The ground-state wave function of an infinite system of fermions is approximated by the state-independent Jastrow ansatz. In order to optimize the pair correlation function, the Euler-Lagrange equations of the variational problem for the energy expectation value are derived. The short- and long-range behaviors of the optimum pair correlation function are discussed. Application of graphical techniques and use of rigorous results on the connection between the slope of the static structure function as $k\ensuremath{\rightarrow}0$+ and long-range Jastrow correlations allow one to prove that the optimum pair correlation function behaves like $1+O({r}^{\ensuremath{-}2})$ as $r\ensuremath{\rightarrow}\ensuremath{\infty}$. A connection is derived between the weight of the long-range correlations and the Landau parameters. As simple examples the limit of Bose statistics and the electron-gas problem are considered. The consequences of these investigations on numerical calculations and their relation to alternative expansion methods are investigated.
We study many-body correlations in the ground state of a general quantum system of bosons or fermions by including an additional Jastrow function in our recently proposed variational coupled-cluster method. Our approach combines the advantages of state-dependent correlations in the coupled-cluster theory and of the strong, short-ranged correlations of the Jastrow function. We apply a generalized linked-cluster expansion for the Jastrow wave function and provide a detailed analysis for practical evaluation of the Hamiltonian expectation value as an energy functional of the Jastrow function and the bare density-distribution functions introduced and calculated in our earlier publications; a simple, first-order energy functional is derived and detailed formulas for the higher-order contributions are provided. Our energy functional does not suffer the divergence as most coupled-cluster calculations often do when applying to Hamiltonians with hardcore potentials. We also discuss possible applications of our technique, including applications to strongly correlated fermion systems.
We present a perturbative treatment of Jastrow-type correlation factors which focus on an accurate description of short-range correlations. Our approach is closely related to coupled cluster perturbation theory with the essential difference that we start from a variational formulation for the energy. Such kind of perturbation theory is especially suited for multiscale bases, such as wavelets, which provide sparse representations for Jastrow factors. Envisaged applications in solid-state physics are confined many-particle systems such as electrons or multiexcitons in quantum dots. The resulting Jastrow factors can be further used as trial wave functions in quantum Monte Carlo calculations for these systems. First applications to a screened homogeneous Fermi gas model demonstrate that first-order Jastrow factors already recover 95% of the correlation energy in variational Monte Carlo calculations over a fairly large range of densities and screening parameters. The corresponding second- and third-order perturbation energies turned out to be more sensitive to the specific choice of the model parameters. Furthermore, we have compared our first-order Jastrow factors with those obtained from Fermi hypernetted chain calculations, where excellent agreement at short and intermediate interparticle distances has been observed.
We study the accuracy of analytical wave-function-based many-body methods derived by energy minimization of a Jastrow-Feenberg ansatz for electrons (the ``Fermi hypernetted chain/Euler Lagrange'' approach). Approximations to avoid the complexity of the fermion problem are chosen to parallel successful boson theories and to be computationally efficient. For the three-dimensional homogeneous electron gas, we calculate the correlation energy, the pair distribution function, and the static structure function in comparison with simulation results. We also present a variant of theory which is interpreted as an approximate, self-consistent sum of ladder and ring diagrams of perturbation theory. The theory performs particularly well in the highly dilute density regime.
The ground state of a many-boson system is studied within the range of the Bijl-Dingle-Jastrow-type description when the radial distribution function
g(r) differs little from its asymptotic value. The treatment of the problem is based on the development of power series in $\ensuremath{\alpha}=1\ensuremath{-}g(0)$ for all physical quantities which depend on the particle density. The
n-particle distribution functions
p(n) are evaluated to order ${\ensuremath{\alpha}}^{4}$ as functionals in the
g(r) function for
n=3and4 using the cluster-expansion procedure outlined by Abe. These results are used in connection with the improvement of the ground-state description when the wave function is not the optimum choice. Using
p(3) function obtained, the Bogoliubov-Born-Green-Kirkwood-Yvon equation is solved, also to order ${\ensuremath{\alpha}}^{4}$, for the two-particle correlation function
U(r), and the first two leading corrections to the hypernetted-chain (HNC) approximation are obtained. The variational calculation along with the series expansion for
U(r) yields formulas for the ground-state properties, including some corrections to known results. For a charged boson gas, numerical values of
U(r),
p(3),
p(4), and the ground-state energy are computed using the Gaussian approximation for
g(r), and the results show that the errors associated with the HNC approximation are small. A brief discussion is presented on the method of determining the general expansion coefficients of the correlation functions of
p(n) in terms of
g(r).
