chemtools.topology.point.EigenValueTool

class chemtools.topology.point.EigenValueTool(eigenvalues, eps=1e-15)

Class of descriptive tools based on eigenvalues.

Initialize class.

Parameters:

• eigenvalues (np.ndarray(N, 3)) – A 2-D array recording the eigenvalues at each \(N\) point.
• eps (float, optional) – The error bound for being a zero eigenvalue.

eigenvalues

Eigenvalues.

ellipticity

Ellipticity.

\[\frac{\lambda_\text{min}}{\lambda_\text{min-1}} - 1\]

bond_descriptor

Bond descriptor which is the ratio of average of positive and negative eigenvalues.

\[\frac{\left(\frac{\sum_{\lambda_k > 0} \lambda_k}{\sum_{\lambda_k > 0} 1}\right)} {\left(\frac{\sum_{\lambda_k < 0} \lambda_k}{\sum_{\lambda_k < 0} 1}\right)}\]

eccentricity

Eccentricity (essentially the condition number).

\[\sqrt{\frac{\lambda_\text{max}}{\lambda_\text{min}}}\]

index

Index which is the number of negative-curvature directions.

\[\sum_{\lambda_k < 0} 1\]

rank

Rank which is the number of positive eigenvalues.

\[\sum_{\lambda_i > 0} 1\]

signature

Signature which is the difference of number of positive & negative eigenvalues.

\[\sum_{\lambda_k > 0.} 1 - \sum_{\lambda_k < 0.} 1\]

morse

Rank and signature.

\[\left(\sum_{\lambda_k > 0} 1, \sum_{\lambda_k > 0.}1 - \sum_{\lambda_k < 0.} 1\right)\]

A system is degenerate if it has a zero eigenvalue and consequently, it’s critical point is said to be “catastrophe”. It returns a warning in this case.