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The Selfconsistent Quasiparticle RPA and Its Description of Thermal Pairing Properties in Nuclei

Nguyen Dinh Dang

1 - Heavy-Ion Nuclear Physics Laboratory, Nishina Center for Accelerator-Based Science RIKEN 2-1 Hirosawa, Wako city, 351-0198 Saitama, Japan

2 - Institute for Nuclear Science and Techniques, Vietnam Atomic Energy Commission Hanoi - Vietnam

The Selfconsistent Quasiparticle RPA (SCQRPA) is constructed [1] to study the effects of fluctuations on pairing properties in finite systems. The theory is applied to nuclei at finite temperature [2] and angular momentum [3]. Particle number projection is taken into account within the Lipkin-Nogami method. Several issues such as the smoothing of superfluid-normal phase transition, thermally assisted pairing in hot rotating nuclei, extraction of the nuclear pairing gap using an improved odd-even mass difference are discussed [4]. Finally, a novel approach of embedding the projected SCQRPA eigenvalues in the canonical ensemble (CE) is proposed (the CE-SCQRPA) [5]. Applied to a doubly-folded equidistant multilevel pairing model, the proposed CE-SCQRPA produces results in good agreement with those obtained by using the exact eigenvalues, whenever the latter are possible, and is workable also for large values of particle number (N>14), where the diagonalisation of the pairing Hamiltonian is impracticable.

References

[1] N. Quang Hung and N. Dinh Dang, 
Phys. Rev. C 76 (2007) 054302 and 77 (2008) 029905(E).

[2] N. Dinh Dang and N. Quang Hung, 
Phys. Rev. C 77 (2008) 064315.

[3] N. Quang Hung and N. Dinh Dang, 
Phys. Rev. C 78 (2008) 064315.

[4] N. Quang Hung and N. Dinh Dang, 
Phys. Rev. C 79 (2009) 054328.

[5] N. Quang Hung and N. Dinh Dang, in preparation

The two faces of RPA: density functional theory and many-body perturbation theory

Xavier Gonze

Unité Physico-Chimie et de Physique des Matériaux (PCPM), Université catholique de Louvain, Place Croix du Sud 1, B-1348 Louvain-la-Neuve, Belgique

The RPA expression for total energy might be derived either within density functional theory, (in the adiabatic-connection fluctuation-dissipation framework, by setting the exchange-correlation kernel to zero), or from many-body perturbation theory (from the Nozières functional in the GW approximation for the self-energy, when the green's function corresponds to an energy-independent one-body Hamiltonian). I will review results in which these two faces of RPA appear. First, the RPA exchange-correlation potential is obtained from a linear-response Sham-Schlüter equation [1,2]. Second, the RPA band gap (I-A expression) corresponds to non-renormalized G_0 W_0 [3]. I will also show that density functional theory within the RPA provides a correct description of bond dissociation for the hydrogen dimer in a spin-restricted Kohn-Sham formalism, i.e., without artificial symmetry breaking, with important static (left-right) correlation [4]. Although exact at infinite separation and accurate near the equilibrium bond length [5], the RPA dissociation curve displays unphysical repulsion at larger but finite bond lengths [4,6].

References

[1] Y.-M. Niquet, M. Fuchs, X. Gonze, J. Chem. Phys. 118, 9504 (2003)

[2] Y.-M. Niquet, M. Fuchs, X. Gonze, Phys. Rev. A 68, 032507 (2003)

[3] Y.-M. Niquet, X. Gonze, Phys. Rev. B 70, 245115 (2004)

[4] M. Fuchs, Y.-M. Niquet, X. Gonze, K. Burke, J. Chem. Phys. 122, 094116 (2005)

[5] M. Fuchs, X. Gonze, Phys. Rev. B 65, 235109 (2002)

[6] M. Fuchs, K. Burke, Y.-M. Niquet, X. Gonze, Phys. Rev. Lett. 90, 189701 (2003)

How approximate is the random phase approximation? Comparing RPA against full configuration-interaction calculations

Calvin W. Johnson

Department of Physics, San Diego State University, USA

I compare the random phase approximation against numerically exact configuration-interaction calculations in the nuclear shell model and find good, not perfect, results. This is of particular importance when one is using a phenomenological interaction. I also discuss the so-called "collapse" of RPA in "phase changes" and illustrate the difference between first- and second-order phase changes and how collapse occurs in only one.

