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A convenient expression is derived for the coefficient, Bₗ₂ (n), which determines the first gradient corrections to the exchange and correlation energy of an inhomogeneous electron gas. The result is exact to all orders in e^2 and is expressed in terms of a single-particle propagator. The lowest-order contributions to Bₗ₂ are evaluated without the simplifying high-density approximation. The formal structure of the derived expression for Bₗ₂ is then used to develop an approximation method for including the contribution of all higher-order graphs within the random-phase approximation. Numerical results are given for Bₗ₂ in the metallic density range. Application to surface energy of metals show an improvement over any previous approximations for Bₗ₂.
Geldart et al. (Sun,) studied this question.
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