Random Phase Approximation Equations Research Articles

Control of plasmons in topological insulators via local perturbations

We use a fully quantum mechanical approach to demonstrate control of plasmonic excitations in prototype models of topological insulators by molecule-scale perturbations. Strongly localized surface plasmons are present in the host systems, arising from the topologically nontrivial single-particle edge states. A numerical evaluation of the random phase approximation equations for the perturbed systems reveals how the positions and the internal electronic structure of the added molecules affect the degeneracy of the locally confined collective excitations, i.e., shifting the plasmonic energies of the host system and changing their spatial charge density profile. In particular, we identify conditions under which significant charge transfer from the host system to the added molecules occurs. Furthermore, the induced field energy density in the perturbed topological systems due to external electric fields is determined.

Finite-temperature-based time-dependent density-functional theory method for static electron correlation systems.

In this study, we developed a time-dependent density-functional theory (TDDFT) with a finite-temperature (FT) scheme, denoted as FT-TDDFT. We introduced the concept of fractional occupation numbers for random phase approximation equation and evaluated the excited-state electronic entropy terms with excited-state occupation number. The orbital occupation numbers for the excited state were evaluated from the change in the ground-state electron configuration with excitation and deexcitation coefficients. Furthermore, we extended the FT formulation to the time-dependent density-functional tight-binding (TDDFTB) method for larger systems, denoted as FT-TDDFTB. Numerical assessment for the FT-(TD)DFT method showed smooth potential curves for double-bond rotation of ethylene in both ground and excited states. Excited-state calculations based on the FT-TDDFTB method were applied to the uniform π-stacking columns composed of trioxotriangulene, possessing neutral radicals in strong correlation systems.

Second-Order Exchange-Dispersion Energy Based on a Multireference Description of Monomers.

We present a method for calculation of the second-order exchange-dispersion energy in the framework of the symmetry-adapted perturbation theory (SAPT) for weakly interacting monomers described with multiconfigurational wave functions. The proposed formalism is based on response properties obtained from extended random phase approximation (ERPA) equations and assumes the single-exchange (S2) approximation. The approach is applicable to closed shell systems where static correlation cannot be neglected or to systems in nondegenerate excited states. We examine the new method in combination with either generalized valence bond perfect pairing (GVB) or complete active space self-consistent field (CASSCF) description of the interacting monomers. For model multireference dimers in ground states (H2···H2, Be···Be, He···H2), exchange-dispersion energies are reproduced accurately. For the interaction between the excited hydrogen molecule and the helium atom we found unacceptably large errors which is attributed to the neglect of diagonal double excitations in the employed approximation to the linear response function.

Generalized Valence Bond Perfect-Pairing Made Versatile Through Electron-Pairs Embedding.

We present an electron-pairs-based method employing a generalized valence bond perfect-pairing (GVB-PP) ansatz that provides a uniformly accurate description of systems where various types of electron correlation play a role and the GVB-PP wave function is a suitable reference. In the proposed EERPA-GVB approach, a GVB-PP energy is amended by adding correlation among electron pairs. The latter is achieved by embedding single pairs or couples of pairs in the environment of the other electron fragments and separately accounting for intra- and interfragment correlation effects. For this purpose, we employ truncated extended random phase approximation equations. Application of EERPA-GVB to systems governed by both short-range (energy barriers) and long-range (molecular interactions) correlation effects proves the good accuracy of the method. Moreover, EERPA-GVB is shown to cure a notorious problem of uncorrelated electron-pair models, namely, spatial symmetry breaking in aromatic molecules, using the example of benzene. We have also successfully applied EERPA-GVB to a challenging problem of a phase transition of the boron chain system, where the correlation changes its character along the reaction path. The accuracy and versatility of EERPA-GVB are accompanied by its attractively low computational cost. By truncation of the extended RPA equations and consideration of only at most two-fragment correlation contributions, the cost of computing the EERPA correlation energy is reduced to scale only quadratically with the number of pairs of electrons.

Second-Order Dispersion Energy Based on Multireference Description of Monomers.

