74 research outputs found
Analyticity of the density of electronic wavefunctions
We prove that the electronic densities of atomic and molecular eigenfunctions
are real analytic in away from the nuclei.Comment: 19 page
ANALYTIC STRUCTURE OF SOLUTIONS TO MULTICONFIGURATION EQUATIONS
Abstract. We study the regularity at the positions of the (fixed) nuclei of solutions to (non-relativistic) multiconfiguration equations (including HartreeâFock) of Coulomb systems. We prove the following: Let {Ď1,..., ĎM} be any solution to the rankâM multiconfiguration equations for a molecule with L fixed nuclei at R1,..., RL â R 3. Then, for any j â {1,..., M}, k â {1,..., L}, there exists a neighbourhood Uj,k â R 3 of Rk, and functions Ď (1) j,k, Ď(2) j,k, real analytic in Uj,k, such that Ďj(x) = Ď (1) (2) j,k (x) + |x â Rk|Ď j,k (x), x â Uj,k. A similar result holds for the corresponding electron density. The proof uses the KustaanheimoâStiefel transformation, as applied in [9] to the study of the eigenfunctions of the SchrĂśdinger operator of atoms and molecules near two-particle coalescence points. 1. Introduction an
The electron density is smooth away from the nuclei
We prove that the electron densities of electronic eigenfunctions of atoms
and molecules are smooth away from the nuclei.Comment: 16 page
The Ground State Energy of Heavy Atoms According to Brown and Ravenhall: Absence of Relativistic Effects in Leading Order
It is shown that the ground state energy of heavy atoms is, to leading order,
given by the non-relativistic Thomas-Fermi energy. The proof is based on the
relativistic Hamiltonian of Brown and Ravenhall which is derived from quantum
electrodynamics yielding energy levels correctly up to order Ry
Analytic structure of solutions to multiconfiguration equations
We study the regularity at the positions of the (fixed) nuclei of solutions
to (non-relativistic) multiconfiguration equations (including Hartree--Fock) of
Coulomb systems. We prove the following: Let {phi_1,...,phi_M} be any solution
to the rank--M multiconfiguration equations for a molecule with L fixed nuclei
at R_1,...,R_L in R^3. Then, for any j in {1,...,M} and k in {1,...,L}, there
exists a neighbourhood U_{j,k} in R^3 of R_k, and functions phi^{(1)}_{j,k},
phi^{(2)}_{j,k}, real analytic in U_{j,k}, such that phi_j(x) =
phi^{(1)}_{j,k}(x) + |x - R_k| phi^{(2)}_{j,k}(x), x in U_{j,k} A similar
result holds for the corresponding electron density. The proof uses the
Kustaanheimo--Stiefel transformation, as applied earlier by the authors to the
study of the eigenfunctions of the Schr"odinger operator of atoms and molecules
near two-particle coalescence points.Comment: 15 page
Enrichment of megabase-sized DNA molecules for single-molecule optical mapping and next-generation sequencing
Abstract Next-generation sequencing (NGS) has caused a revolution, yet left a gap: long-range genetic information from native, non-amplified DNA fragments is unavailable. It might be obtained by optical mapping of megabase-sized DNA molecules. Frequently only a specific genomic region is of interest, so here we introduce a method for selection and enrichment of megabase-sized DNA molecules intended for single-molecule optical mapping: DNA from a human cell line is digested by the NotI rare-cutting enzyme and size-selected by pulsed-field gel electrophoresis. For demonstration, more than 600 sub-megabase- to megabase-sized DNA molecules were recovered from the gel and analysed by denaturation-renaturation optical mapping. Size-selected molecules from the same gel were sequenced by NGS. The optically mapped molecules and the NGS reads showed enrichment from regions defined by NotI restriction sites. We demonstrate that the unannotated genome can be characterized in a locus-specific manner via molecules partially overlapping with the annotated genome. The method is a promising tool for investigation of structural variants in enriched human genomic regions for both research and diagnostic purposes. Our enrichment method could potentially work with other genomes or target specified regions by applying other genomic editing tools, such as the CRISPR/Cas9 system
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