Appendix 1
We apply \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{O}_{i} = 1 - \left| {\varPsi_{i}^{0} } \right\rangle \left( {\left. {\varPhi_{i} } \right|} \right.\) to both sides of equation Eq. 12:
$$- \frac{\hbar }{i}\left| {\dot{\varPsi }^{\left( 1 \right)} \left( t \right)} \right\rangle = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{H}_{0} \left| {\varPsi^{\left( 1 \right)} \left( t \right)} \right\rangle + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{V} \left( t \right)\left| {\varPsi_{i}^{0} } \right\rangle$$
(70)
where we use the expansion given by Eq. 13:
$$\left| {\varPsi^{\left( 1 \right)} \left( t \right)} \right\rangle = \sum\limits_{n} {C_{n} \left( t \right)\exp \left( { - \frac{i}{\hbar }E_{n} t} \right)\left| {\varPsi_{n}^{0} } \right\rangle }$$
(71)
The resulting equation can now be written as
$$- \frac{\hbar }{i}\sum\limits_{n} {\dot{C}_{n} \left( t \right)\exp \left( { - \frac{i}{\hbar }E_{n} t} \right)\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{O}_{i} \left| {\varPsi_{n}^{0} } \right\rangle } = \exp \left( { - \frac{i}{\hbar }E_{i} t} \right)\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{O}_{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{V} \left| {\varPsi_{i}^{0} } \right\rangle ,$$
(72)
where prime denotes that the term with n = i is excluded.
Next, we use the completeness property of the antisymmetrized basis (see Eq. 4) and rewrite the right side of Eq. 72 as
$$\exp \left( { - \frac{i}{\hbar }E_{i} t} \right) \cdot \frac{{f_{0} }}{P}\sum\limits_{n} {\left| {\varPsi_{n}^{0} } \right\rangle \left( {\left. {\varPhi_{n}^{0(0)} } \right|\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{O}_{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{V} \left| {\varPsi_{i}^{0} } \right\rangle } \right.}$$
(73)
Because \(\left( {\varPhi_{i}^{0(0)} } \right.\left| {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{O}_{i} } \right. = \left( {\left. {\varPhi_{i}^{0(0)} } \right|} \right. - \left( {\varPhi_{i}^{0(0)} \left| {\varPsi^{0}_{i} } \right\rangle } \right.\left( {\left. {\varPhi_{i}^{0(0)} } \right|} \right. \equiv 0\), the term at n = i, is excluded, which is signified by the prime on the summation sign.
Rearranging Eq. 72, we obtain
$$\sum\limits_{n} {\left\{ { - \frac{\hbar }{i}\dot{C}_{n} \left( t \right) - \frac{{f_{0} }}{P}\left( {\varPhi_{n}^{0(0)} \left| {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{O}_{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{V} \left| {\varPsi_{i}^{0} } \right\rangle \exp \left( { - i\omega_{in} t} \right)} \right.} \right.} \right\}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{O}_{i} \left| {\varPsi_{n}^{0} } \right\rangle } = 0.$$
(74)
The last equation is fulfilled when the expression in the curly brackets equals zero for all n. This, in turn, leads to an equation for \(\dot{C}_{n} \left( t \right)\):
$$- \frac{\hbar }{i}\dot{C}_{n} \left( t \right) = \frac{{f_{0} }}{P}\exp \left( { - i\omega_{ni} t} \right)\left( {\left. {\varPhi_{n}^{0(0)} } \right|\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{O}_{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{V} \left| {\varPsi_{i}^{0} } \right\rangle } \right.,$$
(75)
where we introduced \(\frac{1}{\hbar }\left( {E_{i} - E_{n} } \right) = \omega_{in} .\)
Appendix 2
Operator \(\hat{T}\) satisfies Eq. 48:
$$\hat{T} = V_{0}^{\mathbb{N}} + V_{0}^{\mathbb{N}} \left( {\frac{{f_{0}^{2} }}{P}} \right)^{ - 1} \left( {E_{i} - H_{p = 0}^{0} + i\eta } \right)^{ - 1} \hat{T},$$
(76)
which can be re-arranged to:
$$V_{0}^{\mathbb{N}} = \left( {\hat{1} - V_{0}^{\mathbb{N}} \left( {\frac{{f_{0}^{2} }}{P}} \right)^{ - 1} \left( {E_{i} - H_{p = 0}^{0} + i\eta } \right)^{ - 1} } \right)\hat{T}$$
$$V_{0}^{\mathbb{N}} = \left( {\left( {E_{i} - H_{p = 0}^{0} + i\eta } \right)\left( {\frac{{f_{0}^{2} }}{P}} \right) - V_{0}^{\mathbb{N}} } \right)\left( {\frac{{f_{0}^{2} }}{P}} \right)^{ - 1} \left( {E_{i} - H_{p = 0}^{0} + i\eta } \right)^{ - 1} \hat{T}.$$
(77)
Taking into account that the total Hamiltonian of the system
$$H = H_{p = 0}^{0} + V_{p = 0} = H_{p}^{0} + V_{p} ,$$
(78)
is invariant with respect to permutations of electrons between atoms:
$$H = H_{0}^{p = 0} + V^{p = 0} = - \frac{{\hbar^{2} }}{{2\mu_{i} }}\nabla_{i}^{2} + H_{i}^{p = 0} (r_{1} ,r_{2} , \ldots )_{i} + V^{p = 0} .$$
(79)
We further re-arrange Eq. 77:
$$V_{0}^{\mathbb{N}} = \left( {E_{i} - H + i\eta } \right)\left( {\frac{{f_{0}^{2} }}{P}} \right)\left( {\frac{{f_{0}^{2} }}{P}} \right)^{ - 1} \left( {E_{i} - H_{p = 0}^{0} + i\eta } \right)^{ - 1} \hat{T}$$
$$\left( {\frac{{f_{0}^{2} }}{P}} \right)\left( {E_{i} - H_{p = 0}^{0} + i\eta } \right)\left( {\frac{{f_{0}^{2} }}{P}} \right)^{ - 1} \left( {E_{i} - H + i\eta } \right)^{ - 1} V_{0}^{\mathbb{N}} = \hat{T}$$
$$\left( {\frac{{f_{0}^{2} }}{P}} \right)\left( {E_{i} - H_{p = 0}^{0} + V_{0} - V_{0} + i\eta } \right)\left( {\frac{{f_{0}^{2} }}{P}} \right)^{ - 1} \left( {E_{i} - H + i\eta } \right)^{ - 1} V_{0}^{\mathbb{N}} = \hat{T}$$
$$\begin{aligned} &V_{0}^{\mathbb{N}} \left( {\frac{{f_{0}^{2} }}{P}} \right)^{ - 1} \left( {E_{i} - H + i\eta } \right)^{ - 1} V_{0}^{\mathbb{N}} \hfill \\ &+ \left( {\frac{{f_{0}^{2} }}{P}} \right)\left( {E_{i} - H_{p = 0}^{0} - V_{0} + i\eta } \right)\left( {\frac{{f_{0}^{2} }}{P}} \right)^{ - 1} \left( {E_{i} - H + i\eta } \right)^{ - 1} V_{0}^{\mathbb{N}} = \hat{T} \hfill \\ \end{aligned}$$
to the final expression for operator \(\hat{T}\):
$$V_{0}^{\mathbb{N}} \left( {\frac{{f_{0}^{2} }}{P}} \right)^{ - 1} \left( {E_{i} - H + i\eta } \right)^{ - 1} V_{0}^{\mathbb{N}} + V_{0}^{\mathbb{N}} = \hat{T}.$$
(80)
Appendix 3
The initial states of the electrons in the Lithium atom are given by [28] where the parameters α and β are taken from [29]
$$\begin{aligned} \phi_{1s} (\vec{R} - \vec{r}_{i} ) & = (\alpha_{1}^{3} /\pi )^{1/2} \exp \,( - \alpha_{1} |\vec{R} - \vec{r}_{i} |),\quad \, i = 1,2 \\ \varphi_{{2s^{1} }} (\vec{R} - \vec{r}_{3} ) & = (\alpha_{2}^{3} /8\pi )^{1/2} (1 - 0.5\alpha_{2} |\vec{R} - \vec{r}_{3} |)\exp \,( - 0.5\alpha_{2} |\vec{R} - \vec{r}_{3} |), \end{aligned}$$
(81)
\(\text{where} \,\alpha_{1} = 2.698,\alpha_{2} = 0.795.\)
The final states of the electrons in the Helium-like Lithium ion are described by
$$\begin{aligned} &\phi^{*}(\vec{R} - \vec{r}_{i} ) = (\alpha ^{*3} /\pi )^{1/2} \exp ( - \alpha ^{*}|\vec{R} - \vec{r}_{i} |),\quad i = 1,2 \\ &{\text{where}}\;\alpha ^{*} = 1.692. \\ \end{aligned}$$
(82)
The final state of the electron in the Hydrogen atom is described by the single-electron wavefunction:
$$\begin{aligned} &\psi_{H} (r) = (\pi )^{ - 1/2} \exp \left( { - \beta r} \right); \\ &\beta = 1 \\ \end{aligned}$$
(83)
Here, α, α*, and β are given in reciprocal Bohr radius units, and R is in Bohr radius units.
