Phase-matrix formalism EMAP

After separation into resonant and background parts, the nonresonant fixed-nuclei phase matrix b0 is converted to a vibronic or rovibronic phase matrix by the energy-modified adiabatic phase-matrix method (EMAP) [409]. This is simply an adaptation of the EMA formalism to the phase matrix f o. The implied vibronic background phase matrix is 

The EMAP method has been used to compute elastic scattering and symmetric-stretch vibrational excitation cross sections for electron scattering by C02 [235], This is one of the first ab initio calculations of vibrational excitation for a polyatomic molecule. The results are in good agreement with experiment, which shows unusually large low-energy cross sections. The theory identifies a near-threshold 

As described above, time-delay analysis [389] of the energy derivative of the phase matrix 4 determines parametric functions that characterize the Breit-Wigner formula for the fixed-nuclei resonant / -matrix R[N(q e). The resonance energy eKS(q), the decay width y(q). and the channel-projection vector y(q) define R and its associated phase matrix such that tan = k(q)R , where 

Using the basic rationale of EMA theory [267], the parametric function e(q) becomes p q) = E—Hn when the kinetic energy of nuclear motion cannot be neglected. However, the operator (erflS(q) — op(q)) has a well-defined c-number value in vibrational eigenstates determined by the eigenvalue equation 

For comparison with projection-operator theory, this corresponds to a Born-Oppenheimer precursor resonance state 

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