The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods

Abstract

The paper is concerned with the practical determination of the characteristic values and functions of the wave equation of Schrodinger for a non-Coulomb central field, for which the potential is given as a function of the distance r from the nucleus.The method used is to integrate a modification of the equation outwards from initial conditions corresponding to a solution finite at r = 0, and inwards from initial conditions corresponding to a solution zero at r = ∞, with a trial value of the parameter (the energy) whose characteristic values are to be determined; the values of this parameter for which the two solutions fit at some convenient intermediate radius are the characteristic values required, and the solutions which so fit are the characteristic functions (§§ 2, 10).Modifications of the wave equation suitable for numerical work in different parts of the range of r are given (§§ 2, 3, 5), also exact equations for the variation of a solution with a variation in the potential or of the trial value of the energy (§ 4); the use of these variation equations in preference to a complete new integration of the equation for every trial change of field or of the energy parameter avoids a great deal of numerical work.For the range of r where the deviation from a Coulomb field is inappreciable, recurrence relations between different solutions of the wave equations which are zero at r = ∞, and correspond to terms with different values of the effective and subsidiary quantum numbers, are given and can be used to avoid carrying out the integration in each particular case (§§ 6, 7).Formulae for the calculation of first order perturbations due to the relativity variation of mass and to the spinning electron are given (§ 8).The method used for integrating the equations numerically is outlined (§ 9).

Quasi-variational coupled cluster theory

“…(32). The additional factor of 2 on the contribution adds B + 2 C instead of 1 2 B + C, correcting the −C acquired by LPFD(+1) at O(T 3 ) to B + C. The corrective transformations not only continue to vanish for two electrons, but also mutually cancel for two holes, so both the 2-electron and 2-hole limiting cases remain correct. The dressing of the corrective transformations with q U matrices ensures that the size of the transformed amplitudes remains comparable to that of the untransformed amplitudes if the latter grow large.…”

Symmetry-improved CJT effective action

“…In this section we apply our symmetry-improved approach to the HF approximation [22][23][24] of the CJT effective action of a O(2) scalar model. We show that the predicted Goldstone boson is massless and the phase-transition is second order already in this approximation.…”

“…Also, a rigorous renormalization-group analysis [21] supports this result. In the first non-trivial truncation of the 2PI effective action, the Hartree-Fock (HF) approximation [22][23][24], one explicitly finds [10][11][12] that the Goldstone boson is massive and the phase transition is first order. Only in the large-N limit of the HF approximation, a consistent prediction is obtained [11,12], where the Goldstone boson is massless and the phase transition is second order.…”

Dissociation dynamics of diatomic molecules in intense laser fields: A scheme for the selection of relevant adiabatic potential curves

“…In BO approximation accurately evaluating the correlation between electrons V ee remains a challenge for numerical calculations. To overcome this difficulty, the so-called Hartree-Fock (HF) SCF method is used [16,17] in order to include the effect of V ee approximately by considering each electron subject to an effective potential that is derived from the averaged charge distribution of all other electrons.…”