We present a finite-temperature extension of the retarded cumulant Green's function for calculations of exited-state, correlation, and thermodynamic properties of electronic systems. The method incorporates a cumulant to leading order in the screened Coulomb interaction W, and improves on the GW approximation of many-body perturbation theory. Results for the homogeneous electron gas are presented for a wide range of densities and temperatures, from cool to warm dense matter regimes, which reveal several hitherto unexpected properties. For example, correlation effects remain strong at high T while the exchange-correlation energy becomes small; also the spectral function broadens and damping increases with temperature, blurring the usual quasiparticle picture. These effects are evident, e.g., in Compton scattering which exhibits many-body corrections that persist at normal densities and intermediate T. The approach also yields exchange-correlation energies and potentials in good agreement with existing methods.
We present a finite-temperature extension of the retarded cumulant Green's function for calculations of exited-state, correlation, and thermodynamic properties of electronic systems. The method incorporates a cumulant to leading order in the screened Coulomb interaction W, and improves on the GW approximation of many-body perturbation theory. Results for the homogeneous electron gas are presented for a wide range of densities and temperatures, from cool to warm dense matter regimes, which reveal several hitherto unexpected properties. For example, correlation effects remain strong at high T while the exchange-correlation energy becomes small; also the spectral function broadens and damping increases with temperature, blurring the usual quasiparticle picture. These effects are evident, e.g., in Compton scattering which exhibits many-body corrections that persist at normal densities and intermediate T. The approach also yields exchange-correlation energies and potentials in good agreement with existing methods.
Thermodynamic properties of the interacting homogeneous electron gas are calculated using a finite-temperature cumulant Green's function approach over a broad range of densities and temperatures up to the warm dense matter regime $T\ensuremath{\sim}{T}_{F}$, where TF is the Fermi degeneracy temperature. These properties can be separated into independent particle and exchange-correlation contributions, and our focus here is on the latter. Our approach is based on the Galitskii-Migdal-Koltun and electron number sum rules from the finite temperature many-body Green's function formalism, together with an extension of the cumulant Green's function to finite temperature. Previously this approach yielded exchange-correlation energies and potentials in good agreement with quantum Monte-Carlo calculations. Here the method is extended for various thermodynamic quantities including the chemical potential, total energy, Helmholtz free-energy, electronic equation of state, specific heat, and isothermal compressibility, which optionally include spin dependence. We find that the exchange-correlation contributions are weakly varying at low temperature but exhibit significant temperature dependence in the WDM regime, as well as a crossover from exchange- to correlation-dominated behavior. In contrast to the T=0+ limit, we also find that renormalization effects are largely but not completely suppressed at finite temperature. Comparisons with other approaches at various levels of approximation are also discussed.
The cumulant expansion is a powerful approach for including correlation effects in electronic structure calculations beyond the GW approximation. However, current implementations are incomplete since they ignore terms that lead to partial occupation numbers and satellites both above and below the Fermi energy. These limitations are corrected here with a generalized cumulant approximation that includes both particle and hole contributions within a retarded Green's function formalism. The computational effort is still comparable to GW, and the method can be extended easily to finite temperature. The approach is illustrated with calculations for the homogeneous electron gas and comparisons to experiment and other methods.
Electronic structure of materials from first principles is essential for one-to-one comparisons between theories and experiments. However, the many-body effects resulting from electron correlation and their interplay with other degrees of freedom pose a great challenge to design an ab initio approximate method with good accuracy and efficiency at the same time. In this thesis, we present our recent developments of ab initio many-body theory for solids based on the finite-temperature Green’s function formalism. We begin by reviewing the basic concepts and definitions of realistic electronic Hamiltonians (Ch. 2) as well as Green’s functions at finite temperature (Ch. 3). Overviews of self-consistent many-body perturbation theories and quantum embedding theories are then presented in Chs. 4 and 5 as complementary materials for the following chapters. In Ch. 6, we present algorithms and implementation details for the finite-temperature self-consistent GW (scGW) method in Gaussian-type orbitals for solids. Our implementation is based on the finite-temperature Green’s function formalism in which all equations are solved on the imaginary axis, without resorting to analytical continuation during the self-consistency. No quasiparticle approximation is employed and all matrix elements of self-energies are explicitly evaluated. We show agreement with other, differently formulated finite-temperature scGW implementations. By migrating computationally intensive calculations to GPUs, we obtain scalable results on large supercomputers with nearly optimal performance. Inclusion of relativistic effects in scGW is discussed in Ch. 7. Our formulation of relativistic scGW is based on the exact two-component formalism with one-electron approximation (X2C1e) and non-relativistic Coulomb interactions. Our theory allows us to study scalar relativistic effects, spin-orbit coupling, and their interplay with electron correlation without adjustable parameters. We examine the effect of the X2C1e by comparison to the established four-component formalism and reach excellent agreement. The simplicity of X2C1e enables construction of higher order theories, such as embedding theories, on top of perturbative calculations. In Ch. 8, we use self-energy embedding theory (SEET) to study the spectra of the prototypical undistorted cubic perovskites SrVO3 and SrMnO3. In the strongly correlated metallic SrVO3, we find that the usual attribution of the satellite peaks at -1.8 eV to Hund or Hubbard physics in the t2g orbitals is inconsistent with our calculations. In the strongly correlated insulator SrMnO3 we recover insulating behavior due to a feedback effect between the strongly correlated orbitals and the weakly correlated environment. Our calculation shows a systematic convergence of spectral features as the space of strongly correlated orbitals is enlarged, paving the way to a systematic parameter-free study of correlated perovskites. Lastly, in Ch. 9, we apply the Green’s function coupled cluster singles and doubles (GFCCSD) impurity solver to realistic impurity problems in strongly correlated solids within the framework of SEET. We describe the implementation details of our Green’s function coupled cluster (GFCC) solver, investigate its performance, and highlight potential advantages and problems on examples of impurities created for antiferromagnetic MnO and paramagnetic SrMnO3. GFCCSD provides satisfactory descriptions for weakly and moderately correlated impurities with sizes that are intractable by existing accurate impurity solvers such as exact diagonalization. However, when correlations become strong, the singles and doubles approximation used in GFCC could lead to instabilities in searching for the particle number present in impurity problems.
