Static Multipole Polarizabilities Research Articles

Multipole polarizabilities of spherically confined hydrogen atom and positronium in screened Coulomb potential

AbstractThe static multipole polarizabilities of the hydrogen atom interacting with a screened Coulomb potential confined in an impenetrable spherical box are calculated in the sum‐over‐states formalism. The system eigenenergies and wave functions are produced by employing the generalized pseudospectral method. High‐precision results are obtained to resolve the discrepancies between previous predictions, and used to analyze the critical behavior of the system with varying the confinement radius and screening parameter. A scaling law of the multipole polarizabilities with respect to the nuclear charge and electron reduced mass is proposed. Based on the scaling law we extend the investigation to the spatially confined positronium atom in a screened Coulomb potential by utilizing a reduced one‐electron model. The results are in good agreement with previous estimates. However, it is conjectured that when the confinement radius is small the confined positronium atom should be treated more reasonably by an effective two‐electron model.

Analytical expressions of non-relativistic static multipole polarizabilities for hydrogen-like ions**Project supported by the National Natural Science Foundation of China (Grant Nos. 11674253, 11474316, and 91636216).

Analytical formulas for the static multipole polarizabilities of hydrogen-like ions are derived by using the analytical wave functions and the reduced Green function and by applying a numerical fitting procedure. Our results are then applied to the studies of blackbody radiation shifts to atomic energy levels at different temperatures. Our analytical results can be served as a benchmark for other theoretical methods.

First-principles study of the binding energy between nanostructures and its scaling with system size

The equilibrium binding energy is an important factor in the design of materials and devices. However, it presents great computational challenges for materials built up from nanostructures. Here we investigate the binding-energy scaling law from first-principles calculations. We show that the equilibrium binding energy per atom between identical nanostructures can scale up or down with nanostructure size. From the energy scaling law, we predict finite large-size limits of binding energy per atom. We find that there are two competing factors in the determination of the binding energy: Nonadditivities of van der Waals coefficients and center-to-center distance between nanostructures. To uncode the detail, the nonadditivity of the static multipole polarizability is investigated. We find that the higher-order multipole polarizability displays ultra-strong intrinsic nonadditivity, no matter if the dipole polarizability is additive or not.

Static polarizability in a combined model

In the framework of the theory of inhomogeneous electron gas, we consider a model to calculate the static multipole polarizability. The analysis shows the possibility of the specific dependence of the electron density perturbation (potential) of an atom in a weak external electrostatic field.

Accurate van der Waals coefficients between fullerenes and fullerene-alkali atoms and clusters: Modified single-frequency approximation

Long-range van der Waals (vdW) interaction is critically important for intermolecular interactions in molecular complexes and solids. However, accurate modeling of vdW coefficients presents a great challenge for nanostructures, in particular for fullerene clusters, which have huge vdW coefficients but also display very strong nonadditivity. In this work, we calculate the coefficients between fullerenes, fullerene and sodium clusters, and fullerene and alkali atoms with the hollow-sphere model within the modified single-frequency approximation (MSFA). In the MSFA, we assume that the electron density is uniform in a molecule and that only valence electrons in the outmost subshell of atoms contribute. The input to the model is the static multipole polarizability, which provides a sharp cutoff for the plasmon contribution outside the effective vdW radius. We find that the model can generate ${C}_{6}$ in excellent agreement with expensive wave-function-based ab initio calculations, with a mean absolute relative error of only $3%$, without suffering size-dependent error. We show that the nonadditivities of the coefficients ${C}_{6}$ between fullerenes and ${\mathrm{C}}_{60}$ and sodium clusters ${\mathrm{Na}}_{n}$ revealed by the model agree remarkably well with those based on the accurate reference values. The great flexibility, simplicity, and high accuracy make the model particularly suitable for the study of the nonadditivity of vdW coefficients between nanostructures, advancing the development of better vdW corrections to conventional density functional theory.

Van der Waals coefficients beyond the classical shell model.