We propose a method for calculating a second-order dispersion energy for weakly interacting multireference systems in arbitrary electronic states. It is based on response properties obtained from extended random phase approximation equations. The introduced formalism is general and requires only one- and two-particle reduced density matrices of monomers. We combine the new method with either generalized valence bond perfect pairing (GVB) or complete active space (CAS) self-consistent field description of the interacting systems. In addition to a general scheme, three approximations, leading to significant reduction of the computational cost, are developed by exploiting Dyall partitioning of the monomer Hamiltonians. For model multireference systems (H2···H2 and Be···Be) the method is accurate, unlike its single-reference-based counterpart. Neither GVB nor CAS description of single-reference monomers improves the dispersion energy with respect to the Hartree-Fock-based results.

Absorption and emission of a collective excitation by a fermionic quasiparticle in a Fermi superfluid

We study the process of absorption or emission of a bosonic collective excitation by a fermionic quasiparticle in a superfluid of paired fermions. From the random phase approximation equation of motion of the bosonic excitation annihilation operator, we obtain an expression of the coupling amplitude of this process, which is limited neither to resonant processes nor to the long wavelength limit. We confirm our result by independently deriving it in the functional integral approach using the gaussian fluctuation approximation, and by comparing it in the long wavelength limit to the quantum hydrodynamic result. Lastly, we give a straightforward application of the coupling amplitude we obtain by calculating the lifetime of the bosonic excitations of an arbitrary wave number. We find a mode quality factor that decreases from its maximum at low wave numbers and vanishes when the bosonic branch hits the continuum of fermionic biexcitations.

An efficient optimization method for geminal-based wave functions

Antisymmetric product of strongly orthogonal geminal (APSG) and its simple variant, generalized valence bond (GVB) perfect pairing are capable of producing the static (intrageminal) electron correlation energy in a chemical entity, and recently it has been proposed how to recover the dynamical (intergeminal) electron correlation energy in terms of the transition density matrix elements by solving the extended random phase approximation equation. The problem of these wave functions is, however, the slow convergence in the optimization. In order to optimize efficiently both orbitals and expansion coefficients of the APSG (or GVB), two different kinds of algorithms are introduced and tested. They employ direct inversion in the iterative subspace along with Broyden–Fletcher–Goldfarb–Shanno methods.

Doorway states in the random-phase approximation

By coupling a doorway state to a sea of random background states, we develop the theory of doorway states in the framework of the random-phase approximation (RPA). Because of the symmetry of the RPA equations, that theory is radically different from the standard description of doorway states in the shell model. We derive the Pastur equation in the limit of large matrix dimension and show that the results agree with those of matrix diagonalization in large spaces. The complexity of the Pastur equation does not allow for an analytical approach that would approximately describe the doorway state. Our numerical results display unexpected features: The coupling of the doorway state with states of opposite energy leads to strong mutual attraction.

Low excitations of 16O using generalized density matrix random phase approximation GDRPA

The random phase approximation (RPA) equations based on the generalized density matrix (GDM), the so-called GDRPA are reformulated in a more compact matrix form, which renders the method especially suitable for realistic nuclear structure calculations. The GDRPA Hamiltonian is expressed in terms of the one-body particle–particle (pp) and hole–hole (hh) density matrices, and the nuclear force contributes not only in the particle–hole (ph) channel, as in normal ph-RPA, but also in the pp and hh channels. The Hamiltonian is diagonalized iteratively starting from initial guess values and the iterating process is carried out until self-consistency is achieved. The calculation in the model space 1p, 1d and 2s using Warburton and Brown interaction WBP is performed for 16 O . The GDRPA in the ph shell model calculations is tested, by comparing the energy eigenvalues and the electron scattering form factors with the results of the normal RPA and with the available experimental data.

Derivation of the RPA (Random Phase Approximation) Equation of ATDDFT (Adiabatic Time Dependent Density Functional Ground State Response Theory) from an Excited State Variational Approach Based on the Ground State Functional.

The random phase approximation (RPA) equation of adiabatic time dependent density functional ground state response theory (ATDDFT) has been used extensively in studies of excited states. It extracts information about excited states from frequency dependent ground state response properties and avoids, thus, in an elegant way, direct Kohn-Sham calculations on excited states in accordance with the status of DFT as a ground state theory. Thus, excitation energies can be found as resonance poles of frequency dependent ground state polarizability from the eigenvalues of the RPA equation. ATDDFT is approximate in that it makes use of a frequency independent energy kernel derived from the ground state functional. It is shown in this study that one can derive the RPA equation of ATDDFT from a purely variational approach in which stationary states above the ground state are located using our constricted variational DFT (CV-DFT) method and the ground state functional. Thus, locating stationary states above the ground state due to one-electron excitations with a ground state functional is completely equivalent to solving the RPA equation of TDDFT employing the same functional. The present study is an extension of a previous work in which we demonstrated the equivalence between ATDDFT and CV-DFT within the Tamm-Dancoff approximation.