References

[1] I. Stetcu and C. W. Johnson, Phys. Rev. C 66, 034301 (2002); Phys. Rev. C 67, 043315 (2003); Phys. Rev. C 69, 024311 (2004). [2]C. W. Johnson and I. Stetcu, Phys. Rev. C 80, 024320 (2009).

RPA and coupled cluster theory & some recent results including range separation

Gustavo E. Scuseria

Department of Chemistry and Department of Physics & Astronomy, Rice University, Houston, Texas, USA

The recent realization that the ground-state correlation energy of the random phase approximation (RPA) is intimately connected to an approximate coupled cluster doubles (CCD) model [1], opens interesting avenues for mixing RPA with density functional theory (DFT) [2]. I will describe some of the recent work done in our research group on RPA, including applications to van der waals and noncovalent interactions [3], the importance of the reference state [4], a simple second-order approximation [5], and a second-order exchange (SOSX) correction that restores antisymmetry [6].

References

[1] The ground state correlation energy of the Random Phase Approximation from a ring Coupled Cluster Doubles approach, G. E. Scuseria, T. M. Henderson, and D. C. Sorensen, J. Chem. Phys. 129, 231101 (2008).

[2] Long-range-corrected hybrids including random phase approximation correlation, B. G. Janesko, T. M. Henderson, and G. E. Scuseria, J. Chem. Phys. 130, 081105 (2009).

[3] Long-range corrected hybrid functionals including random phase approximation correlation: Application to noncovalent interactions, B. G. Janesko, T. M. Henderson, and G. E. Scuseria, J. Chem. Phys. 131, 034110 (2009).

[4] The role of the reference state in long-range random phase approximation correlation, B. G. Janesko and G. E. Scuseria, J. Chem. Phys. 131, 154106 (2009).

[5] Coulomb-only second-order perturbation theory in long-range-corrected hybrid density functionals, B. G. Janesko and G. E. Scuseria, Phys. Chem. Chem. Phys. 11, 9677 (2009).

[6] Hybrid functionals including random phase approximation correlation and second-order screened exchange, J. Paier, B. G. Janesko, T. M. Henderson, G. E. Scuseria, A. Gruneis, and G. Kresse, J. Chem. Phys. submitted.

Promising first results with an RPA correlation functional based on the frequency-dependent Kohn-Sham exchange kernel

Andreas Görling and Andreas Hesselmann

Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Erlangen, Germany

The random phase approximation (RPA) correlation energy is expressed in terms of the exact local Kohn-Sham (KS) exchange potential and corresponding adiabatic and nonadiabatic exchange kernels for density-functional reference determinants. The approach naturally extends the RPA method in which, conventionally, only Coulomb interactions are included. By comparison with the coupled cluster singles doubles with perturbative triples method it is shown for a set of small molecules that the new RPA method based on the KS exchange kernel yields correlation energies more accurate than RPA on the basis of Hartree-Fock exchange.

References

[1] A. Hesselmann and A. Görling, Mol. Phys., in press.

Large-scale Second RPA calculations

Panagiota Papakonstantinou

Institute of Nuclear Physics, T.U. Darmstadt, Darmstadt, Germany

Extended RPA theories, which go beyond first-order RPA, are often used to describe the strength, decay width and fine structure of collective excitations in nuclei and other many-body systems. Second RPA (SRPA) is a simple and straightforward extension of RPA to second order, based on the Hartree-Fock ground state. In the present work, large-scale (i.e., without arbitrary truncations of the 1p1h+2p2h model space), “self-consistent” (i.e., with a single two-body interaction as the sole input) SRPA calculations are performed and analyzed. Divergence problems are avoided thanks to the use of finite-range interactions. It is presented how the large-scale eigenvalue problems that SRPA entails can be treated, and how the method operates in producing self-energy corrections and fragmentation. Besides the usual inconsistency problems (broken symmetries), stability problems are encountered, which are traced back to missing ground-state correlations. Nevertheless, nuclear giant resonances appear rather stable with respect to correlations. The results are discussed in the context of extended RPA theories, as well as of suitable effective interactions.

