Winifred M. Huo

Resonant energy transfer cross sections formulated in terms of form factors: An appraisal of the first Born approximation

In the Born approximation without exchange the differential cross section of energy transfer between two molecules is expressible in terms of the generalized oscillator strengths (f) for the excitation of individual molecules by electron impact. Thus, the Born cross section can be unambiguously calculated using electron impact data. In the absence of such data, an approximate expression of f can still be obtained from experimental values of transition moments, ionization potentials, and excitation energies based on the first term of Lassettre’s series representation of f. It is shown, in the case of dipole–dipole energy transfer, that the cross section calculated with the approximate f corresponds to scattering from a shielded dipole–dipole intermolecular potential, with the correct asymptotic behavior at both large and small r. Hence, it appears that even when approximate f’s are used, the present method is superior to the dipole–dipole theory of Cross and Gordon. The validity of the Born approximation a...

Limiting Form of the Effective Potential for Electron Scattering

In a previous article [J. Chem. Phys. 56, 3468 (1972)], an effective potential Ṽ f0 for direct scattering is defined such that its Fourier transform, multiplied by −(2π)−1, gives the scattering amplitude. Here we study the behavior of Vf0 at the limit q→ 0, with q the distance between the incident electron and the scattering center. It is found that the limiting form of Ṽ f0 is expressible by an ascending series, the first term being a q−1 term. The coefficient of the constant term is also derived. The applicability of the Born approximation under a number of limiting conditions is considered by means of Ṽ f0. For inelastic scattering, it is found that at a fixed incident energy, deviations from the Born approximation will be observed if the momentum transfer is increased. At a fixed scattering angle, deviations from the Born approximation will also be observed if the energy is increased. However, at a fixed momentum transfer, the Born limit will be approached with increasing energy. A number of calculati...

High‐energy approximation for exchange scattering and application to 1 1S → 2 3S excitation of He by electron impact

A high‐energy approximation is derived for the second order term in the T matrix for exchange scattering of electrons by atoms or molecules. A limiting selection rule for exchange scattering at high incident energies is obtained, which states that second‐order contributions are important in forward scattering if the initial and final states of the target system have the same orbital term symbol. Calculations on the zero angle differential cross section, approximated by the cross section at zero momentum change, of the 1 1S→2 3S excitation in He by electron impact are carried out over an energy range 100–500 eV using the present approximation. Reasonable agreement with the experimental data of Skerbele, Harshbarger, and Lassettre [J. Chem. Phys. 58, 4285 (1973)] is obtained, indicating that second‐order contributions are sufficient to account for the differences between Born‐Oppenheimer calculations and the experimental data. The present calculation also shows that, for this transition in He, a minimum exi...

Negative ion contamination of electron impact spectra

Since two particles with the same charges and the same kinetic energies are transmitted equally by electrostatic velocity analyzers regardless of mass, it follows that negative ions produced on electron impact with a gas can interfere with the electron impact spectrum of the gas. Possible contribution of negative ions to the electron impact spectrum of water is studied by theoretical means. It is concluded that H− ions can contribute peaks to the spectrum in the apparent energy loss range 4–6 eV if the spectrum is produced by electrons with kinetic energy in the range 5–7 eV. With incident kinetic energies of 15 eV or higher negative ion contamination is not important. In general, negative ion interference can be important, especially for hydrogen containing compounds, when (a) the kinetic energy of incidence is very low and (b) when the electron spectrometer employed contains only electrostatic lenses and analyzers. These considerations suggest that negative ion spectra might be studied as a means of detecting excited quantum states of radical fragments especially for hydrogen containing substances which give H− on collision with electrons.

Effective Long Range Potential for the Scattering of Electrons by Atoms and Molecules

An effective potential Ṽf0 is defined such that the direct scattering amplitude is expressed by a Borntype integral. It is found that Ṽf0 consists of three types of potentials characterized by their q dependence, where q is the distance between the incident electron and the scattering center. The long range potential, Ṽf0LR, is important at large q. It is expressed as an inverse power series and expressions up to the q−6 term are obtained. The first non‐Born term in Ṽf0LR is just the dipole polarization potential. The next term is proportional to q−5 and complex. The oscillatory potential, Ṽf0OS, is also expressed as an inverse power series but multiplied by eiaq. The leading term in Ṽf0OS is proportional to q−4. Because of the presence the oscillatory factor, Ṽf0OS will decrease faster than Ṽf0LR at large q. At small q, the short range potential Ṽf0SR becomes important. It is characterized by an exponential dependence on q. The symmetry properties of Ṽf0LR and Ṽf0OS are studied and compared with the sele...

A study of the Bonham series representation of the Born–Oppenheimer exchange amplitude and the derivation of a local exchange potential

The convergence and range of applicability of the Bonham series representation of the Born–Oppenheimer exchange amplitude is investigated. Numerical calculations on 1 1S→2 3S and 1 1S→2 3P of He by electron impact demonstrate that the first three terms of the Bonham series can provide an adequate representation of the Born–Oppenheimer amplitude from high energies down to near threshold. The three‐term Bonham series is then used to represent the Hartree–Fock exchange kernel in momentum space, which has been shown by Lassettre to be proportional to the Born–Oppenheimer amplitude. Inverse Fourier transform, plus an additional approximation of replacing the momentum which appears as an expansion parameter in the Bonham series by its averge value, gives us a local exchange potential. If the averge momentum in the Thomas–Fermi model is used, the first term of the local exchange potential is just the electron gas exchange potential. A second term corresponding to a correction for the inhomogeneity in the electro...

A time‐dependent variation–perturbation method for the calculation of transition properties and its relation to the random phase approximation

A time−dependent variation−perturbation (VP) formulation is presented for the simultaneous calculation of the ground state correlation function and the excited state wave function of an atomic or molecular system. The wave functions are chosen for the optimal calculation of transition properties with a relatively small number of molecular integrals. The VP equations so obtained for a singlet system are identical with the equations in the random phase approximation (RPA). On the other hand, if the ground state is a singlet and the excited state a triplet, the VP and RPA equations are different. The latter assumes a ground state correlation function with an open shell component, which is a poor approximation for a closed shell ground state. The orthonormalization condition in the VP scheme is different from that in the RPA. The consequence of this difference is discussed. It is also pointed out that the RPA excitation energy actually contains part of the ground state correlation energy; hence the RPA excita...

Effect of the reverse reaction on the experimental determination of energy transfer rate constants

The effect of the reverse reaction in the experimental determination of energy transfer rate constants is re‐emphasized. It is shown that the pressure independence of the measured rate constant, frequently used experimentally as an indicator of the absence of reverse reactions, may have more complicated sources like the presence of a third channel of dissipating the excited species. Thus, pressure independence alone is not a reliable indicator of the absence of reverse reactions. The quenching of Ar 3P1 by H2 is used as an illustrative example with the energy transfer rate constant and the rate constant of the reverse reaction estimated using the Born approximation [W. M. Huo, J. Chem. Phys. (preceding paper)].

On the equivalence of semiclassical and Born total cross sections. Penning ionization of H(12S) by He(21S)

The equivalence of total cross sections calculated in the Born approximation and the semiclassical straight‐line trajectory method is shown to hold for a potential of arbitrary symmetry if a set of conditions is satisfied. This is used to support a previous argument [W. M. Huo, J. Chem. Phys. 66, 3572 (1977)] on the applicability of the Born approximation to near resonant energy transfer at thermal energies. As an example, the Penning ionization of H(12S) by He(21S) is calculated in the Born approximation using electron impact data.