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# Triple Density Quantum Similarity Measures and the Tensorial Representation of Molecular Quantum Object Sets

DOI link for Triple Density Quantum Similarity Measures and the Tensorial Representation of Molecular Quantum Object Sets

Triple Density Quantum Similarity Measures and the Tensorial Representation of Molecular Quantum Object Sets book

# Triple Density Quantum Similarity Measures and the Tensorial Representation of Molecular Quantum Object Sets

DOI link for Triple Density Quantum Similarity Measures and the Tensorial Representation of Molecular Quantum Object Sets

Triple Density Quantum Similarity Measures and the Tensorial Representation of Molecular Quantum Object Sets book

## ABSTRACT

TDQS Measures ........................................................................... 292 12.4.3 Discrete Representation of MQOS Elements in

TDQS Framework ........................................................................ 294 *1E-mail: [email protected]

12.4.4 TDQS Matrices and Computer Memory Compact Storage ......................................................................................... 295

12.4.5 A Simple Albeit Illustrative Example of the TDQSH Structure ....................................................................................... 296

12.5 Transformation of the TDQS Hyper Vectors: Generalized Carbó Similarity Indices ......................................................................... 298

12.5.1 Preliminary Considerations ............................................ 298 12.5.2 Definitions of the TD Carbó Similarity Index ............... 299 12.5.3 The p-Like GTO Functions Example ............................ 300 12.5.4 Geometric Interpretation of the Generalized CSI .......... 300 12.5.5 Some General Remarks About TDCSI and Higher DF

Order CSI ..................................................................................... 302 12.5.6 Alternative Local CSI Definition ................................... 302 12.5.7 The Local TDCSI Transformation for the p-Like GTO

Functions Example ...................................................................... 302 12.6 MQOS Tensorial Representations and Their Possible Comparison

Relationships ................................................................................ 303 12.6.1 The p-Like GTO Functions Example ............................ 304 12.6.2 DDQS Over the Simplified p-Like GTO Example ........ 306 12.6.3 Some Comments About the Linear Independence of

TDQSH Submatrices ................................................................... 306 12.6.4 Carbó Similarity Indices Over Condensed TDQSH ...... 307 12.6.5 The p-Like Simple Application Example ...................... 308 12.6.6 Distances Using TDQSH Submatrices .......................... 308 12.6.7 Euclidian Distances Between the p-Type GTO Tensor

Representations ............................................................................ 309 12.6.8 Higher Order Carbó Similarity Indices .......................... 310 12.7 Quantum QSPR (QQSPR) and the Origin of TDQS Measures ....311 12.8 Binary States TD Representations and Origin Shifts ................... 313 12.9 Description of a Devoted Program Suite ..................................... 316 12.10 Conclusions ................................................................................. 316 Keywords .............................................................................................. 318 References ............................................................................................. 318

12.1 INTRODUCTION

In this chapter, molecular quantum similarity (QS) measures involving three density functions are studied, providing the necessary algorithms and programming sources for application purposes. Triple density representation of some known molecular quantum object set (MQOS) permits to express each element as a symmetric matrix with dimension equal to the MQOS cardinality. Such matrix formulation appears instead of the vector representations founded on double density QS (DDQS) measures or as a result of the usual classical descriptor parameterization. The whole triple density quantum similarity (TDQS) measures description of a given MQOS corresponds to a third order hypervector or tensor, whose elements are symmetric matrix representations of every molecular structure belonging to the MQOS. Such tensorial representation permits to set up an extended set of procedures in order to study the relationships between the MQOS elements, beyond the usual vector description. For the sake of completeness the quantum mechanical origin of the TDQS integrals is sketched as an introduction. The three p-type Gaussian orbitals are employed along the theoretical development to illustrate, via a concrete and particularly simple case example, the structure of the definitions encountered along the discussion, as well as the ability of QS measures to discriminate between DF belonging to degenerate wave functions. The present study can be considered in this way a first step towards the general theory and computational feasibility of a hypermatricial or tensorial representation of molecular structures associated to any MQOS. Generalized Carbó similarity indices (CSI) are also studied as a way to manipulate the TDQS measures for easy interpretation. Besides the appropriate description of the programs associated to this chapter, here are given several application examples, based essentially on the same background philosophy as the usual DDQS measures of previous papers.