Van der Waals (vdW) coefficients can be accurately generated and understood by modelling the dynamic multipole polarizability of each interacting object. Accurate static polarizabilities are the key to accurate dynamic polarizabilities and vdW coefficients. In this work, we present and study in detail a hollow-sphere model for the dynamic multipole polarizability proposed recently by two of the present authors (JT and JPP) to simulate the vdW coefficients for inhomogeneous systems that allow for a cavity. The inputs to this model are the accurate static multipole polarizabilities and the electron density. A simplification of the full hollow-sphere model, the single-frequency approximation (SFA), circumvents the need for a detailed electron density and for a double numerical integration over space. We find that the hollow-sphere model in SFA is not only accurate for nanoclusters and cage molecules (e.g., fullerenes) but also yields vdW coefficients among atoms, fullerenes, and small clusters in good agreement with expensive time-dependent density functional calculations. However, the classical shell model (CSM), which inputs the static dipole polarizabilities and estimates the static higher-order multipole polarizabilities therefrom, is accurate for the higher-order vdW coefficients only when the interacting objects are large. For the lowest-order vdW coefficient C6, SFA and CSM are exactly the same. The higher-order (C8 and C10) terms of the vdW expansion can be almost as important as the C6 term in molecular crystals. Application to a variety of clusters shows that there is strong non-additivity of the long-range vdW interactions between nanoclusters.

Evaluating interaction energies of weakly bonded systems using the Buckingham-Hirshfeld method.

We present the finalized Buckingham-Hirshfeld method (BHD-DFT) for the evaluation of interaction energies of non-bonded dimers with Density Functional Theory (DFT). In the method, dispersion energies are evaluated from static multipole polarizabilities, obtained on-the-fly from Coupled Perturbed Kohn-Sham calculations and partitioned into diatomic contributions using the iterative Hirshfeld partitioning method. The dispersion energy expression is distributed over four atoms and has therefore a higher delocalized character compared to the standard pairwise expressions. Additionally, full multipolar polarizability tensors are used as opposed to effective polarizabilities, allowing to retain the anisotropic character at no additional computational cost. A density dependent damping function for the BLYP, PBE, BP86, B3LYP, and PBE0 functionals has been implemented, containing two global parameters which were fitted to interaction energies and geometries of a selected number of dimers using a bi-variate RMS fit. The method is benchmarked against the S22 and S66 data sets for equilibrium geometries and the S22x5 and S66x8 data sets for interaction energies around the equilibrium geometry. Best results are achieved using the B3LYP functional with mean average deviation values of 0.30 and 0.24 kcal/mol for the S22 and S66 data sets, respectively. This situates the BHD-DFT method among the best performing dispersion inclusive DFT methods. Effect of counterpoise correction on DFT energies is discussed.

Computational investigation of static multipole polarizabilities and sum rules for ground-state hydrogenlike ions

High precision multipole polarizabilities, $\alpha_{\ell}$ for $\ell \le 4$ of the $1s$ ground state of the hydrogen isoelectronic series are obtained from the Dirac equation using the B-spline method with Notre Dame boundary conditions. Compact analytic expressions for the polarizabilities as a function of $Z$ with a relative accuracy of 10$^{-6}$ up to $Z = 100$ are determined by fitting to the calculated polarizabilities. The oscillator strengths satisfy the sum rules $\sum_i f^{(\ell)}_{0i} = 0$ for all multipoles from $\ell = 1$ to $\ell = 4$. The dispersion coefficients for the long-range H-H and H-He$^+$ interactions are given.

Accurate van der Waals coefficients from density functional theory

The van der Waals interaction is a weak, long-range correlation, arising from quantum electronic charge fluctuations. This interaction affects many properties of materials. A simple and yet accurate estimate of this effect will facilitate computer simulation of complex molecular materials and drug design. Here we develop a fast approach for accurate evaluation of dynamic multipole polarizabilities and van der Waals (vdW) coefficients of all orders from the electron density and static multipole polarizabilities of each atom or other spherical object, without empirical fitting. Our dynamic polarizabilities (dipole, quadrupole, octupole, etc.) are exact in the zero- and high-frequency limits, and exact at all frequencies for a metallic sphere of uniform density. Our theory predicts dynamic multipole polarizabilities in excellent agreement with more expensive many-body methods, and yields therefrom vdW coefficientsC6,C8,C10for atom pairs with a mean absolute relative error of only 3%.

