The set of all electronic states that can be expressed as a single Slater determinant forms a submanifold, isomorphic to the Grassmannian, of the projective Hilbert space of wave functions. We explored this fact by using tools of Riemannian geometry of Grassmannians as described by Absil et. al [Acta App. Math. 80, 199 (2004)], to propose an algorithm that converges to a Slater determinant that is critical point of the overlap function with a correlated wave function. This algorithm can be applied to quantify the entanglement or correlation of a wave function. We show that this algorithm is equivalent to the Newton method using the standard parametrization of Slater determinants by orbital rotations, but it can be more efficiently implemented because the orbital basis used to express the correlated wave function is kept fixed throughout the iterations. We present the equations of this method for a general configuration interaction wave function and for a wave function with up to double excitations over a reference determinant. Applications of this algorithm to selected electronic systems are also presented and discussed.

The Schwarz reflection principle states that a minimal surface S in R 3 is invariant under reflections in the plane of its principal geodesics and also invariant under 180• -rotations about its straight lines. We find new examples of embedded triply periodic minimal surfaces for which such symmetries are all of horizontal type.

In this work we propose an algorithm to find critical points of the inner product between an element of the Grassmannian and a fixed point of the projective space of the exterior algebra where the Grassmannian is embedded. This has interesting applications to electronic structure theory, where the wave functions are represented by elements of the exterior algebra. This method is exemplified for a Grassmannian that is a model for the hydrogen molecule, H2.

Given a triangle ABC let A B C be a Jacobi triangle for ABC. When ∠BA C = ∠CA B = α , ∠AB C = ∠CB A = β and ∠AC B = ∠BC A = γ , the triangle ABC is a Jacobi triangle for A B C . In this case we say that ABC and A B C are reciprocal Jacobi triangles. In 2015, G.T. Vickers presented a necessary condition for two triangles to be reciprocal, but the question whether that condition was also sufficient remained open. In this work we prove it by using basically trigonometric relations.

Resumo. Neste trabalho estudamos a versão do problema isoperimétrico para trapézios, que consiste em determinar o trapézio de máximaárea dentre aqueles de perímetro prescrito. Uma condição necessária da isoperimetria do trapézioé que ele seja isósceles. Para isso, consideramos um trapézio arbitrário não paralelogramo de perímetro L dado e fazemos o estudo de casos de acordo com a natureza de seusângulos. A soluçãoé obtida por meio de uma prova analítica a partir de desigualdades que envolvem o perímetro e aárea do trapézio.