scholarly journals On the use of the Obara–Saika recurrence relations for the calculation of structure factors in quantum crystallography

2020 ◽  
Vol 76 (2) ◽  
pp. 172-179 ◽  
Author(s):  
Alessandro Genoni
Keyword(s):  
Recurrence Relations ◽  
Basis Functions ◽  
Future Perspectives ◽  
Recurrence Formulas ◽  
Atomic Nuclei ◽  
Structure Factors ◽  
Modern Methods ◽  
Molecular Integrals ◽  
Chemical Strategies ◽  
Quantum Crystallography

Modern methods of quantum crystallography are techniques firmly rooted in quantum chemistry and, as in many quantum chemical strategies, electron densities are expressed as two-centre expansions that involve basis functions centred on atomic nuclei. Therefore, the computation of the necessary structure factors requires the evaluation of Fourier transform integrals of basis function products. Since these functions are usually Cartesian Gaussians, in this communication it is shown that the Fourier integrals can be efficiently calculated by exploiting an extension of the Obara–Saika recurrence formulas, which are successfully used by quantum chemists in the computation of molecular integrals. Implementation and future perspectives of the technique are also discussed.

HgH2 meets relativistic quantum crystallography. How to teach relativity to a non-relativistic wavefunction

2021 ◽  
Vol 77 (1) ◽  
pp. 54-66
Author(s):  
Michal Podhorský ◽  
Lukáš Bučinský ◽  
Dylan Jayatilaka ◽  
Simon Grabowsky
Keyword(s):  
Systematic Error ◽  
Structure Factor ◽  
Relativistic Quantum ◽  
Relativistic Effects ◽  
X Ray ◽  
Structure Factors ◽  
Quantum Crystallography

The capability of X-ray constrained wavefunction (XCW) fitting to introduce relativistic effects into a non-relativistic wavefunction is tested. It is quantified how much of the reference relativistic effects can be absorbed in the non-relativistic XCW calculation when fitted against relativistic structure factors of a model HgH2 molecule. Scaling of the structure-factor sets to improve the agreement statistics is found to introduce a significant systematic error into the XCW fitting of relativistic effects.

Change in genetic variance under selection in a self-fertilizing population.

Genetics ◽  
1994 ◽  
Vol 136 (2) ◽  
pp. 693-704
Author(s):  
T Hayashi ◽  
Y Ukai
Keyword(s):  
Linkage Disequilibrium ◽  
Recurrence Relations ◽  
Genetic Variance ◽  
Additive Effect ◽  
Frequency Change ◽  
Genotypic Frequency ◽  
Truncation Selection ◽  
Recurrence Formulas ◽  
Additive Action ◽  
Self Fertilization

Abstract In this study we show how the genetic variance of a quantitative trait changes in a self-fertilizing population under repeated cycles of truncation selection, with the analysis based on the infinitesimal model in which it is assumed that the trait is determined by an infinite number of unlinked loci without epistasis. The genetic variance is reduced not as a consequence of the genotypic frequency change but due to the build-up of linkage disequilibrium under truncation selection in this model. We assume that the order of the genotypic contribution from each locus is n-1/2, where n is the number of loci involved, and investigate the change in linkage disequilibrium resulting from selection and self-fertilization using genotypic frequency dynamics in order to analyze the change in the genetic variance. Our analysis gives recurrence relations of genetic variance among the succeeding generations for the three cases of gene action, i.e., purely additive action, pure dominance without additive effect and the presence of both additive effect and dominance, respectively. Numerical examples are also given as a check on the recurrence formulas.

Statistical Theory of the Strength of Fiber Bundles

1983 ◽  
Vol 50 (3) ◽  
pp. 601-608 ◽  
Author(s):  
L. N. McCartney ◽  
R. L. Smith
Keyword(s):  
Statistical Theory ◽  
Numerical Calculation ◽  
Statistical Models ◽  
Recurrence Relations ◽  
Fiber Bundles ◽  
Single Fibers ◽  
Analytic Approximations ◽  
Recurrence Formulas ◽  
First Time ◽  
Failure Probabilities

Following a review of statistical models of the failure of single fibers and bundles of these fibers, algebraic recurrence formulas are derived that generate expressions for the failure probabilities of bundles of classical fibers. One of these recurrence relations is suitable for the accurate numerical calculation of failure probabilities of bundles consisting of up to 500 single fibers. It is shown how account can be taken of the effect of defect-free fibers having finite strength. Numerical results are compared to three asymptotic analytic approximations, two of which have been proposed in the statistical literature and are now applied to fiber problems for the first time.

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