In atoms, the outermost electrons are weakly bound to the nucleus as compared to the atomic core electrons. Lithium and other atoms have a single electron in their outermost shell; this electron is the weakest bound electron. Zheng et al. developed a theory, known as Weakest Bound Electron Potential Model (WBEPM), for the spectroscopic characteristics of such atoms [16]. The weakest bound electron model theory allows us to deal with many-electron systems as a binary system; the weakest bound electron is considered an electron in a field created by a nucleus dressed up with an atomic core (consists of other electrons surrounding the nucleus). The weakest electron is excited to a higher energy level moves in an orbit that has a larger period; this reduces the coupling between core electrons and the weakest bound electron. Unlike Self consistent field (SCF) method in which each electron moves in average potential arising from other electrons, the WBEPM is based on the consideration of successive dynamic ionization [16]. The potential in which the weakest bound (WB) electron moves is given by
$$\begin{array}{*{20}c} {V\left( {r_{i} } \right) = \frac{A}{{r_{i} }} + \frac{B}{{r_{i}^{2} }}} \\ \end{array}$$
(1)
The radial part of the Schrodinger equation with this potential is as follows
$$\begin{array}{*{20}c} {\frac{{{\text{d}}^{2} R}}{{{\text{d}}r^{2} }} + \frac{2}{r}\frac{{{\text{d}}R}}{{{\text{d}}r}} + 2\left( {E - \frac{A}{{r_{i} }} - \frac{B}{{r_{i}^{2} }} - \frac{{l\left( {l + 1} \right)}}{{r_{i}^{2} }}} \right) = 0} \\ \end{array}$$
(2)
The last two terms have the same denominator to be combined. Here \(A = - Z^{*}\) where \(Z^{*}\) is the effective charge of the nucleus,
$$\begin{array}{*{20}c} {\frac{{{\text{d}}^{2} R}}{{{\text{d}}r^{2} }} + \frac{2}{r}\frac{{{\text{d}}R}}{{{\text{d}}r}} + 2\left( {E + \frac{{Z^{*} }}{{r_{i} }} - \frac{{l^{*} \left( {l^{*} + 1} \right)}}{{r_{i}^{2} }}} \right) = 0} \\ \end{array}$$
(3)
here\({l}^{*}=l-{\delta }_{n} , {n}^{*}=n-{\delta }_{n}\), are effective orbital quantum number and principal quantum number, respectively, \({\delta }_{n}\) is the quantum defect in the quantum numbers and is given as a function of n.
$$\begin{array}{*{20}c} {\delta_{n} = a + \frac{b}{{\left( {n - \delta_{o} } \right)^{2} }} + \frac{c}{{\left( {n - \delta_{o} } \right)^{4} }} + \frac{d}{{\left( {n - \delta_{o} } \right)^{6} }}} \\ \end{array}$$
(4)
\(\delta_{o}\) is the lowest quantum defect, & a, b, c, and d are found by fitting the first few energies of Rydberg levels, the energy (E) of the weakest bound electron is given by
$$\begin{array}{*{20}c} {E = - \frac{{Z^{*2} }}{{2n^{*2} }} = - \frac{{Z^{*2} }}{{2\left( {n - \delta_{n} } \right)^{2} }}} \\ \end{array}$$
(5)
The solution of Eq. (3) gives the radial wavefunction given by
$$\begin{array}{*{20}c} \begin{aligned} R & = \left( {\frac{{2Z^{*} }}{{n^{*} }}} \right)^{{l^{*} + \frac{3}{2}}} \sqrt {\frac{{\left( {n^{*} - l^{*} - 1} \right)!}}{{2n^{*} \Gamma \left( {n^{*} + l^{*} + 1} \right)}}} \\ & \quad \times \exp \left( { - \frac{{Z^{*} r}}{{n^{*} }}} \right)r^{{{ }l^{*} }} L_{{n^{*} - l^{*} - 1}}^{{2l^{*} + 1}} \left( {\frac{{2Z^{*} r}}{{n^{*} }}} \right) \\ \end{aligned} \\ \end{array}$$
(6)
The transition probability \((A_{fi} )\) of a transition for spontaneous emission between levels \(\left( {n_{f} ,l_{f} } \right) \& \left( {n_{i} ,l_{i} } \right)\), is given by
$$\begin{array}{*{20}c} {A_{fi} = 20261 \times 10^{ - 6} \frac{{\left( {E_{f} - E_{i} } \right)^{3} }}{{2l_{i} + 1}}S} \\ \end{array}$$
(7)
In above expression \(E_{f} > E_{i}\) and are the energies of upper and lower states, respectively, S represents electric dipole line strength. In lighter atoms, LS coupling dominates; therefore, line strength can be found by
$$\begin{array}{*{20}c} {S_{LS} = \left[ {J_{f} ,J_{i} ,L_{f} ,L_{i} } \right]\left( {\left\{ {\begin{array}{*{20}c} {L_{f} } & S & {J_{f} } \\ {J_{i} } & 1 & {L_{i} } \\ \end{array} } \right\}\left\{ {\begin{array}{*{20}c} {L_{f} } & {l_{f} } & {L_{c} } \\ 1 & {L_{i} } & {l_{i} } \\ \end{array} } \right\}P_{{l_{i} l_{f} }}^{\left( 1 \right)} } \right)^{2} } \\ \end{array}$$
(8)
The terms in the bracket contain two 6 J symbols and the matrix element \(P_{{l_{i} l_{f} }}^{\left( 1 \right)}\), which is given by
$$\begin{array}{*{20}c} {P_{{l_{i} l_{f} }}^{\left( 1 \right)} = l_{ > } \left\langle {n_{i} ,l_{i} \left| r \right|n_{f} ,l_{f} } \right\rangle = l_{ > } \mathop \smallint \limits_{o}^{\infty } r^{3} R_{{n_{i} l_{i} }} R_{{n_{f} l_{f} }} {\text{d}}r} \\ \end{array}$$
(9)
The integral in the above expression can be found by the formula based on WBEPM theory [11, 12].
