Ramon Carbó

Molecular Quantum Similarity: theoretical Framework, Ordering Principles, and Visualization Techniques12

Publisher Summary This chapter focuses on the practical results of Quantum Similarity Measures (QSM). It defines and interprets a QSM according to quantum mechanical principles and this is accompanied by a construction of a QSM theoretical framework. The set of QSM can be transformed or combined to obtain a new kind of auxiliary terms that can be named “Similarity Indices.” There is a great deal of possible QSM manipulations leading to a variety of Quantum Similarity Index (QSI) definitions. Some of them are cosine-like similarity index, distance similarity index, and the Hodgkin-Richards or Tanimoto indices. The matrix representation of a molecular set can be associated to a set of finite dimensional vectors representing the molecules. This leads to the concept of “point-molecules” collected as a “Molecular Point Cloud.” To construct a solid theoretical body concerning QSM, a set of rules referred to as the Mendeleev Postulates are described in the chapter.

Foundations and recent developments on molecular quantum similarity

A general definition of the Quantum Molecular Similarity Measure is reported. Particular cases of this definition are discussed, drawing special attention to the new definition of Gravitational-like Quantum Molecular Similarity Measures. Applications to the study of fluoromethanes and chloromethanes, the Carbonic Anhydrase enzyme, and the Hammond postulate are presented. Our calculations fully support the use of Quantum Molecular Similarity Measures as an efficient molecular engineering tool in order to predict physical properties, biological and pharmacological activities, as well as to interpret complex chemical problems.

On quantum molecular similarity measures (QMSM) and indices (QMSI)

Quantum molecular similarity measures (QMSM) and the possibility to construct a discreten-dimensional representation of any electronic structure is briefly described. The quantum molecular similarity indices (QMSI) are presented next. They constitute a possible transformation of the initial QMSM, intended to be useful in a great variety of applications. A set of diverse possibilities in QMSI definitions is given. A comparison of the indices obtained directly from electronic density distributions with those derived from the QMSM discrete representation of molecules leads to a handful of useful results, allowing a mathematical connection between the initial description of Carbó and the Hodgkin-Richards QMSIs. From the discussion of this kind of comparative reasoning a description of new index forms can be deduced. A brief numerical example is given.

Triple density molecular quantum similarity measures: A general connection between theoretical calculations and experimental results

A newquantum similarity measure is defined. It is proposed as a way to project density functions from infinite dimensional spaces, where density functions belong, to finite dimensional ones. The procedure allows the representation of a given molecule initially described in terms ofnth order density functions as a point in a finite dimensional Euclidean space: apoint-molecule. Further manipulation of the obtained information permits the representation of a molecular set as a cloud of points: amolecular point cloud, in a variety of graphical manners. A previous experience in the field is compared with this new approach.

Definition, mathematical examples and quantum chemical applications of nested summation symbols and logical Kronecker deltas

Abstract Two mathematical symbols are introduced and their applications to computational chemistry and to artificial intelligence are described. The first symbol is called a nested summation symbol (NSS). After a discussion of its properties, we insist on the significance of this symbol in developing sequential and parallel computational algorithms. Another group of symbols, related to logical expressions, are defined under the name logical Kronecker deltas (LKDs). Application examples of the NSS and LKD technique to several computational quantum chemistry topics are given. It is shown how some standard formulae become shorter and easier to write, generalize, program and evaluate. It is emphasized that both symbols constitute a powerful link between mathematical formalism and programming in high level languages.

Elementary Unitary MO Transformations and SCF Theory

Publisher Summary The rules that can be deduced from the discussion presented in this chapter enable obtaining the mathematical form of the variation upon the rotation of an active pair of any quantum mechanical object in the framework of molecular orbital (MO) theory. As a consequence, a simple energy-variation expression might be obtained, leading to various computational mechanisms with a high degree of control possibilities and producing energy extrema with much less difficulty than the classical self-consistent field (SCF) procedures. The chapter discusses that the procedure that has been called “single rotation” seems to be the most efficient of the computational techniques tested. If a high degree of precision is required, one might choose the double-rotation technique at the risk of needing more CPU time than in single-rotation (SR) techniques. The other possibilities tested are less interesting from a computational point of view, and in many cases, the performances obtained are very similar to SCF procedures. It must be stressed that Jacobi-like rotation techniques need not resemble the formalism of SCF. The best computational performances are obtained when Jacobis rotations are implemented with computational philosophy. Given a structured computational hardware, in which the elementary steps can be performed in parallel fashion and not sequentially, the search for optimal MO coefficients may be reduced to times that will be like some classical extended Hiickel performances. It is not difficult to predict for the near future a rising interest in the Jacobi rotation techniques.