The first-order correction to the S-matrix elements given by Eq. 31 becomes:
$$\begin{aligned} \left\langle {\Psi _{f}^{0} } \right|V_{0} ^{\mathbb{N}} \left| {\left. {\Phi _{i}^{0} } \right)} \right. & = \frac{1}{{f_{0} }}\frac{1}{{f_{{{\text{Li}}}} }}\left( {\sqrt 3 \left\langle {\Psi _{1} (\vec{r}_{1} ,\vec{r}_{2} ,\vec{r}_{3} )} \right|V_{0} ^{\mathbb{N}} \text{e} ^{{\left( {i(\vec{k}_{f} - \vec{k}_{i} ) \cdot \vec{R}_{{I,II}} } \right)}} \left| {\left. {\Psi _{{{\text{Li}}1}} (\vec{r}_{1} ,\vec{r}_{2} ,\vec{r}_{3} )} \right)} \right.} \right. \\ & \quad + \frac{3}{2}\left\langle {\Psi _{2} (\vec{r}_{1} ,\vec{r}_{2} ,\vec{r}_{3} )} \right|V_{0} ^{\mathbb{N}} \text{e} ^{{\left( {i(\vec{k}_{f} - \vec{k}_{i} ) \cdot \vec{R}_{{I,II}} } \right)}} \left| {\left. {\Psi _{{{\text{Li}}2}} (\vec{r}_{1} ,\vec{r}_{2} ,\vec{r}_{3} )} \right)} \right. \\ & = \frac{4}{{\sqrt 3 }}\frac{1}{P}\frac{{f_{0} }}{{f_{{{\text{Li}}}} }}\left\{ {\left\langle {\Phi _{{{\text{Li}}^{ + } }} (\vec{R} - \vec{r}_{1} ,\vec{R} - \vec{r}_{2} )\Phi _{H} (\vec{r}_{3} )} \right|V_{0} \text{e} ^{{\left( {i(\vec{k}_{f} - \vec{k}_{i} ) \cdot \vec{R}_{{I,II}} } \right)}} } \right. \\ \quad \left| {\left. {(\phi _{{1s}} (\vec{R} - \vec{r}_{1} )\phi _{{1s}} (\vec{R} - \vec{r}_{2} )\varphi _{{2s}} (\vec{R} - \vec{r}_{3} )} \right)} \right. \\ & \quad - \left\langle {\Phi _{{{\text{Li}}^{ + } }} (\vec{R} - \vec{r}_{1} ,\vec{R} - \vec{r}_{2} )\Phi _{H} (\vec{r}_{3} )} \right|V_{0} \text{e} ^{{\left( {i(\vec{k}_{f} - \vec{k}_{i} ) \cdot \vec{R}_{{I,II}} } \right)}} \left| {\left. {\phi _{{1s}} (\vec{R} - \vec{r}_{3} )\phi _{{1s}} (\vec{R} - \vec{r}_{2} )\varphi _{{2s}} (\vec{R} - \vec{r}_{1} ))} \right)} \right. \\ & \quad - \left\langle {\Phi _{{{\text{Li}}^{ + } }} (\vec{R} - \vec{r}_{3} ,\vec{R} - \vec{r}_{2} )\Phi _{H} (\vec{r}_{1} )} \right|V_{0} \text{e} ^{{\left( {i(\vec{k}_{f} - \vec{k}_{i} ) \cdot \vec{R}_{{I,II}} } \right)}} \left| {\left. {(\phi _{{1s}} (\vec{R} - \vec{r}_{1} )\phi _{{1s}} (\vec{R} - \vec{r}_{2} )\varphi _{{2s}} (\vec{R} - \vec{r}_{3} )} \right)} \right. \\ & \quad + \left. {\left\langle {\Phi _{{{\text{Li}}^{ + } }} (\vec{R} - \vec{r}_{3} ,\vec{R} - \vec{r}_{2} )\Phi _{H} (\vec{r}_{1} )} \right|V_{0} \text{e} ^{{\left( {i(\vec{k}_{f} - \vec{k}_{i} ) \cdot \vec{R}_{{I,II}} } \right)}} \left| {\left. {\phi _{{1s}} (\vec{R} - \vec{r}_{3} )\phi _{{1s}} (\vec{R} - \vec{r}_{2} )\varphi _{{2s}} (\vec{R} - \vec{r}_{1} ))} \right)} \right.} \right\} \\ \end{aligned}$$
