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.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.

Physical Review AMay 1, 1973264 citations

Electron Correlations in Strongly Interacting Systems

J. J. Quinn, George Simion

WORLD SCIENTIFIC eBooksMay 1, 20101 citation

Coulomb Correlations in Inhomogeneous Many-Particle Systems

Setsuo Ichimaru

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 ${S}_{0}$ . 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 $K$ is the response function of the electron density to an external potential field, and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K}$ its conjugate; $v$ 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.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 ${S}_{0}$. 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 $K$ is the response function of the electron density to an external potential field, and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K}$ its conjugate; $v$ 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.

Physical ReviewOct 11, 196514 citations

Spatial correlations of a two-dimensional electron gas

A. Isihara, Tadashi Toyoda

The European Physical Journal BDec 1, 197611 citations

Long range correlations in the electron gas

João da Providência

Jan 1, 19750 citations

Linked-cluster expansion for Jastrow-type wave functions and its application to the electron-gas problem

James D. Talman

A linked-cluster expansion for the expectation values of operators between correlated wave functions of the Jastrow type is described. The expansion is based on the logarithm of the correlation function and leads to terms that are explicitly proportional to the particle number for infinite systems. The lowest-order terms for a weakly interacting system of spin-1/2 particles are calculated explicitly. From these, the Euler-Lagrange equations for the correlation functions have been calculated. The equation for the correlation function between particles of antiparallel spin can be solved by Fourier transform, and it is shown that for $r\ensuremath{\rightarrow}\ensuremath{\infty}$, the correlation function behaves like $1+\frac{A}{{r}^{2}}$ . In the case of a system of particles interacting by hard-core repulsions, the logarithm of the correlation function does not exist inside the hard-core radius. It is then necessary to sum over ladder diagrams as in the derivation of the reaction matrix in many-body perturbation theory. This is carried out to provide a low-density approximation to the ground-state energy of the system and the Euler-Lagrange equation for the correlation function. It is argued that the correlation function again behaves like $1+\frac{A}{{r}^{2}}$ for large $r$ . The expansion is applied to the problem of calculating the correlation energy of the electron gas. It is shown that a variational calculation based on the lowest-order terms in an expansion of the wave function leads to a divergent result (unbounded from below), but that a convergent result is obtained by summing over ring diagrams, which correspond to long-range polarization of the electron cloud. The result is similar to that of Gaskell obtained by a somewhat different method. It is shown analytically that at high densities the result behaves logarithmically, specifically as $0.0570\mathrm{ln}{r}_{s}\ensuremath{-}0.1324$ Ry. The lowest-order exchange contributions have also been calculated and found to contribute 0.040 Ry to the energy at densities of physical interest.A linked-cluster expansion for the expectation values of operators between correlated wave functions of the Jastrow type is described. The expansion is based on the logarithm of the correlation function and leads to terms that are explicitly proportional to the particle number for infinite systems. The lowest-order terms for a weakly interacting system of spin-1/2 particles are calculated explicitly. From these, the Euler-Lagrange equations for the correlation functions have been calculated. The equation for the correlation function between particles of antiparallel spin can be solved by Fourier transform, and it is shown that for $r\ensuremath{\rightarrow}\ensuremath{\infty}$, the correlation function behaves like $1+\frac{A}{{r}^{2}}$. In the case of a system of particles interacting by hard-core repulsions, the logarithm of the correlation function does not exist inside the hard-core radius. It is then necessary to sum over ladder diagrams as in the derivation of the reaction matrix in many-body perturbation theory. This is carried out to provide a low-density approximation to the ground-state energy of the system and the Euler-Lagrange equation for the correlation function. It is argued that the correlation function again behaves like $1+\frac{A}{{r}^{2}}$ for large $r$. The expansion is applied to the problem of calculating the correlation energy of the electron gas. It is shown that a variational calculation based on the lowest-order terms in an expansion of the wave function leads to a divergent result (unbounded from below), but that a convergent result is obtained by summing over ring diagrams, which correspond to long-range polarization of the electron cloud. The result is similar to that of Gaskell obtained by a somewhat different method. It is shown analytically that at high densities the result behaves logarithmically, specifically as $0.0570\mathrm{ln}{r}_{s}\ensuremath{-}0.1324$ Ry. The lowest-order exchange contributions have also been calculated and found to contribute 0.040 Ry to the energy at densities of physical interest.

Physical Review AOct 1, 197424 citations

{S}_{0}

. 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

K

is the response function of the electron density to an external potential field, and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K}$ its conjugate;

v

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.

1+\frac{A}{{r}^{2}}

. In the case of a system of particles interacting by hard-core repulsions, the logarithm of the correlation function does not exist inside the hard-core radius. It is then necessary to sum over ladder diagrams as in the derivation of the reaction matrix in many-body perturbation theory. This is carried out to provide a low-density approximation to the ground-state energy of the system and the Euler-Lagrange equation for the correlation function. It is argued that the correlation function again behaves like

1+\frac{A}{{r}^{2}}

for large

r

. The expansion is applied to the problem of calculating the correlation energy of the electron gas. It is shown that a variational calculation based on the lowest-order terms in an expansion of the wave function leads to a divergent result (unbounded from below), but that a convergent result is obtained by summing over ring diagrams, which correspond to long-range polarization of the electron cloud. The result is similar to that of Gaskell obtained by a somewhat different method. It is shown analytically that at high densities the result behaves logarithmically, specifically as $0.0570\mathrm{ln}{r}_{s}\ensuremath{-}0.1324$ Ry. The lowest-order exchange contributions have also been calculated and found to contribute 0.040 Ry to the energy at densities of physical interest.