The method of correlated basis functions provides a means of systematically exploiting our intuitive understanding of real physical systems. Here we seek especially to develop a formalism for describing systems of strongly interacting fermions, for example, nuclei and liquid
He3, using a complete set of correlated functions ${\ensuremath{\Psi}}_{\mathrm{m}}=\frac{F{\ensuremath{\Phi}}_{\mathrm{m}}}{{({\ensuremath{\Phi}}_{\mathrm{m}}, {F}^{2}{\ensuremath{\Phi}}_{\mathrm{m}})}^{\frac{1}{2}}}$.
F is taken as a product of two-body factors, $\ensuremath{\Pi}{i<j}^{}[f({r}_{\mathrm{ij}})]$, the
f(rij) embodying at least the correlations arising from strong short-range repulsions; and ${{\ensuremath{\Phi}}_{\mathrm{m}}}$, a set of independent-particle-model functions which might serve as a basis in the absence of highly singular interactions. The matrix elements
Hmn,
Nmn of the Hamiltonian and identity operators with respect to the correlated basis ${{\ensuremath{\Psi}}_{\mathrm{m}}}$ are evaluated by generalization of cluster-expansion techniques developed originally for the calculation of the partition function of a classical imperfect gas and adapted to the quantum-mechanical many-body problem by Iwamoto and Yamada. Contributions to our expansions for the
Hmn are classified according to order of magnitude in the quantity $\ensuremath{\rho}\ensuremath{\omega}$, where $\ensuremath{\omega}=\ensuremath{\int}({f}^{2}(r)\ensuremath{-}1)d\mathrm{r}$ is a "correlation parameter" and $\ensuremath{\rho}$ the average particle density. To the extent that $|\ensuremath{\rho}\ensuremath{\omega}|$ is small, it provides a true expansion parameter. A detailed technical discussion of the cluster expansions is given, with special emphasis on their symmetry properties upon truncation. Although the many-body problem is soluble by simultaneous diagonalization of the matrices (
Hmn) and (
Nmn), it is preferable first to transform to an orthonormal basis of "unperturbed" correlated functions ${{\ensuremath{\Theta}}_{\mathrm{m}}}$ with respect to which the Hamiltonian and identity operators have the matrix representations (
Hmn) and $({N}_{\mathrm{mn}})=({\ensuremath{\delta}}_{\mathrm{mn}})$, and then to attempt the separate diagonalization of (
Hmn). In general, such a transformation is expedited by the L\"owdin procedure carried out to the desired order in $\ensuremath{\rho}\ensuremath{\omega}$. However, a Schmidt orthogonalization scheme might prove more advantageous in specific cases. Field-theoretic techniques (of both perturbative and nonperturbative nature) may be brought to bear in the diagonalization of (
Hmn). The inverse of the unitary transformation which takes the basis ${{\ensuremath{\Phi}}_{\mathrm{m}}}$ into the basis ${{\ensuremath{\Theta}}_{\mathrm{m}}}$, applied to the given singular Hamiltonian, leads---for any
F in the class considered---to a non-singular "effective Hamiltonian." Following this formal cue, we set out to construct explicitly a second-quantized effective Hamiltonian which generates the same matrix elements, to prescribed order in $\ensuremath{\rho}\ensuremath{\omega}$, when operating between independent-particle kets of the model occupation-number representation, as does the given Hamiltonian in the configuration-space representation, between the corresponding correlated functions of the L\"owdin basis. The quasiparticles described by this effective Hamiltonian interact via well-behaved two-body, three-body, \ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{},
Q-body, \ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{} potentials, in contrast to the highly singular two-body interactions of the real particles. For small $|\ensuremath{\rho}\ensuremath{\omega}|$, the effective two-body forces predominate. An avenue to deeper understanding of the applicability of the nuclear-shell model and the dressed-quasiparticle theory of liquid
He3 has been opened. This article treats only the formalism for a uniform extended medium. Numerical calculations, as well as the extension of the method to finite systems, will follow.