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arXiv:nucl-th/9902073 · Physical Review Letters

Integral Representation of the Random-Phase Approximation Correlation Energy

F. Dönau, D. Almehed, R. G. Nazmitdinov

Published Jul 12, 1999• 18 citations
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Abstract

Using the spectral function ${\ensuremath{\varphi}}^{\ensuremath{'}}(z)/\ensuremath{\varphi}(z)$, the random-phase approximation correlation energy and other properties of a finite system can be written as a contour integral in a compact way. This yields a transparent expression and reduces drastically the numerical efforts for obtaining reliable values. For illustration the method is applied to pairing vibrations in rotating nuclei.

DOI
10.1103/physrevlett.83.280

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Using the spectral function \ensuremathφ\ensuremath′(z)/\ensuremathφ(z){\ensuremath{\varphi}}^{\ensuremath{'}}(z)/\ensuremath{\varphi}(z)\ensuremathφ\ensuremath′(z)/\ensuremathφ(z), the random-phase approximation correlation energy and other properties of a finite system can be written as a contour integral in a compact way. This yields a transparent expression and reduces drastically the numerical efforts for obtaining reliable values. For illustration the method is applied to pairing vibrations in rotating nuclei.
We develop a scheme to exactly evaluate the correlation energy in the random-phase approximation, based on linear response theory [Y. R. Shimizu, J. D. Garrett, R. A. Broglia, M. Gallardo, and E. Vigezzi, Rev. Mod. Phys. 61, 131 (1989)]. It is demonstrated that our formula is equivalent to a contour integral representation recently proposed [F. Donau, D. Almehed, and R. G. Nazmitdinov, Phys. Rev. Lett. 83, 280 (1999)] being numerically more efficient for realistic calculations. Examples are presented for pairing correlations in rapidly rotating nuclei.
The random phase approximation (RPA) builds in correlations left out by mean-field theory. In full $0\ensuremath{\Elzxh}\ensuremath{\omega}$ shell-model spaces we calculate the Hartree-Fock (HF)+RPA(\mathrm{HF})+\mathrm{RPA}(HF)+RPA binding energy, and compare it to exact diagonalization. We find that often, but not always, the HF+RPA\mathrm{HF}+\mathrm{RPA}HF+RPA gives a good approximation to the ``exact'' ground state energy. In those cases where the RPA is less satisfactory, however, there is no obvious correlation with properties of the HF state, such as deformation or overlap with the exact ground state wave function. This weakens the reliability of the RPA for computing binding energies.
The correlation energy of an inhomogeneous electron gas is ruled by the density correlation function LLL. It obeys a Bethe-Salpeter equation (BSE), whose kernel is the functional derivative of the particle self-energy. It is solved on equal footing for two-particle and single-particle Green's functions within Hedin's GW approximation. According to the Hartree and exchange-correlation contributions to the kernel, the correlation expression decays into random-phase-approximation (RPA) -like and screened-exchange-like parts. The actual treatment of the screened potential WWW in the BSE for LLL leads to different second-order screened exchange corrections (SOSEX). Including correlation in the kernel even yields approximations beyond SOSEX. The progress beyond RPA is illustrated for spinless and spin-polarized homogeneous electron gases, indicating that the screening treatment is substantial.
The random phase approximation (RPA) provides a qualitatively good approximation of electronic correlation in many systems. It has been successfully applied to the spin-unpolarized homogeneous electron gas (HEG) for decades, especially in the studies of its many properties such as correlation energy, momentum distribution and the spin resolved pair correlation function and structure factor. However, the RPA neglects important contributions to the spin-parallel structure factor originating from exchange effects. We derive an approximation for the spin-parallel structure factor including the lowest order exchange process beyond the RPA analytically by means of contour integration and evaluate it numerically by means of a Monte-Carlo integration. A generalized Hellmann-Feynman theorem with functional derivatives is used.