Science Institute

We establish existence of infinitely many distinct solutions to the multi-configurative Hartree-Fock type equations for N -electron Coulomb systems with quasi-relativistic kinetic energy −α −2 ∆ xn + α −4 −α −2 for the n th electron. Finitely many of the solutions are interpreted as excited states of the molecule. Moreover, we prove existence of a ground state. The results are valid under the hypotheses that the total charge Z tot of K nuclei is greater than N − 1 and that Z tot is smaller than a critical charge Z c . The proofs are based on a new application of the Lions-Fang-Ghoussoub critical point approach to nonminimal solutions on a complete analytic Hilbert-Riemann manifold.

We establish existence of infinitely many distinct solutions to the multi-configurative Hartree-Fock type equations for N -electron Coulomb systems with quasi-relativistic kinetic energy −α −2 ∆ xn + α −4 −α −2 for the n th electron. Finitely many of the solutions are interpreted as excited states of the molecule. Moreover, we prove existence of a ground state. The results are valid under the hypotheses that the total charge Z tot of K nuclei is greater than N − 1 and that Z tot is smaller than a critical charge Z c . The proofs are based on a new application of the Lions-Fang-Ghoussoub critical point approach to nonminimal solutions on a complete analytic Hilbert-Riemann manifold.

Complete Lyapunov functions (CLF) are scalar-valued functions, which are non-increasing along solutions of a given autonomous ordinary differential equation. They separate the phase-space into the chain-recurrent set, where the CLF is constant along solutions, and the set where the flow is gradient-like and the CLF is strictly decreases along solutions. Moreover, one can deduce the stability of connected components of the chain-recurrent set from the CLF.

The minimum mode following method for finding first order saddle points on an energy surface is used, for example, in simulations of long time scale evolution of materials and surfaces of solids. Such simulations are increasingly being carried out in combination with computationally demanding electronic structure calculations of atomic interactions, so it is essential to reduce as much as possible the number of function evaluations needed to find the relevant saddle points. Several improvements to the method are presented here and tested on a benchmark system involving rearrangements of a heptamer island on a close packed crystal surface. Instead of using a uniform or Gaussian random initial displacement of the atoms, as has typically been done previously, the starting points are arranged evenly on the surface of a hypersphere and its radius is adjusted during the sampling of the saddle points. This increases the diversity of saddle points found and reduces the chances of reconverging on previously located saddle points. The minimum mode is estimated using the Davidson method, and it is shown that significant savings in the number of function evaluations can be obtained by assuming the minimum mode is unchanged until the atomic displacement exceeds a threshold value. The number of function evaluations needed for a recently published benchmark (S. T. Chill et al. J. Chem. Theory Comput. 2014, 10, 5476) is reduced to less than a third with the improved method as compared with the best previously reported results.

Complete Lyapunov functions are of much interest in control theory because of their capability to describe the longtime behaviour of nonlinear dynamical systems. The state-space of a system can be divided in two different regions determined by a complete Lyapunov function: the region of the gradientlike flow, where the Lyapunov function is strictly decreasing along solution trajectories, and the chain-recurrent set whose chain-transitive components are level sets of the Lyapunov function. There has been continuous effort to properly identify both regions and in this paper we discuss the extension of our methods to compute complete Lyapunov functions in the plane to the three-dimensional case, which is directly applicable to higher dimensions, too. When extending the methods to higher dimensions, the number of points for collocation and evaluation grows exponentially. To keep the number of evaluation points under control, we propose a new way to choose them, which does not depend on the dimension.