Frequency-dependent Dipole Polarizabilities Research Articles

First-principles calculation of the frequency-dependent dipole polarizability of argon

In this work we report state-of-the-art theoretical calculations of the dipole polarizability of the argon atom. Frequency dependence of the polarizability is taken into account by means of the dispersion coefficients (Cauchy coefficients), which is sufficient for experimentally relevant wavelengths below the first resonant frequency. In the proposed theoretical framework, all known physical effects including the relativistic, quantum electrodynamics, finite nuclear mass, and finite nuclear size corrections are accounted for. We obtained ${\ensuremath{\alpha}}_{0}=11.0775(19)$ for the static polarizability and ${\ensuremath{\alpha}}_{2}=27.976(15)$ and ${\ensuremath{\alpha}}_{4}=95.02(11)$ for the second and fourth dispersion coefficients, respectively. The result obtained for the static polarizability agrees (within the estimated uncertainty) with the most recent experimental data [C. Gaiser and B. Fellmuth, Phys. Rev. Lett. 120, 123203 (2018)] but is less accurate. The dispersion coefficients determined in this work appear to be the most accurate in the literature, improving by more than an order of magnitude upon previous estimates. By combining the experimentally determined value of the static polarizability with the dispersion coefficients from our calculations, the polarizability of argon can be calculated with accuracy of around $10\phantom{\rule{0.16em}{0ex}}\mathrm{ppm}$ for wavelengths above roughly $450\phantom{\rule{0.16em}{0ex}}\mathrm{nm}$. This result is important from the point of view of quantum metrology, especially for a new pressure standard based on thermophysical properties of gaseous argon. Additionally, in this work we calculate the static magnetic susceptibility of argon, which relates the refractive index of dilute argon gas with its pressure. While our results for this quantity are less accurate than in the case of the polarizability, they can provide, via the Lorenz-Lorentz formula, the best available theoretical estimate of the refractive index of argon.

Ground-state frequency-dependent dipole polarizabilities of [formula omitted] three-electron ions

Frequency-dependent dipole polarizabilities of 22S lithiumlike states for 3≤Z≤10 are tabulated for frequencies up to the 32P absorption frequency using fully correlated Gaussian expansions. To account for the frequency-dependence between and near the poles at the 22P and 32P absorption frequencies, the Schrödinger–Coulomb Hamiltonian is diagonalized in a composite basis of L=1 angular momentum Gaussians containing subsets of partially optimized n2P (n=2−5) wavefunctions and both frequency-independent and frequency-dependent first-order perturbed functions. The tabulated results are well-converged and should be considered as benchmark dipole polarizabilities in the non-relativistic limit. The calculated dipole polarizabilities are used to determine a fit to the Cauchy expansion. The 32P and 42P energies calculated using the composite basis set are new upper bounds to the exact non-relativistic energies, as are many of the 52P energies.

Relativistic four-component linear damped response TDDFT for electronic absorption and circular dichroism calculations.

We present a detailed theory, implementation, and a benchmark study of a linear damped response time-dependent density functional theory (TDDFT) based on the relativistic four-component (4c) Dirac-Kohn-Sham formalism using the restricted kinetic balance condition for the small-component basis and a noncollinear exchange-correlation kernel. The damped response equations are solved by means of a multifrequency iterative subspace solver utilizing decomposition of the equations according to Hermitian and time-reversal symmetry. This partitioning leads to robust convergence, and the detailed algorithm of the solver for relativistic multicomponent wavefunctions is also presented. The solutions are then used to calculate the linear electric- and magnetic-dipole responses of molecular systems to an electric perturbation, leading to frequency-dependent dipole polarizabilities, electronic absorption, circular dichroism (ECD), and optical rotatory dispersion (ORD) spectra. The methodology has been implemented in the relativistic spectroscopy DFT program ReSpect, and its performance was assessed on a model series of dimethylchalcogeniranes, C4H8X (X = O, S, Se, Te, Po, Lv), and on larger transition metal complexes that had been studied experimentally, [M(phen)3]3+ (M = Fe, Ru, Os). These are the first 4c damped linear response TDDFT calculations of ECD and ORD presented in the literature.

