Complex coordinate rotation of the electron propagator

Abstract

It is now widely appreciated that the real poles of the electron propagator G(E) yield information on the ionization potentials and electron affinities of the stationary states of an atom or molecule. It is herein shown that application of the Aguilar-Balslev-Combes-Simon coordinate transformation, r-r exp(i8), to G(E) yields an analytically continued complex propagator G(Z, 8) whose complex poles correspond to the complex electron affinities associated with nonstationary, resonance states of an atomic or molecular anion. As an initial application of the coordinate rotation technique we derive and discuss the working equations for a coordinate rotated propagator which is correct to second order in the electron-electron interaction. This is followed by use of the formalism in a model study of a 2p shape resonance in the Be atom. Our second-order results for this system are then compared to those obtained by previous authors employing static exchange, and static-exchange plus cutoff polarization methods.

Characterization of molecular shape resonances using different decouplings of the dilated electron propagator with application to 2Π CO− and 2B2g C2H4− shape resonances

“…The results obtained by us will be seen to be close to those of Donnelly who has compared these values to the energies and widths obtained by other methods. The underlying zeroth order decoupling being real, the DonnellySimons approach 19 is not suitable for isolation of resonances at the level of SCF itself and the correlation and relaxation effects in resonance formation therefore cannot be apportioned conveniently. A comparative investigation of the different decouplings allows a probe of the role of correlation and relaxation and an examination of the chemical content of the resonant FDAs is useful for eliciting mechanistic clues.…”

Application of higher order decouplings of the dilated electron propagator to Π2 CO−, Πg2 N2− and Πg2 C2H2− shape resonances

“…[3][4][5][6][7][8] The dilated 9-11 ͑complex scaled͒ electron propagator method, [12][13][14] where all the electronic coordinates have been scaled by a complex scale factor (ϭ␣e i ) has been quite effective in describing electron attachment shape and electron detachment Auger resonances. 15 In these calculations, the resonances are located by plotting the poles of the dilated electron propagator as a function of ␣ and ͑␣ and trajectories͒ and those showing a semblance of invariance with respect to variations in are associated with resonances.…”

Higher order decouplings of the dilated electron propagator with applications to P Be−2, P Mg−2 shape and S Be+2 (1s−1) Auger resonances

“…It is this exact parallelism of the biorthogonal dilated electron propagator decouplings with those of the undilated real electron propagator decouplings that separates it from other approaches to the construction of the complex scaled electron propagator 9,10 where even the second order decoupling involves more complicated formulation. Higher order and renormalized decouplings from these alternative approaches 9,10 may be still more complicated and have not been attempted.…”

“…12 The calculation of resonance energy and width from the resonant poles of the dilated electron propagator is demanding since these are identified by constructing the dilated electron propagator for a large number of values ͑ϳ5-10 ␣ and ϳ30 values per ␣ being typically representative͒ and associating the poles invariant to changes in with resonances. [7][8][9][10][11] The treatment of resonances using the dilated electron propagator is therefore ϳ150 times more arduous compared to an equivalent real electron propagator calculation of ionization potential and electron affinity. For this reason, although the use of the third order decoupling has become routine in calculation of ionization potentials/electron affinities, the treatment of resonances using the dilated electron propagator method has been limited to the use of second or pseudo-second order ͑diagonal 2ph-TDA͒ decouplings and their quasi-particle variants.…”

“…The dilated 7,8 electron propagator [9][10][11] where all electronic coordinates in the Hamiltonian have been scaled by a complex factor ϭ␣e i has been successfully employed to elicit the energy and width of electron scattering shape and Auger resonances. 12 The calculation of resonance energy and width from the resonant poles of the dilated electron propagator is demanding since these are identified by constructing the dilated electron propagator for a large number of values ͑ϳ5-10 ␣ and ϳ30 values per ␣ being typically representative͒ and associating the poles invariant to changes in with resonances.…”