Linear Density Response Function Research Articles

Time-dependent exchange-correlation hole and potential of the electron gas

The exchange-correlation hole and potential of the homogeneous electron gas have been investigated within the random-phase approximation, employing the plasmon-pole approximation for the linear density response function. The angular dependence as well as the time dependence of the exchange-correlation hole are illustrated for a Wigner-Seitz radius ${r}_{s}=4$ (atomic unit). It is found that there is a substantial cancellation between exchange and correlation potentials in space and time, analogous to the cancellation of exchange and correlation self-energies. Analysis of the sum rule explains why it is more advantageous to use a noninteracting Green function than a renormalized one when calculating the response function within the random-phase approximation and consequently the self-energy within the well-established $GW$ approximation. The present study provides a starting point for more accurate and comprehensive calculations of the exchange-correlation hole and potential of the electron gas with the aim of constructing a model based on the local density approximation as in density functional theory.

Assessing the accuracy of hybrid exchange-correlation functionals for the density response of warm dense electrons.

We assess the accuracy of common hybrid exchange-correlation (XC) functionals (PBE0, PBE0-1/3, HSE06, HSE03, and B3LYP) within the Kohn-Sham density functional theory for the harmonically perturbed electron gas at parameters relevant for the challenging conditions of the warm dense matter. Generated by laser-induced compression and heating in the laboratory, the warm dense matter is a state of matter that also occurs in white dwarfs and planetary interiors. We consider both weak and strong degrees of density inhomogeneity induced by the external field at various wavenumbers. We perform an error analysis by comparing with the exact quantum Monte Carlo results. In the case of a weak perturbation, we report the static linear density response function and the static XC kernel at a metallic density for both the degenerate ground-state limit and for partial degeneracy at the electronic Fermi temperature. Overall, we observe an improvement in the density response when the PBE0, PBE0-1/3, HSE06, and HSE03 functionals are used, compared with the previously reported results for the PBE, PBEsol, local-density approximation, and AM05 functionals; B3LYP, on the other hand, does not perform well for the considered system. Additionally, the PBE0, PBE0-1/3, HSE06, and HSE03 functionals are more accurate for the density response properties than SCAN in the regime of partial degeneracy.

High-frequency limit of spectroscopy.

We consider an arbitrary quantum mechanical system, initially in its ground-state, exposed to a time-dependent electromagnetic pulse with a carrier frequency ω0 and a slowly varying envelope of finite duration. By working out a solution to the time-dependent Schrödinger equation in the high-ω0 limit, we find that, to the leading order in ω0 -1, a perfect self-cancellation of the system's linear response occurs as the pulse switches off. Surprisingly, the system's observables are, nonetheless, describable in terms of a combination of its linear density response function and nonlinear functions of the electric field. An analysis of a jellium slab and jellium sphere models reveals a very high surface sensitivity of the considered setup, producing a richer excitation spectrum than accessible within the conventional linear response regime. On this basis, we propose a new spectroscopic technique, which we provisionally name the Nonlinear High-Frequency Pulsed Spectroscopy (NLHFPS). Combining the advantages of the extraordinary surface sensitivity, the absence of constraints by the traditional dipole selection rules, and the clarity of theoretical interpretation utilizing the linear response time-dependent density functional theory, NLHFPS has a potential to evolve into a powerful characterization method for nanoscience and nanotechnology.

Nuclear response functions with low-momentum interactions

Linear density response functions for nuclear matter are calculated with separable 1S0 and 3S1 2-nucleon interactions obtained from experimental phase-shifts by inverse scattering techniques. It has been shown that results of nuclear matter binding energy Brueckner calculations with these potentials agree closely with those using realistic Bonn interactions. It has also been shown that low-momentum versions give binding energies (nearly) independent of momentum cut-offs $\Lambda \geq \sim 2$ fm-1. Shown here is that the $\Lambda = 2$ rank-one version of the separable interaction (for the S-states) yields a near agreement between a second-order and an all-order (Brueckner) binding energy calculation. Second order calculations of response functions are then made with this version using a method that Kwong and Bonitz used for the Coulomb gas. It is based on time-evolving two-time Green’s functions by Kadanoff-Baym (KB) equations. The effect of correlations, going beyond the conventional HF+RPA method, is included by “dressing” the Green’s function propagators with time-dependent complex self-energies in addition to the real HF-field. It would be of interest to see if the calculated response is also, like the binding energy, independent of cut-offs larger than $\sim 2$ fm-1. That would require using separable potentials of a higher rank, not accommodated by the present computer program. Some results with a 1.5 fm-1 cut-off are however also shown below together with those with a 2.0 fm-1 cut-off. The KB-equations are time-evolved until the nuclear medium is fully correlated. The time-dependent density response to an external perturbation is then registered and Fourier transformed to obtain energy-dependent density response functions. Most previous calculations have been made with in-medium effective (e.g., Gogny or Skyrme) interactions. The medium dependence is in the present calculations contained in the second-order terms.

