Chapter 6

Coupling of angular momenta – the vector

model

In the previous chapter, we introduced the central-field approximation as a way to

obtain separable basis functions for the states of multielectron atoms. These product
states are solution to a Schrödinger equation with a purely radial potential, and they
are annotated with an electron orbital nomenclature (see chapters 1.5 and 5.2.1).
What we need to do now is to apply perturbation theory, using these electron con-
figurations as zeroth-order functions.
When we initially introduced the CFA in chapter 5, and established electron con-
figurations, we ignored two contributions to the total Hamiltonian; namely the spin-
orbit interaction and what’s left of the mutual electron repulsion term, when we have
separated out its radial part. The issue now at hand is to add these contributions as
perturbations. Eventually, we will then be able to further add other interactions, for
example with external fields and electromagnetic moments from the nucleus.
What we have to deal with presently is interactions between the orbital and spin
angular momenta of all electrons. A model for doing this was developed by early
spectroscopists, before quantum mechanics and before the salient features of atomic
structure were known. It is a phenomenological model based on the addition of
vectors, describing the different angular momenta, with constraints concerning the
discrete nature of the vector sums. The latter rules was initially entirely based on
empirical observations.
When studying atomic spectra, it was noted that groups of of energy levels (de-
rived from spectral lines) appeared together, and that some specific types of atoms
always had such groupings with particular multiplicities. For example, alkali atoms
had states appearing in doublets, alkaline earths were shown to have singlets and
triplets, and nitrogen had its energy levels arranged in quartets and doublets. Work-
ing backwards, it was hypothesised that this has to do with interacting angular mo-
menta, and that spin had to be included in the treaties. The model was developed
before quantum mechanics, but it turns out that when the addition of angular mo-
menta was subsequently put on a quantum mechanical footing, this vector model
still gave excellent qualitative results, and it still greatly facilitates a close study of
basic atomic structure and atomic spectroscopy.

73
74 6 Coupling of angular momenta – the vector model

In the following, we will introduce the vector model, and we will quickly proceed
to the application of quantum mechanical perturbation analyses of the same issue.
When doing this, we will have to isolate special cases, where some interactions
dominate over others. The extremes of these special cases are the so called LS-
coupling and j j-coupling schemes. They will be introduced in chapters 6.3 and
6.4, and then treated in detail in chapters 7 and 8. Albeit the above schemes are
limiting cases, they will help us to understand a majority of atomic structure, and
via interpolation schemes, we can get a good grasp also of various intermediated
situations.

6.1 The concept of the vectormodel

The central-field approximation is a key to the justification of the vector model. In

the pre-quantum era, and after the introduction of the atomic models by Rutherford
and Bohr, the formulation was rather along the lines that only a few of the nega-
tive charges in an atom contribute to the finer features of atomic structures. With
an atomic model based on discrete orbitals, this could also be stated as that only
electron in the outermost occupied orbitals give angular contributions to the struc-
ture, whereas the inner ones give a spherically symmetric contribution to the overall
energy of the atom. This hypothesis was for early spectroscopists a postulate based
on observations. Today, we can give rational arguments for it, and we will dwell on
it more in chapter 6.2.
A purely central potential gives a comparatively simple solution to the Schrö-
dinger equation. The observation that atomic spectra can still be very complicated
must then lead to the conclusion that these complications have to to with atoms
outside the symmetric closed shells, i.e. the valence electrons. Inner, fully occupied
orbitals, may combine to a spherical force field, but the remaining valence electrons
will exert torques on each other. From there, we can understand that the number
of valence electrons must have an important bearing on the energy level structure,
and thus it is understandable why atoms found in the same column in the periodic
system have similar spectra.
The said angular effects must, from a Newtonian point of view, be possible to
describe as interactions between different angular momenta that have the effect that
they will all vary with time, whereas their sum will remain constant (for the moment
we rule out dissipation). We have
dL
=τ , (6.1)
dt
and if we take the example of the angular part of the electrostatic interaction between
two valence electrons, it is clear that neither L1 , nor L2 will be constant in time. They
will both precess about their vector sum, as illustrated in fig 6.1
In quantum mechanical language, L21 and L21 will commute with L2 , Lz and the
Hamiltonian describing the interaction, but L1z and L2z will not. Likewise, for an
6.1 The concept of the vectormodel 75

Fig. 6.1 The orbital angular

momenta of two electrons, L1
L1 + L2
and L2 , are coupled trough
electrostatic interaction. As a
result, they will both precess,
whereas their sum is constant
in time.

L1

L2

electron spin interacting with the same electron’s orbital angular momentum, Li
and Si will process around their sum Ji , as described in chapter 4.
It is clear that sums of angular momenta play an important role, and thus we
have to know with what restrictions such additions can be made. To start with: from
classical physics it is clear that a sum of angular momenta is in itself an angular
momentum. Moreover, we know that a general quantum mechanical angular mo-
mentum J must be quantized such that the eigenvalues of J2 are j( j + 1), with j
being either zero, a positive integer, or a positive half-integer, and the eigenvalues
of Jz are m j , whose possible values lie between − j and j and are separated by inte-
ger numbers (see appendix C).
With this, we can illustrate the addition rules with a few practical examples. A
more rigorous justification for these rules is given in appendix I. We begin by con-
sidering a number of valence electrons with spins Si that interact to form a total spin
S = ∑i Si . We know that for every electron, si = 21 , and we assume that the orbital
angular momenta (and/or the principal quantum numbers) are such that we do not
have to bother with the Pauli principle. We write the quantum numbers associated
with S2 and Sz as S and MS respectively (see chapter 1.5), and the possible values of
S are illustrated in figure 6.2.
For a case with two valence electrons, the possible results for the total spin are
S = 0 or S = 1 — the two spins are either aligned or anti-parallel. These two options
correspond to the singlets and triplets described in chapter 2.2. With a third electron,
the third spin-vector must be added to the vector sum of the two first, either increas-
ing or decreasing the total spin, and the possible values are S = 1/2 and S = 3/2.
Those values gives spectral lines appearing in doublets and quartets, and we see that
there is indeed a logical link between the total spin S, and the quantity 2S + 1, which
was assumed to be the multiplicity (see chapter 1.5). It should be noted that some
76 6 Coupling of angular momenta – the vector model

Fig. 6.2 Illustration of the

vector addition of electron S
spins. On the x-axis is the
number of electrons, and
on the y-axis the total spin
(the vector sum). With two
5/2
electrons, only singlets (S =
0) and triplets (S = 1) are 2
possible. Three electrons
produces doublets (S = 1/2)
and (S = 3/2) quartets, and so 3/2
on. Note that is this diagram,
it is assumed that the Pauli
principle is not an impediment 1
(some of the other quantum
numbers are different).
1/2

1 2 3 4 5 Ne

of the addition paths in figure 6.2 may in some instances be inhibited by the Pauli
principle.
Adding orbital angular momenta works in the same way, except that the different
li can be any positive integer, but are never half-integers. Take an example with one
d-electron (l1 = 2) and one p-electron (l2 = 1). The possible values of the quantum
number for the total angular momentum, L, are 3, 2 and 1. If there is a third electron,
its angular momentum is added in the same way to the different sums of the first two.
At this stage, the order of the summation does not matter.
Left to do is to take into account the spin-orbit interaction. Here we are faced
with a choice. We can either first couple all the individual Li and Si to a number of
Ji = Li + Si , whereafter all the individual Ji are summed into a grand total angular
momentum, J = ∑i Ji . Alternatively, we first form L = ∑i Li and S = ∑i Si , and after
that we get to the same J = L + S, having taken a different route. This is illustrated
in figure 6.3
What path we take in order to get to J is actually crucial. Even when we exclude
angular effects from inner, closed shells, we will have two angular momenta per
valence electrons, and they will all interact each other. We have already resigned to
calculate energy contributions with perturbation theory, using CFA electron config-
urations as zero-order states. However, if we take the Hamiltonian from (5.2) and
(5.4), and add the spin-orbit contribution from (4.16):

