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Physical Review B

Perturbative calculation of Jastrow factors: Accurate description of short-range correlations

Hongjun Luo, D. Kolb, Heinz‐Jürgen Flad, Wolfgang Hackbusch

Published Mar 16, 2007• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.75.125111

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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 present a perturbative treatment of Jastrow-type correlation factors which focuses 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, like wavelets, which provide sparse representations for Jatrow factors. Envisaged applications in solid state physics are confined many-particle systems like electrons or multi excitons in quantum dots. The resulting Jastrow factors can be further used as trial wavefunctions in quantum Monte Carlo calculations for these systems. First applications to a screened homogeneous Fermi gas model demonstrate that already first order Jastrow factors 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 inter-particle distances has been observed. 1
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+Ar21+\frac{A}{{r}^{2}}1+r2A​. 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+Ar21+\frac{A}{{r}^{2}}1+r2A​ for large rrr. 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.
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.
With the ability to solve the many-body Schrödinger equation accurately, in principle all physics and chemistry could be derived from first principles. However, exact wave functions of realistic and interesting systems are in general unavailable because they are non-deterministic polynomial-hard to compute (Troyer, 2005). This implies that we need to rely on approximations. The variational Monte Carlo (VMC) method is widely used for ground state studies, but requires a trial wave function ansatz which must trade off between efficiency and accuracy. The method also has many common features with machine learning algorithms, and as neural networks have shown impressive power as function approximators, the idea is to use a neural network as the trial wave function guess. For fermionic systems, like electronic structure systems, the wave function needs to obey Fermi-Dirac statistics. This is typically achieved using a Slater determinant. As a neural network hardly can model this feature, our approach is to replace the single-particle functions in the Slater determinant with restricted Boltzmann machines, getting the RBM ansatz. In addition, we add further correlations via so-called Jastrow factors (Drummond, 2004). The de facto standard trial wave function ansatz for electronic structure calculations is the Slater-Jastrow ansatz, which was implemented as a reference. Our primary focus is on closed-shell circular quantum dots, where we compute the ground state energy and electron density of two-dimensional systems with up to 90 electrons and three-dimensional systems with up to 70 electrons. The energy obtained by the RBM ansatz was reasonably close to experimental results, and it gradually became closer as we added more complex correlation factors. For our most complicated Jastrow factor, the energy was found to be lower than the energy provided by the Slater-Jastrow ansatz for small dots, but for larger dots it was slightly higher. However, the one-body density profile reveals that the RBM ansatz gives more distinctly located electrons compared to the Slater-Jastrow ansatz. This can be explained by the way the RBM ansatz models the correlations. From the two-body density profile, we also observe that the repulsive interactions get more significant as we add a Jastrow factor. Based on the electron densities and the energy distribution between kinetic and potential energy, it is certain that the various methods provide different electron configurations. For low-frequency dots, the electron density becomes more localized with an additional radial peak compared to high-frequency dots. This is reminiscent of what is known as Wigner localization (Ghosal, 2007). The computational time consumption was found to be favorable for the RBM ansatz for small systems and the Slater-Jastrow ansatz for large systems. This can be explained by the exploding number of variational parameters in the RBM ansatz as the system sizes increase. The RBM ansatz with Jastrow factors were notably more computationally intensive than the other ansätze, and evidently, there is no reason to use the simplest Jastrow factor when more complicated Jastrow factors are available.
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.