Real-time Kadanoff-Baym approach to nuclear response functions

Linear density response functions are calculated for symmetric nuclear matter of normal density by time-evolving two-time Green's functions in real time. Of particular interest is the effect of correlations. The system is therefore initially time-evolved with a collision term calculated in a direct Born approximation until fully correlated. An external time-dependent potential is then applied. The ensuing density fluctuations are recorded to calculate the density response. This method was previously used by Kwong and Bonitz for studying plasma oscillations in a correlated electron gas. The energy-weighted sum-rule for the response function is guaranteed by using conserving self-energy insertions as the method then generates the full vertex-functions. These can alternatively be calculated by solving a Bethe -Salpeter equation as done in works by Bozek et al. The (first order) mean field is derived from a momentum-dependent (non-local) interaction while 2nd order self-energies are calculated using a particle-hole two-body effective (or residual) interaction given by a gaussian local potential.We show results of calculations of the response function S(ɷ,q0) for q0 = 0.2, 0.4 and 0.8fm-1. Comparison is made with the nucleons being un-correlated i.e. with only the first order mean field included.We discuss the relation of our work with the Landau quasi-particle theory as applied to nuclear systems by Babu and Brown and followers.

Local and linear chemical reactivity response functions at finite temperature in density functional theory.

We explore the local and nonlocal response functions of the grand canonical potential density functional at nonzero temperature. In analogy to the zero-temperature treatment, local (e.g., the average electron density and the local softness) and nonlocal (e.g., the softness kernel) intrinsic response functions are defined as partial derivatives of the grand canonical potential with respect to its thermodynamic variables (i.e., the chemical potential of the electron reservoir and the external potential generated by the atomic nuclei). To define the local and nonlocal response functions of the electron density (e.g., the Fukui function, the linear density response function, and the dual descriptor), we differentiate with respect to the average electron number and the external potential. The well-known mathematical relationships between the intrinsic response functions and the electron-density responses are generalized to nonzero temperature, and we prove that in the zero-temperature limit, our results recover well-known identities from the density functional theory of chemical reactivity. Specific working equations and numerical results are provided for the 3-state ensemble model.

Linear and nonlinear density response functions for a simple atomic fluid

We use molecular dynamics simulations to investigate the linear and nonlinear density response functions for simple fluids under the influence of spatially periodic external fields. Using a direct Fourier space decomposition of the instantaneous microscopic density for the perturbed fluid we can clearly identify the distinct order of response. Using a single component sinusoidal longitudinal force for a set of wavelengths and amplitudes we show that in the linear response regime the proportionality between the external field amplitude and the density perturbation can be used to determine the linear density response function, and hence the pair correlation function, static liquid structure factor, and lowest order direct correlation function. We show also that for large external field amplitudes a single component external field can be used to determine the form for lowest order and second lowest order nonlinear response functions for restricted regions of the total response function spaces.

Resolution-of-identity approach to Hartree–Fock, hybrid density functionals, RPA, MP2 and GW with numeric atom-centered orbital basis functions