Nested summation symbols and perturbation theory

A new mathematical concept, the nested summation symbol (NSS) has been developed. This concept is attached to a mathematical linear operator directly related to the summation symbols. After a discussion on its properties, we investigate the potential usefulness of this symbol in the developing of sequential and parallel computational algorithms, constituting a powerful link between mathematical formalism and high level languages programming. A NSS is well suited in order to express some kind of mathematical formulae and to implement them in any computational environment. In this sense, NSSs are directly related toartificial intelligence techniques. Nested sums are connected withgeneralized nested do loop (GNDL) structures, a programming concept developed in our laboratory. This paper shows an application of the NSS. The NSS concept has been used to obtain in a compact form the expressions of the general energy and wavefunction corrections associated to the perturbation theory under the Brillouin-Wigner or the Rayleigh-Schrödinger formalisms.

Many Center AO Integral Evaluation Using Cartesian Exponential Type Orbitals (CETO'S)1

Publisher Summary This chapter discusses a general, simple and pedagogical framework based on Cartesian exponential type orbitals (CETOs) functions, to obtain atomic and molecular integrals, which can be potentially used within a LCAO computational system. One and two electron integrals over various operator kinds have been solved and analytical forms found as well, proving in this manner the flexibility and complete possibilities of the proposed methodology. Two center density expansions into separate centers have been employed successfully to overcome the many center integral problem. CETO functions appear in this manner as a plausible alternative to the present Gaussian Type Orbitals (GTO) quantum chemical computational flood, constituting the foundation of another step signaling the path toward Slater type orbital (STO) integral calculation. Many interesting methodological ideas, along the description of CETOs and their alternative forms: spherical exponential type orbitals (SETOs), laplace exponential type orbitals (LETOs), well-oriented CETOs (WO-CETOs), and elementary CETO (E-CETOs), have been also defined: Logical Kronecker delta and nested summation symbols among others. Both concepts permit easily written integrals and related formulae within a compact mathematical formalism, which possess an immediate and intuitive translation to high level programming languages.

Open Shell SCF Theory: An ab Initio Study of Some Interstellar Molecules

Publisher Summary The chapter presents a description of a restricted self-consistent field (SCF) open shell framework, where all the necessary hints are given in order to facilitate implementation. It claims that neither priority nor definitive form, is an attempt to describe in detail a sufficiently general open shell SCF procedure that is—(1) easily programmable, (2) without insuperable difficulties during application, and (3) with such characteristics that the program structure may be implemented in any breed of computer. Some interstellar molecules provide an amount of computational work sufficient for study of the appropriate features of the methodology. The different molecular states are computed by a choice of trial vectors and the freezing of the appropriate state symmetry by means of a shift operator. Furthermore, as it should be expected, computation time is very sensitive to the number of basis-functions used. The number of SCF cycles is increased when going from ground state to excited states, and does not seem to bear a direct relation to basis set size, but to the need to use larger shifts in order to get smooth convergence features.

Generalized Rayleigh-Schrödinger perturbation theory in matrix form

Thenested summation symbols (NSS) formalism is used as a starting point to formulate a completely general Rayleigh-Schrödinger perturbation theory (RSPT) scheme. In order to make the theoretical framework practical from a computational point of view, the matrix form for the theory is given in every case. As a result, an algorithmic iterative recipe to compute eigenvalue and eigenvector corrections up to any order is described. Degenerate systems are also treated. At the same time the described procedure allows the computation of eigenvalue and eigenvector derivatives with respect to a set of parameters.