Dyson orbitals, quasi-particle effects and Compton scattering

doi:10.1016/j.jpcs.2004.08.016

Journal of Physics and Chemistry of Solids

Volume 65, Issue 12, December 2004, Pages 2031-2034

https://doi.org/10.1016/j.jpcs.2004.08.016 Get rights and content

Abstract

Dyson orbitals play an important role in understanding quasi-particle effects in the correlated ground state of a many-particle system and are relevant for describing the Compton scattering cross section beyond the frameworks of the impulse approximation (IA) and the independent particle model (IPM). Here we discuss corrections to the Kohn–Sham energies due to quasi-particle effects in terms of Dyson orbitals and obtain a relatively simple local form of the exchange–correlation energy. Illustrative examples are presented to show the usefulness of our scheme.

Introduction

Dyson orbitals are a set of one-particle orbitals that are associated with many-electron wavefunctions. These orbitals connect the exact ground-state of the N-electron system with excited states containing N−1 or N+1 electrons. The importance of Dyson orbitals in understanding Compton scattering spectra has been emphasized recently by Kaplan et al. [1], who present a general formalism for the Compton cross section, which goes beyond the standard treatment involving the frameworks of the impulse approximation (IA) and the independent particle model (IPM). The breakdown of the IA in describing core Compton profiles is well-known [2]. More recently, high resolution valence Compton profiles (CPs) of Li at relatively low photon energy of 8–9 keV have been found to show asymmetries in shape and smearing of the Fermi surface (FS) features where deviations from the IA have been implicated [3], [4].

Dyson orbitals also give insight into quasi-particle effects in the correlated ground state of the many-body system. In this article, we focus on understanding energies of one-particle excitations of the ground state, which play an important role not only in the formalism of the Compton scattering cross-section, but also in the band structure problem more generally. To this end, a Green's function approach is used to obtain an expression for the exchange–correlation energy in terms of the self-energy operator involved in the description of the Dyson orbitals. A local ansatz for the self-energy is then invoked to obtain a relatively simple expression for the excitation energies. We illustrate our scheme by considering the example of first ionization energies of low Z atoms from Z=1 (H) to Z=6 (C) and find good agreement with the corresponding experimental results. As another example, the measured bandgap in diamond is also reproduced reasonably by our computations.

An outline of this article is as follows. The introductory remarks are followed in Section 2 by a brief overview of the general formalism of Ref. [1] for the Compton scattering cross section. The importance of properly including excitation energies in the computation for describing the asymmetry of the CP around q=0 is stressed. Section 3 presents the Green's function formulation and discusses our scheme for computing excitation energies. Section 4 gives a few illustrative applications of the theory, followed in Section 5 by a few concluding remarks.

Section snippets

Dyson orbitals and Compton scattering

The Dyson spin–orbital g_n can be defined in terms of the many-body ground-state wavefunction Ψ₀ and the wavefunction Ψ_n of the singly ionized excited system characterized by the quantum number n [5], [6], [7], [8] $g_{n} (x_{N}) = \sqrt{N} \int Ψ_{n} {(x_{1} \dots x_{N - 1})}^{*} Ψ_{0} (x_{1} \dots x_{N}) d x_{1} \dots d x_{N - 1},$ where the index x denotes both spatial and spin coordinates and the integration over dx_i implicitly includes a summation over the spin coordinates. The Dyson orbitals thus give generalized overlap amplitudes between the ground state and the

Green's function method

In order to gain a handle on the nature of the excitation energies $E_{b}^{(n)},$ it proves useful to approach the problem through the Green's function method. The orbitals g_n(r) of Eq. (1) satisfy the Dyson equation [9], [10], [11] $(\frac{p^{2}}{2 m} + V_{ext} (r) + V_{H} (r)) g_{n} (r) + \int d^{3} r^{'} Σ_{xc} (r, r^{'}, E_{b}^{(n)}) g_{n} (r^{'}) = E_{b}^{(n)} g_{n} (r)$ where V_ext(r) is the external potential, V_H is the Hartree potential and $\underset{xc}{Σ}$ is the self-energy.

The Green's function can be expressed as [11] $G (r, r^{'}, ω) = \sum_{n} \frac{g_{n} (r) g_{n}^{*} (r^{'})}{ω - E_{b}^{(n)} + i δ sign (E_{b}^{(n)} - μ)},$ where μ is the chemical

Excitation energy calculations

Fig. 1 provides an illustrative example of the usefulness of Eq. (17). The first ionization energy of atoms from H to C (Z=1−6) is considered using relativistic DFT atomic wavefunctions [21]. The exchange–correlation energy ϵ_xc(r) and the potential v_xc(r) have been calculated within the LDA parametrized by Hedin and Lundqvist [22]. The LDA eigenvalues (open circles) are seen to be substantially lower than the experimental values (diamonds). The quasiparticle correction of Eq. (17) brings the

Summary and conclusions

We discuss aspects of Dyson orbitals for gaining insight into quasi-particle effects in the correlated ground state of a many-particle system. The importance of Dyson orbitals, which connect the many-body ground state with its singly ionized excited states, has been emphasized previously for describing Compton scattering profiles beyond the limitations of the IA and the IPM, and we start with a brief review of this earlier study [1]. We focus on delineating corrections to the Kohn–Sham energies

Acknowledgements

This work is supported by the US Department of Energy contract DE-AC03-765F00098 and benefited from the allocation of supercomputer time at NERSC and Northeastern University's Advanced Scientific Computation Center (ASCC).

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Dyson orbitals, quasi-particle effects and Compton scattering

Abstract

Introduction

Section snippets

Dyson orbitals and Compton scattering

Green's function method

Excitation energy calculations

Summary and conclusions

Acknowledgements

Chem. Phys.

Adv. Quant. Chem.

J. Phys. Chem. Solids

Comput. Mater. Sci.

Phys. Rev. B

Phys. Rev. A

Phys. Rev. B

Phys. Rev. B

Adv. Chem. Phys.

Phys. Rev.

Sov. Phys. Solid State

Z. Physikalische Chem.

Phys. Rev.

Solid State Physics

Rev. Mod. Phys.

Phys. Rev. B

Phys. Lett. A