The averaged energy of Eq. (2.1.3) should be further summed over all final and averaged over all initial molecular states. For molecular transitions the initial state i and the final state f are degenerate over the magnetic quantum numbers m i and m f of the angular momentum operators i J and f J of the active molecule, so that additional averaging should be made over these magnetic quantum numbers (and those associated with the virtual states of the perturbing molecule). This removal of degeneracy on the magnetic quantum numbers constitutes an intrinsic Collisional Line Broadening and Shifting of Atmospheric Gases Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 12/27/14. For personal use only. Semi-classical calculation of pressure-broadened line widths b l of its translational motion relative to the active molecule: b b l J l + = . The vectors |Jm〉 and |lm 2 〉 are eigenvectors of the Hamiltonians H a and H b respectively. Collisional Line Broadening and Shifting of Atmospheric Gases Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 12/27/14. For personal use only. l J K + = having the projection M on the space-fixed z-axis. In the vector coupling scheme (see, e.g., Ref. [7]), the vectors f i J J J − = and the vectors f i l l l − = are set to zero for all bath molecules (the vectors b l are separated and treated classically), so that f i J J K − = , where i J and f J are the angular momenta for the initial and the final states of the active molecule. The selection rules for the projection M depend on the operator X (l) : for the electric dipole transitions l = 1 and ∆M = 0, ± 1. To compute the matrix elements Λ ) m J KM m J m J KM m J m J KM m J ∞ ∞ = j J J j , (2.3.5) where J 2 are the rotational quantum numbers for the bath molecule 2 J ρ are the populations of the corresponding energy levels, f(v) is the Maxwell-Boltzmann velocity distribution (normalised to unity) and b is the impact parameter. The matrix element of Eq. (2.3.3) thus becomes [6] 2 2 2Jm lµ m J l µ n l middle J J J pl J W J J J c J J J c J J J c c i J J J c J J J c J J J c i ℏ (2.8.16) Collisional Line Broadening and Shifting of Atmospheric Gases Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 12/27/14. For personal use only. σ A π J J J J W εσ C C J J S c c c f i f i c J J J J f i