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https://github.com/kolaru/normalmodes.jl

Small utilities to deal with normal modes in molecular physics
https://github.com/kolaru/normalmodes.jl

molecular-dynamics molecule normal-modes

Last synced: 27 days ago
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Small utilities to deal with normal modes in molecular physics

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README 

        
# NormalModes.jl



This is a simple package that provides utility function to compute normal modes from a Hessian matrix, and then work with them.



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.



# A proper introduction to 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.



We are interested in the ground state of an equilibrium Hamiltonian $\mathcal{H} = \mathcal{T} + \mathcal{V}$, whith $\mathcal{T}$ the kinetic energy operator and $\mathcal{V}$ the potential energy operator. We note ${\rm \bf x}$ the vector of concatenated position, that is ${\rm \bf x} = {\rm vcat}({\rm \bf r}_1, {\rm \bf r}_2, {\rm \bf r}_3, \dots)$[^julia], with ${\rm \bf r}_A$ the position of atoms $A$.



Then we do a Taylor expansion of the potential, leading to



$$

\mathcal{V}({\rm \bf x}) = V_0 + \nabla \mathcal{V}({\rm \bf x}) + {\rm \bf x}^T {\rm \bf H} {\rm \bf x} + \mathcal{O}({\rm \bf x}^3),

$$



where ${\rm \bf H}$ is the Hessian matrix of the potential[^potential], and $V_0$ a constant shift that we can set to zero. Also since we are assuming to be at equilibrium, the gradient $\nabla \mathcal{V}({\rm \bf x})$ is zero as well. This leads to the approximation



$$

\mathcal{V}({\rm \bf x}) = {\rm \bf x}^T {\rm \bf H} {\rm \bf x}

$$



and the Hamiltonian



$$

\mathcal{H} = \sum_i \frac{-\hbar^2}{2 m_i} \frac{\partial^2}{\partial x_i^2} + {\rm \bf x}^T {\rm \bf H} {\rm \bf x},

$$



where $m_i$ is the mass of dimension $i$[^mass]. Now this is the Hamiltionian of a set of coupled harmonic oscillator, and we will change basis to rewrite it as a system of *uncoupled* harmonic oscillators.



We introduce the diagonal mass weighting matrix ${\rm \bf M}$ with ${\rm \bf M}_{ii} = m_i^{-\frac{1}{2}}$. We do an eigendecomposition of the weighted Hermitian $\rm \bf M H M$ as



$$

{\rm \bf M H M} = {\rm \bf U \Omega U}^T,

$$



where $\rm \bf U$ is the matrix of vector and $\rm \bf \Omega$ the diagonal matrix of eigenvalues, that we write ${\rm \bf \Omega}_{ii} = \omega_i^2$[^posdef].



We have what we need to define the new coordinates $\rm \bf z$ that we need, namely through



$$

{\rm \bf x} = {\rm \bf M U z}.

$$



Inserting it in the Hamiltonian we get the decoupled system of harmonic oscillators (see the annex for the derivation)



$$

\mathcal{H} = \sum_j \frac{-\hbar^2}{2} \frac{\partial^2}{\partial z_j^2} + \omega_j^2 z_j^2.

$$



We can check wikipedia to get whatever we need from it, like for example the Wigner distribution for sampling, and it is how it was implemented in the package.



# The package



The package aim at providing an interface to normal modes, but what are they? That's where it gets confusing. The andidates are



1. The eigen vectors ${\rm \bf u}_j$ that (the columns of $\rm \bf U$). They are normed and orthogonal, but live in the mass weighted space.

2. The columns of $\rm \bf MU$. They live in real space, but they are neither normed nor orthogonal.

3. The columns of $\rm \bf MU$ normalized. Still not orthogonal to each other but at least they have norm 1. In addition all information about $\rm \bf M$ is removed, which is needed to convert between $\rm \bf x$ and $\rm \bf z$ or to perform Wigner sampling. So those are ofter return together with so-called *mode masses*, the norm of the columns of $\rm \bf MU$ before being normed[^lunacy].



