The calculation of rotational energy levels for a ^{3}Σ state
[11] proceeds in much the same fashion as for
^{2}Π states. The basis set functions for the complete problem are
represented by . Since we are dealing with a
^{3}Σ state, Λ = 0 and *S* = 1.
Since *S* = 1, Σ takes on the values + 1, 0, - 1.
(footnote 6)
Note that, even though the spin projection quantum number Σ will not be a
good quantum number in the final ^{3}Σ wave functions, it is
perfectly acceptable, and indeed, from the point of view of the author,
desirable, to use a basis set in which Σ is a good quantum number. Basis
set functions for the problem thus have one of the following forms:
| 0 1 1; 1 *J M* ⟩,
| 0 1 0; 0 *J M* ⟩, or
| 0 1 - 1; - 1 *J M* ⟩.
For *J* = 0 only the second function exists, of course.

The matrix elements of the Hamiltonian in this basis set can again be
represented by (1.16). Since nonvanishing
matrix elements of are again
diagonal in *J* and *M*, the matrix used to determine rotational
energy levels for a ^{3}Σ
electronic state *S* = 1,

Consider now the matrix elements of . If we employ the same reasoning used in obtaining the
energies given in (1.17), we find that all
three components of the nonrotating-molecule ^{3}Σ state lie at
the same energy. (There is no first order spin-orbit interaction
*A*ΛΣ in Σ electronic states because
Λ = 0.) Nevertheless, states with different values of Ω
do have different energies [12], except that pairs of
states related to each other by a change in sign of both Λ and Σ
are degenerate. From these considerations, we find that the matrix of
is given
by

(1.24) |

The two states with Ω = ± 1 have been given the energy
*E* and the state with Ω = 0 has been given the energy
*E* - 2λ.^{2}Π problem, there is only one relevant energy separation.
That separation is represented here by the quantity 2λ. (Experimental
values for λ in the ^{3}Σ ground state of several
molecules are given in table VI of [12]).

Matrix elements of the rotational Hamiltonian ^{2}Π state. The matrix of
for *J* ≥ 1 is
thus

(1.25) |

where we have again used arguments (see chapter 2) to conclude that the matrix elements
of *B*(*L*^{2} - *L*_{z}^{2})
in the basis set under consideration are all equal.

Unfortunately the sum of (1.24) and
(1.25) does not immediately factor into any smaller
diagonal blocks, so we must apparently find the roots of a cubic secular
equation. However, there is one simplifying procedure which has not yet been
employed. For reasons of symmetry, the size of the secular determinant for
rotational energy levels can often be halved by working with the basis set
functions ^{-½}[ ±*S* -Σ; -Ω *J M* ⟩]

(1.26) |

Let us consider the matrices of and in a new basis set consisting of
^{-½}[ | 0 1 1; 1 *J M* ⟩ +*J M* ⟩],*J M* ⟩, and
^{-½}[ | 0 1 1; 1 *J M* ⟩ -*J M* ⟩].

(1.27) |

The sum of the matrices (1.24) and (1.27), i.e.,

*B*, which actually occurs in (1.25). Strictly
speaking, for example, there should be three different values for *B* in
(1.25). Because the internuclear distance will be slightly different for the
states with Ω = ± 1 than it is for the state with
Ω = 0, one value for *B* occurs in the matrix positions
(1, 1) and (3, 3), one in the position (2, 2), and one in the
off-diagonal positions (1, 2), (2, 1), (2, 3), and (3, 2).
(The fact that differences in internuclear distance lead to three and only
three values for *B* in (1.25) can be shown from
symmetry considerations, as indicated in
chapter 2.) In addition to the internuclear
distance effect, all matrix elements of operators involving **S**, which
were evaluated numerically from (1.13),
using *S* = 1 and Σ = + 1,
*S* and Σ
are not quite good quantum numbers. Since these matrix elements were evaluated
numerically, they are somewhat difficult to find in (1.25), but the
2^{½} occuring in the off-diagonal matrix elements, for example,
is not *exactly* 2^{½}, because *S* and Σ, are
not exactly good quantum numbers. Finally, has one value for the
state with Ω = 0, and a slightly different value for the states
with Ω = ± 1. Unfortunately, the extent of the various
deviations mentioned above can only be quantitatively determined at the present
time from experiment. For this reason it is common to introduce, in some way,
additional parameters in (1.25), which are to be determined by fitting the
experimental data. Since *B* is the only parameter occurring in (1.25),
and since each matrix element contains *B,* it is possible to allow for
all of the above-mentioned problems in a purely formal way by replacing the
single parameter *B* by a set of *B*_{eff}'s. It can be shown
by symmetry arguments (see chapter 2) that a
maximum of seven *B*_{eff}'s could be used: three of which would
occur in the positions (1, 1) and (3, 3); three of which would occur
in the position (2, 2); and one of which would occur in the off-diagonal
positions. [A single *B*_{eff} in the off-diagonal positions of
(1.25) corresponds to a single *B*_{eff} in the off-diagonal
positions of (1.27).]

We shall use in our calculations one value for *B* on the diagonal in
(1.25) or (1.27), and one value for
*B*, written as *B* - ½γ,^{-1},
say), then the difference in equilibrium internuclear distance for the two
multiplet components of a ^{3}Σ
state is expected to be very small, and only one coefficient for
*J*(*J* + 1) is needed on the diagonal of
(1.25). On the other hand, contamination of the
^{3}Σ state by other electronic
states through spin-orbit interaction will cause the matrix elements of the
components of **S** to deviate perceptibly from those given in
(1.13). On the diagonal, such deviations, as
well as the differences in , can be allowed for
by adjusting the parameter λ. Off the diagonal, such deviations can be
allowed for by adjusting the parameter *B*.

In any case, the secular equation obtained from the sum of
(1.24) and (1.27) modified as
described above leads to the following energies for the three states of given
*J* ≥ 1:

(1.28) |

For *J* = 0 the energy is
*E* - 2λ + *B J*(*J* + l) + 2*B* +
.

(1.29) |

(The first approximate equality in (1.29) indicates the loss of terms in
γ^{2}. The second approximate equality indicates a power series
expansion of the radical. The third approximate equality represents the
case (b) approximation
*B J* >> | λ |
for the small γ-dependent term.) Dropping the first two
(*J*-independent) terms in (1.28) and making the
substitutions *J* = *N*,
*J* = *N* - 1,*J* = *N* + 1 in the three energy expressions
given in (1.28) and in (1.29), we obtain Schlapp's
expressions

(1.30) |

apart from the small *J*-independent term + ½γ.
(*N* is assigned to the three levels of given *J* such that the level
with *N* = *J* + 1 has the highest energy and the
level with *N* = *J* - 1

It is interesting to note that the parameter γ, which was introduced here
to allow for some slight discrepancies in matrix elements of the spin
operators, is generally introduced as a coupling parameter between the vectors
* N* and