Dynamic Polarizability and Higher-Order Electric Properties of Fluorene, Carbazole, and Dibenzofuran

Static electric properties, from the dipole moment to the second-hyperpolarizability tensor γ, of the 3-membered, isoelectronic ring molecules, fluorene (FL), carbazole (CR), and dibenzofuran (DBF), have been calculated at various levels of approximation. The electron correlation effects have been included at the coupled-cluster (CC) level, using CCSD and CC2 versions of the method. DFT calculations with the CAM-B3LYP functional have also been performed, and the results are compared to the CC values. The electric property-tailored Pol basis set and its more compact Z3Pol version have been employed in all static calculations. Differences between dipole polarizability values computed at the Pol and Z3Pol bases have been found to be almost negligible. Therefore, all components of the frequency-dependent dipole polarizability tensor α(-ω;ω) have been determined at the CAM-B3LYP/Z3Pol level. Divergence occurring at electronic resonances has been eliminated using the complex polarization propagator (CPP) formalism, explicitly introducing an imaginary iΓ parameter to account approximately for the finite lifetime of the excited state. The imaginary part of the dipole polarizability Im α(-ω;ω) has been calculated for a wide range of external radiation energies up to 10 eV, and its maxima have been compared to the calculated vertical electronic excitation energies.

Frozen Virtual Natural Orbitals for Coupled-Cluster Linear-Response Theory.

The frozen-virtual natural-orbital (NO) approach, whereby the unoccupied-orbital space is constructed using a correlated density such as that from many-body perturbation theory, has proven to yield compact wave functions for determining ground-state correlation energies and associated properties, with corresponding occupation numbers providing a guide to the truncation of the virtual space. In this work this approach is tested for the first time for the calculation of higher-order response properties, particularly frequency-dependent dipole polarizabilities using coupled-cluster theory. We find that such properties are much more sensitive to the truncation of virtual space in the NO basis than in the original canonical molecular orbital (CMO) basis, with truncation errors increasing linearly with respect to the number of frozen virtual NOs. The reasons behind this poor performance include the more diffuse nature of NOs with low occupation numbers as well as the reduction in sparsity of the perturbed singles amplitudes in the NO basis and the neglect of orbital response. We tested a number of approaches to improve the performance of the NO space, including the use of a field-perturbed density to define the virtual orbitals and various external-space corrections. The truncation of the CMO space, on the other hand, yields errors in coupled-cluster dipole polarizabilities of less than 2% even after removing as much as 50% of the full virtual space. We find that this positive performance of the CMO space results from a cancellation of errors due to the truncation of the unperturbed and perturbed amplitudes, as well as sparsity of the singles amplitudes. We introduce a simple criterion called a dipole amplitude to use as a threshold for truncating the CMO basis for such property calculations.

Frequency-dependent polarizabilities and shielding factors for confined one-electron systems

Frequency-dependent dipole polarizabilities and shielding factors are calculated for the ground state of spherically symmetric screened one-electron systems embedded in an impenetrable spherical cavity. Coulomb, Yukawa, Hulthén and exponential cosine-screened Coulomb potentials are considered. In contrast to free systems, Dirichlet boundary conditions introduce a contribution to the shielding factor that results from an integral over the surface of the confining boundary. This is a fundamental difference between free and confined systems and results in unexpected modifications to some of the classic relations for free systems. The methods derived also give a simple expression for the polarizability of the confined harmonic oscillator as an example of extending the methods of this work to potentials beyond the four studied.

C6 Coefficients and Dipole Polarizabilities for All Atoms and Many Ions in Rows 1–6 of the Periodic Table

Using time-dependent density functional theory (TDDFT) with exchange kernels, we calculate and test imaginary frequency-dependent dipole polarizabilities for all atoms and many ions in rows 1-6 of the periodic table. These are then integrated over frequency to produce C6 coefficients. Results are presented under different models: straight TDDFT calculations using two different kernels; "benchmark" TDDFT calculations corrected by more accurate quantum chemical and experimental data; and "benchmark" TDDFT with frozen orbital anions. Parametrizations are presented for 411+ atoms and ions, allowing results to be easily used by other researchers. A curious relationship, C6,XY ∝ [αX(0)αY(0)](0.73), is found between C6 coefficients and static polarizabilities α(0). The relationship C6,XY = 2C6,XC6,Y/[(αX/αY)C6,Y + (αY/αX)C6,X] is tested and found to work well (<5% errors) in ∼80% of the cases, but can break down badly (>30% errors) in a small fraction of cases.

Incremental evaluation of coupled cluster dipole polarizabilities.

In this work we present the first implementation of the incremental scheme for coupled cluster linear-response frequency-dependent dipole polarizabilities. The implementation is fully automated and makes use of the domain-specific basis set approach. The accuracy of the approach is determined on the basis of a test suite of 47 molecules and small clusters. The local approximation in the coupled cluster singles and doubles polarizability exhibits a mean error of 0.02% and a standard deviation of 0.32% when using a third-order incremental expansion. With the proposed approach, it is possible to compute polarizabilities with larger basis sets compared to the canonical implementation and thus it is possible to obtain higher total accuracy. The incremental scheme yields the smallest errors for weakly-bound and quasi-linear systems, while two- and three-dimensional (cage-like) structures exhibit somewhat larger errors as compared to the full test set.