Long-range interactions between an atom in its ground S state and an open-shell linear molecule

Theory of long-range interactions between an atom in its ground S state and a linear molecule in a degenerate state with a nonzero projection of the electronic orbital angular momentum is presented. It is shown how the long-range coefficients can be related to the first and second-order molecular properties. The expressions for the long-range coefficients are written in terms of all components of the static and dynamic multipole polarizability tensor, including the nondiagonal terms connecting states with the opposite projection of the electronic orbital angular momentum. It is also shown that for the interactions of molecules in excited states that are connected to the ground state by multipolar transition moments additional terms in the long-range induction energy appear. All these theoretical developments are illustrated with the numerical results for systems of interest for the sympathetic cooling experiments: interactions of the ground state Rb((2)S) atom with CO((3)Π), OH((2)Π), NH((1)Δ), and CH((2)Π) and of the ground state Li((2)S) atom with CH((2)Π).

Casimir potential of a compact object enclosed by a spherical cavity

We study the electromagnetic Casimir interaction of a compact object contained inside a closed cavity of another compact object. We express the interaction energy in terms of the objects' scattering matrices and translation matrices that relate the coordinate systems appropriate to each object. When the enclosing object is an otherwise empty metallic spherical shell, much larger than the internal object, and the two are sufficiently separated, the Casimir force can be expressed in terms of the static electric and magnetic multipole polarizabilities of the internal object, which is analogous to the Casimir-Polder result. Although it is not a simple power law, the dependence of the force on the separation of the object from the containing sphere is a universal function of its displacement from the center of the sphere, independent of other details of the object's electromagnetic response. Furthermore, we compute the exact Casimir force between two metallic spheres contained one inside the other at arbitrary separations. Finally, we combine our results with earlier work on the Casimir force between two spheres to obtain data on the leading order correction to the Proximity Force Approximation for two metallic spheres both outside and within one another.

Accurate description of the optical response of a multilayered spherical system in the long wavelength approximation

The optical response of a multilayered spherical system of unlimited number of layers (a ``matryoshka'') in the long wavelength limit can be accounted for from the knowledge of the static multipole polarizability of the system to first-order accuracy. However, for systems of ultrasmall dimensions or systems with sizes not-too-small compared to the wavelength, this ordinary quasistatic long wavelength approximation (LWA) becomes inaccurate. Here we introduce two significant modifications of the LWA for such a nanomatryoshka in each of the two limits: the nonlocal optical response for ultrasmall systems $(l10\text{ }\text{nm})$, and the ``finite-wavelength corrections'' for systems $\ensuremath{\sim}100\text{ }\text{nm}$. This is accomplished by employing the previous work for a single-layer shell, in combination with a certain effective-medium approach formulated recently in the literature. Numerical calculations for the extinction cross sections for such a system of different dimensions are provided as illustrations for these effects. This formulation thus provides significant improvements on the ordinary LWA, yielding enough accuracy for the description of the optical response of these nanoshell systems over an appreciable range of sizes, without resorting to more involved quantum mechanical or fully electrodynamic calculations.

A universal formula for dispersion coefficients between alkali atoms

A universal formula is developed for dispersion coefficients between alkali atoms. It gives values for C2n for n=3, 4, 5 and depends only on the static multipole polarizabilities and the corresponding first excitation energies. Comparison with available literature values shows that the foumula is highly accurate.

Molecular calculations of excitation energies and (hyper)polarizabilities with a statistical average of orbital model exchange-correlation potentials

An approximate Kohn–Sham exchange-correlation potential νxcSAOP is developed with the method of statistical averaging of (model) orbital potentials (SAOP) and is applied to the calculation of excitation energies as well as of static and frequency-dependent multipole polarizabilities and hyperpolarizabilities within time-dependent density functional theory (TDDFT). νxcSAOP provides high quality results for all calculated response properties and a substantial improvement upon the local density approximation (LDA) and the van Leeuwen–Baerends (LB) potentials for the prototype molecules CO, N2, CH2O, and C2H4. For the first three molecules and the lower excitations of the C2H4 the average error of the vertical excitation energies calculated with νxcSAOP approaches the benchmark accuracy of 0.1 eV for the electronic spectra.

Static multipole polarizability of atoms and ions in the Thomas-Fermi model

A method based on the Thomas-Fermi model for the electrostatic potential is proposed to calculate the multipole polarizabilities α K of atoms and ions. In the case of highly charged ions an equation having a simple analytical form is derived. The dipole α1 and quadrupole α2 polarizabilities of atoms and ions with more than half-filled electron shells are calculated.