(84)
where we used the orthogonality of the spin parts of the wavefunctions.
The matrix element 〈Ψ
0
f
|V
0|Φ
0
i
). in Eq. 64 can be rewritten as
$$\begin{aligned} \left\langle {\varPsi_{f}^{0} } \right|V_{0} \left| {\left. {\varPhi_{i}^{0} } \right)} \right. & = \frac{4}{\sqrt 3 }\frac{1}{P}\frac{{f_{0} }}{{f_{\text{Li}} }}\int {d^{3} R} \text{e}^{{\left( {i(\vec{k}_{f} - \vec{k}_{i} ) \cdot \vec{R}} \right)}} \left\{ {\varDelta_{1s^{*}1s} \left[ {\left( {\frac{6}{R}\varDelta_{1s^{*}1s} S_{1s'2s} - 2A_{1s'2s} \varDelta_{1s^{*}1s} - 4K_{1s^{*}1s} S_{1s'2s} } \right)} \right.} \right. \\ & \quad - \left. {\left( {S_{1s'2s} \left(\frac{3}{R}\varDelta_{1s^{*}1s} - K_{1s^{*}1s} - K_{1s^{*}2s} \right) - A_{1s'1s} \varDelta_{1s^{*}2s} } \right)} \right] \\ & \quad - \left. {\varDelta_{1s^{*}1s} \varDelta_{1s^{*}2s} \left( {\frac{{3S_{1s'1s} }}{R} - A_{1s'1s} } \right) + S_{1s'1s} \left( {K_{1s^{*}1s} \varDelta_{1s^{*}2s} + K_{1s^{*}2s} \varDelta_{1s^{*}1s} } \right)} \right\}, \\ \end{aligned}$$
(85)
where we introduced the following symbols for integrals
$$\begin{array}{*{20}l} {\varDelta_{1s^{*}1s} = \left\langle {\phi ^{*}} \right|\left. {\phi_{{1s^{1} }} } \right\rangle } \hfill \\ {\varDelta_{1s^{*}2s} = \left\langle {\phi ^{*}} \right|\left. {\phi_{{2s^{1} }} } \right\rangle } \hfill \\ {S_{1s'1s} = \left\langle {\psi_{He^{*}} (\vec{r})} \right|\left. {\phi_{1s} (\vec{R} - \vec{r})} \right\rangle } \hfill \\ {S_{1s'2s} = \left\langle {\psi_{He^{*}} (\vec{r})} \right|\left. {\phi_{2s} (\vec{R} - \vec{r})} \right\rangle } \hfill \\ \end{array} ,\quad \begin{array}{*{20}l} {K_{1s^{*}1s} = \left\langle {\phi ^{*}} \right|\frac{1}{r}\left| {\phi_{{1s^{1} }} } \right\rangle } \hfill \\ {K_{1s^{*}2s} = \left\langle {\phi ^{*}} \right|\frac{1}{r}\left| {\phi_{{2s^{1} }} } \right\rangle } \hfill \\ {A_{1s'1s} = \left\langle {\psi_{{{\text{He}}^{*}}} (\vec{r})} \right|\frac{1}{r}\left| {\phi_{1s} (\vec{R} - \vec{r})} \right\rangle } \hfill \\ {A_{1s'2s} = A_{1s'1s} = \left\langle {\psi_{{{\text{He}}^{*}}} (\vec{r})} \right|\frac{1}{r}\left| {\phi_{2s} (\vec{R} - \vec{r})} \right\rangle .} \hfill \\ \end{array}$$
(86)
All these integrals have analytical expressions, and they are listed below.