H = HCF + Hto + HSO , (6.2)

we still need to treat the last two terms one by one. For a perturbative calculation,
that will only work if the largest term is treated first, and if each subsequent term
in the stepwise calculation has an energy contribution significantly smaller than the
6.1 The concept of the vectormodel 77

~s2 ~s2
~j2
J~
~
S J~

~s1 ~l2

~ ~j1
L
~l2 ~s1

~l1 ~l1

Fig. 6.3 Two different ways to couple the four angular momenta of a two electron atom (L1 , S1 ,L2
and S2 ) in order to form the total electronic angular momentum 2. In the left panel, the strongest
interaction is that between the electrons. The individual Li and Si first couple to form L and S, and
the the spin orbit interaction gives us J. This is the LS-coupling approximation. In the right panel,
the spin-orbit interaction is the most pronounces. In this — the j j-coupling case — the individual
Ji are first formed, and then these couple due to the electron-electron interaction and will give us J

preceding one. That is to say that both pictogram in figure 6.3 represents approxi-
mations and represent two limiting cases.
What this means is that we need to know the relative importance of Hto and
HSO , and the answer to that question is not universal. For light atoms, and for a
considerable part of the periodic system, the interaction between electrons is much
more important than the spin-orbit interaction. In that case, we first treat Hto , and
the result are states referred to as atomic terms. For these, the first stage on the path
to J has been to form L and S from the angular momenta of the individual electrons.
The situation is referred to as ‘LS–coupling’, and it will be treated in chapters 6.3
and 7. In the final step (ignoring external fields and nuclear effects), the term HSO
is applied as a perturbation to the atomic terms. This reveals the ‘fine structure’,
which will be quantified in the quantum number J. The representation of the states
in LS-coupling is:
| γ L S J MJ i . (6.3)
The fine-structure Hamiltonian HSO scales as Z 3 — see (4.12) — whereas Hto
has a linear scaling with Z. As a consequence the approximation that the spin-orbit
term is small compared with the Coulomb torque one may not hold for very heavy
atoms.
In the other extreme limit, we will have the case where we can safely inverse the
order in which the two angular Hamiltonians are applied. In these cases, the internal
spin–orbit interaction swamps the coupling between the valence electrons. This situ-
ations is what is illustrated in the right part of figure 6.3. We call this ‘ j j–coupling’,
78 6 Coupling of angular momenta – the vector model

since we begin by forming the individual Ji , and then we add all these to form J.
The quantum numbers L and S no longer commutes with the total Hamiltonian, and
the representation must rather be:

| γ j1 . . . jN J MJ i . (6.4)

We treat j j–coupling in chapters 6.4 and 8.

Obviously both LS–coupling and j j-coupling are approximations. However, with
these descriptions as a basis, we are able to describe also many intermediate cases.
We will look closer at such cases in chapter 9.

6.2 Closed shells

We will take a closer look at closed shells, and the contribution to atomic structure of
electrons in these closed shells. As has been established in chapter 5, a closed shell
is an electron orbital, designated by the quantum numbers nl, which has exactly the
maximum number of electrons in it that is possible.
The fact that orbitals, or shells, can be filled at all is a consequence of the Pauli
principle. We designate each electron with a certain set of quantum numbers n, l, ml ,
and ms . In a stringent point of view, this is incorrect. In the real multielectron atoms,
all electrons are entangled, and when we talk about electron configurations, the re-
ality is that each set of quantum numbers represent occurrences. Assigning specific
quantum-number labels to each electron is strictly speaking unphysical. Neverthe-
less, thinking of electron configuration is this way will work fine for figuring out the
salient features of the atomic structure.
The Pauli principle tells us that no electrons may be identical. That is, they may
not have the same set of quantum numbers — that would make the wave function
exchange symmetric. This means that for a given set of n, l, and ml , there can be only
two electrons — with opposite spins. For each ‘shell’, nl, we can allow 2(2l + 1)
electrons. Thus, and s-shell is closed, or filled, when it has two electrons. A p-shell
takes six electrons, a d-shell ten, and an f-shell fourteen.
This way of filling up means that in a closed shell, for every electron that has a
positive value of ml there must be another that is identical except for a different sign
of ml . Likewise, for every spin-up electron, there must be a spin-down one. This
means that the sums of all ml and all ms for all electrons in all closed shells most be
zero. The projection quantum numbers are the only ones that can contribute to an
orientation, and thus closed shells must be spherically symmetric.
For the spin-orbit Hamiltonian (4.16), where all vectors Li and Si are included in
the summation, we can therefore ignore all electrons in closed shells — their contri-
bution to the spin-orbit couplings will be zero. When formulating the Hamiltonian
(4.16), it will suffice to include valence electrons.
It is intuitively reasonable to assume that the angular contribution of the electron-
electron interaction from closed shells will also be zero, since these filled orbitals
6.3 LS-coupling 83

6.3 LS-coupling

In the LS-coupling approximation, we assume that the angular part of the interac-
tion between electrons is so much stronger than the spin-orbit interaction, intrinsic
for each electron, that the latter can be ignored in a first step of the approximation.
Once we have specified the electronic configuration, the next step is thus to form the
quantum numbers L and S, and we will assume that the corresponding operators L2
and S2 commute with the Hamiltonian. More generally, we will take this as a def-
inition of LS-coupling schemes, that is ones for which the Hamiltonian is diagonal
in L and S.
We saw in chapter 6.2 that we can ignore electrons in closed shells. The process
of adding up the different Li and Si for the valence electrons is done as covered in
chapter 6.1 and as illustrated in figure xxxxxXXX.
The quantum numbers L can be zero or any positive integer. It is annotated in a
way analogous to that on the orbital angular momentum of individual electrons (see
chapter 1.5), but with capital letters. A total angular momentum of L = 0 will be
annotated ‘S’, L = 1 ‘P’, L = 2 ‘D’ and so on. This letter symbol will be the basis
of the LS-coupling term.
The total spin, S, will be half-integer for an odd number of electrons, and integer
or zero for an even number. In an LS-coupling term, the value of S is indicated by
writing the numerical value of 2S +1 as a superscript to the right of the letter symbol
for L. 2S + 1 is called the multiplicity of the term, for reasons that will be clarified
in the following.

6.3.1 Atomic terms

We will look more closely at the total number of quantum numbers needed to de-
scribe an atomic state, and the multiplicity (or degeneracy) of a 2S+1 L-term. For an
individual electron, we need the four quantum numbers n, l, ml and ms to describe
a state, which corresponds to nu numbers of degrees of freedom. For an N-electron
atom, we can therefore assume that we have 4N degrees, and a need for 4N quantum
numbers.
Suppose that the number of atoms in closed shells is Nc and the number of va-
lence electrons is No , with Nc + No = N. Within the CFA, we describe the closed
shells with their corresponding part of the electron configuration, which will gives
us Nc pairs of n and l. In light of the Pauli principle — which is the effect making
these shells ‘closed’ in the first place — this will actually specify all four relevant
quantum numbers for these electrons. This leaves us with another 4No to be deter-
mined.
The valence part of the configuration provides 2No numbers. For the remaining
2No , thing will depend on how many valence electrons we have. If there is only one,
only two degrees of freedom remains, and specifying the ”total quantum numbers
based on L and S, that is either | LSML MS i or | LSLMJ i, gives apparently give us too
84 6 Coupling of angular momenta – the vector model

