This work describes a new procedure to obtain optimal molecular superposition based on quantum similarity (QS): the geometric-quantum similarity molecular superposition (GQSMS) algorithm. It has been inspired by the QS Aufbau principle, already described in a previous work, to build up coherently quantum similarity matrices (QSMs). The cornerstone of the present superposition technique relies upon the fact that quantum similarity integrals (QSIs), defined using a GTO basis set, depend on the squared intermolecular atomic distances. The resulting QSM structure, constructed under the GQSMS algorithm, becomes not only optimal in terms of its QSI elements but can also be arranged to produce a positive definite matrix global structure. Kruskal minimum spanning trees are also discussed as a device to order molecular sets described in turn by means of QSM. Besides the main subject of this work, focused on MS and QS, other practical considerations are also included in this study: essentially the use of elementary Jacobi rotations as QSM refinement tools and inward functions as QSM scaling methods.

Computation of density gradient quantum similarity integrals is analyzed, while comparing such integrals with overlap density quantum similarity measures. Gradient quantum similarity corresponds to another kind of numerical similarity assessment between a pair of molecular frames, which contrarily to the usual up to date quantum similarity definitions are not measures, that is: strictly positive definite integrals. As the density gradient quantum similarity integrals are defined as scalar products of three real functions, they appear to possess a richer structure than the corresponding positive definite density overlap quantum similarity measures, while preserving the overall similarity trends, when the molecular frames are relatively moved in three-dimensional space. Similarity indices are also studied when simple cases are analyzed in order to perform more comparisons with density overlap quantum similarity. Multiple gradient quantum similarity integrals are also defined. General GTO formulae are given. Numerical results within the atomic shell approximation (ASA) framework are presented as simple examples showing the new performances of the gradient density quantum similarity. Fortran 90 programs illustrating the proposed theoretical development can be downloaded from appropriate websites.

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