Molecular quantum similarity operators

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] ... 

Based on the molecular quantum similarity measmes. Molecular Quantum Self-Similarity Measures (MQS-SM) were proposed as molecular descriptors where each molecule is compared with itself and all the others, and appropriate Hermitian operators 2 are associated to each molecular property [Ponec et al, 1999]. 

Before proceeding to other operator types, it should be mentioned that the density functions introduced here are not the only possible way to carry out such analyses in molecular quantum similarity. It has been shown how extended wave functions may be derived that also hold, e.g., the gradient of the wave functions, which means that a new class of wave functions may be derived that are vector-like, much like what is found in relativistic quantum theory. However, it is beyond the scope of the present chapter to discuss this entire field, and the interested reader is referred to the literature, especially Carbo-Dorca et al. where a clear discussion is given. 

The idea of molecular quantum similarity can be extended to other operators, provided they are positive definite. In this sense, they will lead to real, positive definite values for the MQSM evaluated over the density functions of the involved quantum objects. 

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. 

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... 

Similar transformation can be applied to any molecular quantum mechanical operator. The dipole moment operator can be convmiiently expressed as... 

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. 

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]. 

In a quantum chemical calculation on a molecule we may wish to classify the symmetries spanned by our atomic orbitals, and perhaps to symmetry-adapt them. Since simple arguments can usually give us a qualitative MO description of the molecule, we will also be interested to classify the symmetries of the possible MOs. The formal methods required to accomplish these tasks were given in Chapters 1 and 2. That is, by determining the (generally reducible) representation spanned by the atomic basis functions and reducing it, we can identify which atomic basis functions contribute to which symmetries. A similar procedure can be followed for localized molecular orbitals, for example. Finally, if we wish to obtain explicit symmetry-adapted functions, we can apply projection and shift operators. 

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... 

Similar to quantum mechanics, which can be formulated in terms of different quantities in addition to the traditional wave function formulation, in quantum chemistry a number of alternative tools are developed for this purpose, which may be useful in the context of the present book. We have already described different approximate models of representing the electronic structure using (many-electronic) wave functions. The coordinate and second quantization representations were employed to get this. However, the entire amount of information contained in the many-electron wave function taken in whatever representation is enormously large. In fact it is mostly excessive for the purpose of describing the properties of any molecular system due to the specific structure of the operators to be averaged to obtain physically relevant information and for the symmetry properties of the wave functions the expectation values have to be calculated over. Thus some reduced descriptions are possible, which will be presented here for reference. 

In the QM/MM method the system is usually a priori divided into QM (the solute) and classical (MM, the solvent) parts, and an effective operator describes the interaction between the two subsystems. The solvent molecules are treated with a classical force field ( classical meaning that there are no elementary particles or quantum effects ) that opens the possibility to take a much larger number of solvent molecules into account. Optionally, the whole system can be embedded in a continuum, e.g., for taking large-range interactions into account. Similar to the continuum approach, the solute is separated from the solvent and its molecular properties are therefore well defined. The remaining problem is to find an accurate approximate representation of... 

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. 

MQSM are integrals involving two or more DITs attached to molecular systems and an optional operator. In this part we will try to describe a general framework from which the similarity betw n quantum objects can be computed. 

An equation similar to the Eq. (4) holds true for general quantum amplitudes E) and general hamiltonian operators. Eq. (4) is a model constructed from a coordinate projection procedure. The wave function F(x, p) is the projection on coordinate space of the general probability amplitude (x, p) = (x, p ) [14], Since E(x, p) are eigenfunctions of the molecular hamiltonian, the only way to change the state of a system prepared in a given stationary state is via the coupling operator U. 

---