many quantum numbers. This is not actually a problem, since in this case, we will
always have L = l and S = s = 1/2.
For two valence electrons, the sets | LSJMJ i or | LSML MS i will do perfectly
to provide a full description. If we have three or more valence electrons, the LS-
coupling designation is incomplete. This latter situation diverges for cases where
all electrons are in the same orbital or not. If they are not, a ‘parent term’ can be
formed before the final atomic term is specified. If the valence electrons are indeed
in identical ni and li , the actual number of degrees of freedoms will be limited by
the Pauli principle, and the LS-description will be sufficient after all.
In the LS-coupling approximation, we will thus specify an atomic term for an
N-electron atom as:
2S+1
(n1 l1 . . . nNc lNc ) no1 lo1 . . . nN lN L, (6.18)

where the part of the electron configuration inside a parenthesis represents the va-
lence electrons. This is more than often omitted in the representation. If parent terms
are needed, the representation will be (with the valence configuration omitted for
clarity):
no1 lo1 . . . nN−1 lN−1 (2Sp +1 Lp ) nN 2S+1 L . (6.19)
In a case with a many valence electrons (for example a d-orbital with four or more
electrons), more than one parent term may be needed. The the atomic terms above
will be added specification for the remaining quantum numbers, when more pertur-
bations such as the spin-orbit coupling are added.

6.3.2 Two non-equivalent electrons

To give concrete examples, we start by showing the possible atomic terms for a case
of two valence electrons that are not in the same orbitals. That is, we have either or
both of the conditions n1 6= n2 and l1 6= l2 fulfilled (we now ignore the electrons if
the inner shell and simply index the valance electrons beginning with one). This is
the simplest case, since then we do not have to be careful about obeying the Pauli
principle. We can never make these two electrons equivalent my specifying ml1 , ml2 ,
ms1 and ms2 . In the following, we will give to examples — the configurations 2p3p
and 3d4p.
For the 2p3p-configuration (or any n1 pn2 p with n1 6= n2 ), we have l1 = l2 = 1 and
as always s1 = s2 = 1/2. The values for ml1 and ml2 can be +1, 0 or -1, and the spins
can be parallel or anti-parallel with the chosen quantization axis (‘up’ or ‘down’).
With all possible permutations, this should give us 36 different states, where many
may be degenerate in energy. When l1 and l2 couples, the possible values for L are
2, 1 or 0. The two electrons spins can be either mutually parallel or anti-parallel,
giving the possible values of 1 or 0 for S. The possible terms for the configuration
2p3p are thus:
1
D , 3D , 1P , 3P , 1S , 3S . (6.20)
6.3 LS-coupling 85

If we choose to specify the rest of the term with the quantum numbers J and JM
(which will prove to be appropriate in the absence of external fields), we can make
an accounting of the number of state, and show that it is indeed 36. We know the for
each term, the possible values for J are between L + S and |L − S|. This is illustrated
in table 6.1.

Table 6.1 Possible LS-coupling states for a 2p3p-configuration

term J MJ number of
states

1D 2 −2 . . . 2 5

3 −3 . . . 3 7
3D 2 −2 . . . 2 5
1 −1 . . . 1 3
1P 1 −1 . . . 1 3

2 −2 . . . 2 5
3P 1 −1 . . . 1 3
0 0 1
1S 0 0 1
3S 1 −1 . . . 1 3

Total number of states: 36

For the 3d4p-configuration, S is still 0 or 1, and L can vary between 3 and 1. The
possible terms become:
1
F , 3F , 1D , 3D , 1P , 3P , (6.21)
and if we count the number of possible state, we in this case get 60.

6.3.3 Two equivalent electrons

To exemplify this case, we take the configuration 2p2 (the ground state configuration
for atom number 6, C). Had it not been for the Pauli principle, the possible terms
would have been the same as in the case for 2p3p, shown in (6.20). However, in this
case our electrons are equivalent and we must take into account that only exchange
anti-symmetric wave functions are possible. We must exclude atomic terms that are
only possible for electrons equivalent. We must also exclude combinations of ml1 ,
ms1 , ml2 and ms2 that are the same for equivalent electrons.
86 6 Coupling of angular momenta – the vector model

A practical way to illustrate this is to use a table representation of the above indi-
vidual electron quantum numbers, introduced by Condon & Shortley. Such a table
is organised horizontally and vertically by the total angular momentum projection
numbers ML = ml1 + ml2 and MS = ms1 + ms2 . To illustrate this, we first show an
example for the already treated configuration 2p3p in table 6.2. Each entry in this ta-

Table 6.2 ghghhghgh

MS
2p3p
1 0 -1

2 ( 1+ 1+ ) ( 1+ 1− ) ( 1− 1+ ) ( 1− 1− )

( 1+ 0+ ) ( 1+ 0− ) ( 1− 0+ ) ( 1− 0− )
1
( 0+ 1+ ) ( 0+ 1− )( 0− 1+ ) ( 0− 1− )

( 1+ −1+ ) ( 1+ −1− ) ( 1− −1+ ) ( 1− −1− )

ML 0 ( 0+ 0+ ) ( 0+ 0− )( 0− 0+ ) ( 0− 0− )
( −1+ 1+ ) ( −1+ 1− )( −1− 1+ ) ( −1− 1− )

( 0+ −1+ ) ( 0+ −1− ) ( 0− −1+ ) ( 0− −1− )

-1
( −1+ 0+ ) ( −1+ 0− )( −1− 0+ ) ( −1− 0− )

-2 ( −1+ −1+ ) ( −1+ −1− ) ( −1− −1+ ) ( −1− −1− )

ble is a pair of electrons, with the numerical values showing mli and the plus/minus
superscripts indicates mli . We see that ML can vary between 2 and -2, which means
that the maximum value of L is to, and that S = 1 is the maximum total spin, since
MS is between 1 and -1. We also see that we have 36 states in total, which is fully
consistent with what we showed in chapter 6.3.2.
For the case with the 2p3p-configuration, a table such as table 6.2 is superfluous.
However, for the case with equivalent electrons it will help. Consider the same type
of table for the 2p2 -configuration. Initially, we can copy table 6.2, but then a number
of terms have to be deleted. We have to remove pairs of identical electrons, such as
(1+ 1+ ). Furthermore, with identical electrons we have some pairs that identical,
such as (1+ 1− ) and (1− 1+ ); these also have to go. We end up with table 6.3. The
36 states have been reduced to 15. Some terms are no longer feasible. For example
3 D is no longer possible, since we cannot have M = 2 and M = 1 at the same time.
L S
We still need at least one triplet, and we can make out from table 6.3 that we must
have a 3 P-term. If we contemplate the possible values of J and MJ for 3 P, we find
that it corresponds to 9 states. We can then remove 9 electron pairs from table 6.3,
with MS - and ML -values both between 1 and -1. For what remains in the table, we
see that we must also have a term 1 D. This will take care of five states, which means
that the only thing remaining in table 6.3 will be a lone electron pair for ML = 0 and
MS = 0. This has to correspond to 1 S.
We conclude that the possible LS-coupling terms for the 2p2 -configuration are:
6.3 LS-coupling 87

Table 6.3 Diagram for identification of LS-coupling terms not in conflict with the Pauli principle,
for the 2p2 -configuration.

MS
2p2
1 0 -1

2 ( 1+ 1− )

( 1+ 0− )
1 ( 1+ 0+ ) ( 1− 0− )
( 0+ 1− )

( 1+ −1− )
ML 0 ( 1+ −1+ ) ( 0+ 0− ) ( 1− −1− )
( −1+ 1− )

( 0+ −1− )
-1 ( 0+ −1+ ) ( 0− −1− )
( −1+ 0− )

-2 ( −1+ −1− )

1
D , 3P , 1S , (6.22)

and in table 6.4 we show that we get 15 different states, as predicted.

