How to convert $\rm cm^{-1}$ to $\rm eV/Å^2$?

Question

As in the title, how to convert $\rm cm^{-1}$ to $\rm eV/Å^2$? Å stands for angstrom.

Answer 1

This should not be possible. Notice that $cm^{-1}$ has dimensions of 1/length, as does $A^{-1}$.

$eV/A^2$ has dimensions of energy/length$^2$. This makes no sense, given the available information.

Answer 2

The only use I can think of for $\left[eV/Å^2\right]$ is in the quadratic potential constant $\alpha$ such that

$$ V(x)=\alpha x^2 $$

as it appears in a generic hamiltionian. $V$ should be in $eV$ (energy) and $x$ should be in $Å$ (length). It's my assumption that you have something like this

$$ V'(x)=\beta \xi^2(x) $$

where $V'$ is in $cm^{-1}$, and $\xi$ is dimensionless. Thus, I infer you want to convert from $\beta$ to $\alpha$, and I also infer that the dimensionless distance is

$$ \xi(x)=\frac{x}{A_0} $$

where $A_0=a_0/(1\cdot 10^{-10})$ is the Bohr radius in $Å$ (length), and $a_0$ is the Bohr radius in SI units. Proceeding à-la engineer, the conversion is:

$$ V(x)\left[eV\right]=\left[\frac{eV}{cm^{-1}}\right]\cdot V'(x)\left[cm^{-1}\right]\tag{1}\label{one} $$

where the conversion factor $\mathit{CF}$ is obtained from the definition of eV (see this for info):

$$ \left[\frac{eV}{cm^{-1}}\right]=\frac{|e|}{100|h||c|}\cdot hc=\mathit{CF}\cdot hc\approx 8066\cdot hc $$

with $|e|$, $|h|$, $|c|$ are the absolute values of the electron charge, Planck constant and speed of light in vacuum in SI units. Also, $hc$ is in SI units.

The conversion (eq. \ref{one}) becomes, once substituting each expression, as

$$ \alpha x^2=\mathit{CF}\cdot hc\cdot \beta\frac{x^2}{A_0^2} $$

and after simplifying $x$ and expliciting $\alpha$, the following conversion relation is obtained:

$$ \alpha\left[eV/Å^2\right]=\mathit{CF}'\cdot\frac{hc}{a_0^2}\cdot\beta\left[cm^{-1}\right]\qquad \mathit{CF}'=\frac{|e|}{100|h||c|}\cdot10^{-20} $$

where all physical constants are in SI units. Thus, after calculating the dimensionless unit conversion factor $\mathit{CF}'$ and plugging in all the physical constants in their SI units, you should be able to get the desired unit of measure from the initial one.

DISCLAIMER: I am assuming that you're in the middle of converting a hamiltonian from a wavenumber formulation to a atomic-unit formulation, but I could be wrong. The unit $\left[cm^{-1}\right]$ usually refers to an energy, and there are countless usages that could correspond to different physical interpretations of your target unit of measure. I have chosen the simplest one it came to mind, and I've shown how it is possible to connect the two.