Optical and structural characterization of thin films containing metallophthalocyanine chlorides

The structural and optical investigation of thin films containing aluminum and gallium phthalocyanine chlorides is presented. The films were fabricated by Physical Vapor Deposition technique onto quartz substrates and annealed after fabrication in an ambient atmosphere for 24 h at the temperature equal to 150 °C or 250 °C. The experimental results and theoretical calculation of the Third Harmonic Generation process are reported. The third order nonlinear optical properties are expected and can be more or less accurately predicted due to the assembly of the molecules and theoretical calculations of the frequency-dependent dipole polarizabilities, third hyperpolarizabilities, third order susceptibilities, frontier and second frontier molecular orbitals. These parameters were used to understand the relationship of optical properties with the molecular structures. We found that the annealing process causes formation of nanostructures and the value of the third order optical susceptibility makes these materials interesting for future nonlinear optical applications.

Theoretical and experimental investigations on the nonlinear optical properties of gold(III) dithiolene complexes

Degenerate four-wave mixing (DFWM) experiments have been performed to determine the third-order nonlinear optical (NLO) susceptibilities (χ(3)) of gold(III) maleimide dithiolate tetraphenylphosphonium, (PPh4)[Au(midt)2], (Au-P) and gold(III) maleimide dithiolate melamine melaminium hybrid solvate, (C3N6H6)(C3N6H7+)[Au(midt)2]−·2DMF·2H2O, (Au-Mel). Ab-initio quantum mechanical calculations (time-dependent Hartree–Fock (TDHF) method) of Au-P and Au-Mel have been carried out to compute the electric dipole moment (μ), the dispersion-free and frequency-dependent dipole polarizability (α) and the second hyperpolarizability (γ) values. These theoretical calculations are in good agreement with the experimentally obtained results by the DFWM technique. All the investigations show clearly the effect played by the counter ion on the resulting NLO properties of the two gold complexes.

Dynamic dipole polarizability of the helium atom with Debye-Hückel potentials

In the present study, we investigate the effect of Debye-H\uckel potentials on the frequency-dependent dipole polarizability of the helium atom using highly correlated wave functions within the framework of the pseudostate summation technique. The dynamic dipole polarizability of He (1${s}^{2}$ ${}^{1}\phantom{\rule{-0.16em}{0ex}}S$) as a function of the scaled number density of the plasma electrons for arbitrary plasma temperature is presented. Screening effects on the resonance frequencies are also presented. In a free-atomic system, our calculated results are in agreement with the available theoretical and experimental predictions.

Application of Hermitian time-dependent coupled-cluster responseAnsätzeof second order to excitation energies and frequency-dependent dipole polarizabilities

Linear-response methods, based on the time-dependent variational coupled-cluster or the unitary coupled-cluster model, and truncated at the second order according to the M\o{}ller-Plesset partitioning, i.e., the TD-VCC[2] and TD-UCC[2] linear-response methods, are presented and compared. For both of these methods a Hermitian eigenvalue problem has to be solved to obtain excitation energies and state eigenvectors. The excitation energies thus are guaranteed always to be real valued, and the eigenvectors are mutually orthogonal, in contrast to response theories based on ``traditional'' coupled-cluster models. It turned out that the TD-UCC[2] working equations for excitation energies and polarizabilities are equivalent to those of the second-order algebraic diagrammatic construction scheme ADC(2). Numerical tests are carried out by calculating TD-VCC[2] and TD-UCC[2] excitation energies and frequency-dependent dipole polarizabilities for several test systems and by comparing them to the corresponding values obtained from other second- and higher-order methods. It turns out that the TD-VCC[2] polarizabilities in the frequency regions away from the poles are of a similar accuracy as for other second-order methods, as expected from the perturbative analysis of the TD-VCC[2] polarizability expression. On the other hand, the TD-VCC[2] excitation energies are systematically too low relative to other second-order methods (including TD-UCC[2]). On the basis of these results and an analysis presented in this work, we conjecture that the perturbative expansion of the Jacobian converges more slowly for the TD-VCC formalism than for TD-UCC or for response theories based on traditional coupled-cluster models.

Precision Calculations of Atomic Polarizabilities: A Relevant Physical Quantity in Modern Atomic Frequency Standard

Electric dipole polarizabilities of atoms are very important in many different physical applications, such as the precision atomic frequency standard. Calculations of these properties are very important and challenging. We propose a calculation strategy to calculate the frequency dependent dipole polarizabilities with high precision variationally by using a set of high quality orbital bases where the electron correlations can be taken into account adequately. The static polarizabilities of the ground state of Na are calculated accurately by such a method and can be compared with precision experiment measurement directly. The calculation result is in excellent agreement with the available experimental measurements within about 0.1%, which demonstrates the validity of our strategy. Our calculation strategy has a wide usage, not only in polarizibilies, but also in other fields such as theoretical treatment of electron-atom scattering processes. Using the same orbital bases, we carry out precision calculation of Na− affinities. Our calculated affinity is in excellent agreement with precision laser spectroscopy measurements within 0.1%.

