Resolutions of the Coulomb operator: VI. Computation of auxiliary integrals

We discuss the efficient computation of the auxiliary integrals that arise when resolutions of two-electron operators (specifically, the Coulomb and long-range Ewald operators) are employed in quantum chemical calculations. We derive a recurrence relation that facilitates the generation of auxiliary integrals for Gaussian basis functions of arbitrary angular momentum and propose a near-optimal algorithm for its use

The two faces of static correlation

Restricted Hartree-Fock (RHF) and UHF wavefunctions for beryllium-like ions with nuclear charge 3 ≤ Z ≤ 5 are found using a near-complete Slater basis set. The triplet (RHF → UHF) instability and correlation energy are investigated as a function of Z and we find that the instability vanishes for Z > 4.5. We reproduce this surprising behavior using a minimal-basis model and, by comparing with the stretched H(2) molecule, conclude that "static" (also known as nondynamical, near-degeneracy, first-order, or strong) correlation comes in two flavors: one that can be captured by UHF and another that cannot. In the former (Type A), there is an "absolute near-degeneracy"; in the latter (Type B), there is a "relative near-degeneracy." This dichotomy clarifies discussions of static correlation effects.J.W.H. thanks P.-F. Loos for valuable discussions and the National Science and Engineering Research Council of Canada for funding. P.M.W.G. thanks the NCI National Facility for supercomputer time and the Australian Research Council (Grants Nos. DP0771978 and DP0984806)

The nature of electron correlation in a dissociating bond

We have constructed the unrestricted Hartree-Fock (UHF), restricted Hartree-Fock (RHF), and full configuration interaction (FCI) position and momentum intracules and holes for H···H at bond lengths R from 1 to 10 bohrs. We trace the recently discovered inversion of the UHF position hole at intermediate R to over-localization of the spin-orbitals, and support this by a correlation energy component analysis. The RHF and UHF momentum holes are found to be more complicated; however their features are explained through decomposition of electron correlation effects. The UHF momentum hole is also found to invert and exhibits interesting behavior at large R. The RHF (but not UHF) and FCI momentum intracules exhibit Young-type interference patterns related to recent double photoionization experiments. Our analyses yield the most comprehensive picture to date of the behavior of the electrons during homolytic bond fission.J. W. H. thanks the Natural Sciences and Engineering Research Council (NSERC) of Canada for funding. L.K.M. thanks the RSC for a Summer Research Scholarship. P.M.W.G. thanks the NCI National Facility for a generous grant of supercomputer time and the Australian Research Council (Grant Nos. DP0984806 and DP1094170)

Uniform electron gases

We show that the traditional concept of the uniform electron gas (UEG) --- a homogeneous system of finite density, consisting of an infinite number of electrons in an infinite volume --- is inadequate to model the UEGs that arise in finite systems. We argue that, in general, a UEG is characterized by at least two parameters, \textit{viz.} the usual one-electron density parameter $\rho$ and a new two-electron parameter $\eta$. We outline a systematic strategy to determine a new density functional $E(\rho,\eta)$ across the spectrum of possible $\rho$ and $\eta$ values.Comment: 8 pages, 2 figures, 5 table

Distributions of r 1 ⋅r 2 and p 1 ⋅p 2 in atoms

We consider the two-electron position and momentum dot products, α = r1·r2 and β = p1·p2, and present a method for extracting their distributions, A(α) and B(β), from molecular wave functions built on Gaussian basis functions. The characteristics o

Intracule functional models. V. Recurrence relations for two-electron integrals in position and momentum space

The approach used by Ahlrichs [Phys. Chem. Chem. Phys., 2006, 8, 3072] to derive the Obara-Saika recurrence relation (RR) for two-electron integrals over Gaussian basis functions, is used to derive an 18-term RR for six-dimensional integrals in phase space and 8-term RRs for three-dimensional integrals in position or momentum space. The 18-term RR reduces to a 5-term RR in the special cases of Dot and Posmom intracule integrals in Fourier space. We use these RRs to show explicitly how to construct Position, Momentum, Omega, Dot and Posmom intracule integrals recursively

Acute kidney injury in acute Q fever

Coxiella burnetii infection is not known to involve directly the kidneys. Kidney injury associated with Q fever usually manifests in the setting of chronic infection or endocarditis with development of immune complex deposition. Acute kidney injury (AKI) in the context of acute Q fever infection may be more pathologically heterogeneous. We describe two cases of severe AKI secondary to acute Q fever infection, each with marked differences in pathological characteristics, and clinical course

Distributions of <b>r</b>₁·<b>r</b>₂ and <b>p</b>₁·<b>p</b>₂ in Atoms

We consider the two-electron position and momentum dot products, α = <b>r</b>₁·<b>r</b>₂ and β = <b>p</b>₁·<b>p</b>₂, and present a method for extracting their distributions, <i>A</i>(α) and <i>B</i>(β), from molecular wave functions built on Gaussian basis functions. The characteristics of the Hartree–Fock <i>A</i>_HF(α) and <i>B</i>_HF(β) for He and the first-row atoms are investigated, with particular attention to the effects of Pauli exchange. The effects of electron correlation are studied via the holes, Δ<i>A</i>(α)  <i>A</i>(α) – <i>A</i>_HF(α) and Δ<i>B</i>(β)  <i>B</i>(β) – <i>B</i>_HF(β), and the hole structures are rationalized in terms of radial and angular correlation effects. Correlation effects are also examined through an analysis of the first moments of <i>A</i>(α), <i>A</i>_HF(α), <i>B</i>(β), and <i>B</i>_HF(β)

Predictors of decline of residual renal function in new peritoneal dialysis patients

Objective: The aim of this study was to prospectively evaluate the risk factors for decline of residual renal function (RRF) in an incident peritoneal dialysis (PD) population