2 Relativistic quantum defect orbital method

The Relativistic Quantum Defect Orbital (RQDO) method has been described in detail in previous papers (Karwowski & Martín 1991; Martín et al. 1993). Therefore, we shall only briefly summarise its most fundamental aspects.

 

Table 1: Line strengths for the $n{\rm s}\to n^{\prime } {\rm d}$ fine-structure and multiplet transitions

Transition	${\rm {}^2S_{1/2}-{}^2D_{3/2}}$	${\rm {}^2S_{1/2}-{}^2D_{5/2}}$	${\rm {}^2S-{}^2D}$
${\rm 3s \to 3d}$	0.761(-1)	0.114	0.191
	0.74(-1)	0.110	0.184
${\rm 3s \to 4d}$	0.521(-1)	0.778(-1)	0.130
	0.58(-1)	0.86(-1)	0.144
${\rm 3s \to 5d}$	0.690(-2)	0.104(-1)	0.173(-1)
	-	0.11(-1)	0.181(-1)
${\rm 3s \to 6d}$	0.210(-2)	0.315(-2)	0.525(-2)
	-	-	0.532(-2)
${\rm 3s \to 7d}$	0.905(-3)	0.138(-2)	0.229(-2)
	-	-	0.229(-2)
${\rm 3s \to 8d}$	0.489(-3)	0.736(-3)	0.123(-2)
	-	-	0.120(-2)
${\rm 4s \to 4d}$	0.114(+1)	0.171(+1)	0.285(+1)
	0.113(+1)	0.170(+1)	0.283(+1)
${\rm 4s \to 5d}$	0.355	0.530	0.885
	0.393	0.59	0.983
${\rm 4s \to 6d}$	0.471(-1)	0.706(-1)	0.118
	0.50(-1)	0.74(-1)	0.124
${\rm 4s \to 7d}$	0.141(-1)	0.213(-1)	0.354(-1)
	0.15(-1)	0.22(-1)	0.364(-1)
${\rm 4s \to 8d}$	0.614(-2)	0.923(-2)	0.154(-1)
	-	-	0.156(-1)

In this and the remaining tables, A(B) denotes $A \cdot 10^{(B)}$.
S-values are given in atomic units.
The first entry corresponds to RQDO S-values from this work.
The second entry corresponds to comparative values.
For the fine-structure lines: Critical compilation (Fuhr et al. 1988).
For the multiplets: Tull et al. (1972).

 

Table 2: Line strengths for the $n{\rm p}\to n^{\prime }{\rm p}$ fine-structure and multiplet transitions

Transition	${\rm {}^2P_{1/2}^o-{}^2P_{3/2}^o}$	${\rm {}^2P_{3/2}^o-{}^2P_{1/2}^o}$	${\rm {}^2P_{3/2}^o-{}^2P_{3/2}^o}$	${\rm {}^2P^o-{}^2P^o}$
${\rm 3p \to 4p}$	0.383(-1)	0.418(-1)	0.404(-1)	0.120
	0.39(-1)	0.39(-1)	0.39(-1)	0.118
${\rm 3p \to 5p}$	0.355(-2)	0.364(-2)	0.363(-2)	0.108(-1)
	-	-	-	0.105(+1)
${\rm 3p \to 6p}$	0.101(-2)	0.101(-2)	0.102(-2)	0.304(-2)
	-	-	-	0.293(-2)
${\rm 4p \to 5p}$	0.370	0.403	0.390	0.116(+1)
	0.387	0.387	0.387	0.116(+1)
${\rm 4p \to 6p}$	0.300(-1)	0.306(-1)	0.306(-1)	0.912(-1)
	0.30(-1)	0.30(-1)	0.30(-1)	0.899(-1)
${\rm 5p \to 6p}$	0.211(+1)	0.231(+1)	0.222(+1)	0.663(+1)
	0.221(+1)	0.221(+1)	0.221(+1)	0.663(+1)

See footnotes in Table 1.

