A dissertation submitted for the fulfilment of the requirements of the degree of Master of Science to the Faculty of Science, Witwatersrand University, Johannesburg. June 2016. / The combination of effective numerical techniques and scientific intuition to find new and novel types of materials is the process used in the discovery of materials for future technologies. Adding to that, being able to calculate the radiative lifetimes of excitons, exciton properties, and the optical properties by using efficient numerical techniques gives an estimation and identification of the best candidate materials for a solar cell. This approach is inexpensive and stable. Present ab initio methods based on Many-body perturbation theory and density functional theory are capable of predicting these properties with a high enough level of accuracy for most cases. The electronic properties calculated using GaAs as a reference system and the 3D hybird perovskite CH3NH3PbI3 are based on density functional theory. The optical properties are investigated by calculating the dielectric function. The theoretical framework of the radiative lifetime of excitons and calculating the exciton properties are based on Wannier model of the exciton and the Bethe-Salpeter equation. / MT2017

In this thesis, we have studied the electronic and optical properties of solar energy m-terials. The studies are performed in the framework of density functional theory (DFT), GW, Bethe-Salpeter equation (BSE) approaches and Kinetic Monte Carlo (KMC). We present four sets of results. In the first part, we report our results on the band gap engineering issues for BiNbO4and NaTaO3, both of which are good photocatalysts. The band gap tuning is required for these materials in order to achieve the maximum solar to hydrogen conversion efficiency. The most common method for the band gap reduction is an introduction of foreign elements. The mono-doping in the system generates electrons or holes states near band edges, which reduce the efficiency of photocatalytic process. Co-doping with anion and cation or anion and anion can provide a clean band gap. We have shown that further band gap reduction can be achieved by double-hole mediated coupling between two anionic dopants. In the second part, the structure and optical properties of (CdSxSe1x)42nanoclusters have been studied. Within this study, the structures of the (CdS)42, (CdSe)42, Cd42Se32S10, Cd42Se22S20, and Cd42Se10S32 clusters have been determined using the simulated annealing method. Factors influencing the band gap value have been analyzed. We show that the gap is most significantly reduced when strongly under coordinated atoms are present on the surface of the nanoclusters. In addition, the band gap depends on the S concentration as well as on the distribution of the S and Se atoms in the clusters. We present the optical absorption spectra calculated with BSE and random phase approximation (RPA) methods based on the GW corrected quasiparticle energies. In the third part, we have employed the state-of-art computational methods to investigate the electronic structure and optical properties of TiO2high pressure polymorphs. GW and BSE methods have been used in these calculations. Our calculations suggest that the band gap of fluorite and pyrite phases have optimal values for the photocatalytic process of decomposing water in the visible light range. In the fourth part we have built a kinetic model of the first water monolayer growth on TiO2(110) using the kinetic Monte Carlo (KMC) method based on parameters describing water diffusion and dissociation obtained from first principle calculations. Our simulations reproduce the experimental trends and rationalize these observations in terms of a competition between different elementary processes. At high temperatures our simulation shows that the structure is well equilibrated, while at lower temperatures adsorbed water molecules are trapped in hydrogen-bonded chains around pairs of hydroxyl groups, causing the observed higher number of molecularly adsorbed species at lower temperature. / <p>QC 20140603</p>

Dissertation (Ph.D.) -- Worcester Polytechnic Institute. / Keywords: ladder approximation; inelastic process; Bethe-Salpeter; BSE; nucleon; scalar field; pion. Includes bibliographical references (p.161-163).

No description available.

