# Density functional theory, forces and path integrals

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2011

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Brittain, David Raymond Bogumil

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## Abstract

The work in this thesis is aimed, broadly speaking, at developing methods of applying quantum mechanics to molecular systems; methods which are accurate, but which have a modest computational cost. The focus is on two subjects in particular: density functional theory (DFT) and path integral Monte Carlo methods. Firstly, two new density functionals for exchange are developed using reference electron densities for which the spherical average of the exchange hole is known analytically. The Taylor series of the spherical average of the exchange hole is correct to second order for both of these functionals by construction. One functional uses a reference density designed to be representative of the valence regions of atoms. In a combination with some existing functionals, it performs well on a standard test set of atomisation energies. The other new functional uses the density of a hydrogen-like atom with a fractional number of electrons as a reference, and is the only known pure density functional that is exact for both the infinitely separated hydrogen molecular ion and the hydrogen atom. With some modifications, it is the best performing of any known density functional for the dissociation curve of the hydrogen molecular ion, a problem which is known to pose a challenge for density functional methods. Unfortunately, it does not seem to be widely applicable. A method of decomposing the DFT exchange energy in terms of the inter{u00AD}electronic distance is then described. This allows the quantitative comparison of the distance dependence of DFT exchange and Fock exchange, so long as the spherical average of the exchange hole is known analytically for the exchange functional under investigation. The technique is applied to various chemical systems for which DFT exchange is known to perform poorly: the dissociation curve of the hydrogen molecular ion and a series of isodesmic reactions of hydrocarbons. It is shown quantitatively that the DFT exchange functionals tested, including the two new functionals developed in this work, fail to model the true non-local nature of the Fock exchange. The poor performance of DFT exchange functionals for the dissociation curve of the hydrogen molecular ion is then revisited from the perspective of the electron density and electrostatic forces, using the generalised electrostatic theorem. Finally, a new method for performing path integral Monte Carlo simulations of systems of fermions is described. Using the antisymmetry property of the fermionic density matrix, it is shown that it is possible to write the partition function of a system of fermions as a difference of two path integrals, whose integrands are greater than equal to zero everywhere. This allows the application of Metropolis Monte Carlo methods. A method for determining expectation values of observables and free energy differences based on thermodynamic integration is described.

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Thesis (PhD)

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## DOI

10.25911/5d5151ba5ecb2