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BetaFit.lyx
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BetaFit.lyx
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#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
\begin_header
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\end_header
\begin_body
\begin_layout Title
betaFIT: A software for fitting points to functions
\end_layout
\begin_layout Section*
Extendend Morse oscillator
\end_layout
\begin_layout Standard
Consider the EMO function:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{align}
V(r) & =\mathfrak{D}_{e}\left(1-\exp\left(-\beta(r)\left(r-r_{e}\right)\right)\right)^{2}+V_{e}.\label{eq:EMO}
\end{align}
\end_inset
If we know
\begin_inset Formula $V(r)$
\end_inset
,
\begin_inset Formula $\mathfrak{D}_{e}$
\end_inset
and
\begin_inset Formula $r_{e}$
\end_inset
, then we can obtain a corresponding function
\begin_inset Formula $\beta(r)$
\end_inset
.
However, similar to the equation
\begin_inset Formula $4=x^{2}$
\end_inset
having two real-number solutions, for each value of
\begin_inset Formula $r$
\end_inset
, there can be more than one real number
\begin_inset Formula $\beta(r)$
\end_inset
that satisfies Eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:EMO"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
To find the possible values of
\begin_inset Formula $\beta(r)$
\end_inset
, we can start with the following rearrangement:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
\frac{V(r)-V_{e}}{\mathcal{\mathfrak{D}}_{e}}=\left(1-\exp\left(-\beta(r)\left(r-r_{e}\right)\right)\right)^{2}.\label{eq:solvingEMOforBeta_beforeSquareRooting}
\end{equation}
\end_inset
\begin_inset Formula $V(r)\ge V_{e}$
\end_inset
for all
\begin_inset Formula $r$
\end_inset
, because
\begin_inset Formula $V_{e}$
\end_inset
is defined as the minimum value of
\begin_inset Formula $V(r).$
\end_inset
Also,
\begin_inset Formula $\mathfrak{D}_{e}>0$
\end_inset
for bound potentials, so the left
\begin_inset Note Note
status open
\begin_layout Plain Layout
Why do we say left HAND side instead of just left side?
\end_layout
\end_inset
side cannot be negative.
We can then proceed as follows:
\end_layout
\begin_layout Standard
\begin_inset VSpace medskip
\end_inset
\end_layout
\begin_layout Standard
\begin_inset VSpace medskip
\end_inset
\end_layout
\begin_layout Standard
\begin_inset VSpace medskip
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Float table
placement H
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="4" columns="6">
<features tabularvalignment="middle">
<column alignment="right" valignment="top">
<column alignment="left" valignment="top">
<column alignment="center" valignment="top">
<column alignment="right" valignment="top">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<row>
<cell alignment="right" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $4$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\xout off
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\color none
\begin_inset Formula $=x^{2}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset space ~
\end_inset
\begin_inset space ~
\end_inset
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\end_inset
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\end_inset
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\end_inset
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\end_inset
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\end_inset
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\end_inset
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\end_inset
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\end_inset
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\end_inset
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\end_inset
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\end_inset
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\end_inset
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\end_inset
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\end_inset
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\end_inset
\begin_inset space ~
\end_inset
\begin_inset space ~
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="right" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\xout off
\uuline off
\uwave off
\noun off
\color none
\begin_inset Formula $\frac{V(r)-V_{e}}{\mathcal{\mathfrak{D}}_{e}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $=\left(1-\exp\left(-\beta(r)\left(r-r_{e}\right)\right)\right)^{2}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="right" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\pm\sqrt{4}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $=x$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset space ~
\end_inset
\begin_inset space ~
\end_inset
\begin_inset space ~
\end_inset
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\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="right" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\pm\sqrt{\frac{V(r)-V_{e}}{\mathcal{\mathfrak{D}}_{e}}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $=1-\exp\left(-\beta(r)\left(r-r_{e}\right)\right).$
\end_inset
\end_layout
\end_inset
</cell>
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\begin_inset Text
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\begin_inset Text
\begin_layout Plain Layout
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
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</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
Instead of a table array, equation numbering is easier if we can do something
like alignat{4}
\end_layout
\begin_layout Plain Layout
\begin_inset Formula
\begin{align}
4 & =x^{2} & & \frac{V(r)-V_{e}}{\mathcal{\mathfrak{D}}_{e}} & =\left(1-\exp\left(-\beta(r)(r-r_{e})\right)\right)^{2}\nonumber \\
\pm\sqrt{4} & =x & & \pm\sqrt{\frac{V(r)-V_{e}}{\mathcal{\mathfrak{D}}_{e}}} & =1-\exp\left(-\beta(r)(r-r_{e})\right).\label{eq:plusMinus4}
\end{align}
\end_inset
\end_layout
\end_inset
The
\begin_inset Formula $+\sqrt{4}$
\end_inset
in the equation above corresponds to
\begin_inset Formula $x>0$
\end_inset
and the
\begin_inset Formula $-\sqrt{4}$
\end_inset
corresponds to
\begin_inset Formula $x<0,$
\end_inset
and similarly we have:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{align}
+\sqrt{\frac{V(r)-V_{e}}{\mathcal{\mathfrak{D}}_{e}}}=1-\exp\left(-\beta(r)(r-r_{e})\right) & ,\,\,\,\,\,1-\exp\left(-\beta(r)\left(r-r_{e}\right)\right)>0\\
-\sqrt{\frac{V(r)-V_{e}}{\mathcal{\mathfrak{D}}_{e}}}=1-\exp\left(-\beta(r)(r-r_{e})\right) & ,\,\,\,\,\,1-\exp\left(-\beta(r)\left(r-r_{e}\right)\right)<0,
\end{align}
\end_inset
and after considering the behavior of the
\begin_inset Formula $\exp$
\end_inset
function more carefully we have:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{align}
+\sqrt{\frac{V(r)-V_{e}}{\mathcal{\mathfrak{D}}_{e}}}=1-\exp\left(-\beta(r)\left(r-r_{e}\right)\right) & ,\,\,\,\,\,\beta(r)(r-r_{e})>0\\
-\sqrt{\frac{V(r)-V_{e}}{\mathcal{\mathfrak{D}}_{e}}}=1-\exp\left(-\beta(r)\left(r-r_{e}\right)\right) & ,\,\,\,\,\,\beta(r)(r-r_{e})<0.
