An algorithm to delineate and integrate topological basins in a three-dimensional quantum mechanical density function

The growing activity in the area of Quantum Chemical Topology warrants a new algorithm to delineate topological basins in 3D scalar fields other than the electron density. A method based on the "octal tree search algorithm" of computer graphics is proposed to reach this goal. We illustrate the algorithm on the L(r) function, which is the negative of the Laplacian of the electron density. Because of its complicated topology, even in a simple test molecule such as water, it benefits from the octal tree algorithm as a robust, compact, and general technique to find the boundaries of topological basins. For the first time, we are able to compute the population and volume of the core and valence (bonding and nonbonding, i.e., lone pair) basins given by L(r)'s topology.     

Representation of molecular structure using quantum topology with inductive logic programming in structure–activity relationships

The requirement of aligning each individual molecule in a data set severely limits the type of molecules which can be analysed with traditional structure activity relationship (SAR) methods. A method which solves this problem by using relations between objects is inductive logic programming (ILP). Another advantage of this methodology is its ability to include background knowledge as 1st-order logic. However, previous molecular ILP representations have not been effective in describing the electronic structure of molecules. We present a more unified and comprehensive representation based on Richard Bader's quantum topological atoms in molecules (AIM) theory where critical points in the electron density are connected through a network. AIM theory provides a wealth of chemical information about individual atoms and their bond connections enabling a more flexible and chemically relevant representation. To obtain even more relevant rules with higher coverage, we apply manual postprocessing and interpretation of ILP rules. We have tested the usefulness of the new representation in SAR modelling on classifying compounds of low/high mutagenicity and on a set of factor Xa inhibitors of high and low affinity.     

一维函数方法求解原子和分子薛定谔方程

对薛定谔方程的严格数值求解, 尤其是发展标准方法之外的、 包含新功能的解法, 一直是物理学研究的基本关注点. 本文介绍一种近些年发展的一维函数近似解方法, 该方法通过对波函数的不同坐标分量进行处理来求解原子和分子体系的薛定谔方程. 电子的试探波函数被离散化到实空间均匀格点上, 因此可以通过残差矢量校正的方法对其进行改进. 一维函数方法本身的特征决定其非常利于数值积分, 避免了很多由常规的多电子、 多中心势分子积分所带来的问题. 计算中, 最终能量是从严格的能量上限逐渐收敛所获得, 计算出的两电子薛定谔波函数呈现出常规单电子近似方法所含有的电子关联效应. 不同于密度泛函理论及Hartree-Fock的单电子解法, 本方法对电子-电子排斥能的多体效应的处理更加精确.     

Shape‐similarity relations based on topological resolution

Shape‐similarities of electron density clouds of molecules provide important clues concerning chemical and physical properties, including information about their reactivities in biochemical systems. The concept of topological resolution is used for quantifying molecular similarities: within a hierarchy of finer and cruder topologies, the crudest topology that already provides discrimination between two objects (such as two fuzzy electron density clouds) is used to define a measure of their similarities. The finer this topology, the more similar the two objects. This approach, the method of topological resolution‐based similarity measures (TRBSM), can be combined with a geometrically motivated resolution‐based similarity measure (RBSM) within a metric space. Some of the relations between these two approaches are discussed in this contribution, with special emphasis on applications to electron densities. This revised version was published online in July 2006 with corrections to the Cover Date.     

The localized electrons detector as an ab initio representation of molecular structures

The local single particle momentum is proposed as a localized‐electrons detector (LED) that provides a direct three‐dimensional representation of bonding interactions in molecules. It is given exclusively in terms of the electron density and its gradient. We show that the graphical representation of bonding interactions given by LED is consistent with the local curvatures of the electron density as given by the eigenvalues of the Hessian matrix, according to a local symmetry classification of the critical points here introduced. LED consistently complements the topological analysis of the electron density given by the quantum theory of atoms in molecules, by providing a graphical representation of the symmetry of the bonding interactions in molecular systems. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem 110:2418–2425, 2010     

Method and algorithm of obtaining the molecular intrinsic characteristic contours (MICCs) of organic molecules

