Boundary conditions for augmented plane wave methods

The augmented plane wave method uses the Rayleigh-Ritz principle for basis functions that are continuous but with discontinuous derivatives and the kinetic energy is written as a pair of gradients rather than as a Laplacian. It is shown here that this procedure is fully justified from the mathematical point of view. The domain of the self-adjoint Hamiltonian, which does not contain functions with discontinuous derivatives, is extended to its form domain, which contains them, and this modifies the form of the kinetic energy. Moreover, it is argued that discontinuous basis functions should be avoided.Comment: 5 pages, no figur

A many-body approach to crystal field theory

A self-consistent many-body approach is proposed to build a first-principles crystal field theory, where crystal field parameters are calculated ab initio. Many-body theory is used to write the energy of the interacting system as a function of the density matrix of the noninteracting system. A variation of the energy with respect to the density matrix gives an effective Hamiltonian that is diagonalized to determine the density matrix providing the lowest energy. The equations are written in terms of the Hopf algebra of functional derivatives with respect to external fermionic sources. This approach contains the many-body theory of Green functions as a special case, and the usual crystal field theory as a first approximation. Therefore, it is expected to provide good results for strongly-interacting electron systems.Comment: Equation (8) and references are correcte

Combinatorial Hopf algebras from renormalization

In this paper we describe the right-sided combinatorial Hopf structure of three Hopf algebras appearing in the context of renormalization in quantum field theory: the non-commutative version of the Fa\`a di Bruno Hopf algebra, the non-commutative version of the charge renormalization Hopf algebra on planar binary trees for quantum electrodynamics, and the non-commutative version of the Pinter renormalization Hopf algebra on any bosonic field. We also describe two general ways to define the associative product in such Hopf algebras, the first one by recursion, and the second one by grafting and shuffling some decorated rooted trees.Comment: 16 page