How accurate is RI-RPA? Quality of resolution-of-the-identity methods for RPA correlation energies

Henk Eshuis, Julian Yarkony and Filipp Furche

Department of Chemistry, University of California, 1102 Natural Sciences II, Irvine, CA 92697-2025, USA

Recently the random phase approximation (RPA) has seen renewed interest as a way of computing molecular correlation energies in a Kohn-Sham context. The RPA has the attractive feature of describing long-range interactions correctly [1], thus addressing a long-standing problem in density functional theory. Additionally it has been shown recently that the RPA can be cast in a computationally attractive form, making it a promising method for larger molecules (30-100 atoms) [2]. The off-diagonal parts of the orbital rotation Hessian matrices are rank-deficient and can be therefore be represented accurately in a relatively small auxiliary basis set using resolution-of-the-identity (RI) methods, in the same vein as in RI-MP2 [3, 4]. In this work we study the accuracy of RI-RPA correlation energies for several testcases. I will show that the error due to RI is small and comparable to the error of RI-MP2. I will discuss why RI-RPA can make RPA calculations more efficient.

References

[1] Dobson, J. In Time-Dependent Density Functional Theory, Vol. 706; Springer: Berlin Heidelberg, 2006; page 443.

[2] Furche, F. J. Chem. Phys. 2008, 129, 114105.

[3] Feyereisen, M.; Fitzgerald, G.; Komornicki, A. Chem. Phys. Lett. 1993, 208, 359 – 363.

[4] Weigend, F.; Haeser, M. Theor. Chim. Acta 1997, 97, 331–340.

Linear response strength functions with iterative Arnoldi diagonalization

Jacek Dobaczewski

Department of Physics, University of Jyväskylä, P.O. box 35, FIN-40014, Finland; Institute of Theoretical Physics, Warsaw University, ul. Hoża 69, PL-00681, Warsaw, Poland

We report on an implementation of a new method to calculate RPA strength functions with iterative non-hermitian Arnoldi diagonalization method, which does not explicitly calculate and store the RPA matrix. We discuss the treatment of spurious modes, numerical stability, and how the method scales as the used model space is enlarged. We perform the particle-hole RPA benchmark calculations for double magic nucleus 132Sn and compare the resulting electromagnetic strength functions against those obtained within the standard RPA

References

[1] J. Toivanen, B.G. Carlsson, J. Dobaczewski, K. Mizuyama, R.R. Rodriguez-Guzman, P. Toivanen, P. Vesely; arXiv:0912.3234

Collective Excitations within the Second RPA

Danilo Gambacurta,

Dipartimento di Fisica e Astronomia dell'Università di Catania, INFN Sez. Catania

Second Random Phase Approximation [1] (SRPA) is a natural extension of RPA obtained by introducing more general excitation operators where two particle-two hole configurations, in addition to the one particle-one hole ones, are considered. Some applications of SRPA with the phenomenologic Skyrme interaction in nuclei will be shown. Particular attention will be devoted to the issue of the residual interaction to be used in the matrix elements beyond the standard RPA ones. Both in RPA and SRPA use is made of the Quasi Boson Approximation [2] that amounts to use the Hartree-Fock state as reference state. Some problematic aspects due to this approximation will be finally analyzed and an extended SRPA approach, in which ground state correlations are taken into account, will be presented and applied in the context of metallic clusters[3]. Ground state correlations are taken into account either in a perturbative way or by means of a more consistent procedure that allows to obtain better results, especially in the case of the Dipole Plasmon.

References

[1] C. Yannouleas, Phys. Rev. C35 1159 (1987).

[2] P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer-Verlag, Berlin, 1980).

[3] D. Gambacurta and F. Catara, Phys. Rev. B 79, 085403 (2009) and submitted to Phys. Rev. B

An overview of RPA from the Nuclear Physics perspective

Peter Schuck

Institut de Physique Nucleaire, Orsay, France

In the introductory talk from the nuclear side, I will shortly review how RPA is applied in its standard way to nuclear structure using density dependent effective forces. I also will repeat some properties of RPA which makes it an appreciated approximation. I will continue to outline extensions of RPA in use in nuclear physics, like self consistent inclusion of ground state correlations, renormalised RPA, symmetry aspects, etc. Useful literature is listed below.

References

[1] J. Dukelsky, P. Schuck, Towards a variational theory for RPA like correlations and fluctuations, Nucl. Phys. A512 (1990) 466

[2] See also Nucl. Phys. A628 (1998) 17

Extended RPA from time dependent density matrix theory

Peter Schuck

Institut de Physique Nucleaire, Orsay, France

In this short second talk I will replace my collaborator Mitsuru Tohyama and talk about an extension of RPA from the formalism of the Time Dependent Density Matrix (TDDM) approach including higher correlation functions. In this context I also will touch on the problem under which conditions the Goldstone theorem remains fulfilled in higher RPA's and on the strict fulfillement of the Pauli principle.

