There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Electronic excitations lie at the origin of most of the commonly measured spectra. However, the first-principles computation of excited states requires a larger effort than ground-state calculations, which can be very efficiently carried out within density-functional theory. On the other hand, two theoretical and computational tools have come to prominence for the description of electronic excitations. One of them, many-body perturbation theory, is based on a set of Green’s-function equations, starting with a one-electron propagator and considering the electron-hole Green’s function for the response. Key ingredients are the electron’s self-energy S and the electron-hole interaction. A good approximation for S is obtained with Hedin’s GW approach, using density-functional theory as a zero-order solution. First-principles GW calculations for real systems have been successfully carried out since the 1980s. Similarly, the electron-hole interaction is well described by the Bethe-Salpeter equation, via a functional derivative of S. An alternative approach to calculating electronic excitations is the time-dependent density-functional theory (TDDFT), which offers the important practical advantage of a dependence on density rather than on multivariable Green’s functions. This approach leads to a screening equation similar to the Bethe-Salpeter one, but with a two-point, rather than a four-point, interaction kernel. At present, the simple adiabatic local-density approximation has given promising results for finite systems, but has significant deficiencies in the description of absorption spectra in solids, leading to wrong excitation energies, the absence of bound excitonic states, and appreciable distortions of the spectral line shapes. The search for improved TDDFT potentials and kernels is hence a subject of increasing interest. It can be addressed within the framework of many-body perturbation theory: in fact, both the Green’s functions and the TDDFT approaches profit from mutual insight. This review compares the theoretical and practical aspects of the two approaches and their specific numerical implementations, and presents an overview of accomplishments and work in progress.

/pdf/electronic-excitations-density-functional-versus-many-body-2tgqd9dfpy.pdf

Electronic excitations: density-functional versus many-body Green's-function approaches

http://nano-bio.ehu.es/files/articles/Gonze_CPC_2009_590.pdf

ABINIT: First-principles approach to material and nanosystem properties

This article describes the variational and fixed-node diffusion quantum Monte Carlo methods and how they may be used to calculate the properties of many-electron systems. These stochastic wave-function-based approaches provide a very direct treatment of quantum many-body effects and serve as benchmarks against which other techniques may be compared. They complement the less demanding density-functional approach by providing more accurate results and a deeper understanding of the physics of electronic correlation in real materials. The algorithms are intrinsically parallel, and currently available high-performance computers allow applications to systems containing a thousand or more electrons. With these tools one can study complicated problems such as the properties of surfaces and defects, while including electron correlation effects with high precision. The authors provide a pedagogical overview of the techniques and describe a selection of applications to ground and excited states of solids and clusters.

/pdf/quantum-monte-carlo-simulations-of-solids-6ten286iud.pdf

Quantum Monte Carlo simulations of solids

Calculations of ground-state and excited-state properties of materials have been one of the major goals of condensed matter physics. Ground-state properties of solids have been extensively investigated for several decades within the standard density functional theory. Excited-state properties, on the other hand, were relatively unexplored in ab initio calculations until a decade ago. The most suitable approach up to now for studying excited-state properties of extended systems is the Green function method. To calculate the Green function one requires the self-energy operator which is non-local and energy dependent. In this article we describe the GW approximation which has turned out to be a fruitful approximation to the self-energy. The Green function theory, numerical methods for carrying out the self-energy calculations, simplified schemes, and applications to various systems are described. Self-consistency issue and new developments beyond the GW approximation are also discussed as well as the success and shortcomings of the GW approximation.

https://iopscience.iop.org/article/10.1088/0034-4885/61/3/002/pdf

The GW method

A generalized plasmon-pole model is presented to describe the energy dependence of the polarizability of semiconductors and insulators. This model leads in a natural way to a simple prescription for determining plasmon band structures. As an application, the plasmon band structure of silicon is calculated within the random-phase approximation. Our plasmon-pole model offers considerable computational advantages over earlier such models in the context of self-energy calculations within the Hedin GW framework. Furthermore, we comment on the purely classical origin of the f-sum rule for the polarizability.

Generalized plasmon-pole model and plasmon band structures of crystals.

We comment on the controversies that seem to exist about the experimental value of the cohesive energy for silicon. We argue that this can be safely taken to be 4.62\ifmmode\pm\else\textpm\fi{}0.08 eV/atom. We conclude that, whereas in the case of diamond the cohesive energy predicted by recent variational quantum Monte Carlo pseudopotential calculation is in excellent agreement with experiment, for silicon there remains a discrepancy of 0.19 eV/atom. To clarify these points, we mention briefly some fundamental principles of the measurement of cohesive energies and explain how they can be obtained from the thermodynamic tables.

Cohesive energies of crystals

We demonstrate that accurate comparisons can be made of the absolute energies of pseudoatoms and pseudosolids obtained from local density approximation pseudopotential calculations with experimental results.

Energies of atoms and solids within the local-density approximation

The plasmon dispersion in silicon has been determined along the ? and ? axis of the first Brillouin zone, taking local-field effects fully into account. The empirical pseudopotential method has been used to obtain the electron band structure. A very effective method of analytic continuation is employed to obtain the singular integrals involved in the calculation of the random-phase-approximation polarization matrix. Using the analytic continuation of the dielectric matrix across its branch cut, we investigate the plasmon energies and lifetimes. It is shown that the experimental observability of the theoretically obtained plasmon band gap at the L point is questionable due to an unexpectedly large lifetime of plasmons in the second band as compared with those in the first band. The obtained plasmon energies are typically 10% larger than the reported experimental values. The calculated plasmon energies display a stronger dispersion than the experimental values. Moreover, the experimentally observed anisotropy in the dispersion along the ? and ? axis is not reproduced by our calculations.

/pdf/plasmon-dispersion-in-silicon-obtained-by-analytic-v8xakllz3u.pdf

Plasmon dispersion in silicon obtained by analytic continuation of the random-phase-approximation dielectric matrix

Plasmon excitations in crystals.

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