The first example involves obtaining rotational energy expressions for a
^{2}Π state of a diatomic molecule
[10], which is well separated from other electronic
states, and in which the spin-orbit splitting is small compared to such
electronic separations. The rotational energy levels of such an electronic
state correspond to Hund's case (a), Hund's case (b), or some
intermediate case between these two.

The wave functions of the nonrotating molecule are represented by
,
where the quantum numbers Λ, *S*, Σ are all slightly bad. The
quantum number Ω = Λ + Σ, however, is
still good. The basis set functions for the complete (nonrotating plus
rotating) problem are represented by . Since we
are dealing with a ^{2}Π state,
Λ = ± 1 and
*S* = ½. Since *S* = ½, Σ
takes on the values ± ½. The quantum number *M* does not enter
into the calculation of rotational energy levels for molecules unperturbed by
the presence of external electric or magnetic fields. Hence, it will be ignored
(even though the quantum number *M* will be carried along in the
notation). The quantum number *J* will *not* be assigned a numerical
value, since we are interested in the energy levels as a function of *J*.
Basis set functions of interest for this problem thus have one of the following
four forms:
*J M* ⟩,*J M* ⟩,*J M* ⟩,*J M* ⟩.

We must now set up the matrix of the Hamiltonian in the basis set given above. Let us represent a typical matrix element by the expression

(1.16) |

The complete Hamiltonian for a molecule in the absence of external fields is
always diagonal in the quantum numbers *J* and *M*. Thus, for the
problem under consideration, the matrix element given on the left of (1.16)
vanishes unless *J*′ = *J* and
*M*′ = *M*.

It can easily be seen, that if we restrict ourselves to matrix elements
diagonal in *J* and *M*, and to the limited basis set appropriate for
the rotational energy levels of a ^{2}Π electronic state

Consider first the matrix elements of . The quantum numbers of the basis
set have been divided by a semicolon into two groups, reflecting the
factorization of (1.12). Because of this
factorization, matrix elements of must be diagonal in the three rotational quantum
Ω*J M*. Values for matrix elements of diagonal in these three quantum
numbers can be obtained from
eqs (1.7). If we label rows and columns
of a 4 × 4 matrix by the four functions given just before
eq (1.16), then we find that the matrix of
has the
form

(1.17) |

*E* represents the constant given in the third line of
(1.7), and ± ½ A represents
the spin splitting. We observe that only two distinct values for the energy
occur, namely *E* + ½ *A* and
*E* - ½ *A*. This is consistent with the
general phenomenon that energy levels in the nonrotating molecule characterized
by Ω ≠ 0 are doubly degenerate, the two degenerate functions
having equal and opposite values of Ω (in this case
Ω = ± and
Ω = ± ½). To be perfectly general we should allow
the energy pattern to vary somewhat from that determined on the basis of
simple spin-orbit interaction. However, in this case there is really only one
relevant energy parameter in the nonrotating molecule, namely the distance
between the Ω = ± *A*.

Matrix elements of the rotational Hamiltonian
(1.11) in a basis set consisting of the four
functions labeling the matrix (1.17) can be obtained as
follows. The operators *B*(*J*^{2} -
*J _{z}*

We now write down the matrix of analogous to the matrix of given in
(1.17). When
*J* ≥ it has the form

(1.18) |

The desired rotational energy levels are found by solving the secular equation
obtained from the sum of the matrices for and . It can be seen that the sum of

(1.19) |

(The algebraic operations necessary in obtaining (1.19) are simplified if
one-half the trace is subtracted from each *all* the basis set functions for a
given value of *J*, the two (doubly degenerate) energy levels given in
(1.19) represent all the energy levels belonging to that value of *J*. The
first two (*J*-independent) terms in (1.19) are often ignored in
discussions of ^{2}Π rotational energy levels.

When *J* = ½, the basis set functions labeling the first
and third rows and columns of (1.17) and
(1.18) do not exist. For this *J* value the
Hamiltonian matrix factors into two identical 1 × 1 matrices, giving
rise to a doubly degenerate energy level at *E* - ½
*A* + + *B*[*J*(*J* + 1) + ].

Limiting Hund's case (a) and case (b) behavior can be obtained by
expanding the square roots in (1.19) as power series.
Consider first case (a) behavior, which occurs, from a mathematical point
of view, when *A* | >> *B J*.*A* | >> *B J*,

(1.20) |

The approximation (1.20) to the radical must now be substituted into
(1.19). Since we must both add and subtract (1.20), it is
convenient at this time to replace *A* |*A*. The
same two energy levels will be obtained by adding and subtracting
*A* |*A* (though not
necessarily in the same order). Keeping only terms through order
*B*/*A*, and dropping the first two terms in (1.19), we obtain the
two energy levels

(1.21a) | |

(1.21b) |

which, apart from an extra + ½ *B* arising from
,
agree with the familiar [1] (p. 220) Hund's
case (a) energy level expression
*B*[*J*(*J* + l) - Ω^{2}].*B*^{2}/*A*^{2}
in (1.20), we see that the coefficient of
*J*(*J* + 1)*B* value, must be replaced by
*B*(1 + *B*/*A*)*B*(1 -
*B*/*A*)

Hund's case (b) occurs, from a mathematical point of view, when
*B J* >> | *A* |.*A* = 0. If *A* = 0, the last two terms of
eqs (1.19) can be written

(1.22a) | |

(1.22b) |

Both of these equations are of the form
*B*[*N*(*N* + 1) - 1].*N*, which is equal, to *J* + ½
for the higher energy level of given *J*, and equal to
*J* - ½ for the lower energy level. Equations (1.22)
then have the form of the familiar [1]
(pp. 221-224) Hund's case (b) expression
*B*[*N*(*N* + 1) - Λ^{2}].

The significance of the quantum number *N*, which has arisen here in a
somewhat formal way, can best be understood by examining the eigenfunctions of
the matrix sum (1.17) + (1.18)
when *A* = 0, i.e., the case (b) eigenfunctions. The two
eigenfunctions obtained by diagonalizing the 2 × submatrix in the
upper lefthand corner of (1.17) +
(1.18) when *A* = 0 are given on the
right-hand side of (1.23) as linear combinations of the
basis set functions used to label the rows and columns
of (1.17) and (1.18).

(1.23) |

It can be seen that the two basis set functions in a given linear combination
are characterized by the same values of Λ, *S*, *J*, and
*M*, but by different values of Σ and Ω. For this reason we
say that Λ, *S*, *J*, and *M* are good quantum numbers in
Hund's case (b), but that Σ and Ω are not.

Although the linear combinations of functions given on the right-hand side of
(1.23) are not eigenfunctions of *S _{z}*
and

It is sometimes convenient to label the eigenfunctions represented by the linear combinations on the right-hand side of (1.23) by a single label of the form . These labels are found on the lefthand side of (1.23). The absence of a semicolon indicates that the functions cannot be factored into a nonrotating-molecule part and a rotating-molecule part.

Some authors also find it convenient to use functions of the form as
basis set functions for diatomic molecule calculations. In this monograph we
have chosen always to consider the nonrotating-molecule part of the problem
separately from the rotating-molecule part. This decision requires us to use a
basis set in which Ω and Σ, or perhaps just Ω, are defined,
and in which *N* is not defined. Wave functions in which *N* is
defined will therefore appear to arise here somewhat arbitrarily as linear
combinations of the basis set functions, the linear combinations being obtained
by diagonalizing appropriate Hamiltonian matrices.