Holographic electron density shape theorem and its role in drug design and toxicological risk assessment

Fuzzy electron density fragments in macromolecular quantum chemistry, combinatorial quantum chemistry, functional group analysis, and shape-activity relations

Conspectus Just as complete molecules have no boundaries and have "fuzzy" electron density clouds approaching zero density exponentially at large distances from the nearest nucleus, a physically justified choice for electron density fragments exhibits similar behavior. Whereas fuzzy electron densities, just as any fuzzy object, such as a thicker cloud on a foggy day, do not lend themselves to easy visualization, one may partially overcome this by using isocontours. Whereas a faithful representation of the complete fuzzy density would need infinitely many such isocontours, nevertheless, by choosing a selected few, one can still obtain a limited pictorial representation. Clearly, such images are of limited value, and one better relies on more complete mathematical representations, using, for example, density matrices of fuzzy fragment densities. A fuzzy density fragmentation can be obtained in an exactly additive way, using the output from any of the common quantum chemical computational techniques, such as Hartree-Fock, MP2, and various density functional approaches. Such "fuzzy" electron density fragments properly represented have proven to be useful in a rather wide range of applications, for example, (a) using them as additive building blocks leading to efficient linear scaling macromolecular quantum chemistry computational techniques, (b) the study of quantum chemical functional groups, (c) using approximate fuzzy fragment information as allowed by the holographic electron density theorem, (d) the study of correlations between local shape and activity, including through-bond and through-space components of interactions between parts of molecules and relations between local molecular shape and substituent effects, (e) using them as tools of density matrix extrapolation in conformational changes, (f) physically valid averaging and statistical distribution of several local electron densities of common stoichiometry, useful in electron density databank mining, for example, in medicinal drug design, and (g) tools for combinatorial quantum chemistry approaches using fuzzy fragment databanks and rapid construction of a large number of approximate electron densities for large sets of related molecules, relevant in theoretical molecular and nanostructure design.

Electron density shape analysis of a family of through-space and through-bond interactions

A family of styrene derivatives has been used to study the effects of through-space and through-bond interactions on the local and global shapes of electron densities of complete molecules and a set of substituents on their central rings. Shape analysis methods which have been used extensively in the past for the study of molecular property-molecular shape correlations have shown that in these molecules a complementary role is played by the through-space and through-bond interactions. For each specific example, the dominance of either one of the two interactions can be identified and interpreted in terms of local shapes and the typical reactivities of the various substituents. Three levels of quantum chemical computational methods have been applied for these structures, including the B3LYP/cc-pVTZ level of density functional methodology, and the essential conclusions are the same for all three levels. The general approach is suggested as a tool for the identification of specific interaction types which are able to modify molecular electron densities. By separately influencing the through-space and through-bond components using polar groups and groups capable of conjugation, some fine-tuning of the overall effects becomes possible. The method described may contribute to an improved understanding and control of molecular properties involving complex interactions with a possible role in the emerging field of molecular design.

Accelerating materials property predictions using machine learning

The materials discovery process can be significantly expedited and simplified if we can learn effectively from available knowledge and data. In the present contribution, we show that efficient and accurate prediction of a diverse set of properties of material systems is possible by employing machine (or statistical) learning methods trained on quantum mechanical computations in combination with the notions of chemical similarity. Using a family of one-dimensional chain systems, we present a general formalism that allows us to discover decision rules that establish a mapping between easily accessible attributes of a system and its properties. It is shown that fingerprints based on either chemo-structural (compositional and configurational information) or the electronic charge density distribution can be used to make ultra-fast, yet accurate, property predictions. Harnessing such learning paradigms extends recent efforts to systematically explore and mine vast chemical spaces, and can significantly accelerate the discovery of new application-specific materials.

Information carriers and (reading them through) information theory in quantum chemistry