Consideramos nesse trabalho questões relativas a parte inferior do espectro de energia-momento para o modelo de teoria campos na rede com tempo imaginário, associado ao sistema ferromagnético de spins clássicos de $N$-compontentes definido na rede $d$ dimensional: O Modelo de Spin O$(N)$. Esses sistemas são caracterizados por uma distribuição de probabilidade de spin por sítio. Tratamos apenas da região de altas temperaturas. O espectro de energia e momento deste modelo apresenta curvas de dispersão isoladas, que podem ser interpretadas como quasi-partículas. Em particular, estudaremos os estados de uma e duas quasi-partículas. Para o espectro de uma partícula, obteremos a curva de dispersão e a massa de uma partícula. Esse resultado mostra a existência da chamada 'lacuna espectral'. Ainda trabalhando no espectro de uma partícula, demonstraremos a existência de uma banda de espectro contínuo, associada a estados de duas partículas livres, e determinaremos a largura desta banda. Nossa análise de duas partículas é restrita a uma aproximação em escada da equação Bethe-Salpeter. Usando essa aproximação mostraremos que a existência e a localização de estados ligados depende da verificação da dominação gaussiana para a função de correlação de quatro pontos. É sabido que estados ligados de duas partículas aparecem abaixo da banda de duas partículas se não vale a dominação gaussiana. Mostraremos que estados ligados de duas partículas aparecem acima da banda de duas partículas, caso a dominação gaussiana seja verificada. Além disso, mostramos como o padrão espectral de duas partículas para desses modelos podem ser compreendido através da correspondência entre a equação Bethe-Salpeter e um operador hamiltoniano de Schrödinger de duas partículas na rede com potenciais atrativos ou repulsivos do tipo delta e dependentes dos indices de spin. Uma transformação de staggering é utilizada para relacionar os casos de potenciais atrativos e repulsivos e o espectro dos hamiltonianos e suas autofunções. / In this work, we consider the low-lying energy-momentum spectrum for the imaginary-time lattice quantum field model associated with d-dimensional lattice ferromagnetic classical N-component vector spin systems: The O(N) Spin Model. Each system is characterized by a single site 'a priori' spin probability distribution. We work only at high temperature region (0<&#946;<=1). The energy-momentum spectrum exhibits isolated dispersion curves which are identified as single particles and multi-particle bands. In particular, we study states of one and two-particles. For the single particle spectrum, we obtain the dispersion curve and the particle mass. This result show the existence of the so called 'low spectral gap'. Still working with the single particle spectrum, e show the existence of a continuum spectra band, associated to states of two free partciles, and we obtain the band width. Our two-particle bound state analysis is restricted to a ladder approximation of the Bethe-Salpeter equation, and the existence of bound states depend on whether or not Gaussian domination for the four-point function is verified. It is known that two-particle bound states appear below the two-particle band if Gaussian domination does not hold. Here, we show that two two-particle bound states appear above the two-particle band if Gaussian domination is verified. We also show how the complete two-particle spectral pattern for these models can be understood by making a correspondence between the Bethe-Salpeter equation and a two-particle lattice Schrödinger Hamiltonian operator with attractive or repulsive spin-dependent delta potentials at the origin. A staggering transformation is used to relate the attractive and repulsive potential cases, as well as their associated Hamiltonians spectrum and eigenfunctions.

