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Comparison between the correlated-basis-functions method and the density-functional theory for inhomogeneous systems

Zhao-Qing Zhang, Ruibao Tao, Chia-Wei Woo

As a hypothetical inhomogeneous quantum many-body model, we take a weakly interacting Bose gas made inhomogeneous by the presence of a weak external field. The interaction is assumed to be Fourier transformable and characterized by a strength parameter \ensuremath{\lambda}; this is also true for the external field, characterized by a strength parameter \ensuremath{\varepsilon}. The ground-state energy is calculated to order ${\ensuremath{\varepsilon}}^{2}$${\ensuremath{\lambda}}^{2}$ using three methods: diagrammatic perturbation theory, correlated basis functions (CBF), and density-functional theory (DFT). The diagrammatic perturbation theoretical calculation is new. It differs from our earlier approach for homogeneous Bose systems which evolved from the Hugenholtz-Pines theory. We are able to reproduce the same results in an efficient manner. For the inhomogeneous system, the results are used to measure CBF against DFT. It is found that in their simplest forms CBF yields results which are better than the renormalized DFT by Ebner and Saam. In particular, CBF is superior to DFT in the local-density approximation, even though the latter was designed to take care of systems with slow and small density variations such as the model considered here.As a hypothetical inhomogeneous quantum many-body model, we take a weakly interacting Bose gas made inhomogeneous by the presence of a weak external field. The interaction is assumed to be Fourier transformable and characterized by a strength parameter \ensuremath{\lambda}; this is also true for the external field, characterized by a strength parameter \ensuremath{\varepsilon}. The ground-state energy is calculated to order ${\ensuremath{\varepsilon}}^{2}$${\ensuremath{\lambda}}^{2}$ using three methods: diagrammatic perturbation theory, correlated basis functions (CBF), and density-functional theory (DFT). The diagrammatic perturbation theoretical calculation is new. It differs from our earlier approach for homogeneous Bose systems which evolved from the Hugenholtz-Pines theory. We are able to reproduce the same results in an efficient manner. For the inhomogeneous system, the results are used to measure CBF against DFT. It is found that in their simplest forms CBF yields results which are better than the renormalized DFT by Ebner and Saam. In particular, CBF is superior to DFT in the local-density approximation, even though the latter was designed to take care of systems with slow and small density variations such as the model considered here.

Physical review. B, Condensed matterFeb 1, 19883 citations

Density Functional Theory for the Correlated

Robert O. Jones

Jan 1, 20200 citations

Foundations of Density Functional Theory

Eugene S. Kryachko, Eduardo V. Ludeña

Springer eBooksJan 1, 19902 citations

Diagrammatic Analysis of the Method of Correlated Basis Functions. III. Excitation Spectrum of a Weakly Interacting Bose Gas

Archana Bhattacharyya, Chia-Wei Woo

In the continuing effort to establish contacts between the method of correlated basis functions (CBF) and the diagrammatic perturbation theory, we carry out parallel calculations of the excitation spectrum of a weakly interacting Bose gas. The Hugenholtz---Pines (HP) theory is first used to obtain the exact liquid-structure function and the energy spectrum, each to three orders in the strength parameter $\ensuremath{\lambda}$ that characterizes the interaction between a pair of particles in the system. Next we construct sets of CBF using three different correlating factors, and in each case calculate the excitation energy of the system. The results are compared to those obtained from the HP theory. Complete agreement is found to all orders considered. In conjunction with a paper to follow, the analysis demonstrates the flexibility of the CBF method and the relative ease of application in comparison to other methods.In the continuing effort to establish contacts between the method of correlated basis functions (CBF) and the diagrammatic perturbation theory, we carry out parallel calculations of the excitation spectrum of a weakly interacting Bose gas. The Hugenholtz---Pines (HP) theory is first used to obtain the exact liquid-structure function and the energy spectrum, each to three orders in the strength parameter $\ensuremath{\lambda}$ that characterizes the interaction between a pair of particles in the system. Next we construct sets of CBF using three different correlating factors, and in each case calculate the excitation energy of the system. The results are compared to those obtained from the HP theory. Complete agreement is found to all orders considered. In conjunction with a paper to follow, the analysis demonstrates the flexibility of the CBF method and the relative ease of application in comparison to other methods.

Physical Review AJan 1, 197310 citations

Density Matrices and Density Functionals

Robert Erdahl, Vadene H. Smith

Springer eBooksJan 1, 1987422 citations

Diagrammatic Analysis of the Method of Correlated Basis Functions

Hock-Kee Sim, Chia-Wei Woo, J. R. Buchler

The weakly interacting Bose gas is studied on the one hand with the independent-particle perturbation formalism of Hugenholtz and Pines, and on the other hand by a perturbation theory using a set of correlated basis functions. We find by comparison that up to fourth order in energy, a Jastrow correlated wave function sums all rings and ladders, and other diagrams in part; and the simplest correlated basis functions perturbation correction accounts for all remaining discrepancies.

Physical Review LettersMay 18, 197013 citations

A Comparison Between Correlated Basis Functions Method and the Density Functional Theory

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