\end{align}
\end_inset
Rearranging, we get:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{align}
\beta(r)\left(r-r_{e}\right) & =-\ln\left(1-\sqrt{\frac{V-V_{e}}{\mathfrak{D}_{e}}}\right),\,\,\,\,\,\beta(r)(r-r_{e})>0\\
\beta(r)\left(r-r_{e}\right) & =-\ln\left(1+\sqrt{\frac{V-V_{e}}{\mathfrak{D}_{e}}}\right),\,\,\,\,\,\beta(r)(r-r_{e})<0.
\end{align}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
I'm not the sure what the best way to compactify the above equation is,
but here's an attempt:
\end_layout
\begin_layout Plain Layout
\begin_inset Formula
\begin{align}
\beta(r) & =\frac{\ln\left(1\mp\sqrt{\frac{V-V_{e}}{\mathfrak{D}_{e}}}\right)}{r_{e}-r},\begin{cases}
- & \beta(r)(r-r_{e})>0\\
+ & \beta(r)(r-r_{e})<0.
\end{cases}
\end{align}
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
We also know that
\begin_inset Formula $\ln(\cdot)>0$
\end_inset
for arguments greater than 1, and
\begin_inset Formula $\ln(\cdot)<0$
\end_inset
for arguments smaller than 1, and that
\begin_inset Formula $\sqrt{\frac{V-V_{e}}{\mathfrak{D}_{e}}}>0$
\end_inset
, we can conclude the following, which completely agrees with what we derived
above:
\begin_inset Formula
\begin{align}
\ln\left(1-\sqrt{\frac{V-V_{e}}{\mathfrak{D}_{e}}}\right) & <0\\
\ln\left(1-\sqrt{\frac{V-V_{e}}{\mathfrak{D}_{e}}}\right) & >0.
\end{align}
\end_inset
\end_layout
\begin_layout Standard
Rearranging again yields:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{align}
\beta(r) & =\frac{\ln\left(1-\sqrt{\frac{V-V_{e}}{\mathfrak{D}_{e}}}\right)}{r_{e}-r},\,\,\,\,\,\beta(r)(r-r_{e})>0\label{eq:betaWhenBeta(r-re)>0}\\
\beta(r) & =\frac{\ln\left(1+\sqrt{\frac{V-V_{e}}{\mathfrak{D}_{e}}}\right)}{r_{e}-r},\,\,\,\,\,\beta(r)(r-r_{e})<0.\label{eq:betaWhenBeta(r-re)<0}
\end{align}
\end_inset
\end_layout
\begin_layout Standard
We can now :
\end_layout
\begin_layout Itemize
choose to use
\series bold
\emph on
only
\emph default
\series default
Eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:betaWhenBeta(r-re)>0"
plural "false"
caps "false"
noprefix "false"
\end_inset
with the consequence that
\begin_inset Formula $\beta(r)$
\end_inset
will be negative in the inner wall of the potential well, then will switch
signs at
\begin_inset Formula $r_{e}$
\end_inset
; or we can
\end_layout
\begin_layout Itemize
choose to use
\series bold
\emph on
only
\series default
\emph default
Eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:betaWhenBeta(r-re)<0"
plural "false"
caps "false"
noprefix "false"
\end_inset
with the consequence that
\begin_inset Formula $\beta(r)$
\end_inset
will be positive in the inner wall of the potential well, then will switch
signs at
\begin_inset Formula $r_{e}$
\end_inset
; or we can
\end_layout
\begin_layout Itemize
choose
\begin_inset Formula $\beta(r)$
\end_inset
to be always positive, with the consequence that we would have to switch
at
\begin_inset Formula $r_{e}$
\end_inset
from using Eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:betaWhenBeta(r-re)<0"
plural "false"
caps "false"
noprefix "false"
\end_inset
in the inner wall to using Eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:betaWhenBeta(r-re)>0"
plural "false"
caps "false"
noprefix "false"
\end_inset
in the outer wall; or (among maybe other choices?) we can just
\end_layout
\begin_layout Itemize
choose either Eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:betaWhenBeta(r-re)<0"
plural "false"
caps "false"
noprefix "false"
\end_inset
or Eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:betaWhenBeta(r-re)>0"
plural "false"
caps "false"
noprefix "false"
\end_inset
for each individual value of
\begin_inset Formula $r$
\end_inset
depending on which choice gives us a smaller residual, but with the consequence
that
\begin_inset Formula $\beta(r)$
\end_inset
will often be switching signs.
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
This helped me figure out what was going on, but it might not be needed
anymore.
\end_layout
\begin_layout Plain Layout
\begin_inset Float table
wide false
sideways false
status open
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout