Molecular quantum similarity matrix

Bultinck, P. and Carbo-Dorca, R. (2003) Molecular quantum similarity matrix based clustering of molecules using dendrograms. J. Chem. Inf. Comput. Sci., 43, 170-177. 

We have already used similarity matrices to cluster molecules. As such, they provide the necessary data to investigate the construction of a molecular set taxonomy. The most common techniques to do so include the molecular point clouds previously described. There, the columns of the molecular quantum similarity matrix yielded coordinates of the molecules in the N-dimensional space. Often, the N dimensionality cannot yet be used for a graphical representation. However, several techniques exist to reduce the dimensionality of the data, which allow it to be represented graphically on common devices like a computer screen or a plotter. In addition to these plots, several instances have involved Kruskal trees and other algorithms. °... 

Another matter is the consistency of the molecular quantum similarity matrix Z. The MQSM produced by a specific alignment technique for a given molecular pair of the set of molecules that construct Z should not be contradictory with the computed MQSM for the other pairs of molecules. To illustrate this point, consider the Euclidean distance, as defined by the square root of Eq. [13] ... 

Once an operator has been chosen for the calculation of the MQSM for a set of N molecules, one can calculate all MQSMs between every two molecules, which gives rise to the whole N x N array of MQSM. This symmetrical matrix is called the molecular quantum similarity measure matrix (MQSMM), denoted Z. 

Another important concept in molecular quantum similarity is associated with convex conditions. The idea underlying convex conditions, associated with a numerical set, a vector, a matrix, or a function, has been described previously in the initial work on VSS and related issues.Convex conditions correspond to several properties of some mathematical objects. The symbol X(x) means that the conditions (x) = 1 A x V(R ) hold simultaneously for a given mathematical object x, which is present as an argument in the convex conditions symbol. Convex conditions become the same as considering the object as a vector belonging to the unit shell of some VSS. For such kind of elements,... 

Even if other methodologies may provide better results, it must be stated that the methodology presented in this work, and that includes descriptor generation, similarity matrix transformation, and statistical procedure, has not been altered in any way to take into account the nature of the studied system. In this way, the exposed QSAR protocol is potentially capable of handling and characterizing different molecular biological activities from diverse molecular sets without introducing further information than those provided by quantum similarity, which is based on electronic... 

Whenever the commercially available particles do not match the operator s requirements, a variety of possibilities exist in order to modify the particles from company suppliers. Similarly to other doped beads the dyes [92] or quantum dots [107, 108] can be physically entrapped into magnetic beads by swelling or are covalently bound to the surface of the particles. If localization of the dye on the particle surface is desired or if the polarity of dye and/or matrix polymer does not allow the irreversible entrapment of the dye in the bulk polymer, a covalent attachment of the dye is preferable [109, 110]. Even the covalent binding of whole fluorescent nanoparticles to magnetic microparticles is possible, as shown by Kinosita and co-workers who investigated the rotation of molecular motors [111]. 

The fact that matrix elements of the fundamental band are dependent on the rotational quantum numbers j, f cannot be ignored. As a consequence, many more B coefficients must now be evaluated which, in principle, poses no special problem. The volume of data needed renders the task awkward. Molecular spectroscopists have for generations coped with similar problems which were solved with the so-called Dunham expansion in terms of j(j+1). Specifically, for our purpose, the lowest-order expansion for the fundamental band looks like [63]... 

Similar selection rules hold for molecular spectra. In fact, let T and V. be wave functions for two levels in any quantum mechanical system. Then if P is the appropriate operator, a transition between levels i and j is permitted if the matrix element... 

Abstract. The Chebyshev operator is a diserete eosine-type propagator that bears many formal similarities with the time propagator. It has some unique and desirable numerical properties that distinguish it as an optimal propagator for a wide variety of quantum mechanical studies of molecular systems. In this contribution, we discuss some recent applications of the Chebyshev propagator to scattering problems, including the calculation of resonances, cumulative reaction probabilities, S-matrix elements, cross-sections, and reaction rates. 

Warshel and Chu [42] and Hwang et al. [60] were the first to calculate the contribution of tunneling and other nuclear quantum effects to PT in solution and enzyme catalysis, respectively. Since then, and in particular in the past few years, there has been a significant increase in simulations of quantum mechanical-nuclear effects in enzyme and in solution reactions [16]. The approaches used range from the quantized classical path (QCP) (for example. Refs. [4, 58, 95]), the centroid path integral approach [54, 55], and variational transition state theory [96], to the molecular dynamics with quantum transition (MDQT) surface hopping method [31] and density matrix evolution [97-99]. Most studies of enzymatic reactions did not yet examine the reference water reaction, and thus could only evaluate the quantum mechanical contribution to the enzyme rate constant, rather than the corresponding catalytic effect. However, studies that explored the actual catalytic contributions (for example. Refs. [4, 58, 95]) concluded that the quantum mechanical contributions are similar for the reaction in the enzyme and in solution, and thus, do not contribute to catalysis. 

Some recent developments concerning macromolecular quantum chemistry, especially the first linear-scaling method applied successfully for the ab initio quality quantum-chemistry computation of the electron density of proteins, have underlined the importance and the applicability of quantum chemistry-based approaches to molecular similarity. These methods, the linear-scaling numerical Molecular Electron Density Lego Approach (MEDLA) method [6 9] and the more advanced and more generally applicable linear-scaling macromolecular density matrix method called Adjustable Density Matrix Assembler or ADMA method [10,11], have been employed for the calculation of ab initio quality protein electron densities and other... 

---