3.4. Example: Hönl-London Intensity Expressions for a ¹Π - ¹Σ⁺ Transition

(3.10a)

(3.10b)

By application of the symmetry operation σ 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 ¹Π 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 Z component of the dipole moment operator, to determine intensities. The Z component of µ, rather than the X or Y component, is chosen for intensity calculations involving unpolarized light and molecules in the absence of external fields, because nonvanishing matrix elements of the former obey the selection rule ΔM = 0, whereas nonvanishing matrix elements of the latter two obey the selection rule ΔM = ±1. The selection rule ΔM = 0 gives rise to particularly simple summation expressions, e.g., (3.17), thus reducing the amount of algebra required to obtain the final intensity expressions, e.g., (3.20) and (3.21).

The quantity µ_Z, as given in (3.1), can be rewritten in the form

(3.12)

By using the selection rule ΔM = 0 for elements of the direction cosine matrix of the form α_Zs, the selection rule ΔΛ = ±1 for µ_x ± iµ_y, and the selection rule ΔΛ = 0 for µ_z,we see that a general matrix element of µ_Z takes the form

(3.13)

Application of the symmetry operation σ (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 ¹Π 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 ΔJ 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 2J + 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 ΔM = 0, the sum over upper state M values, i.e., the sum over that quantum number which would be M′, contributes nothing until M′ = M. We thus write

(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 ¹Π-¹Σ⁺ transition in a diatomic molecule [1] (pp. 204-211).