Based on a method that apparently seems to be widely used in computational chemistry (cf <https://en.wikipedia.org/wiki/Anisotropic_Network_Model>)


Trying to build a very simple model with 3 atoms linked together in an equilateral triangle, Suppose the 2d coordinates of the 3 atoms are as follows:


$(x\_0, y\_0) = (0, 0)$


$(x\_1, y\_1) = (1/2, \frac{\sqrt{3}}{2})$


$(x\_2, y\_2) = (1, 0)$


Computing the Hessian matrix as described in the wikipedia article above I get the following matrix entries for the Hessian matrix H.





|  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- |
| 0 | 0 | 1/4 | $\sqrt{3}/4$ | 1 | 0 |
| 0 | 0 | $\sqrt{3}/4$ | 3/4 | 0 | 0 |
| 1/4 | $\sqrt{3}/4$ | 0 | 0 | 1/4 | -$\sqrt{3}/4$ |
| $\sqrt{3}/4$ | 3/4 | 0 | 0 | -$\sqrt{3}/4$ | 3/4 |
| 1 | 0 | 1/4 | -$\sqrt{3}/4$ | 0 | 0 |
| 0 | 0 | -$\sqrt{3}/4$ | 3/4 | 0 | 0 |



So I am expecting the matrix to have 3 eigenvalues equal to zero, corresponding to the 3 rigid body motions in 2 dimensions (2 translations one rotation).


However this is not the case, the 3 lowest eigenvalues are 1/2 and a multiplicity 2 eigenvalue around -0.65...


So what is wrong? Did I misinterpret the formula in this wikipedia article?


Isn't this the case that 3 eigenvalues should exactly be equal to 0? Or does this only occur asymptotically with large molecules as the number of atoms gets bigger?


In advance many thanks for any help!



Elastic network model Hessian rigid body motion 0 eigenvalues
Your construction of the stiffness matrix (Hessian or Kirchhoff matrix) is not correct. The $2\times 2$ diagonal blocks should not be zero,$^\ast$ these are minus the sums of the other blocks in the same row, $H\_{ii}=-\sum\_{j\neq i}H\_{ij}$, where $H\_{ij}=\begin{pmatrix}
(x\_j-x\_i)^2&(x\_j-x\_i)(y\_j-y\_i)\\
(y\_j-y\_i)(x\_j-x\_i)&(y\_j-y\_i)^2\end{pmatrix}$ denotes the $2\times 2$ block in position $i,j$.


Hence the correct stiffness matrix $H$ in your case $(x\_0, y\_0) = (0, 0),\;\;(x\_1, y\_1) = (\tfrac{1}{2}, \tfrac{1}{2}\sqrt{3}),\;\;(x\_2, y\_2) = (1, 0)$ is
$$H=\left(
\begin{array}{cccccc}
 -\tfrac{5}{4} & -\tfrac{\sqrt{3}}{4} & \tfrac{1}{4} & \tfrac{\sqrt{3}}{4} & 1 & 0 \\
 -\tfrac{\sqrt{3}}{4} & -\tfrac{3}{4} & \tfrac{\sqrt{3}}{4} & \tfrac{3}{4} & 0 & 0 \\
 \tfrac{1}{4} & \tfrac{\sqrt{3}}{4} & -\tfrac{1}{2} & 0 & \tfrac{1}{4} & -\tfrac{\sqrt{3}}{4} \\
 \tfrac{\sqrt{3}}{4} & \tfrac{3}{4} & 0 & -\tfrac{3}{2} & -\tfrac{\sqrt{3}}{4} & \tfrac{3}{4} \\
 1 & 0 & \tfrac{1}{4} & -\tfrac{\sqrt{3}}{4} & -\tfrac{5}{4} & \tfrac{\sqrt{3}}{4} \\
 0 & 0 & -\tfrac{\sqrt{3}}{4} & \tfrac{3}{4} & \tfrac{\sqrt{3}}{4} & -\tfrac{3}{4} \\
\end{array}
\right)$$
Its eigenvalues are $(-3,-3/2,-3/2,0,0,0)$; indeed, three eigenvalues are equal to zero.




---


$^\ast$ See, for example, equation (3) of [Anisotropic network model: systematic evaluation and a new web interface](https://academic.oup.com/bioinformatics/article/22/21/2619/251345)


