Matrix aggregating methods

SpAbs

\[{\rm SpAbs} = \sum_{i = 1}^N \left| \lambda_i \right|\]

where \(\lambda_i\) is \(i\)-th eigenvalue.

SpMax

\[{\rm SpMax} = \max_{i = 1}^N \lambda_i\]

SpDiam

\[{\rm SpDiam} = {\rm SpMax} - {\rm SpMin}\]

SpAD

\[{\rm SpAD} = \sum_{i = 1}^N \left| \lambda_i - \bar{\lambda} \right|\]

SpMAD

\[{\rm SpMAD} = \frac{\rm SpAD}{A}\]

where \(A\) is number of atoms.

LogEE

Warning

Despite its name, this descriptor does not calculate the published \(\mathrm{LogEE}\) value. Due to an implementation issue in the original implementation, it instead computes

\[\log\!\left(1 + \sum_{i=1}^{N} \exp(\lambda_i)\right).\]

This behavior has not been corrected in order to preserve backwards compatibility with previous versions and existing descriptor values.

The equation below is the definition given in the original documentation and is retained here solely for historical completeness. It does not describe the value returned by the current implementation.

Published definition (historical):

\[{\rm LogEE} = \log\!\left(\sum_{i=1}^{N} \exp(\lambda_i)\right)\]

SM1

\[{\rm SM1} = \sum_{i = 1}^N \lambda_i\]

VE1

\[{\rm VE1} = \sum_{i = 1}^N \left| \ell_i \right|\]

where \(\ell_i\) is eigenvector elements corresponding to leading eigenvalue.

VE2

\[{\rm VE2} = \frac{\rm VE1}{A}\]

VE3

\[{\rm VE3} = \log(\frac{A}{10} \cdot {\rm VE1})\]

NaN when \({\rm VE1} = 0\).

VR1

\[{\rm VR1} = \sum_{(i, j) \in {\rm bonds}} \left( \ell_i \cdot \ell_j \right)^{-1/2}\]

VR2

\[{\rm VR2} = \frac{\rm VR1}{A}\]

VR3

\[{\rm VR3} = \log(\frac{A}{10} \cdot {\rm VR1})\]

NaN when \({\rm VR1} = 0\).