Quantum similarity indices

Until now the molecular quantum similarity measure has primarily been the integral measure Zab- The direct comparison of two values Zab and Zcd does not directly yield an idea of the degree of similarity. Consequently, a numerical transformation must be established, which allows the comparison of the similarity degree between different pairs of molecules. 

It is almost impossible to enumerate and discuss all of the similarity indices that have been published. Furthermore, new indices continue to be published. Therefore, this discussion will be limited to those indices that have been most common in the application in molecular quantum similarity. For a general review of similarity indices and their application, the reader is referred to the work by Willett, Barnard and Downs. 

The most common index in molecular quantum similarity is the generalized cosine, introduced by Carbo et al. in their first paper on quantum 

This is an example of the so-called C-class descriptors, which give a value in the interval [0,1], where a higher value signifies a greater degree of similarity. One of the nice features of these indices is that they can be extended beyond the overlap similarity given in Eq. [70]. So, with any operator fi, a similarity measure can be defined as 

In the Carbo index, the denominator is the geometric mean of the self-similarities. Naturally, we could also think about using the arithmetic mean, which gives rise to the Hodgkin-Richards index originally developed for comparisons of electrostatic potentials  

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By different mathematical transformations. Molecular Quantum Similarity Indices (MQSI) are derived from molecular quantum similarity measures. They are divided into two main classes C-class indices, referred to as correlation-like indices ranging from 0 (maximum dissimilarity) to 1 (maximum similarity), and D-class indices, referred to as distance-like indices ranging from 0 (maximum similarity) to infinity (maximum dissimilarity). C-class indices can be transformed into D-class indices d, by the following ... 

Quantum Self-Similarity Measures quantum similarity quantum similarity = quantum similarity Quantum Similarity Indices quantum similarity Quantum Similarity Measures quantum similarity representation molecular descriptors rigidity —> flexibility indices... 

By different mathematical transformations. Molecular Quantum Similarity Indices (MQSI) are derived from molecular quantum similarity measures. They are divided into two main classes C-class indices, referred to as correlation-like indices ranging from 0 (maximum... 

Besalu, E., Gallegos, A. and Carbo-Dorca, R. (2001) Topological quantum similarity indices and their use in QSAR application to several families of antimalarial compounds. MATCH Commun. Math. Comput. Chem., 44, 41-64. 

Once a set of quantum objects to study is chosen and the operator related to the MQSM definition in Eq. (2) is defined, the MQSM related to the set is unique. But they can be transformed or combined in order to obtain a new kind of auxiliary terms which can be named Quantum Similarity Indices (QSI). A vast quantity of possible MQSM manipulations leading to a variety of QSI definitions exists. The most used QSI are as follows. 

Quantum Similarity Measure has been outlined and several related Quantum Similarity Indices have been defined in Section 5. 

Lobato M, Amat L, Besalu E, Carbo-Dorca R. Structure-activity relationships of a steroid family using quantum similarity measures and topological quantum similarity indices. Quant Struct-Act Relatsh 1997 16 465-472. 

The work mentioned so far relies on quantum similarity indices that in turn rely on the overlap measure of electron densities. Simple overlap between electron density functions is, however, not the only possible choice. Based on the developed theory of vector semispaces and tagged sets, Carbo et al. have... 

Cooper and Allan ° have used momentum density in several studies. A problem remains in obtaining the momentum space densities because most calculations are performed with position space wave functions. In a sense, working in momentum space is yet another way to reduce the overweighting of the core electron density. Most of the following discussions on, e.g., molecular alignment and quantum similarity indices, remain valid when we... 

As discussed earlier, we cannot only derive first-order electron densities, but also we can extend them to higher order electron densities. We have used the second-order electron density p(ri,r2) in lieu of the first-order electron density on several occasions in molecular quantum similarity, because the second-order electron density is in fact the lowest order density where electron correlation becomes apparent. It has been used extensively by Ponec et al. "- in the study of similarity in pericyclic reactions where the second-order electron density offers important advantages over the first-order electron density. In another contribution, Ponec et al. went to the third-order electron density. Again, most of the discussion relating to molecular quantum similarity indices and molecular alignment is also applicable to higher order electron densities, replacing where necessary the first-order electron density by, for example, the second-order electron density. 

What physical meaning can be attached to the molecular quantum similarity indices calculated with some positive definite operator No direct indication exists that any meaning should be attached in general. Eor the... 

The work discussed below is situated in the context of a mathematically rigorous theory of quantum similarity measures (QSM) and quantum similarity indices (QSI) as developed by Carbo [5, 37]. Following Carbo, we define the similarity of two atoms (a and b) as a QSM... 

Fig. 9.4 Quantum similarity indices for noble gases, using the Dirac-delta function as separation...

Lobato, M., Amat, L., Besalil, E. and Carbd-Dorca, R. (1997). Structure-Activity Relationships of a Steroid Family Using Quantum Similarity Measures and Topological Quantum Similarity Indexes. Quant.Struct.-Act.Relat., 16,465-472. 

As a first step to the recovery of the periodic patterns in Mendeleev s table, Carbd s quantum similarity index (9.69) was used, with the Dirac-5 as separation operator. In this case the expression (9.69) reduces to an expression for shape functions (9.66). 

