TDDFT-SOS Studies on the Polarizabilities and Second-order Hyperpolarizabilities of Zn_3O_3 II-VI Semiconductor Clusters

The geometrical structure of semiconductor clusters including Zn3O3 was optimized by the DFT B3LYP method. With the same basis sets, dipole moments, polarizabilities and secondorder hyperpolarizabilities have been calculated and compared with the results obtained by TDDFT B3LYP method combined with sum-over-state （SOS） formula. The calculation results indicate that the dipole moments of the ground state depend on the atom radius and electronegative differences between elements and are their balance point as well. The polarizabilities of the clusters accord with the rule of the corresponding energy transformation from ground to excited state. The results predict an increase of second-order hyperpolarizabilities with increasing the distances between atoms in the clusters as well as a decrease of the polarizabilities and second-order hyperpolarizabilities in the same serial of semiconductor clusters with increasing the dipole moments of the ground states. The changes of dipole moments in ground states are inconsistent with transition moments. Spatial structure, charge transfer and other factors play an important role in composing the transition moments.

Structure,stability,catalytic activity,and polarizabilities of small iridium clusters

At the second order Douglas–Kroll–Hess（DKH2） level, the B3 PW91 functional in conjunction with the relativistic all-electron basis set of valence triple zeta quality plus polarization functions are employed to compute bond lengths, dissociation energies, vertical ionization potentials, and the highest occupied molecular orbital-lowest unoccupied molecular orbital energy gaps of the small iridium clusters（Irn, n≤8）. These results are compared with the experimental and theoretical data available in the literature. Our results confirm the theoretical predictions made by Feng et al. about the catalytic activities of the Ir4 and Ir6 clusters. From the optimized geometries, DKH2 calculations of static electric mean dipole polarizabilities and polarizability anisotropies are also carried out. It is the first time that the polarizabilities of small iridium clusters have been studied. For n≤4, the mean dipole polarizabilities per atom present an odd–even oscillatory behavior,whereas from Ir5 to Ir8, they decrease with the cluster size increasing. The dependence of the polarizability anisotropy on the structure symmetry of the iridium cluster is verified.

Determination of static dipole polarizabilities of Yb atom

We determine the static values of the scalar and tensor dipole polarizabilities of the ground, 6s6p^3P0~o, and 6s6p^3P1~o states of the Yb atom. These results can be useful in many experiments undertaken using this atom. We employed a combined configuration interaction（CI） method and a second-order many-body perturbation theory（MBPT） to evaluate energies and electric dipole（E1） matrix elements of many low-lying excited states of the above atom. These values are compared with the other available theoretical calculations and experimental values. By combining these E1 matrix elements with the experimental excitation energies, we estimate the dominant valence correlation contributions to the dipole polarizabilities of the above states. The core contribution is obtained from the finite field approach. We also compare these values with the other theoretical results as there are no precise experimental values that are available for these properties.

Calculations of the dynamic dipole polarizabilities for the Li^+ ion

The B-spline configuration-interaction method is applied to the investigations of dynamic dipole polarizabilities for the four lowest triplet states(23S,33S,23P,and 33P) of the Li+ion.The accurate energies for the triplet states of n3S,n3P,and n3D,the dipole oscillator strengths for 23S(33S)→n3P,23P(33P)→n3S,and 23P(33P)→n3D transitions,with the main quantum number n up to 10 are tabulated for references.The dynamic dipole polarizabilities for the four triplet states under a wide range of photon energy are also listed,which provide input data for analyzing the Stark shift of the Li+ion.Furthermore,the tune-out wavelengths in the range from 100 nm to 1.2 μm for the four triplet states,and the magic wavelengths in the range from 100 nm to 600 nm for the 23S→33S,23S→23P,and 23S→33P transitions are determined accurately for the experimental design of the Li+ion.

Calculations of dynamic multipolar polarizabilities of the Cd clock transition levels

The pursuit of a systematic frequency uncertainty beyond 10^(-18) clock has triggered a multitude of investigations on the multipolar and higher-order lattice light shifts.The Cd atom has been proposed as a new candidate for the development of a lattice clock because of its smaller blackbody radiation shift at room temperature.Here,we apply an improved combined method of the Dirac-Fock plus core polarization and relativistic configuration interaction methods to calculate the dynamic multipolar polarizabilities of the Cd clock states.The effects of the high-order core-polarization potentials on the energies,reduced matrix elements,and multipolar polarizabilities have been evaluated systematically.The detailed comparison with available literature demonstrates that taking into account of the high-order core-polarization potentials is a simple and effective approach to improve the results of atomic properties for heavy atoms.

