Nuclear Configuration Space Research Articles

Structural homeomorphism between the electron density and the virial field

The virial field V(r) is defined by the local statement of the quantum mechanical virial theorem, as the trace of the Schrodinger stress tensor. This field defines the electronic potential energy density of an electron at r and integrates to minus twice the electronic kinetic energy. It is the most short-ranged description possible of the local electronic potential energy and it exhibits the same transferable behavior over bounded regions of real space (corresponding to the functional groups of chemistry) as does ρ(r). This article establishes a structural homeomorphism between −V(r) and ρ(r), showing that the two fields are homeomorphic over all of the nuclear configuration space. The stable or unstable structure defined by the gradient vector field Δρ(r; X) for any configuration X of the nuclei can be placed in a one-to-one correspondence with a structure defined by the field −ΔV(r; X′). In particular, a molecular graph for ρ(r) defining a molecular structure is mirrored by a corresponding virial graph for V(r) and the lines of maximum density linking bonded nuclei in the former field are matched by a set of lines of maximally negative potential energy density in the latter. The homeomorphism is also geometrically faithful, an equilibrium geometry in general, exhibiting equivalent structures in the two fields. The demonstration that the virial field, whose integrated value equals twice the total energy, is essentially just a locally scaled version of the electron density is suggestive of possible new approaches in density functional theory. © 1996 John Wiley & Sons, Inc.

Information theory and electron density

The intriguing concept of inherent uncertainty of probability schemes in information theory and statistical inference is applied to the molecular electron density. The electron density function is treated as a multimodal, three-dimensional probability density function describing the distribution of the electrons of a molecule in real space. A simple theory is proposed to introduce the amount of information associated with perturbations of the nuclear geometry such as molecular vibrations and reaction paths, in particular. It is shown by computations that the amount of information associated with the normal modes of vibration is related to the reduced mass. The proposed theory also suggests a novel Riemannian nuclear configuration space which is completely defined by the observable electron density of a molecular system.

The metric properties of the reduced nuclear configuration space: Remarks

A definition of the distance on the orbit spaces of topological groups acting continuously on metric spaces is applied to define some metric properties of the reduced nuclear configuration spaces. Straightforward proofs are given and a comparison with former results is carried out.

The fundamental syntopy of quasi‐symmetric systems: Geometric criteria and the underlying syntopy of a nuclear configuration space

AbstractPoint symmetry is a discrete concept; A nuclear configuration for a given stoichiometry either has or has not a particular point symmetry. By contrast, both static and dynamic properties of actual molecules exhibit continuous features. Using the formalism of fuzzy‐set theory, we had previously proposed the concept of syntopy as a continuous extension of the symmetry concept for quasi‐symmetric systems: This was based on an energetic criterion taking into account the energy costs of nuclear rearrangements. This extension of symmetry was necessarily dependent on the considered electronic state: For a given geometric arrangement of the nuclei, the energy cost of some rearrangement is dependent on the actual potential surface, that is, on the electronic state, in the Born–Oppenheimer approximation. In the extension of the syntopy model reported in the present work, we consider a syntopy criterion that is common to all electronic states. The syntopy thus defined—called the fundamental syntopy of the reduced nuclear configuration space—is independent of the potential surface and of the electronic state: It is defined only in terms of a geometric condition, which makes it more appropriate to rationalize mesoscopic structures. This new syntopy model provides a connection between all possible syntopies generated by the various potential‐energy surfaces supported by the considered family of atomic nuclei. © 1993 John Wiley & Sons, Inc.

Relations among functional groups within a stoichiometry: A nuclear configuration space approach

AbstractA general scheme is proposed for the study of the interrelations and transformations of flexible functional groups within the configuration space M of all possible chemical species of a fixed overall stoichiometry S. The methods presented provide a concise but sufficiently detailed description of the occurrence of various functional groups in different families of nuclear arrangements, as well as their transformations into one another during chemical reactions and conformational changes. The approach uses experimental information on the importance of various functional groups, topological criteria for the flexibility of these functional groups, and the fundamental pattern of distribution of configurational families within the nuclear configuration space. The resulting schemes have simple, easily programmable representations. © 1992 John Wiley & Sons, Inc.