References

[1] M. Tohyama, P. Schuck, Extended RPA with ground state correlations, Eur. Phys. J. A121 (2004) 217.

[2] See also Phys. Rev. C70 (2004) 057307.

RPA selfconsistency and the killing condition

Jorge Dukelsky

Instituto de Estructura de la Materia. CSIC. Serrano 123. 28006 Madrid. Spain.

The Lipkin-Meshkov-Glick (LM) model [1] is a schematic two-level model based on the SU(2) algebra of particle-hole operators. It has been extensively used in nuclear physics as a benchmark model to test many-body approximations. We have used it to test the Self Consistent RPA (SCRPA) [2] by explicitly finding the RPA vacuum that satisfies the killing condition within the particle-hole collective subspace. On a different context, it has been shown that number Projected BCS (PBCS) also satisfies the killing condition [3]. The PBCS state requires the extension of the SU(2) LM to the O(5) Agassi model [4] that incorporates the pair algebra. In this talk I will review our derivation of the SCRPA within the LM model. By comparing both RPA vacuums I will pose the problem of whether PBCS is a unique RPA vacuum, and whether it contains enough particle-hole correlations to describe appropriately systems of interest.

References

[1] H. J. Lipkin, N. Meshkov and A. J. Glick, Nucl. Phys. 62 (1965) 188.

[2] J. Dukelsky and P. Schuck, Nucl. Phys. A 512 (1990) 466.

[3] Y. Ohrn and J. Linderberg, Int. J. Quantum Chem. 15 (1979) 1109.

[4] D. Agassi, Nucl. Phys. A 116 (1968) 49.

Extensions of the RPA: Are they correct?

Osvaldo Civitarese

Departamento de Fisica, Universidad de La Plata, Argentina

Several extensions of the RPA have been proposed, to account for the failure of the approximation in the presence of phase transitions. The problem is particularly severe for the case of proton-neutron correlations, where the attractive two-particle channels of the interaction may induce the collapse of the RPA. In this talk we review the situation and show that most of the proposed extensions are unjustified on theoretical grounds.

Consistent ground states and renormalizations in ab initio propagator theory of molecules

J. V. Ortiz

Department of Chemistry, Auburn University, USA

1. Antisymmetrized geminal power (AGP) wavefunctions satisfy certain ground-state consistency requirements for the random phase approximation (RPA) of the polarization propagator. The geminal in natural form has coefficients for its component Slater determinants that are not unique, for they are determined only up to a phase factor. A procedure for expressing the geminal uniquely in terms of its occupation numbers and canonical, general spin-orbitals (GSOs) is described. The AGP total energy is thus a functional of its occupation numbers and canonical GSOs. 2. The effective electron-electron interaction that emerges from RPA calculations has been incorporated in self-energy approximations for the electron propagator. Calculations of small-molecule ionization energies have been obtained with Hartree-Fock reference orbitals and with Kohn-Sham orbitals produced by various, approximate exchange-correlation functionals. To obtain reasonable results, these approximations must be improved with ladder corrections in the self-energy.

Real-time calculations for RPA response and nonlinear dynamics

Kazuhiro Yabana

Center for Computational Sciences, University of Tsukuba, Japan

It has been well known that the RPA for excited states is equivalent to the linearized time-dependent mean-field theory. I report our recent work solving the time-dependent mean-field equation in real-time to describe electron dynamics in linear and nonlinear regimes. In the linear response regime, I will show that a simple adiabatic time-dependent density-functional theory is capable of describing very accurately the oscillator strength distributions of small and large size molecules [1,2]. Then I will move to a description of nonlinear electron dynamics in crystalline solid induced by a uniform field of intense laser pulse. The basic equation we solve is the time-dependent Kohn-Sham equation which couples with the polarization field. In a weak field limit, it describes a dielectric response of electrons, while it describes the optical dielectric breakdown when an external laser pulse is sufficiently intense [3].

References

[1] K. Yabana, T. Nakatsukasa, J.-I. Iwata, G.F. Bertsch, Phys. Stat. Sol. (b)243, 1121 (2006).

[2] Y. Kawashita, K. Yabana, M. Noda, K. Nobusada, T. Nakatsukasa, J. Mol. Struct. THEOCHEM 914, 130 (2009).