The cumulant expansion is a powerful approach for including correlation effects in electronic structure calculations beyond the GW approximation. However, the expansion is not generally valid, as current implementations ignore terms that mix particle and hole states and lead to partial occupation numbers of one-electron states. These limitations are corrected here using a cumulant expansion of the retarded one-electron Green's function that includes both particle and hole contributions. The approach provides a consistent framework that improves on the GW approximation to the spectral function without additional computational effort. The method is illustrated with results for the homogeneous electron gas and comparisons to experiment and other methods.
In this lecture we present many-body perturbation theory as a method to determine quasiparticle excitations in solids, especially electronic band structures, accurately from first principles.The main ingredient is the electronic self-energy that, in principle, contains all many-body exchange and correlation effects beyond the Hartree potential.As its exact mathematical expression is unknown, approximations must be used in practical calculations.The approximation is obtained using a systematic algebraic approach on the basis of Green function techniques.It constitutes an expansion of the self-energy up to linear order in the screened Coulomb potential, which describes the interaction between the quasiparticles and includes dynamic screening through the creation of exchange-correlation holes around the bare particles.The implementation of the approximation relies on a perturbative treatment starting from density functional theory.Besides a detailed mathematical discussion we focus on the underlying physical concepts and show some illustrative applications.
Description of electron correlation is crucial to chemical accuracy in quantum chemical calculations. However, the interaction of electrons in a system is an insoluble many-body problem. Constructing approximations to describe electron correlation is a challenging task, but there has been success in the quantum chemistry community using wavefunction methods and density functional theory. However, there are still challenges to be overcome in areas such as theoretical solid state chemistry, which requires description of large systems and the use of finite temperature. Large system sizes and finite temperature can be difficult to treat solely with the currently available methods. Therefore, a new class of methods, based on the temperature-dependent Green's function, is implemented and explored. This thesis is toward investigating the use of temperature-dependent Green's functions for ab-initio quantum chemistry. That is, we are treating a quantum chemical Hamiltonian with realistic electron interactions. While this formalism has been applied to model systems in the condensed matter community, its has been used much less by the quantum chemistry community. Therefore, the numerical behavior and accuracy of Green's function methods for quantum chemical calculations is relatively unknown. This work investigates the ability of the temperature-dependent Green's function, which is an ensemble formalism, to give access to temperature dependent thermodynamic quantities such as free energy and entropy when calculated in a second-order and perturbative manner (GF2). We find that this method is able to give good accuracy for lower temperatures and excellent accuracy for higher temperatures for a molecular case and is able to qualitatively describe a simple model of a solid. The results of this work are presented in chapter 3. Although Green's functions have a clear connection with spectra at zero temperature, it is not straight forward to obtain spectra from the finite temperature Green's function, which is calculated on the imaginary frequency axis. Therefore, we must investigate methods to obtain spectral quantities in a consistent and reliable manner from the imaginary axis. Chapter 4 investigates several methods to do so and we compare our results with experimental and highly accurate benchmark data. We find that it is possible to obtain spectral quantities that can differ by several electron volts, even if the same level of theory is used to obtain the Green's function. This reiterates that finding a spectrum from the imaginary axis is nontrivial and that one must exercise caution when comparing spectral quantities that were calculated using different techniques, even if they are treated with the same theoretical accuracy. Accessing larger systems with the Green's function requires the use of quantum embedding. Quantum embedding describes nontrivial electron interaction between a highly accurate ``active space'' or ``impurity'' and a larger, lower level ``environment''. It is challenging to construct an impurity solver that is reliable at low temperatures. In chapter 5 of this work, we implement and test a temperature dependent configuration interaction impurity solver for quantum embedding. This solver can be used in quantum embedding schemes such as dynamical mean field theory and self-energy embedding theory for larger systems. Overall, this thesis has made progress toward using Green's functions for ab-initio quantum chemistry at finite temperature. Groundwork has been laid for using this formalism to calculate thermodynamics and spectra using a Green's function with realistic electron interactions and to explore quantum embedding using an impurity solver at low temperatures.