The efficient implementation of electronic structure methods is essential for first principles modeling of molecules and solids. We present here a particularly efficient common framework for methods beyond semilocal density-functional theory (DFT), including Hartree–Fock (HF), hybrid density functionals, random-phase approximation (RPA), second-order Møller–Plesset perturbation theory (MP2) and the GW method. This computational framework allows us to use compact and accurate numeric atom-centered orbitals (NAOs), popular in many implementations of semilocal DFT, as basis functions. The essence of our framework is to employ the ‘resolution of identity (RI)’ technique to facilitate the treatment of both the two-electron Coulomb repulsion integrals (required in all these approaches) and the linear density-response function (required for RPA and GW). This is possible because these quantities can be expressed in terms of the products of single-particle basis functions, which can in turn be expanded in a set of auxiliary basis functions (ABFs). The construction of ABFs lies at the heart of the RI technique, and we propose here a simple prescription for constructing ABFs which can be applied regardless of whether the underlying radial functions have a specific analytical shape (e.g. Gaussian) or are numerically tabulated. We demonstrate the accuracy of our RI implementation for Gaussian and NAO basis functions, as well as the convergence behavior of our NAO basis sets for the above-mentioned methods. Benchmark results are presented for the ionization energies of 50 selected atoms and molecules from the G2 ion test set obtained with the GW and MP2 self-energy methods, and the G2-I atomization energies as well as the S22 molecular interaction energies obtained with the RPA method.

Linear density response function in the projector augmented wave method: Applications to solids, surfaces, and interfaces

We present an implementation of the linear density response function within the projector-augmented wave (PAW) method with applications to the linear optical and dielectric properties of both solids, surfaces, and interfaces. The response function is represented in plane waves while the single-particle eigenstates can be expanded on a real space grid or in atomic orbital basis for increased efficiency. The exchange-correlation kernel is treated at the level of the adiabatic local density approximation (ALDA) and crystal local field effects are included. The calculated static and dynamical dielectric functions of Si, C, SiC, AlP and GaAs compare well with previous calculations. While optical properties of semiconductors, in particular excitonic effects, are generally not well described by ALDA, we obtain excellent agreement with experiments for the surface loss function of the Mg(0001) surface with plasmon energies deviating by less than 0.2 eV. Finally, we apply the method to study the influence of substrates on the plasmon excitations in graphene. On SiC(0001), the long wavelength $\pi$ plasmons are significantly damped although their energies remain almost unaltered. On Al(111) the $\pi$ plasmon is completely quenched due to the coupling to the metal surface plasmon.

Linear density response function within the time-dependent exact-exchange approximation

We have calculated the frequency-dependent exact-exchange (EXX) kernel of time-dependent (TD) density-functional theory employing our recently proposed computational method based on cubic splines. With this kernel we have calculated the linear density response function and obtained static polarizabilites, van der Waals coefficients, and correlation energies for all spherical spin-compensated atoms up to argon. Some discrete excitation energies have also been calculated for Be and Ne. As might be expected, the results of the TDEXX approximation are close to those of TD Hartree-Fock theory. In addition, correlation energies obtained by integrating over the strength of the Coulomb interaction turn out to be highly accurate.

Dynamic response of a strongly perturbed electron gas

The theoretical description of many-body systems in their ground state has nowadays become an affordable task because of recent advances in the numerical implementation of ab initio methods, as well as huge improvements in computing capabilities. However, an accurate evaluation of excitedstates properties in such systems is still a major theoretical challenge. The problem is of obvious importance as most experimental techniques are based on testing of excited state. Apart from its intrinsic fundamental interest, the knowledge of the system excitation spectrum is useful to understand a large variety of physical phenomena, such as ~i! photoabsorption of atoms 1,2 and plasmas, 3,4 ~ii! optical response of clusters, 5‐ 8 ~iii! core polarization and dielectric response in solids, 9,10 ~iv! resonant photoemission, 11 or ~v! neutralization of ions at surfaces. 12,13 Common solid-state approaches to study excited states are many-body perturbation and time-dependent densityfunctional theories. The application of these methods to real systems often relies on system symmetries, such as twodimensional or three-dimensional periodicity. The theoretical description of electronic excitations turns out to be more intricate when a localized impurity breaks the spatial symmetry. Numerical methods based on system periodicity are not efficient in this case. Furthermore, localized perturbations in solids, such as impurities in bulk, inner-shell holes, or adsorbates at surfaces strongly modify their environment. The appearance of new states bound to the impurity 14 and/or resonances in the continuum 15 can change dramatically the local properties of the system. The influence of impurity atoms in the excitation spectrum of a 2D electron gas ~where any attractive potential has a bound state! has been considered recently. 16 Kondo-type effects arise for single magnetic impurities as well. 17,18 In metallic systems, valence-band electrons are highly mobile and the presence of an atomic impurity introduces a strong distortion in the electronic density. Our purpose in this work is to show that this rearrangement of electronic charge determines the local dynamic response of the valence electrons to external perturbations. We study the case of an atomic impurity embedded in a free electron gas ~FEG!, which represents the valence band of a simple metal ~such as Na or Al!. Therefore, band-structure effects are not considered as we are interested in determining the local dynamic response of the system. We use density-functional theory to calculate the ground state of the system and obtain the valence-electron response function in linear theory afterwards. We show that the dynamic screening of the Coulomb interaction is highly modified in the vicinity of the impurity and has a complex behavior determined by two competing effects. As an example, we calculate transition probabilities for hole-filling processes in the embedded impurity. We compare some of our results with those obtained in calculations which do not include the presence of the impurity to remark the specific effects introduced by the latter. For small external dynamic perturbations of the system, the many-body response is well described by the linear density response function x(r,r8,v). The self-consistent response function x(r,r8,v) can be obtained at the randomphase approximation ~RPA! level in terms of the response function of a system of independent particles x 0 (r,r8,v) ~atomic units are used throughout unless otherwise stated!: x~ r,r8,v!5x 0 ~ r,r8,v!