Consideramos nesse trabalho questões relativas a parte inferior do espectro de energia-momento para o modelo de teoria campos na rede com tempo imaginário, associado ao sistema ferromagnético de spins clássicos de $N$-compontentes definido na rede $d$ dimensional: O Modelo de Spin O$(N)$. Esses sistemas são caracterizados por uma distribuição de probabilidade de spin por sítio. Tratamos apenas da região de altas temperaturas. O espectro de energia e momento deste modelo apresenta curvas de dispersão isoladas, que podem ser interpretadas como quasi-partículas. Em particular, estudaremos os estados de uma e duas quasi-partículas. Para o espectro de uma partícula, obteremos a curva de dispersão e a massa de uma partícula. Esse resultado mostra a existência da chamada 'lacuna espectral'. Ainda trabalhando no espectro de uma partícula, demonstraremos a existência de uma banda de espectro contínuo, associada a estados de duas partículas livres, e determinaremos a largura desta banda. Nossa análise de duas partículas é restrita a uma aproximação em escada da equação Bethe-Salpeter. Usando essa aproximação mostraremos que a existência e a localização de estados ligados depende da verificação da dominação gaussiana para a função de correlação de quatro pontos. É sabido que estados ligados de duas partículas aparecem abaixo da banda de duas partículas se não vale a dominação gaussiana. Mostraremos que estados ligados de duas partículas aparecem acima da banda de duas partículas, caso a dominação gaussiana seja verificada. Além disso, mostramos como o padrão espectral de duas partículas para desses modelos podem ser compreendido através da correspondência entre a equação Bethe-Salpeter e um operador hamiltoniano de Schrödinger de duas partículas na rede com potenciais atrativos ou repulsivos do tipo delta e dependentes dos indices de spin. Uma transformação de staggering é utilizada para relacionar os casos de potenciais atrativos e repulsivos e o espectro dos hamiltonianos e suas autofunções. / In this work, we consider the low-lying energy-momentum spectrum for the imaginary-time lattice quantum field model associated with d-dimensional lattice ferromagnetic classical N-component vector spin systems: The O(N) Spin Model. Each system is characterized by a single site 'a priori' spin probability distribution. We work only at high temperature region (0<&#946;<=1). The energy-momentum spectrum exhibits isolated dispersion curves which are identified as single particles and multi-particle bands. In particular, we study states of one and two-particles. For the single particle spectrum, we obtain the dispersion curve and the particle mass. This result show the existence of the so called 'low spectral gap'. Still working with the single particle spectrum, e show the existence of a continuum spectra band, associated to states of two free partciles, and we obtain the band width. Our two-particle bound state analysis is restricted to a ladder approximation of the Bethe-Salpeter equation, and the existence of bound states depend on whether or not Gaussian domination for the four-point function is verified. It is known that two-particle bound states appear below the two-particle band if Gaussian domination does not hold. Here, we show that two two-particle bound states appear above the two-particle band if Gaussian domination is verified. We also show how the complete two-particle spectral pattern for these models can be understood by making a correspondence between the Bethe-Salpeter equation and a two-particle lattice Schrödinger Hamiltonian operator with attractive or repulsive spin-dependent delta potentials at the origin. A staggering transformation is used to relate the attractive and repulsive potential cases, as well as their associated Hamiltonians spectrum and eigenfunctions.

This thesis focuses on developing new approaches to solving the ground state properties of quantum many-body Hamiltonians, and the goal is to develop a systematic approach which properly balances efficiency and accuracy. Two new formalisms are proposed in this thesis: the Variational Discrete Action Theory (VDAT) and the Off-Shell Effective Energy Theory (OET). The VDAT exploits the advantages of both variational wavefunctions and many-body Green's functions for solving quantum Hamiltonians. VDAT consists of two central components: the Sequential Product Density matrix (SPD) and the Discrete Action associated with the SPD. The SPD is a variational ansatz inspired by the Trotter decomposition and characterized by an integer N, and N controls the balance of accuracy and cost; monotonically converging to the exact solution for N → ∞. The Discrete Action emerges by treating the each projector in the SPD as an effective discrete time evolution. We generalize the path integral to our discrete formalism, which converts a dynamic correlation function to a static correlation function in a compound space. We also generalize the usual many-body Green's function formalism, which results in analogous but distinct mathematical structures due to the non-abelian nature of the SPD, yielding discrete versions of the generating functional, Dyson equation, and Bethe-Salpeter equation. We apply VDAT to two canonical models of interacting electrons: the Anderson impurity model (AIM) and the Hubbard model. We prove that the SPD can be exactly evaluated in the AIM, and demonstrate that N=3 provides a robust description of the exact results with a relatively negligible cost. For the Hubbard model, we introduce the local self-consistent approximation (LSA), which is the analogue of the dynamical mean-field theory, and prove that LSA exactly evaluates VDAT for d=∞. Furthermore, VDAT within the LSA at N=2 exactly recovers the Gutzwiller approximation (GA), and therefore N>2 provides a new class of theories which balance efficiency and accuracy. For the d=∞ Hubbard model, we evaluate N=2-4 and