where α ij and β ij are two corresponding matrix elements in the matrices A and
B that represent each molecule in the pair.
After the electronic structure calculation yields a wavefunction file,
AIMAll/AIMStudio program [18] is used to calculate the localization and delocalization indices. A Python program (AIMLDM), developed by Sumar et al. [25],
extracts the localization and delocalization indices from AIMAll’s output and
calculates the matrix invariants as well as the Frobenius distances.
3.2.3 Some Limitations of LDMs and Possible Solutions
LDMs share well-known limitations with all matrix representatives of molecular
graphs when used as a tool for comparing different molecules. These limitations are
briefly outlined along with possible solutions. In this Sect. (3.2.3), only the proposed solutions are outlined leaving examples of their actual usages in a subsequent
Sect. (3.4) below.
3.2.3.1 Ambiguity of Atomic Labelling
Any matrix representation of the molecular graph, complete or incomplete, is
labelling-dependent since there exists n! ways to label the n-atoms composing a
given molecule. Unless all compared molecules have very similar graphs and can
be given consistent atomic labelling, e.g. benzoic acids substituted, say, at the paraposition by monoatomic substituents such as halogens, one must rely on “matrix
invariants”.
Labelling-independent invariants extracted from a matrix representation of a
molecular graph include, for example, the characteristic polynomial, the eigenvalues, the trace, and the determinant.
LDMs, by being real and symmetric, are diagonalizable by a similar
transformation:
P
À1
fP ¼ D;
ð3:12Þ
where D is the diagonalized LDM. The eigenvalues can then be organized as a
vector sorted in a consistent order of, say, increasing value.
60
C.F. Matta et al.
B that represent each molecule in the pair.
After the electronic structure calculation yields a wavefunction file,
AIMAll/AIMStudio program [18] is used to calculate the localization and delocalization indices. A Python program (AIMLDM), developed by Sumar et al. [25],
extracts the localization and delocalization indices from AIMAll’s output and
calculates the matrix invariants as well as the Frobenius distances.
3.2.3 Some Limitations of LDMs and Possible Solutions
LDMs share well-known limitations with all matrix representatives of molecular
graphs when used as a tool for comparing different molecules. These limitations are
briefly outlined along with possible solutions. In this Sect. (3.2.3), only the proposed solutions are outlined leaving examples of their actual usages in a subsequent
Sect. (3.4) below.
3.2.3.1 Ambiguity of Atomic Labelling
Any matrix representation of the molecular graph, complete or incomplete, is
labelling-dependent since there exists n! ways to label the n-atoms composing a
given molecule. Unless all compared molecules have very similar graphs and can
be given consistent atomic labelling, e.g. benzoic acids substituted, say, at the paraposition by monoatomic substituents such as halogens, one must rely on “matrix
invariants”.
Labelling-independent invariants extracted from a matrix representation of a
molecular graph include, for example, the characteristic polynomial, the eigenvalues, the trace, and the determinant.
LDMs, by being real and symmetric, are diagonalizable by a similar
transformation:
P
À1
fP ¼ D;
ð3:12Þ
where D is the diagonalized LDM. The eigenvalues can then be organized as a
vector sorted in a consistent order of, say, increasing value.
60
C.F. Matta et al.