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Progress of Theoretical Physics

Cluster Development Method in the Quantum Mechanics of Many Particle System, I

Fumiaki Iwamoto, Masami Yamada

Published Apr 1, 1957• 185 citations
Publisher

Abstract

In order to investigate the many particle system by the variational principle, the cluster development method proposed by Jastrow is systematically formulated. It is a direct generalization of the Hartree method, in which two particle functions are included in the trial function to represent the correlations between particles. The energy expectation value is expanded in a series, each term of which is expressed in cluster integrals. It is found that the series converges rapidly when the correlations are marked only for small particle separations, so that this method is expected to be useful especially for the system of particles interacting through short range potentials with repulsive cores.

DOI
10.1143/ptp.17.543

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In order to investigate the many particle system by the variational principle, the cluster development method proposed by Jastrow is systematically formulated. It is a direct generalization of the Hartree method, in which two particle functions are included in the trial function to represent the correlations between particles. The energy expectation value is expanded in a series, each term of which is expressed in cluster integrals. It is found that the series converges rapidly when the correlations are marked only for small particle separations, so that this method is expected to be useful especially for the system of particles interacting through short range potentials with repulsive cores.
A new variational approach is developed for studying the properties of systems of particles interacting through singular short-range repulsions that give rise to strong two-particle correlations. The correlated trial function ${\ensuremath{\Psi}}_{\ensuremath{\gamma}}={e}^{S}{\ensuremath{\Phi}}_{\ensuremath{\gamma}}$ (state $\ensuremath{\gamma}$) results, with proper choice of SSS, in a simple form for the energy expectation value 〈H〉〈H〉〈H〉---as well as for other matrix elements of interest---which is devoid of all reference to the strong repulsions except through e2S{e}^{2S}e2S factors and hence is particularly suited to calculation. In many cases an independent-particle type ${\ensuremath{\Phi}}_{\ensuremath{\gamma}}$ seems appropriate. The cluster evaluation of this form for 〈H〉〈H〉〈H〉 is discussed, both in the few-particle and many-particle cases. Using the techniques of Iwamoto and Yamada, simplified convergent cluster expansions for the energy expectation value are derived for many-fermion and many-boson systems. A program for application of this method to nuclear problems is being initiated.
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.
The cluster variation method, which was originally devised for quantum-mechanical many-body systems at zero temperature, is extended to treat finite-temperature systems. The new method is based on a variational principle that the partition function is the maximum of a certain functional, APF (approximate partition function). This method includes the conventional cluster variation method at zero temperature as a limiting case. A procedure is given for calculating APF by cluster expansion for fermion systems. In this procedure, use is made of single-particle energies which should be determined self-consistently. The non-orthogonality problem which comes from the presence of correlations among particles is solved by the use of linear combinations of (generalized) Jastrow wave functions. Further development of this method is briefly discussed.
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.
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.
A new cluster expansion technique is developed for the calculation of the energy expectation value of the many-fermion systems with singular potentials.The wave function, which is studied in this paper, is of the formwhich, in a special case, coincides with Jastrow wave function.The formula for the energy expectation value is written in terms of the complex cluster integrals.The method introduced in the derivation of this formula has an analogy to the classical cluster expansion method in the theory 0f an imperfect classical gas of multicomponents.