[3] T. Otobe, M. Yamagiwa, J.-I. Iwata, K. Yabana, T. Nakatsukasa, G.F. Bertsch, Phys. Rev. B77, 165104 (2008).

Adiabatic-connection fluctuation-dissipation density-functional theory based on range separation

Julien Toulouse

UPMC Paris 06, Laboratoire de Chimie Théorique, 4 place Jussieu, 75005, Paris

One difficulty in the Kohn-Sham formulation of density functional theory using local density and generalized-gradient approximations is the description of nonlocal correlation effects, such as those involved in weak van der Waals complexes, bound by London dispersion forces [1]. Adiabatic-connection fluctuation-dissipation theorem (ACFDT) approaches, such as the random phase approximation (RPA), are one the most promising ways of constructing highly non-local correlation functionals [2]. In a similar spirit as in the work of Kohn et al. [3], we propose an ACFDT approach based on a range separation of electron-electron interactions [4]. It involves a rigorous combination of a short-range density functional with one of the possible long-range generalizations of the RPA. This method corrects several shortcomings of the standard RPA and it is particularly well suited for describing weakly-bound van der Waals systems, as demonstrated on several chemical systems. This method is expected to supersede a previously-proposed approach based on long-range second-order perturbation theory [5] for systems with small electronic gap.

References

[1] J. F. Dobson, K. McLennan, A. Rubio, J. Wang, T. Gould, H. M. Le, B. P. Dinte, Aust. J. Chem. 54, 513 (2001)

[2]F. Furche, Phys. Rev. B 64, 195120 (2001)

[3] W. Kohn and Y. Meir and D. E. Makarov, Phys. Rev. Lett. 80, 4153 (1998)

[4] J. Toulouse, I. C. Gerber, G. Jansen, A. Savin, J. G. Angyan, Phys. Rev. Lett. 102, 096404 (2009)

[5] J. G. Angyan, I. C. Gerber, A. Savin, J. Toulouse, Phys. Rev. A 72, 012510 (2005)

Failure of the random phase approximation correlation energy

Paula Mori-Sánchez, Aron J. Cohen and Weitao Yang

Department of Chemistry, Duke University, Durham, North Carolina 27708, USA

The random phase approximation (RPA) to the correlation energy is investigated for the dissociation of simple molecules. The observed successes and failures are rationalized by extending the RPA method to fractional occupations and examining its performance for exact conditions on fractional charges and fractional spins. RPA satisfies the constancy condition for fractional spins that leads to correct bond dissociation and no static correlation error for H2, but massively fails for fractional charges, with an enormous delocalization error even for a one-electron system such as H2+. Other methods such RPA including the Hartree-Fock response (RPAE) or range-separated RPA can reduce this delocalization error but only at the cost of increasing the static correlation error. None of the RPA methods seem to have the discontinuous nature required to satisfy both exact conditions and the full unified condition, emphasizing the need to go beyond smooth functionals of the orbitals.

Testing various RPA ground state energy expressions for atomic correlation energies

Georg Jansen

Faculty of Chemistry, University Duisburg-Essen, Germany

The electron correlation energies of atoms and atomic ions [1,2] will be used to illustrate the performance of RPA correlation energy expressions to reproduce this fundamental property. RPA functionals with and without exact exchange contributions and corresponding second-order approximations will be studied, employing standard functionals and exchange-correlation potentials which were obtained from electron densities as determined by ab initio calculations.

References

[1] E.R. Davidson, S.A. Hagstrom, S.J. Chakravorty, V.M Umar, C.F. Fischer, Phys. Rev. A 44 (1991) 7071.

[2] S.J. Chakravorty, S.R. Gwaltney, E.R. Davidson, F.A. Parpia, C.F. Fischer, Phys. Rev. A 47 (1993) 3649.

RPA correlation energy expressions based on adiabatic connection fluctuation-dissipation theorem

János G. Ángyán

CRM2, Nancy-University & CNRS, Vandoeuvre-lès-Nancy, F-54506, France

A variety of ground state correlation energy expressions can be obtained from the the ACFDT (adiabatic connection fluctuation-dissipation theorem) approach by using various forms of the RPA, like direct RPA (dRPA) or RPA with exchange (RPAx). Although the naive form of the ACFDT/RPA correlation energy expression involves both coupling-strength and frequency integrations, one can perform at least one (or both) of these integrals analytically. Our analysis is an attempt to understand the relationship between sometimes quite differently-looking correlation energy formulas.

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