Theory of two shells of atomic electrons using non‐orthogonal radial orbitals

Abstract

AbstractsA theory for handling non-orthogonal radial orbitals of two shells of atomic electrons based on the mathematical apparatus of irreducible tensor operators is presented. The general expressions for one-and two-electron operator matrix elements are given.On prtsente une thtorie pour traiter deux couches d'klectrons atomiques en employant des orbitales radiales non-orthogonales, qui est baste sur l'appareil mathtmatique des opkrateurs tensoriels irrtductibles. Les expressions gtntrales pour les tlkments de matrice des optrateurs a un et deux electrons sont donntes.Es wird eine Theorie fur die Behandlung zwei Elektronenschalen eines Atoms mit Anwendung von nicht-orthogonalen Radialorbitalen gegeben, die auf dem mathematischen Formalismus der irreduziblen Tensoren gegrundet ist. Die allgemeinen Ausdrucke fur die Matrixelemente der Ein und Zweielektronen-Operatoren sind angefuhrt.

Coupled tensorial form for atomic relativistic two-particle operator given in second quantization representation

“…It is convinient to explain such transformation by examining the schema ( Fig. 1) which arises when applying a graphical method of angular momentum theory [21]. In schematic form we can write…”

“…The irreducible tensorial product (the diagram A 2 ) composed of creation and annihilation operators was produced by using Jucys theorems of graphical angular momentum theory [21]. To obtain the desired form of the irreducible product, we carried out several recouplings of angular momenta and made several changes of positions of creation and annihilation operators in (1).…”

Extended method of calculation for two shells of atomic electrons having the same orbital quantum number

“…In equation (50) of [3], after calculating the two-electron reduced matrix elements and summing in respect of the two-electron quantum numbers L and S, one obtains the {A'}'s as was mentioned in [ 3 ] . In going over to the extended method of calculation these {A'}'s must be replaced by {A"}'s just as for oneelectron operator matrix elements.…”

“…-4[( 1s' I 2s') (Is" I 2s') + (Is' 1 2s") (1s" I 2s")] (Is' I Is") -4[(ls' \2s')(ls" 12s") + (1s' \2s")(ls" \2s')](ls' I~s " ) (~s ' 12s") -4[( 1s' I 2s') (1s' I 2s") + (Is" 1 2s') (Is" 1 2s")] (2s' I 2s") + 4[(ls'I 2~' )~( 1 s " 1 2~" )~ + 2(ls'I 2s')(ls' I2s")(ls" 12s')(ls" 12s") + (1s' I2s")yl.C" I2s')2] Both normalization integrals in (37) and (50) of [3] are equal to each other, because the configuration ls22s2 allows only one term IS.…”

“…In this respect this paper represents the generalization of methods given in [2] and at the same time the extension of the methods of [3], since the ideas of [2] are generalized for solving the more complicated problem of [3].…”