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Physical Review A

Short-Range Correlations and Electron-Gas Response Functions

John C. Kimball

Published May 1, 1973• 264 citations
Publisher

Abstract

The behavior of interacting electrons is dominated by the Coulomb repulsion in the limit of zero interparticle separation. Hence, in this limit the derivative of the two-particle correlation function is equal to the two-particle correlation function itself divided by the Bohr radius. Because correlation functions can be obtained from linear-response functions, the relation between the correlation function and its derivative can be used as a criterion for the validity of approximate linear-response functions. Some of the more familiar dielectric functions and spin susceptibilities are discussed in this context. For these response functions it is shown that the electric and magnetic response should approach the same ($q\ensuremath{\rightarrow}\ensuremath{\infty}$) asymptotic limit.

DOI
10.1103/physreva.7.1648

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The behavior of interacting electrons is dominated by the Coulomb repulsion in the limit of zero interparticle separation. Hence, in this limit the derivative of the two-particle correlation function is equal to the two-particle correlation function itself divided by the Bohr radius. Because correlation functions can be obtained from linear-response functions, the relation between the correlation function and its derivative can be used as a criterion for the validity of approximate linear-response functions. Some of the more familiar dielectric functions and spin susceptibilities are discussed in this context. For these response functions it is shown that the electric and magnetic response should approach the same (q\ensuremath→\ensuremath∞q\ensuremath{\rightarrow}\ensuremath{\infty}q\ensuremath→\ensuremath∞) asymptotic limit.
The fluctuation-dissipation theorem is applied to the study of the pair correlation functions of the system containing charged particles which is not invariant under spatial translation. The results are expressed in terms of the longitudinal response functions and involve correlation functions which depend only on screened short-range Coulomb interactions S0{S}_{0}S0​. In the case of the electron gas, the density-density correlation function is $S(\mathbf{r}{\mathbf{r}}^{\ensuremath{'}};\ensuremath{\omega})=[\ensuremath{\delta}(\mathbf{r}\ensuremath{-}\mathbf{R})\ensuremath{-}K(r{\mathbf{R}}^{\ensuremath{'}\ensuremath{'}};\ensuremath{\omega})v({\mathbf{R}}^{\ensuremath{'}\ensuremath{'}}\mathbf{R})]{S}_{0}(\mathbf{R}{\mathbf{R}}^{\ensuremath{'}};\ensuremath{\omega})[\ensuremath{\delta}({\mathbf{R}}^{\ensuremath{'}}\ensuremath{-}{\mathbf{r}}^{\ensuremath{'}})\ensuremath{-}v({\mathbf{R}}^{\ensuremath{'}}{\mathbf{R}}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}})\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K}({\mathbf{R}}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}}{\mathbf{r}}^{\ensuremath{'}};\ensuremath{-}\ensuremath{\omega})]$where KKK is the response function of the electron density to an external potential field, and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K}$ its conjugate; vvv represents the Coulomb interaction; and the repeated spatial coordinates are integrated over the volume. It is shown that this expression implies the principle of superposition of dressed particles (the bare particles plus their associated screening clouds); for the calculations of the correlation functions, the dressed particles may be superposed without further consideration of the long-range parts of Coulomb interactions. Similar reductions are carried out for the various correlation functions in the electron-ion system, and the superposition principle is confirned. A possible extension of the results to a nonequilibrium stationarysystem is suggested and discussed. The interrelationship among various longitudinal response functions, which is characteristic of the inhomogeneous system, is discussed and clarified. Simplifications arising from the homogeneity and discussions from diagrammatic considerations are given in Appendices.
A discussion of the various properties of pair correlation functions of a magnetic electron gas is presented. The cusp conditions obeyed by the many-electron wave functions for a pair of electrons with parallel and antiparallel spins in close proximity are expressed as derivative conditions on the corresponding parallel-and antiparallel-spin pair correlation functions. In particular, it is shown that in a fully ferromagnetic gas the parallel spin correlations determine the correlation energy whereas in the paramagnetic case, this function is of less importance in comparison to the antiparallel-spin correlation function. In the high-momentum limit, it is found that the momentum distribution of electrons is dominated by antiparallel-spin correlations in the partially magnetic case whereas in the fully ferromagnetic case, it is dominated by parallel-spin correlations but is smaller by a factor of ${q}^{\ensuremath{-}2}$. A calculation of the parallel-spin correlation function for the ferromagnetic gas in perturbation theory is presented. A model for the exchange correlation energy of the system which takes into account the exact conditions given here is proposed. Corresponding results for the two-dimensional electron gas are also stated.
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A microscopic formula for the effective two-body interaction between electrons or holes in an ideal metal is derived within the linear response theory. Both the zero-range Hubbard interaction U0 and the long-range Coulomb interaction, denoted by ΔV(q), are included. The effective interaction, which is necessarily spin-dependent, is expressed in terms of the exact charge and spin density-density correlation functions and of their higher order mixtures. These correlation functions are analyzed diagrammatically. Attention is paid to clarifying the different roles played by U0 and ΔV(q). We also display the corresponding new formula for the free energy. Comparison is made with previous theories.