Subspace quantum mechanics and the variational principle

Abstract

This work is a continuation and extension of the delineation of the properties of a quantum subspace—a region of the real space of a molecular system bounded by a surface through which the flux in the gradient of the (observable) charge density is zero. Such subspaces are of interest as they constitute a basis for theoretical definitions of chemical concepts as obtained through experiment. The subspace stationary state variational principle, which was previously used to define this particular class of subspaces, reveals a close interrelationship between the hypervirial theorem and the variational property of the energy. By exploiting this interrelationship, one may obtain a variational solution to Schrödinger’s equation with the zero-flux surface requirement serving as a variational constraint. The connection between the statement of the subspace variational principle and the result of perturbation theory is established at the level of the first-order correction to the total energy. The statement of the generalized time-dependent subspace variational principle is derived using a modified form of Hamilton’s principle. The principle is also a variational statement of a time-dependent hypervirial theorem, generalized to systems bounded by surfaces of zero-flux in the gradient of the charge density. It, therefore, enables one to describe the time dependence of subspace averaged properties. The use of virial sharing operators, which permit the definition of single particle potential energies, in a quantum analogue of Hamilton’s extended principle, leads to a unique variational definition of the electronic energy within the Born–Oppenheimer approximation. In terms of this extended principle, both the classical and the quantum equations of motion are obtained for systems in which forces of constraint are operative by demanding that the integral of the variation of the kinetic energy plus the virtual work involved in the variation be zero. As a result of this discussion, the total energy of a subspace of a molecular system is defined in terms of separate electronic and nuclear contributions. This definition of the energy of a subspace applies whether or not applied forces are acting on the nuclei of the system.