Table 6.4 Possible LS-coupling states for a 2p2 -configuration

term J MJ number of
states

1D 2 −2 . . . 2 5

2 −2 . . . 2 5
3P 1 −1 . . . 1 3
0 0 1
1S 0 0 1

Total number of states: 15

6.3.4 More than two valence electrons

Also with three or more valence electrons, it will make a big difference if all or
some of the electrons are equivalent. The best way to show how LS-coupling terms
can be identified is via examples.
88 6 Coupling of angular momenta – the vector model

Suppose we have three identical p-orbital electrons, such as the ground state
configuration in N, 2p3 . Making a diagram such as tables 6.2 or 6.3 may at first
glance appear unattractive, due to the large number of combination. If the electrons
had been inequivalent (such as the exotic doubly excited configuration 2p3p4p), we
would have had 256 combinations of the different mli and msi , and thus a big and
cumbersome table.
However, with identical electrons, a vast majority of these trios of quantum num-
bers must be excluded. Moreover, the symmetry of a diagram like table 6.3 means
that suffices to construct one quadrant of, while including the ML = 0 row and the
possible MS = 0 column. For the 2p3 example, we get a diagram as in table 6.5.
From this, we can extract that the only possible terms are:

Table 6.5 ghghhghgh

MS
2p3
3/2 1/2

2 ( 1+ 1− 0+ )

ML 1 ( 1+ 1− −1+ )
( 1+ 0+ 0− )

( 1+ 0+ −1− )
0 ( 0+ −1+ −1+ ) ( 1+ 0− −1+ )
( 1− 0+ −1+ )

2
D , 2P , 4S , (6.23)

If we instead study an excited configuration of N, such as 2p2 3s, we have to first

couple two of the electrons, and then add the third, at every step taking into account
the Pauli principle as necessary. The final result is the same no matter which electron
pair one starts with, but the most logic procedure, and the standardised one, is to
being with the most tightly bound electrons, and then successively add the more
loosely bound.
For 2p2 3s, we form the parent terms from the two equivalent 2p electrons, which
we found in (6.22). To these three parent terms, we add the 3s-electron, with l3 = 0
and s3 = 1/2, with all possible projections of ml3 and ms3 . We find four possible
LS-coupling terms:
6.3 LS-coupling 89

2p2 (1 D) 3s 2 D
2p2 (3 P) 3s 4 P
2p2 (3 P) 3s 2 P (6.24)
2 1 2
2p ( S) 3s S .

6.3.5 Fine structure

We will finish this introduction to the LS-coupling scheme by adding the spin-orbit
Hamiltonian of 4.16. To recapitulate; we have first applied the CFA in order to get a
separable Schrödinger equation and an electron configuration zero-order state. We
have then applied what is left of the electron-electron repulsion, the Coulomb torque
Hamiltonian of 5.4, as a perturbation. That has split and/or shifted the configuration
states into LS-coupling terms, with a degeneracy in energy of (2L + 1)(2S + 1). We
will now remove a part of that degeneracy by applying the spin-orbit interaction as
yet another Hamiltonian.
We begin by slightly reformulating 4.16:
" #
N N
HSO = ∑ f (ri ) Li · Si = ∑ ξ (ri ) L·S . (6.25)
i i

We now restrict the summation to the valence electrons (based on the arguments
given in chapter 6.2), and we have taken the LS-coupling approximation that the
individual spins and orbital angular momenta couple to S and L so strongly that
other interaction only perturb this coupling marginally.
The Hamiltonian given in the last line of (6.25) commutes with J. This means
that is will be diagonal in J and MJ , and this can also be seen as the essence of
the LS-coupling approximation. Non-diagonal terms in J would signal the depar-
ture from the LS-coupling, and would make the last equality in (6.25) invalid. It
should, however, be noted that by its very nature, the spin-orbit Hamiltonian does
not commute with L and S. ML and MS will not provide good quantum numbers,
but within the approximation that L2 and S2 do commute with (6.25), the diagonal
representation is:
| LSJMJ i . (6.26)
The energy correction to to the perturbation (6.25) will be:
*" #+
N
ESO = ∑ ξ (ri ) hL·Si . (6.27)
i

The first factor in (6.27) can only be calculated analytically if we know the wave
function, which we do not. It can however, be derived empirically from spectroscop-
ical data, and it can also be estimated by approximative and/or numerical methods
90 6 Coupling of angular momenta – the vector model

Fig. 6.4 ghghghhghghg 1S

1S
E / cm-1 0 : (21 648.01)

20 000

15 000

1D
1D
10 000 2 : (10 192.63)

5 000

2p2
3P : (43.40)
3P 2
3P : (16.40)
1
0 3P : (0.00)
0

(see chapters 7 and 14). We define the ‘fine structure factor’ for a certain term thus:
* " # +
 N 
A(γLS) ≡ γLS  ∑ ξ (ri )  γLS . (6.28)
 
 i 

Here, γ is as usual short for the electron configuration. We see that the fine structure
factor depends on L and S, making it specific for each atomic term, but that it is
independent of J.
The second factor of (6.27) can be calculates as was done in 4.9. That is, we
have:
A(γLS)
ESO = [J(J + 1) − L(L + 1) − S(S + 1)] . (6.29)
2
This means that every LS-term will be split up in (2S + 1) ‘fine-structure levels’ (or
2L + 1 if S > L). This will be annotated by adding the quantum number J to the
LS-coupling term:
2S+1
LJ . (6.30)
The (2J + 1)-fold degeneracy in MJ will remain.
In figures 6.4 and 6.5, we illustrate in partial energy level diagrams how this can
play out. These examples are for the 2p2 and 2p3 configurations respectively, where
the terms were found in (6.22) and (6.23). The energy values given are experimental
values for neutral C (2p2 ) and N (2p3 ) respectively.
For the doublet 2 D5/2 and 2 D3/2 in figure 6.5, we see that the fine structure fac-
tor can be negative. The small splittings in the multiplets compared to the energy
6.4 j j-coupling 91

Fig. 6.5 ghghghhghghg E / cm-1

30 000 2P
2P
3/2 : (28 839.306)
2P
1/2 : (28 838.920)

20 000 2D
3/2 : (19 233.177)
2D
2p3
2D
5/2 : (19 224.464)

10 000

4S
0 4S : (0.00)
3/2

difference between the term shows that LS-coupling is a good approximation in this
case.We will defer a discussion on the energy order of the terms until chapter 7.
For every term with a multiplicity of two or larger, the fine structure factor
ζ (γLS) can be deduced from experimental data and (6.29). In figure 6.4, we see
that the interval between 3 P2 and 3 P1 is close to twice as great as that between 3 P1
and 3 P0 . This is roughly consistent with (6.29).
We can take a closer look at the energy interval between fine-structure levels in
LS-coupling. From (6.29), we can calculate this interval as:

ESO (γ, L, S, J) − ESO (γ, L, S, J − 1) = A(γLS) J . (6.31)

This is known as ‘Landé’s interval rule’, which shows that the interval between
two adjacent levels is directly proportional to the one of them with the highest J.
It is useful in empirical analyses of spectra, and it can be used as a measure of the
goodness of the LS-coupling approximation.

6.4 j j-coupling

The LS-coupling described in the preceding chapter is a limiting case, and at the
other extreme end of this scale we have j j-coupling. This occurs when the energy
contribution of HSO in (6.2) is so large that Hto can at first be neglected. As we
have seen, the Coulomb torque Hamiltonian scales linearly with the atomic number,
while the spin-orbit Hamiltonian is proportional to its fourth power.
In terms of the vector model, this means that we first have to couple the Li and Si
of each individual valence electron (we can still take the average effect of all closed
92 6 Coupling of angular momenta – the vector model

shells as spherically symmetric). That is, we form the quantum numbers ji and m ji
from all the li , mli , si and msi , following the rules for quantum mechanical vector
addition, outlined in chapter 6.1. For the case of two valence electrons, this is what
is illustrated in the right panel on figure 6.3. The quantum numbers mli and msi will
no longer be ‘good’, since their corresponding angular momentum projections are
not constants of motion under the interaction.
The Hamiltonian that will be used in j j-coupling is:
1
H = HCF + HSO = ∑ − ∇2i + ∑ VCFi (ri ) + ∑ ξ (ri ) Li · Si , (6.32)
i 2 i i

The first two terms will give the zero-order solutions, which also here are the elec-
tronic configuration. The CFA is still applied in the same way. The solutions will
be degenerate in mli and msi , and after forming Slater determinant combinations of
these, the spin-orbit Hamiltonian can be applied as a perturbation.
An important difference between (6.4) and the Hamiltonian used in LS-coupling
is the absence of cross-couplings between the electrons. Equation (6.4) is a sum of
single-electron Hamiltonians, and we can designate the solutions as:

(n1 , l1 , j1 , . . . , nN , lN , jN ) , (6.33)

where N is used for the number of electrons outside closed orbitals. Such sets will
be the j j-coupling terms. The fine structure in j j-coupling will appear when the
Hamiltonian Hto is applied as a subsequent perturbation. In the vector image, the
Ji will combine to a J, and we will get fine structure levels described by quantum
numbers J.
We have seen that LS-coupling is a good approximation for light atoms. In con-
trast, j j-coupling rarely appears in pure form. Even for very heavy atoms, we often
see a mixed case. Some of these will be presented in chapter 9.
One set of elements that display almost pure j j-coupling is highly charged heavy
ions. In these, the strong binding energy of the valence electron will diminish the
relative importance of the Coulomb exchange term. Even for elements and atomic
states that are neither LS-coupled, nor j j-coupled, an understanding of the two lim-
iting case will facilitate the understanding of the atomic structure.

6.4.1 j j-coupling terms

The terms in j j-coupling will be made up of the ensemble of all the ji among the
valence electrons. Even though the electrons do not interact at this level of interac-
tion, there will still some combinations of quantum numbers that will be prohibited
by the Pauli principle, and the procedure to follow in order to identify these will
resemble that used for LS-coupling.
The notation for j j-coupling has not become as standardised as is the case for LS-
coupling. The one we will use in this book is where the electrons in open shells are
6.4 j j-coupling 93

divided into groups with common values of ni , li and ji . The electron configuration
of closed shells is notated in the same way as for LS-coupling. Examples of how the
valence electrons can be annotated are as follows:

[6p21/2 ]
[5d25/2 5d3/2 ] . (6.34)

In the first example, there are two electrons, with n1 = n2 = 6, l1 = l2 = 1, and

j1 = j2 = 1/2. Note that this combination does not violate the Pauli principle. We
could have ml1 = 1, ms1 = −1/2, m j1 = 1/2; and ml2 = −1, ms2 = 1/2, m j2 = −1/2.
In the second example, we have three 5d-electrons. Two of them have j1 = j2 = 5/2,
and the third has j3 = 3/2.
For non-equivalent electrons, identifying the possible j j-coupling terms is an
easy task. As an example, we take the configuration 6p7p. The possible terms will
be:
[6p3/2 7p3/2 ] , [6p3/2 7p1/2 ] , [6p1/2 7p3/2 ] , [6p1/2 7p1/2 ] . (6.35)
For the sake of comparison with the LS-couples 2p3p-configuration shown in table
6.1, we can tabulate the terms in (6.35), with the respective possible J-values, and
keep a record of the possible number of states. This is done in table 6.6. We find

Table 6.6 Possible j j-coupling states for a 6p7p-configuration

term J MJ number of
states

3 −3 . . . 3 7
2 −2 . . . 2 5
[6p3/2 7p3/2 ]
1 −1 . . . 1 3
0 0 1

2 −2 . . . 2 5
[6p3/2 7p1/2 ]
1 −1 . . . 1 3

2 −2 . . . 2 5
[6p1/2 7p3/2 ]
1 −1 . . . 1 3

1 −1 . . . 1 3
[6p1/2 7p1/2 ]
0 0 1

Total number of states: 36

that the number of states is consistent with what it was when we coupled two non-
equivalent p-electrons with LS-coupling.
94 6 Coupling of angular momenta – the vector model

6.4.2 Equivalent electrons

An example of two equivalent electrons, where j j-coupling plays an important role

is the ground configuration of lead: 6p2 . Even in this case, the j j-coupling is not
quite pure, but we will use this as an example nevertheless. We know from table 6.6
that had the electrons been non-equivalent, we would have had MJ -values between
3 and -3, and we would have had 36 states altogether. In order to figure out which
states that are not forbidden by the Pauli principle, we can make a diagram similar
to those used in chapter 6.3.3. This is shown in table 6.7, where the entries in the
table are pairs of the quantum numbers m j1 och m j2 . In agreement with the case for

Table 6.7 Diagram for identification of j j-coupling terms not in conflict with the Pauli principle,
for the 6p2 -configuration.

j1 j2
6p2
3/2 3/2 3/2 1/2 1/2 1/2

2 ( 3/2 1/2 ) ( 3/2 1/2 )

( 3/2 -1/2 )
1 ( 3/2 -1/2 )
( 1/2 1/2 )
MJ
( 3/2 -3/2 ) ( 1/2 -1/2 )
0 ( 1/2 -1/2 )
( 1/2 -1/2 ) ( -1/2 1/2 )

( -1/2 -1/2 )
-1 ( -1/2 3/2 )
( -3/2 1/2 )

-2 ( -3/2 -1/2 ) ( -3/2 -1/2 )

-3

possible J 2 , 0 2 , 1 0

the 2p2 -configuration in table 6.3, we get 15 possible states.

A conclusion from table 6.7 is that the allowed j j-terms for the 6p2 -configuration
are:

[6p23/2 ]2 , [6p23/2 ]0 , [6p3/2 6p1/2 ]2 , [6p3/2 6p1/2 ]1 , [6p21/2 ]0 . (6.36)

In (6.36), we have also added the values of J (the j j-coupling fine structure) as a
subscript to the terms. In chapter 8 we will give more a more complete account of
which configurations that can lead to which j j-coupling terms.
6.4 j j-coupling 95

Fig. 6.6 ghghghhghghg (E/hc) / cm-1

30 000
[6p23/2]0 : (29 466.83)

[6p23/2]

[6p23/2]2 : (21 457.80)

20 000

6p2

[6p3/2 6p1/2] [6p3/2 6p1/2]2 : (10 650.33)

10 000

[6p3/2 6p1/2]0 : (7 819.26)

[6p21/2]
0 [6p21/2]0 : (0.00)

6.4.3 Fine structure in j j-coupling

In order to make the j j-coupling complete, we must finally add the last term of the
Hamiltonian in (6.2). This time, the last component to be added as a perturbation is
the Coulomb torque Hamiltonian — the electron-electron repulsion.
The Hamiltonian Hto commutes with J2 and Jz , which means that a good diagonal
representation will now be:

| n1 l1 j1 . . . nN ln jN JMJ i (6.37)