Structure effects on the rate of selective or total oxidation of saturated and unsaturated hydrocarbons and their correlations have been used successfully in the exploration of the reaction mechanisms. Adams 150) has shown that the oxidation of alkenes to aldehydes or alkadienes on a BijOj-MoOj catalyst exhibits the same influence of alkene structure on rate as the attack by methyl radicals an excellent Type B correlation has been gained between the rate of these two processes for various alkenes (series 135, five reactants, positive slope). It was concluded on this basis that the rate-determining step of the oxidation is the abstraction of the allylic hydrogen. Similarly, Uchi-jima, Ishida, Uemitsu, and Yoneda 151) correlated the rate of the total oxidation of alkenes on NiO with the quantum-chemical index of delo-calizability of allylic hydrogens (series 136, five reactants). 

Field-based similarities are usually evaluated by the cosine or correlation function similarity measure employed initially by Carbo and co-workers (67) to compute molecular similarities based upon quantum mechanical wavefunctions. Such a measure, which is usually called a Carbo similarity index, is given by... 

Meyer-Richards similarity index quantum similarity... 

Molecular Similarity and QSAR. - In a first contribution on the design of a practical, fast and reliable molecular similarity index Popelier107 proposed a measure operating in an abstract space spanned by properties evaluated at BCPs, called BCP space. Molecules are believed to be represented compactly and reliably in BCP space, as this space extracts the relevant information from the molecular ab initio wave functions. Typical problems of continuous quantum similarity measures are hereby avoided. The practical use of this novel method is adequately illustrated via the Hammett equation for para- and me/a-substituted benzoic acids. On the basis of the author s definition of distances between molecules in BCP space, the experimental sequence of acidities determined by the well-known a constant of a set of substituted congeners is reproduced. Moreover, the approach points out where the common reactive centre of the molecules is. The generality and feasibility of this method will enable predictions in medically related Quantitative Structure Activity Relationships (QSAR). This contribution combines the historically disparate fields of molecular similarity and QSAR. 

If the functions f and g are correlated, then C = 1. By omitting the factors -1 in numerator and denominator of Eq. [38], we obtain the index proposed by Carbo et al.226 The basic features of this parameter are contained in the numerator, which is essentially an overlap integral between f(t) and g(r). This integral with first-order density functions can be seen as the simplest example of a more general family of quantum similarity measures.23.236... 

Maximizing the Carbo index has been described in several previous scientific reports. McMahon and King described the use of gradient methods in 1997, and in the same year, Parretti et described the use of Monte Carlo techniques. In many studies, including the two just mentioned, these maximizations do not refer to quantum similarity, but instead they refer to maximizing the similarity in molecular electrostatic potentials, which is different. 

This new quantum similarity-dissimilarity index formulation does not necessarily have to coincide with the original similarity-dissimilarity indices (Equations 17.4 and 17.5). Therefore, both matrices (the original metric of the QOS DF Z and the metric of the DQOS tag set elements Z ) might provide complementary geometrical and topological information about the associated QO point cloud. A discussion on the nature of the QS metric matrices has been recently published. More information on this QS feature can be obtained. 

The fluorescent components are denoted by I (intensity) followed by a capitalized subscript (D, A or s, for respectively Donors, Acceptors, or Donor/ Acceptor FRET pairs) to indicate the particular population of molecules responsible for emission of/and a lower-case superscript (d or, s) that indicates the detection channel (or filter cube). For example, / denotes the intensity of the donors as detected in the donor channel and reads as Intensity of donors in the donor channel, etc. Similarly, properties of molecules (number of molecules, N quantum yield, Q) are specified with capitalized subscript and properties of channels (laser intensity, gain, g) are specified with lowercase superscript. Factors that depend on both molecular species and on detection channel (excitation efficiency, s fraction of the emission spectrum detected in a channel, F) are indexed with both. Note that for all factorized symbols it is assumed that we work in the linear (excitation-fluorescence) regime with negligible donor or acceptor saturation or triplet states. In case such conditions are not met, the FRET estimation will not be correct. See Chap. 12 (FRET calculator) for more details. 

Also other Type B and C series from Table II are consistent with the above elimination mechanisms. The dehydration rate of the alcohols ROH on an acid clay (series 16) increased with the calculated inductive effect of the group R. For the dehydrochlorination of polychloroethanes on basic catalysts (series 20), the rate could be correlated with a quantum-chemical reactivity index, namely the delocalizability of the hydrogen atoms by a nucleophilic attack similar indices for a radical or electrophilic attack on the chlorine atoms did not fit the data. The rates of alkylbenzene cracking on silica-alumina catalysts have been correlated with the enthalpies of formation of the corresponding alkylcarbonium ions (series 24). Similar correlations have been obtained for the dehydrosulfidation of alkanethiols and dialkyl sulfides on silica-alumina (series 36 and 37) in these cases, correlation by the Taft equation is also possible. The rate of cracking of 1,1-diarylethanes increased with the increasing basicity of the reactants (series 33). 

King, J.W. (1993) The inverse molecular transform index a descriptor for molecular similarity analysis. Int.J. Quantum Chem. Quant. Biol. Symp., 20, 139-145. 

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