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Encyclopedia of Complexity and Systems Science pp 7422–7480Cite as

Quantum Similarity and Quantum Quantitative Structure-Properties Relationships (QQSPR)

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Definition of the Subject

The concept of Quantum Similarity (QS) was introduced for the first time in 1980 in a paper by Carbó et al. [1] entitled: How Similar is a Molecule to Another? There the basic aspects ofthe theory were set up. The backbone of QS was constituted by the conceptual support of the QSM.

QSM.

A QSM between two quantum objects (QO): associated to the density function (DF) tags: \( { \{\rho_A ,\rho_B\} } \) was defined as the density overlap integral:

$$ Z_{AB} =\int_D \rho_A (\mathbf{r}) \rho_B (\mathbf{r})\mskip2mu\mathrm{d}\mathbf{r} = \langle\rho_A\rho_B\rangle =\langle\rho_B\rho_A\rangle = Z_{BA}\:, $$

which is always a positive real number, because the involved DF are non-negative definite real functions. Self-similarity measure integrals were defined in the same manner:

$$ I=A,B\colon Z_{II} =\int_D \rho_I (\mathbf{r}) \rho_I(\mathbf{r})\mskip2mu\mathrm{d}\mathbf{r} = \big\langle\rho_I^2\big\rangle\:. $$

QS Matrix.

A simple example, which remains formally valid...

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Notes

  1. 1.

    The Hadamard product (sometimes also called Schur or Kronecker product) is related to the multiplication result of two sums and constructed by the sum of the resultant diagonal cross-terms only. In this way, the inward (or Hadamard) product of two sums can be specified by the following algorithm:

    $$ \bigg(\sum\limits_I^N a_I\bigg)\ast \bigg(\sum\limits_I^N b_I\bigg) = \sum\limits_I^N a_I b_I\:. $$

    Both sums shall have the same number of terms N, for the IMP being feasible.

  2. 2.

    By the symbol: \( { \mathbf{A}\ast > 0 } \), applied to an arbitrary matrix \( { \mathbf{A}=\{a_{ij}\} } \), is meant that \( { \forall i,j\colon a_{ij} > 0 } \).

  3. 3.

    In the case where the nature of the operator or the set of operators, active in the Sobolev norm definition, has to be specified, then the notation Hilbert–Sobolev ?-, O- or ?-spaces, can be obviously employed.

Abbreviations

Glossary:

There is a brief description of the terms used herein. The defined items appear in alphabetical order. When in a definition a term already defined in the glossary is mentioned, it is written in bold face; then, the reader has to refer to the corresponding glossary item, where more information is given.

Carbó (similarity) index:

The Carbó (Similarity) Index is a QS index, which corresponds to the cosine of the angle subtended by the density function (DF) tags of any pair of quantum objects. The values of the Carbó index lie within the interval \( { (0,1] } \). The lower bound corresponds to a complete dissimilarity, while the unit value is encountered when comparing a quantum object with itself. See also: Euclidian Distance (Similarity) Index.

Core set:

A QOS with the additional information of a known and well-defined property value for every QO.

(First order) density function (DF):

The first order DF is a quantum theory concept, associated to a known electronic submicroscopic system. To be understood by this term is a wave function squared module or the full DF, reduced by integration over the space and spin coordinates to a function of the space coordinates of a unique electron. As it is customary in the literature, the name will be shortened to DF. Such a function can be also obtained via direct computation within Density Functional Theory (DFT). DF can be employed as tags associated to well-defined quantum objects. DF are non-negative functions. According to the usual quantum mechanical interpretation, within the DF collection there is contained all the information one can extract from the system. This last statement is the basic postulate which appears in first place when developing QQSPR theory.

(Full) density function:

A quantum theoretical concept, associated to a known electronic (or other particle sets) system. It corresponds to the system's wave function squared module.

Density function (DF) tags:

The tag set, (see: tagged set) collecting the quantum mechanical density functions used as descriptors in a QOS.

DQOS:

Discrete Quantum Object Set. A QOS whose tags are finite dimensional vectors, whose elements, in turn, are computed as quantum similarity measures.

Euclidean distance (similarity) index:

Like the Carbó Index the Euclidean Distance Index is a quantum similarity index, involving the Euclidean distance between two density functions. The range of this index is \( { [0,\infty) } \), thus their minimal values correspond to two identical quantum objects, while the index grows in relation to the difference in the compared quantum objects.

FQQSPR equation:

Fundamental Quantum Quantitative Structure-Properties Relationships equation. The subject of study and analysis of the present contribution. A non-empirical equation, which can be deduced by quantum mechanical theoretical means, serving as the basic tool to obtain QQSAR or Quantum QSAR. Here, only the linear FQQSPR equations are deeply discussed, as they are the main source of QQSPR studies, but the FQQSPR equation can easily be extended to any order.

Molecular descriptors:

Parameters of varied origin: empirical, theoretical or experimental, or functions obtained from quantum mechanical manipulations like the Density Function or the Electrostatic Molecular Potential, or by empirical considerations, associated to a given molecule, which represent the molecular structure environment and can be employed to obtain QSPR. Any set of parameters or functions, which can be used as tags in a tagged set, whose objects are molecules.

