#### Date of Award

7-1980

#### Degree Type

Thesis

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Chemistry

#### Supervisor

Professor R.F.W. Bader

#### Abstract

This thesis is concerned with the rigorous definitions of the two central concepts embodied in the notion of molecular structure, namely, the concepts of atoms and bonds. The basis for the present approach is provided by the topological properties of the charge distribution in a given molecular system. The essential observation is that the only local maxima of a ground state distribution occur at the positions of the nuclei. The nuclei are, therefore, identified as point attractors of the gradient vector field of the charge density. The associated basins partition the molecule into atomic fragments. Each atom is a stable structural unit defined as the union of an attractor and its basin. The common boundary of two neighbouring atomic fragments, the interatomic surface, contains a particular critical point, which generates a pair of gradient paths linking the two neighbouring attractors. The union of this pair of gradient paths and their endpoints is called a bond path. The network of bond paths defines a molecular graph of the system.

Having defined a unique molecular graph for any molecular geometry, the total configuration space is partitioned into a finite number of regions. Each region is associated with a particular structure defined as an equivalence class of molecular graphs. A chemical reaction in which chemical bonds are broken and/or formed is, therefore, a trajectory in configuration space which must cross one of the boundaries between two neighbouring structural regions. These boundaries form the catastrophe set of the system which, like a phase diagram in thermodynamics, denotes the points of "balance" between neighbouring structures. A general analysis of the structural changes in an ABC type system is given in detail, together with specific examples of all possible structural elements in a molecular system.

The topological definition of an atom as the union of a nuclear attractor and its basin is equivalent to its definition in terms of its boundary. An atomic boundary consists of the union of a number of surfaces through which the flux of the gradient vector of the charge density is zero. Such a surface is called a zero-flux surface. In general, the zero-flux surfaces define subsystems of a molecular system which represent atomic fragments or functional groupings of such fragments. The properties of these subsystems are delineated through a variational approach to the quantum mechanics of molecular subsystems. In both the time-dependent and the stationary state case, the variational equations for a subsystem are generalizations of the corresponding expressions for a total system. The stationary-state variational principle relates the first-order change in the subsystem energy functional to the flux, through the boundary of the subsystem, of the current density generated by an infinitesimal arbitrary variation in the state function. From this principle, the generalizations of the important hypervirial and virial theorems are obtained, which apply equally to a subsystem as to the total system. In particular, a definition of the subsystem energy, which exhibits the important additivity property, is obtained through the subsystem virial theorem. The time-dependent variational equations generalize the stationary state variational principle; they are shown to follow from Schwinger's quantum action principle and principle of stationary action. The application of these principles to a subsystem requires that the subsystem quantum action integral be stationary in the sense that its first variation, ensuing from arbitrary variations in the state function between two given time endpoints, contains only contributions from these time end-points and from the spatial boundary of the subsystem, at intermediate times. When the aforementioned variations are produced by an infinitesimal unitary transformation, the subsystem variational principle relates that change in the subsystem Lagrangian to the commutator of the generator of the transformation with the Hamiltonian of the system. From this statement of the time-dependent variational principle, a force law is obtained, which generalizes Ehrenfest's theorem, and can be regarded as the quantum analogue of Newton's equation of motion. The variational principle also leads to a generalization of the continuity equation, which governs the time change of an atomic population, and an extension of the virial theorem to describe a subsystem - an atom, or a functional grouping of atoms - in a time-dependent molecular environment.

#### Recommended Citation

Nguyen-Dang, Thanh-Tung, "Topological and Quantum Mechanical Study of Molecular Charge Distributions" (1980). *Open Access Dissertations and Theses.* Paper 622.

http://digitalcommons.mcmaster.ca/opendissertations/622