Force constants and dipole‐moment derivatives of molecules from perturbed Hartree–Fock calculations

General expressions for the force constants and dipole‐moment derivatives of molecules are derived, and the problems arising in their practical application are reviewed. Great emphasis is placed on the use of the Hartree–Fock function as an approximate wavefunction, and a number of its properties are discussed and re‐emphasised. The main content of this paper is the development of a perturbed Hartree–Fock theory that makes possible the direct calculation of force constants and dipole‐moment derivatives from SCF–MO wavefunctions. Essentially the theory yields ∂ϕi / ∂RJα, the derivative of an MO with respect to a nuclear coordinate.

Force constants and dipole moment derivatives of molecules from perturbed Hartree-Fock calculations II. Applications to limited basis set SCF-MO wavefunctions

The perturbed Hartree–Fock theory developed in the preceding paper is applied to LiH, BH, and HF, using limited basis‐set SCF–MO wavefunctions derived by previous workers. The calculated values for the force constant ke and the dipole‐moment derivative μ(1) are (experimental values in parentheses): LiH, ke  =  1.618(1.026)mdyn/Å,μ(1)  =  −18.77(−2.0±0.3)D/ÅBH,ke  =  5.199(3.032)mdyn/Å,μ(1)  =  −1.03(−)D/Å;HF,ke  =  12.90(9.651)mdyn/Å,μ(1)  =  −2.15(+1.50)D/Å. The values of the force on the proton were calculated exactly and according to the Hellmann–Feynman theorem in each case, and the discrepancies show that none of the wavefunctions used are close to the Hartree–Fock limit, so that the large errors in ke and μ(1) are not surprising. However no difficulties arose in the perturbed Hartree–Fock calculation, so that the application of the theory to more accurate wavefunctions appears quite feasible.

Determination of the vibrational transition moment ratio for the Coriolis interacting v9 and v10 fundamentals of allene

Comparison between observed and calculated infrared band contours has been made to determine the vibrational transition moment ratio |M10/M9| for the Coriolis interacting ν9 and ν10 perpendicular fundamentals of allene-h4. The ratio obtained is appreciably lower than that of a previous estimate and the result obtained by integrated band intensity measurements of Overend and Crawford. From the best estimate of the ratio, the dipole moment derivatives of the two bands are determined; the value for the weaker band ν9 is subject to a large uncertainty.

Improved arthritic knee health in a pilot RCT of phytotherapy

Although practitioner-prescribed 'western' herbal medicine (phytotherapy) is a popular complementary therapy in the UK, no clinical studies have been reported on patient-orientated outcomes. The objective of this pilot study was to investigate the effects of phytotherapy on symptoms of osteoarthritis of the knee. A previous study of Chinese herbal medicine for the treatment of irritable bowel syndrome, published in the Journal of the American Medical Association,(1) acted as a model in the development of the protocol of this investigation. Twenty adults, previously diagnosed with osteoarthritis of the knee, were recruited from two Inner London GP practices into this randomized, double-blind, placebo-controlled, pilot study carried out in a primary-care setting. All subjects were seen in consultation three times by a herbal practitioner who was blinded to the randomization coding. Each subject was prescribed treatment and given lifestyle advice according to usual practice: continuation of conventional medication where applicable, healthy-eating advice and nutrient supplementation, Individualized herbal medicine was prescribed for each patient, but only dispensed for those randomized to active treatment - the remainder were supplied with a placebo. At baseline and outcome (after ten weeks of treatment), subjects completed a food frequency questionnaire and the Western Ontario and McMaster Universities Osteoarthritis Index (WOMAC) knee health and Measure Yourself Outcome Profile (MYMOP) wellbeing questionnaires. Subjects completing the study per protocol (n = 14) reported an increased intake of wholegrain foods (p = 0.045) and oily fish (p = 0.039) compared to baseline, but no increase in fruit and vegetables and dairy products intakes. There was no difference in the primary outcome measure of knee health assessed as the difference in the mean response (baseline-week 10) in WOMAC score between the two treatment groups. However, there was, compared with baseline, improvement in the active group (n = 9) for the mean WOMAC stiffness sub-score at week 5 (p = 0.035) and week 10 (p = 0.060) but not in the placebo group (n = 5). Furthermore, for the active, but not the placebo group, the mean WOMAC total and sub-scores all showed clinically significant improvement (>= 20%) in knee symptoms at weeks 5 and 10 compared with baseline. Moreover, the mean MYMOP symptom 2 sub-score, mostly relating to osteoarthritis (OA), showed significant improvement at week 5 (p = 0.02) and week 10 (p = 0.008) compared with baseline for the active, but not for the placebo group. This pilot study showed that herbal medicine prescribed for the individual by a herbal practitioner resulted in improvement of symptoms of OA of the knee.

