On Factorization of Molecular Wavefunctions

Recently there has been a renewed interest in the chemical physics literature of factorization of the position representation eigenfunctions \{$\Phi$\} of the molecular Schr\"odinger equation as originally proposed by Hunter in the 1970s. The idea is to represent $\Phi$ in the form $\varphi\chi$ where $\chi$ is \textit{purely} a function of the nuclear coordinates, while $\varphi$ must depend on both electron and nuclear position variables in the problem. This is a generalization of the approximate factorization originally proposed by Born and Oppenheimer, the hope being that an `exact' representation of $\Phi$ can be achieved in this form with $\varphi$ and $\chi$ interpretable as `electronic' and `nuclear' wavefunctions respectively. We offer a mathematical analysis of these proposals that identifies ambiguities stemming mainly from the singularities in the Coulomb potential energy.Comment: Manuscript submitted to Journal of Physics A: Mathematical and Theoretical, May 2015. Accepted for Publication August 24 201

A new proof of the analyticity of the electronic density of molecules

We give a new, short proof of the regularity away from the nuclei of the electronic density of a molecule obtained in [1,2]. The new argument is based on the regularity properties of the Coulomb interactions underlined in [3,4] and on well-known elliptic technics. [1] S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, T. Oe stergaard Soerensen: The electron density is smooth away from the nuclei. Comm. Math. Phys. 228, no. 3 (2002), 401-415. [2] S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, T. Oestergaard Soerensen: Analyticity of the density of electronic wave functions. Ark. Mat. 42, no. 1 (2004), 87-106. [3] W. Hunziker: Distortion analyticity and molecular resonances curves. Ann. Inst. H. Poincar\'e, s. A, t. 45, no 4, 339-358 (1986). [4] M. Klein, A. Martinez, R. Seiler, X.P. Wang: On the Born-Oppenheimer expansion for polyatomic molecules. Comm. Math. Phys. 143, no. 3, 607-639 (1992). The paper is published in Letters in Mathematical Physics 93, number 1, pp. 73-83, 2010. The original publication is available at " www.springerlink.com "

Design of Sectoral Antennas Using a Metallic EBG Structure and Multiple Sources Feeding for Base Station Applications

This work aims to study and design base station antennas with metallic electromagnetic band gap (EBG) materials able to create a sectoral radiation pattern presenting at least a 60∘ beamwidth. The use of metallic structures offers a new approach to industrial partners seeking to reduce costs and facilitate design procedures. A new method allowing the improvement of both the directivity and the bandwidth by using a printed antenna array is studied

Regularity for eigenfunctions of Schr\"odinger operators

We prove a regularity result in weighted Sobolev spaces (or Babuska--Kondratiev spaces) for the eigenfunctions of a Schr\"odinger operator. More precisely, let K_{a}^{m}(\mathbb{R}^{3N}) be the weighted Sobolev space obtained by blowing up the set of singular points of the Coulomb type potential V(x) = \sum_{1 \le j \le N} \frac{b_j}{|x_j|} + \sum_{1 \le i < j \le N} \frac{c_{ij}}{|x_i-x_j|}, x in \mathbb{R}^{3N}, b_j, c_{ij} in \mathbb{R}. If u in L^2(\mathbb{R}^{3N}) satisfies (-\Delta + V) u = \lambda u in distribution sense, then u belongs to K_{a}^{m} for all m \in \mathbb{Z}_+ and all a \le 0. Our result extends to the case when b_j and c_{ij} are suitable bounded functions on the blown-up space. In the single-electron, multi-nuclei case, we obtain the same result for all a<3/2.Comment: to appear in Lett. Math. Phy

Comprehensive Cancer-Predisposition Gene Testing in an Adult Multiple Primary Tumor Series Shows a Broad Range of Deleterious Variants and Atypical Tumor Phenotypes.

Multiple primary tumors (MPTs) affect a substantial proportion of cancer survivors and can result from various causes, including inherited predisposition. Currently, germline genetic testing of MPT-affected individuals for variants in cancer-predisposition genes (CPGs) is mostly targeted by tumor type. We ascertained pre-assessed MPT individuals (with at least two primary tumors by age 60 years or at least three by 70 years) from genetics centers and performed whole-genome sequencing (WGS) on 460 individuals from 440 families. Despite previous negative genetic assessment and molecular investigations, pathogenic variants in moderate- and high-risk CPGs were detected in 67/440 (15.2%) probands. WGS detected variants that would not be (or were not) detected by targeted resequencing strategies, including low-frequency structural variants (6/440 [1.4%] probands). In most individuals with a germline variant assessed as pathogenic or likely pathogenic (P/LP), at least one of their tumor types was characteristic of variants in the relevant CPG. However, in 29 probands (42.2% of those with a P/LP variant), the tumor phenotype appeared discordant. The frequency of individuals with truncating or splice-site CPG variants and at least one discordant tumor type was significantly higher than in a control population (χ2 = 43.642; p ≤ 0.0001). 2/67 (3%) probands with P/LP variants had evidence of multiple inherited neoplasia allele syndrome (MINAS) with deleterious variants in two CPGs. Together with variant detection rates from a previous series of similarly ascertained MPT-affected individuals, the present results suggest that first-line comprehensive CPG analysis in an MPT cohort referred to clinical genetics services would detect a deleterious variant in about a third of individuals.JW is supported by a Cancer Research UK Cambridge Cancer Centre Clinical Research Training Fellowship. Funding for the NIHR BioResource – Rare diseases project was provided by the National Institute for Health Research (NIHR, grant number RG65966). ERM acknowledges support from the European Research Council (Advanced Researcher Award), NIHR (Senior Investigator Award and Cambridge NIHR Biomedical Research Centre), Cancer Research UK Cambridge Cancer Centre and Medical Research Council Infrastructure Award. The University of Cambridge has received salary support in respect of EM from the NHS in the East of England through the Clinical Academic Reserve. The views expressed are those of the authors and not necessarily those of the NHS or Department of Health. DGE is an NIHR Senior Investigator and is supported by the all Manchester NIHR Biomedical Research Centre

