46
in the oxygen to 2-0-1 in the Po is a consequence of the change in
the coupling scheme from LS to jj. One more example of the better description provided by the
Another fact that can be appreciated in Figure 5 is the behavior jj coupling in the case of the heavier elements is presented in
of the lowest energy terms for the excited electronic configuration Figure 7, which shows the energy of the terms for the excited
np3(n+1)s1. These terms can be indicated as the singlets 5S2o and 3S1o electronic configuration ns 2np 1(n+1)s 1 for the ions +1 of the
(they have the symbols of quintet and triplet, respectively, but they elements of group 15. As can be seen in this figure, the lowest energy
have only one J value and because of this they can be called singlets). LS triplet 3P2,1,0o is very close to the singlet 1P1o for the elements N and
As we move from oxygen to Po, these terms keep close to each other P. However, starting from As, these terms split into two jj doublet like,
and can be better considered like a doublet (3/2,1/2,1/2;1/2)2,1. This (1/2,1/2)0,1o and (3/2,1/2)2,1o. In addition, it is worth to note that for
6 Dias et al. Quim. Nova

electronic configurations with non-equivalent electrons and excited Table 2. Landé interval rule test for the elements of group 14 (p2). Theoretical
states, Hund’s rules do not apply, and the lowest energy term is the energy ratio (3P2 – 3P1)/(3P1 – 3P0) = 2. Percentage deviation from the energy
(1/2,1/2)0o, which has the lowest J value, and not the term (1/2,1/2)1o. ratio equal to 2 is given between parenthesisa
3
P0 3
P1 = λ 3
P2 (3P2 – 3P1)/(3P1 – 3P0)
C 0 16.416713 43.4134567 1.6 (18%)
Si 0 77.115 223.157 1.9 (5.3%)
Ge 0 557.1341 1409.9609 1.5 (23%)
Sn 0 1691.806 3427.673 1.0 (49%)
Pb 0 7819.2626 10650.3271 0.4 (82%)
a3
P0 is taken as the reference state. Experimental values from NIST, in cm-1.46

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.

Table 3. Landé interval rule test for the elements of group 16 (p4). Theoretical
energy ratio (3P2 – 3P1)/(3P1 – 3P0) = 2. Percentage deviation from a ratio equal
to 2 is given between parenthesisa
Figure 7. Connection between LS and jj terms for the ions +1 of the elements 3
P2 3
P1 = 2λ 3
P0 (3P2 – 3P1)/(3P1 – 3P0)
of the group 15 in the electronic configuration ns2np1(n + 1)s1. Energy levels
O 0 158.265 226.977 2.3 (15%)
from NIST46
S 0 396.055 573.64 2.2 (12%)
THE LANDÉ INTERVAL RULE AND THE SPIN-ORBIT Se 0 1989.497 2534.36 3.7 (83%)
COUPLING Te 0 4750.712 4706.495 –107.4 (5472%)
Po 0 16831.61 7514.69 –1.8 (190%)
As indicated above, if the LS spin-orbit coupling is followed, a3
P2 is taken as the reference state. Experimental values from NIST, in cm-1.46
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 Differently from main group elements, lanthanides and transition
between the terms with J = 0 and J = 1 is equal λ, the difference elements may present small, medium, or large deviations from the
between the terms with J = 1 and J = 2 must be equal to 2λ. This is Landé interval rule, without a clear pattern as we move along a group
another way to verify if the LS spin-orbit coupling is followed by an or period in the periodic table. In this way, the discussion of the most
element. As can be seen from the percentages in the last column of suitable coupling schemes for these elements must be done with great
Table 2, we can say that the Landé interval rule (and the LS coupling) caution and will not be addressed here.
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 IMPACT ON TEACHING
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, Teaching spectroscopic terms to undergraduate and graduate
above that the LS coupling scheme is not acceptable but a deviation students is a hard work. It is a very abstract subject and the thinking
higher than 30% is clearly a reasonable limit. based on the total energy of atoms requires to give up of the thinking
In the case of group 16, as shown in Table 3, the energy ratio based on electronic configurations and electrons in atomic orbitals.
(3P2 – 3P1)/(3P1 – 3P0) is close to 2 for the elements O and S but departs 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
Figure 8. Separation between different energy levels (spectroscopic terms), terms and their significance, especially for the heavier elements of
for electronic configurations p2, p4, and d2, following the Landé interval rule the periodic table. By illustrating the assignment of LS and jj terms
and the LS coupling graphically, it becomes clear that the jj coupling scheme is much
Vol. 48, No. 1 The role of jj coupling on the energy levels of heavy atoms 7

more appropriate to describe the heavier atomic systems, while still 17. Figgs, B. N.; Introduction to Ligand Fields, reprinted ed.; Interscience
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