Foresee, For C, For E

For the factor $C$, we consider the potential energy in bringing a charge from infinity to its location ${r}_{eo}$ in the configuration around the positive charge along the line joining it and the positive charge.

In the case of electrostatic consideration,

$PE=\lim_{r\rightarrow\infty}{\cfrac { q^{ 2 }A }{ 4\pi \varepsilon _{ o } } (\cfrac { 1 }{ { r }_{ eo } } -\cfrac { 1 }{ r } )}$

For the case developed in the post "Like Wave, Like Particle, Not Attracted to Electrons"

$PE={ m }_{ e }c^{ 2 }\lim _{ r\rightarrow \infty  }{ \{ ln{ (\cfrac { { r } }{ { r_{ eo } } } ) }+C({ r }_{ eo }-{ r })\}  }$

Equating the two,

${ m }_{ e }c^{ 2 }\lim _{ r\rightarrow \infty  }{ \{ ln{ (\cfrac{ { r } } { { r_{ eo } } } ) }+C({ r }_{ eo }-{ r })\}  } =\lim_{r\rightarrow\infty}{\cfrac { q^{ 2 }A }{ 4\pi \varepsilon _{ o } } (\cfrac { 1 }{ { r }_{ eo } } -\cfrac { 1 }{ r } )}$

$\lim _{ r\rightarrow \infty  }{ \{ ln{ (\cfrac { { r } } { r_{ eo } }  ) }+C({ r }_{ eo }-{ r })\}  } =\lim_{r\rightarrow\infty}{\cfrac { q^{ 2 }A }{ 4\pi \varepsilon _{ o }{ m }_{ e }c^{ 2 } } (\cfrac { 1 }{ { r }_{ eo } } -\cfrac { 1 }{ r } )}$

$C=\lim _{ r\rightarrow \infty  }{ \{  } \cfrac { 1 }{ ({ r }_{ eo }-{ r }) } \cfrac { q^{ 2 }A }{ 4\pi \varepsilon _{ o }{ m }_{ e }c^{ 2 } } (\cfrac { 1 }{ { r }_{ eo } } -\cfrac { 1 }{ r } )-\cfrac { 1 }{ ({ r }_{ eo }-{ r }) } ln{ (\cfrac{ { r } } { { r_{ eo } } } ) }\}$

$C=\lim _{ r\rightarrow \infty  }{ \{  } \cfrac { -q^{ 2 }A }{ 4\pi \varepsilon _{ o }{ m }_{ e }c^{ 2 } } \cfrac { 1 }{ { r }_{ eo }r } -\cfrac { 1 }{ ({ r }_{ eo }-{ r }) } ln{ (\cfrac { { r } }{ { r_{ eo } } } ) }\}$

$C=\lim _{ r\rightarrow \infty  }{ \{  } -\cfrac { 1 }{ ({ r }_{ eo }-{ r }) } ln{ (\cfrac{ { r } } { { r_{ eo } } } ) }\}=0$

For the case developed in the post "Like Wave, Like Particle, Not Attracted to Electrons II"

${ r }_{ ef }e^{ -C.{ r }_{ ef } }={ r }_{ eo }e^{ -C.{ r }_{ eo } }e^{ \cfrac { -q^{ 2 }A }{ 4\pi \varepsilon _{ o }{ m }_{ e }c^{ 2 } } \cfrac { 1 }{ { r }_{ eo } }  }$

When $C$ = 0,

${ r }_{ ef }={ r }_{ eo }e^{ \cfrac { -q^{ 2 }A }{ 4\pi \varepsilon _{ o }{ m }_{ e }c^{ 2 } } \cfrac { 1 }{ { r }_{ eo } }  }$

Which means ${r}_{ef}$ can be determined theoretically once the electron configuration is known (for $A$), because ${r}_{eo}$ can also be calculated given the electron configuration. And the ionization energy that was developed in the post "Like Wave, Like Particle, Not Attracted to Electrons", can also be obtained,

${E}_{s}={ m }_{ e }c^{ 2 } \{ln{( \cfrac{{r}_{eo}}{{r }_{ e f}})}\}$

since, ${r}_{ef}\lt{r}_{eo}$, ${E}_{s}$ is positive as expected.