Theoretical approach to reactions of polyatomic molecules

AbstractA scheme for systematic reduction of the theoretical treatment of elementary reactions involving polyatomic molecules is described; it consists of (1) limitation to the energetically relevant regions of the nuclear configuration space (the reaction path and its near environs) and (2) restriction to the dynamically relevant subspace of the nuclear configuration space (the active modes). Starting from a generalized reaction path Hamiltonian of Nauts and Chapuisat allowing for the use of arbitrary curvilinear coordinates and several large‐amplitude modes, the realization of the above‐sketched scheme is discussed. A compilation of recent work along these lines, mostly based on the simplified Miller‐Handy‐Adams reaction path Hamiltonian, is given with particular emphasis on applications of a statistical adiabatic model.

The concept of ‘syntopy’

Point symmetry is essentially a discrete concept, whereas molecular structures and their conformational changes display continuous features, even within semi-classical models. Using the formalism of fuzzy-set theory, we propose a continuous extension of the point-symmetry concept for quasi-symmetric structures. For each point k of the reduced nuclear configuration (metric) space M of the set of all molecular systems having a specified stoichiometry, and for an energy criterion ϵ suggested by the statics or the dynamics of the systems, we define membership functions μ i (k, ϵ) which generate fuzzy subsets Si (ϵ) of the metric space M. These fuzzy subsets are used to obtain a continuous extension of the symmetry point groups of formal, classical, nuclear configurations.

From symmetry to syntopy : a fuzzy-set approach to quasi-symmetric systems

Point symmetry is essentially a discrete concept, whereas molecular structures and their conformational changes display continuous features. Using the formalism of fuzzy-set theory, we propose a continuous extension of the symmetry concept for quasi-symmetric systems. For each point k of the reduced nuclear configuration space M of all N-atom systems having a specified stoichiometry, and for an energy criterion e suggested by the statics or the dynamics of these systems, we define membership functions µ i (k, e) which generate 'syntopy' fuzzy subsets S i , in the metric space M ; these latter are used to obtain a continuous extension s(k, e) of the symmetry point groups g i . Illustrations of the procedures used are given on simple examples, and applications of the new concept are outlined.

A method for the characterization of molecular conformations

In the course of conformational motions of molecules the changes in shapes of electronic charge distributions follow that of the nuclear framework. However, this coupling between the changes in the nuclear geometry and electron density may depend on the actual nuclear displacement; the coupling may be weak or strong for a given conformational motion. It is of some interest to analyze how faithfully the charge density variations follow the nuclear displacements in a family of conformational rearrangements. In certain cases small conformational changes may induce large changes in the shape of charge density distributions, while in other cases large and qualitatively important conformational changes may involve qualitatively inessential distortions in the shape of electron distributions. In this article we describe a new classification of conformations based on those domains of nuclear configuration space within which the „shape groups” (symmetry independent homology groups) of the electric charge density remain invariant. Such an analysis might be valuable when seeking correlations between molecular structure and certain biological or biochemical activities.

Reflection properties of reaction paths in the reduced nuclear configuration space

Chemical reactions and conformational changes of N-atom systems can be described as displacements in a (3N-6)-dimensional metric configuration space M provided with a global metric. Although space M has a metric, it is not in general a vector space; it is a topological space. In contrast to the commonly used internal configuration spaces based on bond length/bond angle internal coordinates, and having no global metrics, within space M each internal configuration of the nuclei of the molecule corresponds to one and only one point of the space. This property of M is advantageous when analyzing chemical reactions. The global metric of M ensures that differences between any two internal configurations can be interpreted as a distance in this space that allows one to provide M with coordinate systems by turning M into a manifold with boundary. Certain formal reaction paths show some counterintuitive behavior within this space: they may undergo a formal reflection at some points of M. A condition, the tangent criterion, is used for the diagnosis of such reaction paths and for the determination of special nuclear configurations where such reflections occur.

Homology of Potential Energy Surfaces

It is shown that all the adjacency relations for the basins of attraction of stable chemical species and transition states can be derived from the topology of the pattern of intrinsic-reaction-coordinate- and-separatix trajectories in the nuclear configuration space. The results are applied to thermal [1,3] sigmatropic rearrangements and they show that even the symmetry-forbidden path proceeds concertedly. The corresponding homological formulas giving the adjacency relations are derived.

