I have the following Gaussian function:
$\rho(r) = q_i (\alpha/\pi)^{3/2} \exp(-\alpha r^2)$
Qualitatively, the "width" of this Gaussian is related to $\frac{1}{\alpha}$: the larger the value of $\alpha$, the smaller the "width" of the Gaussian.
This Wikipedia article uses this definition of a Gaussian function:
$f(x) = a\exp \left(-\frac{(x-b)^2}{2c^2}\right)$
and says that one way to define the width is to consider the full width at half maximum:
$\text{FWHM} = 2 \sqrt{2 \ln 2} c \approx 2.35482c$
In other words, the width of $f(x)$ is proportional to $c$. But, in my function $\rho(r)$, $\alpha$ appears in two places: in the exponential and as a coefficient of the exponential. How should I define the width of $\rho(r)$?