The shape resonance in electron-formaldehyde scattering: an investigation using the dilated electron propagator method
Introduction
Resonances are known to arise from a pronounced change in electron scattering cross-section due to a temporary trapping of the impinging electron in one of the unoccupied orbitals of the target. These anions are short lived and decay by ejecting the electron into the continuum in times of 10−12 to 10−14 s in the gas phase [1]. Electron scattering off formaldehyde is of particular interest as it is the simplest molecule containing a highly polar carbonyl group, and a low-energy shape resonance corresponding to a quasibound state of HCHO− has been observed in elastic electron-formaldehyde collisions 2, 3as well as in vibrational excitation [4]at collision energies near 1 eV.
Ab initio calculations of the HCHO− attributes are few 5, 6, 7with those using the complex Kohn variational method 6, 7providing the most reasonable description of this system. Due to the polarity of the carbonyl bond and the reorganisational effects accompanying the metastable electron attachment, strong correlation effects occur between the incident electron and the electrons of the target 6, 7and a need for a correlated treatment of the HCHO− attributes is highlighted by the unusually high value for the resonance energy calculated using the static exchange approximation [7]. Even after substantial incorporation of correlation effects, both the vibrational structure and the peak positions calculated using the complex Kohn variational method [7]differ from the experimental results, and the further investigation of this important prototypical resonance, involving a nonlinear polyatomic molecule, using other correlated techniques is of obvious importance.
The electron propagator theory 8, 9, 10, 11has emerged as a potent tool for the direct calculation of the electron attachment energy, and when coupled with complex scaling 12, 13of all electronic coordinates, the resulting dilated electron propagator method 14, 15, 16has emerged as an effective tool for the treatment of atomic and molecular shape resonances [17]. In this method, all the electronic coordinates in the Hamiltonian are scaled by a complex scale factor (η=αeiθ) and the orbital energies and amplitudes are the zeroth-order poles and Feynman Dyson Amplitudes (FDAs) of the bi-orthogonal dilated electron propagator 17, 18which can be systematically improved using the second (Σ2) and other higher order perturbative decouplings, or renormalised decouplings such as the diagonal two particle one hole-Tamm Dancoff Approximation (2ph-TDA) 8, 9, 10, 11, 19, to incorporate a greater extent of correlation and relaxation effects. The quasi-particle approximation 20, 21, 22offers a more economic, yet reasonably accurate, computational approach.
A detailed study of these decouplings in the treatment of molecular resonances has established their effectiveness [17]. However, the applications so far have been on diatomic or linear nonpolar polyatomic molecules. It is therefore useful to employ the dilated electron propagator technique for calculating the shape resonance attributes of this prototypical nonlinear polar polyatomic system. The resonant poles of the dilated electron propagator furnish fairly accurate values for the resonance energy and the corresponding FDAs provide an orbital picture of resonance formation and decay. A study of HCHO− could therefore provide insights of general significance in designing feasible decouplings for the larger polyatomics.
Toward this end, it is our purpose in this Letter to present the main results obtained from an application of the zeroth order (Σ0), the second order (Σ2), the diagonal 2ph-TDA (Σ2ph-TDA) and their quasi-particle variants the (Σq2) and the (Σ2ph-TDAq) decouplings of the dilated electron propagator to the investigation of the HCHO− shape resonance, employing fairly balanced and systematically augmented basis sets.
The bi-orthogonal dilated electron propagator and its implementation have been discussed in detail elsewhere [17]and in the following section, we only offer the main formulae of immediate interest. The resonant poles and FDAs of HCHO are discussed in Section 3and a brief summary of the main results concludes this Letter.
Section snippets
Method
As in the case of the real, undilated one electron propagator 8, 9, 10, 11, the bi-orthogonal dilated matrix electron propagator may also be expressed as [17]where is the zeroth order propagator for the uncorrelated motion, here chosen as given by the bi-variational self-consistent field (SCF) approximation 23, 24. The self energy matrix contains the relaxation and correlation effects [17].
Solution of the bi-variational SCF equations 23, 24for the N
Results and discussion
The Gaussian basis sets employed in our calculations utilize the 4s,2p contracted Gaussian type orbitals (CGTOs) for carbon and oxygen and 2s CGTOs for hydrogen from Dunning's compilation [25]. This basis was augmented with four additional p-type and one d-type function {α(p)=0.0382, 0.01232, 0.004107, 0.001325; α(d)=0.0653} on carbon, {α(p=0.0689, 0.0222, 0.00717, 0.000746; α(d)=1.211} on oxygen and one p-type function {α(p)=0.07} on hydrogen atoms to obtain the 66 CGTO basis (I) with 4s,6p,1d
Concluding remarks
Our investigation of the shape resonance in the electron-formaldehyde scattering using the dilated electron propagator method has led to confirmation of the >C=O orbital of formaldehyde as the resonant orbital. The unoccupied orbitals are extremely susceptible to basis set variations and the consistency in the resonance energies and widths, and also the orbital topology from different decouplings and bases, inspires confidence in the effectiveness of the dilated electron propagator
Acknowledgements
We are pleased to acknowledge financial support from the Department of Science and Technology, India (grant No. SP/S1/H26/96).
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