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

The S‐Matrix Formulation of the Cluster Expansion in Statistical Mechanics

Pietro Fré

Fortschritte der PhysikJan 1, 19779 citations

Implicit purification for temperature-dependent density matrices

Anders M. N. Niklasson

An implicit purification scheme is proposed for calculation of the temperature-dependent, grand canonical single-particle density matrix, given as a Fermi-Dirac operator expansion in terms of the Hamiltonian. The computational complexity is shown to scale with the logarithm of the polynomial order of the expansion, or equivalently, with the logarithm of the inverse temperature. The system of linear equations that arise in each implicit purification iteration is solved efficiently by a conjugate gradient solver. The scheme is particularly useful in connection with linear scaling electronic structure theory based on sparse matrix algebra. The efficiency of the implicit temperature expansion technique is analyzed and compared to some explicit purification methods for the zero temperature density matrix.An implicit purification scheme is proposed for calculation of the temperature-dependent, grand canonical single-particle density matrix, given as a Fermi-Dirac operator expansion in terms of the Hamiltonian. The computational complexity is shown to scale with the logarithm of the polynomial order of the expansion, or equivalently, with the logarithm of the inverse temperature. The system of linear equations that arise in each implicit purification iteration is solved efficiently by a conjugate gradient solver. The scheme is particularly useful in connection with linear scaling electronic structure theory based on sparse matrix algebra. The efficiency of the implicit temperature expansion technique is analyzed and compared to some explicit purification methods for the zero temperature density matrix.

arXiv:cond-mat/0311469Physical review. B, Condensed matterDec 30, 200339 citations

Density matrices of a many-Boson system at low temperatures

I. O. Vakarchuk

Theoretical and Mathematical PhysicsMay 1, 197517 citations

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.