This Perspective discusses the reduction of the electronic wave function via the second-order reduced density matrix to the electron density ρ(r), which is the key ingredient in density functional theory (DFT) as a basic carrier of information. Simplifying further, the 1-normalized density function turns out to contain essentially the same information as ρ(r) and is even of preferred use as an information carrier when discussing the periodic properties along Mendeleev's table where essentially the valence electrons are at stake. The Kullback-Leibler information deficiency turns out to be the most interesting choice to obtain information on the differences in ρ(r) or σ(r) between two systems. To put it otherwise: when looking for the construction of a functional F(AB) = F[ζ(A)(r),ζ(B)(r)] for extracting differences in information from an information carrier ζ(r) (i.e. ρ(r), σ(r)) for two systems A and B the Kullback-Leibler information measure ΔS is a particularly adequate choice. Examples are given, varying from atoms, to molecules and molecular interactions. Quantum similarity of atoms indicates that the shape function based KL information deficiency is the most appropriate tool to retrieve periodicity in the Periodic Table. The dissimilarity of enantiomers for which different information measures are presented at global and local (i.e. molecular and atomic) level leads to an extension of Mezey's holographic density theorem and shows numerical evidence that in a chiral molecule the whole molecule is pervaded by chirality. Finally Kullback-Leibler information profiles are discussed for intra- and intermolecular proton transfer reactions and a simple S(N)2 reaction indicating that the theoretical information profile can be used as a companion to the energy based Hammond postulate to discuss the early or late transition state character of a reaction. All in all this Perspective's answer is positive to the question of whether an even simpler carrier of information than the electron density function ρ(r) can be envisaged: the shape function, integrating to 1 by construction fulfils this role. On the other hand obtaining the information (or information difference) contained in one (or two) systems from ρ(r) or σ(r) can be most efficiently done by using information theory, the Kulback-Leibler information deficiency being at the moment (one of) the most advisable functionals.

Communications on quantum similarity (2): A geometric discussion on holographic electron density theorem and confined quantum similarity measures

The so-called holographic electron density theorem (HEDT) is analyzed from an algebraic perspective, and a brief analytical point of view is also given. The connection of the HEDT with quantum similarity measures (QSM) over electronic density functions (DF) is studied using GTO functions, atomic ASA DF, and promolecular ASA DF. Restricted integration of QSM over a box of finite side length is discussed for all this DF. This work emphasizes the geometric aspects of HEDT, but for the sake of completeness, some analytical insight based on a general Taylor series expansion is also given at the end.

A Fundamental Relation of Molecular Informatics on the Information-Carrying Properties of Density Functions

The molecular electron density carries the complete information about the molecule. This information is stored in the shape and more general topological features of molecular electron densities. A fundamental relation of molecular informatics, building on the Hohenberg-Kohn theorem, is the holographic electron density theorem: any nonzero volume part of a molecular electron density in a non-degenerate electronic ground state contains the complete information about all properties of the entire molecule. This fundamental feature of all molecules applies to all exhibited and also to all latent molecular properties, where latent properties are those not normally exhibited, only in response to some external stimulus. Recently it has become feasible to compute ab initio quality electron densities and approximate forces acting on individual nuclei in large molecules, even those beyond the thousand atom range, such as proteins. The newly expanded size range where reliable modelling methods can be also applied extends the role of detailed molecular shape analysis to macromolecules. In this context, it has become possible to study how the fundamental information-carrying properties of electron density take a newly recognized role influencing the predominance of specific nuclear conformations within the family of astronomically many potentially stable conformations of some macromolecules. Some special problems and results are discussed.

On the importance of the “density per particle” (shape function) in the density functional theory

The central role of the shape function sigma(r) from the density functional theory (DFT), the ratio of the electron density rho(r) and the number of electrons N of the system (density per particle), is investigated. Moreover, its relationship with DFT based reactivity indices is established. In the first part, it is shown that an estimate for the chemical hardness can be obtained from the long range behavior of the shape function and its derivative with respect to the number of electrons at a fixed external potential. Next, the energy of the system is minimized with the constraint that the shape function should integrate to unity; the associated Lagrange multiplier is shown to be related to the electronic chemical potential micro of the system. Finally, the importance of the shape function for both molecular structure, reactivity, and similarity is outlined.

Linear scaling approaches to quantum macromolecular similarity: evaluating the similarity function

The evaluation of the electron density based similarity function scales quadratically with respect to the size of the molecules for simplified, atomic shell densities. Due to the exponential decay of the function's atom-atom terms most interatomic contributions are numerically negligible on large systems. An improved algorithm for the evaluation of the Quantum Molecular Similarity function is presented. This procedure identifies all non-negligible terms without computing unnecessary interatomic squared distances, thus effectively turning to linear scaling the similarity evaluation. Presented also is a minimalist dynamic electron density model. Approximate, single shell densities together with the proposed algorithm facilitate fast electron density based alignments on macromolecules.

Distributions and averages of molecular conformations

A computational technique is proposed for the study of electron density variations within a distribution of molecular conformations. These variations are defined in terms of the deviations of individual electron densities from the average density associated with the average of conformations within a conformational range of a molecule.