^{1,a)}and Giulia Galli

^{1,2}

### Abstract

We present a study of the binding energy (BE) curves of rare gas and alkaline-earth dimers using an energy functional that includes exact exchange (EXX) and correlationenergies within the random phase approximation (RPA). Our results for the equilibrium positions and long range behavior of the potential energy curves show great improvements over those obtained at the density functional theory level, within local and semilocal approximations. BEs are improved as well in the case of rare gas dimers. For Ar and Kr, the accuracy of our results is comparable to that of so-called van der Waals density functionals, although EXX/RPA yields BE curves that agree better with experiment for large separation distances, as expected. We also discuss shortcomings of the EXX/RPA perturbative approach and analyze possible sources of error in the description of the potential energy curve of alkaline-earth dimers, in particular, , exhibiting an unphysical maximum at large separations. We suggest that the lack of self-consistency in current EXX/RPA approaches might be largely responsible for most of the observed shortcomings. Finally, we present a tight-binding approach to obtain the eigenvalues of the dielectric matrix entering the calculation of the RPA correlationenergy that greatly improves the efficiency of EXX/RPA calculations.

This work was funded by DOE SciDAC under Grant No. DEFC02-06ER25794. We would like to thank C. Umrigar for making available the accurate KS potentials of He, Be, and Ne atoms, and P. Giannozzi for his useful advice on pseudopotential generation. We thank Deyu Lu for many useful discussions.

I. INTRODUCTION

II. COMPUTATIONAL APPROACH

A. Total energy in the EXX/RPA scheme

B. Plane-wave pseudopotential implementation of the EXX/RPA total energy

C. Tight-binding approach to dielectric band structures

III. EXX/RPA STUDIES OF A FEW SELECTED SYSTEMS

A. Argon and krypton dimers

B. Beryllium and calcium dimers

C. Atomic systems

IV. CONCLUSION

### Key Topics

- Dielectrics
- 22.0
- Density functional theory
- 19.0
- Eigenvalues
- 15.0
- Laser Doppler velocimetry
- 15.0
- Wave functions
- 7.0

## Figures

Correlation energy obtained in the RPA as a function of the imaginary frequency calculated from 400 eigenvalues of the dielectric matrix for , at the distance of 7.20 bohr (close to the equilibrium bond length). Circles are obtained by explicit diagonalization; the solid line is from TB calculations (see text). The figure clearly shows that full and TB calculations give almost identical results.

Correlation energy obtained in the RPA as a function of the imaginary frequency calculated from 400 eigenvalues of the dielectric matrix for , at the distance of 7.20 bohr (close to the equilibrium bond length). Circles are obtained by explicit diagonalization; the solid line is from TB calculations (see text). The figure clearly shows that full and TB calculations give almost identical results.

Comparison of BE curves for Ar dimer obtained using different DFT functionals (see text) and experiment. LDA (dashed) and GGA (dashed-dotted) functionals give very poor result as expected. EXX/RPA (red) performs better than vdW-DF (blue, from Ref. 33). Near the equilibrium position, the EXX/RPA curve is not as good as the one obtained from a recently proposed VV09 (purple) functional (Ref. 44), but it does approach the experimental curve (black solid) faster than any other ones in the asymptotic region.

Comparison of BE curves for Ar dimer obtained using different DFT functionals (see text) and experiment. LDA (dashed) and GGA (dashed-dotted) functionals give very poor result as expected. EXX/RPA (red) performs better than vdW-DF (blue, from Ref. 33). Near the equilibrium position, the EXX/RPA curve is not as good as the one obtained from a recently proposed VV09 (purple) functional (Ref. 44), but it does approach the experimental curve (black solid) faster than any other ones in the asymptotic region.

BE curve of the Kr dimer as obtained using the EXX/RPA technique (see text), compared to those obtained from experiment and vdW-DF functional (from Ref. 33). The inset shows the same curves over a wider range of interatomic distances.

BE curve of the Kr dimer as obtained using the EXX/RPA technique (see text), compared to those obtained from experiment and vdW-DF functional (from Ref. 33). The inset shows the same curves over a wider range of interatomic distances.

Dissociation energy curves of the Be dimer calculated using the EXX/RPA technique (see text) with different input wave functions: LDA (red squares), GGA (black circles), HF (blue diamonds), and hybrid functional PBE0 (purple triangles up). The curve denoted experiment is obtained by fitting an Expanded Morse oscillator function to recent experimental data (Ref. 46). Theoretical results from QMC and CI methods are also shown for comparison. Inset: total energies from EXX/RPA with exact-exchange only (top panel), i.e., no correlation, and RPA correlation energies (bottom panel). When a curve in the top panel is combined with a corresponding correlation energy part in the bottom one, the full EXX/RPA binding curve in the figure is recovered.

Dissociation energy curves of the Be dimer calculated using the EXX/RPA technique (see text) with different input wave functions: LDA (red squares), GGA (black circles), HF (blue diamonds), and hybrid functional PBE0 (purple triangles up). The curve denoted experiment is obtained by fitting an Expanded Morse oscillator function to recent experimental data (Ref. 46). Theoretical results from QMC and CI methods are also shown for comparison. Inset: total energies from EXX/RPA with exact-exchange only (top panel), i.e., no correlation, and RPA correlation energies (bottom panel). When a curve in the top panel is combined with a corresponding correlation energy part in the bottom one, the full EXX/RPA binding curve in the figure is recovered.

EXX of Be dimer as a function of Be–Be distances evaluated from the HF expression with different input wave-functions: LDA (red squares), GGA (black circles), and HF (blue diamonds).

EXX of Be dimer as a function of Be–Be distances evaluated from the HF expression with different input wave-functions: LDA (red squares), GGA (black circles), and HF (blue diamonds).

Exact-exchange (left) and RPA correlation (right) energies evaluated from different input one-particle orbitals including the exact KS ones for He, Be, and Ne.

Exact-exchange (left) and RPA correlation (right) energies evaluated from different input one-particle orbitals including the exact KS ones for He, Be, and Ne.

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