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Exploring connections between statistical mechanics and Green’s functions for realistic systems: Temperature dependent electronic entropy and internal energy from a self-consistent second-order Green’s function
Including finite-temperature effects from the electronic degrees of freedom in electronic structure calculations of semiconductors and metals is desired; however, in practice it remains exceedingly difficult when using zero-temperature methods, since these methods require an explicit evaluation of multiple excited states in order to account for any finite-temperature effects. Using a Matsubara Green’s function formalism remains a viable alternative, since in this formalism it is easier to include thermal effects and to connect the dynamic quantities such as the self-energy with static thermodynamic quantities such as the Helmholtz energy, entropy, and internal energy. However, despite the promising properties of this formalism, little is known about the multiple solutions of the non-linear equations present in the self-consistent Matsubara formalism and only a few cases involving a full Coulomb Hamiltonian were investigated in the past. In this study, to shed some light onto the iterative nature of the Green’s function solutions, we self-consistently evaluate the thermodynamic quantities for a one-dimensional (1D) hydrogen solid at various interatomic separations and temperatures using the self-energy approximated to second-order (GF2). At many points in the phase diagram of this system, multiple phases such as a metal and an insulator exist, and we are able to determine the most stable phase from the analysis of Helmholtz energies. Additionally, we show the evolution of the spectrum of 1D boron nitride to demonstrate that GF2 is capable of qualitatively describing the temperature effects influencing the size of the band gap.
The dyadic green's function for a moving isotropic medium
Dyadic Green function for moving isotropic medium
GFCCLib: Scalable and efficient coupled-cluster Green's function library for accurately tackling many-body electronic structure problems
Coupled-cluster Green’s function (GFCC) calculation has drawn much attention in the recent years for targeting the molecular and material electronic structure problems from a many-body perspective in a systematically improvable way. However, GFCC calculations on scientific computing clusters usually suffer from expensive higher di- mensional tensor contractions in the complex space, expensive inter-process communi- cation, and severe load imbalance, which limits it’s routine use for tackling electronic structure problems. Here we present a numerical library prototype that is specifically designed for large-scale GFCC calculations. The design of the library is focused on a systematically optimal computing strategy to improve its scalability and efficiency. The performance of the library is demonstrated by the relevant profiling analysis of running GFCC calculations on remote giant computing clusters. The capability of the library is highlighted by computing a wide near valence band of a fullerene C60 molecule for the first time at the GFCCSD level that shows excellent agreement with the experimental spectrum.
Monte Carlo Second- and Third-Order Many-Body Green’s Function Methods with Frequency-Dependent, Nondiagonal Self-Energy
We fully develop the Monte Carlo many-body Green’s function (MC-GF) method with the following enhancements: (1) The truncation order of the perturbation expansion of the Dyson self-energy is raised from the second order (MC-GF2) to the third order (MC-GF3) with the aid of a computerized procedure to enumerate and transform all 84 third-order Goldstone diagrams into Monte Carlo integrable expressions and then into central processing unit (CPU)/graphical processing unit (GPU)-parallel computer codes. (2) An efficient algorithm is proposed that computes all off-diagonal and diagonal elements of the MC-GF2 and MC-GF3 self-energy matrices by common subexpression elimination. (3) The frequency-independent approximation is lifted by introducing a method that computes frequency derivatives of the MC-GF2 and MC-GF3 self-energies up to any arbitrarily high order at nearly no additional computational cost. (4) The imaginary-time integration in the Laplace-transformed expressions of the self-energy is carried out stochastically (instead of using a quadrature in the previous implementations), resulting in a 50- to 200-fold speedup. (5) The efficiency of the redundant-walker convergence acceleration scheme is analyzed numerically, and the guidelines are established to select an optimal number of walkers for maximal efficiency. When such an optimal number is used, the cost per sample is constant of molecular size on either many CPUs or many GPUs. (6) The computational cost to obtain a binding energy within a given statistical uncertainty is observed to increase as (tentatively) O(n 4 ) and O(n 5 ) of molecular size (n is the number of electrons) with and without the diagonal approximation, respectively, as opposed to O(n 5 ) and O(n 6 ) of the corresponding deterministic algorithms. With this method applied to the electron binding energies of C 60 , we show that the third-order corrections to the self-energies are much greater in electron binding energies than in ground-state energies. As a result, they display a sign of oscillatory convergence toward experimental results, not necessarily improving the agreement with increasing perturbation order, justifying MC-GF3 and motivating even higher-order methods.
