In modelling the energetics of molecules and solids, the need for practical electron density functionals that seamlessly include the van der Waals interaction is growing. Such functionals are still in their infancy, and there is yet much experimentation to be performed in the formulation and numerical testing of the requisite approximations. A ground-state density functional approach that uses the exact relations of the adiabatic connection formula and the fluctuation-dissipation theorem to obtain the xc energy from the density-density response function seems promising, though a direct local density approximation for the interacting susceptibility will fail to yield the vdW interaction. Significant nonlocality can be built into the interacting susceptibility by screening a 'bare' susceptibility, for which a carefully chosen constraint-obeying local approximation is sufficient to yield a non-trivial van der Waals energy [6]. The constraints of charge conservation, and no response to a constant potential, are guaranteed by expressing the bare susceptibility in terms of the double gradients of a nonlocal bare polarisability. for which it should be easier to make an approximation based on physical principles than it would be for the susceptibility. The 'no-flow' condition is also deemed important. In this work, a simple delta-function approximation for the nonlocal polarisability is fully constrained by a new version of a recently-discovered force theorem (sum rule), requiring the additional input of the independent-electron Kohn-Sham potential. This constrained polarisability cannot be used as input for the seamless vdW scheme, which requires a non-delta-function bare polarisability, and is instead applied to systems containing spherical fragments in a perturbative/asymptotic fashion for calculation of the widely-separated van der Waals interaction. The main thrust of this work is an investigation of the efficacy of the force theorem to constrain simple approximations for response quantities. Many recent perturbative vdW density functionals are based on response functions that are electron-hydrodynamical approximations to the response of the uniform electron gas. These schemes require their response functions to be 'cut off' at low density and high density-gradient, where the approximation overestimates the true response. The imposition of the cut-off is crucial to the success of such schemes. Here, we replace the cut-off with an exact theorem (the force theorem) which naturally 'ties down' the response, based on the potential- and density-functions of the system. This is the first time that the force theorem has been directly applied as a constraint upon a model response function (its original use, by Vignale and Kohn (7), was as an exact identity in time-dependent DFT). Also new in this work is the orbital-by-orbital Kohn-Sham version of the force theorem, and its proof (differing significantly from Vignale's original derivation (8) of the interacting theorem) by directly appealing to the Kohn-Sham orbitals makes its first appearance here. For quantum dots, our constrained response-approximation exactly recovers the net linear dipole response, due mainly to the force theorem's ideal applicability to harmonically confined systems. For angularly-averaged atoms, reasonable static dipole polarisabilities are obtained for the independent-electron Kohn-Sham (bare) case. The results are poor for the fully-interacting case, attributable to the local nature of the approximation. This lends weight to the assertion that it is better to approximate a bare quantity, then screen it, than it is to directly approximate a fully-interacting quantity. Dynamic net polarisabilities constrained by the force theorem are guaranteed to have the correct high-frequency asymptotic convergence to the free electron response. It is seen that the calculated dynamic polarisabilities for atoms are too small at intermediate frequencies, since the calculated vdW C6 coefficients (Hamaker constants) of atomic dimers are up to an order of magnitude too small, even without the use of a low-density cutoff. It is seen that our constrained local model response is non-analytic along the imaginary-frequency axis, and this is very detrimental to the C6 calculations, even though the integrated net polarisability is analytic. Improvement of the polarisability ansatz is indicated, perhaps to a non-deltafunction uniform-gas-based approximation. The use of pseudopotentials may improve the force theorem results, by softening the extreme nature of the bare Coulomb potential.

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 189-196).

Thesis (M. Phil.)--University of Hong Kong, 2010. / Includes bibliographical references (leaves 64-67). Also available in print.

Thesis (M. Phil.)--University of Hong Kong, 2010. / Includes bibliographical references (leaves 64-67). Also available online.

Thesis (M. Sc.)--University of Alberta, 2009. / Title from pdf file main screen (viewed on Dec. 11, 2009). "A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Master of Science in Statistics, Department of Mathematical and Statistical Sciences, University of Alberta." Includes bibliographical references.

published_or_final_version / Physics / Master / Master of Philosophy

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Density functional theory is a successful theory used in physics, chemistry and nanoscience to describe the ground state properties of solids and molecules. It calculates ground state energies and related properties by using the density of the valence electrons as a fundamental variable. In a system of interacting electrons, the electrons will correlate due to the Pauli exclusion principle, as well as their coulomb repulsion. This interaction energy is known as the exchange-correlation energy and is approximated in density functional theory because it is the only unknown in the energy as a functional of density. The simplest model to approximate this exchangecorrelation energy is the local density approximation, which only relies on the local density of the valence electrons at every point. Generalized gradient approximations are approximations which build upon the local density approximation by also using the gradient of the local density. Recently, many new versions of the generalized gradient approximation have been developed to attempt to obtain better energetic and structural properties either at the same time, or at the expense of the other. In this study, we examine the performance of these models by calculating the atomization energy of the AE6 test set. The cohesive energy, lattice constant and bulk modulus of a four solid test set was also calculated. These calculations were done using ABINIT, a density functional theory code that uses a pseudopotential model with plane waves to examine molecules and solids. One of the more recently developed generalized gradient approximation models, the SOGGA, is tested to compare with the standard models. The accuracy of using a pseudopotential model is also tested. It was found that by using a generalized gradient approximation that was better for energy calculations, the structural property calculations would not be as accurate. The SOGGA is a functional that approximates structural properties of solids accurately but does not calculate energies as well. It was also found that using a pseudopotential model resulted in a 1% difference from the all electron calculations. / Density functional theory -- Molecular data -- Solids -- Second order GGA -- Discussions and conclusions. / Department of Physics and Astronomy