Molecular quantum similarity basis functions

The overall form of each of these equations is fairly simple, ie, energy = a constant times a displacement. In most cases the focus is on differences in energy, because these are the quantities which help discriminate reactivity among similar stmctures. The computational requirement for molecular mechanics calculations grows as where n is the number of atoms, not the number of electrons or basis functions. Immediately it can be seen that these calculations will be much faster than an equivalent quantum mechanical study. The size of the systems which can be studied can also substantially ecHpse those studied by quantum mechanics. 

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. 

As to the content of Volume 25, the Editors thank the authors for their contributions, which give an interesting picture of part of the current state of the art of the quantum theory of matter From nonlinear-optical calculations, over a study of ion motion in molecular channels, a treatment of molecular integrals over Gaussian basis functions, and an investigation of soliton dynamics in franr-polyacetylene, to applications of quantum molecular similarity measures. 

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. 

The choice and generation of basis sets has been addressed by many authors [190,192,528,554-563]. While we consider here only the basic principles of basis-set construction, we should note that this is a delicate issue as it determines the accuracy of a calculation. Therefore, we refer the reader to the references just given and to the review in Ref. [564]. In Ref. [559] it is stressed that the selection of the number of basis functions used for the representation of a shell riiKi should not be made on the grounds of the nonrelativistic shell classification nj/j but on the natural basis of j quantum numbers resulting in basis sets of similar size for, e.g., Si/2 and pi/2 shells, while the p /2 basis may be chosen to be smaller. As a consequence, if, for instance, pi/2 and p /2 shells are treated on the tijli footing, the number of contracted basis functions may be doubled (at least in principle). The ansatz which has been used most frequently for the representation of molecular one-electron spinors is a basis expansion into Gauss-type spinors. 

Atomic basis functions in quantum chemistry transform like covariant tensors. Matrices of molecular integrals are therefore fully covariant tensors e.g., the matrix elements of the Fock matrix are F v = (Xn F Xv)- In contrast, the density matrix is a fully contravariant tensor, P = (x IpIx )- This representation is called the covariant integral representation. The derivation of working equations in AO-based quantum chemistry can therefore be divided into two steps (1) formulation of the basic equations in natural tensor representation, and (2) conversion to covariant integral representation by applying the metric tensors. The first step yields equations that are similar to the underlying operator or orthonormal-basis equations and are therefore simple to derive. The second step automatically yields tensorially correct equations for nonorthogonal basis functions, whose derivation may become unwieldy without tensor notation because of the frequent occurrence of the overlap matrix and its inverse. 

Similarly, expanding the KS potential in an LCAO expansion makes molecular density-functional calculations practical [9]. For metals and similar crystalline solids, it is best to expand the Kohn-Sham potential in momentum space via Fourier coefficients. For molecular solids various real-space method are under investigation. For molecules studied with the big, well-chosen Gaussian basis sets of quantum chemistry, it is undoubtedly best to expand the KS potential in linear-combination-of-Gaussian-type-orbital (LCGTO) form [10]. 

Apparently, the concept of similarity plays an important role in the chemistry of functional groups. Motivated by the recent revival of interest in molecular similarity [7-39], we shall present a systematic approach towards a quantum chemical description of functional groups. There are two main components of the approach described in this report. The first component is shape-similarity, based on the topological shape groups and topological similarity measures of molecular electron densities[2,19-34], whereas the second component is the Density Domain approach to chemical bonding [4]. The topological Density Domain is a natural basis for a quantum... 

Despite the interest to obtain AO integral algorithms over cartesian exponential orbitals or STO fimctions [43] in a computational universe dominated by GTO basis sets [2], this research was started as a piece of a latter project related to Quantum Molecular Similarity [44], with the concurrent aim to have the chance to study big sized molecules in a SCF framework, say, without the need to manipulate a huge number of AO functions. 

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