show that N=3 provides a truly minimal yet precise description of Mott physics with a cost similar to the GA. VDAT provides a flexible scheme for studying quantum Hamiltonians, competing both with state-of-the-art methods and simple, efficient approaches all within a single framework. VDAT will have broad applications in condensed matter and materials physics. In the second part of the thesis, we propose a different formalism, off-shell effective energy theory (OET), which combines the variational principle and effective energy theory, providing a ground state description of a quantum many-body Hamiltonian. The OET is based on a partitioning of the Hamiltonian and a corresponding density matrix ansatz constructed from an off-shell extension of the equilibrium density matrix; and there are dual realizations based on a given partitioning. To approximate OET, we introduce the central point expansion (CPE), which is an expansion of the density matrix ansatz, and we renormalize the CPE using a standard expansion of the ground state energy. We showcase the OET for the one band Hubbard model in d=1, 2, and ∞, using a partitioning between kinetic and potential energy, yielding two realizations denoted as K and X. OET shows favorable agreement with exact or state-of-the-art results over all parameter space, and has a negligible computational cost. Physically, K describes the Fermi liquid, while X gives an analogous description of both the Luttinger liquid and the Mott insulator. Our approach should find broad applicability in lattice model Hamiltonians, in addition to real materials systems. The VDAT can immediately be applied to generic quantum models, and in some cases will rival the best existing theories, allowing the discovery of new physics in strongly correlated electron scenarios. Alternatively, the OET provides a practical formalism for encapsulating the complex physics of some model and allowing extrapolation over all phase space. Both of the formalisms should find broad applications in both model Hamiltonians and real materials.

In the first two chapters, we provide a new way to think about the Green’s function-basedGW approximation and Bethe-Salpeter equation (BSE). The former is the most popular beyond-mean-field method for band structures of solids and an increasingly popular one for ionization potentials and electron affinities of molecules. The latter is widely used to compute neutral excitation energies and spectra for solids as well as, increasingly, molecules. Inspired by quantum chemistry approaches, we obtain a computational scaling reduction and avoid approximating certain dynamical quantities. The new formalism suggests further improvements to the GW and BSE methods. In chapters four and five, we derive and test a cheap, approximate version of the GW and BSE for large molecules and then extend the strategy to periodic systems. In chapter six, we assess another Green’s function-based method, the constrained random phase approximation with exact diagonalization, usually applied to solids. This method allows one to treat electron correlation within an active space of important orbitals while also including some of the external orbital space effects. In chapters seven and eight, we implement the BSE in the PySCF software package for periodic systems using Gaussian density fitting and then apply it to a challenging system, the superatomic solid Re₆Se₈Cl₂.

This dissertation documents the results of two related efforts. Firstly, a model of nucleon-nucleon (NN) interactions is developed based on scalar field theory. Secondly, the relativistic 2-body Bethe-Salpeter equation (BSE) is generalized to handle inelastic processes in the ladder approximation. Scalar field theory describes the behavior of scalar particles, particles with spin 0. In the present work scalar field theory is used to describe NN interactions mediated by pion exchange. The scalar theory is applied to nucleons despite the fact that nucleons are fermions, spin 1/2 particles best described by fourcomponent Dirac spinor fields. Nevertheless, the scalar theory is shown to give a good fit to experiment for the total cross sections for several reactions [1]. The results are consistent with more elaborate spinor models involving one boson exchange (OBE). The results indicate that the spin and isospin of nucleons can to some extent be ignored under certain conditions. Being able to ignore spin and isospin greatly reduces the complexity of the model. A limitation of the scalar theory is that it does not distinguish between particle and anti-particle. Consequently one must decide how to interpret the s-channel diagrams generated by the theory, diagrams which involve particle creation and annihilation. The issue is resolved by extending the scalar theory to include electric charge, and formulating NN interactions in terms of complex scalar fields, which are able to describe both particles and anti-particles. A generalized Bethe-Salpeter equation (GBSE) is developed to handle inelastic processes in the ladder approximation. The GBSE, formulated using the scalar theory, is new, and introduces a systematic method for analyzing families of coupled reactions. A formalism is developed centered around the amplitude matrix M' defined for a given Lagrangian. M' gives the amplitudes of a family of reactions that arise from the Lagrangian. The formalism demonstrates how these amplitudes, to 2nd order, segregate into independent groups of coupled BSE's. The GBSE formalism is applied to the coupled BSE (CBSE) of Faassen and Tjon (FT) [2] for the reaction N+N->N+Delta, showing that the CBSE is missing a coupling channel, and in the expansion, under counts ladder diagrams. A proof is given of the equivalence of the series of ladder diagrams generated by M' and the S-matrix. A section on future work discusses several projects for further development and application of the GBSE.