Sum rule for the dynamical response of a confined Bose-Einstein condensed gas

We extend a sum rule, resulting from the covariance properties of the Schroedinger equation under transformation in an accelerated frame and originally derived for an inhomogeneous electron system, to the linear-density response functions of a confined Bose-condensed gas at finite temperature. We apply the sum rule to test some approximate theories and to show that in the case of a harmonic confinement the random-phase approximation includes the Kohn modes in its spectrum.

Lattice vibrations and elastic constants of crystalline [formula omitted] and [formula omitted] near low-temperature melting

The phonon dispersion curves and the elastic constants are evaluated for crystalline 4 He and 3 He near melting at low temperature in the deep quantal regime. The structures considered are the hexagonal close-packed and the body-centred and face-centred cubic ones for 4 He and the body-centred cubic one for 3 He . The calculations use a density functional approach to construct a field of effective force constants in the strongly anharmonic solid near melting from its Debye–Waller factor and from the linear density response function of the fluid phase at freezing. The results of the calculations are compared with the available experimental evidence: good agreement is generally found for transverse phonon frequencies and shear elastic constants, and phenomenological adjustments of the force-constant field are proposed for a quantitative description of the longitudinal modes. A relationship between the dispersion curves of phonons in crystalline 4 He and that of elementary collective excitations in superfluid 4 He is displayed and discussed in the light of current interpretations of atomic dynamics in the superfluid phase, with main emphasis on the region of the roton minimum.

Lattice vibrations and elastic constants of three- and two-dimensional quantal Wigner crystals near melting

The phonon dispersion relations and the elastic constants are evaluated for Wigner electron crystals near the critical coupling strength for melting in the fully quantal regime at zero temperature. The structures considered are the body-centred and face-centred cubic lattices in dimensionality D = 3 and the triangular lattice in D = 2. The calculations are based on a density functional approach requiring as input the linear density response function of the fluid phase at freezing and the Debye - Waller factor of the crystal at melting. These are known from quantal Monte Carlo simulations both for D = 3 and for D = 2. Comparison with earlier results of harmonic calculations shows appreciable softening from anharmonicity, which is mainly associated with the exchange and correlation contributions to the effective force constants of the quantal crystal near melting. Mechanical stability of the body-centred cubic electron crystal at melting is demonstrated through a self-consistent calculation of the lattice vibrations and the mean square particle displacement entering the Debye - Waller factor, as well as by calculations of the elastic constants using the methods of long waves and of homogeneous deformations. Finally, a relationship is displayed between phonon dispersion curves in the triangular Wigner crystal near melting and plasmon excitations in the two-dimensional electron fluid near freezing.

Linear-response theory for partly quenched systems

The matrix of linear density response functions is constructed within density functional theory for a monatomic fluid adsorbed in a quenched material. The structural Ornstein-Zernike relations obtained by Given and Stell with the replica method are regained in the appropriate limit. The usefulness of the present reformulation of their theory is illustrated by deriving from it the free-energy functional and the structural closure relations in the hypernetted-chain approximation. The extension to dynamical properties of partly quenched systems is indicated.