Excitation energies from extended random phase approximation employed with approximate one- and two-electron reduced density matrices

Starting from Rowe's equation of motion we derive extended random phase approximation (ERPA) equations for excitation energies. The ERPA matrix elements are expressed in terms of the correlated ground state one- and two-electron reduced density matrices, 1- and 2-RDM, respectively. Three ways of obtaining approximate 2-RDM are considered: linearization of the ERPA equations, obtaining 2-RDM from density matrix functionals, and employing 2-RDM corresponding to an antisymmetrized product of strongly orthogonal geminals (APSG) ansatz. Applying the ERPA equations with the exact 2-RDM to a hydrogen molecule reveals that the resulting (1)Σ(g)(+) excitation energies are not exact. A correction to the ERPA excitation operator involving some double excitations is proposed leading to the ERPA2 approach, which employs the APSG one- and two-electron reduced density matrices. For two-electron systems ERPA2 satisfies a consistency condition and yields exact singlet excitations. It is shown that 2-RDM corresponding to the APSG theory employed in the ERPA2 equations yields excellent singlet excitation energies for Be and LiH systems, and for the N(2) molecule the quality of the potential energy curves is at the coupled cluster singles and doubles level. ERPA2 nearly satisfies the consistency condition for small molecules that partially explains its good performance.

Stellar electron-capture rates on nuclei based on a microscopic Skyrme functional

We compute electron-capture rates for ${}^{54,56}$Fe and Ge isotopes using a self-consistent microscopic approach. The single-nucleon basis and the occupation factors in the target nucleus are calculated in the finite-temperature Skyrme Hartree-Fock model, and the ${J}^{\ensuremath{\pi}}={0}^{\ifmmode\pm\else\textpm\fi{}}$, ${1}^{\ifmmode\pm\else\textpm\fi{}}$, ${2}^{\ifmmode\pm\else\textpm\fi{}}$ charge-exchange transitions are determined in the finite-temperature random-phase approximation (RPA). The scheme is self-consistent; i.e., both the Hartree-Fock and the RPA equations are based on the same Skyrme functional. Several interactions are used in order to provide a theoretical uncertainty on the electron-capture rates for different astrophysical conditions. Comparing electron-capture rates obtained either with different Skyrme sets or with different available models indicates that differences up to one to two orders of magnitude can arise.

Elementary excitations in superfluidH3e-H4emixtures

We have studied the dynamic structure function of superfluid $^{3}\text{H}\text{e-}^{4}\text{H}\text{e}$ mixtures at zero temperature as a function of pressure and $^{3}\text{H}\text{e}$ concentration. Results obtained in the full random-phase approximation (RPA) plus density-functional theory and in a generalized Landau-Pomeranchuk approach are presented and compared with experiment. Analytic expressions for several sum rules of the dynamic structure functions have been determined, and have been used to obtain average energies of the collective excitations. In the RPA approach, the dispersion relation of the collective modes shows typical features of level repulsion between zero-soundlike and phonon-rotonlike excitations. The structure of the coupled RPA equations for the mixture leads in a natural way to the hybridization of the collective modes. The mixed $^{3}\text{H}\text{e-}^{4}\text{H}\text{e}$ dynamic structure function quenches the zero-soundlike mode before it crosses the phonon-roton branch, causing that the former mode only appears with enough strength after the crossing.

The ground state correlation energy of the random phase approximation from a ring coupled cluster doubles approach

We present an analytic proof demonstrating the equivalence between the random phase approximation (RPA) to the ground state correlation energy and a ring-diagram simplification of the coupled cluster doubles (CCD) equations. In the CCD framework, the RPA equations can be solved in O(N(4)) computational effort, where N is proportional to the number of basis functions.

Evaluation of pairwise entanglement in translationally invariant systems with the random phase approximation

We discuss a general mean field plus random phase approximation (RPA) for describing composite systems at zero and finite temperature. We analyze in particular its implementation in finite systems invariant under translations, where for uniform mean fields it requires just the solution of simple local-type RPA equations. As test and application, we use the method for evaluating the entanglement between two spins in cyclic spin 1/2 chains with both long and short range anisotropic XY-type couplings in a uniform transverse magnetic field. The approach is shown to provide an accurate analytic description of the concurrence for strong fields, for any coupling range, pair separation or chain size, where it predicts an entanglement range which can be at most twice that of interaction. It also correctly predicts the existence of a separability field together with full entanglement range in its vicinity. The general accuracy of the approach improves as the range of the interaction increases.