The molecular intrinsic characteristic contour (MICC) is defined as the set of all the classical turning points of electron movement in a molecule. Studies on the MICCs of some medium organic molecules, such as dimethylether, acetone, and some homologues of alkanes, alkenes, and alkynes, as well as the electron density distributions on the MICCs, are shown for the first time. Results show that the MICC is an intrinsic approach to shape and size of a molecule. Unlike the van der Waals hard-sphere model, the MICC is a smooth contour, and it has a clear physical meaning. Detailed investigations on the cross-sections of MICCs have provided a kind of important information about atomic size changing in the process of forming molecules. Studies on electron density distribution on the MICC not only provide a new insight into molecular shape, but also show that the electron density distribution on the boundary surface relates closely with molecular properties and reactivities. For the homologues of alkanes, Rout(H), Dmin, and Dmax (the minimum and maximum of electron density on the MICC), all have very good linear relationships with minus of the molecular ionization potential. This work may serve as a basis for exploring a new reactivity indicator of chemical reactions and for studying molecular shape properties of large organic and biological molecules.     

Logical Models of Molecular Shapes and Their Families

A new discrete mathematical model of molecular shape is proposed, making use of the partition property of a representation of molecular shape. According to its geometrical and topological structure, a molecular surface can be partitioned into unbounded two-dimensional subsets (domains) and some common subsets of closures of two or more domains. The sets of these domains as a base of a finite topology, containing the Boolean n-cube as a lower Boolean sub-lattice of this topology, defines the domain of the proposed logical model. A logical function can be obtained that reflects the properties of the topological domains as well as the interrelations on the set of domains. Based on classical or quantum-chemical representations of molecular shape, these models allow one the implementation of methods of logical diagnostics in chemistry, and the definition of a metric on the set of molecular shape equivalence classes. The families of molecular shapes can be considered as sets of logical models. The proposed model is unified in the sense that the structures of differentiable and non-differentiable surfaces can be represented in the same mathematical framework. These logical models will also work for interpenetrations of the above types of surfaces.     

Lone pairs mapping by Laplacian of 3He NMR chemical shift

Results of approbation of a new quantum mechanical approach of lone pairs (LPs) visualization, its optimization and testing on a range of model molecules are presented. The main idea of proposed methodology is using ³He atom as a probe for investigating electronic shells of species with LPs. As model objects, we consider “classical” examples of hydrogen cyanide, methanimine, ammonia, phosphine, formaldehyde, water, and hydrogen sulfide. It is shown that LPs can be visualized by means of 3D maps of Laplacian of ³He chemical shift ∇²δ_He. NMR calculations could be performed using level of theory as low as B3LYP/6-31G, allowing for the reduction of computational time without significant loss of quality. Advantages of our approach are discussed in comparison with usual methods of lone pairs visualization (electron localization function, molecular electrostatic potential).     

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

Each complete, boundaryless molecular electron density is fully determined by any nonzero volume piece of the electron density cloud. This inherent feature of molecules, called the "holographic" property of molecular electron densities, provides a strong foundation for the local, quantum chemical shape analysis of various functional groups, pharmacophores, and other local molecular moieties. A proof is presented for the relevant molecular shape theorem, the "holographic electron density shape theorem", and the role of this theorem in quantum chemical, quantitative shape-activity relations (QShAR) is discussed. The quantum chemical methods of molecular shape analysis can be extended to ab initio quality electron densities of macromolecules, such as proteins, as well as to local molecular moieties, such as functional groups or pharmacophores, based on the transferability and additivity of local, fuzzy density fragments and the associated local density matrixes within the framework of the ADMA (Adjustable Density Matrix Assembler) approach. In addition to new results on chemical bonding and the development of macromolecular force methods, the new methodologies are also applicable to QShAR studies in computer-aided drug discovery and in toxicological risk assessment.     