Homology of a Structurally Stable Chemical Rearrangement

The structural stability in the sense of Adronov and Pontriagin is proved to be a property of any rearrangement occurring in a single electronic state sheet of the potential energy surface if such a rearrangement involves two internal degrees of freedom. It is then shown that features of no dynamical significance (the paths of steepest descent joining critical configurations) determine completely those of dynamical significance: The topology of the set of paths of steepest descent determines the triangulation of the nuclear configuration space which is rigorously derived from homology theory.

Catchment regions as “molecular loges” on potential energy hypersurfaces

Analogies between three dimensional electronic loges of molecular electronic charge distributions and higher dimensional catchment regions of potential energy hypersurfaces, representing various chemical species in reaction topology, are discussed. Connectedness properties of both regular and pathological catchment regions are analysed. Just as the electronic loges in three space, catchment regions are necessarily connected subsets of a nuclear configuration space. (A formal proof is presented here for both regular and pathological cases.) The closures of these two types of sets, however, may show remarkably different features.

The metric properties of the reduced nuclear configuration space

AbstractA proof is given for the triangle inequality of distance function d in the reduced nuclear configuration space M, used in the differentiable manifold model of potential‐energy hypersurfaces [P. G. Mezey, Int. J. Quantum Chem. Quantum Chem. Symp. 17, 137 (1983)]. This result completes the proof that M is indeed a metric space with metric d.

Constraints on electronic energy hypersurfaces of higher multiplicities

Upper and lower bounds are derived for electronic energy hypersurfaces of various multiplicities over a general nuclear configuration space. Whereas, in relations derived earlier, only energy hypersurfaces of the same multiplicity could be used as upper or lower bounds for each other, this restriction is now removed in the more general relations proposed in the present study.

Molecular structure and reaction mechanism: A topological approach to quantum chemistry

Reaction topology and the differentiable manifold model of the nuclear configuration space and potential energy hypersurfaces provide a mathematical framework for computer-aided quantum chemical synthesis planning. Relations between the geometrical and topological models of molecules and reaction mechanisms are analysed and properties of induced topologies in various product spaces are discussed.

Classification schemes of nuclear geometries and the concept of chemical structure. Metric spaces of chemical structure sets over potential energy hypersurfaces

A general concept of chemical structure based on topologies of the nuclear configuration space is proposed. A metric, describing chemically significant structural differences, is introduced and two recently proposed definitions of molecular structure are compared within the general topological framework.

Reaction topology of excited state potential energy hypersurfaces

Topology ("rubber geometry") is an exceptionally suitable mathematical framework for a global quantum chemical description of the fundamental relations between nonrigid molecular systems, chemical reactions, and many photochemical processes. In Reaction Topology the concept of nuclear geometry is replaced by open subsets of an abstract nuclear configuration space nR. The ground and various excited state energy expectation value functionals induce a sequence of topologies in the nuclear configuration space nR, leading to a consistent topological description of molecular structure, excimers, exciplexes, and reaction mechanisms. Utilizing earlier results on topological properties of the abstract nuclear charge space wZ, a theorem is proven on the ordering of excited state potential energy hypersurfaces for sequences of isoelectronic molecules.

Level set topology of the nuclear charge space and the electronic energy functional

AbstractElectronic energy level set topologies in the abstract nuclear charge space Z of molecular systems are defined and analyzed. Two theorems, one on the general convexity of level sets in Z, another on homotopies of boundaries of level sets, induced by nuclear geometry variations in the nuclear configuration space R, are proven. The applications of the two theorems are illustrated by examples of various molecules and ions.

Quantum chemical reaction networks, reaction graphs and the structure of potential energy hypersurfaces

Global properties of the Born-Oppenheimer energy expectation value functional, defined over the nuclear configuration space R, are analyzed. Quantum chemical reaction graphs and reaction networks are defined in terms of intersection graphs of connected sets of nuclear geometries, representing various chemical structures. The set of all possible reaction mechanisms on the given energy hypersurface and the associated activation energy conditions are analyzed using reachability matrices defined over digraphs Ds(λ) and Ds(λ, E).