A Close Match

Two functions $f(x)$ and $g(x)$ were plotted on the same axes, where
$$f(x) =\left(\frac{a}{x}\right)^x\quad \quad g(x) = b\exp\left(-\frac{(x-c)^2}{d}\right)$$
I chose the coefficients $a, b, c$ and $d$ so as to make the function $g(x)$ match $f(x)$ 'as closely as possible' for points past the maximum of $f(x)$. My resulting charts were as follows.


Is it possible to approximately work out the values I chose? Can you choose values to obtain a closer match between the two?