THE ENERGY LEVELS OF A CONFIGURATION

CHAPTER 3
THE ENERGY LEVELS OF A CONFIGURATION
THE number of independent wave functions of a given electron con-
figuration is its number of states. For a partly filled n, /-shell, con-
taining q electrons, the number of states is
These states may be assembled in energy levels, according to different
sorts of coupling schemes. The number of states in a given level is
called its degeneracy number e. In spherical symmetry, the energy
levels are characterized by the total angular momentum quantum
number J, having e = 2J + 1. If we can neglect the electrodynamic
and relativistic effects, the energy levels J will be assembled in
multiplet terms, each characterized by the spin quantum number S
and the orbital angular momentum quantum number L, producing a
degeneracy number e = (2S + 1)(2L + 1). If L and S are well-
defined "good" quantum numbers, we have Russell-Saunders
coupling. The main reason for the energy differences between the
multiplet terms is the difference in the potential energy of inter-
electronic repulsion, arising from the different distribution of the
electrons of the partly filled shell on the angular functions A. In the
Slater description of atomic energy levels, as given in Condon and
Shortley's Theory of Atomic Spectra, the different terms of a given
configuration are assumed, as a first approximation, to have the
same radial functions of the orbitals corresponding to the same
attraction by the central field and the same orbital energies (eqn. 5),
but to have energy differences dependent on interelectronic repulsion
parameters representing the mutual repulsions between the electrons
in the partly filled shell. We shall discuss the phenomenon below in
the general terminology of perturbation theory.
It is rather complicated to calculate the number of multiplet
terms belonging to a given configuration and their values of L and S.
If the configuration contains more than one partly filled shell, the
(11)
30

THE ENERGY LEVELS OF A CONFIGURATION 31
number of states will be the product of the binomial expressions (11)
for each partly filled shell, and the L- and .S-values are produced by
unrestricted vector sum coupling of the L- and S-values of each shell.
However, we shall not treat this problem in general, but rather con-
centrate our attention on the case of one partly filled shell. The
multiplet terms will be designated by names of the L-values analo-
gous to eqn. (1):
L-0123456789 10... (12)
SPDFGHIKLMN
and the "multiplicity" 25 + 1 is indicated as a superscript at the
left-hand side of the symbol. The single state of a completely filled
or empty shell corresponds to one term X S having S = 0 and L = 0.
The 4/ + 2 states of one electron in a shell corresponds to one term
2
S,
2
P, . . . with S = \ and L = /. In any type of coupling, the
hole-equivalency principle of Pauli holds that q "holes" (e.g. 41+2 —
q electrons in a shell) and q electrons produce the same number and
species of terms. Thus, both 2 electrons in a partly filled shell
(p2 , d2 , . . .) and 41 electrons (p4 , d8 , f12 , . . .) have their (21 +
l)(4l + 1) states (eqn. 11) distributed on the series of multiplet
terms
!S,
3
P, *D, 3 F, 1 G, 3 H, . . . (L < 21) (13)
terminating at the highest L-value possible by simple addition of the
two equal /-values. According to a proposal by Mulliken, we shall use
capital letters for the quantum numbers of systems and small letters
for the quantum numbers of orbitals.
If at least three electrons occur in a partly filled shell, more than
one multiplet term may have the same combination of L and S.
This is not effective for p3 (having the terms 4 S, 2 D and 2 P), but d3
has two terms 2 D in addition to the terms 4 F, 4 P, 2 P, 2 F, 2 G and 2 H.
The more complicated d
n
- and fw -systems have a long series of
terms. If we use the words singlet (S = 0), doublet (J), triplet (1),
quartet (f ), quintet (2), and sextet (S = f ), d4 has one quintet term
Ö
D, seven triplet and eight singlet terms, and d 5 has one sextet term
6
S, four quartet terms 4 G, 4 P, 4 D and 4 F, and eleven doublet terms.
According to Hund's rule, the lowest term in a given configuration
has the maximum value of S, and the highest value of L compatible

32 THE ENERGY LEVELS OF A CONFIGURATION
with this condition. Hund's rule does not consider the order of the
other values of L and S, which may be distributed on the energy scale
in a rather irregular way.
The orbitals with room for two electrons of opposite spin direction
are often symbolized by boxes containing arrows in upward or
downward direction. However, this representation often leads to
incorrect results when applied uncritically, e.g. to the number of
energy levels. It can be safely used for two types of argument only :
the maximum value of S can always be found as half the number of
upward arrows minus half the number of downward arrows, but
the distribution of the other S values cannot be simply recognized.
Further, the number of states is given correctly by applying the per-
mutation theory to the boxes. However, a fundamental difficulty can
be seen in a simple example: one electron in each of two orbitals
corresponds obviously to four states. Of these, the two states with
parallel spins belong to a triplet state with S = 1, but it cannot be
concluded that the two states with "compensated" spins both
belong to the singlet state with S = 0. Actually, they are mixed in
such a way that a triplet and a singlet state results.
Most text-books describe perturbation theory in a number of steps
which leads to some confusion in terminology. These steps are:
(1) "Zero-order" perturbation, where a set of "degenerate"
wave functions Ψ Ά , Ψκ, . . . ¥n having the same energy before the
perturbation are used for forming linear combinations Ψ 1 = ka^a +
. . . + kn^n appropriate for step (2) (securing vanishing non-
diagonal elements between two such different linear combinations
under the perturbation).
(2) First-order perturbation of the energy, where the diagonal
elements E
n
= ^Ψ^Ψχ of each wave function is calculated, changing
the energy from its original value Ex to Ex + En (the latter quantity
is often referred to as the diagonal element).
(3) First-order perturbation of the wave function, where the non-
diagonal elements E12 = J^iP^ between two wave functions is
calculated, changing the wave function from the original Ψ χ to
n>
multiplied by a normalization factor.

THE ENERGY LEVELS OF A CONFIGURATION 33
(4) Second-order perturbation of the energy, where the non-
diagonal elements E in change the energy by the last summation term
in
Ε = E„ + E„„ - y E?un p- (14)
/ { JE*m ~r I-'m m — LTL — ϋηη
In the writer's opinion, it is somewhat artificial to separate out the
behaviour of energy and wave functions, and it seems important to
discuss two concepts, the steps (1) + (2) called here, simply, first-
order perturbations and the steps (3) + (4) second-order per-
turbations.
The first-order perturbations are characterized by considering the
same wave function before and after the perturbation Ρ in the in-
tegral (we use here an integral sign without variables to indicate
an integration over all pertinent variables in their whole domain;
other authors use the brackets <¥Ί|Ρ| Ψ{> proposed by Dirac).
We are here considering the fact that Ρ might contain opera-
tors, affecting the kinetic energy. The total energy of a normalized
wave function is J^H^ where the Hamiltonian function Η partly
produces kinetic energy, necessitating in its definition the indication
of a wave function before Η and one after Η ; and partly potential
energy, which is simply the integration of the product of with a
potential. In nearly all cases met in practice, Ρ is only a potential.
Thus the diagonal elements of energy are simply the integrals of
P
1
/7 !2 . This is the change of energy expected in the classical theory of
electricity by putting an extended charge distribution Ψ-f down in a
potential Ρ (calculated for electrons, not for positive charges).
On the other hand, the second-order perturbations concern two
wave functions Ψ1 and Ψ2 . If Ρ is a pure potential, the non-diagonal
elements will be the integrals of Ρ^Ί^· If Ρ was a constant, this
integral would be zero, since $ΨχΨ
2
of two orthogonal wave functions
vanish. Thus, the non-diagonal elements are produced by differing
values of Ρ at the places in (the multi-dimensional) space, where
ΨΧ Ψ2 is positive and where it is negative. It is one of the principal
purposes of group theory to indicate which non-diagonal elements
must necessarily be zero; this occurs in most cases when the quan-
tum numbers Γη assigned to each of the wave functions Ψ 1 and Ψ2
are different, as we shall discuss later.
Generally, the second-order perturbations change a given wave

34 THE ENERGY LEVELS OF A CONFIGURATION
function Ψ1 by intermixing it with small parts of other wave functions
Ψ2 , ... with which it has non-diagonal elements. The influence on
the wave function Ψχ is inversely proportional to the distance on the
energy scale of the other wave functions, while the influence
on the energy is proportional to the squares of the non-diagonal
elements. On the other hand, only the energy and not the wave
function itself is changed by a first-order perturbation. Thus, there is
a close connexion between the chemical concepts of covalent bonding
and second-order perturbations; and between electrovalent bonding
and first-order perturbations. We shall justify the general use of
this notation in the chapters on molecular orbitals and ligand field
theory.
The perturbation theory is closely connected with the theory of
secular determinants. If q different wave functions fulfil the necessary
conditions for intermixing (having the same Γη , which, for example,
may signify the same combination of L and S in spherical symmetry),
the energies of the resulting mixed functions will be the q roots Ε
(the eigenvalues) of the polynomial of qth degree, written as a
symmetrical determinant:
Ί1 - Ε
E
12
Ei3 . Εια
E12
E2 2 — Ε
E23 E2q
Ei3 E2
3 E33 Ε .
E3q -0 (15)
Eiq E
2
q Eeq
Eqq — Ε
explaining the terms "diagonal elements" for Enn and "non-diagonal
elements" for E
n
m- The diagonal sum rule for such determinants states
that the sum of the q eigenvalues always equals the sum of the q
diagonal elements.
If one of the diagonal elements, E n n, is separated from all the other
diagonal elements by an energy which is considerably greater than
the order of magnitude of the non-diagonal elements, it will be a valid
approximation that one of the roots Ε has the approximate value
Enn + y p-^r- (16)
En
representing a sort of additive, repulsive potential, depressing Ε
when Emm is larger than Enn , and increasing Ε in the opposite case.
This is analogous to eqn. (14).