NIST: Atomic Spectros. - Eigenvector Composition of Levels

10. Eigenvector Composition of Levels

The wave functions of levels are often expressed as eigenvectors that are linear combinations of basis states in one of the standard coupling schemes. Thus, the wave function Ψ(α J) for a level labeled α J might be expressed in terms of normalized LS coupling basis states Φ(γSLJ):

$\Psi(\alpha J)=\sum_{\gamma\,S\,L} \|c(\gamma\,S\,L\,J)\,\Phi(\gamma\,S\,L\,J)\quad.$	(4)
The c(γSLJ) are expansion coefficients, and
$\sum_{\gamma\,S\,L} ~\|c(\gamma\,S\,L\,J)\|^2 = 1\quad.$	(5)

The squared expansion coefficients for the various γSL terms in the composition of the α J level are conveniently expressed as percentages, whose sum is 100 %. Thus the percentage contributed by the pure Russell-Saunders state γSLJ is equal to 100 × |c(γSLJ)|². The notation for Russell-Saunders basis states has been used only for concreteness; the eigenvectors may be expressed in any coupling scheme, and the coupling schemes may be different for different configurations included in a single calculation (with configuration interaction). "Intermediate coupling" conditions for a configuration are such that calculations in both LS and jj coupling yield some eigenvectors representing significant mixtures of basis states.

The largest percentage in the composition of a level is called the purity of the level in that coupling scheme. The coupling scheme (or combination of coupling schemes if more than one configuration is involved) that results in the largest average purity for all the levels in a calculation is usually best for naming the levels. With regard to any particular calculation, one does well to remember that, as with other calculated quantities, the resulting eigenvectors depend on a specific theoretical model and are subject to the inaccuracies of whatever approximations the model involves.

Theoretical calculations of experimental energy level structures have yielded many eigenvectors having significantly less than 50 % purity in any coupling scheme. Since many of the corresponding levels have nevertheless been assigned names by spectroscopists, some caution is advisable in the acceptance of level designations found in the literature.

11. Ground Levels and Ionization Energies for the Neutral Atoms

Table of Ground Levels and Ionization Energies for the Neutral Atoms
When using the above table, be sure to use your "Back" button to return to this document.

The ground-state electron configurations of elements heavier than neon are shortened in the table by using rare-gas element symbols in brackets to represent the corresponding electrons. The ground levels of all neutral atoms have reasonably meaningful LS-coupling names, the corresponding eigenvector percentages lying in the range from ~55 % to 100 %. These names are listed in the table, except for Pa, U, and Np; the lowest few ground-configuration levels of these atoms comprise better 5f ^N(L₁S₁J₁), 6d_j7s² (J₁j) terms than LS-coupling terms. The relatively large spin-orbit interaction of the 6p electrons produces jj-coupling structures for the (6p²_1/2)₀, (6p²_1/26p_3/2)^o_3/2, and (6p²_1/26p²_3/2)₂ ground levels of the 6p², 6p³, and 6p⁴ configurations of neutral Pb, Bi, and Po, respectively. As noted in the section jj Coupling of Equivalent Electrons, the jj-coupling names are more appropriate for these atoms than the alternative LS-coupling designations in the table.

The ionization energies in the table are from recent compilations [13, 14]. The uncertainties are mainly in the range from less than one to several units in the last decimal place, but a few of the values may be in error by 20 or more units in the final place; i.e., the error of some of the two place values could be greater than 0.2 eV. Although no more than four decimal places are given here, values for both the neutral and singly-ionized atoms are given to their full accuracies in [14].