Submitted by Cristian Gutierrez (cristian@ift.unesp.br) on 2016-11-25T17:35:07Z No. of bitstreams: 1 Cristian_Gutierrez_PhD_Thesis.pdf: 2056100 bytes, checksum: 98402a9e05e7c393491419def7ff3ca9 (MD5) / Approved for entry into archive by Felipe Augusto Arakaki (arakaki@reitoria.unesp.br) on 2016-11-30T13:24:29Z (GMT) No. of bitstreams: 1 gutierrezgomez_cl_dr_ift.pdf: 2056100 bytes, checksum: 98402a9e05e7c393491419def7ff3ca9 (MD5) / Made available in DSpace on 2016-11-30T13:24:29Z (GMT). No. of bitstreams: 1 gutierrezgomez_cl_dr_ift.pdf: 2056100 bytes, checksum: 98402a9e05e7c393491419def7ff3ca9 (MD5) Previous issue date: 2016-10-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O trabalho apresentado nessa tese foi dedicado em explorar soluções de estado ligado para a equação de Bethe-Salpeter, obtidas diretamente no espaço de Minkowski. Para isso, consideramos um procedimento que combina a representação integral de Nakanishi para a amplitude Bethe-Salpeter, desenvolvido por N. Nakanishi na década de sessenta, em conjunto com a projeção da amplitude de Bethe-Salpeter no plano nulo, também conhecida como a projeção na frente de luz. Este método, além de permitir calcular as energias de ligação, que são acessíveis a partir de cálculos bem conhecidos no espaço Euclidiano, permite que se obtenha a amplitude Bethe-Salpeter no espaço de Minkowski e a função de onda de valência na frente de luz. A verificação da validade desse procedimento foi confirmada através de comparação da amplitude de Bethe-Salpeter obtida diretamente no espaço Euclidiano com a amplitude correspondente derivada da equação de Bethe-Salpeter, usando a representação integral de Nakanishi, uma vez a rotação de Wick é realizada. O sucesso dessa abordagem, quando aplicado ao problema do estado ligado de duas partículas escalares trocando uma outra partícula escalar no estado fundamental, assim como o estudo correspondente no limite de energia zero, nos motivou a ampliar a aplicação do procedimento para o estudo de outros problemas de interesse. Em particular, o método foi estendido para o estudo de sistemas com duas dimensões espaciais e uma temporal (2+1), considerando o interesse crescente que surgiu em Física da matéria condensada, onde podemos destacar o caso de elétrons de Dirac no grafeno. Nessa análise preliminar, nos restringimos ao modelo escalar que nos permitiu acessar as principais dificuldades que deverão ser enfrentadas ao estudar o problema do estado ligado entre dois férmions. Dessa forma, este tratamento pode ser considerado como um primeiro passo para a implementação de um método mais realístico em um problema fermiônico. Os cálculos anteriores que consideramos em nossos estudos foram realizados através da aproximação de escada para o kernel de interação irredutível para os estados de onda-s. Portanto, uma das extensões que exploramos nesta tese foi o efeito de se introduzir a contribuição de ordem seguinte no kernel de interação, conhecida como a contribuição de escada-cruzada (cross-ladder). Os efeitos nas energias de ligação e na função de onda na frente de luz é foram analisados de forma detalhada, através dos resultados apresentados. Um estudo particularmente interessante, que foi extensivamente estudado nesta tese, se refere ao problema do espectro da equação Bethe-Salpeter para o estado ligado escalar-escalar. O espectro de estados excitados foi obtido com a abordagem da representação integral Nakanishi, sendo comparado com o obtido no espaço Euclidiano. Além disso, as raçoes excitado/fundamental do espectro relativístico foram reduzidas para às não-relativístico através da escolha de energias de ligação pequenas e considerando a massa do bóson trocado sendo próxima de zero. A função de onda de valência na frente de luz e a função de onda no parâmetro de impacto são apresentadas mostrando as principais características dos estados excitados conhecidos da estrutura não relativística. Na análise do espectro, também são estudadas as amplitudes de momentum-transverso para o estado fundamental e o primeiro estado