Quantum mechanical close coupling approach to molecular collisions: Averaged definite parity j z approximation with Clebsch–Gordan weights

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Abstract
The new averaged definite parity jz(ADPjz) approximation is considered in detail from the standpoint of the exact body frame CC cross section equations. These cross section equations are decomposed into contributions from the various possible orbital angular momenta and projections of the total angular momentum J onto the scattering vector. This decomposition then permits a clearer examination of the quantities actually approximated in jz-conserving approximations. The various contributions to the exact cross sections are regrouped into components having the same form as the jzCCS and DPjz approximate equations. It is found that the labeling of jzCCS amplitude densities by an orbital angular momentum quantum number enters in a very natural way and in fact that only labeling by l̄ is permitted for consistency. Further, it is shown that only the amplitude density is approximated and this does not require the body frame T-matrix to be diagonal in jz. After the approximate amplitude density is substituted into the exact partial cross section expression, the resulting equations for the partial cross sections σJ(j0→j) lead to an average over the effective orbital angular momenta with weights wJlλ(j ‖j0) =〈l0jλ‖Jλ〉2[(2l+1)/(2J+1)]. A similar analysis of the definite parity partial cross section leads to an average over the effective orbital angular momenta of the proper parity with weights wJplλ(j ‖j0) =〈l0jλ‖Jλ〉2[(2l+1)/(2J+1)][2/(1+δλ,0)]. This enables one to include effects of the various turning points associated with values of l satisfying ‖J−j‖⩽l ⩽J+j. The present method of including parity through the values of l averaged also can correctly describe situations where the odd parity partial cross section is larger than the even parity partial cross section. The result is an approximation which describes definite parity partial cross sections, integral cross sections, and differential cross sections with very high accuracy. Finally, the ADPjz is a general method for which other averaging weights are possible. The previously obtained constant weight version is compared to the present Clebsch–Gordan weight version by applying both to He+H2 collisions. Results for definite parity partial cross sections and integral cross sections are used as the basis of the comparison.