The relativistic quantum defect orbitals are determined by solving analytically the quasi-relativistic scalar second-order Dirac-like equation, obtained after decoupling the radial, two-component Dirac equation, through a non-unitary transformation.

$$% \left[- \frac{\rm d^2}{{\rm d}r^2}\, + \frac{\Lambda(\Lambd... ...ht] {\psi_k}^{\rm RD} = 2 {\rm e}^{\rm RD} {\psi_k}^{\rm RD},$$ (1)

with

$$% \Lambda = \eta - n + l - \delta' + c,$$ (2)

$$% \ Z'_{\rm net} = Z_{\rm net} ( 1 + {\alpha}^2 E^x ),$$ (3)

$$% {\rm e}^{\rm RD} = - \frac {(Z'_{\rm net})^2 } { 2 ( \eta -... ...c {( 1 + {\alpha}^2 E^x / 2 )}{( 1 + {\alpha}^2 E^x )^2 }\cdot$$ (4)

$\eta$ is the relativistic principal quantum number, n and l are the principal and orbital angular momentum quantum numbers, $\delta'$ is the relativistic quantum defect, c is an integer chosen to ensure the normalizability of the wavefunction and its correct nodal structure; $Z'_{\rm net}$ is the scaled nuclear charge acting on the valence electrons at large radial distances; E^x is the experimentally measured energy, and $\alpha$ is the fine structure constant. Atomic units are used throughout. Since the effective Hamiltonian in Eq. (1) includes a screening term, the quantum defect orbitals are approximately valid in the core region of space. Core polarization effects are implicitly included in the calculations given that they are accounted for in the ${\Lambda}$ parameter of the model Hamiltonian. On the other hand, given the one-electron nature of the RQDO formalism, it can be expected to perform better in highly excited states, where the active electron interacts less with the core and other valence electrons, than in low-lying energy states. The relativistic quantum defect orbitals lead to closed-form analytical expresions for the transition integrals. This allows us to calculate transition probabilities and oscillator strengths by simple algebra and with a high effectiveness/cost ratio.

Our methodology supplies one-electron radial wavefunctions, characterized by the n, l and j quantum numbers (Martín & Karwowski 1991), that we employ in the transition matrix elements for the initial and final states of the active electron. These correspond to levels of a given L, S and J symmetry in many-electron atoms. We take care of the presence of the remaining electrons by including the appropriate angular factors in the line strengths. Thus, the electric quadrupole line strength for a transition between two states, is given by the equation

$$% S(nlj,n'l'j') = \frac { 2J+1 } { 2S+1}\, {(2J'+1)}\, < L \parallel C^{(2)}\parallel L' >^2$$

$$\times \vert< R_{nlj}\vert Q(r)\vert R_{n'l'j'}>\vert^2$$ (5)

where Q(r) is the quadrupole transition operator, r², and $< L \parallel C^{(2)} \parallel L' >$ is the pertinent reduced matrix element.

The relationships between the line strength S (in atomic units, $e^2 a_{\rm o}^4$), the oscillator strength f (dimensionless), and the transition probability A (in ${\rm s}^{-1}$) are given by

$$% g'A = (8\pi ^2 \hbar \alpha / m \lambda ^2) \,gf = (6.6703 \, 10^{15}/ \lambda ^2) \,gf$$ (6)

$$% g'A = (32 \pi ^5 \alpha c a_{\rm o}^4 / 15 {\lambda}^5) \, S_{\rm E2} = (1.11995\, 10^{18}/ \lambda ^5) \, S_{\rm E2}$$ (7)

where $\lambda$ is the transition wavelength (in Å), and $g^{\prime}$ and g are the degeneracies of the upper and lower states, respectively.