Bromelain as an adjunctive treatment for moderate-to-severe osteoarthritis of the knee: a randomized placebo-controlled pilot study

Background: Osteoarthritis (OA) of the knee is the most prevalent joint disorder. Previous studies suggest that bromelain, a pineapple extract, may be a safer alternative/adjunctive treatment for knee OA than current conventional treatment. Aim: To assess the efficacy of bromelain in treating OA of the knee. Design: Randomized, double-blind placebo-controlled trial. Methods: Subjects (n=47) with a confirmed diagnosis of moderate to severe knee OA were randomized to 12 weeks of bromelain 800 mg/day or placebo, with a 4-week follow-up. Knee (pain, stiffness and function) and quality-of-life symptoms were reported monthly in the WOMAC and SF36 questionnaires, respectively. Adverse events were also recorded. The primary outcome measure was the change in total WOMAC score from baseline to the end of treatment at week 12. Longitudinal models were used to evaluate outcome. Results: Thirty-one patients completed the trial (14 bromelain, 17 placebo). No statistically significant differences were observed between groups for the primary outcome (coefficient 11.16, p=0.27, 95%CI-8.86 to 31.18), nor the WOMAC subscales or SF36. Both treatment groups showed clinically relevant improvement in the WOMAC disability subscale only. Adverse events were generally mild in nature. Discussion: This study suggests that bromelain is not efficacious as an adjunctive treatment of moderate to severe OA, but its limitations support the need for a follow-up study.

Characterising the variability and extremes of the stratospheric polar vortices using 2D moment analysis

The mean state, variability and extreme variability of the stratospheric polar vortices, with an emphasis on the Northern Hemisphere vortex, are examined using 2-dimensional moment analysis and Extreme Value Theory (EVT). The use of moments as an analysis to ol gives rise to information about the vortex area, centroid latitude, aspect ratio and kurtosis. The application of EVT to these moment derived quantaties allows the extreme variability of the vortex to be assessed. The data used for this study is ECMWF ERA-40 potential vorticity fields on interpolated isentropic surfaces that range from 450K-1450K. Analyses show that the most extreme vortex variability occurs most commonly in late January and early February, consistent with when most planetary wave driving from the troposphere is observed. Composites around sudden stratospheric warming (SSW) events reveal that the moment diagnostics evolve in statistically different ways between vortex splitting events and vortex displacement events, in contrast to the traditional diagnostics. Histograms of the vortex diagnostics on the 850K (∼10hPa) surface over the 1958-2001 period are fitted with parametric distributions, and show that SSW events comprise the majority of data in the tails of the distributions. The distribution of each diagnostic is computed on various surfaces throughout the depth of the stratosphere, and shows that in general the vortex becomes more circular with higher filamentation at the upper levels. The Northern Hemisphere (NH) and Southern Hemisphere (SH) vortices are also compared through the analysis of their respective vortex diagnostics, and confirm that the SH vortex is less variable and lacks extreme events compared to the NH vortex. Finally extreme value theory is used to statistically mo del the vortex diagnostics and make inferences about the underlying dynamics of the polar vortices.

Harringtonian virtue: Harrington, Machiavelli, and the method of the 'Moment'

This article presents a reinterpretation of James Harrington's writings. It takes issue with J. G. A. Pocock's reading, which treats him as importing into England a Machiavellian ‘language of political thought’. This reading is the basis of Pocock's stress on the republicanism of eighteenth-century opposition values. Harrington's writings were in fact a most implausible channel for such ideas. His outlook owed much to Stoicism. Unlike the Florentine, he admired the contemplative life; was sympathetic to commerce; and was relaxed about the threat of ‘corruption’ (a concept that he did not understand). These views can be associated with his apparent aims: the preservation of a national church with a salaried but politically impotent clergy; and the restoration of the royalist gentry to a leading role in English politics. Pocock's hypothesis is shown to be conditioned by his method; its weaknesses reflect some difficulties inherent in the notion of ‘languages of thought’.

First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solution's fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.