The normalization factor in Eq. 64 is found as follows:
$$\begin{aligned} f_{0} & = \frac{4}{3} \times \left\langle {4\phi ^{*}(\vec{R} - \vec{r}_{1} ) \cdot \phi ^{*}(\vec{R} - \vec{r}_{2} ) \cdot \psi_{He^{*}} (r_{3} ) - } \right.3\phi ^{*}(\vec{R} - \vec{r}_{1} ) \\ & \quad \cdot \phi ^{*}(\vec{R} - \vec{r}_{3} ) \cdot \psi_{He^{*}} (r_{2} ) - \phi ^{*}(\vec{R} - \vec{r}_{3} ) \cdot \phi ^{*}(\vec{R} - \vec{r}_{2} ) \cdot \left. {\psi_{He^{*}} (r_{1} )} \right| \\ & \quad \left| {\text{e}^{{\left( {i(\vec{k}_{f} - \vec{k}_{i} ) \cdot \vec{R}} \right)}} (\phi_{1s} (\vec{R} - \vec{r}_{1} )\phi_{1s} (\vec{R} - \vec{r}_{2} )\varphi_{2s} (\vec{R} - \vec{r}_{3} ) - \phi_{1s} (\vec{R} - \vec{r}_{3} )\phi_{1s} (\vec{R} - \vec{r}_{2} )\varphi_{2s} (\vec{R} - \vec{r}_{1} ))} \right\rangle \\ & = \frac{20}{3}\int {d^{3} R\text{e}^{{\left( {i(\vec{k}_{f} - \vec{k}_{i} ) \cdot \vec{R}} \right)}} } \{ \varDelta_{1s^{*}1s} \varDelta_{1s^{*}1s} S_{1s'2s} - \varDelta_{1s^{*}1s} \varDelta_{1s^{*}2s} S_{1s'1s} \} \\ \end{aligned}$$
(87)
$$\begin{aligned} \varDelta_{1s^{*}1s} & = \left\langle {\phi ^{*}} \right|\left. {\phi_{{1s^{1} }} } \right\rangle = \frac{{8\left( {\alpha \alpha ^{*}} \right)^{3/2} }}{{\left( {\alpha + \alpha ^{*}} \right)^{3} }} \\ \varDelta_{1s^{*}2s} & = \left\langle {\phi ^{*}} \right|\left. {\phi_{{2s^{1} }} } \right\rangle = \frac{{8\left( {\frac{\alpha }{2}\alpha ^{*}} \right)^{3/2} }}{{\left( {\frac{\alpha }{2} + \alpha ^{*}} \right)^{3} }}\left( {1 - \frac{3\alpha }{\alpha + 2\alpha ^{*}}} \right) \\ S_{1s'1s} & = \left\langle {\psi_{He^{*}} (\vec{r})} \right|\left. {\phi_{1s} (\vec{R} - \vec{r})} \right\rangle \\ &= \frac{{8\left( {\alpha \beta } \right)^{3/2} }}{{\beta^{2} - \alpha^{2} }}{\text{sh}}\left[ {\frac{R}{2}(\alpha - \beta )} \right]e^{{ - \frac{R}{2}(\alpha + \beta )}} \\ & \quad \times\left( {\frac{1}{\alpha + \beta }} \right. + \frac{{{\text{cth}}\frac{R}{2}(\alpha - \beta )}}{\alpha - \beta } - \left. {\frac{8\alpha \beta }{{R\left( {\alpha^{2} - \beta^{2} } \right)^{2} }}} \right) \\ \end{aligned}$$
(88)
