Quantum Similarity Maximization—MaxiSim and QSSA

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. 

To overcome this computational dilemma, Constans et al. created an efficient algorithm that proceeds in different steps. Using the ASA method, the authors succeed in replacing the true integral measure of Eq. [21] by a summation, which was accomplished by introducing a function in the ASA density expression that effectively compresses the atomic shells. In the limit, the atomic electron densities are infinitely compacted into the nuclei. To illustrate the procedure, consider first expression [42]. A compression parameter 1 is introduced in the different shells such that 

The molecular electron density then becomes in the limit of t approaching infinity  

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. 

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