Exchange representations in Kohn-Sham theory

Kohn-Sham density functional theory (DFT) is the most widely used method in quantum chemistry. It has the potential to provide accurate results at low computational cost. The quality of a DFT calculation is determined by the exchange-correlation energy functional. Hybrid functionals, which contain a fraction of exact orbital exchange, are extensively used due to their accuracy in a variety of applications. However, as commonly implemented, these functionals are outside the Kohn-Sham scheme, since the exchange operator is not a local multiplicative potential. In order to handle orbital dependent functionals correctly, schemes which determine a local multiplicative potential must be employed. The implementation and application of several such methods is the focus of this thesis. In Chapter 1 we outline the Hartree-Fock scheme, which defines the exchange energy, and overview wavefunction based procedures that recover correlation energy. Alternative theories based on the electron density are then considered and the foundations of modern DFT are reviewed. The formalism of the optimized effective potential (OEP) method is introduced, which is the rigorous way to handle orbital dependent functionals. A number of approximations to the exchange only OEP method are outlined in Chapter 2 and their implementation is described. The methods are applied to the calculation of NMR shielding constants, highlighting differences between the approximations; their use in the construction of multiplicative hybrid functionals is also considered. In Chapter 3 these approximations are further investigated in the calculation of excited states and structural perturbations. In Chapter 4, the theory and implementation of a direct optimization procedure to determine OEPs is outlined, along with an implementation of the constrained search procedure, which allows the determination of the Kohn-Sham exchange-correlation potential from any input density. Chapter 5 compares the performance of the approximate exchange potentials with those of OEP, highlighting the presence of correlated character in some of the approximate methods. The OEP implementation is extended to include hybrid exchange-correlation functionals in Chapter 6. The performance of these methods for the calculation of NMR shielding constants, rotational g tensors and transition metal NMR chemical shifts is investigated. In all cases, substantial improvements over conventional results are obtained. In Chapter 7 DFT is used to investigate an interaction of relevance in organic chemistry. Concluding remarks are given in Chapter 8

Alternative separation of exchange and correlation energies in multi-configuration range-separated density-functional theory

The alternative separation of exchange and correlation energies proposed by Toulouse et al. [Theor. Chem. Acc. 114, 305 (2005)] is explored in the context of multi-configuration range-separated density-functional theory. The new decomposition of the short-range exchange-correlation energy relies on the auxiliary long-range interacting wavefunction rather than the Kohn-Sham (KS) determinant. The advantage, relative to the traditional KS decomposition, is that the wavefunction part of the energy is now computed with the regular (fully-interacting) Hamiltonian. One potential drawback is that, because of double counting, the wavefunction used to compute the energy cannot be obtained by minimizing the energy expression with respect to the wavefunction parameters. The problem is overcome by using short-range optimized effective potentials (OEPs). The resulting combination of OEP techniques with wavefunction theory has been investigated in this work, at the Hartree-Fock (HF) and multi-configuration self-consistent-field (MCSCF) levels. In the HF case, an analytical expression for the energy gradient has been derived and implemented. Calculations have been performed within the short-range local density approximation on H2, N2, Li2 and H2O. Significant improvements in binding energies are obtained with the new decomposition of the short-range energy. The importance of optimizing the short-range OEP at the MCSCF level when static correlation becomes significant has also been demonstrated for H2, using a finite-difference gradient. The implementation of the analytical gradient for MCSCF wavefunctions is currently in progress.Comment: 5 figure

The choice of basic variables in current-density functional theory

The selection of basic variables in current-density functional theory and formal properties of the resulting formulations are critically examined. Focus is placed on the extent to which the Hohenberg--Kohn theorem, constrained-search approach and Lieb's formulation (in terms of convex and concave conjugation) of standard density-functional theory can be generalized to provide foundations for current-density functional theory. For the well-known case with the gauge-dependent paramagnetic current density as a basic variable, we find that the resulting total energy functional is not concave. It is shown that a simple redefinition of the scalar potential restores concavity and enables the application of convex analysis and convex/concave conjugation. As a result, the solution sets arising in potential-optimization problems can be given a simple characterization. We also review attempts to establish theories with the physical current density as a basic variable. Despite the appealing physical motivation behind this choice of basic variables, we find that the mathematical foundations of the theories proposed to date are unsatisfactory. Moreover, the analogy to standard density-functional theory is substantially weaker as neither the constrained-search approach nor the convex analysis framework carry over to a theory making use of the physical current density

Robust High-Dynamic-Range Vector Magnetometry via Nitrogen-Vacancy Centers in Diamond

We demonstrate a robust, scale-factor-free vector magnetometer, which uses a closed-loop frequency-locking scheme to simultaneously track Zeeman-split resonance pairs of nitrogen-vacancy (NV) centers in diamond. Compared with open-loop methodologies, this technique is robust against fluctuations in temperature, resonance linewidth, and contrast; offers a three-order-of-magnitude increase in dynamic range; and allows for simultaneous interrogation of multiple transition frequencies. By directly detecting the resonance frequencies of NV centers aligned along each of the diamond's four tetrahedral crystallographic axes, we perform full vector reconstruction of an applied magnetic field