$$\begin{aligned} S_{1s'2s} & = \left\langle {\psi_{He^{*}} (\vec{r})} \right|\left. {\phi_{2s} (\vec{R} - \vec{r})} \right\rangle = \frac{{\left( {\alpha \beta } \right)^{3/2} }}{\sqrt 2 }\frac{{R \cdot {\text{sh}}\left[ {\frac{R}{2}\left( {\frac{\alpha }{2} - \beta } \right)} \right]e^{{ - \frac{R}{2}\left( {\frac{\alpha }{2} - \beta } \right)}} }}{{\beta^{2} - \left( {\frac{\alpha }{2}} \right)^{2} }} \\ & \quad \times \left\{ {\frac{4}{{R\left( {\frac{\alpha }{2} + \beta } \right)}}\frac{8\alpha \beta }{{R^{2} \left( {\left( {\frac{\alpha }{2}} \right)^{2} - \beta^{2} } \right)^{2} }}} \right. + \frac{{4{\text{cth}}\left[ {\frac{R}{2}\left( {\frac{\alpha }{2} - \beta } \right)} \right]}}{{R\left( {\frac{\alpha }{2} - \beta } \right)}} \\ & \quad - \frac{{\alpha \left( {1 - {\text{cth}}\left[ {\frac{R}{2}\left( {\frac{\alpha }{2} - \beta } \right)} \right]} \right)}}{{R\left( {\frac{\alpha }{2} + \beta } \right)}} - \frac{{2\alpha \left( {3 - {\text{cth}}\left[ {\frac{R}{2}\left( {\frac{\alpha }{2} - \beta } \right)} \right]} \right)}}{{R\left( {\frac{\alpha }{2} + \beta } \right)^{2} }} \\ & \quad - \frac{12\alpha }{{R^{2} \left( {\left( {\frac{\alpha }{2}} \right) + \beta } \right)^{3} }} + \frac{12\alpha }{{R^{2} \left( {\left( {\frac{\alpha }{2}} \right) - \beta } \right)^{3} }} + \frac{{\alpha \left( {1 - {\text{cth}}\left[ {\frac{R}{2}\left( {\frac{\alpha }{2} - \beta } \right)} \right]} \right)}}{{R\left( {\frac{\alpha }{2} - \beta } \right)}} \\ & \quad - \frac{{2\alpha \left( {1 + {\text{cth}}\left[ {\frac{R}{2}\left( {\frac{\alpha }{2} - \beta } \right)} \right]} \right)}}{{R\left( {\left( {\frac{\alpha }{2}} \right)^{2} - \beta^{2} } \right)}} + \left. {\frac{{2\alpha \left( {1 - 3{\text{cth}}\left[ {\frac{R}{2}\left( {\frac{\alpha }{2} - \beta } \right)} \right]} \right)}}{{R\left( {\frac{\alpha }{2} - \beta } \right)^{2} }}} \right\} \\ \end{aligned}$$
(89)
$$\begin{aligned} K_{1s^{*}1s} = & \left\langle {\phi ^{*}\left| \frac{1}{r} \right|\phi_{{1s^{1} }} } \right\rangle = \frac{{4\left( {\alpha \alpha ^{*}} \right)^{3/2} }}{{\left( {\alpha + \alpha ^{*}} \right)^{2} }}{\text{sh}}\left[ {\frac{R}{2}(\alpha + \alpha ^{*})} \right]e^{{ - \frac{R}{2}(\alpha + \alpha ^{*})}} \\ & \quad \times \left( {1 + \frac{4}{{R\left( {\alpha + \alpha ^{*}} \right)}} - {\text{cth}}\left[ {\frac{R}{2}(\alpha + \alpha ^{*})} \right]} \right) \\ \end{aligned}$$
(90)
$$\begin{aligned} A_{1s'1s} & = \left\langle {\psi_{He^{*}} (\vec{r})} \right|\frac{1}{r}\left| {\phi_{1s} (\vec{R} - \vec{r})} \right\rangle = \frac{{4\left( {\alpha \beta } \right)^{3/2} }}{{\left( {\alpha^{2} - \beta^{2} } \right)}}{\text{sh}}\left[ {\frac{R}{2}(\alpha - \beta )} \right]e^{{ - \frac{R}{2}(\alpha + \beta )}} \\ & \quad \times\left( {1 + \frac{4\alpha }{{R\left( {\alpha^{2} - \beta^{2} } \right)}} - {\text{cth}}\left[ {\frac{R}{2}(\alpha - \beta )} \right]} \right) \\ \end{aligned}$$