This time, the electrostatic mutual repulsion couples the different Ji , which has the
effect that the individual m ji will no longer be good quantum numbers. Filled or-
bitals will have a total angular momentum of zero, as before.
To form the quantum number J, one has to follow the standard rules for addition
of quantum mechanical angular momenta, outlined in chapter 6.1. If we again take
the example of a j j-coupled p2 -configuration, as illustrated in table 6.7.
For two non-identical p-electrons, both the individual ji can be either 3/2 or 1/2.
This gives three different combinations. For j1 = j2 = 3/2, J can be 3, 2, 1 or
0. For j1 = 3/2 , j2 = 1/2 (or inversely), the possible values are 2 and 1. For the
last combination, j1 = j2 = 3/2, we have to have either J = 1 or J = 0. When the
electrons are identical, we saw in the analysis accompanying table 6.7 that some of
these combinations will be excluded, and we end up with the terms shown in (6.36).
In figure 6.6, we show the lowest energy states in Pb, which has the ground state
configuration 6p2 . It should be noted that the j j-coupling is not pure in the ground
state of Pb. The energy levels to the right of the figure are the experimentally mea-
sured energy levels. The two intermediate stages are here just statistical averages. In
96 6 Coupling of angular momenta – the vector model

a LS-coupling scheme (see figure 6.4), the three lowest states would belong to the
term 3 P, and the two upper ones would have been 1 D and 1 S. In many treaties of the
lead atom, these levels are annotated with the LS-coupling notation, simply because
that label is easier and more well known.
If the j j-coupling had been purer in lead, the three groups of levels in figure 6.6
would have been more distinct, and the within each term, the fine-structure splitting
would have been smaller.
In chapter 8, we will look into this coupling scheme in more detail. We will study
more complex electronic configurations — with more than two electrons — and we
will add quantitative analyses.

6.5 LS- and j j-coupling

The
Include a comparison between LS and jj. Maybe a progressive graph between
2p2 and 6p2
Chapter 7
LS–coupling

The LS-coupling scheme was introduced in chapter 6 as a limiting example of the

central field approximation. Each individual electron has both orbital angular mo-
mentum and a spin, and when we have two electrons or more, we will in total have
at least four different angular moment that all interacts with each other via electro-
magnetic couplings. This means that the full wave function of the system will always
be an entangled state.
When we try to disentangle this complication by first applying the CFA, and then
treating the interactions between angular momenta in decreasing order of magni-
tude, we get different limiting cases. LS-coupling is the one where we assume that
we first can apply only the angular contribution of the electrostatic interaction be-
tween the electrons as a perturbation, and only after that the spin-orbit interactions.
In this chapter, we will do several things. We will expand the initial treatment,
introduced in chapter 6.3, to more complex situations, we will put the theory on a
more rigorous theoretical footing,and we will go into much more detail. Initially, we
will focus on LS-coupling atomic terms. After that, we will look at fine structure,
and eventually we will explore the functional form of the LS-coupling states.

7.1 Atomic terms

The LS-coupling atomic terms are what we get when we apply the first perturbation
term, the Coulomb torque Hamiltonian, to the electronic configuration. The latter
will provide the zero-order states, and as we have seen in previous chapters, they
have the great advantage that they give wave functions that are separable in individ-
ual electron coordinates.
Even though an electron configuration is an assignment of N hydrogenlike or-
bitals to N electrons, it does not make any sense to say that each individual electron
is in one specified orbital. The crossterm means that the complete state will always
be an entangled one, and that will be even more obvious when we apply degenerate

97
98 7 LS–coupling

perturbation theory — and are forced to use superposition states as the zero-order
states in the calculation.
The Coulomb torque Hamiltonian (5.4) commutes with all the angular momen-
tum operators of the entire atom, L2 , Lz , S2 , Sz , J2 , and Jz . If (or rather when) we
start take into account the spin orbit interaction, (4.16), the total Hamiltonian will
no longer commute with Lz and Sz . Presently however, with HSO neglected, there is
no torque between L and S, and hence the perturbation Hamiltonian that we use to
produce the atomic terms, Hto , will not have any non-diagonal elements in ML or
MS .

7.1.1 Allowed LS-coupling terms

We recall the notation for atomic terms, introduced in chapter 1.5, and used in chap-
ter 6.3. The perturbation Hto will break the degeneracy in the quantum numbers L
and S, which means that these two numbers will be enough to assign a term. The
number L is for historic reasons annotated with a letter — S, P, D, F, . . . for L = 0,
1, 2, 3, . . . . To indicate S, one writes the multiplicity — 2S + 1 — as a superscript to
the right of the letter indicating L.
Added to that, atomic data tables often indicate whether a term has odd or even
parity. An odd term gets a lowercase ‘o’ as a superscript to the right of the L-letter.
For example, the ground state terms in C is the even term 2p2 3 P, whereas an excited
state is the odd term 2p3s 3 P o .
Using diagrams like the examples in tables 6.2, 6.3 and 6.5, we can deduce the
possible atomic terms for every conceivable electron configuration. If all electrons
are non-equivalent, the task is easy. When they are equivalent, and we have to cater
for the Pauli principle, it is slightly more challenging. It is also often more relevant,
since most ground state configurations will have exclusively equivalent electrons.
We saw in tables 6.3 and 6.5 that is gets more complicated to find the allowed
term for more valence electrons. This problem is alleviated due to a symmetry con-
sideration. Consider Nv equivalent valence electrons in an orbital that can accommo-
date mximum No before it is full. The possible terms for N0 − Nv valence electrons
will be exactly the same as for Nv electrons. For example, a p2 -configuration will
have the same terms as a p6 one, a d7 -configuration will yield an identical group of
terms as d3 , and so on.
This can be found by constructing charts like tables 6.3 and 6.5, but instead of
entering the ml - and ms -values of the occupied orbitals, one does that for the unoc-
cupied ones instead. The only difference in the table is that positive values of ml -
and ms will then correspond to negative values of ML - and MS , but that does not
change the conclusion.
In table 7.1, we give a table with the allowed LS-coupling terms for s- p- and
d-configurations. Separately, in table 7.2, we show f-configurations. We point out
yet again that only the valence electrons have to be included in this analysis, since
all closed shell will only contribute with a 1 S contribution (see chapter 6.2). As
7.1 Atomic terms 99

Table 7.1 ghghhghgh

electron allowed
configuration terms

s 2S

s2 1S

p , p5 2P o

p2 , p4 1S , 1D 3P

p3 2P o , 2D o 4S o

d , d9 2D

1S , 1D
d2 , d8 1G
3P , 3F

2P , 2 D (2)
d3 , d7 2F , 2G 4P , 4F
2H

1S 1D 3P , 3D
(2) , (2)
d4 , d6 1F , 1G (2)
3F
(2) , 3G 5D

1I 3H

2S , 2P
2D , 2 F (2) 4P , 4D
(3)
d5 2G 2H 4F
6S
(2) , , 4G
2I

far as energy is concerned, the contribution from filled inner shells will change the
energy, but only with an additive quantity that will be the same for all terms in the
configuration. It will not affect the complexity of the structure.
Rather few of the terms in tables 7.1 and 7.2 will occur for atoms without con-
siderable perturbations. To start with, the LS-coupling approximation will fail when
the spin-orbit term of the total Hamiltonian gains importance, which will increas-
ingly be the case as we go to heavier atoms. For example, this will affect essentially
all atoms with 4f- and 5f-configuration ground states — the lanthanides and the
actinides. Moreover, there will be many cases where different configurations are al-
most degenerate, which will in a strict sense invalidate the CFA. Notwithstanding,
it will still be useful in most cases to identify terms in the LS-coupling scheme, and
to then describe real observed energy levels as superpositions of these.
In tables 7.1 and 7.2, the numbers in brackets after some terms indicate that the
term appears more than once for the configuration concerned. This can only occur
for case with at least three valence electrons. In those cases, the quantum numbers
L, S, ML , and MS are not sufficient for uniquely cover all the permutations of mli and
msi
100 7 LS–coupling

Table 7.2 ghghhghgh

electron allowed
configuration terms

f , f13 2F o

1S , 1D 3P , 3F
f2 , f12 1G , 1I 3H

2P o , 2D o
(2) 4S o , 4D o
2F o , 2G o
f3 , f11 (2) (2) 4F o , 4G o
2H o , 2I o
(2) 4I o
2K o , 2L o