Superconductivity in the Vicinity of Charge Ordered State in Organic Conductor β-(meso-DMBEDT-TTF)2PF6

We study theoretically competition between the charge ordering and the superconductivity in two-dimensional organic conductor beta-(meso-DMBEDT-TTF)_2PF_6. We analyze the extended Hubbard model on a weakly-dimerized lattice based on the random phase approximation and Eliashberg equations. We found reentrant behavior of the checkerbord-type charge-ordered phase in the phase diagram, and the triplet superconductivity due to the charge fluctuation in the neighboring region. In the low temperature and high pressure region, the singlet superconductivity also appears due to the enhancement of the spin fluctuation.

Contour integral method for thermal and quantal fluctuations

The partition function by means of the static path approximation (SPA) plus the random-phase approximation (RPA) treatment can be written as a contour integral form without solving the RPA equations for a separable interaction. This method is an efficient way to evaluate numerically the partition function for realistic calculations. As an illustration, we adopt the pairing model at finite temperature.

The RPA equation embedded into infinite-dimensional Fock spaceF∞

To clear up both algebraic and geometric structures for integrable systems derived from self-consistent field theory, in particular, a geometrical aspect of the random phase approximation (RPA) equation is presented from the viewpoint of symmetry of the evolution equation. The RPA equation for an infinite-dimensional Grassmannian is constructed. It gives us a simple geometrical interpretation that the collective submanifold is a rotator on a curved surface.

Real versus artifactual symmetry-breaking effects in Hartree–Fock, density-functional, and coupled-cluster methods

We have examined the relative abilities of Hartree-Fock, density-functional theory (DFT), and coupled-cluster theory in describing second-order (pseudo) Jahn-Teller (SOJT) effects, perhaps the most commonly encountered form of symmetry breaking in polyatomic molecules. As test cases, we have considered two prototypical systems: the 2Sigmau+ states of D( infinity h) BNB and C3+ for which interaction with a low-lying 2Sigmag+ excited state leads to symmetry breaking of the nuclear framework. We find that the Hartree-Fock and B3LYP methods correctly reproduce the pole structure of quadratic force constants expected from exact SOJT theory, but that both methods appear to underestimate the strength of the coupling between the electronic states. Although the Tamm-Dancoff (CIS) approximation gives excitation energies with no relationship to the SOJT interaction, the random-phase-approximation (RPA) approach to Hartree-Fock and time-dependent DFT excitation energies predicts state crossings coinciding nearly perfectly with the positions of the force constant poles. On the other hand, the RPA excited-state energies exhibit unphysical curvature near their crossings with the ground (reference) state, a problem arising directly from the mathematical structure of the RPA equations. Coupled-cluster methods appear to accurately predict the strength of the SOJT interactions between the 2Sigmau+ and 2Sigmag+ states, assuming that the inclusion of full triple excitations provides a suitable approximation to the exact wave function, and are the only methods examined here which predict symmetry breaking in BNB. However, coupled-cluster methods are plagued by artifactual force constant poles arising from the response of the underlying reference molecular orbitals to the geometric perturbation. Furthermore, the structure of the "true" SOJT force constant poles predicted by coupled-cluster methods, although correctly positioned, has the wrong structure.

Extension of the random phase approximation including the self-consistent coupling to two-phonon contributions

A microscopic formalism is developed that includes the coupling to two particle-hole phonons in the particle-hole propagator by extending the dressed random phase approximation (DRPA) equation for a finite system. The resulting formalism is applied to study the low-lying excitation spectrum of 16O. It is observed that the coupling to two-phonon states at low energy generates excited states with quantum numbers that cannot be obtained in the DRPA approach. Nevertheless, the two-phonon states mix weakly with particle-hole configurations and participate only partially in the formation of the lowest-lying positive-parity excited states. The stability of the present calculation is tested vs. the truncation of model space. It is demonstrated that when single-particle strength fragmentation is properly considered, the present formalism exhibits convergence with respect to the chosen model space within the confines of the chosen approximation scheme.