$$\begin{aligned} K_{{1s^{*}2s}} & = \left\langle {\phi ^{*}} \right|\frac{1}{r}\left| {\phi _{{2s^{1} }} } \right\rangle = \frac{{\sqrt 2 \left( {\alpha \alpha ^{*}} \right)^{{3/2}} }}{{\left( {\frac{\alpha }{2} + \alpha ^{*}} \right)^{2} }}{\text{sh}}\left[ {\frac{R}{2}\left( {\frac{\alpha }{2} + \alpha ^{*}} \right)} \right]e^{{ - \frac{R}{2}\left( {\frac{\alpha }{2} + \alpha ^{*}} \right)}} \\ & \quad \times \left( {1 + \frac{4}{{R\left( {\frac{\alpha }{2} + \alpha ^{*}} \right)}} - {\text{cth}}\left[ {\frac{R}{2}\left( {\frac{\alpha }{2} + \alpha ^{*}} \right)} \right] - \frac{{R\alpha }}{2}\left( {1 - {\text{cth}}\left[ {\frac{R}{2}\left( {\frac{\alpha }{2} + \alpha ^{*}} \right)} \right]} \right)} \right. \\ & \left. {\quad - \frac{{2\alpha \left( {1 - {\text{cth}}\left[ {\frac{R}{2}\left( {\frac{\alpha }{2} + \alpha ^{*}} \right)} \right]} \right)}}{{\left( {\frac{\alpha }{2} + \alpha ^{*}} \right)}} - \frac{{6\alpha }}{{R\left( {\frac{\alpha }{2} + \alpha ^{*}} \right)^{2} }}} \right) \\ A_{{1s'2s}} & = A_{{1s'1s}} = \left\langle {\psi _{{He^{*}}} (\vec{r})\left| {\frac{1}{r}} \right|\phi _{{2s}} (\vec{R} - \vec{r})} \right\rangle = \frac{{\sqrt 2 \left( {\alpha \beta } \right)^{{3/2}} }}{{\left( {\frac{\alpha }{2}} \right)^{2} - \beta ^{2} }}{\text{sh}}\left[ {\frac{R}{2}\left( {\frac{\alpha }{2} - \beta } \right)} \right]e^{{ - \frac{R}{2}\left( {\frac{\alpha }{2} + \beta } \right)}} \\ & \quad \times \left\{ {1 - {\text{cth}}\left[ {\frac{R}{2}\left( {\frac{\alpha }{2} - \beta } \right)} \right] - \frac{{R\alpha }}{2}\left( {1 - {\text{cth}}\left[ {\frac{R}{2}\left( {\frac{\alpha }{2} - \beta } \right)} \right]} \right) - \frac{{\alpha \left( {1 - {\text{cth}}\left[ {\frac{R}{2}\left( {\frac{\alpha }{2} - \beta } \right)} \right]} \right)}}{{\left( {\frac{\alpha }{2} + \beta } \right)}}} \right. \\ & \left. {\quad - \frac{{\alpha \left( {1 - {\text{cth}}\left[ {\frac{R}{2}\left( {\frac{\alpha }{2} - \beta } \right)} \right]} \right)}}{{\left( {\frac{\alpha }{2} - \beta } \right)}} - \frac{{2\alpha }}{{R\left( {\frac{\alpha }{2} + \beta } \right)^{2} }} - \frac{{2\alpha }}{{R\left( {\frac{\alpha }{2} - \beta } \right)^{2} }}} \right\} \\ \end{aligned}$$
(91)
