Efficient Algorithm for Two-Center Coulomb and Exchange Integrals of Electronic Prolate Spheroidal Orbitals

We present a fast algorithm to calculate Coulomb/exchange integrals of prolate spheroidal electronic orbitals, which are the exact solutions of the single-electron, two-center Schr\"odinger equation for diatomic molecules. Our approach employs Neumann's expansion of the Coulomb repulsion 1/|x-y|, solves the resulting integrals symbolically in closed form and subsequently performs a numeric Taylor expansion for efficiency. Thanks to the general form of the integrals, the obtained coefficients are independent of the particular wavefunctions and can thus be reused later. Key features of our algorithm include complete avoidance of numeric integration, drafting of the individual steps as fast matrix operations and high accuracy due to the exponential convergence of the expansions. Application to the diatomic molecules O2 and CO exemplifies the developed methods, which can be relevant for a quantitative understanding of chemical bonds in general.Comment: 27 pages, 9 figure

Two-Center Integrals for r_{ij}^{n} Polynomial Correlated Wave Functions

All integrals needed to evaluate the correlated wave functions with polynomial terms of inter-electronic distance are included. For this form of the wave function, the integrals needed can be expressed as a product of integrals involving at most four electrons

An Input-to-State Stability Approach to Verify Almost Global Stability of a Synchronous-Machine-Infinite-Bus System

Conditions for almost global stability of an operating point of a realistic model of a synchronous generator with constant field current connected to an infinite bus are derived. The analysis is conducted by employing the recently proposed concept of input-to-state stability (ISS)–Leonov functions, which is an extension of the powerful cell structure principle developed by Leonov and Noldus to the ISS framework. Compared with the original ideas of Leonov and Noldus, the ISS–Leonov approach has the advantage of providing additional robustness guarantees. The efficiency of the derived sufficient conditions is illustrated via numerical experiments