A Toy Model of Chemical Reaction Networks - TBI - Universität Wien

3.4. WAVE FUNCTION ANALYSIS 27
Derwent Fragmentation Codes, which are often used in patent search
systems. The idea is to subdivide the structure into fragments that are
then represented by three-letter codes of the form Genus / Species /
Subspecies, e.g. EAD = Alcohol/free alcohol/free aromatic alcohol.
This systems is rather complicated, hard to adapt to new chemistries,
and very hard if not impossible to convert into a canonical representation.
• Connection Tables. A molecule is represented as a list of atoms and a
list of bonds connecting these atoms. There is a plethora of file formats.
MOL and SDF Files are simple flat format files, and the most used ones,
see http://www.mdli.com/downloads/literature/ctfile.pdf.
3.4 Wave function analysis
The generalized eigenvalue problem can be transformed into a standard (symmetric)
eigenvalue problem using a congruence transformation by factorizing
the metric, here overlap matrix [124, sect. 1.31, 5.71]. Because the basis set
is orthogonalized, the transformation is called orthogonalization.
In the Toy Model, Löwdin’s symmetric orthogonalization with S = S 1/2 S 1/2
is used. S is assumed to be positive-definite (it should be noted that too big
overlaps (> 0.8) may lead to a non-positive-definite overlap matrix). Eq. 3.11
then transforms to the symmetric form:
(S −1/2 HS −1/2 )C ′ = C ′ E, (3.14)
where C ′ = S 1/2 C.
Using this method, the overlap matrix stays exactly symmetric and fewer
numerical errors than in the canonical method are introduced. In canonical
orthogonalization, a Cholesky decomposition S = LL T is used. This method
is more efficient than the calculation of S −1/2 but more error-prone. Numerical
errors may also break the symmetry of a molecule. However, there are
further developments leading to linear scaling algorithms for those matrix
computations [19].
The total electronic energy of the molecule is obtained by the formula
E = ∑ i
n i E i , (3.15)
where n i is the occupation number of the MO φ i .
The electron distribution is derived from eq. 3.9 by using a partitioning,
e.g. isolating one summand for each atom. The only way of identifying in the