(3.10a) |

(3.10b) |

By application of the symmetry operation σ_{υ} as described in
chapter 2, we find the parities of the
functions (3.10a) and (3.l0b) to be

(3.11a) |

(3.11b) |

respectively, if we arbitrarily assume a value of *L* = 1 for
the ^{1}Π state, and arbitrarily
assume correlation with a united atom state of odd parity [see
eqs (2.11)].

Because we are interested in intensity formulas which are valid for unpolarized
light and for molecules in the gas phase in the absence of external fields,
all three directions in space are equivalent, and it is sufficient to calculate
the matrix elements of *µ _{Z}*, the laboratory-fixed

The quantity *µ _{Z}*, as given in
(3.1), can be rewritten in the form

(3.12) |

By using the selection rule Δ*M* = 0 for elements of
the direction cosine matrix of the form α_{Zs}, the
selection rule ΔΛ = ±1 for
*µ _{x}* ±

(3.13) |

Application of the symmetry operation
σ_{υ} (see
sect. 2.9) indicates that the right-hand side
of (3.13) is equal to zero if the upper sign is used for
*J*′ = *J*, or if the lower sign is used for
*J*′ = *J* ± 1. If the opposite sign choice
is made in each case, the right-hand side of (3.13) is equal to

(3.14) |

Note that we again assume *L* = 1 for the ^{1}Π
state, and take the lower sign choice in
(2.11a), in agreement with the choice made
in obtaining (3.11b). The fact that the matrix element
given in (3.13) vanishes for certain sign choices and for certain
Δ*J* is consistent with the parities given in
(3.11) and with the parity selection rule
for electric dipole transitions.

The matrix element (3.14) can be further simplified by recalling that the elements of the direction cosine matrix do not contain the variables of the nonrotating-molecule problem, whereas the molecule-fixed components of the dipole moment operator do not contain the rotational variables. Since the complete basis set functions are products of a function containing only the variables of the nonrotating-molecule problem and a function containing only the rotational variables, we can write (3.14) as

(3.15) |

The second factor in (3.15) represents a matrix element of the type given in
table 6 above. The first factor
represents a matrix element which cannot be calculated from symmetry
considerations alone. Hence we shall treat it as a parameter, which is to be
determined by fitting the calculated intensity expressions to the experimental
data. For simplicity we define a quantity *µ*_{⊥}

(3.16) |

where *µ*_{perp;} is, of course, independent of the rotational
quantum numbers. In addition, we choose the phase factors for the two wave
functions |1 0 0⟩ and |0^{+} 0 0⟩
such that *µ*_{⊥} is real and positive. (Such a choice is
possible at this point, since we have not yet considered the phase of any
matrix element connecting these two states.)

Spectral line intensities are actually proportional to the square of the dipole
moment matrix elements, i.e., to the square of the quantity first given in
(3.13) and later rewritten in
(3.15). Furthermore, we are considering molecules in the
absence of external fields, so that the 2*J* + 1 states having
the same *J* but different *M* are all degenerate. Thus, the total
intensity *I* is obtained by summing over all *M* values for the
upper state and over all *M* values for the lower state under
consideration. Since nonvanishing matrix elements of *µ _{Z}* obey
the selection rule Δ

(3.17) |

Consider now the intensity of an *R* branch
(*J*′ = *J* + 1). We find from
table 6 that (3.17) becomes

(3.18) |

where Ω has been given its value of zero. Using the summation expressions

(3.19) |

we obtain

(3.20) |

In a similar fashion we can obtain

(3.21) |

The relative intensities given in (3.20) and (3.21) agree with the well-known
Hönl-London expressions for a
^{1}Π-^{1}Σ^{+} transition in a diatomic
molecule [1] (pp. 204-211).