chemtools.topology.eigenvalues.EigenValueTool

class chemtools.topology.eigenvalues.EigenValueTool(eigenvalues, eps=1e-15)[source]

Class of descriptive tools based on eigenvalues.

Initialize class.

Parameters:
  • eigenvalues (np.ndarray) – A two-dimensional array recording the eigenvalues of each point in a row.
  • eps (float, optional) – The error bound for being a zero eigenvalue.
eigenvalues

Eigenvalues of points.

ellipticity

Ellipticity of electron density.

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

Bond descriptor defined as 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) of the set of eigenvalues.

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

Index of critical point which is the number of negative-curvature directions.

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

Rank of the critical point.

The rank of a critical point is the number of positive eigenvalues. This is used to classify critical points on it’s stability (trajectories going in or out).

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

Signature of point(s) which is the difference of number of positive & negative values.

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

Rank and signature of the critical point.

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

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.