Angular Momentum in the Helium Atom

We now use lower-case letters for the angular momentum quantities of a single electron and use capital letters for the angular momentum quantities of an atom. Let li and Si be the orbital and spin angular momenta of electron 1, let h and S2 be the orbital and spin angular momenta of electron 2, and so on. The total orbital angular momentum L and total spin angular momenta S of the electrons of the helium atom are vector sums of the contributions of the individual electrons  

The total electronic angular momentum of the atom is denoted by J  

There are c eratore for all of these quantities and their z components. The eigenvalues of the and 5 operators follow the same pattern as other angular 

In Russell-Saunders coupling, each energy level corresponds to a particular value of L and a particular value of S and is called a term. A Russell-Saunders term symbol is assigned to each term. The main part of the symbol is a letter giving the value of L, as follows  

After F the symbols are in alphabetical order. Notice the similarity with the subshell designations for a hydrogen-like atom. 

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The development of a full angular momentum, three dimensional, smooth exterior complex dilated, finite element method for computing bound and resonant states in a wide class of quantum systems is described. Applications to the antiprotonic helium system, doubly excited states in the helium atom and to a model of a molecular van der Waals complex are discussed. 2001 by Academic Press. 

In order to appreciate the size of the basis sets required for fully converged calculations, consider the interaction of the simplest radical, a molecule in a electronic state, with He. The helium atom, being structureless, does not contribute any angular momentum states to the coupled channel basis. If the molecule is treated as a rigid rotor and the hyperfine structure of the molecule is ignored, the uncoupled basis for the collision problem is comprised of the direct products NMf ) SMg) lnii), where N = is the quantum number... 

As described in Ref. [25], the Hartree approach has been applied to get energies and density probability distributions of Br2(X) 4He clusters. The lowest energies were obtained for the value A = 0 of the projection of the orbital angular momentum onto the molecular axis, and the symmetric /V-boson wavefunction, i.e. the Eg state in which all the He atoms occupy the same orbital (in contrast to the case of fermions). It stressed that both energetics and helium distributions on small clusters (N < 18) showed very good agreement with those obtained in exact DMC computations [24],... 

States of individual atoms are usually described by quantum numbers L, S, and for the electronic orbital, spin, and total angular momenta, respectively. However, in scattCTing and bound-state problems involving pairs of atoms or molecules it is common to use lower-case letters for quantum numbers of individual collision partners and reserve capital letters for quantities that refer to the collision system (or complex) as a whole. Thus, in this subsection we will use I and s for the quantum numbers of a single helium atom and reserve L and S for the end-over-end angular momentum of the atomic pair and the total spin, respectively. 

Table 7.1 The conveigence of the ground-state energy (in E, ) in the partial-wave expansion of the helium atom. is the Cl energy calculated in a basis of AOs of angular momentum I

Insufficient experimental data is available to demonstrate either the occurrence or lack of selection rules. In Table 15, some of the fastest reactions do correspond to transitions allowed for both atoms. Singlet helium transfers to Ne at a rate consistent with the data in Fig. 26, whereas triplet helium transfers comparatively slowly neither of the He transitions is allowed. However, there appear to be other cases, discussed earlier, where a forbidden transition is preferred to an allowed transition. Stepp and Anderson146 have suggested that there is partial conservation of electronic angular momentum accompanying energy transfer between atoms, and interpreted experiments on mercury fluorescence by means of the steps... 

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