excitado, que podem ser obtidos, de forma equivalente, no espaço de Minkowski assim como no espaço Euclidiano. Finalmente, focamos o estudo nos fatores de forma eletromagnéticos elásticos na abordagem da Bethe-Salpeter. Consciente de que o cálculo correto dos fatores de forma deve ser feito no espaço de Minkowski, o fator de forma elástico foi calculado levando-se em consideração a aproximação de impulso padrão. Além disso, foi também estudado o efeito da contribuição de ordem superior no fator de forma. / The work presented in this thesis was dedicated in exploring bound-state solutions of the Bethe-Salpeter equation directly in the Minkowski space. For that, we consider a method that combines the Nakanishi integral representation for the Bethe-Salpeter amplitude, developed by Noboru Nakanishi in the sixties, together with the projection of the Bethe-Salpeter amplitude onto the null-plane, also known as the light-front projection. This approach, besides of allowing to compute the binding energies, which are accessible from the usual Euclidean calculation, enables to obtain the Bethe-Salpeter amplitude in the Minkowski space and the light-front wave function. The feasibility of such an approach is further verified by comparing the Bethe-Salpeter amplitude obtained directly in the Euclidean space with the corresponding amplitude obtained by solving the Bethe-Salpeter equation, using the Nakanishi integral representation, once the Wick rotation is performed to this latter. The success of the approach when applied to study the bound state problem of two-scalar particles exchanging another scalar particle in the ground state, as well as the corresponding study at the zero-energy limit, has encouraged us to extend this method to another interesting problems. In particular, we start by extending the method to study problems in (2+1) dimensions due to the increasing interest in the condensed-matter physics, like the study of Dirac electrons in graphene. In this initial examination we restrict to the scalar model, which enables us to access to the main difficulties that we will face when studying the fermion-fermion bound state problem. Hence, this calculation can be considered as the first step towards the implementation of the method to real fermionic problems. The previous calculations have been performed by considering the ladder approximation for the irreducible interacting kernel for s-wave states. Therefore, one of the extensions that is explored in this thesis is the effect of introducing the next contribution in the interacting kernel, known as the scalar-scalar cross-ladder contribution. The effects in the eigenvalues and the light-front wave functions are analyzed in detail, by considering the computed results. A particular interesting subject, extensively studied in this thesis, is concerned to the spectrum of the Bethe-Salpeter equation for the scalar-scalar bound-state problem. The spectrum of excited states obtained with the Nakanishi integral representation approach is compared with that obtained in the Euclidean calculation. Besides, the ratio energies excited/ground of the relativistic spectrum is reduced to the non-relativistic one by choosing small binding energies and the mass of the exchanged boson approaching to zero. The valence light-front wave function and the impact-parameter space valence wave function are displayed, revealing the main features of excited states known from the non-relativistic framework. In the analysis of the spectrum, we also studied the transverse-momentum amplitudes for the ground and first-excited state, which can be equivalently obtained in the Minkowski or Euclidean spaces. Finally, we focus on the study of electromagnetic elastic form factors within the Bethe-Salpeter approach. Aware that the correct calculation of form factors should be performed in the Minkowski space, the calculation of the elastic form factor is carried out with the standard impulse approximation and in addition the effect of the next contribution to the form factor is studied.