1S 1D 3P , 3 D (2)
(2) , (4) (3)
1F , 1G 3F 3 5S , 5D
(4) (4) , G (3)
f4 , f10 1H
(2) , 1I
(3)
3H 3
(4) , I (2)
5F , 5G
1 K, 1 L 3K 3 5I
(2) (2) , L
1N 3M

2P o , 2D o
(4) (5) 4S o , 4 P o(2)
2F o , 2G o
(7) (6) 4D o , 4F o
2H o , 2I o (3) (4) 6P o , 6F o
f5 , f9 (7) (5) 4G o , 4H o
(4) (3)
2K o , 2L o 6H o
(5) (3) 4I o , 4K o
2M o , 2N o (3) (2)
(2) 4L o , 4M o
2O o

1S 1 3P , 3 D (5)
(4) , P (6) 5S
1D 1 3F
, 5P
(6) , F (4) (9) , 3 G (7) 5D
1G 1 3H (3) , 5 F (2)
(8) , H (4) (9) , 3 I (6) 5G
f6 , f8 1I 1 3K (3) , 5 H (2) 7F
(7) , K (3) (6) , 3 L (3) 5I 5K
1L 1 3M (2) ,
(4) , M (2) (3) , 3N 5L
1N 1 3O
(2) , Q
2S o , 2P o
(2) (5) 4S o , 4P o
2D o , 2F o (2) (2)
(7) (10) 4D o , 4F o
2G o 2 o (6) (5) 6P o
(10) , H (9) 4G o , 4H o , 6D o
f7 2I o , 2K o (7) (5) 6F o , 6G o
8S o
(9) (7) 4I o , 4K o
2L o , 2M o (5) (3) 6H o
(5) (4) 4L o , 4M o , 6I o
2N o , 2O o (3)
(2) 4N o
2Q o
7.1 Atomic terms 101

Table 7.3 ghghhghgh

MS
d3
3/2 1/2

5 ( 2+ 2− 1+ )

4 ( 2+ 2− 0+ ) ( 2+ 1+ 1− )

( 2+ 2− −1+ ) ( 2+ 1+ 0− )
3 ( 2+ 1+ 0+ )
( 2+ 1− 0+ )( 2− 1+ 0+ )

( 2+ 2− −2+ ) ( 2+ 1+ −1− )
ML 2 ( 2+ 1+ −1+ ) ( 2+ 1− −1+ )( 2− 1+ −1+ )
( 2+ 0+ 0− )( 1+ 1− 0+ )

( 2+ 1+ −2− ) ( 2+ 1− −2+ )
( 2+ 1+ −2+ ) ( 2− 1+ −2+ )( 2+ 0+ −1− )
1
( 2+ 0+ −1+ ) ( 2+ 0− −1+ )( 2− 0+ −1+ )
( 1+ 1− −1+ )( 1+ 0+ 0− )

( 2+ 0+ −2− ) ( 2+ 0− −2+ )
( 2+ 0+ −2+ ) ( 2− 0+ −2+ )( 2+ −1+ −1− )
0
( 1+ 0+ −1+ ) ( 1+ 1− −2+ )( 1+ 0+ −1− )
( 1+ 0− −1+ ) ( 1− 0+ −1+ )

As an example, we give the term chart for a d3 -configuration in table 7.3. If we

pick out the terms from this, with the procedure described in chapter 6.3, the first
term we identify is 2 H. When we proceed, we get 2 G, 4 F and 2 F, but then we find
that we have to use 2 D twice in order to account for all entries in the table. Finally,
we get single occurrences of 4 P and 2 P, in accordance with table 7.1.

7.1.2 Genealogy of atomic terms

When a term occurs more than once for an electron configuration, the level can
be more specified if it is possible to discriminate between different paths to the
final vector coupling term. As an example, we can consider three non-equivalent
electrons with the l-quantum numbers 0,1 and 2.
An analysis of the what possible terms that emanates from this spd-configuration
yields the result: 4 F, 2 F (2) , 4 D, 2 D (2) , 4 P, 2 P (2) . That is, all three doublets occur
twice. In this case, we can differentiate between the terms that have the same nota-
tion.
102 7 LS–coupling

If we first add the s- and p-electron, this gives the two possible terms 3 P and
1 P.If we now add the d-electron, we find the following possible terms for the spd-
configuration (these terms are all of odd parity, but here we omit the superscript):

sp(3 P)d ⇒ 4 F,2 F,4 D,2 D,4 P,2 P

sp(1 P)d ⇒ 2 F,2 D,2 P . (7.1)

The terms in the brackets after sp are called parent terms. In most cases, the terms
with the same final term, but with different parents, will have differing energies, and
the way the quantum numbers have been specified in (7.1) is the representation that
diagonalises the Hamiltonian.
We could also have begun by adding the s- and d-electron. That had resulted in
the parent terms (3 D) and (1 D), and then of course the same set of final terms. The
logical order to chose in which order to add the electron angular momentum vectors
is to begin with the most tightly bound electrons. In that case, the final electron is
in a manner of speaking added to an ionic term that is further specified by a parent
term.
In the example in table 7.3 — with a d3 -configuration — it is not possible to
specify parentage in this way. The three electrons are identical, and we have now
way to discriminate between the two 2 D-terms.
A pure d3 -configuration is actually rare. Looking at the periodic system in figure
5.2, and the ground state configurations in table 5.1, it appears as if vanadium might
have this configuration in its ground state. However, the tabulated configuration is
3d3 4s2 , which is because the 3d- and 4s-electrons have very similar binding energy.
It is therefore more proper to treat V as an electron with five valence electrons, even
if in the lowest energy state two of the five electrons fill up an s-orbital. Indeed, the
lowest energy LS-coupling term of the excited configuration 3d4 4s will lie lower that
most terms in 3d3 4s2 . To illustrate this further, we show a partial grotrian diagram
for V in figure 7.1.
In figure 7.1, we have omitted fine structure. There are many other things we can
learn from the program. One is that terms of different configurations overlap. This
shows that in this case, the CFA is not a good approximation. The angular part of
the electrostatic repulsion between electrons is to strong to treat as a perturbation
with great accuracy. Nevertheless, the LS-coupling notation is still pertinent, but the
assignment of the quantum numbers should be seen as what the leading term is in a
superposition. Most of the terms in the diagram are strongly mixed.
As pointed out earlier, we are not able to dedicate parent terms in the 3d3 4s2 , and
this is also the case for 3d5 . For the 3d4 4s-configuration, there is first a term given
for the 3d4 configuration in the positive vanadium ion, and to that a 4s-electron is
added. For 3d3 4s4p we identify two parent terms — one for 3d3 and one for 4s4p.
7.1 Atomic terms 103

(E / hc) / cm-1

20 000 3d5 6F
3d4(3F2)4s 2F
3d4(3H)4s 2H
3d4(3P2)4s 2P 3d3(4F)4s4p(3P) 6F
3d3(4F)4s4p(3P) 6D
3d4(3G)4s 4G
3d3(4F)4s4p(3P) 6G
3d4(3F2)4s 4F
15 000 3d34s2 2H 3d4(3P2)4s 4P
3d34s2 2D2 3d4(3H)4s 4H
3d34s2 2P

3d34s2 2G
10 000
3d34s2 4P
3d4(5D)4s 4D

5 000

3d4(5D)4s 6D

3d34s2 4F
0
3d34s2 3d44s 3d34s4p 3d5

Fig. 7.1 no fine structure, no odd/even

7.1.3 Energies of LS-coupling terms

Calculating absolute energies within the CFA is a daunting task, and for the radial
part one frequently resorts to heavy numerical calculations (see chapter 14). What
we will do in this section is to look specifically at the energy contribution that arises
when we apply perturbation theory in order to compute relative energies of LS-
coupling terms.
If we consider a certain configuration of electrons in open shells, this is specified
of a set of quantum numbers ni and li . This leaves a number of possibilities for mli
and msi . If there is only one orbital outside filled shells that is populated, there is
place for No = 2(2l − 1) electrons. If the number of (equivalent) electrons in the
orbital is Nv , the number of ways that these can be arranged is:
No !
. (7.2)
Nv !(No − Nv )!