QO:

Quantum Object. An element of a QOS, a tagged set where every element is constructed as a composite of a submicroscopic system description (the object) and a density function (the tag). The possibility to represent all the information contained within any quantum system by the full or reduced DF, permits one to describe such an entity formed by the structure of the system and its DF together; this can be called a QO. Plural: QO's.

QOS:

Quantum Object Set. A tagged set made of QO's.

QS:

Quantum Similarity: A discipline dealing with similarity measures between submicroscopic systems. Such kinds of quantum similarity measures (QSM) can be computed using the quantum theoretical description of such kinds of objects. According to quantum theory all the information one can obtain about a quantum system is contained in the state DF and the set of possible reduced forms, hence quantum similarity is part of this information contained in the DF. Usually, for computational convenience, the (first order reduced) DF has been employed as a universal descriptor for comparison purposes and it is the employed tag of a QO. Similarity comparisons become possible by means of comparing the QO DF tags.

QSAC:

Quantum Similarity Aufbau Condition: A condition that the QS Matrix has to comply with in order to be positive definite and admit Choleski decomposition.

QSAR & QSTR:

QSAR are QSPR employed to estimate biological molecular activity values via molecular descriptors. When QSPR are employed to estimate molecular toxicity, they can be called QSTR.

Quantum similarity matrices (QS matrices):

[Depending on the context so as not to cause confusion with the term QS Measures the abbreviation QSM can also be used to denote QS Matrices]: Any matrix which contains computed QSM results involving several QO. Usually the QS Matrix is symmetric and square when the QSM on the involved elements of a unique QOS are ordered in pairs. In this case the QSAC must hold. The QS Matrix can be rectangular when computing and ordering into a matrix the QSM of two different QOS. QS Hypermatrices appear when higher order QSM associated to more than two DF are involved.

Quantum similarity measures (QSM) and indices (QSI):

A positive definite integral, computed employing the QO DF tags as integrands and, if necessary, including a positive definite operator which can be supposed to act as a weight. Possessing the structure of a measure, such an integral can be interpreted as a generalized volume. QSM between two or several QO correspond to integrals, which can be constructed primarily with integrands made with the product of the density tags of the compared quantum objects plus a positive definite operator. Such a definition ensures the positive definite nature of the QS integrals. A quantum self-similarity measure corresponds to the QSM computed with the density function tags of a unique QO. QSI are manipulations of the QSM in order to obtain QS comparisons within an adequate range.

QSPR:

(Classical) Quantitative Structure-Properties Relationships. This term refers to any empirical relationship permitting the connection between molecular structure, represented by a set of parameters (molecular descriptors) of any origin and molecular properties. Usually, by a classical QSPR can be understood a non-causal multilinear relationship, obtained via statistical reasoning and procedures. This is the generic name given to any functional (usually linear) connecting the properties of a molecule with the attached molecular descriptors. The QSPR functionals are empirically obtained by statistical analysis, usually employing (non-)linear regression techniques or any variant of it.

QQSPR:

Quantum Quantitative Structure-Properties Relationships. These are non-empirical functionals derived from the structure of a FQQSPR equation. Thus, this kind of relationship, if it exists, to some extent can be considered causal. A QQSPR permits us to compute the molecular properties of U-molecules, just employing non-empirical considerations and parameters or molecular descriptors of quantum mechanical origin. In this sense, these relationships can be considered universal, applicable to any molecular structure. However, due to the quantum mechanical nature of the QQSPR, one can obtain structure-properties relationships between QO's of any kind: nuclei, atoms, molecules. By QQSPR are here understood QSPR obtained by means of the application of canonical quantum theoretical methods to obtain expectation values of a Hermitian operator, which within QQSPR theory is described by functionals of the density function tags of a given QOS, acting as a basis set. QQSPR are more general than classical QSPR as they can be obtained for any QO, incorporating information difficult to take into account with classical molecular descriptors. The descriptors in the QQSPR framework are the DF tags of the QO.

QS matrix:

Quantum Similarity Matrix. The matrix possessing positive definite elements, constructed by quantum similarity measures using the DF tags of the elements of a QOS. See: Quantum Similarity Matrices.

Tagged set:

A set constructed as the Cartesian product of two separate sets. One of them, the object set is composed of well-defined elements of any nature, called objects. The other set in the tagged set is the tag set, made of some descriptors attached to every object, which is supposed to contain information about the objects; tags can be mathematical elements made by bit strings, vectors, matrices, functions…

U-m:

Unknown (properties) molecule. A molecule which can be described as a QO, but lacks the property information associated to the molecules belonging to the Core Set. U-m properties act as the unknowns to be evaluated in the QQSPR theory developed here.

U-molecule:

A U-m.

U-m set:

A QOS containing U-m's as elements.

Wave function:

A by-product of solving the Schrödinger equation. When studying stationary submicroscopic systems by means of classical quantum mechanics, the pair energy – wave function correspond to the eigenvalue – eigenfunction pairs of the Hermitian Hamiltonian operator constructed for the system, which substitutes the classical Hamilton function. The squared module of the stationary wave functions for each system state is customarily interpreted, since Born times, as the probability density function to find the system as a whole in some space infinitesimal element of volume.

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    Ramon Carbó-Dorca & Ana Gallegos

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Carbó-Dorca, R., Gallegos, A. (2009). Quantum Similarity and Quantum Quantitative Structure-Properties Relationships (QQSPR). In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_440

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