 

Table 3: Line strengths for the $n{\rm p}\to n^{\prime } {\rm f}$ fine-structure and multiplet transitions

Transition	${\rm {}^2P_{1/2}^o-{}^2F_{5/2}^o}$	${\rm {}^2P_{3/2}^o-{}^2F_{5/2}^o}$	${\rm {}^2P_{3/2}^o-{}^2F_{7/2}^o}$	${\rm {}^2P^o-{}^2F^o}$
${\rm 3p \to 4f}$	0.207	0.608(-1)	0.365	0.633
	0.201	0.57(-1)	0.345	0.603
${\rm 3p \to 5f}$	0.684(-2)	0.186(-2)	0.112(-1)	0.199(-1)
	-	-	0.12(-1)	0.208(-1)
${\rm 3p \to 6f}$	0.717(-3)	0.180(-3)	0.109(-2)	0.199(-2)
	-	-	-	0.233(-2)
${\rm 3p \to 7f}$	0.121(-3)	0.271(-4)	0.165(-3)	0.313(-3)
	-	-	-	0.430(-3)
${\rm 3p \to 8f}$	0.245(-4)	0.445(-5)	0.256(-4)	0.545(-4)
	-	-	-	0.102(-3)
${\rm 4p \to 4f}$	0.104(+1)	0.298	0.179(+1)	0.313(+1)
	0.105(+1)	0.300	0.180(+1)	0.315(+1)
${\rm 4p \to 5f}$	0.141(+1)	0.416	0.250(+1)	0.432(+1)
	0.138(+1)	0.394	0.237(+1)	0.414(+1)
${\rm 4p \to 6f}$	0.982(-1)	0.277(-1)	0.167	0.293
	0.95(-1)	0.27(-1)	0.163	0.285
${\rm 4p \to 7f}$	0.195(-1)	0.537(-2)	0.323(-1)	0.572(-1)
	0.19(-1)	-	0.32(-1)	0.564(-1)
${\rm 4p \to 8f}$	0.629(-2)	0.169(-2)	0.101(-1)	0.181(-1)
	-	-	0.10(-1)	0.182(-1)

 

Table 3: continued

Transition	${\rm {}^2P_{1/2}^o-{}^2F_{5/2}^o}$	${\rm {}^2P_{3/2}^o-{}^2F_{5/2}^o}$	${\rm {}^2P_{3/2}-{}^2F_{7/2}}$	${\rm {}^2P^o-{}^2F^o}$
${\rm 5p \to 5f}$	0.100(+2)	0.286(+1)	0.172(+2)	0.301(+2)
	0.100(+2)	0.287(+1)	0.172(+2)	0.301(+2)
${\rm 5p \to 6f}$	0.602(+1)	0.179(+1)	0.107(+2)	0.185(+2)
	0.60(+1)	0.170(+1)	0.102(+2)	0.179(+2)
${\rm 5p \to 7f}$	0.509	0.145	0.873	0.153(+1)
	0.493	0.141	0.85	0.148(+1)
${\rm 5p \to 8f}$	0.115	0.322(-1)	0.192	0.339
	0.110	0.32(-1)	0.189	0.331
${\rm 6p \to 6f}$	0.528(+2)	0.151(+2)	0.908(+2)	0.159(+3)
	0.53(+2)	0.15(+2)	0.91(+2)	0.159(+3)
${\rm 6p \to 7f}$	0.198(+2)	0.593(+1)	0.356(+2)	0.613(+2)
	0.199(+2)	0.57(+1)	0.341(+2)	0.597(+2)
${\rm 6p \to 8f}$	0.181(+1)	0.520	0.312(+1)	0.545(+1)
	0.176(+1)	0.50	0.301(+1)	0.527(+1)

See footnotes in Table 1.

The total line strength of the multiplet is equal to the sum of the line strengths of all the multiplet lines

$$% S({\gamma} L, {\gamma}' L') = {\sum} \, S({\gamma} J, {\gamma}' J').$$ (8)