Dipole (hyper)polarizabilities of neutral silver clusters

At the Douglas-Kroll-Hess （DKI4） level, the B3PW91 functional along with the all-eleclron relativistic basis sets of valence triple and quadruple zeta qualities are used to determine the structure, stability, and electronic properties of the small silver clusters （Agn, n ≤ 7）. The results presented in this study are in good agreement with the experimental data and theoretical values obtained at a higher level of theory from the literature. Static polarizability and hyperpolarizability are also reported. It is verified that the mean dipole polarizability per atom exhibits an odd-even oscillation and that the polarizability anisotropy is directly related to the cluster shape. In this article, the first study of hyperpolarizabilities of small silver clusters is presented. Except for the monomer, the second hyperpolarizabilities of the silver clusters are significantly larger than those of the copper clusters.

Electric Multipole Polarizabilities of Quantum Bound Systems in the Transition Matrix Formalism

A new general formalism for determining the electric multipole polarizabilities of quantum (atomic and nuclear) bound systems based on the use of the transition matrix in momentum space has been developed. As distinct from the conventional approach with the application of the spectral expansion of the total Green’s function, our approach does not require preliminary determination of the entire unperturbated spectrum;instead, it makes possible to calculate the polarizability of a few-body bound complex directly based on solving integral equations for the wave function of the ground bound state and the transition matrix at negative energy, both of them being real functions of momenta. A formula for the multipole polarizabilities of a two-body bound complex formed by a central interaction potential has been derived and studied. To test, the developed t-matrix formalism has been applied to the calculation of the dipole, quadrupole and octupole polarizabilities of the hydrogen atom.

Ab initio calculations for the absorption spectra and polarizabilities of small sulfur clusters

Absorption spectra for Sn clusters （n=2...8） are calculated using an adiabatic time-dependent density functional formalism within the local density approximation （LDA）. We compare the calculated spectra with those computed using a simple LDA approach. The time-dependent LDA （TDLDA） spectra display a significant blue shift with respect to the LDA spectra. The calculated spectra present a variety of features that can be used for comparison with future experimental investigations. We also obtain a significant threshold absorption, which can distinguish between different ground states of the sulfur clusters. In addition, the polarizabilities of the clusters are calculated by using the higherorder finite-difference pseudopotential density functional method in real space. We find that the polarizabilities of the clusters considered are higher than the value estimated from the ＇hard sphere＇ model using the bulk static dielectric constant. The computed polarizabilities per atom tend to decrease with increasing cluster size. The polarizabilities are closely related to the HOMO-LUMO gaps and the geometrical configurations.

Static dipole polarizabilities of Sc_n (n ≤15) clusters

The static dipole polarizabilities of scandium clusters with up to 15 atoms are determined by using the numerically finite field method in the framework of density functional theory. The electronic effects on the polarizabilities are investigated for the scandium clusters. We examine a large highest occupied molecular orbital -- the lowest occupied molecular orbital （HOMO-LUMO） gap of a scandium cluster usually corresponds to a large dipole moment. The static polarizability per atom decreases slowly and exhibits local minimum with increasing cluster size. The polarizability anisotropy and the ratio of mean static polarizability to the HOMO-LUMO gap can also reflect the cluster stability. The polarizability of the scandium cluster is partially related to the HOMO-LUMO gap and is also dependent on geometrical characteristics. A strong correlation between the polarizability and ionization energy is observed.

Dynamic polarizabilities of the clock states of Al^(+)

The dynamic polarizabilities of 3s^(21)S_(0)and 3s3p^(3)P_(0)^(0) states of Al+are calculated using the hybrid configuration interaction and many-body perturbation theory method,and multiconfiguration Dirac-Hartree-Fock method in this work.Five ultraviolet magic wavelengths for the Al^(+)clock transition 3s^(21)S_(0)-3s3p^(3)P_(0)^(0) are predicted.Although the suitable lasers are not available presently,the potential precision measurement on these magic wavelengths for the Al^(+)clock transition would be used to extract the ratios of several certain transition matrix elements with high accuracy,and then help to improve the precision and reliability of the estimate of the BBR shift of the Al+clock transition.The differential dynamic polarizabilities at certain wavelengths are evaluated,which are useful to assess the ac Stark shift of the Al^(+)clock transition frequency and helpful in the clock experiments to suppress the ac Stark shift of the clock transition as possible as it can.

