Quantum similarity maximization

Two objects are similar and have similar properties to the extent that they have similar distributions of charge in real space. Thus chemical similarity should be defined and determined using the atoms of QTAIM whose properties are directly determined by their spatial charge distributions [32]. Current measures of molecular similarity are couched in terms of Carbo s molecular quantum similarity measure (MQSM) [33-35], a procedure that requires maximization of the spatial integration of the overlap of the density distributions of two molecules the similarity of which is to be determined, and where the product of the density distributions can be weighted by some operator [36]. The MQSM method has several difficulties associated with its implementation [31] ... 

Molecular quantum similarity measures, as formulated in integral (1), are dependent on the relative position of both studied molecules in space. Consequently, a procedure capable of arranging the molecular coordinates needs to be established. Two methodologies have been implemented to deal with this question the maximal similarity rule (MSR) [71], which considers that the optimal orientation corresponds to the one that maximizes the value of integral (1) and the topo-geometrical superposition algorithm... 

From a mathematical point of view, we can propose maximization of the MQSM as a function of a set of molecular alignment parameters in such a way as to obtain a superposition procedure with a general scope. Several algorithms aimed directly at maximizing the MQSM have been published. The first example is the MaxiSim algorithm developed by Constans et al. A second one is the quantum similarity superposition algorithm (QSSA) by Bultinck et al. An algorithm developed by Stefanov and Cioslowski is similar to the MaxiSim and QSSA ideas, and it will therefore not be described explicitly because it invokes similar points of view. 

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. 

It is beyond the scope of this chapter to discuss the range of structure-based methods that chemists can use for molecular afignment. This field of research has been, and continues to be, very active. One algorithm, called TGSA, will be presented here in some detail, however, because of its popularity in molecular quantum similarity studies. Structure-based techniques differ from the aforementioned techniques in several respects. First, they not attempt to maximize the MQSM for a pair of molecules. Second, they do not make a specific reference to molecular quantum similarity as such, they are aimed at a wider range of applications. Third, they are not based on electron density in a formal way, but instead they take a more familiar approach based on chemical topology. Consequently, they apply well-known concepts such as chemical bonds and try to overlap the most similar and largest common structure elements in both molecules. 

When the zero-quantum 1% transition is greater than double-quantum Wi, the nOe enhancements will be negative. Similarly, when is greater than 1%, the resultant nOe will have a positive sign. The predominance of Wi and over one another depends on the molecular motion. It is known that the Wo transition is maximal when the molecule tumbles at a rate of about 1 KHz, while the Wi transition is fastest at a tumbling rate of about 800 MHz. On this basis, a rough idea of the sign of nOe can be obtained. For example, small molecules in nonviscous... 

Within the context of the current systems, only the weak fluorescing features of C6o that are maximized at 710 nm should be mentioned explicitly. A representative emission spectrum is illustrated in Fig. 8.2a. Regarding, the phenylene-acetylene building blocks, the major features include strong visible light fluorescence with quantum yields close to unity. Similarly to the absorption maxima, the fluorescence... 

The specific structure of the states for Hp was described in detail in [79], where it is mentioned as a well-known physical effect. For example, it was noted in the theory of disordered semiconductors that a similar "ladder" structure of states is realized for the system where the Coulomb potential is modified within a sphere as a constant potential (see [86,87] for a qualitative discussion and analytical solution of the problem). For quantum chemistry, the situation is interesting, as was shown in a series of publications of Connerade, Dolmatov and others (see e.g. [19,88-91] note that the series of publications on confined many-electron systems by these authors is much wider). The picture described is realized to some extent for the effective potential of inner electrons in multi-electron atoms, as it is defined by orbital densities with a number of maximal points. The existence of a number of extrema generates a system of the type described above [89]. This situation was modeled and described for the one-electron atom in [88] it is similar to that one described in Sections 5.2 and 5.3. 

Type-A measures are appropriate for measuring the similarity between one or, at most, a few pairs of structures that are characterized by quantum-mechanical descriptions of various sorts. Following the pioneering studies of Carbo et al.io a large number of similarity measures have been described that fall in this class (see, e.g.. Refs. 11—14). Here, a molecule is described by an electron probability density function, and the similarities between pairs of molecules are calculated with measures that describe the overlap of their density functions. Work in this area is exemplified by that of Manaut et al.i" who describe a procedure to align two molecules so as to maximize the similarity... 

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