509 research outputs found
A Primer on Variational Laplace (VL)
This article details a scheme for approximate Bayesian inference, which has underpinned thousands of neuroimaging studies since its introduction 15 years ago. Variational Laplace (VL) provides a generic approach to fitting linear or non-linear models, which may be static or dynamic, returning a posterior probability density over the model parameters and an approximation of log model evidence, which enables Bayesian model comparison. VL applies variational Bayesian inference in conjunction with quadratic or Laplace approximations of the evidence lower bound (free energy). Importantly, update equations do not need to be derived for each model under consideration, providing a general method for fitting a broad class of models. This primer is intended for experimenters and modellers who may wish to fit models to data using variational Bayesian methods, without assuming previous experience of variational Bayes or machine learning. Accompanying code demonstrates how to fit different kinds of model using the reference implementation of the VL scheme in the open-source Statistical Parametric Mapping (SPM) software package. In addition, we provide a standalone software function that does not require SPM, in order to ease translation to other fields, together with detailed pseudocode. Finally, the supplementary materials provide worked derivations of the key equations
Two electrons on a hypersphere: a quasi-exactly solvable model
We show that the exact wave function for two electrons, interacting through a
Coulomb potential but constrained to remain on the surface of a
-sphere (), is a polynomial in the
interelectronic distance for a countably infinite set of values of the
radius . A selection of these radii, and the associated energies, are
reported for ground and excited states on the singlet and triplet manifolds. We
conclude that the model bears the greatest similarity to normal
physical systems.Comment: 4 pages, 0 figur
Ground state of two electrons on concentric spheres
We extend our analysis of two electrons on a sphere [Phys. Rev. A {\bf 79},
062517 (2009); Phys. Rev. Lett. {\bf 103}, 123008 (2009)] to electrons on
concentric spheres with different radii. The strengths and weaknesses of
several electronic structure models are analyzed, ranging from the mean-field
approximation (restricted and unrestricted Hartree-Fock solutions) to
configuration interaction expansion, leading to near-exact wave functions and
energies. The M{\o}ller-Plesset energy corrections (up to third-order) and the
asymptotic expansion for the large-spheres regime are also considered. We also
study the position intracules derived from approximate and exact wave
functions. We find evidence for the existence of a long-range Coulomb hole in
the large-spheres regime, and infer that unrestricted Hartree-Fock theory
over-localizes the electrons.Comment: 10 pages, 10 figure
Excited states of spherium
We report analytic solutions of a recently discovered quasi-exactly solvable
model consisting of two electrons, interacting {\em via} a Coulomb potential,
but restricted to remain on the surface of a -dimensional sphere.
Polynomial solutions are found for the ground state, and for some higher
() states. Kato cusp conditions and interdimensional degeneracies are
discussed.Comment: 6 pages, 2 figures, to appear in Mol. Phy
Invariance of the correlation energy at high density and large dimension in two-electron systems
We prove that, in the large-dimension limit, the high-density correlation
energy \Ec of two opposite-spin electrons confined in a -dimensional space
and interacting {\em via} a Coulomb potential is given by \Ec \sim -1/(8D^2)
for any radial confining potential . This result explains the observed
similarity of \Ec in a variety of two-electron systems in three-dimensional
space.Comment: 4 pages, 1 figure, to appear in Phys. Rev. Let
Harmonically trapped jellium
We discuss the model of a -dimensional confined electron gas in which the
particles are trapped by a harmonic potential. In particular, we study the
non-interacting kinetic and exchange energies of finite-size inhomogeneous
systems, and compare the resulting Thomas-Fermi and Dirac coefficients with
various uniform electron gas paradigms. We show that, in the thermodynamic
limit, the properties of this model are identical to those of the
-dimensional Fermi gas.Comment: 6 pages, 2 figures, 1 table, invited paper for Peter Taylor's 60th
anniversary, submitted to Molecular Physic
Recommended from our members
Embodied Standpoints in Gender Difference Graphs and Tables: When, Where and Why are Men Still Prioritized?
Gender difference graphs and tables typically present data representing males first, ahead of data representing females. The APA Publication Manual in 2010 advised authors against this ‘bias’ when reporting gender differences. An experiment examined how this preference is related to embodied cognition, and two content analytic studies examined its persistence despite APA’s advice against it. In Study 1, 256 students drew bar graphs of gender differences and power differences. Participants spontaneously arrayed men first and higher power groups first most often, even when graph axes were placed to cue the opposite order. These results suggest that male-first order preferences follow from embodied biases to position agentic groups left and higher up in graphs and tables. Two content analyses systematically sampled psychology articles in four journals over a decade (Study 2) or 70 journals in a recent year (Study 3) to examine the impact of the APA manual’s advice on authors. The male-first preference remains prevalent in psychology publications, has reversed in Psychology of Women Quarterly (Study 2) and become polarized by author gender in social psychology (Study 3). These findings suggest that embodied cognition affects the visual representation of gender differences in psychology and is variably moderated by recent injunctions
- …