Since the two nonrotating-molecule wave functions of a
^{1}Π state are characterized by
Ω = ±1, while the
nonrotating-molecule wave function of a
^{1}Σ^{+} state is characterized by
Ω = 0^{+}, there can
be no interaction between these states in the nonrotating molecule. A
*homogeneous* perturbation of one of these states by the other is not
possible.

However, the rotating-molecule operator of eq (1.10) has
in general nonvanishing matrix elements between states with
ΔΛ = ± 1 and Δ*S* = 0.
Consequently, a *heterogeneous* perturbation of one of these states by the
other is possible. This heterogeneous perturbation can be treated theoretically
as follows.

Consider a 3 × 3
Hamiltonian matrix with rows and columns labeled by the three interacting wave
functions =
|1 0 0; 1 *J M*⟩,
**|** -1 0 0; -1 *J M*⟩
and **|** 0^{+} 0 0; 0 *J M*⟩.
We find, as outlined below, that this matrix has the form

(4.1) |

where the upper left-hand 2 × 2 diagonal block represents the
^{1}Π state by itself (arbitrarily placed at the energy origin in
the nonrotating molecule), where the lower right-hand 1 × 1 diagonal
block represents the ^{1}Σ^{+} state by itself (located
at energy *E*_{Σ} in the nonrotating molecule), and where
the off-diagonal elements represent the heterogeneous perturbation (Coriolis
interaction) between the two states. Matrix elements for the
2 × 2 and 1 × 1 diagonal blocks were obtained as described
in chapter 1. Off-diagonal matrix elements
were obtained as follows.

The first line of terms in when
is written as in
eq (1.11), contains operators which
give rise to nonvanishing matrix elements only if ΔΩ = 0.
Consequently, the first line of (1.11) cannot be responsible for an interaction
between the ^{1}Π and the ^{1}Σ^{+} states.
The first and third terms of the second line of (1.11) give rise to
nonvanishing matrix elements only if ΔΣ = ± 1.
Consequently, these two terms can also not be responsible for the interaction
under consideration here. The second term of the second line of (1.11) gives
rise to nonvanishing matrix elements if
ΔΛ = ± 1 and
Δ*S* = ΔΣ = 0. This term thus does
connect the ^{1}Π and ^{1}Σ^{+} states under
consideration.

The second term of the second line of (1.11) contains four operators:
*J*_{+}, *J*_{-},
*L*_{+}, *L*_{-}. Matrix elements of
*J*_{+} and *J*_{-} can be obtained from expressions
of the form (1.13) and
(1.14), since both *J* and Ω are
good quantum numbers in the basis set. Matrix elements of *L*_{+}
and *L*_{-} cannot be found from such expressions, since *L*
is not a good quantum number. However, matrix elements of
*L*_{±} still obey the selection rule
ΔΛ = ± 1, because Λ is a good quantum
number. It can be shown by using symmetry arguments involving
σ_{υ} (see
chap. 2), that

(4.2) |

where the + or - sign on the right-hand side of (4.2) allows for various
possible sign choices and values of *L* in applying the transformation
equation (2.11a) to the
^{1}Π state. Neither sign choice in (4.2) leads to inconsistencies
elsewhere in the calculation, since no matrix elements between the two
components of the ^{1}Π state are introduced. The relative phases
of the electronic orbital wave functions
|Λ = +1⟩ and |Λ = -1⟩ are
fixed once and only once by the choice of sign in (4.2). The + sign was chosen
in writing (4.1), where the quantity
*B*⟨Π| *L*_{+} |Σ⟩, given by

(4.3) |

is considered to be an unknown adjustable (real) parameter. Energy levels can be obtained by diagonalizing (4.1).

Actually, (4.1) can be factored by symmetry into a
2 × 2 diagonal block and a 1 × 1 diagonal block, by using
the functions 2^{-1/2}[ | +1 0 0; +1 *J M*⟩ ±
| -1 0 0; -1 *J M*⟩]
as a basis set for the ^{1}Π state, but this factorization will not
be performed here.

Van Vleck [25] has introduced the phrase "pure
precession" in connection with heterogeneous perturbations. From a quantum
mechanical point of view, pure precession is said to occur whenever *L* is
a good quantum number. If *L* is a good quantum number, it is possible to
use (1.13) to evaluate the quantity
⟨Π| *L*_{+} |Σ⟩ occurring in
(4.1). Since this quantity is really a matrix element of
the form ⟨Λ = + 1|
*L*_{+} |Λ = 0⟩ we find

(4.4) |

The value [*L*(*L* + 1)]^{1/2} occurs when the
Σ and Π states belong to the same *L* complex, i.e., when the
Σ and Π states represent different projections of the same *L*
along the internuclear axis. The value 0 occurs when the Σ and Π
states belong to different *L* complexes.

It is interesting to note that the matrix (4.1) describes
several apparently rather different situations.
If the ^{1}Σ^{+} state and the ^{1}Π state are
very nearly degenerate in the nonrotating molecule
(*E*_{Σ} ≅ 0), and if *L* is an
approximately good quantum number equal to unity (*L* = 1), then
the rotational energy levels which result from diagonalizing (4.1) correspond
to those of a case (d) *p*-complex. (Pure case (d) occurs, from
a quantum mechanical point of view, when *L* is a perfectly good quantum
number, and when all 2*L* + 1 states corresponding to different
projections of this *L* along the internuclear axis are exactly degenerate
in the nonrotating molecule.) If the ^{1}Σ^{+} state and
the ^{1}Π state are separated by a distance large compared to the
distance between neighboring rotational levels
(|*E*_{Σ}| >> *B J*), then the
rotational energy levels which result from diagonalizing (4.1) correspond to
those of a normal ^{1}Σ^{+} state and those of a normal
^{1}Π state with some Λ-doubling. If the rotational energy
levels of the ^{1}Σ^{+} state cross those of the
^{1}Π state for some value *J _{c}* of the rotational
quantum number [i.e., if