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.