Polarizabilities of Impurity Doped Quantum Dots under Pulsed Field: Role of Additive White Noise

We make a rigorous exploration of the profiles of a few diagonal and off-diagonal components of linear (αxx, αyy, αxy andαyx), first nonlinear (βxxx, βyyy, βxyy andβyxx), and second nonlinear (γxxxx, γyyyy, γxxyyandγyyxx) polarizabilities of quantum dots under the influence of external pulsed field. Simultaneous presence of additive white noise has also been considered. The quantum dot contains dopant described by a Gaussian potential. The numbers of pulse and the dopant location have been found to fabricate the said profiles jointly. The β components display greater complexity in their profiles in comparison with the α and γ counterparts. The presence of noise prominently enhances the influence of dopant coordinate on the polarizability profiles, particularly for α and γ components. However, for β components, the said influence becomes quite evident both in the presence and absence of additive noise. The study reveals some means of achieving stable, enhanced, and often maximized output of noise-driven linear and nonlinear polarizabilities.

Study of Molecular Polarizabilities and Orientational Order Parameter in the Nematic Phase of 6.O12O.6 and 7.O12O.7

As a part of our systematic study of the physical properties of the compounds of symmetric liquid crystal dimer homologous series, α,ω-bis-(4-n-alkylaniline benzylidene-4’-oxy) alkanes (m.OnO.m), we report here an optical study of two nematogenic compounds of this series viz., 6.O12O.6 and 7.O12O.7 by carrying out the measurement of extraordinary and ordinary refractive indices using modified spectrometer. The molecular polarizability anisotropies are evaluated using Lippincott δ-function model, the molecular vibration method, Haller’s extrapolation method and scaling factor method. The molecular polrizabilities, αe and αo are calculated using Vuks’ isotropic field model and Neugebauer’s anisotropic field model. The orientational order parameter, S, is estimated by using the molecular polarizability values calculated from refractive index and denstity data as well as polarizability anisotropy values calculated as a function of temperature in the nematic phase of both of the dimers. These values are discussed with reference to the order parameter values obtained directly from the optical birefringence, Δn, data without assuming any internal field. A comparison of the order parameter values from all of these methods was carried out in the light of available literature data.

Infrared Polarizabilities of 3d-Transition and Rare-Earth Metals

A transition or rare-earth metal is modeled as the atom immersed in a jellium at intermediate electron gas densities specified by? rs=4.0. The ground states of the spherical jellium atom are constructed based on the Hohenberg-Kohn-Sham density-functional formalism with the inclusion of electron-electron self-interaction corrections of Perdew and Zunger. Static and dynamic polarizabilities of the jellium atom are deduced using time-dependent linear response theory in a local density approximation as formulated by Stott and Zaremba. The calculation is extended to include the intervening elements In, Xe, Cs, and Ba. The calculation demonstrates how the Lindhard dielectric function can be modified to apply to non-simple metals treated in the jellium model.

Polarizabilities and Hyperpolarizabilities for Lithium Atoms in Debye Plasmas

The effects of plasma environments on energies, oscillator strengths, polarizabilities and hyperpolarizabilities for lithium atom have been calculated by combining the l-dependent model potential of free lithium atom and linear variation method based on B-spline basis functions. The influence of plasma on lithium atom is represented by the Debye screened potential, which describes effectively the averaged effect of the plasma environment on atomic spectra. The results are in agreement with other reported ones.

Dipole Polarizabilities of the Ground States for Berylliumlike Ions

Dipole polarizabilities of the ground states for berylliumlike ions with nuclear charge Z = 4 to 10 are calculated by using the Rayleigh-Ritz variational method with multiconfiguration interaction wave functions. The representative models of convergence are listed and compared with other theoretical data for nuclear charge Z = 4 to6. The present dipole polarizabilities are in good agreement with previous accurate theoretical values available in the literature. For results with greater nuclear charge number, the present calculations may provide benchmarked data for future theoretical and experimental studies. Dynamic dipole polarizabilities of the ground state for the beryllium atom at selected frequencies are also calculated and compared with other theoretical values in the literature.

Molecular multipoles and (hyper) polarizabilities from the Buckingham expansion:revisited

The Buckingham expansion is important for understanding molecular multipoles and(hyper)polarizabilities.In this study,we give a complete derivation of the Buckingham expansion in the traced form using successive Taylor series.Based on the derivation results,a general Buckingham expansion in the traced form is proposed,from which highly accurate numerical calculations using the finite field method can be achieved.The transformations from the traced multipoles and multipole-multipole polarizabilities to the corresponding traceless counterparts are realized with an auxiliary traced electric field gradient.The applications of thefinite field method in this study show good agreements with previous theoretical calculations and experimental measurements.