As far as I know the last one is what, despite my strong feelings, is known as *normal modes*. To avoid users a slow descent into madness, the package provides a specific object type `NormalDecomposition(hessian)`, that stores the useful information and encapsulate the inner confusing parts. Available are

- `normal_modes` : The normal modes (in real space), useful for example to plot them.

- `sample` : Perform Wigner sampling, and get the displacements from the mean for positions and momenta.

- `frequencies` : Frequencies of the modes (in GHz).

- `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.



# Appendix



## Derivation of the uncoupled equation



We want to substitute ${\rm \bf x} = {\rm \bf M U z}$ into



$$

\mathcal{H} = \sum_i \frac{-\hbar^2}{2 m_i} \frac{\partial^2}{\partial x_i^2} + {\rm \bf x}^T {\rm \bf H} {\rm \bf x}.

$$



By the definition of the diagonal matrix ${\rm \bf M}$ and the orthogonal matrix ${\rm \bf U}$ we have



$$

{\rm \bf x}^T {\rm \bf H x} =

    {\rm \bf z}^T {\rm \bf U}^T {\rm \bf M} {\rm \bf HMUz} = {\rm \bf z}^T {\rm \bf \Omega z},

$$



where ${\rm \bf \Omega}$ is diagonal.



For the other term, we use Leibnitz chain rule, starting with the first derivative only



$$

\frac{\partial}{\partial x_i} =

    \sum_j \frac{\partial z_j}{\partial x_i} \frac{\partial}{\partial z_j}.

$$



Next we use ${\rm \bf z} = {\rm \bf U}^T {\rm \bf M}^{-1} {\rm \bf x}$ to compute



$$

\frac{\partial z_j}{\partial x_i} =

    \frac{\partial}{\partial x_i} {\rm \bf e}_j^T {\rm \bf U}^T {\rm \bf M}^{-1} {\rm \bf x} =

    {\rm \bf e}_j^T {\rm \bf U}^T {\rm \bf M}^{-1} {\rm \bf e}_i,

$$



where ${\rm \bf e}_i$ are the unit vectors of the canonical basis. Since no new

dependency in ${\rm \bf x}$ or ${\rm \bf z}$ appear, we can apply it

twice. We transpose one of them however, to make the vector with index $i$

appear together[^transpose_trick]



$$

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

    -\frac{\hbar^2}{2} \sum_{i, j, k} \frac{1}{m_i}

        \frac{\partial z_k}{\partial x_i}

        \left[\frac{\partial z_j}{\partial x_i} \right]^T

    = -\frac{\hbar^2}{2} \sum_{j, k}

        {\rm \bf e}_k^T {\rm \bf U}^T {\rm \bf M}^{-1}

        \left[ \sum_i \frac{1}{m_i} {\rm \bf e}_i {\rm \bf e}_i^T \right]

        {\rm \bf M}^{-1} {\rm \bf U} {\rm \bf e}_j

        \frac{\partial}{\partial z_j} \frac{\partial}{\partial z_k}.

$$



Now it would be really beautiful if



$$

\sum_i \frac{1}{m_i} {\rm \bf e}_i {\rm \bf e}_i^T = {\rm \bf M}^2,

$$



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, we get, as expected,



$$

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

    = -\frac{\hbar^2}{2} \sum_j \frac{\partial^2}{\partial z_j^2}.

$$



## Footnotes



[^julia]: I am indeed using the julia function `vcat` to describe this mathematical operation, I have never found a better way to express it.



[^potential]: This makes sense because in the position representation the potential is not an operator but just a scalar.



[^mass]: Which is the mass of the atom from which this component comes from. Essentially in the vector of mass $m_i$ each mass is repeated 3 times, once for each spatial dimension of each atom.



[^posdef]: Eigenvalue are in theory positive or zero, because the Hamiltonian is semi-positive definite and symmetric. Numerically the smallest eigenvalues can be negative instead of exactly zero.



[^annex]: The correct transformation of the derivative requires a bit of work to properly simplify, but trust me you can do it.



[^lunacy]: Which is pure lunacy if you ask me, the code I was using was norming the component in one function, and then immediately unnorming them in the next one. 



[^transpose_trick]: We transpose a scalar, which is a trick I like very much and

which is surprisingly useful.