An accurate method for solution of one-electron perturbation equations

An accurate nonvariational method for solution of one-electron nonhomogeneous equations arising in time-dependent perturbation theory is presented. The frequency-dependent dipole polarizability of hydrogen is calculated as an example.

Static and dynamic polarizabilities of Mg-like ions

In the present study, we investigate the static and frequency-dependent dipole polarizabilities for the ground state of neutral Mg and four ions of the magnesium sequence: Al+, Si2+, P3+ and S4+. The wavefunctions are written in terms of configuration interaction (CI) expansions. Radial functions of orbitals up to n = 9 are used in the calculations, some optimized on the polarizability, some on the energies of low-lying states. We present a detailed study of the convergence of the results as the size of the CI expansions is increased. A very satisfactory degree of stability is achieved.

CI calculations of long-range C 6 dispersion coefficients for BH–BH

Large scale CI calculations of frequency-dependent dipole polarizabilities (FDPs) of ground state BH at the experimental distance of R = 2.329 a 0 have been performed using two sets of contracted GTO functions involving up to 316 × 10 6 symmetry-adapted Slater determinants, comparing recently developed interpolation and variational techniques. Final calculation of BH–BH dispersion coefficients was done using the larger basis set of 268 AOs in the frozen-core FCI approximation and the lower bounds resulting from the variational approach.

Numerical solution of Dalgarno-Lewis equations by a mapped Fourier grid method

Inhomogeneous radial differential equations emerging in applications of standard perturbation theory are numerically solved by a novel approach making use of Fourier grid methods in conjunction with a simple mapping scheme. The proposed algorithm is applied along the lines of the Dalgarno-Lewis method [Proc. R. Soc. London, Ser. A 223, 70 (1955)] to the calculation of the static dipole polarizabilities and hyperpolarizabilities of 1s, 2s, and 2p states of hydrogen atom and their frequency dependent dynamic dipole polarizabilities. The high efficiency and accuracy of the algorithm are demonstrated for the above test cases, where exact values are available. Then, the frequency dependent dipole polarizability of the ground state of lithium atom is computed by a variationally stable method combining an effective local potential approach with a second-order energy correction. The obtained results are in perfect agreement with other elaborate theoretical approaches.

The frequency-dependent dipole polarizability of the mercury dimer from four-component relativistic density-functional theory

The frequency-dependent dipole polarizability of Hg(2) is calculated using response theory within four-component relativistic density-functional theory [using the local-density approximation (LDA) and the hybrid functional B3LYP] including corrections for the basis-set superposition error. The anisotropic component of the polarizability tensor agrees well with the values obtained from collision-induced Raman spectroscopy carried out at a wavelength of 488 nm. The values obtained from the two density functionals agree closely with the experimentally derived anisotropy component of the dipole polarizability, despite their rather large differences in the dimer potential-energy curves (LDA is strongly overbinding while B3LYP is purely repulsive). The first two refractivity virial coefficients for the generalized Clausius-Mossotti function are derived.

Frozen-core full-CI calculation of imaginary frequency-dependent dipole polarizabilities of ground state BeH 2 and the C6 dispersion coefficients of its homodimer

Frozen-core full-CI calculations of frequency-dependent dipole polarizabilities (FDPs) of ground state BeH 2 at the experimental distance of R = 2.506 a 0 have been performed using an extended set of 208 contracted GTO functions [9s9p5d3f] on Be and [9s8p6d] on H involving about 58 × 10 6 symmetry-adapted Slater determinants. The Casimir–Polder integral was then evaluated analytically using eight optimized imaginary frequencies chosen according to a recently developed interpolation technique which allows for the evaluation of the three dipole dispersion constants for the BeH 2–BeH 2 homodimer, from which isotropic C 6 and anisotropy γ 6 coefficients are derived for the first time.

Method of the reduced-added Green function in the calculation of atomic polarizabilities

The Green function in the quantum defect theory provides an exact account for high-excited and continuum electronic states. We modify it by taking into account the ground and low-excited states using their wave functions calculated ab initio. As an application, we present a simple and efficient semianalytical method for the calculation of atomic electric frequency-dependent scalar dipole polarizability, for both real and imaginary frequencies. The polarizabilities calculated for some atoms (Li, Na, K, Be, Mg, Ca, Si, P, S, O, Al, Ge, C, N, F, He, Ne, Ar, Kr, and Xe) are compared with existing methods of computational quantum chemistry and with experiments; good accuracy of the proposed method is demonstrated.