The SBA survey 2008

Most readers will already know that in the late spring of this year the SBA once again conducted a survey of members. This latest survey was a follow-up to the survey conducted in 2006 which produced some interesting results about the pattern of beekeeping in Scotland and the longer term effects of Varroa infestation on managed honeybees in Scotland

Efficient calculation of molecular integrals over London atomic orbitals

The use of London atomic orbitals (LAOs) in a non-perturbative manner enables the determination of gauge-origin invariant energies and properties for molecular species in arbitrarily strong magnetic fields. Central to the efficient implementation of such calculations for molecular systems is the evaluation of molecular integrals, particularly the electron repulsion integrals (ERIs). We present an implementation of several different algorithms for the evaluation of ERIs over Gaussian-type LAOs at arbitrary magnetic field strengths. The efficiency of generalized McMurchie-Davidson (MD), Head-Gordon-Pople (HGP) and Rys quadrature schemes is compared. For the Rys quadrature implementation, we avoid the use of high precision arithmetic and interpolation schemes in the computation of the quadrature roots and weights, enabling the application of this algorithm seamlessly to a wide range of magnetic fields. The efficiency of each generalised algorithm is compared by numerical application, classifying the ERIs according to their total angular momenta and evaluating their performance for primitive and contracted basis sets. In common with zero-field integral evaluation, no single algorithm is optimal for all angular momenta thus a simple mixed scheme is put forward, which selects the most efficient approach to calculate the ERIs for each shell quartet. The mixed approach is significantly more efficient than the exclusive use of any individual algorithm

Predicting patient-reported stroke outcomes: a validation of the six simple variable prognostic model.

BACKGROUND: Case-mix represents the range of disease severity and baseline characteristics that may be the cause of variation in outcomes between individuals and populations. Adjustment for case-mix is therefore important to allow meaningful comparison of healthcare outcomes. The best available case-mix adjustment model for stroke (the Six Simple Variable [SSV] model) was developed to adjust the hard endpoints of independent survival, survival and alive and living at home. There is increasing interest in the measurement of patient-reported outcomes through self-completed questionnaires, though there are currently no robust adjustment models for any such outcome. We aimed to determine whether the SSV prognostic model derived to predict 6-month post-stroke independent survival has wider utility in case-mix adjustment of a patient-reported functional outcome measure, the Subjective Index of Physical and Social Outcome (SIPSO), collected by post 6 months after stroke onset. METHODS: We examined data from 176 patients admitted following an acute stroke and recruited into a prospective cohort study in three participating acute hospitals in Yorkshire, UK. Patients in receipt of palliative care or with transient ischaemic attack were excluded. Using the beta coefficients from the published SSV model to predict independent survival, individual probabilities of 'good' outcome as measured with the dichotomised SIPSO collected by post 6 months after stroke onset were calculated. The ability of the SSV case-mix adjustment model to discriminate patients with 'good' over 'poor' outcome was assessed through calculation of C statistics. Correct predictions were visualised with calibration plots. RESULTS: The C statistics for the SSV model to predict the physical and social subscales of the SIPSO outcome measure were 0.73 (95% CI 0.65-0.79) and 0.66 (0.58-0.82), respectively. Inclusion of patients who died prior to follow-up and ascribing them a score of 0 improved the discrimination (0.76 [0.70-0.82] and 0.70 [0.64-0.76], respectively). Calibration plots demonstrated a tendency to over-optimistic predictions, although confidence limits were wide. CONCLUSIONS: The SSV model predicts adequately the physical component of the SIPSO patient-reported outcome measure and may be useful to adjust this outcome for case-mix following stroke in survivors to follow-up. This could be of benefit in observational studies, stratified randomisation for trials, and in comparison of between-institution clinical trials. Further exploration of the generalizability of the model to adjust other patient-reported stroke outcomes may be warranted

Expression and distribution of Toll-like receptors 11–13 in the brain during murine neurocysticercosis

The functions of Toll-like receptors (TLRs) 11–13 in central nervous system (CNS) infections are currently unknown. Using a murine model of neurocysticercosis, we investigated the expression and distribution of TLRs 11–13 by using both gene specific real-time PCR analysis and in situ immunofluoresence microscopy in both control and neurocysticercosis brains. In the mock infected brain, mRNAs of TLRs 11–13 were constitutively expressed. Parasite infection caused an increase of both mRNAs and protein levels of all three TLRs by several fold. All three TLR proteins were present in both CNS and immune cell types. Among them TLR13 was expressed the most in terms of number of positive cells and brain areas expressing it, followed by TLR11 and TLR12 respectively. Among the nervous tissue cells, TLRs 11–13 protein levels appeared highest in neurons. However, TLR13 expression was also present in ependymal cells, endothelial cells of pial blood vessels, and astrocytes. In contrast, infiltrating CD11b and CD11c positive myeloid cells predominantly produced TLR11 protein, particularly early during infection at 1 wk post infection (~50% cells). TLRs 12 and 13 proteins were present on approximately 5% of infiltrating immune cells. The infiltrating cells positive for TLRs 11–13 were mostly of myeloid origin, CD11b+ cells. This report provides a comprehensive analysis of the expression of TLRs 11–13 in normal and parasite infected mouse brains and suggests a role for them in CNS infections