Green's functions for Sturm-Liouville problems
Green function for inhomogeneous Strum-Liouville differential equation problem
Theory of solar cell light trapping through a nonequilibrium Green's function formulation of Maxwell's equations
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The dyadic green's function for an infinite moving medium
Derivation of dyadic Green function for electromagnetic field in moving medium using Minkowski theory and method of Fourier analysis
Static Green's function for elastic electron- hydrogen scattering and resonances.
Static Green function for elastic electron scattering by hydrogen atoms, using integrodifferential equations to determine resonance energies
Cumulant Green's function calculations of plasmon satellites in bulk sodium: Influence of screening and the crystal environment
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A Green's function approach to the vibration of thin spherical shell segments
Green function approach to vibration of thin spherical shell segments
Time-dependent Green's function for a moving isotropic nondispersive medium.
Time-dependent Green function for moving isotropic nondispersive medium
Tartarus: A relativistic Green's function quantum average atom code
A relativistic Green’s Function quantum average atom model is implemented in the Tartarus code for the calculation of equation of state data in dense plasmas. We first present the relativistic extension of the quantum Green’s Function average atom model described by Starrett [1]. The Green’s Function approach addresses the numerical challenges arising from resonances in the continuum density of states without the need for resonance tracking algorithms or adaptive meshes, though there are still numerical challenges inherent to this algorithm. We discuss how these challenges are addressed in the Tartarus algorithm. The outputs of the calculation are shown in comparison to PIMC/DFT-MD simulations of the Principal Shock Hugoniot in Silicon. Finally, we also present the calculation of the Hugoniot for Silver coming from both the relativistic and nonrelativistic modes of the Tartarus code.
Green's functions applied to the theory of spectroscopy
Green’s functions are a powerful analytical and computational tool for ab initio calculations of X-ray spectra. For example, Green’s functions provide an efficient means for calculations over broad energy ranges since many-body effects can be incorporated naturally in terms of the electron self-energy. Here, their role in the theory of X-ray absorption and related spectroscopies is discussed, with particular focus on many-body effects such as quasi-particle energy shifts and lifetimes, core-hole interactions and electron–phonon interactions. Additionally, the cumulant expansion for the Green’s function is reviewed and compared with the usual GW Dyson equation approach for including multielectron excitations that produce satellite features in X-ray spectra.
Direct calculation of second-order density matrix. 1 - Theory of the Green's function technique
Second order density matrix calculations using Green function
Equation-of-Motion Coupled-Cluster Cumulant Green’s Function for Excited States and X-ray Spectra
Green’s function methods within many-body perturbation theory provide a general framework for treating electronic correlations in excited states and spectra. Conventional methods using the Dyson equation or the cumulant expansion are typically based on the GW self-energy approximation. In order to extend this approximation in molecular systems, a non-perturbative real-time coupled-cluster cumulant Green’s function approach has been introduced, where the cumulant is obtained as the solution to a set of coupled first order, non-linear differential equations. This approach naturally includes non-linear corrections to conventional cumulant Green’s function techniques where the cumulant is linear in the GW self-energy. The method yields the spectral function associated with the core Green’s function, which is directly related to the x-ray photoemission spectra (XPS) of molecular systems. The approach also yields very good results for binding energies and satellite excitations. The x-ray absorption spectrum (XAS) is then obtained as a convolution of the core spectral function and an effective one-body absorption spectrum. Here this approach is extended to include the full coupled-cluster-singles (CCS) core Green’s function by including the complete form of the non-linear contributions to the cumulant as well as all single, double, and triple cluster excitations in the CC amplitude equations. This approach naturally builds in orthogonality and shake-up effects analogous to those in the Mahan-Noizeres-de Dominicis edge singularity corrections that enhance the XAS near the edge. Themethod is illustrated for the XPS and XAS of NH 3 .
Green's function methods for multiphysics simulations (Final Report)
Green's functions are important tools for analyzing mathematical properties of partial differential equations (PDEs), and for numerically solving PDEs, especially when equations for the same operator but with multiple right hand sides need to be solved simultaneously. This proposal aims at developing efficient and accurate numerical methods for computing Green's functions, which can be used to tackle a challenging question in multiphysics simulation of DOE-mission science problems: how to couple quantum physics with classical physics. The key mathematical difficulty is properly formulate a "boundary condition" for the region described by quantum physics, and conventional approaches often model such boundary conditions in an empirical way. The proposed Green's function methods use the Dirichlet-to-Neumann map to formulate a boundary condition that is non-empirical and can couple the quantum and classical regions in an in principle exact way. The key ingredient of the new methods is to construct the Dirichlet-to-Neumann map in an efficient, accurate and versatile manner. The new methods have provably low complexity and are ideally suited for massively parallel and emerging many-core computational systems.