Theses possibilities correspond to combinations of the different mli and msi , exactly
as was shown in the tables in chapter 6.3 and also in table 7.3.
104 7 LS–coupling

All such table entries correspond to unique wave functions that are degenerate in
energy in the zero-order approximation — they all belong to the same configuration
within the CFA. This means that the perturbation theory has to be of the degener-
ate kind, and that in order to determine energy contributions relatively large secular
problems have to be solved. If we take the d3 -configuration in table 7.3 as an exam-
ple, the first order energy correction can be found by taking the matrix element of
the perturbation Hamiltonian for the Slater determinant that holds all permutations
of electron coordinates between all spin-orbital entries in table 7.3 — 120 terms for
the complete version of the table.
Solving secular problems of as high dimensions as 120 or higher is a daunting
task, but it can be greatly facilitated. Fist of all because the perturbation is diagonal
in ML and MS , and secondly by using the sum-diagonal rule for matrices. We will
look at the former of these simplifications first.
The degeneracy in ML and MS is a consequence of the fact that the operators Lz
and Sz commute with the Hamiltonian, as long as the spin-orbit coupling is ignored.
We can then consider the matrix element of the commutation relation, for example
between Lz and the Hamiltonian, between two spin-orbital wave functions, such as
the entries in table 7.3.

{mli , msi }0 | [Lz , H] | {mli , msi } = ML0 MS0 | [Lz , H] | ML MS = 0 .



(7.3)

The states | ML MS i are eigenvectors to Lz , with eigenvalue ML , so (7.4) reduces to:

(ML0 − ML ) HML0 M0 :ML MS = 0 , (7.4)

S

where HML0 M0 :ML MS is the matrix element of the Hamiltonian between two states
S
| ML0 MS0 i and | ML MS i. Thus, the only elements of the secular problem that will be
non-zero are this for which ML0 = ML0 . Exactly the same argument will be valid for
Sz and MS .
The fact that we only have to retain terms that are diagonal in ML and MS means
that the for the example with d3 , we will, instead of a secular problem with dimen-
sion 120, have 36 separate ones, where no one is of higher dimension than eight.
If we take a p2 -configuration as an other example (see table 6.3), a 15-dimensional
quest is reduced to 11 separate secular equation, with no higher dimension than
three.
Determinants as large as 3 × 3, or 8 × 8, can still be cumbersome. However, the
general mathematical theorem that the the sum of diagonal elements in a square
matrix always equal the sum of the eigenvalues means the we typically only need to
solve integrals for single wavefunctions, and the determinants we still have to take
on are of low dimensions. We will illustrate this in a way introduced by Slater.
Table 7.4 is essentially the same as table 6.3, except that the spin-orbital expres-
sions have been replaced by the corresponding terms. The way to understand the
table is that although we have 15 different spin-orbitals, there will only be three
energies — the ones corresponding to 1 S, 3 P and 1 D. Nine eigenvalues will be de-
7.1 Atomic terms 105

Table 7.4 ghghhghgh

MS
p2
1 0 -1

2 1D

1 3P 3P , 1D 3P

ML 0 3P 1S , 3P , 1D 3P

-1 3P 3P , 1D 3P

-2 1D

generate and will all give the energy of the 3 P-term. Five will have the energy of 1 D,
and just one will have the non-degenerate energy of 1 S.
We can start by solving the integral for ML = 2 and MS = 0. The actual procedure
for the integration will be explained in next section. For the moment, we just assume
that we can compute single wavefunction integrals. We have:

E2,0 = (1+ , 1− ) | H | (1+ , 1− ) = h2, 0 | H | 2, 0i = E1 D .


(7.5)

Here, the numbers in the first subscript, and also in the second matrix element, are
the values of ML and MS respectively. In the first matrix element, we instead use
the spin-orbitals, represented as in table 6.3. The state | 2, 0 i must be one of the five
that has the energy of 1 D, which means that we can determine the latter by solving
a single integral.
To find the energy on the term 3 P, we can use the fact that there are six occur-
rences in table 7.4 where 3 P appears alone. This means that also the energy E3 P can
be found with help of one integral. For example,

E3 P = E1,1 = (1+ , 0+ ) | H | (1+ , 1+ ) .


(7.6)

The final remaining term, 1 S can only be found in one place in the table, namely
for ML = 0, MS = 0. For those values, we have a cubic secular problem, but two of
the eigenvalues have already been found. The diagonal sum rule says that:

E1 S + E3 P + E1 D
= (1+ , −1− ) | H | (0+ , 10 ) + (0+ , 0− ) | H | (0+ , 0− )




+ (−1+ , 1− ) | H | (−1+ , 1− ) . (7.7)



As a consequence, we can get the final energy by solving three integrals, and we
have completed the task in hand without having to solve any secular problems, let
alone one of dimension 15.
106 7 LS–coupling

There are cases where the diagonal sum rule does not work all the way, but it
will always simplify matters considerably. As an example where one determinant
cannot be avoided, we will look again at the d3 -configuration, treated in table 7.3.
We know that there are seven different terms, 2 H, 2 G, 4 F, 2 F, 2 D, 4 P and 2 P, with 2 D
appearing twice. We accordingly have eight energies to compute. This is illustrated
in table 7.5.

Table 7.5 ghghhghgh

MS
p2
3/2 1/2 -1/2 -3/2

5 2H 2H

4 2G , 2H 2G , 2H

3 4F 2F , 4F , 2G , 2H 2F , 4F , 2G , 2H 4F

2 D1 , 2 D2 , 2 F 2 D1 , 2 D2 , 2 F
2 4F 4F
4F , 2G , 2H 4F , 2G , 2H
ML
4P 2P , 4 P , 2 D1 , 2 D2 2P , 4 P , 2 D1 , 2 D2 4P
1 4F 2F 4F 2G 2H 2F 4F 2G 2H 4F
, , , , , ,
4P 2P , 4 P , 2 D1 , 2 D2 2P , 4 P , 2 D1 , 2 D2 4P
0 4F 2F 4F 2G 2H 2F 4F 2G 2H 4F
, , , , , ,
4P 2P , 4 P , 2 D1 , 2 D2 2P , 4 P , 2 D1 , 2 D2 4P
-1 4F 2F 4F 2G 2H 2F 4F 2G 2H 4F
, , , , , ,
2 D1 , 2 D2 , 2F 2 D1 , 2 D2 , 2F
-2 4F 4F
4F , 2G , 2H 4F , 2G , 2H

-3 4F 2F , 4F , 2G , 2H 2F , 4F , 2G , 2H 4F

-4 2G , 2H 2G , 2H

-5 2H 2H

With single matrix elements, we can calculate E2 H and E4 F . The former together
with the diagonal sum rule helps us to get E2 G . The three we have then helps us to
calculate E2 F . However, when we get to the terms present for ML = 2, MS = 1/2, we
have a secular problem of dimension six, but we have only managed to predetermine
four eigenvalues. To proceed, we will have to diagonalise the remaining matrix in
order to find E2 D1 and E2 D2 , but it is only quadratic. When that is done, we can
obtain the remaining two energies, E4 P and E2 P , with the sum rule.