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Kolaru authored Aug 18, 2023
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Expand Up @@ -4,7 +4,7 @@ This is a simple package that provides utility function to compute normal modes

The other use of this package is to give me a space to ramble about normal modes, and I will shamelessly use this README for this purpose.

# Proper introduction
# A proper introduction of normal modes

Because I find the subject confusing, and the introductions describing the problem as well, I think it is worth laying down the basics here.

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- `wave_number` : Wave numbers of the modes (in inverse cm).
- `reduced_mass` : Reduced masses of the modes (in AMU), following [this question](https://physics.stackexchange.com/questions/401370/normal-modes-how-to-get-reduced-masses-from-displacement-vectors-atomic-masses) and (I believe) similar to what Gamess does.

The 3 last are returned as Unitful quantity.

# Caveats

Currently all internal calculations are performed in atomic units. If a return is returned without units, it is either unitless (like the modes) or in atomic units, as the use of Unitful quantities is not finished yet.

The code still only half convince myself, is poorly documented and tested, and is missing a bunch of nice API.

# Appendix

## Derivation of the uncoupled equation
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as it would make everything collapse. Thankfully with our choice of ${\rm \bf M}$ it is exactly what happens.

Putting it in and using the fact that ${\rm \bf e}_j^T {\rm \bf e}_k$ is a Kroenecker delta $\delta_{j k}$, we get, as expected,
Putting it in and using the fact that ${\rm \bf e}_j^T {\rm \bf e}_k$ is a Kroenecker delta $\delta{j k}$, we get, as expected,

$$
\sum_i \frac{-\hbar^2}{2 m_i} \frac{\partial^2}{\partial x_i^2}
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