Yoram Tal

A topological theory of molecular structure

A theory of molecular structure is presented. The theory demonstrates that the concepts of atoms and bonds may be rigorously defined and given physical expression in terms of the topological properties of the observable distribution of charge for a molecular system. As a consequence of these definitions, one in turn obtains a definition of structure and a predictive theory of structural stability. The theory is linked to quantum mechanics by demonstrating that the atoms so defined represent a class of open quantum subsystems with a unique set of variationally defined properties.

Quantum topology of molecular charge distributions. II. Molecular structure and its change

This paper illustrates how the concepts of atoms and bonds may be given definite expression in terms of the topological properties of the charge density, ρ (r), and how, as a consequence of these identifications, one is led to a definition of structure and to a phenomenological analysis of structural stability. This approach finds its natural expression in Rene Thom’s general analysis of structural stability as it applies to a system whose behavior is describable in terms of the gradient of some scalar field. Chemical observations are made in real space, and thus chemical behavior is determined by the morphology of a system’s charge distribution and its evolution with time. The analysis of the topological properties of ρ (r) via the associated gradient vector field ∇ρ (r), reduces to the identification of the critical points in ρ (r). Two types of critical points assume special roles in the analysis. A (3,−3) critical point, a maximum in ρ (r), is an attractor and is identified with the position of a nucl...

Calculation of the average properties of atoms in molecules

An algorithm for calculating atomic properties, based on the topological theory of molecular structure, is described and applied. For any given molecular system atoms are rigorously defined in terms of the topological properties of the systems charge distribution rho (x) in three-dimensional space. The essential feature of this distribution is that its only local maxima, the attractors of the associated gradient vector field grad rho (x), occur at the positions of the nuclei. The region of space occupied by an atom, the atomic basin, is the space traversed by all gradient paths of grad rho (x) which terminate at its nucleus. This property is used to construct a coordinate transformation which maps each atomic basin onto three-dimensional space. The Jacobian J, of this transformation depends solely on the second derivatives of rho (x). The determinant of J is obtained by solving a first-order differential equation governed by grad2 rho (x). Any atomic property may then be calculated by an integration of an associated single-particle density over the basin of the atom. Numerical integrations of some atomic properties of diatomic and polyatomic molecules are reported.

Quantum topology. IV. Relation between the topological and energetic stabilities of molecular structures

The relation between the structural stability of a molecular system as determined by the topological properties of its charge distribution and the energetic stability of the same system as determined by the properties of its potential energy hypersurface is studied. In general, it is found that one may associate a given molecular structure with an open neighborhood of an energetically stable geometry of the system. A change in molecular structure is an abrupt process, and examples are given in which the change in structure is found to occur in the immediate neighborhood of the transition state geometry. These observations suggest that topologically unstable structures correspond to energetically unstable geometries of a system. Such a correspondence can be rationalized in terms of the Hellmann–Feynman theorem which is shown to relate the gradients of the energy hypersurface to the gradient vector field of the charge density—the field whose instabilities determine the instabilities in molecular structure.

On the Z−1 and N−1/3 expansions of Hartree–Fock atomic energies

Numerical Hartree–Fock (HF) calculations for all atoms and ions in the range 2⩽N⩽86 and N⩽Z⩽N+20 were performed and analyzed in terms of the Z−1 perturbation series. Using the theoretical values of the leading expansion coefficient e0(N), we obtain e1(N), e2(N), and ek(N), k⩾3, by a least squares fitting of the HF results. It is found that the difference E−(e0Z2+ e1Z+e2) is, in general, not governed by e3Z−1 but by a higher order term involving ek(N) above. The value of k increase from k=3 for N=2 to k=8 for 60⩽N⩽86. Next, we study the N‐dependence of the expansion coefficients en(N), 0⩽n⩽2, and of the total energies E(Z,N) by means of an N−1/3 power series expansion. It is noted that the extension of these functions over the set of nonintegral values of N is not unique, and consequently their N−1/3 expansions are, in part, model‐dependent. We use our numerical data to extract the model‐independent part of y≡E/Z7/3q1/3, q≡N/Z, which may be expressed in terms of the universal expansion y = F00(q)+ F10(q)N−...

The hydrogenic limit of many‐electron atoms

The properties of many‐electron atoms in the limit of infinite nuclear charge Z and a constant degree of ionization 1−N/Z (N being the total number of electrons) are investigated. The hydrogenic limit of such atoms, defined as a system of N noninteracting electrons moving about a nucleus of charge Z, is studied in detail. In addition to the well‐known formulas for the energy components we derive closed form expressions for all the moments of the charge density ∫dτrkρ(r) where k⩾−1. These results serve as a starting point for the comparison of two density functional theories: the Thomas–Fermi (TF) theory and the local density functional (LDF) theory. It is shown that in both cases the energy components, as well as the moments of the charge density, converge to the hydrogenic limit as N/Z→ 0. We further compare the TF and LDF predictions for the total energy E over the whole range 0⩽N/Z⩽1. It is shown that LDF may closely approach the TF theory for an appropriate choice of the electronic repulsion parameter.

Universal scaling relations for free and bonded atoms

Scaling relations, relating the energy and expectation value of r−1 in free and bonded atoms, are studied. In the limit of a large number of electrons these relations are functions of the ratio q = N/Z, where N is the number of electrons and Z is the nuclear charge. It is shown that the q dependence of such relations may be rather accurately predicted from the Thomas–Fermi and local density functional theories even for small values of N. The validity of the theoretical prediction is demonstrated by a comparison between the theoretical and the calculated Hartree–Fock values for a large number of atoms and ions. The same comparison is made for bonded atoms as defined by the theory of quantum topology. It is found that both free and bonded atoms obey the proposed relations to a similar degree of accuracy.

Correlation Energies from Hartree-Fock Electrostatic Potentials at Nuclei and Generation of Electrostatic Potentials from Asymptotic and Zero-Order Information

It has been recognized 1-16 that electrostatic potentials at the nuclei of atoms and molecules play key roles in determining total energies. (For an overview, see the preceeding chapter by Politzer). It has also been recognized that the Hellmann-Feynman theorem provides a fundamental link between an energy change and the electrostatic potential at the nucleus caused by the electrons. The latter will be called the EPN.