Toward similarity measures for macromolecular bodies: Medla test calculations for substituted benzene systems

A new method is proposed for the evaluation of numerical similarity measures for large molecules, defined in terms of their electron density (ED) distributions. The technique is based on the Molecular Electron Density Lego Assembler (MEDLA) approach, proposed earlier for the generation of ab initio quality electron densities for proteins and other macromolecules. The reliability of the approach is tested using a family of 13 substituted aromatic systems for which both standard ab initio electron density computations and the MEDLA technique are applicable. These tests also provide additional examples for evaluating the accuracy of the MEDLA technique. Electron densities for a series of 13 substituted benzenes were calculated using the standard ab initio method with STO-3G, 3-21G, and 6-31G** basis sets as well as the MEDLA approach with a 6-31G** database of electron density fragments. For each type of calculation, pairwise similarity measures of these compounds were calculated using a point-by-point numerical comparison of the EDs. From these results, 2D similarity maps were constructed, serving as an aid for quick visual comparisons for the entire molecular family. The MEDLA approach is shown to give virtually equivalent numerical similarity measures and similarity maps as the standard ab initio method using a 6-31G** basis set. By contrast, significant differences are found between the standard ab initio 6-31G** results and the standard ab initio results obtained with smaller STO-3G and 3-21G basis sets. These tests indicate that the MEDLA-based similarity measures faithfully mimic the actual, standard ab initio 6-31G** similarity measures, suggesting the MEDLA method as a reliable technique to assess the shape similarities of proteins and other macromolecules. The speed of the MEDLA computations allows rapid, pairwise comparisons of the actual EDs for a series of molecules, requiring no more computer time than other simplified, less detailed representations of molecular shape. The MEDLA method also reduces the need to store large volumes of numerical density data on disk, as these densities can be quickly recomputed when needed. For these reasons, the proposed MEDLA similarity analysis technique is likely to become a useful tool in computational drug design. © 1995 John Wiley & Sons, Inc.     

Critical point representations of electron density maps for the comparison of benzodiazepine-type ligands

A procedure for the comparison of three-dimensional electron density distributions is proposed for similarity searches between pharmacological ligands at various levels of crystallographic resolution. First, a graph representation of molecular electron density distributions is generated using a critical point analysis approach. Pairwise as well as multiple comparisons between the obtained graphs of critical points are then carried out using a Monte Carlo/simulated annealing technique, and results are compared with genetic algorithm solutions.     

New rules on potential surface topology and critical point search

The topology of potential energy surfaces provides a unified framework for the study of individual molecular properties, all conformational changes as well as chemical reactions. Molecular behavior, electronic and vibrational properties, conformational freedom, reactivity bond formation and bond breaking are all energy dependent, and the potential energy surface approach provides an elegant, conceptually convenient, although rather complicated representation of this energy dependence. Topology as a mathematical tool is exceptionally suitable for the extraction of the most essential features of complicated representations. By applying topological methods for potential surface analysis, a new, global perspective of many aspects of chemistry emerges. Some of these topological results also have important practical, computational significance. A family of new topological rules and symmetry relations will be adapted for applications in low dimensional relaxed cross-sections of configuration spaces, with a special emphasis on their role in the search for critical points, primarily energy minima and saddle points of transition structures of potential energy surfaces and hypersurfaces.     

First Experimental Quantitative Charge Density Studies of Advanced Intermediate of Vitamin D Analogues

Vitamins D are a group of fat-soluble secosteroids which play a regulatory role in the functioning of most cells. Rational design of new vitamin D analogs, of increased therapeutic potency and lowered calcemic side effects, requires high-resolution initial structures and a deep understanding of interactions with the molecular targets. In this paper, using quantum crystallography, we present the first determination of the experimental quantitative charge density of an advanced intermediate of vitamin D analogues as well as a reconstruction of the theoretical electron density of final vitamin D analogues. Application of these methods allows for topological and electrostatic interaction energy analysis. We showed that the A-ring chair conformation has a significant influence on the topological properties of vitamin D compounds. Moreover, the interactions between the CD-ring and side-chain additionally stabilize the crystal structure. These results are supported by our theoretical calculations and previous biological studies.     