Calculations on the Nonlinear Second-Order Optical Polarizabilities for Series of Donor-C_(60) Molecules

The equilibrium geometries and UV visible spectra of a series of donor C 60 molecules were obtained by means of the AM1 and INDO/CI method, on the basis of accurate geometric and electronic structures. The nonlinear second order optical polarizabilities were calculated using the method INDO/SDCI combined with the Sum Over States (SOS) expression. The calculated β ( λ =1 34 μm) values are 28 81, 48 56, 57 33, 66 99, 70 85, 85 84, and 142 14 (×10 -30 esu) for the molecules A, B, C, D, E, F and G, respectively. The frontier orbitals were plot for the representative molecules in order to exhibit the intramolecular charge transfer. The results indicate that introduction of thienylethylene can enhancethe NLO response and the dimethyl^aniline substituted di^thienyl^ethylene C 60 (molecule G) possesses the largest NLO second order optical polarizability. The large β values can be attributed to the charge transfer between the substituents and C 60 , as well as within the three dimensional conjugated sphere of C 60 .

Magic wavelengths for 6s_(1/2)→5d_(3/2,5/2)transitions of Yb~+ions

The wave functions,energy levels and matrix elements of Yb+ions are calculated using the relativistic configuration interaction plus core polarization(RCICP)method.The static and dynamic electric dipole polarizabilities of the ground state and low-lying excited states are determined.Then,the magic wavelengths of the magnetic sublevel 6s_(1/2,m=1/2)→5d_(3/2,m=±3/2,±1/2)and 6s_(1/2,m=1/2)→5_(d5/2,m=±5/2,±3/2,±1/2)transitions in the linearly,right-handed,and left-handed polarized light are further determined.The dependence of the magic wavelengths upon the angle between the direction of magnetic field and the direction of laser polarization is analyzed.

A simple semiempirical model for the static polarizability of ions

A concise analytical model for the static dipole polarizability of ionized atoms and molecules is created for the first time.As input,it requires,alongside the polarizability of neutral counterpart of a given ion,only the charge and elemental composition.This physically motivated semiempirical model is based on a number of established regularities in polarizability of charged monatomic and polyatomic compounds.In order to adjust it,the results of quantum chemistry calculations and gas-phase measurements available for a broad range of ionized multielectron species are employed.To counteract the appreciable bias in the literature data toward polarizability of monoatomic ions,for some molecular ions of general concern the results of the authors'own density functional theory calculations are additionally invoked.A total of 541 data points are used to optimize the model.It is demonstrated that the model we suggested has reasonable(given the substantial uncertainties of the reference data)accuracy in predicting the static isotropic polarizability of arbitrarily charged ions of any size and atomic composition.The resulting polarizability estimates are found to achieve a coefficient of determination of 0.93 for the assembled data set.The created analytic tool is universally applicable and might be advantageous for some applications where there is an urgent need for rapid low-cost evaluation of the static gas-phase polarizability of ionized atoms and molecules.This is especially relevant to constructing the complex models of nonequilibrium chemical kinetics aimed at precisely describing the observable refractive index(dielectric permittivity)of plasma flows.The data sets that support the findings of this study are openly available in Science Data Bank at https://doi.org/10.57760/sciencedb.07526.

A simple semiempirical model for the static polarizability of electronically excited atoms and molecules

We present a semiempirical analytical model for the static polarizability of electronically excited atoms and molecules,which requires very few readily accessible input data,including the ground-state polarizability,elemental composition,ionization potential,and spin multiplicities of excited and ground states.This very simple model formulated in a semiclassical framework is based on a number of observed trends in polarizability of electronically excited compounds.To adjust the model,both accurate theoretical predictions and reliable measurements previously reported elsewhere for a broad range of multielectron species in the gas phase are utilized.For some representative compounds of general concern that have not yet attracted sufficient research interest,the results of our multireference second-order perturbation theory calculations are additionally engaged.We show that the model we developed has reasonable(given the considerable uncertainties in the reference data)accuracy in predicting the static polarizability of electronically excited species of arbitrary size and excitation energy.These findings can be useful for many applications,where there is a need for inexpensive and quick assessments of the static gas-phase polarizability of excited electronic states,in particular,when building the complex nonequilibrium kinetic models to describe the observed optical refractivity(dielectric permittivity)of nonthermal reacting gas flows.