$$\begin{aligned} \left\langle {n_{i} ,l_{i} |r|n_{f} ,l_{f} } \right\rangle & = \mathop \smallint \limits_{o}^{\infty } r^{3} R_{{n_{i} l_{i} }} R_{{n_{f} l_{f} }} {\text{d}}r = \left( { - 1} \right)^{{n_{i} + n_{f} + l_{i} + l_{f} }} \left( {\frac{{2Z_{i}^{*} }}{{n_{i}^{*} }}} \right)^{{l_{i}^{*} }} \left( {\frac{{2Z_{f}^{*} }}{{n_{f}^{*} }}} \right)^{{l_{f}^{*} }} \left( {\frac{{Z_{f}^{*} }}{{n_{f}^{*} }} - \frac{{Z_{i}^{*} }}{{n_{i}^{*} }}} \right)^{{ - l_{f}^{*} - l_{i}^{*} - 4}} \\ & \quad \times \left[ {\frac{{n_{f}^{*4} \Gamma \left( {n_{f}^{*} + l_{f}^{*} + 1} \right)}}{{4Z_{f}^{*3} \left( {n_{f}^{*} - l_{f}^{*} - 1} \right)}}} \right]^{{ - \frac{1}{2}}} \left[ {\frac{{n_{i}^{*4} \Gamma \left( {n_{i}^{*} + l_{i}^{*} + 1} \right)}}{{4Z_{i}^{*3} \left( {n_{i}^{*} - l_{i}^{*} - 1} \right)}}} \right]^{{ - \frac{1}{2}}} \mathop \sum \limits_{{m_{1} = 0}}^{{n_{f}^{*} - l_{f}^{*} - 1}} \mathop \sum \limits_{{m_{2} = 0}}^{{n_{i}^{*} - l_{i}^{*} - 1}} \frac{{\left( { - 1} \right)^{{m_{2} }} }}{{m_{1} !m_{2} !}} \\ & \quad \left( {\frac{{Z_{f}^{*} }}{{n_{f}^{*} }} - \frac{{Z_{i}^{*} }}{{n_{i}^{*} }}} \right)^{{m_{1} + m_{2} }} \left( {\frac{{Z_{f}^{*} }}{{n_{f}^{*} }} + \frac{{Z_{i}^{*} }}{{n_{i}^{*} }}} \right)^{{ - m_{1} - m_{2} }} \Gamma \left( {l_{f}^{*} + l_{i}^{*} + m_{1} + m_{2} + 4} \right) \\ & \quad \times \begin{array}{*{20}c} {\mathop \sum \limits_{{m_{3} = 0}}^{S} \left( {\begin{array}{*{20}c} {l_{i}^{*} - l_{f}^{*} + m_{2} + 2} \\ {n_{f}^{*} - l_{f}^{*} - 1 - m_{1} - m_{3} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {l_{f}^{*} - l_{i}^{*} + m_{1} + 2} \\ {n_{i}^{*} - l_{i}^{*} - 1 - m_{2} - m_{3} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {l_{i}^{*} + l_{f}^{*} + m_{1} + m_{2} + m_{3} + 3} \\ {m_{3} } \\ \end{array} } \right)} \\ \end{array} \\ \end{aligned}$$
(10)
in which \(S = {\text{min}}\left( {n_{f}^{*} - l_{f}^{*} - 1 - m_{1} , n_{i}^{*} - l_{i}^{*} - 1 - m_{2} } \right).\)