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Physical review. B, Condensed matter

Theory of electron liquids. II. Static and dynamic form factors, correlation energy, and plasmon dispersion

Naoki Iwamoto, E. Krotscheck, David Pines

Published Apr 1, 1984• 102 citations
Publisher

Abstract

We use the Ceperley and Alder Green's-function Monte Carlo data for the static form factor, $S(q)$, of an electron liquid to calculate the static local-field correction, $G(q)$, to the dielectric function which takes into account short-range electron correlations. The resulting local-field correction in the long-wavelength limit is obtained by smoothly fitting to the results of Iwamoto and Pines, which are exact in this limit. We use the $G(q)$ thus obtained to calculate the static and dynamic form factors, the contribution to the correlation energy from different momentum transfers, plasmon dispersion, and the plasmon-dispersion coefficient. We compare these theoretical quantities with the available experimental data, and use the ${\ensuremath{\omega}}^{3}$-sum rule (which we also calculate from the Monte Carlo data) to discuss the limitations arising from the use of the static local-field correction.

DOI
10.1103/physrevb.29.3936

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We use the Ceperley and Alder Green's-function Monte Carlo data for the static form factor, S(q)S(q)S(q), of an electron liquid to calculate the static local-field correction, G(q)G(q)G(q), to the dielectric function which takes into account short-range electron correlations. The resulting local-field correction in the long-wavelength limit is obtained by smoothly fitting to the results of Iwamoto and Pines, which are exact in this limit. We use the G(q)G(q)G(q) thus obtained to calculate the static and dynamic form factors, the contribution to the correlation energy from different momentum transfers, plasmon dispersion, and the plasmon-dispersion coefficient. We compare these theoretical quantities with the available experimental data, and use the \ensuremathω3{\ensuremath{\omega}}^{3}\ensuremathω3-sum rule (which we also calculate from the Monte Carlo data) to discuss the limitations arising from the use of the static local-field correction.
The parametrized form of local-field corrections is introduced to construct the dielectric function and the nonlocal spin susceptibility that describe the dielectric properties of a paramagnetic degenerate electron liquid in two dimensions beyond the random-phase approximation (RPA). The parameters that take into account the spin-parallel and spin-antiparallel electron-electron interactions at short distances are determined in such a way that the response functions correctly give the Monte Carlo values for the compressibility and the spin susceptibility, respectively, in the long-wavelength limit. The local-field correction thus obtained is compared with that in the Hubbard approximation; the static dielectric function and the spin susceptibility are also compared with those in the RPA. It is found that the sign of the static dielectric function depends on both the dimensionless coupling constant and the wave number, which is a feature markedly different from the RPA results.
Electron correlations of an electron liquid at metallic densities are studied by means of a diagrammatic method, the primary aim being the interpretation of the spectral structure observed for several metals in recent inelastic electron and x-ray scattering experiments. Several sum rules and requirements such as the positivity of the pair distribution function, the compressibility sum rule, the frequency moment sum rules, and the conservation law of the local electron number which have hitherto been regarded as guiding principles in theoretical formulation are first examined and discussed in a critical manner. Detailed examinations are made of the proper polarization function within the Hartree-Fock approximation. The lowest-order perturbation expressions giving two-pair excitations and one-pair---one-plasmon excitations are next examined. It is pointed out that these expressions are indeed divergent in the usual one-pair excitation region; the renormalization of the one-particle Green's function is indispensable there. From the above considerations and further discussions, the quasi-one-pair excitation approximation is presented for the purpose of describing the spectral shape in the dynamical structure factor $S(q,\ensuremath{\omega})$ in the intermediate wave-number and intermediate frequency regions. In the quasi-one-pair excitation approximation the proper polarization function is constructed from a coherent part of the renormalized one-particle Green's function and a static part of the irreducible particle-hole interaction including strong short-range correlations.
We have evaluated the density-density response of the two-dimensional electron gas at zero temperature by solving the Dyson equation for the particle-hole Green's function, including exchange Coulomb matrix elements and short-range contributions in the ladder approximation. We study the effect of these correlations on the total energy, compressibility per particle, local field factor G ( q ), static structure factor and pair-correlation function. Results are compared with the normal random-phase approximation, local field theories and quantum Monte Carlo calculations.
The effects of short-range correlations on the properties of the electron gas are studied using an approach phrased in the polarization potential language of Pines and Nozi\`eres. It was argued by Lowy and Brown that two-body terms make up the dominant contribution to the effective interaction for large momentum transfers. In the region of small-momentum transfers it is known that the random-phase approximation provides an adequate picture of the electron gas. With these two regions of large- and small-momentum transfers in mind, a polarization propagator is constructed to interpolate between them. A number of sum rules, among them the fff-sum rule, involving the imaginary part of the polarization propagator, were found to be satisfied. The failure of our theory to satisfy the compressibility sum rule is discussed in detail. We present here the results for the local ttt-matrix, the dynamic and static form factors, and the pair correlation function. That our pair-correlation function remains positive and is essentially the same as in the paper of Lowy and Brown indicates that the physics and not the formalism is responsible for the results we obtain.
We present an analytical study of the dynamical local-field factors associated with the response of a homogeneous two-dimensional interacting electron liquid as functions of momentum, frequency, and density. We derive sum rules that constrain their asymptotic forms (in momentum and frequency) for both the spin-symmetric and spin-antisymmetric cases. Parametrized expressions for the local-field factors are proposed, based on all available sum rules and on many-body perturbation theory, and these are found to be in good agreement with quantum Monte Carlo calculations. Finally, these expressions are used to evaluate the effective electron-electron interaction in a local approximation for two-dimensional systems. It is shown that both the quantitative and qualitative behaviors of the interaction are sensitive to the inclusion of dynamical correlations.
The Thomas-Fermi (TF) approximation for the static dielectric function of a three-dimensional (3D) electron liquid can be derived by minimizing the TF local-density approximation for the kinetic-energy functional. Here we show that this connection between energy functionals and model dielectric functions is not an artifact, but a general paradigm. Four examples are worked out in detail, by calculating the model dielectric functions that follow, respectively, from (i) exchange corrections to TF theory in 3D, i.e., TF-Dirac theory, (ii) further correlation corrections to TF-Dirac theory in 3D, (iii) TF theory in two dimensions (2D), and (iv) exchange corrections to TF theory in 2D. Each of these cases has certain interesting features, revealing connections between independent many-body methods, that are discussed in some detail. As a by-product of these investigations we also find that a common textbook statement about the long-wavelength $(k\ensuremath{\rightarrow}0)$ limit of the random-phase approximation is not fully correct.