Static Multipole Polarizabilities and Shielding Factors for the S‐Electron Moving in the Screened Coulomb Potential

AbstractThe variational perturbation approach of Rivail and Rinaldi as generalized by Shukla and Easa is utilized to derive a 2L‐pole static polarizability and a shielding factor for the electron in the ground state moving in a screened Coulomb potential of Debye form. The final formulas are reduced to power series of the screening parameter D in the asymptotic region. The special cases discussed by Fowler can readily be obtained from the present generalized form by direct substitution of the corresponding L values.

Dispersion coefficients for alkali-metal dimers.

Knowledge of the long-range interaction between atoms and molecules is of fundamental importance for low-energy and low-temperature collisions. The electronic interaction between the charge distributions of two ground-state alkali-metal atoms can be expanded in inverse powers of R, the internuclear distance. The coefficients ${\mathit{C}}_{6}$, ${\mathit{C}}_{8}$, and ${\mathit{C}}_{10}$ of, respectively, the ${\mathit{R}}^{\mathrm{\ensuremath{-}}6}$, ${\mathit{R}}^{\mathrm{\ensuremath{-}}8}$, and ${\mathit{R}}^{\mathrm{\ensuremath{-}}10}$ terms are calculated by integrating the products of the dynamic electric multipole polarizabilities of the individual atoms at imaginary frequencies, which are in turn obtained by solving two coupled inhomogeneous differential equations. Precise one-electron model potentials are developed to represent the motion of the valence electron in the field of the closed alkali-metal positive-ion core. The numerical results for the static multipole polarizabilities for the alkali-metal atoms and the coefficients ${\mathit{C}}_{6}$, ${\mathit{C}}_{8}$, and ${\mathit{C}}_{10}$ for homonuclear and heteronuclear alkali-metal diatoms are compared with other calculations.

Small GTO pseudostate calculations of multipole polarizabilities and dispersion coefficients between light atoms

Second order variational calculations of static multipole polarizabilities are reported for the ground states of H, He, Li and Be using Hartree—Fock Ψ 0 and the whole atomic Hamiltonian as H 0. Correlation is admitted in the valence shell of He and Be for both the ground and the excited states. The calculations make use of gaussian bases and are devised to follow the lines of our previous work done with a Slater basis. Use of 2-term optimized gaussian orbitals including scale factors for Li and Be gives ab initio results which are close to those obtained with a single optimized Slater orbital for each excited pseudostate. Polarizabilities and excitation energies allow for the one-centre calculation of the C 2 n dispersion coefficients between all pairs of interacting atoms with an absolute accuracy which is the same as that obtained in the corresponding STO calculations.

Long-range coefficients for molecular interactions

General formulae for the coefficients C n shaping the long-range behaviour of electrostatic, induction and dispersion interactions between molecular systems, are given in terms of the spherical tensor theory in its real form. The coefficients depend on a function P, describing in a compact way the relative orientation of the partners, and the permanent and induced moments of the individual molecules. Use of a hypergeneralized London formula allows the molecular dispersion coefficients to be expressed in terms of static multipole polarizabilities and their related excitation energies. Explicit formulae are presented for the first few coefficients of the linear water dimer (H 2O 2) 2. The transformation properties of the real spherical tensors under rotation and translation of the origin of the body-fixed reference frame are given in appendices, together with tables containing the coefficients needed in the expansion of the electric potential due to a non-uniform external field as well as the explicit relations connecting real spherical and cartesian forms for molecular moments and polarizabilities up to l = 3.

One-centre calculation of dispersion coefficients between ground state H atoms from pseudostate decomposition of static multipole polarizabilities

N-term pseudostate decomposition of the exact H atom static polarizabilities allows for one-centre calculation of two-body and three-body dispersion coefficients between ground state H atoms to an accuracy remarkably higher than the previously reported two-centre results. Starting from a simple power basis in the radial variable, the first-order wavefunction of the atom in the external field is expanded into a finite number N of linear pseudostates which diagonalize the matrix of the excitation energies. The expansion is rapidly convergent, and if for N = 5 the numerical results are exact to 6 significant figures, the 20-term approximation gives coefficients which are accurate to the 15th decimal place. The method can be viewed as a generalization of the Slater-Hasse-Kirkwood approximation in the context of a hypergeneralized London formula.