Comprehensive Rare Variant Analysis via Whole-Genome Sequencing to Determine the Molecular Pathology of Inherited Retinal Disease

Inherited retinal disease is a common cause of visual impairment and represents a highly heterogeneous group of conditions. Here, we present findings from a cohort of 722 individuals with inherited retinal disease, who have had whole-genome sequencing (n = 605), whole-exome sequencing (n = 72), or both (n = 45) performed, as part of the NIHR-BioResource Rare Diseases research study. We identified pathogenic variants (single-nucleotide variants, indels, or structural variants) for 404/722 (56%) individuals. Whole-genome sequencing gives unprecedented power to detect three categories of pathogenic variants in particular: structural variants, variants in GC-rich regions, which have significantly improved coverage compared to whole-exome sequencing, and variants in non-coding regulatory regions. In addition to previously reported pathogenic regulatory variants, we have identified a previously unreported pathogenic intronic variant in $\textit{CHM}$ in two males with choroideremia. We have also identified 19 genes not previously known to be associated with inherited retinal disease, which harbor biallelic predicted protein-truncating variants in unsolved cases. Whole-genome sequencing is an increasingly important comprehensive method with which to investigate the genetic causes of inherited retinal disease.This work was supported by The National Institute for Health Research England (NIHR) for the NIHR BioResource – Rare Diseases project (grant number RG65966). The Moorfields Eye Hospital cohort of patients and clinical and imaging data were ascertained and collected with the support of grants from the National Institute for Health Research Biomedical Research Centre at Moorfields Eye Hospital, National Health Service Foundation Trust, and UCL Institute of Ophthalmology, Moorfields Eye Hospital Special Trustees, Moorfields Eye Charity, the Foundation Fighting Blindness (USA), and Retinitis Pigmentosa Fighting Blindness. M.M. is a recipient of an FFB Career Development Award. E.M. is supported by UCLH/UCL NIHR Biomedical Research Centre. F.L.R. and D.G. are supported by Cambridge NIHR Biomedical Research Centre

Bi-allelic Loss-of-Function CACNA1B Mutations in Progressive Epilepsy-Dyskinesia.

The occurrence of non-epileptic hyperkinetic movements in the context of developmental epileptic encephalopathies is an increasingly recognized phenomenon. Identification of causative mutations provides an important insight into common pathogenic mechanisms that cause both seizures and abnormal motor control. We report bi-allelic loss-of-function CACNA1B variants in six children from three unrelated families whose affected members present with a complex and progressive neurological syndrome. All affected individuals presented with epileptic encephalopathy, severe neurodevelopmental delay (often with regression), and a hyperkinetic movement disorder. Additional neurological features included postnatal microcephaly and hypotonia. Five children died in childhood or adolescence (mean age of death: 9 years), mainly as a result of secondary respiratory complications. CACNA1B encodes the pore-forming subunit of the pre-synaptic neuronal voltage-gated calcium channel Cav2.2/N-type, crucial for SNARE-mediated neurotransmission, particularly in the early postnatal period. Bi-allelic loss-of-function variants in CACNA1B are predicted to cause disruption of Ca2+ influx, leading to impaired synaptic neurotransmission. The resultant effect on neuronal function is likely to be important in the development of involuntary movements and epilepsy. Overall, our findings provide further evidence for the key role of Cav2.2 in normal human neurodevelopment.MAK is funded by an NIHR Research Professorship and receives funding from the Wellcome Trust, Great Ormond Street Children's Hospital Charity, and Rosetrees Trust. E.M. received funding from the Rosetrees Trust (CD-A53) and Great Ormond Street Hospital Children's Charity. K.G. received funding from Temple Street Foundation. A.M. is funded by Great Ormond Street Hospital, the National Institute for Health Research (NIHR), and Biomedical Research Centre. F.L.R. and D.G. are funded by Cambridge Biomedical Research Centre. K.C. and A.S.J. are funded by NIHR Bioresource for Rare Diseases. The DDD Study presents independent research commissioned by the Health Innovation Challenge Fund (grant number HICF-1009-003), a parallel funding partnership between the Wellcome Trust and the Department of Health, and the Wellcome Trust Sanger Institute (grant number WT098051). We acknowledge support from the UK Department of Health via the NIHR comprehensive Biomedical Research Centre award to Guy's and St. Thomas' National Health Service (NHS) Foundation Trust in partnership with King's College London. This research was also supported by the NIHR Great Ormond Street Hospital Biomedical Research Centre. J.H.C. is in receipt of an NIHR Senior Investigator Award. The research team acknowledges the support of the NIHR through the Comprehensive Clinical Research Network. The views expressed are those of the author(s) and not necessarily those of the NHS, the NIHR, Department of Health, or Wellcome Trust. E.R.M. acknowledges support from NIHR Cambridge Biomedical Research Centre, an NIHR Senior Investigator Award, and the University of Cambridge has received salary support in respect of E.R.M. from the NHS in the East of England through the Clinical Academic Reserve. I.E.S. is supported by the National Health and Medical Research Council of Australia (Program Grant and Practitioner Fellowship)