A complete shape characterization for molecular charge densities represented by Gaussian-type functions

An algorithm for a detailed 3-D characterization of the shapes of molecular charge distributions is implemented, tested and applied for a family of AB₂ molecules. The characterization is performed by computing a number of topological invariants (“shape groups”) associated with a continuum of molecular surfaces: the complete family of all electronic isodensity contours for the given molecules. These shape groups (the homology groups of truncated surfaces derived from isodensity contours) depend continuously on two parameters: a density value defining the density contour, and a reference curvature value, to which the local curvatures of the isodensity contours are compared. The electronic charge distribution is modeled by means of Gaussian-type functions. The method employs an explicit form of the charge density function in order to compute the curvature properties for the molecular surfaces analytically, from which the shape groups are derived by the algorithm. No visual inspection is required for the characterization and comparison of shapes of molecular charge densities, as these are done algorithmically by the computer. However, visual inspection of the results of the shape analysis is a possible option. For a given molecule, in a given nuclear configuration, the technique provides a two-dimensional shape map, displaying the distribution of shape groups as a function of the local curvature and the level set value (the value of the charge density at the contour). The computer program GSHAPE performs the analysis of shape maps automatically. This feature makes it potentially useful in the context of computer-aided drug design, where unbiased, automated shape characterization methods are valuable tools. As examples, several two-dimensional shape maps for simple systems are discussed. The changes induced in these maps by a change in the nuclear geometry, as well as by the changes of the nuclear charge, are also analyzed. The method is applicable to large biomolecules of interest if charge density information is available.     

Shape groups of the electronic isodensity surfaces for small molecules: Shapes of 10-electron hydrides

An algorithm for a detailed 3-D characterization of the shapes of molecular charge distributions is implemented in the form of a comprehensive package of computer programs, GSHAPE, and applied to a series of 10-electron hydrides to critically evaluate the methodology. Attention is paid to the effects of nuclear geometry and the size of basis on the molecular shape. The characterization is performed by computing a number of topological invariants (“shape groups”) associated with a continuum of molecular surfaces: the complete family of all electronic isodensity contours for the given molecules. These shape groups (the homology groups of truncated surfaces derived from isodensity contours) depend on two continuous parameters: a density value defining the density contour and a reference curvature value, to which the local curvatures of the isodensity contours are compared. The electronic charge distribution is calculated at the ab initio level using basis sets ranging from STO-3G to 6-31G**. No visual inspection is required for the characterization and comparison of shapes of molecular charge densities, as these are done algorithmically by the computer. However, visualization of the results is one option of our program using Application Visualization Software (AVS). For a given molecule, in a given nuclear geometry, the technique provides a 2-D shape map, displaying the distribution of the shape gruops as a function of the local curvature and the level set value (the value of the charge density at the contour). The computer program GSHAPE performs the analysis automatically. This feature makes it potentially useful in the context of computer-aided drug design, where unbiased, automated shape characterization methods are valuable tools. As examples, a variety of 2-D shape maps are discussed. © John Wiley & Sons, Inc.     

Shape analysis of macromolecular electron densities

Macromolecular shape analysis, an important aspect of the interpretation and prediction of biochemical behavior, pharmaceutical drug action, and the properties of advanced materials, is a task of a high level of complexity, where both global shape features and detailed, local shape features are relevant. The methods developed for the study of shapes of small molecules, in terms of molecular isodensity contours (MIDCOs), are not ideally suited for large molecular systems. In particular, the shape analysis of ab initio quality macromolecular electron densities, obtained by the MEDLA (molecular electron density lego assembler) method, requires a new approach. In this contribution, the adaptation of the earlier shape group approach to the electron densities of large molecules, and two new techniques, the shape globe folding map and the self-avoiding MIDCO methods, will be described.     

杂氮硼三环类化合物的UPS及化学键的理论研究

报道了六种杂氮硼三环类化合物的紫外光电子能谱(UPS).采用RHF/3-21G优化了各分子的优势构型,根据化合物UPS的谱带特征结合RHF/6-31G^*的计算结果对化合物的UPS进行了解析和指认,精确给出了各化合物中σN-B配键电子的电离峰位置.利用电子密度拓扑分析方法对各化合物的成键情况的研究显示:在该类化合物中B原子具有较为明显的阳离子的特征,N,B原子间均存在键鞍点.从实验和理论上证实了该类体系中σN-B的存在.各化合物的UPS,SCFMO计算和电子密度拓扑分析都表明,在该类体系中环上CH~3,CH~2的引入削弱了B,N间的成键作用;环上羰基的引入增强了B,N间的成键作用。