Joseph C. Y. Chen

THEORY OF SUBEXCITATION ELECTRON SCATTERING BY MOLECULES. II. EXCITATION AND DE-EXCITATION OF MOLECULAR VIBRATION

The proposed mechanism for the excitation and de‐excitation of molecular vibration by slow electron impact involving compound negative‐ion states as intermediates is studied in some detail in light of the theory presented in Paper I of this series. The absolute orders of magnitude of the cross sections for such collisional processes are calculated for the (N2, e) system, where detailed measurements have recently been made for the excitation of the ground N2 molecules to various excited vibrational states. The observed multipeak structures for the excitation functions are accounted for. The energy dependence of the cross sections for the excitation and de‐excitation of a vibrationally excited N2 molecule exhibits similar multipeak structures in the same energy region. An interpretation of such multipeak structures is offered.

Off—Diagonal Hypervirial Theorem and Its Applications

A hypervirial operator, by definition, is a time‐independent linear operator with an arbitrary functional structure expressed in terms of the dynamical variables of the system under consideration. In the energy representation, the diagonal matrix elements of the hypervirial operator are constant in time; this is known as the hypervirial theorem. The off‐diagonal matrix elements of the hypervirial operator in the energy representation oscillate in time with frequencies related to the energy differences between their corresponding stationary states of the system. This is known as the off‐diagonal hypervirial theorem.The latter theorem is as powerful as the former for dealing with many quantum‐mechanical problems. Several aspects of the possible applications of the theorem to the problem of construction and improvement of wavefunctions are considered. As a special example, the restricted fourth alternative expression for dipole transition matrix elements is derived for pair functions not belonging to states ...

Theory of Subexcitation Electron Scattering by Molecules. I. Formalism and the Compound Negative‐Ion States

The problem of slow‐electron scattering by molecules is treated formally with explicit consideration of the possible effects of the compound negative‐ion states. The formulation is based on the treatments for nuclear reactions and scattering, in particular the unified treatment recently advanced by Feshbach. The transition matrix for both direct and resonance scattering, describing the excitation of the nuclear motion of the target molecule, is obtained. The influence of the compound negative‐ion states on the equivalent scattering potential for resonance scattering is investigated in some detail.

Unitary Operator Formalism of the Eigenvalue Problems

Utilization of the parameters involved in the system under consideration leads to a unitary operator formalism for dealing with eigenvalue problems. The existence of a unitary linear operator which transfers a known function, being predetermined for a special choice of the parameter, to the general eigenfunction is demonstrated and its properties are discussed. The related hypervirial theorems for variational Hellmann—Feynman wavefunction are discussed. The scaling problem and virial theorem are treated in some detail by means of the unitary operator techniques.

Differentiated Schrodinger Equation

It is shown that the generalized Hellmann-Feynman (H-F) theorem and formulas for evaluating an infinite sum can be obtained from a unified matrix equation derived from the differentiated Schrodinger equation. Some consequences and applications of the matrix equation are also discussed.

Translational factors in eikonal approximation and their effect on channel couplings

The coupled equations of the adiabatic-state expansion formulated in the context of the stationary-state scattering theory are examined. The translational factors are introduced in the eikonal approximation through coordinate transformation. The equations are consistent with those of Chen and Watson (1968) in that they are free from the spurious long-range interaction even in the absence of the translational factors. The cancellation of the spurious R-1 long-range interaction of the 2p sigma u-2p pi u rotational coupling in the (p+,H) system is given as an example. The effect of the translational factor on the channel coupling interaction is discussed.

Study of Approximate Wavefunctions Using Weighting Operators

The method of weighting operators is developed for the purpose of studying approximate wavefunctions. The physical significance of the weighting operators is to weight the wavefunction in different regions in the configuration space. Approximate wavefunctions which are made stable with respect to a set of weighting operators are accurate for the calculation of expectation values of operators other than the energy, to the first order in the double perturbation theory. Differential properties of eigenvalue with respect to parameters are investigated using weighting‐operator techniques. It is suggested that the extra sensitivity of the derived expression with respect to weighting operators can be used as a convergence test for variational wavefunctions. A comparative study of several energy‐optimized wavefunctions by the method of weighting operators is carried out for the ground states of the helium isoelectronic sequence. Their relative regional goodness is discussed.

On a Set of Coupled Second‐Order Differential Equations

An alternative method for solving a set of coupled second‐order differential equations, which often appears in theoretical treatments of many‐body problems, is proposed. This method makes use of both mathematical relations derived in matrix theory and physical properties of the potentials provided by the set of equations to be solved.

General Relation of Fundamental Molecular Constants between Two Electronic States

The general relation of fundamental molecular constants between electronic states is investigated in some detail. Formulas stating such relations are found, and the interaction matrix elements, which relate the corresponding electronic states, are examined. Calculations of the quadratic, cubic, and quartic force constants for the hydrogen molecule ion H2+ are carried out, as an example, by utilizing these formulas. The calculated values agree very satisfactorily with those obtained from spectroscopic data.

Differential cross sections in the p+ - H collisions

Elastic, electron-transfer and (2p,)-excitation differential cross sections for the p +-H collisions are calculated in the lso,-2p.~,-2pa, three-state expansion. The results are in satisfactory agreement with the newly obtained experimental data. The spurious long range nonadiabatic interaction found in previous calculations does not appear in our calculation.